by Alex Man Hon Wong - University of Toronto T-Space · Alex Man Hon Wong Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto 2014

Sub-Diffraction Imaging Using Superoscillatory Electromagnetic Waves

by

Alex Man Hon Wong

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Electrical and Computer EngineeringUniversity of Toronto

c© Copyright 2014 by Alex Man Hon Wong

Abstract

Sub-Diffraction Imaging Using Superoscillatory Electromagnetic Waves

Alex Man Hon Wong

Doctor of Philosophy

Graduate Department of Electrical and Computer Engineering

University of Toronto

2014

This thesis presents an investigation on imaging beyond the diffraction limit with superoscillatory elec-

tromagnetic waves. It begins with a survey on sub-diffraction imaging, which motivates the need for

a new sub-diffraction imaging device which is free from drawbacks in devices which have heretofore

been proposed. This work addresses this need through the design and demonstration of superoscillatory

wave-based focusing and imaging devices. Through establishing a relationship between superoscilla-

tion and superdirectivity, a design procedure is formulated whereby antenna array theory is leveraged

to design 1D and 2D superoscillatory waves. These waves are then physically synthesized or imple-

mented as a filter in focusing and imaging systems at frequencies ranging from microwave to optical.

Experimental characterization of these systems has led to successful demonstrations of sub-diffraction

focusing and imaging, with working distances orders of magnitude farther than most other electromag-

netic wave-based sub-diffraction imaging devices. In particular, the Optical Super-Microscope which

achieves far-field sub-diffraction focusing has attractive merits to become a tool for general-purpose

sub-diffraction optical microscopy. An investigation has also been conducted on superoscillatory waves

in the time domain, which has resulted in the first reported demonstration of a superoscillatory tem-

poral waveform, as well as a demonstrated improvement of radar range resolution beyond the Fourier

transform limit.

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My utmost for His Highest

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Acknowledgements

The completion of this degree is, in many sense, a joint effort of countless people and encounters. I

would like to express my heartfelt gratitude towards some such people who have contributed, directly

or otherwise, to this degree.

Let me begin by thanking my parents, whose love, respect, support and patience have sustained me

throughout my lengthy (but soon-to-end) days as a student. I could not have asked for more from any

earthly parent!

Surely, I must thank my supervisor Prof. George V. Eleftheriades. It has been my privilege and

pleasure to study under his supervision for both my master and doctoral degrees. His inspiration,

integrity, insightful perspectives and timely encouragements provided tremendous help throughout my

degree. I have learnt but a fraction of his knowledge and professionalism through several years of study

under him, but am positive that what I have learnt will serve me in good stead.

I also sincerely thank Prof. Costas D. Sarris and Prof. Sean V. Hum for their guidance as my

thesis committee members: they have provided expert guidance and valuable discussions which directly

contributed to this thesis. The same is true with my defense committee members, which also included

Prof. Piero Trivero and Prof. Adrian Nachman, as well as my external appraiser Prof. Christos G.

Christodoulou. I respectfully thank them for their time and contribution, and for triggering insightful

discussions in both defenses which ultimately improved the quality of my thesis from its state at the

initial defense.

Through several years as a member of the electromagnetics research group, I’ve very much enjoyed

interactions — academic or otherwise — with fellow graduate students, within the EM and photonics

groups and beyond. Particularly, I thank Loıc Markley and Yan Wang, my co-authors, for invaluable

collaborations on research projects. Loıc and Yan, along with senior members Ashwin Iyer, Marco

Antoniades, Jiang Zhu and Rubaiyat Islam have provided generous help, from simulation support to

insightful discussions to wise perspectives, especially in the early stage of my degree. I am also thankful

for the company of many colleagues throughout my degree. I’ve started to name some of them in this

acknowledgement, only to realize this space wouldn’t be sufficient to recollect our interactions over the

years. I wish them all the best in their own studies and their endeavors thereafter.

I’m grateful to have worked with undergraduate student Nima Yasrebi over the summer of 2012,

who helped marvelously in building the Eleftheriades optical lab and the Optical Super-Microscope.

I thoroughly enjoyed working with Nima and with other undergraduate students, through teaching,

mentorship and research involvements. I benefit tremendously from these interactions too: they helped

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keep me sane when my research isn’t working!

Last but most importantly I impart praise and thanksgiving to my Lord and Saviour Jesus Christ,

who has helped me learn and grow throughout this degree. His wondrous works often goes beyond my

imagination and comprehension. Nevertheless, I thank His loving and wise bestowment throughout my

degree, and look upon Him for guidance and provision beyond this degree.

With Gratefulness,

Alex M. H. Wong

April, 2014.

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Contents

1 Introduction 1

1.1 Sub-Diffraction Imaging: Historical Perspectives and Motivation . . . . . . . . . . . . . . 1

1.2 Avenues to Sub-Diffraction Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Slightly Sub-Diffraction Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Evanescent Field Based Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.3 Non-Linear and Non-Electromagnetic Imaging Devices . . . . . . . . . . . . . . . . 7

1.3 Superoscillation and Sub-diffraction Imaging . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Objectives and Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Mathematical Preludes 13

2.1 The Plane Wave Formulation of Electromagnetic Fields . . . . . . . . . . . . . . . . . . . 13

2.2 The Diffraction Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Abbe’s Diffraction Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2 Rayleigh’s Diffraction Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.3 Summary and Comparison on Diffraction Limits . . . . . . . . . . . . . . . . . . . 18

2.2.4 Angular Diffraction Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Evanescent-Wave Based Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Superdirectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.1 Schelkunoff’s Superdirectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.2 A Fourier Perspective on Superdirectivity . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.3 Superdirectivity and the Spatial Diffraction Limit . . . . . . . . . . . . . . . . . . 24

2.5 Concluding Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 An Antenna-Based Theory on Superoscillatory Waves 29

3.1 An introduction to Spatial Superoscillations . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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3.1.1 Characterizing Spatial Superoscillations . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.2 Relating Superoscillations to Superdirectivity . . . . . . . . . . . . . . . . . . . . . 31

3.2 Superdirectivity-Inspired Superoscillation Design . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.1 A Schelkunoff Approach to Superoscillation Design . . . . . . . . . . . . . . . . . . 32

3.2.2 Design Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.3 Comparing Superoscillation with Superdirectivity . . . . . . . . . . . . . . . . . . . 35

3.2.4 Connection with Prolate Spheroidal Wave Functions . . . . . . . . . . . . . . . . . 36

3.3 Controlling Sidebands of Superoscillatory Waveforms . . . . . . . . . . . . . . . . . . . . . 37

3.4 Deducing the Limit to Sideband Suppression . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.1 Mathematical Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43


4 Superoscillatory 1D Sub-Diffraction Focusing Devices 45

4.1 Sub-Diffraction-Focused Waveform Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 A Superoscillatory Sub-Diffraction Focusing Screen . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Superoscillatory Sub-Diffraction Focusing in a Waveguide Environment . . . . . . . . . . 52

4.3.1 Waveguide Excitation Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.2 Fabrication and Experimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3.3 Simulation and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4.1 Salient Features in Comparison to Evanescent-Wave-Based Devices . . . . . . . . . 62

4.4.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4.3 Comparison to other Superoscillatory Focusing Devices . . . . . . . . . . . . . . . 66

4.4.4 Implementing the Superoscillatory Focusing Screen . . . . . . . . . . . . . . . . . . 66


5 Superoscillatory 2D Sub-Diffraction Imaging: The Optical Super-Microscope 68

5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2 Conceptualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.1 From Focusing to Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2.2 Implementing a Superoscillatory PSF . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3 Designing a 2D Superoscillation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.4 OTF Implementation with an SLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

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5.5 Constructing the Optical Super-Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.5.1 Numerical Parameters and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.5.2 Assembling the Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . 82

5.6 Calculation and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.6.1 Point Imaging Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.6.2 Two-Point Resolution Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.6.3 Real-Time Imaging for Moving Objects . . . . . . . . . . . . . . . . . . . . . . . . 88

5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.7.1 Connection to Super-Resolving Apertures . . . . . . . . . . . . . . . . . . . . . . . 90

5.7.2 Connection to Resolution Restoration Methods . . . . . . . . . . . . . . . . . . . . 90

5.7.3 Comparison to Concurrent Developments . . . . . . . . . . . . . . . . . . . . . . . 93

5.7.4 Imaging in the Presence Superoscillatory Sidebands . . . . . . . . . . . . . . . . . 93

5.7.5 Further Resolution Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.7.6 Extending the Field of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.7.7 Sub-wavelength Optical Super-Microscopy . . . . . . . . . . . . . . . . . . . . . . . 96

5.8 Summarizing Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6 Temporal Superoscillations 98

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.2 Designing Temporal Superoscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.2.1 A Design Procedure for Temporal Superoscillations . . . . . . . . . . . . . . . . . . 99

6.2.2 The Rapid Oscillation Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.2.3 The Sharp Pulse Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3 Synthesizing Superoscillatory Electromagnetic Waveforms . . . . . . . . . . . . . . . . . . 102

6.3.1 Rapid Oscillation Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.3.2 Superoscillatory Pulse Compression . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.4 Superoscillatory Radar Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.4.2 Superoscillatory Radar Pulse Design . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.4.3 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.4.4 Single Scatterer Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.4.5 Resolving Two Scatterers in Close Range . . . . . . . . . . . . . . . . . . . . . . . 115

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6.4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.5 Summarizing Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7 Conclusion 119

7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

A The Heisenberg Uncertainty Principle 124

A.1 Defining Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

A.2 Deriving the Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

B The Design of Dolph-Tschebyscheff Superdirective Antenna Arrays 126

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

C Calibrating the Spatial Light Modulator 132

C.1 Visibility for Grating Diffraction Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

C.2 Performing SLM Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Bibliography 136

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List of Tables

3.1 Summary of comparison between superdirectivity and spatial superoscillation . . . . . . . 36

4.1 Summary of parameters for the Superoscillatory Sub-Diffraction Focusing Screen . . . . . 49

4.2 Summary of parameters for the SFW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Calculated current excitation coefficients for the Superoscillatory Sub-Diffraction Focusing

Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1 Summary of parameters for a test 2D superoscillatory waveform . . . . . . . . . . . . . . . 73

5.2 Summary of superoscillatory filter design parameters for OSM . . . . . . . . . . . . . . . . 81

5.3 Comparison between the OSM and two concurrent superoscillation-based imaging devices

[85,86]. In this table ∆wdl and ∆sdl respectively denote the spot width and two-point res-

olution distance of a conventional, diffraction-limited imaging system. Table annotations

are hereby given. 1. For [86], NAi represents the numerical aperture of the superoscilla-

tory illumination, which is relevant for evaluating the focal quality, while NAo represents

the numerical aperture of the light collection component, relevant for evaluating the imag-

ing performance of the device. 2. The term ROI (region of interest) adopts the definition

taken in this work, not the slightly different definition used by [85]. 3. Since an explicit

comparison was not given in [86], the “improvement” figures are derived by comparison

with the calculated diffraction limits of a simple optical system with illumination wave-

length λ and numerical aperture NA. 4. Th resolution improvement is calculated and

quoted as the center-to-center distance between two apertures. . . . . . . . . . . . . . . . 92

6.1 Summary of parameters for the rapid oscillation waveform . . . . . . . . . . . . . . . . . . 102

6.2 Summary of parameters for a designing a superoscillatory sharp temporal pulse . . . . . . 104

6.3 Summary of parameters for a designing a superoscillatory sharp temporal pulse . . . . . . 110

6.4 Single object range measurement using a superoscillatory radar pulse and a sinc radar pulse115

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B.1 Zero locations and excitation weights for the 11-element superdirective antenna . . . . . . 130



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List of Figures

2.1 Image profiles of two deeply sub-wavelength circular apertures, separated at varying dis-

tances. The apertures were illuminated at normal incidence (i.e. in phase) by a coherent

laser source. The legend shows the center-to-center aperture separations of the corre-

sponding image profiles. The figure shows the coherent Sparrow limit at 0.75λ/NA and

the coherent Rayleigh limit at 0.82λ/NA. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Evanescent-field-based sub-diffraction imaging. Colors represent phase progression in (b)

and intensity level in other plots. (a) The field pattern at a plane λ/20 away from a

z-directed small dipole. This pattern is tightly localized to within a circle of radius be-

low λ/100. (b) Electric field progression for a line source that is imaged by a superlens.

Solid white lines denote the location of the superlens, while dash white lines denote the

locations of the source (to the left of the lens) and image (to the right of the lens). A

region of rapid field oscillations surrounding the superlens’ output facet indicates the dom-

inance of evanescent waves. (c) The spatial spectrum corresponding to (a), showing large

evanescent-wave content. The black dashed circle separates the propagating spectrum (in-

side) from the evanescent spectrum (outside). (d) The spectral evolution corresponding

to (b), showing large evanescent field components near the output facet of the superlens. . 20

2.3 Superdirective antenna arrays and their antenna patterns. (a) Current Excitations (array

factors) for 3 antennas arrays of length 2λ, with 11, 21 and 31 elements respectively. (b)

The corresponding far-field angular distributions, compared to that of a uniform array. . . 21

2.4 Superdirective antenna arrays and their plane wave spectra. (a) Current Excitations

(array factors) for 3 antennas arrays of length 2λ, with 11, 21 and 31 elements respectively,

for which the antenna patterns are displayed in Fig. 2.3 (b) The plane wave spectra plotted

in log-scale, showing the existence of large evanescent wave (|kx| > k) components. . . . . 24

2.5 Field evolutions from three antenna arrays, of electrical sizes 5λ, 2λ and λ respectively,

which are designed to have similar far-field antenna patterns. The antenna axis is at z = 0. 26

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2.6 A plot of the far-field distance, and the corresponding beamwidth at that distance, for

Tschebyscheff superdirective antennas with varying numbers of elements, with fixed elec-

trical lengths of 2λ, and target sidelobes ripples at 20% the field-strength of the main

beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1 Zero placements of spatial profiles of common waveforms. (a) Zero placement and (b)

Spatial profile for a cosine waveform; (c) Zero placement and (d) Spatial profile for a

periodic sinc waveform. Both waveforms are formed by placing 6 zeros over a design

period of 3λ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Zero placements of spatial profiles of example superoscillatory waveforms. (a) Zero place-

ment for a cosine waveform, obtained by multiplying the phase of zeros in Fig. 3.1a by

a factor of 0.75. (b) The corresponding spatial profile of the cosine waveform. (c) Zero

placement for a sinc waveform, obtained by multiplying the phase of zeros in Fig. 3.1c by

a factor of 0.75. (d) The corresponding spatial profile of the sinc waveform. . . . . . . . . 35

3.3 Suppressing the sideband amplitude by placing zeros within the sideband of the super-

oscillatory waveform. (a) Zero locations of the waveform featured in Fig. 3.2c (b) The

corresponding spatial profile (featured in Fig. 3.2d). (c) Zero locations of the same

waveform, with N − 1 plane waves added, but their amplitude set to zero. (d) The

corresponding spatial profile — which as expected is the same as that of (b). (e) Zero

locations after the sideband zeros are distributed to minimize the sideband amplitude. (f)

The corresponding spatial profile, which features a sideband whose amplitude is clearly

suppressed compared to the waveforms in parts (b) and (d). In subfigures (b), (d) and

(f), the red rectangle denotes the extend of the ROI. . . . . . . . . . . . . . . . . . . . . . 39

4.1 The first five Tschebyscheff polynomials of the first kind. . . . . . . . . . . . . . . . . . . . 47

4.2 Zero locations and the corresponding plane-wave spectrum for the superoscillation sub-

diffraction focusing screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Superoscillatory Waveform Design. (a) One period of the calculated Esrc(x) (black,

dashed) and Eimg(kx) (blue, dotted). (b) The image waveform Eimg(kx), plotted across

the ROI alongside the diffraction-limited sinc waveform (red, dashed). . . . . . . . . . . . 51

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4.4 Full-wave simulation results for the sub-diffraction focusing screen. (a) The simulated

electric field distribution overlaid atop a schematic of the computation domain. Blue

solid lines indicate PMC boundaries; dashed lines (black and green) indicate the source

and image planes respectively. (b) A closeup of the imaged electric field in the ROI

|x| ≤ λ/2 (blue, solid), normalized alongside the source electric field (black, solid) and the

diffraction-limited sinc function (red, dashed). (c) The simulated (magenta, circles) and

calculated (blue, solid) electric fields at the image plane show excellent agreement with

one another. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.5 Design plots for the SFW, showing (a) zero locations, (b) spectral amplitudes, (c) the

designed waveform, and (d) a closeup across the ROI, compared alongside the diffraction-

limited sinc (red, dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.6 Schematic diagrams for the SFW. (a) A schematic of the SFW, showing key design dimen-

sions, the source and image planes, wave terminations and the measurement probe. (b)

The feed network for the SFW. Numbers to the right of the sub-figure indicate connections

to the corresponding ports in (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.7 A photograph of the sub-diffraction focusing waveguide. . . . . . . . . . . . . . . . . . . . 58

4.8 Electric field profiles near the image plane (denoted by white dash lines). (a) Simulation

with absorbers covering the entire waveguide end facets. (b) Simulation with a partial

gap in the absorber at the +z end of the waveguide (see Fig. 4.6 for pictorial depiction.

(c) Measured electric field magnitude. The colour bar to the left applies for simulated

field profiles, while the bar to the right applies for the measured field profile. . . . . . . . 59

4.9 A comparison of the simulated (blue, solid) and calculated (black, dashed) Eimg(x), show-

ing excellent agreement apart from slight deviations surrounding the outer pair of nulls.

The right panel is a closeup of the left across the ROI. . . . . . . . . . . . . . . . . . . . . 60

4.10 A comparison on the measured and simulated superoscillatory foci at the image plane

z = 500 mm. The simulated field profiles are taken at the design image plane, while the

measured field profile is taken at z = 480 mm. . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.11 A close up of Fig. 4.10 over the ROI, comparing the measured and simulated super-

oscillatory foci across the design interval. The diffraction-limited focus in this waveguide

environment is also included for comparison. As in Fig. 4.10, the simulated field profiles

are taken at z = 500 while the measured field profile is taken at z = 480mm. . . . . . . . 62

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4.12 A comparison between superoscillatory-wave-based and evanescent-wave-based sub-wavelength

focusing. (a) the calculated superoscillatory electric field profile on the image plane (red,

solid), as designed for the SFW, plotted alongside a sub-wavelength sinc distribution

(blue, dashed). The latter is generated with uniform superposition of the TE10, TE30,

TE50, TE70 and TE90 modes of the waveguide. (b) The fraction of the total energy on

the screen plane which appears within the sub-wavelength focus at the focal plane, as a

function of the focal distance. For the evanescent-wave-based focusing device, the inten-

sity of the focal profile is generated by integrating |Ey|2 across the entire waveform at the

focal plane; for the superoscillatory wave, it is generated by integrating the same quantity

across the superoscillatory region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.13 Diagram for introducing variations in the sensitivity analysis of the superoscillation focal

spot. The solid arrow depicts a sample excitation or spectral coefficient. The soft blue

circle represents possible signal values after the addition of a variation. A deeper colour

tone represents a higher probability of occurrence. Dashed arrows depict possible variation

vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.14 Sensitivity analysis of the superoscillation focal spot. (a) Typical waveforms obtained

when excitation currents are varied by 1%. (b) Typical waveforms obtained when waveg-

uide mode excitations are varied by 2.5%. In both waveforms, the target waveform is

plotted in red for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.1 A diagram showing concepts of operation for (a) A conventional imaging device for which

both the illumination and the system are diffraction limited. (b) A scanned imaging

device which relies on superoscillatory illumination, and (c) The proposed OSM, which

uses a superoscillatory PSF. The illumination waveform to subfigure (b) is adapted by

permission from Macmillan Publishers Ltd: Nature Materials [B], copyright 2012. . . . . . 70

5.2 Schematic of a 4F imaging system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.3 Designing 2D superoscillatory functions by matching spectral amplitudes. (a) Plane wave

coefficients for the 1D design. Spectral components highlighted by the red rectangle are

used as Bessel function weights for the 2D design. (b) A comparison of the 1D waveform

and a cross-section of the 2D waveform at y = 0, showing noticeable differences and a

displacement of nulls. (c) The comparison as in (b), plotted over the Bloch period of the

1D waveform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

xv

5.4 Designing 2D superoscillatory functions by matching null locations. (a) Plane wave coef-

ficients for the 1D design (blue) and Bessel function coefficients for the 2D design (red).

(b-c) A comparison of the 1D waveform and a cross-section of the 2D waveform at y = 0,

showing well agreement in key performance metrics. (b) is the closeup of (c) across the

ROI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.5 A polar plot of reflection coefficient from the SLM. The arrow indicates the direction of

increasing drive signal level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.6 Superpixel approach to generating arbitrary reflection coefficient from the SLM. (a) Schematic

of the superpixel configuration. Co-ordinates are given by the green arrows. (b) A plot

of the resultant reflection coefficient. The blue and red curves represent reflection coeffi-

cients from pixels A and B, the solid and dashed arrows represent reflection coefficients

for a specific signal set (la, lb), and the green arrow — being the vector summation of the

black arrows — represent the superpixel reflection coefficient rsp(la, lb). . . . . . . . . . . 78

5.7 Superoscillation PSF design for the OSM. (a) Zero locations for the 1D PSF design. Circles

denote zeros within the ROI, Crosses denote zeros without the ROI. (b) The corresponding

plane wave coefficients. (c) Bessel function coefficients for a 2D PSF, sharing the same

nulls as the 1D design. (d) The corresponding reflection filter to be synthesized by the

SLM. (e) Comparison of the spatial profile of the 1D waveform, and the a cross-section of

the 2D waveform at y = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.8 The target spatially varying filter at the Fourier plane. . . . . . . . . . . . . . . . . . . . . 82

5.9 A schematic of the optical super-microscope (OSM). The distance between the five labeled

planes: P1, L1, P2, L2 and P3 are 400 mm apiece. . . . . . . . . . . . . . . . . . . . . . . 83

5.10 A photograph of the OSM prototype, with annotations showing the light path within the

device. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.11 A comparison of measured PSFs for the OSM and the diffraction-limited system. (a) An

image of a 10 µm hole obtained by the OSM, which approximates the system’s PSF. (b)

A close up of the focal point (top), which clearly shows a reduced size compared to the

diffraction-limited focal point (bottom). (c) The horizontal cross-section of the intensity

profile, plotted alongside theoretical calculation. (d) A closeup of (c) across the ROI,

compared against the diffraction limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.12 A comparison of a diffraction-limited imaging system’s PSF with an image it generates of

a small circular aperture. The diffraction-limited imaging system has λ = 632.8 nm and

NA = 0.00864. The comparison shows that the two waveforms are nearly identical. . . . . 86

xvi

5.13 Resolving two closely-spaced objects using the OSM (I). The main panel shows that the

OSM resolves two 15 µm apertures, separated 55 µm center-to-center. The inset shows

the corresponding (unresolved) image for the diffraction-limited system. . . . . . . . . . . 87

5.14 Calculated Result of the 2-point Resolution Experiment. The aperture dimensions and

separation are the same as the same as in Fig. 5.13. The field strength is magnified

fivefold within the white square located in the middle of the ROI. . . . . . . . . . . . . . . 87

5.15 Resolving two closely-spaced objects using the OSM (II). Close ups of two-point resolution

images, for two 15 µm apertures, separated by various distance distances, in (a) horizontal,

(b) vertical and (c) diagonal configurations. Horizontal and vertical separations are 40

µm, 45 µm, 50 µm, 55 µm and 60 µm respectively, from left to right; diagonal separations

measure 35.3 µm, 42.4 µm, 49.5 µm, 56.5 µm and 63.7 µm respectively, from left to right.

The top row shows a close up of the superoscillatory image, while the bottom row shows

the corresponding view with the diffraction-limited system. . . . . . . . . . . . . . . . . . 89

6.1 A schematic of the experimental apparatus for synthesizing temporal superoscillatory EM

waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.2 Design for a rapid temporal superoscillatory waveform. (a) Zero locations, with open

circles denoting zeros inside the design region. (b) The spectral amplitude distribution.

(c) The measured (green, with dots) and calculated (blue) temporal waveforms over one

period, showing excellent agreement. (d) A closeup of (c) across the design region, showing

good experiment-calculation agreement, as well as conformity to the 650 MHz sinusoid

(red, dashed). Black squares label temporal signal inputs to the AWG. . . . . . . . . . . . 103

6.3 Design for a superoscillatory sharp pulse. (a) Zero locations, with open circles denoting

zeros inside the design region. (b) Spectral amplitude distribution. (c) The measured

(green, with dots) and calculated (blue) temporal waveforms over one period, show-

ing excellent agreement. (d) A closeup of (c) across the design region, showing good

experiment-calculation agreement. The calculation and experimental waveforms are com-

pressed 55% and 47% respectively respectively beyond the transform-limited sinc function

(red, dashed). Black squares label temporal signal inputs to the AWG. . . . . . . . . . . . 105

6.4 Typical pulse waveform variations in the presence of 1.5% spectral noise. The original

waveform is the highlighted in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.5 Frequency response of the superoscillatory pulse synthesizing apparatus. . . . . . . . . . . 107

xvii

6.6 The spectrum of waveforms which are truncated at the edge of the design region. This plot

compares the simulated (blue, thick line) and measured (green, with dots) superoscillatory

sharp pulse, as well as the spectrum of a sinc function truncated to the same interval (red,

dashed). It also displays the spectrum of the untruncated periodic waveform (black) for

comparison. Shaded regions denote ranges beyond the bandwidth of the original pulses. . 108

6.7 Design for a sharp temporal superoscillatory radar pulse. (a) Resulting zero locations,

with open circles denoting zeros inside the design region. (b) The spectral amplitude

distribution. (c) The calculated temporal waveform over a period. (d) The measured

temporal waveform, when one period of the waveform is sent to the AWG. (e) Comparison

between the calculated (blue) waveform, the measured (green, with dots) waveform and

the Fourier transform-limited sinc (red, dashed) over the design region. . . . . . . . . . . . 111

6.8 A schematic of the test radar system. Solid arrows denote the signal path of the radar

pulse V (t); the dashed arrow denots the signal path for the impulse I(t) . . . . . . . . . . 112

6.9 Reflection traces of the sinc radar pulse at four image distances: 2.50±0.02 m, 3.00±0.02

m, 3.50± 0.02 m and 4.00± 0.02 m. (top to bottom) . . . . . . . . . . . . . . . . . . . . . 113

6.10 Reflection traces of the superoscillatory radar pulse at four image distances: 2.50 ± 0.02

m, 3.00± 0.02 m, 3.50± 0.02 m and 4.00± 0.02 m. (top to bottom) . . . . . . . . . . . . 113

6.11 Radar range resolution of the superoscillatory pulse (blue) and the sinc pulse (green,

dashed). The scatterer is placed at 3.00 m away from the horn. The 3dB resolutions

are 101 mm and 164 mm respectively. Hence the superoscillatory pulse achieves a range

resolution 38%-improved from that of the sinc pulse. . . . . . . . . . . . . . . . . . . . . . 114

6.12 Simulated radar signals from a pair of object closely spaced in range. (a) Simulated radar

signatures from a sinc radar pulse. (b) Simulated radar signatures from a superoscillatory

radar pulse. In both plots, the black lines denote object locations. . . . . . . . . . . . . . 115

6.13 Simulated radar signatures from a pair of objects spaced 140 mm apart. The top panel

shows the reflection trace for the superoscillatory radar pulse; the bottom panel shows

the reflection for the sinc radar pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.14 Experimental radar signatures from a pair of objects spaced 140 mm apart. The top panel

shows the reflection trace for the superoscillatory radar pulse; the bottom panel shows

the reflection for the sinc radar pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

B.1 A diagram of the antenna which denotes the co-ordinates and defines the angle θ. Each

antenna is represented by a black node, and is separated by distance d from its neighbors. 127

xviii

B.2 Superdirective antenna arrays and their antenna patterns. (Reproduced from Fig. 2.3.)

(a) Current Excitations (array factors) for 3 antennas arrays of length 2λ, with 11, 21 and

31 elements respectively. (b) The corresponding far-field angular distributions, compared

to that of a uniform array. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

xix

Chapter 1

Introduction

1.1 Sub-Diffraction Imaging: Historical Perspectives and Mo-

tivation

The diffractive nature of light poses a fundamental limitation on the resolution of electromagnetic-

wave-based imaging systems. This “diffraction limit”, which restricts the resolution of a lightwave to

roughly half its wavelength, has been well understood since the times and contributions of Abbe and

Rayleigh [1,2]. While overcoming the diffraction limit has been a problem of great interest for more than

a century, it also takes on increasing importance as scientific instrumentation and fabrication reaches

ever deeper into the nanoscale, and the scientific community requires imaging tools with unprecedented

resolution. As thus, the challenge to overcome the diffraction limit has attracted attention and interest

from both industry and academia, and led to various proposals on performing sub-diffraction imaging

— imaging beyond the diffraction limit.

This introductory chapter briefly surveys existing techniques in sub-diffraction imaging. Emphasis

will be given to electromagnetic-wave-based devices which physically form a sub-diffraction image (or

a focal spot) through linear interactions involving the electromagnetic field. Brief mentions will be

given to devices which achieve sub-diffraction resolution relying upon non-linear electromagnetic effects,

material interactions and/or post-processing computations. I will discuss the salient features, advantages

and drawbacks of each emphasized class of devices, and motivate the present work by showcasing the

attractiveness of a sub-diffraction imaging alternative which operates unhindered by limitations present

in imaging devices proposed thus far. This chapter will then state the main thesis objectives and

summarize the organization of this thesis, which documents an endeavor to perform sub-diffraction

1

Chapter 1. Introduction 2

imaging with superoscillatory electromagnetic waves.

1.2 Avenues to Sub-Diffraction Imaging

1.2.1 Slightly Sub-Diffraction Devices

This survey on sub-diffraction imaging begins with a discussion on devices which bring small, but

appreciable, resolution improvements over the diffraction limit. As will be discussed in Chapter 2, it

is somewhat arbitrary to pin the diffraction limit at a specific proportion of the electromagnetic wave,

say half the electromagnetic wavelength, since the exact resolution limitation for an imaging system

is dependent upon many other factors, some of which are difficult to quantify. Such factors include,

but are not limited to, the sidelobe characteristic of an imaging system’s Green’s function, the angle of

illumination of an incident electromagnetic wave and the medium in which the imaged object resides. In

particular, it has been recognized that by accepting larger sidelobe levels, or sweeping an illumination

wave across a range of angles, one can obtain resolution improvements over the diffraction limit [3, 4].

However, such improvements are limited. It has been verified that only slight gains in resolution can be

achieved before sidelobes reach levels comparable to the main peak; likewise, the resolution improvement

from a sweep in illumination angle is limited by the corresponding expansion of the transverse k-space,

which in theory leads to a 50% reduction of the smallest resolvable feature. While resolution improvement

by immersing an object in a high-index medium does not suffer a similar limitation, it represents a method

which reduces the diffraction limit instead of one which images beyond it. This is because the wavelength

of the imaging illumination shrinks accordingly inside this medium. Moreover, the practical difficulty

to find a suitable, matched, high-index material system and the need to immerse an object in such a

material precludes this method from achieving dramatic resolution improvements in a practical system.

As thus while the aforementioned methods provide modest resolution improvements over the diffraction

limit, the search remains for devices which would dramatically improve the imaging resolution beyond

the diffraction limit.

1.2.2 Evanescent Field Based Devices

Of various proposals to surpass the diffraction limit, a very important class achieves this through the use

of evanescent waves. Evanescent waves decay along one or more directions, but oscillate quickly along

directions in which they do not decay, and thus can better capture higher resolution details compared to

propagation waves. Chapter 2 of this thesis will provide a mathematical description of evanescent waves,


while this subsection aims to describe major imaging devices which achieve sub-diffraction imaging using

evanescent waves.

Near-field Scanning Optical Microscopes

In the past century, the near-field scanning optical microscope (NSOM) has become an essential technol-

ogy in obtaining sub-wavelength image resolution. The original idea behind NSOM was first proposed by

Synge [5] and Oseen [6]. In 1972 Ash and Nichols proposed a predecessor of an NSOM, which operated

at microwave frequencies [7]. Finally in the 1980’s Pohl and Betzig [8, 9] introduced the NSOM in its

current form. The NSOM obtains its high resolution by scanning a sharp optical probe across a plane

nanometers away from an object. This probe emanates and/or collects a highly localized electromagnetic

wave, which interacts with the object in the extreme near-field, in which the field localization remains

deeply sub-wavelength. This sub-wavelength localized field contains large amounts of evanescent waves,

which enables this probe to obtain a sub-wavelength image of the object, upon raster scanning it across

a plane hovering over the object. While the NSOM has a typical resolution of about 50 nm, it has two

major drawbacks — a long scan time and the tightly-controlled presence of the optical tip in the extreme

near-field, about 10nm (approximately λ/50) from the object. Recently, other evanescent field based

sub-diffraction imaging devices are proposed, which employed evanescent waves in cleaver manners to

perform sub-diffraction imaging at relaxed working distances. The following surveys a few such methods.

Superlenses

The development of the superlens has been an important and intriguing chapter of evanescent-wave-

based sub-diffraction imaging devices. In year 2000, a seminal work by Pendry [10] showed that a

negative index slab (henceforth called the superlens) would not only give rise to negative refraction, but

could actually amplify evanescent waves, and thus restore high-resolution details inaccessible to classical

imaging systems. Pendry hence conjectured that the superlens could image an object with perfect

clarity, unhindered whatsoever by the diffraction limit. Further, he suggested methods through which the

superlens could be implemented for microwave and optical frequencies. In 2004, Grbic and Eleftheriades

demonstrated superlensing for the first time with a 2D microwave transmission-line metamaterial [11]. A

year later, N. Fang, H. Lee, C. Sun and X. Zhang demonstrated optical sub-diffraction imaging, following

Pendry’s suggestion to use a thin silver slab as a “poor man’s superlens” for TE waves [12].

While the initial superlenses were restricted in terms of the object wave’s polarization and direction

of incidence, much effort has been directed towards making a 3D, isotropic, polarization independent

superlens. Along this front, Iyer and Eleftheriades were first to successfully extend the transmission-line


superlens into 3D in a volumetric fashion [13]; more recently Rudolph and Grbic used a transmission-

line inspired platform and a 3D stereolithographic fabrication technique to demonstrate the first 3D,

fully isotropic, polarization independent superlens [14]. In a related front, Kehr et al. applied the

superlensing effect to improve imaging resolution at mid-IR frequencies [15]. By inserting a perovskite-

based superlens into a near-field scanning scheme, the authors improved the scan resolution compared to

when the superlens was absent. Besides the aforementioned examples, research abounds to develop the

capability of the superlens, extend their frequency and bandwidth coverage, and apply them practically

in building state-of-the-art imaging devices.

While the NRI metamaterials superlens represents a conceptual breakthrough in sub-wavelength

focusing, it suffers a major practical drawback. In particular, the existence of a negative index is

necessarily accompanied by temporal dispersion, which, as dictated by causality, is in turn accompanied

by resonant loss. Dispersion and loss place rather stringent practical limits on the superlens’ imaging

distance and resolution [16–18]. Typical experimental results from sub-wavelength focusing devices have

imaging distances a quarter wavelength from the output facet of the superlens, and resolutions a few

times that of the diffraction limit [11, 14, 19–22]. Active metamaterials may present a way to mitigate

this problem of loss, but working prototypes of improved focusing performances have yet to demonstrate

the practicality of this method.

Hyperlenses and Related Developments

While the superlens forms an image of an object in the near field, the hyperlens images a near-field object

into the far-field with sub-diffraction resolution. Independently proposed in 2006 by Jacob, Alekseyev

and Narimanov [23], and by Salandrino and Engheta [24], the hyperlens is an anisotropic metamaterial

exhibiting a hyperbolic dispersion property, which enables it to convert an object’s evanescent waves

into propagating waves, then image them into the far-field. A year after the concept is proposed,

experimental demonstrations of the first 1D hyperlenses were reported by Smolyaninov, Hung, and

Davis [25] for optical frequencies and Liu et al. for UV frequencies [26].

Since its inception, the hyperlens has seen various conceptual and experimental developments. In

2010 Rho et al. [27] demonstrated a spherical hyperlens capable of 2D sub-diffraction imaging at visible

frequencies, fully integrable with conventional far-field optics. Several variants of the hyperlens, including

the impedance-matched [28], planar [29], oblate [30], and immersion [31] hyperlens, have also been

proposed as sub-diffraction imaging and sub-wavelength focusing devices. A worthwhile mention must

also be given to the cylindrical superlens, which was first proposed by Pendry [32] in 2003 through

applying a co-ordinate transformation to the superlens. Since the cylindrical superlens magnifies a TEM


object wave, it can magnify a sub-wavelength near-field object and hence enable its viewing in the far-

field. In this respect, one can credit it as a first device which features the “hyperlens-like” property —

that of a far-field image distance. More recently, alternative transformations on superlenses yielded the

aforementioned oblate hyperlens [30] as well as a cylindrical superlens design with practically realizable

material parameters [33].

Besides the hyperlens, other sub-diffraction imaging devices have been proposed which convert evanes-

cent object waves into propagating ones, but not necessarily through a direct spatial magnification of the

object waveform. It has been known that a grating generates a linear shift in the transverse spatial fre-

quency spectrum of a waveform; hence its appropriate application can transform evanescent waves into

propagating ones, and thereby allow their transmission into the far-field. The use of a sub-diffraction

(though not necessarily sub-wavelength) grating has been proposed by Lukosz in 1966, in one of the

first works which related image formation to information theory [34]. An imaging system which used

a sub-wavelength grating to convert evanescent waves into propagating counterparts had been demon-

strated by Eckhouse et al. in 2004 [35]. In 2007 Liu et al. reported the so called “far-field optical

superlens” [36], which employed grating and plasmonic effects to convert evanescent waves with about

three times the transverse spatial frequency of red light, and propagated them for far-field observation.

In parallel, a series of works from the research group of Brueck et al. combined imaging views from

different orders of a sub-wavelength grating. These combined views are incorporated with conventional

techniques of angled illumination and high-index immersion media to synthesize a sub-diffraction image

with half-pitch several times below the free-space diffraction limit [37–39]. In a related effort, Malyuskin

and Fusco demonstrated the sub-wavelength focusing capability of a phase conjugation plane, when used

in conjunction with two identical planes of scatterers placed close to the object and image planes [40].

Similarly, Lerosey et al. demonstrated sub-wavelength resolution systems by placing random scatterers

close to a radiating object and afterwards applying a time-reversal operations on the signal received,

either by external detectors or back reflections into the source antennas [41,42]. These proposed devices

concurred in that they achieved sub-diffraction imaging using sub-wavelength scatterers which convert

evanescent to propagating waves.

Indeed the hyperlens and related imaging devices allow one to perform sub-diffraction imaging with

a far-field observation distance. A major limitation nonetheless remains: to faithfully image an object,

the hyperlens (or a scatterer) must be placed at deep sub-wavelength proximity to the object. This

heavily restricts the use of the hyperlens as a sub-diffraction imaging device.


Metascreens

Besides efforts towards achieving sub-diffraction imaging using metamaterials, there is also prevalent

work towards forming sub-diffraction focal spots — and subsequently images — with sub-wavelength

structured surfaces, which one can will collectively term “metascreens”. A metascreen is an electro-

magnetic surface which launches an electromagnetic wave that, after a predefined propagation distance,

evolves into a desired waveform — which is, for most metascreens, a sub-diffraction focal spot. A

metascreen achieves sub-diffraction focusing by generating large components of evanescent waves which

compensate for their decay from the device to the focal plane; it also adjusts the phase of an illumination

wave so the emanating plane waves properly superimpose at the focal plane. Thereafter, a sub-diffraction

image can be formed by raster-scanning the sub-diffraction focal spot across an object. A theoretical

proposal for the metascreen was first reported by Merlin [43], who termed the phenomenon “radiationless

interference”; a working device was first proposed by Wong, Sarris and Eleftheriades [44] based on a

near-field holographic point of view. The field of metascreens has since seen constant development. The

research group of Grbic et al. detailed their design method on the metascreen, based on the plane wave

back propagation method and a method of moments impedance screen design operation [45, 46]. They

further proposed and demonstrated prototypes involving metallic grooves and rings [47,48]. The research

group of Eleftheriades et al. approached metascreen design from a “spatially shifted beams” perspective:

they determined excitation weights to a sub-wavelength spaced antenna array, which, through mutual

coupling and near-field interference effects, would produce a waveform satisfying a set of parameters at

the focal plane [49]. Through this paradigm, they designed metascreens with arrays of slot, dipole and

monopole antennas [49–51], and performed sub-diffraction imaging on metallic objects in free-space and

buried within a dielectric [51,52]. They also extended their work to the optical domain, with plasmonic

slot and monopole arrays designed for near-field optical imaging modalities [53, 54]. Other propos-

als of metascreens include Gordon’s plasmonic waveguide array [55] and Monzon’s varying impedance

sheet [56].

The metascreen directly generates the electromagnetic fields necessary to produce an arbitrary wave-

form. This renders it simpler in construction than the superlens, and makes it immune to resolution

degradation through dispersion and loss. However, the hurdle remains in regards to its limited working

distance. Although in principle nothing prevents one from generating sufficient evanescent waves on

a metascreen to retain the sub-diffraction features of a waveform for any working distance, practical

power and noise considerations limit the working distances of proposed metascreens to at most a quar-

ter of a wavelength [51] — similar to those proposed for superlenses. Thus the search continues for a


sub-diffraction imaging tool with a working distance beyond the evanescent near-field of the device.

Sub-diffraction GRIN Lenses?

Another interesting development in the field of electromagnetic imaging has been the controversy over

whether graded-index (GRIN) lenses can provide sub-wavelength resolution, despite having only positive

refraction indices. Leonhardt et al. theorized [57], then demonstrated [58], sub-diffraction imaging in

a Maxwell’s Fish-Eye GRIN lens when a measurement probe was placed at the pre-calculated point of

image formation. However, Merlin and Zhang, amongst others, argued that the sub-diffraction focusing

effect observed in Leonhardt’s experiments was an artifact of the presence of the measurement probe,

which should not be counted as part of the imaging system [59, 60]. This latter viewpoint corresponds

well to a fundamental corollary of the dispersion relationship — that all components of the wavevector

for a plane wave cannot simultaneously exceed its spatial frequency. It is also worthwhile to note that

similar discussions have arisen a few years prior, in regards to whether the superlens can achieve a sub-

diffraction focus in 3D. This earlier dispute has been resolved by Mesa et al. [61], whose conclusion aptly

applies to the present discussion on GRIN lenses: 3D sub-diffraction imaging is not possible without

prior information on the source or object, and corresponding design of the detector on the image plane.

Nonetheless, sub-diffraction imaging with tailored detectors can become a worthwhile direction of pursuit

in the design of sub-diffraction imaging systems.

1.2.3 Non-Linear and Non-Electromagnetic Imaging Devices

Though the present review concentrates on sub-diffraction imaging devices which operate in the linear

electromagnetic regime, a brief mention is in order regarding other sub-diffraction imaging devices. A

first class of such devices, which include the two photon microscope [62] and various modalities of the Ra-

man spectroscopic microscope [63, 64], achieve sub-diffraction imaging using non-linear electromagnetic

phenomena. In these devices, a signal is released based on non-linear electromagnetic interactions of the

incident photons and resonant levels of the object or its fluorescent label. Since the strength of non-linear

interactions depends on the higher powers of the field strength (instead of just the field-strength itself,

which would be the case for the linear electromagnetic regime), the image formed by these devices can

attain resolution beyond that of the diffraction limit.

In related developments, a second class of imaging device achieves sub-diffraction imaging by isolating

fluorescent markers. Fluorescent markers are often used to tag different parts of a specimen in biomedical

imaging schemes. When illuminated, these markers undergo a physical or chemical process, and reradiate


light at a different (usually lower) frequency which forms the image. It has been proposed that if one

engineers these markers to radiate only from a selective region much smaller than the diffraction limit,

then even though the radiation would form a blurred (diffraction-limited) image, one can still identify

its peak location with sub-diffraction accuracy and deduce that peak to be the location of the radiation.

This is similar to deducing the location of isolated Hertzian dipoles from looking at the far-field antenna

pattern. To this end, methods like stimulated emission depletion microscopy (STED) [65] and ground

state depletion microscopy (GSD) [66] illuminate the object with control beams which selectively switch

off fluorescent markers with sub-wavelength accuracy. To do this, the control beams are shaped such

that they exceed a saturation threshold intensity in regions immediately surrounding a sub-wavelength

area of interest. In this manner, fluorescent markers are saturated outside the sub-wavelength area of

interest, hence achieving the desired isolation. More recently, the demonstrations of stochastic optical

reconstruction microscopy (STORM) [67] and photoactivatable localization microscopy (PALM) [68]

show that photoactivatable fluorescent markers can be prepared and selected individually by tuning the

wavelength of a control beam. These works have attracted great interest in the field of single molecule

imaging.

Yet another class of imaging device achieves sub-diffraction resolution using prior information about

the object. Lukosz [34] proposed early examples of such imaging devices, which traded off some degrees

of freedom on the object field to improve imaging resolution. In another words, if the electromagnetic

field scattered by the object belonged to a particular sub-class of object fields, it could be imaged by

a corresponding device with increased resolution. While the concept explored by Lukosz applies in

principle, more practical implementations emerged in recent years from the field of compressive sensing.

In 2012, Szameit et al. demonstrated a system which reconstructed sub-diffraction images of a set of 100

nm (λ/5) circular apertures, using a post-processing algorithm involving methods of compressive sensing

[69]. In their work the reconstruction algorithm has full knowledge that the images were composed of

100 nm circular apertures. While this subtracts from the attractiveness of the work, their formulation,

which relied on compressive sensing concepts, should generally apply. This leads to hopeful speculation

that sub-diffraction imaging can be performed on sparse objects when one identifies and uses the correct

sampling basis.

While the imaging devices reviewed in this sub-section represent exciting developments useful in

their respective fields of applications, they require combinations of sample preparation, specific material

systems, a priori information (or restriction) on the object field, and extensive data post-processing.

These requirements prevent their use as a tool for general purpose imaging. Hence the development of a

far-field sub-diffraction imaging device with linear electromagnetic operation will be of great use towards


general purpose scientific microscopy.

1.3 Superoscillation and Sub-diffraction Imaging

Whereas the sub-diffraction imaging devices described in section 1.2.2 — namely the near-field probe,

the superlens, the hyperlens (and related devices) and the metascreen — employ evanescent waves

which limit their working distances, a newly emergent class of superoscillation-based sub-diffraction

imaging devices do not suffer this limitation. Superoscillation is a phenomenon whereby over a finite

interval, a waveform oscillates faster than its highest constituent frequency component. With the local

availability of superoscillatory wave components, one can potentially design a device which stores high-

resolution details of an object waveform and reconstructs them at an image plane. In this manner

the need for evanescent waves is removed; hence superoscillatory electromagnetic waves hold promise

for bringing sub-diffraction imaging capabilities into the far-field. The following briefly surveys recent

theoretical developments in the field of superoscillations, as well as experimental developments towards

sub-diffraction imaging.

The mathematical groundwork for superoscillatory waves was laid in a series of works by Slepian et

al. [70], and later emphasized by Aharonov et al. [71] and Berry [72], citing its manifestation in quantum

physical and optical systems. The field has since undergone theoretical developments, with works which

suggest ways to construct superoscillations [72–74] and analyze their properties [75–77]. In particular,

Ferreira and Kempf [73] studied superoscillations in relation to the Nyquist and Shannon limits, and

found that while superoscillatory signals oscillate at a faster rate than the Nyquist limit, they still

abide by the Shannon limit. They also proved that high energy sidebands must accompany the desired

superoscillatory features. The sideband energy (normalized with that of the superoscillatory region)

varies polynomially with the apparent spectral width of the superoscillatory features, and exponentially

with its duration. By suggesting a need for very high sensitivity, this work and others [78] undermined

the practicality of synthesizing superoscillatory waves and using them in imaging systems.

Notwithstanding, sub-wavelength superoscillatory features, vortices, hotspots and imaging devices

have been demonstrated in several works in parallel to this thesis (2008-2013). In 2008, a work by

Dennis, Hamilton and Courtial reported the existence of phase superoscillations in random speckle

patterns. The group showed that a 2D random Gaussian speckle pattern contains a superoscillating

phase in 13 of its area of coverage [79]. Subsequently, in 2010, Brunet, Thomas and Marchiano used

helical beams (which attain superoscillatory phases near their vortices) to measure the location and

size of circular apertures [80]. Although their experiment was done with acoustic waves, similar physics


applies in principle to electromagnetic waves. In regards to amplitude superoscillations, superoscillatory

hotspots were first reported in 2007 by Huang et al., who generated them by passing light through a

quasi-periodic nanohole array [81]. In 2011, Makris and Psaltis theoretically and numerically studied the

formation of diffraction-free superoscillatory waveforms, which directly beamed sub-diffraction features

into the far-field [82]. In the same year, Mazilu et al. and Baumgartl et al. demonstrated sub-wavelength

super-oscillatory focusing with spot size 50% reduced from the diffraction-limited Gaussian beam [83,84].

These foci, however, are immediately surrounded by superoscillatory sidebands, which could hinder the

feasibility of their usage in an imaging device. Notwithstanding, Kosmeier et al. [85], in a work later on

in 2011, reported the use of such a hot spot to bring slight improvements in a sub-diffraction confocal-like

imaging apparatus.

Subsequently in 2012, Rogers et al. reported the so called “super-oscillation lens” [86], which used

diffraction from a patterned metallic plate to generate a sub-wavelength superoscillatory hot spot of

the type reported by [83, 84], at a distance of 10.3 µm away from the plate. This illumination spot

scanned across an object in sub-wavelength steps of 20 nm and 50 nm. An image for each scan step was

obtained with a CCD camera, and then post-processed to reconstruct a super-resolution image. The

operational principle of this super-oscillatory lens was thus similar to that of the NSOM. However the

probe-object proximity requirement was lifted since the sub-wavelength illumination was achieved using

superoscillatory waves. Rogers et al. used an illumination wavelength of 640 nm, a superoscillatory

beam of 185 nm in diameter (intensity FWHM), and reported an experimental resolution of 105 nm.

The fact that the reported resolution was less than 60% of the illumination spot size appears somewhat

unusual even without considering the high amplitude sideband. This can be explained by a few reasons:

(i) An edge-to-edge distance was quoted for circular apertures, which lead to lower minimal resolvable

distances (this will be explained in more detail in Chapter 4); (ii) Advantageous objects (apertures) are

chosen which block sideband penetration along directions crucial to resolution measurements, thereby

reducing possible sideband leakage effects; and (iii) The sub-wavelength scanning operation and the

accompanying post-processing procedure might have produced slight resolution improvements beyond

the diffraction limit for objects chosen in this work. Notwithstanding these caveats, the super-oscillation

lens demonstrates the feasibility of using a superoscillation focus to somewhat improve the resolution of

an imaging system beyond the diffraction limit.

In another interesting recent development is the numerical superoscillation sub-diffraction imaging

method reported in 2012 by Piche et al [87]. In this work the authors reconstructed a complex image

field with a holographic imaging apparatus, then used a superoscillation-based numerical post-processing

procedure to recover sub-diffraction details which were seemingly been lost in the imaging process. More


recently Amineh and Eleftheriades numerically investigated the feasibility of a similar post-processing

procedure using a Tschebyscheff polynomial basis first developed in work leading to this thesis, and pro-

posed applications in microwave imaging [88]. In both cases, the recovery of sub-wavelength information

followed the general framework of an inverse scattering method proposed in a much earlier theoretical

work by Barnes [89]. While Barnes’ numerical recovery method was deemed of limited practicality by

a subsequent work [90], contemporary achievement of high SNR imaging systems may render numerical

superoscillation methods useful in providing sub-diffraction resolution.

The aforementioned developments are a testament to strong research interest in superoscillation-

based sub-diffraction imaging. This strong interest brings both encouragement and motivation. The

author believes a perspective from antenna design would complement well the existing body of re-

searchers and their work on superoscillatory waves. Hence leveraging expertise in electromagnetism and

antenna design, this thesis aims to bring novel perspectives and make unique contributions to the field

of superoscillation-based sub-diffraction imaging. The following section outlines the major objectives of

this thesis and provides a brief organizational preview for the remainder of the thesis report.

1.4 Objectives and Organization

The major objectives of this thesis are threefold. Firstly, this thesis aims to make clear the relationship

between the phenomenon of superoscillation and that of superdirectivity — a phenomenon well known

to antenna designers which shares similar properties with superoscillation. Secondly, this thesis aims to

design and demonstrate superoscillation-based focusing devices which form sub-diffraction foci at working

distances beyond the evanescent near-field, while maintaining practicality from an SNR perspective.

Thirdly, building upon the previous objectives, this thesis aims to demonstrate a superoscillation-based

device capable of sub-diffraction far-field imaging. The present chapter has introduced the topic of

sub-diffraction imaging and motivated this thesis. The following chapters are outlined as below.

Chapter 2 provides mathematical backgrounds which will prove useful to the rest of the thesis.

It begins with an introduction to the plane wave formulation of electromagnetic waves. Upon this

framework it discusses the Abbe, Rayleigh and Sparrow diffraction limits in both the spatial and the

angular domain. It then introduces superdirectivity as a phenomenon which surpasses the angular

diffraction limit.

Chapter 3 elucidates the relationship between superoscillation and superdirectivity. It begins with a

mathematical definition of superoscillation, then by juxtaposing it against superdirectivity, it clarifies the

relationship between the two phenomena. Upon relating these two phenomena, a detailed investigation is


presented which results in the formulation of a design procedure that leverages antenna design techniques

to design superoscillatory waveforms. A theoretical investigation is also presented on the degree to which

one can control, or suppress, the sideband of a superoscillatory waveform.

Chapter 4 reports superoscillation-based 1D sub-wavelength focusing devices. Using a design tech-

nique introduced in the previous chapter, two sub-wavelength superoscillatory focusing devices are de-

signed. The first device utilizes a current screen to focus EM waves to a sub-wavelength spot five

wavelengths away from the screen. The second device performs a similar operation in a rectangular

waveguide. This device was fabricated and tested to obtain a focus 75% the width of a diffraction-

limited focus.

Chapter 5 reports the Optical Super-Microscope — a superoscillation-based 2D sub-diffraction imag-

ing device. It begins by overviewing the concept for a superoscillation-based imaging system. Thereafter,

it extends from 1D into 2D the superoscillatory waveform design procedure reported in Chapters 3 and

4. This design procedure results in spatial domain modulations which are implemented in the Fourier

plane of an optical 4F system. Such spatial domain modulations provide an imaging device with a

superoscillatory point spread function. The chapter details the filter implementation procedure and

the construction of the Optical Super-Microscope, and reports experimental results which demonstrate

sub-diffraction imaging at about 30% improved beyond the diffraction limit.

Chapter 6 diverges from spatial superoscillations to provide a related investigation on temporal su-

peroscillations. Through the same theoretical framework developed in the previous chapters, a design

methodology is presented and utilized to generate temporal superoscillatory electromagnetic waveforms,

which are synthesized and used to improve the range resolution of a radar imaging system. Experimen-

tal demonstrations with one object and two object radar imaging show a 38% improvement in range

resolution beyond the Fourier transform limit.

Finally Chapter 7 summarizes the results of this thesis, outlines its major contributions and suggests

future directions of research.

Chapter 2

Mathematical Preludes

2.1 The Plane Wave Formulation of Electromagnetic Fields

Maxwell’s equations, in phasor form, describe a monochromatic electromagnetic field in a linear, homo-

geneous, isotropic, reciprocal and source-free medium:

∇×E = −jωµH,

∇×H = jωεE,

∇ ·E = 0,

∇ ·B = 0.

(2.1)

From these, one extracts the Helmholtz wave equations, which are written as follows in Cartesian space:

(∇2 + k2)E = 0,

(∇2 + k2)H = 0,

where k = ω√µε =

2π

λ

(2.2)

Here k and λ respectively represent the spatial frequency (also known as wavenumber) and wavelength

of the plane wave. Hereafter, the present discussion concentrates on the electric field, though one

may follow similar operations to define plane waves based on the magnetic field. For simplicity, this

discussion assumes a specialized 3D environment with longitudinal direction z, transverse direction x

and an invariant direction y, in which the electric field points. (If the need arises for an extension into

a fully 3D environment, one can follow a similar treatment but replace x with the transverse direction

(x, y).) In this simplified case one can write the solution to equation (2.2) as

13

Chapter 2. Mathematical Preludes 14

E = E0 exp(−jk · r)y

= E0 exp(−jkxx) exp(−jkzz)y,(2.3)

where kx and kz are related by the dispersion relationship

k2x + k2

z = k2. (2.4)

Equation (2.3) provides a powerful tool for projecting an object field at a selected transverse plane,

say Eobj(x) at z = 0, onto the surrounding sourceless simple medium. To do this, one first converts the

field distribution into its transverse spectrum through a Fourier transform

Eobj(kx) =

∫ ∞−∞

Eobj(x)exp(jkxx)dx. (2.5)

Physically, Eobj(kx) represents the amplitude values of a continuum of plane waves at the object plane

z = 0. In a sourceless region surrounding the object plane, Eobj(kx) can be mapped onto a basis of 2D

plane waves, which superimpose to form the electric field in xz-plane

E(x, z) =1

2π

∫ ∞−∞

Eobj(kx)exp(−jkxx)exp(−jkzz)dkx

=1

2π

∫ ∞−∞

Eobj(kx)exp(−jkxx)exp(−√k2x − k2z)dkx

=1

2π

∫ ∞−∞

E(kx, z)exp(−jkxx)dkx,

(2.6)

where

E(kx, z) = Eobj(kx)exp(−√k2x − k2z). (2.7)

E(kx, z0), where kx ∈ [−∞,∞], represents the plane wave spectrum, or angular spectrum, along

a plane z = z0 ≥ 0 (choosing z0 < 0 requires a change in sign convention in (2.6)). Two types of

plane waves — propagating and evanescent — form the plane wave spectrum. When (|kx| ≤ k), one can

notice from the integrand of (2.6) that corresponding plane wave components undergo phase delay in the

+z-direction. Plane waves which satisfy this condition are called propagating waves. Conversely, when

(|kx| > k) the integrand of (2.6) shows that the corresponding plane wave components exponentially

attenuate in the +z-direction. Such plane waves are called evanescent waves. In theory, if one measures

the electric field at an observation plane z = z0, he could deduce the field at the object plane z = 0. One

way to do this would be to invert the operations in equations (2.5) to (2.7). In practice, however, while


retrieving propagating waves will indeed be possible, it becomes impossible to retrieve evanescent waves

once their amplitudes have dropped below the noise floor. This precludes the retrival of evanescent waves

for z0 & λ for most practical imaging systems. Due to the absence of evanescent waves, conventional

imaging systems attempt to reconstruct the original object waveform — either physically or numerically

— using only propagating waves from the plane wave spectrum. The next section describes how this

leads to the diffraction limit.

2.2 The Diffraction Limit

This section derives a few variations of the diffraction limit, which illustrate how a limited bandwidth

in the spatial frequency domain affects waveform variation in the spatial domain. While the discussion

focuses upon the particular case of electromagnetic wave diffraction, the general principle which underlies

the diffraction limit applies to any pair of reciprocal domains. This more general principle is called the

uncertainty principle. A mathematical derivation for a common expression of the uncertainty principle

— the Heisenburg uncertainty principle — is shown in Appendix A.

2.2.1 Abbe’s Diffraction Limit

As mentioned in the previous section, the unavailability of evanescent waves directly leads to the diffrac-

tion limit. Since plane waves form an orthogonal basis of a waveform in a simple medium, in the absence

of evanescent waves the closest approximation of a waveform is given by

E′obj(x) =1

2π

∫ k

−kEobj(kx)exp(−jkxx)dx

=1

2π

∫ ∞−∞

DEobj(kx), k

exp(−jkxx)dx,

(2.8)

where D· denotes the delimitation operation

D f(x), x0 =

f(x) for |x| ≤ x0

0 otherwise.(2.9)

This is the inverse transform to (2.5), except its integration limits have been truncated to exclude

evanescent waves. The mathematical uncertainty principle [91] postulates that a roughly reciprocal

relation exists between a function’s width in one domain and its minimum achievable width in the

reciprocal domain. As such, since DEobj(kx), k

has compact support, E′obj(x) contains a minimal

width inversely related to the support of DEobj(kx), k

. This minimal width requirement on E′obj(x)


constitutes the diffraction limit. In regards to imaging with electromagnetic waves, Ernst Abbe is

credited with the derivation of the diffraction limit [1, 92]. Abbe recognized that the spatial bandwidth

of an imaging system has support from [−NAk,NAk], where NA = sin θ denotes the numerical aperture,

and θ represents the half angle of the effective aperture of the imaging system. Assuming all propagating

plane waves in this section add in equal proportions and aligned phase, one obtains a waveform known

as the diffraction-limited sinc function:

EDLS(x) =1

2π

∫ NAk

−NAkexp(−jkxx)dkx

=sinNAkx

πx

=NAk

πsinc(NAkx).

(2.10)

The width of this sinc function, as measured from its peak to its first null, is

∆xAbbe =π

NAk=

λ

2NA. (2.11)

Equation (2.11) is commonly known as Abbe’s diffraction limit. In applications where a waveform’s

null may not be well defined, it becomes useful to measure the width of the waveform by its 6dB width

— its full width measured from the points when its electric field attains half its peak amplitude. For

the diffraction-limited sinc, this measurement yields

∆xF ·FWHM =0.603λ

NA. (2.12)

Alternatively, one can also measure the waveform’s 3dB width — its full width at the point when its

intensity (proportional to the electric field squared) attains half the of its peak value. For the diffraction-

limited sinc, this measurement comes to

∆xI·FWHM =0.443λ

NA. (2.13)

For cleaner notation, the subscripts ∆x will hereafter be abbreviated in reference to the aforementioned

diffraction limits. Instead, the quantity involved will be communicated through context and by referral

to the corresponding equation in this section.


−100 −50 0 50 1000

0.5

1

x [µm]

Nor

mal

ized

Inte

nsity

0.82λ / NA

0.80λ / NA

0.75λ / NA

0.70λ / NA

0.61λ / NA

Figure 2.1: Image profiles of two deeply sub-wavelength circular apertures, separated atvarying distances. The apertures were illuminated at normal incidence (i.e. in phase)by a coherent laser source. The legend shows the center-to-center aperture separationsof the corresponding image profiles. The figure shows the coherent Sparrow limit at0.75λ/NA and the coherent Rayleigh limit at 0.82λ/NA.

2.2.2 Rayleigh’s Diffraction Limit

While Abbe, with his equation on the diffraction limit, described an imaging system’s resolution through

the spot size of the smallest possible “pixel”, Rayleigh described resolution through the minimum re-

solvable spacing between two objects [2]. Rayleigh noted that in incoherent imaging systems, when two

point objects are separated such that the peak of the image waveform of point object 1 coincides with the

first null of the image waveform of point object 2, then their intensity superposition will include a small

(27%) dip at the mid point between the supposed location of two objects. This would characterize the

minimal resolvable separation for the imaging system. Since Rayleigh considered a 3D imaging system

with a 2D transverse plane, the delimitation in the kx-domain becomes that of a circular aperture. The

corresponding image waveform is hence the 2D Fourier transform of the circular aperture — the 2D Airy

function, for which the peak-to-null width is commonly used to quote Rayleigh’s resolution criterion:

∆xRayleigh =0.61λ

NA. (2.14)

For imaging devices with high SNR, one can resolve two point objects, as long as their intensity super-

position curve shows any kind of a dip. The Sparrow’s criterion [93] describes this minimal resolvable

distance, which for two circular apertures read

∆xSparrow =0.47λ

NA. (2.15)

For coherent imaging systems, imaged waveforms superimpose in vectorial fashion, hence the resolution


should be characterized by adding electromagnetic fields instead of waveform intensities. Fig. 2.1 shows

such superpositions for two small apertures spaced set distances apart. This figure shows the coherent

equivalents of the Rayleigh and Sparrow limits at

∆xC·Rayleigh =0.82λ

NA;

∆xC·Sparrow =0.75λ

NA.

(2.16)

2.2.3 Summary and Comparison on Diffraction Limits

Thus far in this section we have introduced the Abbe, Rayleigh and Sparrow diffraction limits. All

aforementioned limits are due to the fact that the waveform has limited support in spatial frequency

domain. Abbe concluded from this that the waveform must have a minimal width in the spatial do-

main; Rayleigh concluded that this minimal width would result in a minimal resolvable features in an

object. However, these diffraction limits differ in their scopes of consideration: the Abbe diffraction

limit represents the theoretical ideal, while the Rayleigh and Sparrow diffraction limits take into account

the imaging system. Hence the Rayleigh and Sparrow diffraction limits are particular for the imaging

system being considered. Specifically, these limits would change if, instead of a circular (angular) aper-

ture, and the imaging system contains a rectangular or a slit aperture. This imaging system-dependent

property of the Rayleigh (and Sparrow) diffraction limit make it more practical from the perspective of

evaluating system performance, but more complicated in consideration of a theoretical study. Therefore

most of this work facilitates comparison using the Abbe diffraction limit, which remains irrespective of

limitations arising from an imaging system. Notwithstanding, from the functional similarity in equations

(2.11) to (2.16), it is clear that an imaging system which breaks the diffraction limit the Abbe sense

should also break the diffraction limit in the Rayleigh and Sparrow sense when the system employs a

circular aperture.

The previous subsections have introduced the Abbe and Rayleigh diffraction limits as resolution

limits in the spatial sense. This perspective on the diffraction limit can be more specifically called the

spatial diffraction limit. Hereafter this work will use interchangeably the terms diffraction limit and

spatial diffraction limit; in particular the latter will be used when a distinction needs to be clarified with

respect to the angular diffraction limit — a concept to be introduced in the proceeding subsection.

2.2.4 Angular Diffraction Limit

In the above subsections we discussed the diffraction limit as a physical effect which limits resolution

in the spatial domain. In parallel to this, the diffraction limit can also be understood as a resolution


limitation in the angular domain. Through a process very similar to the previous subsections, one can

conclude that when the extent of the electric field E(x) is limited (i.e. to the effective antenna aperture),

the resulting plane wave spectrum E(kx) will also have a minimal width. Mathematically,

EDLS(kx) =

∫ ∆x2

−∆x2

exp(jkxx)dx

=2 sin kx∆x

2

kx

= ∆x sinckx∆x

2.

(2.17)

In a far-field observation plane, the diffraction-limited plane wave spectrum EDLS(kx) gets mapped

into an angular distribution

Eff (θ) = CEDLS(k cos θ),

= C ′ sinck∆x cos θ

2,

(2.18)

where C is a complex scale factor, C ′ = C∆x and θ is the angle between the observation direction and

the x-axis. The angular width of the beam, measured in the Abbe sense, is hence some ∆θ for which

k∆x cos(π2 −∆θ)

2= π

⇒ cos(π

2−∆θ) = sin ∆θ =

λ

∆x

⇒ ∆θ = sin−1λ

∆x∼=

λ

∆x,

(2.19)

where the last step uses a small angle approximation, which usually applies when one is solving for a

minimum resolvable angular spread. Hence under a far-field, small beam angle scenario, equation (2.19)

is the angular version of Abbe’s diffraction limit; clearly the minimal angular spread of the waveform is

inversely proportional to its extent in the spatial domain, consistent with our discussion on the spatial

diffraction limit. In similar spirit, one can derive an angular version of the Rayleigh diffraction limit

following our discussion in subsection 2.2.2.

2.3 Evanescent-Wave Based Imaging

After our preceding mathematical treatment on the diffraction limit, it becomes easy to understand

how evanescent-wave based imaging devices achieve sub-diffraction imaging. To see this, one can begin

by writing Abbe’s diffraction limit (2.11) in a way to explicitly depict the waveform’s width in the


Figure 2.2: Evanescent-field-based sub-diffraction imaging. Colors represent phase pro-gression in (b) and intensity level in other plots. (a) The field pattern at a plane λ/20away from a z-directed small dipole. This pattern is tightly localized to within a circleof radius below λ/100. (b) Electric field progression for a line source that is imaged bya superlens. Solid white lines denote the location of the superlens, while dash whitelines denote the locations of the source (to the left of the lens) and image (to the rightof the lens). A region of rapid field oscillations surrounding the superlens’ output facetindicates the dominance of evanescent waves. (c) The spatial spectrum correspondingto (a), showing large evanescent-wave content. The black dashed circle separates thepropagating spectrum (inside) from the evanescent spectrum (outside). (d) The spec-tral evolution corresponding to (b), showing large evanescent field components near theoutput facet of the superlens.

kx-domain:

∆x =2π

∆k, where ∆k = 2NAk. (2.20)

Abbe diffraction places a hard limit on ∆k and hence restricts the confinement on ∆x:

NA ≤ 1⇒ ∆k ≤ 2k =4π

λ⇒ ∆x ≥ 2π

4π/λ=λ

2. (2.21)

However, evanescent-wave based imaging devices allow one to access regions of the kx domain where

|kx| > 1, hence freeing ∆k and ∆x from the above limitations. Fig. 2.2 shows two sub-wavelength field

distributions alongside their plane wave spectra to illustrate the fact that evanescent waves accompany


Figure 2.3: Superdirective antenna arrays and their antenna patterns. (a) CurrentExcitations (array factors) for 3 antennas arrays of length 2λ, with 11, 21 and 31elements respectively. (b) The corresponding far-field angular distributions, comparedto that of a uniform array.

sub-wavelength field confinement. Fig. 2.2a shows a sub-wavelength near-field source; Fig. 2.2b shows

the transverse field distribution of a sub-wavelength source imaged through a superlens. Their corre-

sponding plane wave spectra are shown in Fig. 2.2c-d. This figure verifies a fact which holds in general

for evanescent-wave based imaging devices: that the dominance and proper deployment of evanescent

waves provide sub-diffraction imaging capabilities for these devices.

2.4 Superdirectivity

One can view superdirectivity, a concept from antenna array design, as a method to break the angular

diffraction limit introduced in Section 2.2. Up till the discovery of superdirectivity, traditional belief

held that an antenna array produced the narrowest (most directive) beam when driven in uniform

fashion, because in such a case radiation from all elements would interfere constructively at broadside

(where the effective aperture is widest), and destructively in all other directions. However, in a seminal

work in 1943, Schelkunoff [94] proposed a method for designing current excitations on an antenna


array which allows one to arbitrary squeeze the main radiation beam without increasing the overall

antenna size. Fig. 2.3a shows current excitations from three arrays of isotropic antennas, all of overall

length 2λ; Fig. 2.3b compares their beam patterns with that from a uniform antenna array. (These

current excitations are designed using a method based on Tschebyscheff functions first proposed by

Dolph [3]. The interested reader may refer to Appendix B for more details and numerical examples

on designing Dolph-Tschebyscheff antenna arrays.) Since the antenna beam is made arbitrarily narrow

without changing the size of the antenna array, (2.19) has seemingly been violated. Hence the angular

diffraction limit is clearly overcome with superdirective antennas.

2.4.1 Schelkunoff’s Superdirectivity

A mathematical understanding on the design of a superdirective antenna would help us understand how

it squeezes an antenna beam beyond the angular diffraction limit. The electric field of an antenna array

at the antenna axis is given by the convolution of the field radiated by an individual antenna and the

structure of the array, as weighted by excitation currents:

Eant(x) = Eelem(x) ∗ g(x), where g(x) =

N−1∑n=0

anδ(x− nd− x0), (2.22)

where Eelem(x), N , an, d and x0 represent respectively the electric field from one antenna element,

the number of antennas in the array, the current excitation coefficients, the elemental spacing and the

location of the first array element. The far-field antenna pattern, essentially a mapped Fourier Transform

derived in a similar process as equations (2.17) and (2.18), is given by

Eff (θ) = CEff ·elem(θ)AF (θ), (2.23)

where C is a complex scale factor, Eff ·elem(θ) and AF (θ), termed the elemental and array factors, are

Fourier transforms of Eelem(x) and g(x) respectively, upon the mapping

θ = cos−1(kx/k). (2.24)

Since the elemental factor varies much slower than the array factor, the latter often defines the

antenna pattern. The array factor can be expanded as follows:


AF (θ) =

N−1∑n=0

an exp(jk cos θ(nd− x0))

= ejkx0

N−1∑n=0

anzn,

where z = ejkd cos θ.

(2.25)

Alternatively, the polynomial in (2.25) can be written in factored form:

AF (θ) = C ′N−1∏n=1

(z − zn), (2.26)

where C ′ = aN−1ejkx0 and zn are the zeros of the polynomial. Hence, one can design an N -element

antenna array by choosing N − 1 zeros in the complex z-plane, then obtain the corresponding excitation

constants an by simply expanding the right side of (2.25). Further, the portion on the z-plane of

relevance to the antenna pattern forms an arc on the unit circle, stretching from z = e−jkd to z = ejkd.

Specifically, when d < λ/2, this arc subtends an angle less than 360 from the origin of the complex

z-plane. This arc marks the visible region of the antenna array; the remainder of the unit circle forms

the invisible region of the antenna array. Schelkunoff, upon proposing this zero-based method of antenna

array design, mathematically demonstrated that by concentrating zeros along the visible region, one can

generate an antenna beam sharper than that generated with uniform current excitation. Antennas which

generate such antenna beams have since been known as superdirective antennas.

2.4.2 A Fourier Perspective on Superdirectivity

Examining superdirectivity from the Fourier perspective (i.e. with respect to the plane wave spectrum)

clearly elucidates how the superdirective antenna overcomes the angular diffraction limit. Applying

the transformation (2.24), it is simple to see that the visible region, which describes the angular range

θ ε [0, 180], corresponds to the region of propagating waves (|kx| ≤ k) in the spatial frequency domain.

Conversely, the invisible region corresponds to the region of evanescent waves (|kx| > k) in the spatial

frequency domain. Mindful of this, it is now fruitful to examine Fig. 2.4, which shows the transverse

spatial spectra alongside current excitations for the three superdirective antennas displayed in Fig. 2.3.

While the spectra in Figure 2.4b contain tight peaks and low sidelobes in the region of propagating

waves (which correspond to the antenna pattern), they also contain huge sidebands in the region of

evanescent waves. It can be reasoned that the huge sideband in the evanescent wave region results

from the concentration of zeros within the propagation region. Hence their existence is inevitable for


Figure 2.4: Superdirective antenna arrays and their plane wave spectra. (a) CurrentExcitations (array factors) for 3 antennas arrays of length 2λ, with 11, 21 and 31elements respectively, for which the antenna patterns are displayed in Fig. 2.3 (b) Theplane wave spectra plotted in log-scale, showing the existence of large evanescent wave(|kx| > k) components.

superdirective antennas. Therefore, superdirective antennas have wide waveform widths (large ∆k) in

the transverse spatial frequency domain within accordance to the uncertainty principle (see Appendix

A). However, in the formation of an antenna’s far-field, only the propagation spectrum is mapped;

evanescent waves are invisible to the far-field. In this manner, the narrow peaks in the propagating

spectra map into highly directive beams whose angular widths surpass the diffraction limit.

2.4.3 Superdirectivity and the Spatial Diffraction Limit

While it is well known that a superdirective antenna can focus the direction of electromagnetic radiation

beyond the angular diffraction limit, one might speculate whether it can also overcome the spatial

diffraction limit. Such speculation might stem from two observations: that the superdirective antenna has

a large evanescent spectrum, and that the antenna beam angle can, at least conceptually, be arbitrarily

squeezed in the antenna pattern. This section examines these two observations and show that they do

not give superdirective antennas the ability to image beyond the spatial diffraction limit.

This discussion begins with the first observation. It is indeed true that superdirective antennas


contain large evanescent spectra, and that sub-wavelength field variations might be observed within its

near-field, in similarity with previously discussed evanescent-field-based imaging devices. However, dis-

tinction must be made in the evanescent fields purpose of existence in these two cases of sub-diffraction

imaging: in near-field imaging schemes, evanescent-field-dominated near-field profiles perform the imag-

ing, whereas for superdirective antennas, evanescent fields can be viewed as by-products of the antenna

pattern design process, which become invisible in the regime where the image (the desired antenna pat-

tern) is formed. Due to this difference in purpose of existence, evanescent waves present at the near-field

of a superdirective antenna are not optimized for near-field sub-diffraction imaging purposes. Of course,

one can alter the current excitations of a superdirective antenna to achieve near-field sub-wavelength

imaging or vice versa. For example, Ludwig et al. [95] showed numerically that, when a slot antenna

metascreen [49,53] undergoes a slight shift in illumination frequency, the metascreen shifts its operation

from a superdirective mode to a near-field sub-wavelength focusing mode. However, the slot array ceases

to be superdirective in the latter mode, and the device becomes a sub-wavelength spaced antenna array

used purely for near-field sub-diffraction imaging.

The second observation can be cast as the following question: can a superdirective antenna be

designed to form a sub-wavelength far-field focal spot? The antenna far-field is nominally described by

the relation,

rff ≥2D2

λ, (2.27)

where D ' Nd represents the overall antenna size. The fractional term on the right side of the inequality

is named the far-zone distance rff . At this distance the spatial beamwidth is given by

∆xff = rff∆θ ≥ 2D2∆θ

λ. (2.28)

A caution must be given at this point: in relation to our earlier equations on the angular diffraction

limit, the antenna size D in equations (2.27) and (2.28) correspond to ∆x in equations (2.17) to (2.19);

the quantity ∆xff in (2.27) and (2.28) represent the spatial width of the waveform at a distance rff

away from the antenna — not the extent of the field at the antenna plane or axis.

Since, in principle, the beam angle ∆θ of a superdirective antenna can be arbitrarily squeezed without

changing the antenna size D, one might suppose the spatial beamwidth can also be arbitrarily squeezed

by taking ∆θ to zero. This is, however, in direct contradiction with (2.19). The apparent contradiction

between (2.19) and (2.28) is resolved when one realizes the fact that the far zone distance is defined

in a relative sense: that phase errors reaching this distance from the center and from the edge of this


Figure 2.5: Field evolutions from three antenna arrays, of electrical sizes 5λ, 2λ and λrespectively, which are designed to have similar far-field antenna patterns. The antennaaxis is at z = 0.


antenna do not exceed ∆φ = π/8 [96]. (An equivalent derivation from the optics community begins

with ∆φ = π/2, and ends up with an equation similar to (2.27), but with D replaced by R = D/2 [97].)

However, as noted in [98], this relation does not suffice for superdirective antennas, which contain

rapid spatial phase variations on the antenna plane, and hence require better phase agreement then

the figures mentioned above. This intolerance to phase difference renders inadequate the traditional

Rayleigh distance as recorded on the right-hand side of (2.27); instead one needs to travel even further

from the antenna before the far-field antenna pattern is formed. Hence, as one increases the gain of

the superdirective antenna, ∆θ decreases but at the same rff increases. As a result ∆x does not

get arbitrarily squeezed. This fact is illustrated in Fig. 2.5, which shows field evolutions from three

superdirective antennas of lengths 5λ, 2λ and λ, but nevertheless are designed to have similar far-field

beam angles (hence they have different degrees of superdirectivity). If one qualitatively defines the onset

of the far-field as the place where the first sidelobe decreases to half the field amplitude of the main

beam, then it can be seen from Fig. 2.5 that for these three antennas the far-field region begins at

about 20λ away from the antenna. This clearly shows the inapplicability of (2.27); furthermore, it shows

that increasing an antenna’s superdirective gain also pushes out its far-field regime. Fig. 2.6 shows the

onset of the far-field, as well as the spatial width (the electric field full width at half maximum) for

superdirective antennas of varying directivities. We see from this figure that squeezing an antenna’s

angular beamwidth does not necessarily squeeze its spatial width at the onset of the far-field; it might

actually widen it. This analysis agrees with the uncertainty principle: one should not be able to obtain

sub-wavelength field localization with a waveform limited to propagating waves. We are thus left to

conclude that while a superdirective antenna overcomes the angular diffraction limit, it does not also

overcome the spatial diffraction limit.

2.5 Concluding Summary

This chapter reviewed electromagnetic concepts which relate closely with the analysis of superoscilla-

tory waves. It began with the plane wave spectrum, which provides a convenient tool-set for solving

imaging type electromagnetic problems, and lends directly to a Fourier perspective on electromagnetic

waves. Within this formulation a discussion was given regarding the origin of the diffraction limit,

various ways of measuring this limit, and distinguished between spatial and angular diffraction limits.

Upon this mathematical understanding of the diffraction limit, the chapter revisited evanescent-wave

based sub-diffraction imaging devices, and explained how they operated to achieve resolution beyond

(or “side-stepping”) the diffraction limit. Finally, a review was conducted on Schelkunoff’s method of


Figure 2.6: A plot of the far-field distance, and the corresponding beamwidth at thatdistance, for Tschebyscheff superdirective antennas with varying numbers of elements,with fixed electrical lengths of 2λ, and target sidelobes ripples at 20% the field-strengthof the main beam.

superdirective antenna design.

Along with the above-mentioned reviews the chapter presented a Fourier perspective on superdirective

antennas, and investigated the plausibility for a superdirective antenna to surpass the spatial diffraction

limit. The conclusions was to the negative: that a superdirective antenna cannot surpass the spatial

diffraction limit. These represent two contributions from my study towards the present Ph.D. degree.

While superdirectivity does not provide a solution to imaging beyond the spatial diffraction limit, it does

suggest the existence of a spatial domain-analogue — a phenomenon which allows one to image beyond

the spatial diffraction limit. The following chapter will show that this very phenomenon is none other

than superoscillation in the spatial domain.

Chapter 3

An Antenna-Based Theory on

Superoscillatory Waves

3.1 An introduction to Spatial Superoscillations

Whereas the superdirective antenna fails to bring propagating waves into a sub-wavelength focus, super-

oscillations have the potential to do just this. Superoscillation is the phenomenon whereby a waveform

locally oscillates faster than its highest constituent frequency component. Hence by superimposing a

series of these waveforms, one forms a composite waveform which achieves, locally, a widened effective

spectrum. This allows the formation of foci and other wave patterns with sub-diffraction spatial features.

However, along with the desired superoscillatory features come high energy sidebands outside the region

of superoscillation, the energy for which varies polynomially with the superoscillatory region’s apparent

spectral width, and exponentially with its duration [73].

3.1.1 Characterizing Spatial Superoscillations

Superoscillation functions can be broadly classified into two classes: phase superoscillations and am-

plitude superoscillations. Phase superoscillation describes a waveform which becomes superoscillatory

due to rapid phase transitions. For example, along a circular path of radius r, which centers upon an

electromagnetic vortex of order n, every traverse around the vortex incurs a phase change

∆ψ = 2nπ. (3.1)

29

Chapter 3. An Antenna-Based Theory on Superoscillatory Waves 30

The local frequency, as seen along the path of this traverse, is described by

klocal =∂ψ

∂s' ∆ψ

∆s=

2nπ

2πr=n

r, (3.2)

where s indicates distance along the direction of traverse. Equation (3.2) clearly diverges as one takes

r → 0 by shrinking the path along the vortex. Hence superoscillations are found near electromagnetic

vortices and electromagnetic points of singularity. In general, phase superoscillation occurs in a waveform

whenever klocal = ∂ψ∂s is greater than the largest constituent wavenumber which comprises the waveform.

The existence of amplitude superoscillations is less well defined. It has been proposed [92] that both

amplitude and phase superoscillations be measured as a derivative with distance. For example, if a

function is described by u(s) = A(s)ejψ(s) where A(s) and ψ(s) denote the amplitude and phase profiles

of the waveform, then it is proposed that its spatial frequency be described by

kcplx·local =∂

∂slog u(s) =

1

A(s)

∂A(s)

∂s+ j

∂ψ(s)

∂s. (3.3)

Here the real part of kcplx·local describes the local spatial frequency due to amplitude modulation, while

its imaginary part descries the local spatial frequency due to phase modulation. It is suggested that a

superoscillation occurs whenever the magnitude of kcplx·local goes beyond the highest constituent fre-

quency which forms the waveform. While this elegant description leads to a straightforward determina-

tion regarding the existence of a superoscillation, it fails to characterize waveforms with large amplitude

modulations. In particular, the amplitude term diverges when A(s)→ 0 but ∂A(s)∂s 9 0. A consensus has

not been reached on an optimal way to measure an amplitude superoscillation. Notwithstanding, meth-

ods exist which sensibly measure the local frequency for selected classes of waveforms. Firstly, the local

frequency of a waveforms which oscillates quickly within a region of interest (ROI) can be determined by

a sinusoid fitting function. Secondly, the local spectrum of a waveform with a ROI can be calculated by

first truncating the waveform with an appropriate windowing function, then applying a Fourier trans-

form. Thirdly, for sharp waveform features which approach the diffraction limit, a practical indication of

the existence of superoscillations would be the narrowing of such waveforms features beyond that of the

diffraction limited sinc function, without corresponding trade-offs in parameters such as sidelobe ripple

levels within the ROI. In these three conditions, the superoscillation can be quantitatively estimated by

the ratio


S =

klocalkmax

for Method 1

∆klocal∆k for Method 2

∆x∆xlocal

for Method 3

, (3.4)

where the subscript local denotes effective parameters within the ROI, and kmax, ∆k and ∆x, describe

respectively the maximum constituent spatial frequency, the spatial bandwidth and the spatial width

of a waveform. A waveform superoscillations in a region in which S > 1, moreover the magnitude

of S determines the degree to which a waveform superoscillates. These three criteria apply well to

superoscillatory waveforms developed in this work.

3.1.2 Relating Superoscillations to Superdirectivity

How do superoscillations occur when they seemingly violate the uncertainty principle [91], which requires

∆K∆x ≥ C? (3.5)

Here C is a constant, usually of order unity, whose exact value depends on the way in which the spatial

and spectral widths are measured. The answer to this question becomes clear with a simultaneous glance

at a superoscillation profile in the spatial and spatial frequency domains. While the superoscillatory

waveform occupies a limited spectral width — namely the region of propagating waves, they can generate

arbitrarily fast oscillations and narrow peaks within a certain stretch of the spatial domain, as long as

one tolerates high energy sidebands to occur outside the spatial stretch. In the terminology introduced

in the previous subsection, superoscillation can help reduce ∆xlocal without changing ∆K, such that

∆K∆xlocal ≥ C. (3.6)

However, due to the high-energy sidebands outside the ROI (3.5) is not violated. This process is

exactly analogous to superdirectivity: only now the spatial (x) and spatial frequency (kx) domains

are flipped. Conversely, one can claim that superdirectivity is actually a superoscillation phenomenon

in the spatial frequency domain, which is somewhat disguised as high energy sidebands are pushed into

the evanescent region and remain invisible to the antenna far-field. As spatial superoscillations lack

an equivalent of the far-field transformation process applicable for superdirective antennas, their high

energy sidebands remain visible. Nonetheless, in this manner one can generate a sub-wavelength focus

employing propagation waves, and thus extend spatial sub-diffraction imaging capabilities to working


distances beyond the evanescent near-field of the object and the imaging device.

3.2 Superdirectivity-Inspired Superoscillation Design

Having established a dual relationship between superdirectivity and superoscillation, it would now prove

advantageous to design superoscillatory waveforms using well established techniques in antenna array

design. In particular, this section formulates the design of superoscillatory waveforms through an adap-

tation of Schelkunoff’s theory of antenna design as introduced in Chapter 2.

3.2.1 A Schelkunoff Approach to Superoscillation Design

One can choose to form the superoscillatory waveform by superimposing a set of N plane waves within

the propagation band, equispaced in transverse spatial frequency domain, with a spacing of ∆k. In this

manner, the plane wave spectrum reads

Eso(kx) =

N−1∑n=0

anδ(kx − n∆k − kx0), (3.7)

where kx0 denotes the location of the plane wave with the least (most negative) transverse spatial

frequency. To keep this wave a propagating wave, one simply requires that these plane waves spread

across the allowed propagation band. For example, for an imaging system with a set numerical aperture,

one might maximize the usage of this aperture by requiring

kx0 = −NAk; (N − 1)∆k = 2NAk. (3.8)

In the spatial domain this superposition results in the following waveform:

Eso(x) = C

N−1∑n=0

anzn = CaN−1

N−1∏n=1

(z − zn), where C =e−jxkx0

2π, and z = e−jx∆k. (3.9)

From (3.9), it is clear Eso(x) can be designed by placing the zeros zn of the polynomial. Analogous

to the case of superdirective antennas, the portion of the complex function Eso(z) which is of ultimate

interest upon the unit circle, because the unit z-circle maps to real values of x, as evident from the

definition of z in (3.9). Since the target waveform is constructed as a discrete sum of plane waves, the

resultant waveform is periodic with Bloch period

ΛB =2π

∆k. (3.10)


This corresponds to the change in x in one traversal around the unit z-circle.

The aforementioned construction somewhat simplifies the analysis by limiting the degrees of freedom.

It also corresponds well to most implementation scenarios which generate superoscillatory wavefront

through a finite number of excitation points, or by a filter with finite degrees of freedom. Moreover,

a discrete analysis will in many ways approach a continuous one, when one increases the number N

of plane waves involved in the superposition and correspondingly decreases the spectral separation ∆k

between each plane wave. In this limit, the Bloch period approaches infinity, and for finite values of x the

discrete-space Fourier transform achieves a mapping which resembles that of a continuous-space Fourier

transform. Hence while a development has been adopted which is discrete in the spatial frequency

domain, this does not restrict one from studying or designing superoscillatory waveforms which are

aperiodic in the spatial domain.

3.2.2 Design Examples

To illustrate the principle of function design using the Schelkunoff method, this subsection examines two

well known functions in regards to their zero locations, and apply principles inspired by superdirectivity

to turn these functions into corresponding superoscillating functions. The first function is the oscillating

cosinusoid. In the polynomial representation of (3.9), the fastest oscillating cosinusoid will be given by

a set of excitation parameters an = 1, 0, 0, ..., 1, thus the waveform can be written as

Ecos(z) = C(zN−1 + 1), (3.11)

which factors to give the set of zeros

zn = e−jπ(2n−1)N−1 where n = 1 .. N − 1. (3.12)

As one can expect, for this fast-oscillating cosinusoid all available zeros are placed on the unit circle,

in an evenly spaced manner. A second function which we will examine is the function formed when all

N plane waves are excited in uniform amplitude and phase. This function approaches the diffraction-

limited sinc function introduced in Chapter 2, as N → ∞ and ∆k → 0. For this second function the

waveform is

Esinc(z) = C

N−1∑n=0

zn = CzN − 1

z − 1, (3.13)

which features the set of zeros


−1 0 1−1

0

1

ℜ z

ℑz

(a)

−1 0 1−1

0

1

ℜ z

ℑz

(c)

−1.5 −1 −0.5 0 0.5 1 1.5−1

0

1

x/λ(b)

−1.5 −1 −0.5 0 0.5 1 1.5

0

1

x/λ(d)

Figure 3.1: Zero placements of spatial profiles of common waveforms. (a) Zero placementand (b) Spatial profile for a cosine waveform; (c) Zero placement and (d) Spatial profilefor a periodic sinc waveform. Both waveforms are formed by placing 6 zeros over adesign period of 3λ.

zn = e−j2πnN where n = 1 .. N − 1. (3.14)

Thus the spacing of zeros for this periodic sinc function is once again equidistant across the period, apart

from a removed zero at x = 0 which forms the main beam of the since waveform. Fig. 3.1 shows the

zero placement along with a period of both waveforms.

When one concerns himself with only a region of interest (ROI) within a period of the waveform,

one can concentrate zeros to shape the waveform within the ROI. This results in a superoscillation

waveform, which locally contains oscillations more rapid than those in Esin(x) or a peak narrower than

that in Esinc(x). For example, should one be interested to obtain fast oscillations or sharp peaks within

three-quarters of the desired period, the zeros in (3.12) and (3.14) can be close-packed through the

mapping

z′n = z3/4n , (3.15)

where all roots are mapped from the principal branch arg(zn) ε [−π, π], and chosen within the phase

range arg(z′n) ε [− 3π4 ,

3π4 ]. Fig. 3.2 shows the amended zero placement and the resultant superoscilla-

tory waveforms. It is evident that the superoscillation waveforms generated from the mapping (3.15)


−1 0 1−1

0

1

ℜ z

ℑz

(a)

−1 0 1−1

0

1

ℜ z

ℑz

(c)

−1.5 −1 −0.5 0 0.5 1 1.5

−25

0

x/λ(b)

−1.5 −1 −0.5 0 0.5 1 1.5

−5

0

x/λ(d)

Figure 3.2: Zero placements of spatial profiles of example superoscillatory waveforms.(a) Zero placement for a cosine waveform, obtained by multiplying the phase of zerosin Fig. 3.1a by a factor of 0.75. (b) The corresponding spatial profile of the cosinewaveform. (c) Zero placement for a sinc waveform, obtained by multiplying the phaseof zeros in Fig. 3.1c by a factor of 0.75. (d) The corresponding spatial profile of the sincwaveform.

achieve faster oscillations and sharper features than their non-superoscillating counterparts, but high

energy sidebands exist which overwhelm the wavefront in the ROI. Upon comparison with Fig. 2.4,

one would realize the similarity between the superoscillating sub-diffraction peak and the spectrum of a

superdirective antenna.

3.2.3 Comparing Superoscillation with Superdirectivity

Given the close relationship between spatial superoscillation and superdirectivity, I end this section with

a summarizing comparison of these two phenomena. For a spatial superoscillation, a waveform is of

limited support in the spatial frequency domain, over the interval kx ∈ [−NAk,NAk] where NA ≤ 1.

In the spatial domain, one can can define a piecewise continuous ROI consisting of a finite interval (or

finite intervals) in x. Within this ROI, one can achieve rapid oscillations or sub-diffraction sharp peaks,

unlimited by the width limitation in the spatial frequency domain. However, sidebands will exist outside

the ROI, whose energy content is dependent on the degree to which the waveform super-oscillated within

the ROI.

Superdirectivity, on the other hand, involves a waveform whose width in the spatial domain is limited


Superdirectivity Spatial Superoscillation

Domain of Limited Extent x kx

Domain of superoscillation kx x

Region of Interest Propagating wave region (|kx| ≤ k) Designer defined

Sideband Location Evanescent wave region (|kx| > k) Outside ROI

Far-field mapping θ = cos−1(kx/k0)θ ε [0, 180]

N/A

Breaks Diffraction Limit? Angular Spatial

Sideband visible? No Yes

Physical Effects of Sideband Increases Q of antenna

Increases sensitivity of antenna

Incurs metallic loss

Take up a portion of

the waveform’s energy

Visible sideband may

decrease functionality

Table 3.1: Summary of comparison between superdirectivity and spatial superoscillation

to the size of the antenna. This waveform is a superoscillation waveform in the spatial frequency domain,

for which the ROI is the region of propagating waves kx ∈ [−k, k], and the sideband lies in the region of

evanescent waves. Again, within the ROI, one can achieve rapid oscillations or sharp peaks, unlimited by

the width limitation in the spatial domain. Upon a physical far-field mapping (2.24), the superoscillation

in spatial frequency domain maps to a waveform in angular domain, which breaks the angular diffraction

limit (2.19). This mapping renders the evanescent sideband invisible from in the far-field. However,

large sidebands physically manifest themselves as evanescent (or reactive) near-fields, which increase

the required quality factor (Q) for the superdirective antenna. This renders the antenna lossy and very

sensitive and hence prohibits practical antenna arrays from achieving superdirectivity.

Table 3.1 summarizes the above comparison for quick referral.

3.2.4 Connection with Prolate Spheroidal Wave Functions

A final portion of this section compares superdirectivity-inspired superoscillation design with the better

known method of construction using a basis of prolate spheroidal wave functions (PSWFs).

Prolate spheroidal wave functions have been reported and proposed as a suitable class of functions

for constructing superoscillatory waveforms [70, 99]. A set of PSWFs possesses three unique properties

which are helpful for this purpose:

1. They are orthogonal to each other both (a) across a set interval and (b) across the entire 1D space;

2. They form a complete basis across the entire 1D space;

3. As the order of PSWFs increase, the function oscillates quicker within the set region of interest,

but a higher proportion of the waveform’s energy reside outside the region of interest.


These three facts make PSWFs an ideal basis for constructing superoscillatory functions. A super-

oscillatory waveform constructed by a superposition of (relatively) low order PSWFs have the benefit of

minimized sideband energy, while the usage of higher order PSWFs allow for more rapid superoscillations

at the expense of higher energy sidebands. In this manner, constructing a superoscillatory waveform

using PSWFs renders explicit the tradeoff between waveform fidelity and energy efficiency. The inter-

ested reader is referred to the works cited above, as well as [87,89], for further details on superoscillatory

waveform construction with PSWFs.

On the other hand, the superdirectivity-inspired superoscillation design method described in this

section, and used hereafter in the thesis, constructs superoscillatory waveforms using complex expo-

nentials. As complex exponentials also form a complete orthogonal basis in 1D-space, superoscillatory

waveform synthesis with complex exponentials is rather straightforward. As shown in this section,

adapting Schelkunoff’s antenna design theory allows one to obtain useful handles to the superoscillatory

behaviour within the ROI, which for example include a minimized spot width and controls on sidelobe

levels. Moreover, as will be shown in the following section, the zero-placement procedure proposed in

this thesis allows one to control and minimize the sideband amplitude level, which in many cases has

higher practical importance than the sideband energy level. This holds true in particular for applications

which are limited by sensitivity rather than energy efficiency. Finally, handles on energy are not lost,

as one can straightforwardly compute the Strehl ratio for any superposition of complex exponentials,

which represents the proportion of waveform energy within the superoscillatory ROI. Mindful of the

aforementioned advantages, this thesis adopts the superdirectivity-inspired approach introduced in this

section to design and implement of superoscillatory waveforms.

3.3 Controlling Sidebands of Superoscillatory Waveforms

As observed in section 3.2.2, superoscillatory waveforms tend to have sidebands which are more pro-

nounced than features within the ROI. A theoretical study by Ferreira and Kempf [73] shows that, in

conformity to the Shannon limit, the sideband energy for a superoscillatory waveform (normalized with

that of the superoscillatory region) must vary polynomially with the local spectral width of the ROI,

and exponentially with its duration. This seemingly suggests that superoscillatory waveforms are highly

sensitive, and hence need to be generated or processed by systems with very high dynamic range [73,78].

However, this thesis concludes to the contrary: notwithstanding the energy limitation reported in [73],

one can avoid high-amplitude sidebands which render the waveform sensitive, and arrive at superoscilla-

tory waveforms amenable to practical implementation in a sub-diffraction imaging device. This section


explains how this can be accomplished.

Essentially, to achieve some degree of control in the sidebands of a superoscillatory waveform, one

can judiciously place a portion of available zeros outside the ROI. This section will show that, when

performed appropriately, this operation does not compromise the degree of superoscillation within the

ROI. The key to doing this is to increase the number of plane waves used within the allowed spectral

width. This correspondingly increases the period of the waveform and the number of zeros involved. Fig.

3.3 illustrates this approach. Fig. 3.3a-b shows the set of zeros and the corresponding superoscillating

sinusoid, which has been previously shown in Fig. 3.2. A red rectangle in subfigure (b) denotes the ROI

for this waveform (similar denotations apply for subfigures (d) and (f)). Fig. 3.3c-d shows the result

when plane wave components are added such that ∆k — the separation between adjacent plane waves on

the kx-plane — is halved, but the NA remains the same. This means a plane wave component is added in

between adjacent existing plane wave components; however the additional plane wave components are of

zero amplitude. Clearly, the waveform remains unchanged; however the number of zeros have doubled,

and their locations are rearranged, such that each traverse along the unit circle from arg z = −π to

arg z = π is now equivalent to two periods of the waveform. Finally, Fig. 3.3e-f shows zero locations and

the corresponding waveform when zeros outside the ROI are rearranged to maintain a constant amplitude

in the sideband region. This is equivalent to changing the weights of the plane wave components while

retaining their positions in the k-domain. In this work, this rearrangement of zero locations is achieved by

providing a sensible seed (in this case, uniformly spaced zeros within the sideband region) to an iterative

optimization procedure. Since this optimization problem is single-dimensioned and contains no local

extrema, it can be resolved rather straight-forwardly with simple optimization tools, the discussion of

which will be avoided to concentrate attention on factors which in the first place allow this optimization.

Comparing Fig. 3.3f with Fig. 3.3d results in two key observations. Firstly, zeros in the sideband region

for the waveform in sub-figure f have been rearranged more evenly than those in sub-figure d, which has

reduced the maximum amplitude of the sideband. Secondly, this reduction was made possible in large

part by the removal of a superoscillation region centered at an angular position of arg z = π. In general,

one can control the amplitude of the sideband and yet maintain superoscillatory features of a waveform

by the following steps:

1. Maintain the density of zeros within the ROI;

2. Increase the sideband duration; and

3. Arrange zeros to obtain uniform amplitude peaks within the sideband.


Figure 3.3: Suppressing the sideband amplitude by placing zeros within the sideband ofthe superoscillatory waveform. (a) Zero locations of the waveform featured in Fig. 3.2c(b) The corresponding spatial profile (featured in Fig. 3.2d). (c) Zero locations of thesame waveform, with N−1 plane waves added, but their amplitude set to zero. (d) Thecorresponding spatial profile — which as expected is the same as that of (b). (e) Zerolocations after the sideband zeros are distributed to minimize the sideband amplitude.(f) The corresponding spatial profile, which features a sideband whose amplitude isclearly suppressed compared to the waveforms in parts (b) and (d). In subfigures (b),(d) and (f), the red rectangle denotes the extend of the ROI.

3.4 Deducing the Limit to Sideband Suppression

At this point, it stands to reason to ask the following question: is there a limit beyond which one

cannot further suppress the sidebands of a superoscillatory waveform? In particular, if such a limit does

not exist, or is sufficiently low, one can envision the addition of zeros to the point where the sideband

reduces to an acceptable level where they can be ignored much like the sidelobes of an antenna beam.

Unfortunately, such a limit does exist. This section suggests a way to arrive at an estimate for the

sideband amplitude, and show that this lowerbound amplitude exceeds the peak waveform amplitude


within the ROI whenever the waveform contains an appreciable superoscillation effect.

Following the procedure outlined in the previous section, one can always decrease the sideband level

with a further addition of zeros. Hence a limit to sideband suppression, if it exists, could be found by

taking the number of zeros to approach infinity. This operation effectively turns the waveform into an

integration of a continuous (but bandlimited) spatial spectrum of plane waves, which renders the function

aperiodic in the spatial domain. The rigorous representation of this function and its complex extension

has been studied in the continuous domain in under the topic of entire functions of exponential type

(EFETs), for example in [100]. While this section makes use of conclusions formed or deducible from

[100], the discussion will be carried in the z-transform (discrete) domain, where waveform characteristics

are examined as the number of zeros M approach infinity.

3.4.1 Mathematical Derivation

Let us consider the extent of the ROI x ∈ [−xROI , xROI ]. As we take M →∞, the following describes

the tendency of waveform parameters:

• ∆k = ∆KM → 0 (reminder: ∆k is the spectral separation between plane waves, while ∆K is the

total spectral width);

• Λ = 2π∆k →∞ (the waveform’s period approaches infinity); and

• z = ej∆kx → 1 for finite x

From [100] it can be deduced that, if one relocates M zeroes in an EFET f(z) to form a new function

fM (z), the new function fM (z) would have the same bandwidth as f(z). Further, fM (z) would be given

by

fM (z) = f(z) ·M∏n=1

z − zn·newz − zn·orig

, (3.16)

where zn·orig and zn·new respectively represent the original and relocated locations of the n’th zero.

Consider the formation of a superoscillatory waveform by relocating zeros within a standard waveform

cos(kmaxz). Such a relocation will produce a waveform of bandwidth ∆k = [−kmax, kmax]. To produce

a superoscillatory waveform, all zeros within the set zn·orig which reside within the ROI of the target

superoscillatory waveform will be relocated to desired locations within the ROI. Furthermore, as the

generation of superoscillations require the density of zeros within the ROI to be higher than that of the

cosine function, some zeros in zn·orig which lie without, but in close proximity to the ROI will also

be moved to appropriate locations within the ROI. M1 denotes the (fixed) number of zeros which end

up within the ROI. To achieve a uniform amplitude sideband, the location of remaining M −M1 = M2


zeros are shifted slightly towards the origin. M2 will be kept even as it approaches infinity, such that

z = −1 corresponds to a peak in the middle of the sideband, whose amplitude represents the sideband

amplitude. The following evaluates (3.16) for z = −1 and another comparison point within the ROI,

say at z = 1, to examine amplitude changes resultant from the relocation of zeros.

Now consider the relocation of zeros which end up outside the ROI. It can be assumed (as justified in

the following subsection) that sideband zeros are uniformly distributed. In this case the angular distance

between adjacent zeros are

∆φorig =2π

M; ∆φnew =

2π − 2∆kxROIM2

. (3.17)

Since M2 is kept even, the distribution of zeros is symmetric about the horizontal z-axis, with z = −1

sitting at the mid-point of two zeros. Hence the zeros locations in the upper half plane can be described

by

zn = exp(jφn·orig/new) for n = 1..M2

2, where φn = π − (n− 1

2)∆φorig/new. (3.18)

Zero locations of the lower half plane are given by the complex conjugates of the set zn found in

(3.18). Hence, a substitution into (3.16) gives the ratio of the sideband amplitude before and after the

relocation as follows:

Asb(z = −1) = limM2→∞

fM2(z = −1)

f(z = −1)

= limM2→∞

M2/2∏n=1

∣∣∣∣1 + exp(φn·new)

1 + exp(φn·orig)

∣∣∣∣2

= limM2→∞

M2/2∏n=1

cos2(φn·new/2)

cos2(φn·orig/2).

(3.19)

Taking the logarithm on both side yields

logAsb(−1) = limM2→∞

log

M2/2∏n=1

cos2(φn·new/2)

cos2(φn·orig/2)

= limM2→∞

2

M2/2∑n=1

log

(cos(

φn·new2

)

)− log

(cos(

φn·orig2

)

).

(3.20)

As M2 grows towards infinity and ∆φ decreases reciprocally (as seen from (3.17)), the summations

above can be rewritten as integrals


logAsb(−1) = limM2→∞

(4

∆φnew

∫ π/2

∆kxROI/2π

log cosψ dψ − 4

∆φorig

∫ π/2

πM1/M

log cosψ dψ

)

= 4

(∫ π/2

0+

log cosψ dψ

)lim

M2→∞

(1

∆φnew− 1

∆φorig

).

(3.21)

Here the integration index is ψ =φorig/new

2 . The integral log cosψ behaves nicely across the integration

interval and can be evaluated. Thus we direct our attention to the factor which remains within the limit.

From (3.17) one can see that ∆φnew → ∆φorig as M2 →∞, so the middle term evaluates to zero. Thus

one can concluded that

logAsb(−1) = 0⇒ Asb(−1) = 1, (3.22)

which means the infinitesimal movements of zeros in the sideband do not change the sideband amplitude,

even though there are an infinite number of these zeros.

A similar derivation can be carried out for the wave amplitude at z = 1 (corresponding to x = 0

within the centre of the ROI). In this derivation (3.19) the zero factors will feature a negative sign, which

leads to the emergence of sin terms:

Asb(z = 1) = limM2→∞

M2/2∏n=1

∣∣∣∣1− exp(φn·new)

1− exp(φn·orig)

∣∣∣∣2

= limM2→∞

M2/2∏n=1

sin2(φn·new/2)

sin2(φn·orig/2).

(3.23)

The subsequent mathematics remain the same, except in this case the integral features log sinψ, which

evaluates to the same constant over the interval of interest. It can thus be concluded that in the case as

M →∞ shifts in the zeros outside the ROI do not change the waveform amplitude at the sideband and at

x = 0. Therefore, the change in the ratio between sideband amplitude (compared to the superoscillatory

features) can be solely attributed to zeros within the ROI.

Now consider how movements of zeros in the ROI would impact the waveform amplitude at z = −1

and z = −1. Since a) the number of zeros M1 within the ROI is finite, and b) the ROI is mapped to

z → 1 in z-domain, the relocation of zeros within the ROI do not appreciably change the waveform

amplitude at infinity. However they do change the amplitude of the superoscillatory features; their

change on the amplitude of a superoscillatory peak at x = 0 can be straightforwardly calculated with

(3.16):


AROI(z = 1) = limM→∞

fM1(z = 1)

f(z = 1)

= limM→∞

M1∏n=1

1− zn·new1− zn·orig

= limM→∞

M1∏n=1

1− exp(jφn·new)

1− exp(jφn·orig)

=

M1∏n=1

1− (1 + jφn·new)

1− (1 + jφn·orig)

=

M1∏n=1

φn·newφn·orig

(3.24)

Here the set φn·orig represents the phases the original locations of theM1 zeros which lie closest to z = 1.

The fourth step is written with a Taylor series expansion, applicable since φn π ∀ n when M → ∞.

Equation (3.24) gives an estimate on the ultimate suppression of the sideband of a superoscillatory

waveform, achieved when an infinite number of zeros are included in the superoscillatory sideband.

3.4.2 Discussion

Three remarks are due regarding the estimated sideband suppression. Firstly, (3.24) shows that the su-

peroscillatory sideband cannot be suppressed to an amplitude below that of the superoscillatory features

within the ROI. This result is reasonable, since the creation of superoscillatory features require the close-

packing of zeros within the ROI, which reduces the waveform amplitude within the ROI. Conversely, the

density of zeros decreases in the sideband regions, causing a rise in sideband amplitude. Secondly, to

achieve a uniform sideband amplitude, a nonuniform distribution of zeros is usually required in the side-

band to compensate for a displacement in zero locations within the ROI. This non-uniform distribution

will cause the two integrals to differ in the first step of (3.21). Notwithstanding, one can reason that as

M →∞, the angular shift vanishes for zeros within the ROI, hence their affect also vanishes in regards

to reshaping the waveform across the unit z-circle. Hence a uniform placement of zeros as assumed

in the previous subsection is sensible in the limit which was considered. Thirdly, one must exercise

judgment towards the degree of sideband suppression within a superoscillatory waveform, because the

ultimate level of sideband suppression as an asymptotic limit as the waveform period goes to infinity.

Thus at some point one must balance the benefits of decreasing the sideband amplitude with other costs,

such as complexity of the system, energy within the sideband and sideband duration. In this light, it

should be noted that while the sideband amplitude is suppressed by the addition of sideband zeros, the

sideband energy is clearly increased (with respect to waveform energy within the ROI). Notwithstand-


ing this caveat, placing zeros into the sideband of a superoscillatory waveform offers a way to control

the sideband amplitude and achieve waveforms amenable to practical implementation in sub-diffraction

imaging systems.


This chapter introduced the phenomenon of spatial superoscillation, suggested how it can be defined, and

related it to superdirectivity. Leveraging connections between these two phenomena, a superdirectivity-

inspired method was introduced to design spatial superoscillations. The method was described in detail

and its operation was demonstrated in two design examples: the superoscillating cosine waveform and

the superoscillating sinc waveform. Further, this chapter showed that by properly placing zeros one can

suppress the sideband amplitude of a superoscillatory waveform. This comes contrary to claims from

previous works on superoscillations, and opens new possibilities for designing practical superoscillation-

based imaging systems. A mathematical analysis was performed which estimated the ultimate levels of

sideband suppression.

Besides recounting the concept of superoscillation, all theoretical work reported in this chapter rep-

resent novel contributions which form part of the present thesis. The theoretical ground work reported

in this chapter will be used to design and demonstrate superoscillatory focusing and imaging devices in

the following chapters.

Chapter 4

Superoscillatory 1D Sub-Diffraction

Focusing Devices

The previous chapters summarized the need for a superoscillation-based sub-diffraction imaging tool and

developed a theory through which practical superoscillatory waveforms can be designed. This chapter

will suggest and demonstrate proof-of-principle devices which form a superoscillatory sub-diffraction

focus. Section 4.1 describes the general formulation for designing excitations which will launch a wave

that superimposes at the focal distance to form a sub-diffraction and sub-wavelength peak. Section

4.2 introduces a periodic transmission screen which performs sub-wavelength focusing at an imaging

distance of 5 wavelengths — a distance tenfold increased from most near-field sub-wavelength imaging

systems. Section 4.3 reports the design and demonstration of an in-waveguide superoscillatory focusing

device based on a slight modification of plane wave components used in Section 4.2. This focusing device

forms a focus 25%-reduced from the diffraction limit at a focal distance of 4.8λ. Section 4.4 discusses the

salient properties the work presented in this chapter, compares them with other sub-diffraction focusing

devices, and highlights important progresses made towards achieving superoscillatory sub-diffraction

imaging devices.

4.1 Sub-Diffraction-Focused Waveform Design

Whilst the previous chapter has described in a general sense the zero-based formulation for super-

oscillatory wave design, this section uses it to design a specific waveform — a sub-diffraction focus,

surrounded by constant ripples at an arbitrarily set level within a region of interest (ROI) of arbitrary

45

Chapter 4. Superoscillatory 1D Sub-Diffraction Focusing Devices 46

duration. Antenna engineers have derived array excitation coefficients which synthesize this waveform

based on an expansion of Tschebyscheff polynomials. A Tschebyscheff filter — a filter constructed from

a Tschebyscheff function — features ripples of uniform power level, which makes it the ideal response

to many electronics and communication systems. Specifically, for antenna systems, an expansion of

Tschebyscheff polynomials will yield excitation currents which allow an antenna array to form the nar-

rowest radiation beam, and yet have its sidelobes ripple conform to a pre-specified level [3]. When

the array spacing is less than half-wavelength, this narrow beam even surpasses the angular diffraction

limit [101]. Leveraging the connection between superdirectivity and superoscillation, this section de-

scribes an adapted method for the synthesis of superoscillatory waveforms which are optimized to form

the narrowest peak given a pre-specified sidelobe level (within the ROI).

A Tschebyscheff polynomial Tn(x) of the first kind is an n’th order polynomial whose value oscillates

between -1 and 1 across the domain x ∈ [−1, 1]. Its mathematical definition is as follows:

Tn(x) = cosh(n cosh−1 x)

= cos(n cos−1 x) (for |x| ≤ 1).

(4.1)

It can also be defined recursively by the following relations:

Tn(x) =

2xTn−1(x)− Tn−2(x) for n ≥ 2

x for n = 1

1 for n = 0

. (4.2)

For instructive purposes, Fig. 4.1 plots the first five Tschebyscheff polynomials of the first kind.

It will prove fruitful to express the superoscillatory waveform which one desires to synthesize as a

series of weighted Tschebyscheff polynomials. To do this, first assume a case where the number of plane

waves N is odd, hence the number of zeros M = N − 1 is even (the case of even N , odd M will be

discussed toward the end of this section). Further, require that the waveform be symmetric about the

superoscillatory peak x = 0. In this case the target waveform can be expressed as

E(x) =

bN/2c∑n=−bN/2c

cne−jn∆kx

= c0 +

bN/2c∑n=1

2cn cos(n∆kx).

(4.3)

Here the coefficient set cn only differs in index from the set an used elsewhere in this thesis: i.e.

an = cn−bN/2c, for n = 0, 1, ..., N − 1. Defining


Figure 4.1: The first five Tschebyscheff polynomials of the first kind.

z = e−j∆kx;u = <z = cos(∆kx) (4.4)

allows one to rewrite (4.3) as

E(u) = c0 + 2

bN/2c∑n=1

cn cos(n cos−1(u))

= c0 + 2

bN/2c∑n=1

cnTn(u).

(4.5)

This work aims to design a waveform E(x) with the narrowest possible superoscillatory peak in the

center of an ROI which stretches the ROI, which is the interval x ∈ [−L,L]. In u-space this ROI is

mapped to u ∈ [cos(∆kL), 1], where the desired waveform E(u) attains the peak amplitude at u = 1.

This work also aims to suppress sidelobes within the ROI to amplitude levels below 1/P that of the

superoscillatory peak. A family of waveforms which satisfy both objectives would be

Em(u) = Tm(v), (4.6)

where

v = αu+ β =1 + v0

1− cos(∆kL)u− 1 + v0 cos(∆kL)

1− cos(∆kL). (4.7)


Here m is a non-negative integer representing the degree of the polynomial (both in u-domain and v-

domain), while α and β are chosen such that the ROI is mapped to v ∈ [−1, v0]. The peak location

u = 1 is mapped to v = v0, a location for which Tm(v0) = P . Using (4.5) to (4.7) one can arrive at a

waveform design in three simple steps:

1. Choose m = bN/2c in (4.6).

2. Use (4.6) and (4.7) to find Em(u), which should now be a polynomial of degree bN/2c.

3. Calculate the corresponding coefficient set cn by decomposing Em(u) into a combination of

Tschebyscheff polynomials as per (4.5).

Alternatively, instead of solving for the coefficient set cn, one can opt to find the location of zeros

which will lead to the desired superoscillatory waveform. This could be achieved methodically as follows.

Upon choosing m = bN/2c as noted in step 1 above, the zero locations in Tm(v) are then given as

vzq = cos

(π

2

2q − 1

bN/2c

), for q = 1, ..., bN/2c. (4.8)

Using (4.7) one finds the corresponding zeros locations for EbN/2c(u) as

uzq =vzq − βα

=

(vzq +

1 + v0 cos(∆kL)

1− cos(∆kL)

)(1− cos(∆kL)

1 + v0

)=

(1− cos(∆kL))vzq + (1 + v0 cos(∆kL))

1 + v0

for q = 1, ..., bN/2c.

(4.9)

Thereafter, noting the fact that all zeros of an even function in x are located on the unit z-circle, one

finds the set of zero locations wn for EbN/2c(z) as

<wn = uz|n| ; |wn|2 = 1⇒ wn = uz|n| + sgn(n)√u2z|n| − 1,

where n = ±1, ...,±bN/2c.(4.10)

A brief mention on the waveform design procedure is due for the case where N is even, and hence

M = N − 1 is odd. In this case, one can follow the above design procedure to place all zero pairs within

the ROI of the waveform. The even nature of E(x) dictates that the remaining zero be placed on the

unit circle and on the real axis, which limits the zero location to either z = 1 or z = −1. Since the

procedure aims to form a focus at x = 0⇒ z = 1, the remaining zero must be placed at z = −1. Thus

the unpaired zero does not contribute to the high-resolution shaping of the ROI, but helps suppress the


Design Parameter Values

N 5

NA 4k/5

∆k = 2NAkN−1

2k/5

Period (Λ = 2π/∆k) 2.5λ

# of zeros (M = N − 1) 4

Focal distance (s) 5λ

Sidelobe Level (1/P ) 0.2 (for E-field)

ROI x ∈ [−λ/2, λ/2]

Table 4.1: Summary of parameters for the Superoscillatory Sub-Diffraction FocusingScreen

sideband amplitude.

Upon finding either the polynomial coefficients or the zero locations of a superoscillatory waveform,

one can construct the waveform using for example (3.9), or otherwise utilize it to design sub-diffraction

focusing and imaging devices.

4.2 A Superoscillatory Sub-Diffraction Focusing Screen

This section reports the design and simulation of a Superoscillatory Focusing Screen (SFS) which gen-

erates a superoscillatory waveform with a sub-diffraction peak at a focal distance of five wavelengths.

Table 4.1 summarizes parameters of this design:

−1 0 1−1

0

1

ℜ z

ℑz

(a)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

kx/k

Spe

ctra

l Am

plitu

de [n

.u.]

(b)

Figure 4.2: Zero locations and the corresponding plane-wave spectrum for the super-oscillation sub-diffraction focusing screen

In this design, all available zeros are placed into the ROI and arranged to generate the narrowest

possible peak within the ROI, while maintaining the sidelobes at the prescribed level. Fig. 4.2 shows the


resultant zero locations and the corresponding plane-wave spectrum, Eimg(kx), which is mathematically

described as follows:

Eimg(kx) =

N−1∑n=0

anδ(kx − kxn), (4.11)

where an and kxn respectively represent the complex coefficient and transverse wavenumber for the n’th

plane wave in the summation. The set an is calculated from the designed zero locations using (3.9).

One can back-propagate this superposition of plane waves for any distance s using (2.7):

Esrc(kx) = E(kx,−s) = Eimg(kx) exp(−j√k2x − k2z)

=

N−1∑n=0

[an exp(js

√k2 − k2

xn)]δ(kx − kxn)

=

N−1∑n=0

bnδ(kx − kxn).

(4.12)

Here bn describes the complex coefficient of the n’th plane wave component. As expected for propa-

gating waves, it is merely phase-shifted from the corresponding an. After obtaining the set bn, the

corresponding waveform at the source plane is simply given by

Esrc(x) =

N−1∑n=0

bn exp(−jkxnx). (4.13)

The sub-diffraction focusing screen is designed as a spatial filter passing Esrc(x), which will result in the

desired superoscillatory focus Eimg(kx) at distance s = 5λ away from the source plane. Fig. 4.3 shows

a period of the calculated Esrc(x) and Eimg(kx), and compares the latter a with the diffraction-limited

sinc waveform corresponding to NA = 1.

These calculation results are validated by full-wave simulation with Comsol Multiphysics — a com-

mercial simulation package which solves the vectorial Maxwell’s equations using the Finite Element

Method (FEM). Fig. 4.4a shows the simulated electric field distribution overlaid atop the computation

domain. A periodic electric field function Ey(x) = Esrc(x) is excited along the z = 0 plane, which

forms the sub-diffraction focus at the image plane at z = 5λ. The simulation region is reduced to one

period using periodic boundaries (implemented here by perfect magnetic conductors) in the x-direction.

Fig. 4.4b shows a close up of the imaged electric field in the ROI |x| ≤ λ/2, normalized alongside the

source electric field and the diffraction-limited sinc function. Whilst the waveform at the source plane

contains no sub-wavelength variations, a sub-diffraction peak appears in the image whose E-field full-

width half-maximum (FWHM) measures 0.37λ. This is clearly improved over the FWHM of 0.60λ for


Figure 4.3: Superoscillatory Waveform Design. (a) One period of the calculated Esrc(x)(black, dashed) and Eimg(kx) (blue, dotted). (b) The image waveform Eimg(kx), plottedacross the ROI alongside the diffraction-limited sinc waveform (red, dashed).

Figure 4.4: Full-wave simulation results for the sub-diffraction focusing screen. (a)The simulated electric field distribution overlaid atop a schematic of the computationdomain. Blue solid lines indicate PMC boundaries; dashed lines (black and green)indicate the source and image planes respectively. (b) A closeup of the imaged electricfield in the ROI |x| ≤ λ/2 (blue, solid), normalized alongside the source electric field(black, solid) and the diffraction-limited sinc function (red, dashed). (c) The simulated(magenta, circles) and calculated (blue, solid) electric fields at the image plane showexcellent agreement with one another.


a diffraction-limited sinc function with NA = 1. The sub-diffraction focus is hence also sub-wavelength,

despite the fact that the focal plane is located 5λ from the source — an order of magnitude further than

focal distances achieved by evanescent-field-based focusing devices. Fig. 4.4c compares the simulated

and calculated electric fields at the image plane. The near-perfect agreement between calculation of

simulation verifies the validity of the theory. This plot of Eimg(x) reveals that the sub-wavelength peak

has an amplitude 6.6% that of the sidebands. This level of sideband amplitude should be within reason

of design tolerances. Moreover, designs which will be presented later in this thesis will demonstrate

sideband suppression through the method explained in section 3.3.

4.3 Superoscillatory Sub-Diffraction Focusing in a Waveguide

Environment

The Superoscillatory Focusing Screen, introduced in the previous section, can be practically implemented

with planar electromagnetic structures which shall be discussed in the following section. In this section,

however, attention is diverted to demonstrating superoscillation-based sub-diffraction focusing within

a waveguide. The waveguide environment is ideal for simple implementation for superoscillation-based

sub-diffraction focusing, because its waveguide walls act as mirrors (or “anti-mirrors”) for electromag-

netic fields, which (a) reduce the device’s lateral dimension to as small as one Bloch period of the image

waveform, and (b) greatly reduce the number of excitation elements required. This section reports a

Superoscillatory Focusing Waveguide (SFW), which interferes propagating waveguide modes to form a

sub-wavelength focus five wavelengths away from the plane of excitation. It first explains the design

process, then describes the function, fabrication and calibration of components of the experimental ap-

paratus, then reports the measurement procedure and discuss results which demonstrate sub-wavelength

focusing at a five wavelength working distance.

4.3.1 Waveguide Excitation Design

The host medium for the SFW is an air-filled rectangular waveguide with a cross-section created by four

metallic walls, which at microwave frequencies can be well approximated by perfect electric conductors

(PECs). The longitudinal direction will be denoted z, the transverse direction x and the invariant

direction y. Invariance in the y-direction is achieved by choosing a waveguide height h to be below one

half wavelength along the y-direction, and then enforcing, through the waveguide excitations, a TEy

EM-field polarization (i.e. non-zero field components for Ey, Hx, Hz). In this configuration, waveguide


boundaries at x = ±h/2 mirror the electric field in the y-direction, causing an effect similar to periodic

extension; further, since only the zero-order mode is allowed in the y-direction, Ey is kept uniform along

the waveguide height hence achieving invariance.

Mimicking periodic repetition in the x-direction, however, takes some thought. Since the electric field

runs parallel to the waveguide walls at x = ±w/2, these walls image electric fields with reversed polarity.

In particular, a null must be present at the waveguide wall where the field and its image cancels one

another. To accommodate for this, I modify the waveform design introduced in the previous section by

including an extra zero at z = −1. This operation satisfies the aforementioned boundary requirements

because: (a) a null is explicitly placed at the edge of the Bloch period which drives the electric field to

zero at the edge of the waveguide; and (b) having an odd number of zeros means each Bloch period of

the waveform is π-phase-shifted from adjacent periods — which well befits the anti-mirroring property

of waveguide walls at x = ±w/2.


N 6

NA 5k/6

∆k = 2NAkN−1

k/3

Bloch Period (Λb = 2π/∆k) 3λ

Waveguide Width (w) 3λ (w = λb)

# of zeros (M = N − 1) 5

Focal distance (s) 5λ

Sidelobe Level (1/P ) 0.2 (for E-field)

ROI x ∈ [−λ/2, λ/2]

Table 4.2: Summary of parameters for the SFW

Table 4.2 tabulates the design specifications for the SFW. Fig. 4.5 shows the obtained zero locations,

the corresponding spectral distribution, the resultant waveform design, and a closeup along the ROI,

compared alongside the diffraction-limited sinc for a system with NA = 1. It should be noted that the

spectral distribution can be viewed as the superposition of three waveguide modes: TE10, TE30 and

TE50, hence a proper excitation of these modes will lead to a superoscillatory sub-diffraction focus. The

remainder of this subsection describes the a scheme for exciting these modes and derives the proportions

in which these modes are excited.

First an is derived from the Tschebyscheff expansion procedure described in section 4.1, then

bn is obtained through waveform back propagation using (4.12). Though derived for a plane wave

environment, the plane wave coefficient set bn remains valid in the present waveguide environment

because it has been established bn is the superposition of three waveguide modes, which individually


−1 0 1−1

0

1

ℜ zℑ

z

(a)

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

kx/k

Spe

ctra

l Am

plitu

de [n

.u.]

(b)

−1 0 10

0.5

1

x/λ

Fie

ld A

mpl

itude

[n.u

.]

(c)

−0.5 0 0.50

0.5

1

x/λ

Fie

ld A

mpl

itude

[n.u

.]

(d)

Figure 4.5: Design plots for the SFW, showing (a) zero locations, (b) spectral ampli-tudes, (c) the designed waveform, and (d) a closeup across the ROI, compared alongsidethe diffraction-limited sinc (red, dashed).

satisfy the boundary conditions of the waveguide. The SFW excites plane waves according to the

coefficient set bn using a collection of current line sources. The field emanating from a line source is

described by a zero order Hankel function of the second kind. This field and its corresponding spatial

spectrum is written as follows [102]:

Els(x) = H(2)0 (x);

Els(kx) =2

kz=

2j√k2x − k2

.(4.14)

The plane wave spectrum of our desired waveform can be decomposed into the product of the elemental

spectrum and the array spectrum (which are respectively equivalent to the elemental and array factors

from antenna array theory):

Esrc(kx) = Els(kx)Csrc,array(kx). (4.15)

Therefore, dividing the desired plane-wave spectrum by the elemental spectrum, one arrives at the


spectral distribution of the necessary array excitations:

Csrc,array(kx) =Esrc(kx)

Els(kx)

=

(N−1∑n=0

bnδ(kx − kxn)

)(kz2

)

=

N−1∑n=0

b′nδ(kx − kxn)

where b′n =bn√k2 − k2

xn

2.

(4.16)

The array coefficient function is therefore

Csrc,array(x) =1

2π

N−1∑n=0

b′n exp(jkxnx). (4.17)

The choice is made to place line sources at half wavelength intervals. Namely, the locations chosen are

xm =(m− 3)λ

2for m = 1 to N − 1. (4.18)

The Nyquist criterion guarantees this spacing is of sufficient density to synthesize a purely propagating

waveform. For these locations, the excitation coefficients are simply obtained as

h[m] = Csrc,array(xm). (4.19)

Table 4.3 lists the calculated excitation coefficients for the present design, normalized to the component

at x = 0. Exciting five line sources with at locations xm with currents h[m] will result in a waveform

converging to a sub-diffraction focus at a distance s = 5λ from the excitation plane.

Element (m) xm h[m] h′[m]

1 −λ 0.2930 ∠− 2.769 rad 0.3272 ∠− 2.845 rad

2 −λ/2 0.2732 ∠− 1.044 rad 0.3548 ∠− 1.127 rad

3 0 1 ∠ 0 rad 1 ∠ 0 rad

4 λ/2 0.2732 ∠− 1.044 rad 0.3548 ∠− 1.127 rad

5 λ 0.2930 ∠− 2.769 rad 0.3272 ∠− 2.485 rad

Table 4.3: Calculated current excitation coefficients for the Superoscillatory Sub-Diffraction Focusing Waveguide


4.3.2 Fabrication and Experimentation

Figure 4.6a shows a schematic of the SFW, showing key design dimensions, the source and image planes,

as well as wave terminations and the measurement apparatus. To fabricate the waveguide, a 1/8” (3.2

mm) sheet of stainless steel is bent into a waveguide section of width w = 297 mm, height h = 12 mm

and length l = 1 m. Due to a fabrication error, the waveguide width measures slightly narrower than

the intended 300 mm, which would represent 3λ at an experimental frequency of f = 3 GHz. To reflect

this change, the calculations above is slightly modified to obtain an adjusted array of excitation currents

h′[m], which are tabulated alongside the original excitation currents in Table 4.3.

To synthesize the necessary line sources, five metallic posts (1.2 mm diameter) are embedded into

the waveguide, each spaced 50 mm apart along the x-direction. They are placed 200 mm into the −z

end of the waveguide to form the source excitation plane. Each post is formed by connecting a pair

of inner conductor rods which extrude from their respective SMA receptacles which are welded to the

exterior of the top and bottom waveguide walls. Holes of sufficient sizes are drilled through the top

and bottom waveguide walls at the post locations, such that the inner conductors, when fitted through

from both sides, connect without forming electrical contact with the waveguide. The SMA receptacles

provide access ports for our experiment: they are used to feed current into the metallic posts and to

monitor their current levels. The aforementioned array of line sources is excited using the feed network

schematized in Fig. 4.6b. The microwave source originates from port 1 of a programmable network

analyzer (PNA), and splits equally into three signal paths through a 1-to-3 power splitter. One output

path from this power splitter directly feeds port 3 of the waveguide (see Fig. 4.6a for port enumeration);

outputs from the other two paths each go through a variable attenuator and a variable phase shifter,

then get power divided and fed into ports 1 and 5, and 2 and 4. In this manner, the feed network gives

one the freedom to tune the complex current inputs into ports 1 to 5; ports 6 to 10, which exist opposite

to port 1 to 5 on the bottom of the waveguide, are normally terminated with a 50 Ω load, but are used

in the experiment to monitor current levels on the line sources when such need arises.

Since the SFW features with closely-spaced antenna elements embedded in a waveguide environment,

one needs to account for mutual coupling effects amongst antenna elements. To accomplish this, the

attenuators and phase shifters are tuned in a 2-step process. Firstly, with port 1 of the PNA connected

to the feeding network as shown in Fig. 4.6b, port 2 is sequentially connected to output ports 1 to 5

of the feeding network. In this initial step, the attenuators and phase shifters are tuned such that the

S21 readings for these five ports (divided by the S21 reading for port 3) follow the vector h[m] displayed

in Table 4.3, for the experimental frequency 3 GHz. Subsequently, while port 1 of the PNA remains


Figure 4.6: Schematic diagrams for the SFW. (a) A schematic of the SFW, showing keydesign dimensions, the source and image planes, wave terminations and the measure-ment probe. (b) The feed network for the SFW. Numbers to the right of the sub-figureindicate connections to the corresponding ports in (a).

connected to the feeding network, the output ports of the feeding network are connected to the similarly

numbered input ports to the SFW. Port 2 of the PNA is then sequentially connected to ports 6-10 to

monitor the currents coupled into them. With such monitors, one can write the S-parameter observations

as Sab(i), where a and b describe PNA Ports, and i denote the waveguide port connected to port 2 of

the PNA. Since the currents picked up by the monitor are, by and large, proportional to the currents

on the posts, the attenuators and phase shifters are fine-tuned such that the set of values S21(i)/S21(8)

for i = 6 to 10 once again match those for h[m]. This fine-tuning procedure allows one to account for

mutual coupling, and thereby correctly synthesize the desired current excitation on the source array.

Fig. 4.7 shows a photograph of the experimental apparatus. The aforementioned feeding network

drives the predesigned currents into the source array and excite the TE10, TE30 and TE50 waveguide

modes in appropriate proportions. (The TE20 and TE40 modes are not excited due to symmetry.)

Microwave absorbers are placed at the ends of the waveguide to quench reflected components, but a

small gap is left at the end in the +z-direction, through which a coaxial cable probe is inserted to

measure the y-directed electric field at the image plane s = 500 mm, five wavelengths away from the

source.


Figure 4.7: A photograph of the sub-diffraction focusing waveguide.

4.3.3 Simulation and Experimental Results

The focusing experiment is conducted as follows. With the source located at z = 0 mm, the probe is

scanned to measure the electric field at a series of cross-sections from z = 450 mm to z = 530 mm,

with a spacing of 10 mm. For each cross-section measurements are taken from x = −130 mm to x =

130 mm in step sizes of 2.5 mm; a space of 18.5 mm remains unscanned on either side of the waveguide

cross-section to avoid collision with the waveguide walls. The measured electric field profile across this

region is shown in Fig. 4.8c.

The measured electric field is compared with full-wave simulation results obtained using Ansoft HFSS

— a 3D full-wave FEM-based electromagnetic simulation software. In the simulation, one applies wave

port excitations to SMA connectors, which in turn couple currents into the metallic posts to form the

source array. To account for mutual coupling effects within the simulation, I set the wave port excitation

weights to

w[n] = S−1mch[n] =

S61 S62 S63 S64 S65

S71 S72 S73 S74 S75

S81 S82 S83 S84 S85

S91 S92 S93 S94 S95

S10,1 S10,2 S10,3 S10,4 S10,5

−1

h[1]

h[2]

h[3]

h[4]

h[5]

, (4.20)


Figure 4.8: Electric field profiles near the image plane (denoted by white dash lines).(a) Simulation with absorbers covering the entire waveguide end facets. (b) Simulationwith a partial gap in the absorber at the +z end of the waveguide (see Fig. 4.6 forpictorial depiction. (c) Measured electric field magnitude. The colour bar to the leftapplies for simulated field profiles, while the bar to the right applies for the measuredfield profile.

where Smc describes a selected portion of the simulated 10-port S-parameter from which we distill

information on mutual coupling effects (again, see Fig. 4.6a for port enumeration). In the simulation, the

currents w[n] excite the three relevant waveguide modes in appropriate proportions, which propagate

through to both ends of the stainless steel waveguide where they are terminated by perfect matching

layers (PMLs). These PML blocks mimic absorbers in the experiment, and span the entire cross-section

at both ends of the waveguide.

A driven mode analysis is performed at f = 3 GHz. Fig. 4.9 compares the simulated waveform at

the image plane alongside design calculations, and shows excellent agreement between the two, apart

from slight deviations surrounding the outer pair of nulls. Fig. 4.8a shows the electric-field amplitude

across the measurement grid. One can observe that the sub-diffraction central peak, as well as the high

energy sidebands, are visible across this entire region; the low sidelobe amplitude ROI, on the other

hand, appears for a smaller longitudinal interval around the designed image plane at z = 500 mm.

To reconcile the apparent difference between the simulated and experimental results (Figs. 4.8a and


−150 −100 −50 0 50 100 1500

2

4

6

8

10

x [mm]|E

y| [n.

u.]

−50 −25 0 25 500

0.2

0.4

0.6

0.8

1

x [mm]

Figure 4.9: A comparison of the simulated (blue, solid) and calculated (black, dashed)Eimg(x), showing excellent agreement apart from slight deviations surrounding the outerpair of nulls. The right panel is a closeup of the left across the ROI.

4.8c), a modified simulation is conducted where a 6 mm gap is introduced in the y-direction for the

PML situated at the end facet of the waveguide, in similarity to the experimental apparatus depicted in

Fig. 4.6a. Fig. 4.8b shows the electric-field amplitude of this modified simulation. Here, the presence

of a reflected wave generates observable standing-wave patterns. As a result, the sideband amplitude

is decreased — particularly at z = 500 mm — and the central peak is widened. In this spirit, Fig.

4.8c resembles Fig. 4.8b in that all major peaks in Fig. 4.8b are observed in Fig. 4.8c. However,

the seemingly more significant reflected wave components cause a sideband cancellation at z = 500 mm,

producing a central peak more pronounced in amplitude in comparison to calculation and simulation, but

no longer sub-wavelength. Nevertheless, a sub-wavelength pattern is formed in the immediate vicinity

at z = 480 mm. The experimental focal quality at this distance will be examined more closely in the

following figures. Some experimental factors which lead to reflected waves, or otherwise disturb the

superoscillatory wave interference pattern, include the intrusion of the coaxial cable probe, the receiving

characteristic of the small, but finite-sized, probe tip, rounded waveguide corners, slight asymmetry

in feed network distribution and source array construction. In retrospect, raising the height of the

waveguide would most probably reduce affects of rounded corners and slight curvatures of waveguide

walls; while inserting the measurement probe through a network of holes drilled through the horizontal

waveguide wall may enable the full closure of absorber termination, and hence enable the measurement

of an uncontaminated superoscillation waveform. Notwithstanding, the field measured at cross-sections

slightly before the designed imaging distance remains superoscillatory despite interference with reflected

waves.

Figs. 4.10 and 4.11 plot and compare the measured and simulated waveform cross-sections at z = 500

mm (5λ) and a nearby plane z = 480 mm (4.8λ). Simulation with a full (PML) absorber at the +z end of

the waveguide leads to a superoscillation focus with an electric field full-width-half-maximum (FWHM) of


−150 −75 0 75 1500

2

4

6

8

10

x [mm]

|Ey| [

n.u.

]

Experiment

Sim: Full Absorber

Sim: Partial Absorber

Figure 4.10: A comparison on the measured and simulated superoscillatory foci at theimage plane z = 500 mm. The simulated field profiles are taken at the design imageplane, while the measured field profile is taken at z = 480 mm.

37 mm (0.37λ). However, when we employ the partial absorber at the +z end of the waveguide, the image

profile widens to a FWHM of 45 mm (0.45λ). Whereas reflected wave components and slight asymmetry

in the +x-direction obscure our superoscillatory focusing measurement at z = 500 mm, the measured

electric field cross-section at z = 480 mm (4.8λ) clearly demonstrates superoscillatory behaviour. This

profile is plotted in Figs. 9 and 10 for comparison purposes. Here the electric field FWHM of the

central peak is squeezed to 45 mm (0.45λ) — comparable to the partial absorber simulation. Both

waveforms squeeze the central peak to 75% of the diffraction limit of 61 mm (0.61λ), formed from a

uniform, in-phase superposition of all symmetric modes within the waveguide. Even though experimental

imperfections prevented the formation of an extended region of low sidelobe ripples, the experiment has

nevertheless successfully demonstrated superoscillatory sub-diffraction focusing at a multi-wavelength

image distance.

4.4 Discussion

This section consolidates a few remarks regarding the superoscillatory sub-diffraction focusing devices

introduced in this chapter, namely the SFS and the SFW. These remarks will discuss salient properties

of the aforementioned devices, and compare them with other sub-diffraction focusing devices, including

ones which also operate based on superoscillation.


−50 −25 0 25 500

0.2

0.4

0.6

0.8

1

x [mm]

|Ey| [

n.u.

]

Experiment

Sim: Full Absorber

Sim: Partial Absorber

Diffraction Limit

Figure 4.11: A close up of Fig. 4.10 over the ROI, comparing the measured and sim-ulated superoscillatory foci across the design interval. The diffraction-limited focus inthis waveguide environment is also included for comparison. As in Fig. 4.10, the simu-lated field profiles are taken at z = 500 while the measured field profile is taken at z =480mm.

4.4.1 Salient Features in Comparison to Evanescent-Wave-Based Devices

The first remark compares and contrasts to evanescent-wave-based sub-wavelength imaging and focus-

ing devices. Whereas evanescent-field-based devices form sub-wavelength focuses within about half-

wavelength proximity of the device, where evanescent field components dominate the field profile, the

superoscillatory sub-diffraction focusing screen and waveguide presented in this chapter can form a sub-

wavelength focus beyond the evanescent near-field of the focusing device. This chapter has experimentally

demonstrated in-waveguide focusing (and numerically shown free-space focusing) at a focal distance of

5 wavelengths — which already represents a tenfold increase from working distances of evanescent-field-

based imaging devices. While this focal distance still resides within the radiating near-field, since the

waveguide walls render the source dimension effectively infinite, the freedom to extend working distances

to an order of magnitude or even greater beyond the evanescent near-field of a device represents great

progress in electromagnetic-wave-based super-resolution imaging.

For vivid comparison, Fig. 4.12a displays an evanescent-wave-based sub-wavelength field profile with

a similar focal width to that achieved by the SFW. This field profile is formed by superimposing equal

proportions of the TE10, TE30, TE50, TE70 and TE90 modes of the waveguide, the latter two of which

are evanescent. For different focal distances, Fig. 4.12b plots the proportion of the waveform intensity


−50 −25 0 25 500

0.5

1

x [mm]

|Ey| [

n.u.

]

(a)

0 1 2 3 4 5 610

−40

10−30

10−20

10−10

100

x/λ

I foc /

I gen

(b)

Figure 4.12: A comparison between superoscillatory-wave-based and evanescent-wave-based sub-wavelength focusing. (a) the calculated superoscillatory electric field profileon the image plane (red, solid), as designed for the SFW, plotted alongside a sub-wavelength sinc distribution (blue, dashed). The latter is generated with uniform su-perposition of the TE10, TE30, TE50, TE70 and TE90 modes of the waveguide. (b) Thefraction of the total energy on the screen plane which appears within the sub-wavelengthfocus at the focal plane, as a function of the focal distance. For the evanescent-wave-based focusing device, the intensity of the focal profile is generated by integrating |Ey|2across the entire waveform at the focal plane; for the superoscillatory wave, it is gener-ated by integrating the same quantity across the superoscillatory region.

at the screen plane which contributes to forming the sub-wavelength focus at the desired focal plane.

Clearly, the intensity of this field profile decays sharply with focal distance. In contrast, the fractional

intensity for the sub-wavelength focal spot produced by the superoscillatory waveform — which is also

plotted in Fig. 4.12b for comparison — remains invariant with respect to focal distance, and is orders of

magnitude higher than the corresponding intensity of the evanescent-wave-based focal spot at a distance

of 5 wavelengths. This comparison leads to two strong conclusions: (a) the phenomenon experimentally

observed within the SFW must be a result of superoscillatory waves, not evanescent waves; and (b)

despite the unavoidable existence of high-energy sidebands, superoscillatory waves can still generate

much stronger focal fields compared to evanescent-wave-based devices at focal distances of about half

wavelength and beyond.

There does not exist an obvious barrier which prevents one from designing a device with a larger

focal distance. Since the SFS and SFW do not involve evanescent waves, the focal distance can be

made arbitrarily far by adjusting the distance s in the back-propagation operation. However, since the

numerical aperture is given as NA = tan−1(D/2s), where D is the size of the device, an arbitrary increase

in the focal distance s without a corresponding increase in D will reduce the numerical aperture of the

device. In the case where NA 1, it will no longer be practically feasible to obtain a sub-wavelength

superoscillatory focus. Nonetheless one can still obtain a sub-diffraction superoscillatory focus, and


hence a sub-diffraction resolution, as defined by (2.11).

The potential is obvious to improve not only the focal length of a superoscillatory sub-diffraction

focusing device, but also the waveform performance, such as the width of the main beam, the duration of

the ROI, the sidelobe level within the ROI, the sideband amplitude and the sideband energy. Performance

in these five aspects can be traded off against one another: [73] and [77], for example, characterize

the dynamics of such trade-offs and suggest methods to construct optimal superoscillations given pre-

assumed waveform specifications. This work has taken a different direction: it has demonstrated that

practical superoscillatory sub-diffraction waveforms can be designed and physically synthesized, which

can ultimately be used in a sub-diffraction imaging system.

4.4.2 Sensitivity Analysis

A second remark relates to the sensitivity of the achieved superoscillatory waveforms. It is well known

that sensitivity issues prevent superdirective antennas from arbitrarily squeezing the width of a beam

emanating from an antenna. A sensitivity analysis is therefore conducted on the superoscillatory wave-

form generated in this chapter, and its results confirm that they tolerate practical noise levels at the

excitation plane. To characterize the sensitivity of the proposed method, the current excitation coeffi-

cients and the mode excitation coefficients for the SFW are separately varied by the vectorial addition

of a randomly phased white Gaussian component with a mean amplitude specified as a percentage of

the strongest excited antenna or spectral component. Fig. 4.13 pictorially shows how this variation

was introduced. Fig. 4.14a plots typical resulting waveforms after an addition of 1% current excitation

variation to the sub-diffraction focusing waveguide; Fig. 4.14b plots these waveforms after an addition

of 2.5% mode excitation variation. Despite the appearance of an uneven and increased sidelobe level,

the achieved sub-wavelength focal width is unaffected by these levels of perturbation. In practice, the

precise excitation of current elements (and hence the relevant waveguide modes) comes easier for super-

oscillatory waveforms than it does for superdirective antennas [103], since no radiative energy is stored,

and since relatively wide antenna spacings (d ≤ λ/2) reduce the significance of mutual coupling. It is

therefore reasonable to conclude that the aforementioned level of precision can be achieved without much

difficulty in current imaging systems. This analysis shows that the superoscillatory waveforms designed

in this section are reasonably tolerant to small perturbations at the excitation plane — which bodes well

for practical application in sub-diffraction imaging.


Figure 4.13: Diagram for introducing variations in the sensitivity analysis of the super-oscillation focal spot. The solid arrow depicts a sample excitation or spectral coefficient.The soft blue circle represents possible signal values after the addition of a variation. Adeeper colour tone represents a higher probability of occurrence. Dashed arrows depictpossible variation vectors.

−0.5 0.50

0.5

1

x/λ

Fie

ld A

mpl

itude

[n.u

.]

(a)

−0.5 0.50

0.5

1

x/λ

(b)

Figure 4.14: Sensitivity analysis of the superoscillation focal spot. (a) Typical waveformsobtained when excitation currents are varied by 1%. (b) Typical waveforms obtainedwhen waveguide mode excitations are varied by 2.5%. In both waveforms, the targetwaveform is plotted in red for comparison.


4.4.3 Comparison to other Superoscillatory Focusing Devices

A third remark compares the superoscillatory focusing screen presented in this chapter with other

superoscillation-based focusing structures developed in parallel to this work. The superoscillatory fo-

cusing screen has unique features in comparison to other superoscillation-based focusing structures. It

generates a superoscillatory focus deterministically, which allows one to design the location of the fo-

cal spot, and removes speckles present in the earliest demonstrations of superoscillatory electromagnetic

hotspots [79,81]. Moreover, it operates on a basis of sinusoids instead of prolate spheroidal wave functions

(PSWFs) introduced by Slepian and adopted by Huang an Zheludev [99], or the Optical Eigenmodes

(OEis) introduced by Mazilu et al. [83]. Compelling advantages exist for adopting a sinusoidal basis for

superoscillatory waveform design. First, it provides a simpler function set compared to both the PSWFs

and the OEis, for which function values often need to be pre-tabulated. Second, while the PSWF opti-

mizes the energy ratio within the ROI, and the OEi optimizes a physical parameter such as momentum

or spot size, using a sinusoidal basis allows one to design waveforms with handles familiar to electrical

engineers, such as beam width, sidelobe ripples and sideband amplitudes. These handles are less directly

related to the physics-related properties of the waveform, but very relevant to the performance of the

waveform when used in an imaging system. Using these handles, this chapter presented sub-diffraction

waveforms which have much lower sideband amplitudes compared to [99] and an ROI (in units of λ/NA)

much larger then [83] and related works [84–86]. The ease of use and functionality of a sinusoidal basis

distinguishes this work from other superoscillatory focusing devices.

4.4.4 Implementing the Superoscillatory Focusing Screen

Fourthly, as this chapter has presented the design and simulation result on the SFS, a remark is due to

suggest methods in which it can be built. The realization of the SFS relies on the implementation of

a periodic waveform Esrc(x) across a source plane, which, as observable from Fig. 4.4c, does not itself

contain superoscillatory variations. Since this waveform contains only propagating wave components, it

could be generated using reflectarray [104] or transmitarray [105] technologies, with elemental spacing

on the order of half-wavelength. Alternatively, newly emergent metasurfaces [106–108], which allow one

to control both the phase and amplitude of electric and magnetic fields, would make ideal candidates

for flexible implementations of superoscillatory focusing screens.



This chapter has described a procedure for designing sub-diffraction focused waveforms with a uni-

form sidelobe ripple within the ROI. Using this procedure, superoscillatory sub-wavelength focusing

electromagnetic waveforms have been designed and subsequently implemented using a SFS and a SFW.

Experimental results for the waveguide, as well as simulation results for both devices confirm the achieve-

ment of a sub-diffraction and sub-wavelength focus at a distance five wavelengths away from the source

plane — a distance tenfold improved from those achieved by evanescent-field-based imaging devices. A

discussion has elaborated on the salient properties of these sub-diffraction focusing devices, compared

them to other focusing devices, and suggested implementations for the sub-diffraction focusing screen.

While this chapter presents the design, simulation and implementation of sub-diffraction focusing

devices, the next chapter will present an effort towards transforming this device into one which performs

sub-diffraction imaging.

Chapter 5

Superoscillatory 2D Sub-Diffraction

Imaging: The Optical

Super-Microscope

5.1 Motivation

The previous chapter introduced superoscillatory sub-diffraction focusing devices capable of bringing a

sub-diffraction focus beyond the evanescent near-field. This chapter builds upon this pre-established

foundation to develop a device useful for sub-diffraction imaging. The foremost objective for such a

development is to find an elegant solution to transform the focusing device of the previous chapter into

an imaging device. Beyond this, it would also be of great practical advantage to image in 2D instead

of 1D, and image in free-space, rather than within the confines of a waveguide. Moreover, to exploit

full value for superoscillation-based imaging, one would do well to extend the image distance from a

multi-wavelength range to a distance millions of wavelengths away from the device, well into the far-field

of the imaging device. Finally, to demonstrate the scalability of superoscillatory imaging, it would be

ideal to perform free-space, far-field sub-diffraction imaging in the optical regime. This chapter reports

research progress in all aforementioned aspects, which culminate in the demonstration of the Optical

Super-Microscope (OSM), which becomes a unique imaging tool with great potential for a variety of

applications in sub-diffraction imaging.

68

Chapter 5. The Optical Super-Microscope 69

5.2 Conceptualization

5.2.1 From Focusing to Imaging

Traditionally, studies on imaging and focusing have been closely related. As an image is constructed

by object impulses, the resolution at which a focus can be produced determines the resolution of the

constructed wavefront. Conversely, the precision to which one can detect a point object’s location

also determines the resolution of the recorded image of an object. Hence one often simplifies an imaging

problem by first analysing the corresponding focusing problem. In this spirit, scanned imaging represents

the most straightforward way of performing sub-diffraction resolution imaging with a sub-diffraction

focusing device. A scan-based imaging system scans a sub-diffraction focal spot across the object,

collects the scattered electromagnetic wave at each scan location. The scattered waves contain sub-

diffraction information because (a) at each scan step, the superoscillatory waveform illuminates an area

on the object which is sub-diffraction; and (b) the step size — the location difference of the focal

peak between successive scan-steps — is also sub-diffraction. These scattered waves are numerically

processed to reconstruct the object at sub-diffraction resolution. Scan-based imaging has been popular

with evanescent-field-based imaging systems [5, 8, 52]. In parallel to this thesis, sub-diffraction scan-

based imaging has also been demonstrated by scanning sub-diffraction hotspots which reside in very

close proximity to high-amplitude sidebands [85,86]. Hence it stands to reason to envision a scan-based

sub-diffraction imaging system featuring the superoscillatory waveform obtained in the previous chapter;

furthermore one can reasonably expect such a system to achieve superior performances to [85, 86], due

to the expanded low amplitude ROI achieved by waveforms introduced in the previous chapter.

However, the present work seeks to find a more elegant solution to superoscillation-based sub-

diffraction imaging — one which does not rely on a scanning operation. Fig. 5.1 illustrates the operation

conception for the a conventional imaging system, the superoscillation-based scanned imaging system

and the present work. Fig. 5.1a depicts a conventional diffraction-limited imaging scheme: a diffraction-

limited pulse illuminates a diffraction-limited system (i.e. a linear system with a diffraction-limited

PSF). The convolution of the input pulse and the system PSF results in a diffraction-limited image. Fig.

5.1b illustrates the superoscillation-based imaging system reported in [85] and [86]: a superoscillatory

sub-diffraction illumination pulse is scanned across a diffraction-limited system, hence achieving super-

resolution as described in the paragraph above. Fig. 5.1c represents the concept adopted in this work.

The proposed OSM works does away with scanning operations by seeking to render superoscillatory the

Green’s function (or point spread function (PSF)) of a diffraction-limited imaging device. Beyond this

difference, the device operates same as a linear imaging system: the image is formed by the convolution


Figure 5.1: A diagram showing concepts of operation for (a) A conventional imagingdevice for which both the illumination and the system are diffraction limited. (b) Ascanned imaging device which relies on superoscillatory illumination, and (c) The pro-posed OSM, which uses a superoscillatory PSF. The illumination waveform to subfigure(b) is adapted by permission from Macmillan Publishers Ltd: Nature Materials [B],copyright 2012.

of the object (input) waveform and the system PSF. This way, the device images a point object into

a superoscillatory waveform with a sub-diffraction spot size; by extension, the device images a more

complicated object into an image waveform with sub-diffraction resolution. This avoids the need for

scanning and hence brings multiple advantages. First, equipment and computation cost is dramatically

reduced. Second, an image can be conveniently obtained in real-time: one no longer needs to wait while

the device collects and processes the data. Third, the ability to conduct real-time sub-diffraction imaging

allows one to image moving objects. These three advantages, coupled with the far-field imaging property

inherent to superoscillation-based methods, make the OSM a potentially compelling candidate for many

imaging applications.

5.2.2 Implementing a Superoscillatory PSF

This work implements the desired optical imaging system by passing the optical wave through a multi-

plicative filter in the spatial frequency domain. This operation can be accomplished by the so called 4F

system, which is depicted in Fig. 5.2, and works as follows. In far-field optics, a convex lens is known to

perform a Fourier transform on an object field one focal length away from the lens, and project its spec-


Figure 5.2: Schematic of a 4F imaging system.

trum onto an image plane one focal length away from the lens [109]. In this physical Fourier transform,

the spectral components (kx, ky) of a waveform maps into physical space (x, y) via the transformation

x =λf

2πkx; y =

λf

2πky, (5.1)

where f denotes the focal length of the lens. A 4F system includes two such lenses — the input and

output lenses — separated two focal lengths from one another. In this manner, light from an object plane

one focal length away from the input lens gets Fourier transformed into its spatial spectrum at the Fourier

plane, situated midway between the two lenses. The second lens then performs inverse Fourier transform

on this spatial spectrum to form an image at the image plane, one focal length away from the output

lens. In this system, a multiplicative spatial filter can be inserted at the Fourier plane to implement the

optical transfer function (OTF) of the system. This desired OTF forms a Fourier transform pair with

the desired superoscillatory PSF. The proceeding section discusses desirable qualities for the PSF and

its design procedure.

5.3 Designing a 2D Superoscillation Function

The conception of the OSM hinges on the design and implementation of a linear optical system, for which

the PSF is a desired 2D superoscillation function. This subsection reports progress towards designing

an appropriate PSF for the OSM.

The previous chapter reported the design of a 1D superoscillatory waveform. An easy way to extend

this design into 2D is to construct a separable function, namely

Asep(x, y) = A1D(x)A1D(y), (5.2)

where A1D(·) represents a 1D superoscillation function designed in the previous chapter, and Asep(x, y)

represents the resultant 2D superoscillation function. This separable superoscillation function retains


the ROI of its 1D counterpart: if A1D(x) has an ROI of x ∈ [−L,L], then the ROI of Asep(x, y) is a

square of sides 2L, centered upon the origin. However, this function is undesirable since its maximum

sideband amplitude is much more pronounced than its 1D counterpart: if A1D(·) attains a maximum

sideband amplitude of Asb at x = ±xmax (while its central peak is normalized), Asep(x, y) would feature

a maximum sideband amplitude of A2sb — found at (x, y) = (±xmax,±xmax). This pronounced side-

band amplitude render the waveform undesirable from both energy and noise performance perspectives.

Notwithstanding, an observation on Asep(x, y) offers some hope: its sideband distribution contains a

large degree of radial asymmetry. It is plausible to believe that one can achieve a lower Asb if one de-

signs a waveform with radial symmetry (or near-symmetry), since such a waveform spreads its sideband

energy more or less evenly in all directions.

Here the present work seeks to construct a radially symmetric 2D superoscillatory function with a

sub-diffraction focal peak, surrounded first by a circular ROI with low amplitude ripples, then by a

radial sideband. This radially symmetric function can be constructed as a series of zeroth order Bessel

functions, with uniformly increasing spatial frequencies:

g(r) =

P−1∑p=0

bpB0(kpx), where kp =

(p+

1

2

)∆k, (5.3)

where P represents the number of Bessel functions in this superposition. This family of functions is

in many ways similar to the family of cosine functions which are used in Chapter 4 to construct 1D

superoscillatory waveforms. In particular, both feature

1. Non-singular peaks at the origin;

2. Somewhat even spacing between successive nulls; and

3. A delta function (or a delta function pair) as its Fourier / Hankel transform.

To elaborate on the last point, a 2D Fourier transform of a radially symmetric function can be

simplified into a zeroth-order Hankel transform [109]:

F (kρ) = F2Df(r)

= 2πH0f(r)

= 2π

∫ ∞0

f(r)J0(kρr) r dr.

(5.4)

In particular, if f(r) = J0(kρ0r) (i.e. it represents a zeroth-order Bessel function in the r domain), its

Fourier transform is [110]


F (kρ) = 2π

∫ ∞0

J0(kρ0r)J0(kρr) r dr

=2π

kρ0δ(kρ − kρ0).

(5.5)

Hence a zero-order Bessel function Fourier transforms into a delta function in the kρ-domain, which is

to say its spectrum is a circular ring in the transverse k-domain (i.e. the (krho, kθ)-domain), centered at

the origin. It will become clear in a later section how this fact has a bearing on the implementation of

the 2D superoscillatory filter.

Observing the similarity between sinusoids and zero-order Bessel functions, one is encouraged to

attempt to design the Bessel function coefficients bp through slight modifications from the 1D design

procedure reported in the previous chapter. Two attempts will be presented below for generating a

sample waveform with parameters as listed in Table 5.1:


# of Bessel Functions (P ) 21

∆k 4πλ(2P−1)

Location of Bessel Functions (kρ) (p+ 1/2)∆k for p = 0 to P − 1

# of Nulls on +x axis (Q = P − 1) 20

# of Nulls within ROI (Q1) 5

NA 1

Sidelobe Level (Asl) 0.2 (for E-field)

ROI x ∈ [−1.2λ, 1.2λ]

Table 5.1: Summary of parameters for a test 2D superoscillatory waveform

The second method has proven successful and will be used to generate the 2D superoscillatory filter for

the OSM.

Method 1: Spectral Amplitude Matching

As a first attempt, a waveform is examined for which where the coefficients bp match those of an

equivalent design in 1D. First a 1D superoscillation function is designed which is comprised of N = 2P

plane waves, and with other parameters matching those listed in Table 5.1. The set kxn — which

denotes the locations of the delta functions in the kx domain — is chosen symmetrically across the kx

axis, such that K+ = kxn|kxn > 0 = kp; K− = kxn|kxn < 0 = −kp, where kp represents the

set of delta function locations in the kρ-domain. After the coefficient set an is found using the 1D

design procedure, bp is set to match the subset of an which corresponded to the location set K+:


bp = aP+p

for p = 1 to P. (5.6)

The resultant plane-wave coefficients an are shown in Fig. 5.3a; the weights adapted as bp are

highlighted in red. A cross-section of the resultant 2D waveform, g(x, y = 0), is plotted in Fig. 5.3b, along

with the 1D superoscillatory waveform. Clearly the two waveforms do not match and the 2D waveform

does not conform to the sidelobe and sideband performances of the 1D waveform. In particular, the nulls

are displaced going from the 1D to the 2D waveform. From a zero-based paradigm, this displacement of

nulls directly alters the amplitude profile of the waveform.

Figure 5.3: Designing 2D superoscillatory functions by matching spectral amplitudes.(a) Plane wave coefficients for the 1D design. Spectral components highlighted by thered rectangle are used as Bessel function weights for the 2D design. (b) A comparison ofthe 1D waveform and a cross-section of the 2D waveform at y = 0, showing noticeabledifferences and a displacement of nulls. (c) The comparison as in (b), plotted over theBloch period of the 1D waveform.

Method 2: Null Matching

As a second attempt, a waveform g(r) is examined which has the same zero locations rp in the

r-domain as the those on the positive x-axis of a corresponding 1D superoscillation waveform. A 1D

superoscillation function is designed which has 2Q = 40 zeros on the real axis (which requires a waveform

with N = 2Q + 1 = 41 plane waves) and follows other design parameters listed in Table 5.1. The zero


Figure 5.4: Designing 2D superoscillatory functions by matching null locations. (a)Plane wave coefficients for the 1D design (blue) and Bessel function coefficients for the2D design (red). (b-c) A comparison of the 1D waveform and a cross-section of the2D waveform at y = 0, showing well agreement in key performance metrics. (b) is thecloseup of (c) across the ROI.

locations in the +x-axis for the principle period of the waveform form the set xn for n = 1 to Q.

Requiring the 2D waveform share the same nulls leads to set of Q linear equations which allows one to

determine the coefficient set bp up to a constant b0:

g(r = xn) =

Q∑p=0

bpB0(kpxn) = 0

⇒Q∑p=1

bpB0(kpxn) = −b0B0(k0xn),

for n = 1 to Q.

(5.7)

As a final step b0 is determined through a normalization process, for example by setting the focal peak

amplitude of the waveform.

Fig. 5.4 compares the resultant waveforms in 1D and 2D. Fig. 5.4a shows the spectral amplitudes

(plane wave coefficients) of the 1D waveform as well as the spectral amplitudes of the 2D waveform.

While these spectral weights slightly differ, Fig. 5.4b-c show that the cross-section g(x, y = 0) of the

2D waveform matches all nulls of the 1D waveform, and attains similar performances key parameters

such as the focal width, the ROI, the sidelobe ripple level and the sideband amplitude level. The 2D


waveform features a gradual amplitude decay in the direction of increasing r, which is caused in part by

amplitude decay in all Bessel functions with increasing r. Marginal sideband reduction can be obtained

by slightly shifting nulls to the direction of increasing r, but such an optimization will not be pursued

in this work.

Since the null matching method allows one to extend my 1D sub-diffraction superoscillatory waveform

into 2D while maintaining key waveform performance metrics, this method will be adopted to design

the superoscillatory PSF of the OSM.

5.4 OTF Implementation with an SLM

The PSF designed in the previous section can be implemented through a spatially varying multiplicative

filter placed in the Fourier plane of a 4F optical system. This spatial filter will be implemented by a

reflective spatial light modulator (SLM) which features a 1028×724 pixel array and a pixel pitch of 9 µm.

The SLM used in this work is a liquid crystal on silicon (LCoS) device, which in essence features a layer

of liquid crystal above a reflective surface, which can be divided into an uniformly-spaced rectangular

array of pixels, each of which is driven by separate electronic circuitry. When a user-defined voltage

signal (discretely variable from 0 to 255) is applied to an SLM pixel, liquid crystal molecules on the SLM

pixel twist according to the voltage strength, which in turn induces an optical birefringence on the liquid

crystal within the specific SLM pixel. This induced optical birefringence modulates an incoming light

wave by giving it a spatially varying polarization shift. When a pair of polarization filters precede and

proceed the SLM, the polarization shift in individual pixels results in a spatially-varying modulation of

the light wave’s amplitude and in phase [109,111]. This collective modulation will hereafter be referred

to as the reflection coefficient of the SLM.

In the proposed OSM architecture to be discussed in detail later in this chapter, a polarization beam

splitter (PBS) controls the flow of optical waves, and serves as input and output polarization filters to the

OSM. With the PBS in place, a polar plot of the SLM’s normalized reflection coefficient, as a parametric

function of the voltage signal, is displayed in Fig. 5.5. Appendix C explains how this normalized

reflection coefficient can be obtained through a calibration procedure. One can observe two facts from

this plot: firstly, the variation in reflection amplitude and phase are coupled to one another; secondly, the

SLM achieves only a limited range of variations, which is obviously insufficient for performing arbitrary

modulations to the complex electric field. In particular, these limitations prove too restrictive for the

present cause, since the previous section has shown that the ideal superoscillatory OTF requires the

synthesis of reflection coefficients values which are positively and negatively real. To achieve the desired


Figure 5.5: A polar plot of reflection coefficient from the SLM. The arrow indicates thedirection of increasing drive signal level.

reflection coefficient, the remainder of this section describes an adaptation of a superpixel approach first

reported in [112].

Fig. 5.6a illustrates the superpixel approach. In this approach, two adjacent pixels combine to form

a superpixel, as labelled in the illustration. The SLM is tilted slightly along the x-axis from the Fourier

plane, such that for a normally incident light wave, the reflection path from pixel B (r(lb)) is longer than

the reflection path from pixel A (r(la)) by half-wavelength. In this manner, the reflection coefficient

(with respect to the Fourier plane) from pixel B is π-phase shifted from that of pixel A as shown in Fig.

5.6b, and the reflection coefficient from the superpixel,

rsp(la, lb) = r(la)− r(lb), (5.8)

can be varied with much increased freedom. Three facts are true in particular:


Figure 5.6: Superpixel approach to generating arbitrary reflection coefficient from theSLM. (a) Schematic of the superpixel configuration. Co-ordinates are given by the greenarrows. (b) A plot of the resultant reflection coefficient. The blue and red curves rep-resent reflection coefficients from pixels A and B, the solid and dashed arrows representreflection coefficients for a specific signal set (la, lb), and the green arrow — being thevector summation of the black arrows — represent the superpixel reflection coefficientrsp(la, lb).


1. A set of la’s and bs can be found such that =r(la) = =r(lb) ⇒ rsp(la, lb) ∈ <;

2. rsp(lb, la) = −rsp(la, lb); and

3. rsp(la, la) = 0.

(5.9)

These facts show the sufficiency of the superpixel configuration in generating positive and negative real

reflection coefficients.

Clearly the superpixel approach is not without its drawbacks, the most notable of which is that the

resolution of the SLM has decreased by half. Additionally, tilting the SLM at the Fourier plane may

result in imperfect spectral modulation. Nonetheless, experimentation has shown that the decreased

resolution remains sufficient and the tilt insignificant for the purpose of this work. Hence the superpixel

approach is found reliable for generating real reflection coefficients needed for the OSM.

5.5 Constructing the Optical Super-Microscope

After laying groundwork on conceptualization and design, the time comes to assemble the pieces devel-

oped thus far to construct the OSM. This section describes numerical parameters chosen for the OSM

prototype, presents the superoscillatory filter designed, and explains the construction of the OSM.

5.5.1 Numerical Parameters and Design

A HeNe laser source, which has a wavelength of 632.8 nm, is chosen to illuminate the OSM. The working

distance, which is equal to the focal length of the lenses L1 and L2 within the 4F optical system, is

chosen to be 400 mm. Dividing this into the effective half-aperture of the OSM (set by the width of the

SLM pixel array) gives a numerical aperture of

NA =3.456 mm

400 mm= 0.00864. (5.10)

These filter design parameters, along other factors which regulate PSF performance, are tabulated in

Table 5.2.

With these specifications, the 2D superoscillation PSF is designed following the procedure outlined

in section 5.3. Fig. 5.7a-b show the zero location, and plane wave coefficients for the 1D superoscil-

lation function design; Fig. 5.7e shows its waveform distribution. This design is extend into 2D using

the null matching method. Fig. 5.7c shows the Bessel function coefficients; Fig. 5.7d shows the cor-


Figure 5.7: Superoscillation PSF design for the OSM. (a) Zero locations for the 1D PSFdesign. Circles denote zeros within the ROI, Crosses denote zeros without the ROI.(b) The corresponding plane wave coefficients. (c) Bessel function coefficients for a 2DPSF, sharing the same nulls as the 1D design. (d) The corresponding reflection filter tobe synthesized by the SLM. (e) Comparison of the spatial profile of the 1D waveform,and the a cross-section of the 2D waveform at y = 0.


Design Parameter Value

λ 632.8 nm

NA 0.00864

Focal Length (f) 400 mm

# of Bessel Functions (P ) 32

# of Nulls on +x axis (Q = P − 1) 31

# of Nulls within ROI (Q1) 6

∆k = NAk0P

2.7× 10−3k0 = 2681 [rad/m]

Lowest frequency component (kp=0) 12∆k = 1340 [rad/m]

Sidelobe Level (Asl) 0.45 (for E-field)

ROI r ∈ [−1.9λ/NA, 1.9λ/NA]

Table 5.2: Summary of superoscillatory filter design parameters for OSM

responding spectrum of the 2D superoscillation waveform; and Fig. 5.7e shows a cross-section of the

resultant waveform across the x-axis. The comparison in Fig. 5.7e shows the successful extension of

the superoscillatory waveform from 1D to 2D. The 2D waveform matches the ROI duration of the 1D

counterpart, and features similar sidelobe and sideband levels to its 1D counterpart. Fine adjustment

in zero locations can parlay the rolling sideband envelope into slight improvements in PSF performance.

Nonetheless, this work has forgone a pursuit of such improvement to concentrate on the demonstration

of sub-diffraction imaging with the OSM.

To achieve the designed 2D PSF, one needs to implement the 2D spectrum depicted in Fig. 5.7d as

the OTF of the OSM. While it is not possible to implement a filter with circular rings of infinitesimal

thickness, it has been experimentally found that rings of thickness ∆k can instead be used. This

effectively approximates each delta function in the kρ-domain with a corresponding rectangular function,

which holds a constant value from kp−∆k = p∆k to kp + ∆k = (p+ 1)∆k, and vanishes elsewhere. (As

a reminder, kp has been defined in (5.3); while the kp=0 and ∆k for this design have been tabulated in

Table 5.2.) This modified filter is mapped spatially onto the Fourier plane via the relation

r =λf

2πkρ, (5.11)

which is the radial equivalent of (5.1). From (5.11) one finds the spatial width of the rings:

∆r =λf

2π∆k. (5.12)

Fig. 5.8 shows a plot of the 2D reflection filter. The filter values are sampled over the 2D grid of

1024 × 384 superpixels comprising the SLM. Then, for each superpixel, the corresponding set (la, lb) is


Figure 5.8: The target spatially varying filter at the Fourier plane.

determined provide the required reflection coefficient at the corresponding location. Finally, the resultant

signal array is programmed into the SLM to synthesize the desired filter.

5.5.2 Assembling the Experimental Apparatus

Fig. 5.9 shows a schematic of the OSM. It shares great similarity with the example 4F system shown

in Fig. 5.2, except (1) it becomes a “folded” system to implement a reflective filter; (2) it features a

polarization beam splitter which directs light flow within the system; and (3) it includes a diaphragm in

the light path which blocks spurious beams to improve the noise performance of the system. The light

path within the OSM is labelled by arrows which are overlaid atop the optical beam.

Fig. 5.10 shows a photo of the OSM prototype assembled on an optical table. A HeNe laser of

wavelength 632.8 nm illuminates the object. The first lens L1 transforms the light field at the object

into its spectrum at the Fourier plane, where an SLM performs reflective filtering. Thereafter, the

second lens L2 performs an inverse Fourier transform, producing a final image in the spatial domain.

The working distance of the OSM is 400 mm — determined by the focal length of the lens used. As

listed in Table 5.2 the OSM as presently built has a numerical aperture NA = 0.00864. Hence applying

equation (2.11) it has a corresponding diffraction-limited spot width of

D =λ

2NA= 36.7 µm. (5.13)

While obviously a high NA system would be desired for high-resolution imaging, A low NA system is

chosen for this proof-of-principle experiment, so that sub-diffraction images can be well resolved without


Figure 5.9: A schematic of the optical super-microscope (OSM). The distance betweenthe five labeled planes: P1, L1, P2, L2 and P3 are 400 mm apiece.

further magnification optics by a CMOS camera placed at the image plane.

5.6 Calculation and Experimental Results

This section reports imaging results using the OSM. First, a comparison is made between the PSF of

the OSM and that of a diffraction-limited imaging system of the same numerical aperture. Second,

a comparison is made between the images of two 15 µm circular holes, with varied orientations and

separation distances, as obtained from both systems. In both cases experimental measurements agree

well with calculations, and demonstrate the OSM’s achievement of 2D far-field sub-diffraction imaging,

beyond the Abbe and Rayleigh diffraction limits.

5.6.1 Point Imaging Experiment

Fig. 5.11a-b compare the PSF of the OSM with a diffraction-limited imaging system of the same NA. The

PSFs are obtained by imaging a small circular aperture 10 µm in diameter; the PSF of the diffraction-

limited system is obtained by replacing the superoscillatory filter with a uniform reflection filter. Fig.


Figure 5.10: A photograph of the OSM prototype, with annotations showing the lightpath within the device.

5.12 shows that for an optical system with the current NA, a 10 µm aperture is sufficiently small to have

negligible effect on the PSF. The PSF of the superoscillation imaging system contains a sub-diffraction

spot size 28 µm in intensity FWHM. This is clearly reduced from the spot size of the diffraction-limited

system, which measures 39 µm, in agreement with (5.13) to within the camera pixel pitch of 5.3 µm. This

superoscillatory sub-diffraction spot is 150 µm separated from the high-intensity sideband. The inner

half of the region of silence forms the OSM’s field of view (FOV): as long as the objects spatial extent

fall within this FOV, a sub-diffraction image can be achieved without contamination from overlapping

with high-intensity sidebands of the superoscillatory PSF. Fig. 5.11c compares the horizontal cross-

section of the intensity profile with theoretical calculation, while Fig. 5.11d displays a closeup of the

comparison across the ROI. Good agreement has been achieved between calculation and experimental

measurement with the exception of two notable differences: (i) the sideband amplitude is clearly lower in

experiment compared to calculation, and (ii) the focal width is slightly widened in experiment compared

to calculation. Both differences are likely caused by the discrete and rectangular pixelation of the

SLM which is used to implement the desired (circular) filter. As also observed in the sub-diffraction

focusing waveguide experiment, slight imperfections in mode superposition can lead to a slight widening

of the superoscillatory waveform, as well as a notable decrease in sideband amplitude. Nonetheless, the

measured focal width remained clearly narrower than the diffraction-limited peak. This improvement

would translate into sub-diffraction imaging for more complicated objects.


Figure 5.11: A comparison of measured PSFs for the OSM and the diffraction-limitedsystem. (a) An image of a 10 µm hole obtained by the OSM, which approximates thesystem’s PSF. (b) A close up of the focal point (top), which clearly shows a reducedsize compared to the diffraction-limited focal point (bottom). (c) The horizontal cross-section of the intensity profile, plotted alongside theoretical calculation. (d) A closeupof (c) across the ROI, compared against the diffraction limit.


Figure 5.12: A comparison of a diffraction-limited imaging system’s PSF with an imageit generates of a small circular aperture. The diffraction-limited imaging system hasλ = 632.8 nm and NA = 0.00864. The comparison shows that the two waveforms arenearly identical.

5.6.2 Two-Point Resolution Experiment

Next the two-point resolution capability of the OSM. The inset of Fig. 5.13 shows a diffraction-limited

image obtained for two 15 µm circular apertures, separated 55 µm (0.75λ/NA) center-to-center. At this

separation the two apertures cannot be resolved with a diffraction-limited imaging system, in agreement

to the coherent Rayleigh and Sparrow limits (2.16) derived in Chapter 2. However, Fig. 5.13 shows

that the same apertures are well resolved by the OSM and are well separated from the high-intensity

sidebands. An interesting interference pattern forms at the high-intensity sidebands which has caused

an angular modulation on the sideband intensity. Monitoring such sideband interference patterns can

perhaps yield sub-wavelength information for a selected class of known objects, and might warrant an

in-depth investigation for selected applications.

To validate the above experimental result, Fig. 5.14 shows the calculated results for two-point

resolution with and without the OSM. The calculation simulates the OSM by convolving the object field

with the PSF of the OSM. Clearly, the experimental image contains a diffraction ring very similar to

the one obtained in calculation. Features within the ROI are obscured in Fig. 5.14 and their proper

observation requires a fivefold field magnification. This is because the calculated PSF features a more

pronounced sideband compared to the experimental one, as shown in Fig. 5.11c-d. Nonetheless, the

inset, which shows features within the ROI with a field strength which is magnified fivefold, shows the


Figure 5.13: Resolving two closely-spaced objects using the OSM (I). The main panelshows that the OSM resolves two 15 µm apertures, separated 55 µm center-to-center.The inset shows the corresponding (unresolved) image for the diffraction-limited system.

Figure 5.14: Calculated Result of the 2-point Resolution Experiment. The aperturedimensions and separation are the same as the same as in Fig. 5.13. The field strengthis magnified fivefold within the white square located in the middle of the ROI.


resolving of the two apertures, as well as noise profile somewhat similar to that observed in Fig. 5.13.

Hence reasonable agreement has been reached between calculation and experiment.

Fig. 5.15 compares the imaging capability of the OSM and the diffraction-limited system for two

circular apertures separated horizontally, vertically or diagonally by a varied set of distances. To improve

clarity, closeups of the waveforms are displayed with exclude the high-intensity sideband. From Fig. 5.15,

one can see that the OSM resolves two apertures spaced 45 µm apart horizontally and vertically, and

42 µm apart diagonally. This is clearly improved from the diffraction-limited system, which has an

experimental two-point resolution distance of 60 µm. Further resolution improvements can be obtained

with filters with finer pixelation, which will faithfully implement PSF designs containing more degrees

of freedom. The present work nonetheless successfully demonstrates the OSMs capability to perform

optical sub-diffraction imaging at a long working distance of 40 cm.

5.6.3 Real-Time Imaging for Moving Objects

The OSM’s ability to directly image an object without scanning brings attractive advantages and enables

novel functionalities in comparison to scanned-imaging solutions to sub-diffraction imaging [8, 9, 49,86].

Firstly, this dramatically simplifies the imaging system by alleviating needs for sub-wavelength scanning

stages and subsequent post-processing overhead. Furthermore, since an image can be formed in real-

time, the system can be used to image time-varying objects. To demonstrate this, Video 5.1, submitted

with this thesis as Wong Alex MH 201404 video.wmv, shows a real-time capture of our object as

it laterally moves in the object plane. Two points should be emphasized with regards to this video.

Firstly, each frame in this video came directly from a camera capture: data processing has not been

performed. Secondly, the video plays in real-time — at the rate at which camera captures happened

— which rate can be much improved with better data-acquisition hardware and a streamlined image

capture program. Hence the OSM has the capability to perform real-time imaging on moving objects.

This functionality of the OSM further distinguishes it from previously-proposed superoscillation-based

imaging systems [86, 99], and opens doors to attractive potential applications such as in vivo far-field

sub-diffraction biomedical microscopy.

5.7 Discussion

This section includes remarks in which situate the OSM with respect to past and concurrent imaging

devices, and discuss prospective directions for further development.


Figure 5.15: Resolving two closely-spaced objects using the OSM (II). Close ups oftwo-point resolution images, for two 15 µm apertures, separated by various distancedistances, in (a) horizontal, (b) vertical and (c) diagonal configurations. Horizontal andvertical separations are 40 µm, 45 µm, 50 µm, 55 µm and 60 µm respectively, from leftto right; diagonal separations measure 35.3 µm, 42.4 µm, 49.5 µm, 56.5 µm and 63.7µm respectively, from left to right. The top row shows a close up of the superoscillatoryimage, while the bottom row shows the corresponding view with the diffraction-limitedsystem.


5.7.1 Connection to Super-Resolving Apertures

A first remark situates the OSM’s implementation of a super-resolving PSF with respect to previous

works on super-resolving apertures. In this work, the PSF of the OSM is implemented by synthesizing

a multiplicative filter equivalent to the OTF, which can be viewed as a class of super-resolving aper-

tures. Super-resolving apertures were first proposed by Toraldo di Francia through inspiration drawn

from superdirective antennas [113], but later theoretical and numerical works [114] have deemed them

impractical for the same reasons which appeared to undermine the practicality of superoscillations. How-

ever, this work has bypassed this hurdle by careful considerations from the perspective of antenna array

design: by experimentally demonstrating sub-diffraction imaging with the OSM, this work has proven

that super-oscillation systems with practical trade-offs can be constructed using properly designed super-

resolving apertures.

5.7.2 Connection to Resolution Restoration Methods

Secondly, a remark is in order to link the OSM to resolution restoration methods, particularly those

proposed in the numerical domain. Numerical algorithms exist to restore the high-resolution informa-

tion lost due to the illumination wavelength and numerical aperture of the imaging system. Based on

the analyticity of the object field’s spatial spectrum, one can in principle perform analytical continu-

ation to restore high spatial frequency components which are lost in the imaging process [115]. [116]

reports such a restoration method whereby the image field is sampled at low spatial frequency compo-

nents, then extrapolated to restore higher spatial frequency components using the Shannon-Whitaker

sampling theorem. However, while this scheme works in principle, it does not guarantee whether the

involved multiplicative matrix (resulting from a system of linear equations, each of which comes from

an application of the sampling theorem) is sufficiently well-conditioned, and does not predict when a

reliable solution can be reached. Further, in such cases it is also unclear how one might best choose

the required low-frequency sample points to enhance the accuracy of the high-resolution energy being

restored. These unanswered questions help lead to the proposal of other resolution restoration methods,

including one which explicitly involves post-processing using superoscillating prolate spheroidal wave

functions (PSWFs).

Soon after the initial report on PSWFs [70], Barnes proposed a method to restore an object’s ap-

parently lost resolution using superoscillating PSWFs. A PSWF is a scaled version of itself after two

successive delimitations — first in spatial domain then in spatial frequency domain. Therefore, it serves

as an eigenfunction to the lowpass imaging system (which, as explained in Chapter 2, effectively forms


the diffraction limit). Furthermore, a PSWF (of sufficiently high order) serves as an eigenfunction which

carries high resolution information on an object through its region of superoscillation. Hence, the EM

field pertaining to the original object can be restored by decomposing the lowpassed image into PSWFs,

amplifying each PSWF through division by its respective eigenvalue, then superimposing the amplified

PSWFs. In principle, such a procedure restores the object field with infinite resolution; in practice, the

resolution is limited by the number of PSWFs included in the summation.

Barnes’ work bridges the numerical concept of resolution restoration and the OSM presented in

this chapter. If one imagines applying Barnes’ restoration procedure to a lowpassed image of a point-

object, one would obtain, as a resolution-restored image, a superoscillation waveform, featuring a uniform

combination of all PSWFs used in the summation. Since Barnes’ restoration system is linear, one can

characterize it with a PSF equal to the uniform summation of the set of involved PSWFs. Hence, one can

physically implement this system in a structure similar to the OSM, only with the multiplicative filter

at the Fourier plane replaced by a reflection filter which represents the Fourier transform of the PSF.

Conversely, one can say that the OSM physically performs resolution restoration in a similar way to the

PSWF system hereby mentioned, but with a carefully designed PSF for optimized performances in peak

width, sidelobe level and sideband level. In this light one can understand the OSM as a device which, in

a round-about manner, performs sub-diffraction imaging by locally restoring an object’s high-resolution

information.

This subsection ends with a side note on a numerical versus physical implementations of optical

super-resolution systems. In 2012, Piche et al. [87] reported such a system in which the phase and

amplitude of an optical image wavefront were measured, at which point sub-diffraction was restored

through a numerical method akin to that reported in [89]. However, the authors expended much care in

optical experimentation to obtain holographic measurements which allowed them to deduce the phase

of the optical wavefront. As it is usually difficult to measure the phase of an optical wavefront, the

aforementioned numerical resolution restoration methods are often inapplicable in the (coherent) optical

regime; instead, non-linear methods are used. In this regard, systems which achieve sub-diffraction

resolution in a physical manner proves advantageous: they avoid the difficult task of measuring the

optical phase. Moreover, as a physical system which achieves sub-diffraction resolution, the OSM brings

added advantages of instantaneous image formation and freedom from computation load, which cannot

be similarly said for numerical super-resolution methods.


Kosmeier et al. [85] Rogers et al. [86] This work

General Specifications

Illumination Wavelength 633 nm 640 nm 632.8 nm

Working Distance 1 m 10.3 µm 40 cm

Numerical Aperture1 NA = 4.32× 10−3NAi = 0.889;

NAo = 1.4NA = 8.64× 10−3

Scanning Required? Yes Yes No

Scan Step 6.875 µm 50 nm ——

Design and Implementation of Illumination / Filter

Design Method Optical EigenmodesParticle Swarm Opti-mization

Superdirectivity In-spired Zero Placement

Implementation Two 1920× 1080 SLMsFIB-fabricated Al-filmon glass substrate

One 1024× 768 SLM

Minimum feature size 8 µm 200 nm 9 µm

Focal Quality of Illumination / PSF

ROI2 122.4 µm = 0.835λ/NA 454 nm = 0.630λ/NAi 300 µm = 4.10λ/NA

Focal Width (Intensity FWHM) 44.71 µm = 0.305λ/NA 185 µm = 0.257λ/NAi 28 µm = 0.382λ/NA

Improvement over Diff. Lim. 0.44∆wdl 0.50∆wdl3 0.61∆wdl

Imaging Performance: Two-Aperture Resolution

Min. Resolvable Distance:

Center-to-Center 65 µm = 0.44λ/NA 331 nm = 0.72λ/NAo 45 µm = 0.72λ/NA

Edge-to-Edge 55 µm = 0.38λ/NA 121 nm = 0.26λ/NAo 30 µm = 0.41λ/NA

Improvement over Diff. Lim.4 0.78∆sdl 0.92∆sdl3 0.75∆sdl

Table 5.3: Comparison between the OSM and two concurrent superoscillation-basedimaging devices [85,86]. In this table ∆wdl and ∆sdl respectively denote the spot widthand two-point resolution distance of a conventional, diffraction-limited imaging system.Table annotations are hereby given. 1. For [86], NAi represents the numerical apertureof the superoscillatory illumination, which is relevant for evaluating the focal quality,while NAo represents the numerical aperture of the light collection component, relevantfor evaluating the imaging performance of the device. 2. The term ROI (region ofinterest) adopts the definition taken in this work, not the slightly different definitionused by [85]. 3. Since an explicit comparison was not given in [86], the “improvement”figures are derived by comparison with the calculated diffraction limits of a simple opticalsystem with illumination wavelength λ and numerical aperture NA. 4. Th resolutionimprovement is calculated and quoted as the center-to-center distance between twoapertures.


5.7.3 Comparison to Concurrent Developments

Thirdly, it is instructive to compare the OSM with two superoscillation-based imaging devices developed

concurrently to this thesis, namely the work by Kosmeier et al. [85] and that by Rogers et al. [86].

Table 5.3 compares the salient features of these three superoscillation-based imaging devices. It should

be noted that such a comparison is by nature imperfect, since different criteria have used to design,

implement and report the respective imaging devices. In particular, comparison with [86] is difficult,

since it features two relevant NA’s, and since its two-aperture resolution experiment has been conducted

with relative large apertures (0.46λ/NA, as opposed to 0.14λ/NA for this work and 0.068λ/NA for [85]).

In regards to the latter point, since it has been verified that for small apertures up to 0.5λ/NA in

diameter, the aperture size causes negligible effects on the diffraction pattern, apart from an amplitude

constant corresponding to the amount of light which gets through the aperture. Due to this fact, the

improvement factor for two-point resolution has been calculated using the distance between the centers

of the respective apertures, as opposed to their edge-to-edge distances.

From the first two sections of the table, the OSM stands out with its scan-less operation, its relatively

simple implementation and its generous minimum feature size. As shown in the previous section, the

OSM’s scan-less operation enables the fast capture of an image and allows one to image of moving

objects. The generous minimum feature size is also advantageous, in that one can potentially reduce

it to obtain further resolution improvements. This point will be addressed in the following subsection.

In terms of the focal quality and the imaging performance, it can be seen that the OSM achieves a

PSF focal width 72% that of the diffraction limit, and a minimum resolvable distance 75% that of the

diffraction limit. These figures are comparable to the works by Rogers et al. and Kosmeier et al.. The

achievement of comparable resolutions to concurrent superoscillation-based imaging devices, along with

the demonstration of a much larger ROI and the capability to perform single-shot imaging, give the

OSM attractive merits for diverse imaging applications. The following subsections will discuss ways

which may bring significant improvements to the OSM presented in this work.

5.7.4 Imaging in the Presence Superoscillatory Sidebands

A discussion should now be given on performing imaging in the presence of superoscillatory sidebands,

which inevitably exist alongside sub-diffraction features in a superoscillatory waveform. One immediate

effect of the existence of the superoscillatory sideband is that the majority of optical power emanating

from the object is imaged into the sideband. The Strehl ratio — which measures the fractional power

within the superoscillatory region — of the PSF of the OSM measures SOSM = 0.0064. This means that


when an intensity level of 1 mW emanates from the object and passes through the OSM, roughly 6.4 µW

contributes to the formation of the image, while the rest of the power contributes to the superoscillatory

sideband. (The exact proportion depends on the object’s diffraction characteristics.) While this figure

may seem inefficient, it is well within limits of acceptability for optical microscopy, where detectors can

pick up weak signal levels of a few photons pre second, while sources are commonly available in the

range of 1 W — on the order of 1019 photons per second.

As for potential damage caused to biomedical specimens by a relatively high-powered excitation

beam (to compensate for a relatively low power efficiency), an examination at prevalent super-resolution

microscopic modalities should at once alleviate this concern. STED microscopy, for example, uses a

powerful “STED” beam whose peak power intensity reaches 1 GW/cm2

to deplete carriers from all

fluorescent particles outside a sub-diffraction area of interest [65]. Two photon microscopy, which probes

the second-order non-linearity of biological specimen, typically involves excitation pulses at even higher

intensities. Notwithstanding, light pulses of such intensity levels do not cause damage to most biological

specimens as long as they are injected at sufficiently low duty cycles [65]. Comparatively, the OSM

requires excitation pulses with much lower intensity levels than the aforementioned imaging modalities,

since it does not involve non-linear optical interactions between the excitation beam and the specimen

under view. A crude estimate of the required excitation intensity for OSM can be obtained as follows.

As an example, let us assume a scenario whereby one wants to image a point object to a sub-diffraction

width of λ/3NA (roughly the resolution limit of the OSM reported in this work), and hence require the

reception of 100 photons per second across the image area of Aimg = π(λ/6NA)2. This rate of photon

reception would provide enough signal to overcome the detection threshold and allow one to properly

determine the brightness of the point object. Further, assume that (a) the point object scatters uniformly

the incident illumination, such that the amount of scattered light which reaches the image plane can be

approximated by the ratio Θ = πNA2/2π2 = NA2/2π, and (b) half the photons are lost within the OSM,

either from the filtering operation or through spurious reflections across optical components. Under such

a scenario, the rate at which photons must scatter off point object can be estimated as

Rp,obj =2× 100

SOSMΘ. (5.14)

Now the energy of a photon is given by

Ephoton = hf =hc0λ, (5.15)

where h = 6.626× 10−34 is the Planck’s constant and c0 is the speed of light in free space. Combining


these two facts, the optical intensity of the incident light pulse can be estimated as

Iobj =Rp,obj × Ephoton

Aimg=

14400hc0SOSMλ3

. (5.16)

Substituting the wavelength and Strehl ratio for the reported OSM, the intensity comes to Iobj =

177 [µW/cm2] — which is many orders of magnitude smaller than intensities required for superresolution

modalities based on optical non-linear interactions.

Besides power considerations, the presence of a high-energy superoscillatory sideband also places a

limit on the FOV of the imaging system. A later subsection addresses how one can increase the FOV

of the OSM; the present discussion elaborates the care which one needs to take to eliminate spurious

scattering from outside the FOV. As can be reasoned from a glance at the PSF, shown experimentally in

Fig. 5.11, an object from outside the FOV images in such a way that its superoscillatory sideband comes

within the FOV. This will seriously compromise the image. Care is hence needed to eliminate spurious

sources of light from entering the OSM from outside the FOV on the object plane. In experiments

reported in this chapter, an opaque screen is used to block optical illumination (and spurious light)

from passing through the object plane, at all places other than the location(s) of the object aperture(s).

Other approaches, which might prove more practical in implementation in a microscopy system, include

forming a flat-top illumination beam which illuminates only the FOV, and/or constructing an aperture

(or effective aperture) which limits the OSM’s field of light-wave intake to exactly the FOV on the object

plane. Practical implementations of these approaches should be straightforward, considering that the

extent of the FOV is several times larger than the diffraction limit.

5.7.5 Further Resolution Improvements

Experimental results on PSF measurement and two-point resolution investigation have convincingly

proven the viability of performing sub-diffraction imaging using the OSM. The OSM features a reduced

spot width at 72% of the Abbe diffraction limit, and a minimal resolvable distance 75% of the equivalent

Rayleigh criterion. While this work has successfully provided a proof of principle for the proposed concept

of optical super-microscopy, further resolution improvements can be achieved by increasing degrees of

freedom in the spectral domain. While the finite pixelation of the SLM (which measures 18 µm× 9 µm

per superpixel) has limited this work from achieving higher degrees of freedom, such a limitation can be

alleviated by using a custom fabricated mask as the superoscillation filter. Contemporary fabrication

technologies can reliably produce masks with feature sizes on the order of 100 nm. Using masks with a

feature size on this length scale will enable one to encode more degrees of freedom into the superoscillation


filter, and thereby achieve further resolution improvement with the OSM.

5.7.6 Extending the Field of View

While the FOV of the OSM can be restrictive in certain cases, two methods can be employed to dra-

matically increase the FOV of the OSM. Firstly, with the usage of a custom fabricated mask with finer

features, a portion of the increase in degrees of freedom can be used to design a waveform with a larger

ROI, hence increasing the device’s FOV. Secondly, to image a large object, an illumination beam occupy-

ing the OSM’s FOV (which for our case is a circular area with a diameter of approximately 2λ/NA) can

be scanned across the object, and the image of adjacent fields of view can be stitched together, much like

stitching together views from a conventional microscope. It should be emphasized that this large-step

scanning procedure differs fundamentally from a sub-wavelength scanning procedure in near-field probes,

spatially shifted-beam based imaging or the super-oscillation lens [9, 49, 86], in that the step size is on

the order of the FOV, which for our case is more than six times the resolution limit of the system. Thus

this scan modality does not contribute to resolution enhancement; it only increases the overall FOV of

the system. Further, with a large-step scan, a sub-diffraction image of a large object can be obtained in

less time, avoiding mechanical drifts and other slow time artifacts such as those observed in [86].

5.7.7 Sub-wavelength Optical Super-Microscopy

Besides large object imaging, OSM also has the potential to perform sub-wavelength imaging. A low

NA is chosen in this work to enable the measurement of sub-diffraction features with a CMOS camera.

Notwithstanding, one can adapt the OSM to perform far-field imaging at sub-wavelength resolution

simply by increasing the NA to close to, or beyond unity. This can be accomplished by including

conventional magnification stages before or after the OSM, and/or using Fourier transforms lenses with

higher numerical apertures. In the latter case, to maintain waveform fidelity in the spectral domain, one

may want to replace the SLM with a custom-fabricated transmission or reflection filter, which contains

finer feature sizes than the 18 µm superpixel pitch resolution for the SLM used in this work. As explained

above, a spatial amplitude and phase filter with much reduced feature sizes can be reliably produced

with present custom fabrication methods. Further, with the combined amplitude and phase modulation

capability of a custom-fabricated mask, one might be able to do away with Fourier transformation lenses,

and arrive at a single-element super-resolution imaging device. Such enhanced simplicity can lead to

widespread applicability.


5.8 Summarizing Conclusion

This chapter has presented an OSM capable of linear, far-field imaging beyond the diffraction limit. It

provided motivation for the development of a microscopic tool that, in departure from focusing devices

introduced last chapter, operates in free space, in the optical regime, and features an extended working

distance of millions of wavelengths. It has conceptualized and constructed such an imaging device;

in particular it has reported the design and implementation of a 2D superoscillatory PSF which has

subsequently proven suitable for the OSM. The constructed OSM successfully achieved sub-diffraction

imaging: it featured a PSF spot width 72% that of a diffraction-limited system with the same NA, and

a minimal resolvable separation 75% that of a diffraction-limited system. A real-time video capture has

also demonstrated the ability of the OSM to perform direct imaging, without needs of scanning or data

post-processing. The operation of the OSM has been related to earlier research efforts on super-resolving

apertures, and several remarks have been made on how one can further improve the OSM. To this point,

the work has have successfully developed a sub-diffraction imaging device which features linear operation,

a far-field working distance, and avoids undesirable operations of pre-labeling, scanning and data post-

processing. The OSM satisfies all criteria for an ideal imaging device listed in the introductory chapter

of this thesis; its unique properties should make it an attractive tool for general purpose imaging beyond

the diffraction limit.

Chapter 6

Temporal Superoscillations

6.1 Introduction

Since superoscillation relates waveform oscillations in one domain with its spectral width in the reciprocal

domain, its manifestation is unrestricted to the space-spatial-frequency domain which we have examined

thus far into this thesis. In fact, superoscillations can arise in any pair of reciprocal domains, including

the time-frequency domains which are of paramount importance to many aspects of electrical engineering.

The earliest work on superoscillation naturally discussed the topic from a time-frequency perspective [70];

more recently, works which analyzed superoscillations from an information theory point of view also

naturally considered temporal superoscillation signals [73,75]. However, an experimental demonstration

of a temporal superoscillation waveform has yet to be reported (aside from this work), probably due

to the presumed difficulties and apparently impractical dynamic range requirements associated with

synthesizing such waveforms.

This chapter reports an investigation on time domain superoscillations. It reports an intuitive pro-

cedure for designing practical temporal superoscillations, and uses this method to design and synthesize

temporal superoscillatory electromagnetic reforms. It reports basic investigations on these waveforms,

then demonstrates their use in improving the range resolution in a radar imaging system.

6.2 Designing Temporal Superoscillations

The general procedure for designing a superoscillatory waveform, as introduced in Chapter 3, applies

just as well to temporal superoscillations as it does to spatial superoscillations. In this section, the

aforementioned procedure will be recast into the time domain, to obtain an efficient tool for designing

98

Chapter 6. Temporal Superoscillations 99

practical temporal superoscillatory waveforms. Design strategies will also be described for two specific

types of superoscillatory waveforms — the rapid oscillation waveform and the sharp pulse waveform.

6.2.1 A Design Procedure for Temporal Superoscillations

The design procedure for temporal superoscillations begins with defining the waveform spectrum as a

sum of sinusoidal harmonics — or, equivalently, as a sum of equi-spaced delta functions in the frequency

domain:

V (ω) =

N−1∑n=0

anδ(ω − ωn), where ωn = ω0 + n∆ω. (6.1)

Here ω is the angular frequency, ω0 represents the location of the lowest (most negative) frequency delta

function, ∆ω is the frequency spacing between adjacent tones, and an is the weight for the n’th delta

function. The temporal waveform is then written as

V (t) =1

2π

∫ ∞−∞

V (ω)ejωtdω =ejω0t

2π

N−1∑n=0

anzn,

where z = ej∆ωt.

(6.2)

In relation to spatial superoscillation design, (6.2) is analogous to the spatial superoscillation wave-

form, which can be written as the product of zero factors, while (6.1) is analogous to its plane wave

spectrum. As in the spatial case, one can design zeros in (6.2), and thereby design a waveform that is

periodic in time with the Bloch period TB = 2π∆ω . In particular, close-packing zeros into an interval Td

within the Bloch period will lead to superoscillations within that period.

The above formulation will be demonstrated through two design examples: a waveform featuring

rapid superoscillatory oscillations and one featuring superoscillatory pulse compression.

6.2.2 The Rapid Oscillation Waveform

As a review, let us first consider the fastest non-superoscillatory oscillation that can be generated with

a given bandwidth. This oscillation will be a sinusoid of the highest constituent frequency, which, as

shown in Section 3.2, corresponds to evenly distributing the zeros of (6.2) across the unit z-circle:

zn = ejn∆φ′ , where ∆φ =2π

N − 1. (6.3)

To obtain superoscillations at a frequency S times this highest frequency ωN−1, the zeros within the


design region must appear S times as frequent:

zn = ejn∆φ′ , where ∆φ′ =2π

(N − 1)S. (6.4)

Close-packing zeros in such a manner within the design interval will lead to the desired rapid oscillation.

Obviously one must ensure that the number of available zeros is sufficient for the proposed operations

before one can generate the rapid oscillation for the desired time interval. On the other hand, reserving

zeros for placement outside the design region can aid the design of practical superoscillatory waveforms,

since these zeros provide handles for minimizing the sideband amplitude of the waveform, as described

in Chapter 3 and demonstrated in Chapters 4 and 5.

6.2.3 The Sharp Pulse Waveform

The strategy for designing the sharp temporal pulse waveform hereby presented parallels the design of

the spatial sub-diffraction waveform, which has been presented in Section 4.1. The interested reader is

hence referred there for full details on the method. This section aims to present the method for the

temporal case as succinctly as possible.

Having defined a temporal waveform V (t) as a series of complex exponentials, one can write it also

as a series of Tschebyscheff polynomials by defining

u = <z = cos(∆ωt). (6.5)

Using (6.5) on (6.2) allows us to write

V (t) = c0 +

bN/2c∑n=1

2cn cos(n∆ωt)

V (u) = c0 + 2

bN/2c∑n=1


= c0 + 2

bN/2c∑n=1

cnTn(u),

(6.6)

where Tn(u) is the n’th degree Tschebyscheff polynomial of the first kind, while the coefficient set cn

relates to the set an through a shift in index: i.e. an = c|n−bN/2c|, for n = 0, 1, ..., N −1. As explained

from the discussion in Chapter 4, we can design a narrow pulse with uniform sidelobe levels by mapping

the design region (t ∈ [−Td/2, Td/2]) to a properly scaled and shifted Tschebyscheff polynomial with a

degree which matches the degree of the summation in (6.6). This polynomial resides in a space v linearly


mapped to u, such that the design region is mapped to v ∈ [−1, v0], where TbN/2c(v0) = 1/Asl, and

where Asl is a allowable sidelobe level within the design region. One can analytically determine the zero

locations for this Tschebyscheff polynomial, TbN/2c(v):

vzq = cos

(π

2

2q − 1

bN/2c

), for q = 1, ..., bN/2c. (6.7)

Given a target sidelobe level Asl, one can map these zero locations from v-space into u-space through

the linear mapping

uzq =vzq − βα

=

(vzq +

1 + v0 cos(∆ω Td2 )

1− cos(∆ω Td2 )

)(1− cos(∆ω Td2 )

1 + v0

)

=(1− cos(∆ω Td2 ))vzq + (1 + v0 cos(∆ω Td2 ))

1 + v0

for q = 1, ..., bN/2c.

(6.8)

Thereafter, noting the fact that all zeros of an even function in x are located on the unit z-circle, one

finds the set of zero locations for V (z) as


where n = ±1, ...,±bN/2c.(6.9)

As in previous designs, placing zeros within the design region following (6.5) to (6.9) designs the

temporally sharp waveform; placing zeros outside the design region allows one to control the sideband

and minimize its amplitude.

Figure 6.1: A schematic of the experimental apparatus for synthesizing temporal super-oscillatory EM waves.


6.3 Synthesizing Superoscillatory Electromagnetic Waveforms

Following the formulation presented in the previous section, two temporal superoscillatory signals are

numerically designed, then synthesized as time-varying voltage waveforms V (t) using an arbitrary wave-

form generator (AWG) with a bandwidth of 500 MHz. A schematic of the experiment is shown in Fig.

6.1. A computer program samples V (t) at a rate of 1.25 GHz, and inputs the sampled and normal-

ized voltage sequence V [n] to Channel 1 of the AWG, which performs sinc-interpolation on the sample

sequence supplied. At the same time, an “impulse” sequence I[n] — a sample of value 1, followed by

an array of value 0 — forms the input to Channel 2 of the AWG for triggering purposes. The AWG

outputs V (t) and I(t), the reconstructed continuous time signals from the sinc-interpolation process,

to an oscilloscope through RF coaxial cables. Using I(t) as a trigger, an oscilloscope observes the re-

constructed superoscillatory test waveform V (t) with an enhanced sampling rate of 8 GHz, sufficient to

clearly resolve the superoscillatory features of the test waveforms.

The following subsections provide parameters used to design the waveforms, displays the designed

and synthesized waveforms and comments on the experimental results.

6.3.1 Rapid Oscillation Waveform


Bandwidth (B) 500 MHz

# of complex exponentials (N) 31

# of zeros (M = N − 1) 30

# of zeros in design region (M1) 11

# of zeros in sideband region (M2) 19

∆ω = 2BN−1

33.3 MHz = π15

Grad/s

Bloch Period (TB = 2π/∆ω) 30 ns

Superoscillation Ratio (S) 1.3

Table 6.1: Summary of parameters for the rapid oscillation waveform

The first waveform aims to demonstrate the reliable generation of oscillations surpassing the high

frequency bandlimit. A waveform is constructed with five cycles of superoscillations at 650 MHz — 1.3

times the waveform bandwidth of 500 MHz. Table 6.1 shows design parameters for the rapid super-

oscillation waveform. This waveform contains N = 31 complex exponentials, which allows one to place

M = 30 zeros onto the z-plane. 11 zeros are placed according to (6.4) (for n = −5 to 5); the remaining

19 zeros are placed to minimize the sideband amplitude. Fig. 6.2 shows the resulting zero locations,

the spectral amplitude distribution, and the temporal profile of the waveform. The synthesized volt-


age waveform is plotted atop the calculated waveform in Fig. 6.2c-d, which shows excellent agreement

between the calculated and measured temporal waveforms. Fig. 6.2d shows a closeup of the waveform

alongside a sinusoid of frequency 650 MHz — 1.3 times the bandwidth limitation of 500 MHz. From this

figure, it could be seen that the designed and synthesized waveforms indeed super-oscillate at 650 MHz

across the range t ∈ [−4 ns, 4 ns]. Particularly, it approximates a 650 MHz sinusoid across the range

t ∈ [−2 ns, 2 ns]. To the best of the author’s knowledge this has been the first reported experimental

construction of a wave which superoscillates in the time domain.

Figure 6.2: Design for a rapid temporal superoscillatory waveform. (a) Zero locations,with open circles denoting zeros inside the design region. (b) The spectral amplitudedistribution. (c) The measured (green, with dots) and calculated (blue) temporal wave-forms over one period, showing excellent agreement. (d) A closeup of (c) across thedesign region, showing good experiment-calculation agreement, as well as conformity tothe 650 MHz sinusoid (red, dashed). Black squares label temporal signal inputs to theAWG.


6.3.2 Superoscillatory Pulse Compression




# of zeros (M = N − 1) 26



∆ω = 2BN−1

38.5 MHz = 241.7 Mrad/s


Design Region ([−Td/2, Td/2]) [−1.5 ns, 1.5 ns]

Sideband Amplitude (Asl) 0.2

Table 6.2: Summary of parameters for a designing a superoscillatory sharp temporalpulse

Having demonstrated the generation of rapid oscillations beyond a waveform’s bandwidth limit, the

following waveform aims to construct a superoscillatory pulse that is temporally compressed beyond the

Fourier transform limit. Table 6.2 shows parameters for the present design. The waveform contains

N = 27 complex exponentials, which allows one to place M = 26 zeros onto the z-plane. 6 zeros are

placed according to the Tschebyscheff design procedure outlined above; 20 remaining zeros are placed

to minimize the sideband amplitude. Fig. 6.3 shows the resulting zero locations, the spectral amplitude

distribution, and the temporal profile of the waveform. This waveform is synthesized in experiment,

and the measured voltage waveform is plotted with the calculated waveform Fig. 6.3c-d. Once again

the calculated and experimental waveforms align with excellent agreement. The pulse widths at half

maximum (FWHM) are 0.78 ns (calculation) and 0.82 ns (experiment), which are respectively improved

by 55% and 47% beyond a transform-limited sinc waveform, for which the FWHM measures 1.21 ns.

This pulse width can be further improved through tradeoffs with the duration of the design interval and

the sideband amplitude.

6.3.3 Discussion

The main result of the work presented in this section is the experimental demonstration of time-domain

superoscillation, and the realization of pulse compression beyond the Fourier transform limit. However

the following elaborates on a few implications of the foregoing formulation and experiment.


Figure 6.3: Design for a superoscillatory sharp pulse. (a) Zero locations, with opencircles denoting zeros inside the design region. (b) Spectral amplitude distribution. (c)The measured (green, with dots) and calculated (blue) temporal waveforms over one pe-riod, showing excellent agreement. (d) A closeup of (c) across the design region, showinggood experiment-calculation agreement. The calculation and experimental waveformsare compressed 55% and 47% respectively respectively beyond the transform-limitedsinc function (red, dashed). Black squares label temporal signal inputs to the AWG.


Figure 6.4: Typical pulse waveform variations in the presence of 1.5% spectral noise.The original waveform is the highlighted in red.

A. Sensitivity Analysis

In similarity to the case with spatial superoscillations, it is important to know the degree of sensitivity

of a superoscillatory waveform, and its degree of tolerance to system imperfections. This work pursues

such an analysis from both the theoretical and experimental perspectives.

Firstly, to theoretically investigate the tolerable sensitivity of superoscillatory pulse synthesis, a noisy

channel is synthesized whereby the spectral weights of the superoscillatory pulse are superimposed with

randomly phased white Gaussian noise, with mean amplitude 1.5% of the strongest spectral component.

Fig. 6.4 shows typical pulse magnitude variations when this spectral noise is added to the waveform.

Amid this level of noise, one can observe the emergence of stronger and asymmetric sidelobes within

the design region, but the sub-transform-limit pulse width is preserved. This shows that the designed

waveform achieves a level of reasonable robustness required for practical waveform generation.

Secondly, a linear characterization has been performed of the experimental system, for which the

frequency response is displayed in Fig. 6.5. The frequency response is obtained by applying a Fourier

transform to the impulse response, obtained by equating the voltage sequence V [n] with the impulse se-

quence I[n]. As shown, the spectral phase response is mostly linear across the range |f | ≤ 500 MHz. The

spectral amplitude, however, contains a spike at DC, and slight modulations across the AWG’s operation

bandwidth. In the experiment, this DC offset is removed through the application of an opposing DC

shift to the sample sequence V [n]. On the other hand, slight spectral amplitude modulations observed


Figure 6.5: Frequency response of the superoscillatory pulse synthesizing apparatus.

in Fig. 6.5 do not present appreciable variations to the waveform observed, and hence do not require

adjustments to the sample sequence. The fact that the designed superoscillatory waveforms tolerate

these slight modulations highlights their robustness and practicality. Nevertheless, pre-compensation

– adjusting the input through dividing the source spectrum by the system’s transfer function – can

further improve waveform fidelity, and help construct superoscillatory waveforms with higher sensitivity

requirements.

B. Fast Varying Signal Synthesis

One can observe from Figs. 6.2d and 6.3d that sample points sent to the AWG are sparse in the su-

peroscillatory region relative to variations of the waveform. From a conventional perspective (formed

in view of the Nyquist limit), these sample points, which are spaced more than half an oscillation cy-

cle apart, appear inadequate at capturing the rapid variations within the waveforms. However, due to

the superoscillatory nature of the waveform, the variations are faithfully restored by the AWG upon

the digital-to-analog conversion process which essentially acts as a lowpass filter. This demonstrated

capability of locally generating fast varying temporal waveforms beyond the Nyquist limit enables the

generation of sharp pulses and other waveforms traditionally deemed impossible for a specific band-

width. There are, of course, two caveats to this scheme. Firstly, high-energy sidebands accompany the

superoscillatory waveform: their presence contributes to the construction of superoscillations beyond the

Nyquist limit. Secondly, the strength of the superoscillation forms a tradeoff with waveform sensitivity


– as required by the Shannon limit [75]. Notwithstanding, this section has demonstrated the realization

of superoscillatory sharp pulses with reasonable sensitivity. This scheme of superoscillatory pulse gener-

ation should see potential application in situations where one desires the fastest varying waveform from

a system with stringent band limitation.

Figure 6.6: The spectrum of waveforms which are truncated at the edge of the designregion. This plot compares the simulated (blue, thick line) and measured (green, withdots) superoscillatory sharp pulse, as well as the spectrum of a sinc function truncatedto the same interval (red, dashed). It also displays the spectrum of the untruncatedperiodic waveform (black) for comparison. Shaded regions denote ranges beyond thebandwidth of the original pulses.

C. Bandwidth Extension

A further extension to synthesizing superoscillatory fast varying signals is to truncate the sidebands

altogether. Fig. 6.6 shows the spectra of the calculated and experimental waveforms, after they are

truncated to include only the design region. While temporal truncation itself can be seen as a non-linear,

and hence, bandwidth extension process, one observes, from Fig. 6.6, that truncating a superoscillatory

signal expands the signal bandwidth much more dramatically. From this comparison, one can imme-

diately conclude that the local effective bandwidth of a superoscillatory waveform is expanded within

the design region. Leveraging this fact, one can construct a bandwidth extension system which first

generates a superoscillatory waveform with a device with maximum operation bandwidth ∆B, then,

by passing it through a mixer or an RF switch, extends the bandwidth of the resultant waveform to

∆B′ > B. This simple bandwidth extension procedure should find practical applications, for example


for ultra-wideband wireless communication and wide-bandwidth spectroscopic applications. Admittedly

much waveform power will be discarded in a simple truncation process. Nonetheless, the corresponding

bandwidth extension can be of great use in aforementioned applications, where power amplification is

available, and where one can afford to tradeoff power efficiency for a gain in effective bandwidth. On

the other hand, it would be a valuable direction of research to investigate whether parts of the trun-

cated power from the sidebands can be reintegrated back into the system, perhaps through a feedback

mechanism.

6.4 Superoscillatory Radar Imaging

Besides being an endeavour of scientific interest, superoscillatory temporal pulse generation should also

prove applicable to realms such as time domain measurement and synchronization, probing ultrafast

dynamics, and radar imaging. This section presents an investigation on improving the range resolution

of a radar imaging system using superoscillatory radar pulses.

6.4.1 Introduction

In a conventional radar imaging system, the distance between an object and the location of the radar

system is determined by the time it takes for a radar pulse to make a round trip from the radar system

to the object and back:

d =c0t

2, (6.10)

where c0 is the speed of an electromagnetic wave in free-space. As previously discussed, the temporal

width of a waveform is inversely related to its bandwidth. Hence in a radar imaging system, the

bandwidth of a radar pulse limits the sharpness of the pulse, which in turn limits the precision in

determining the distance of an object between an object and the radar system. This precision limitation

is termed the range resolution of the radar system. When one uses a sinc pulse (or a chirped sinc pulse)

as the radar pulse (as is common in radar imaging), the range resolution, quoted from the 3dB signal

point, is given as

R3dB =c0T3dB

2=

0.443c02B

, (6.11)

where T3dB represents the temporal 3dB width of the radar pulse, while B represents its bandwidth.

In light of the inverse relation which generally occurs between the pulse bandwidth and the achievable


range resolution, recent research effort is directed towards using ultra wideband (UWB) pulses for range

detection in radar systems. Nonetheless, as bandwidth has become a precious commodity in modern

communication systems, a way to circumvent the limit expressed in (6.11) will prove useful in high-range-

resolution radar systems of various kinds. This section investigates an alternative route to increase radar

range resolution through using a superoscillatory pulse for radar imaging.

6.4.2 Superoscillatory Radar Pulse Design




# of zeros (M = N − 1) 20



∆ω = 2BN−1

50 MHz = 314.2 Mrad/s


Design Region ([−Td/2, Td/2]) [−1.5 ns, 1.5 ns]

Sideband Amplitude (Asl) 0.2

Table 6.3: Summary of parameters for a designing a superoscillatory sharp temporalpulse

To design a superoscillatory radar pulse, the waveform design procedure presented in Section 6.2

is followed, with parameters listed in Table 6.3. The resultant waveform is largely similar to the one

designed in Section 6.3, with the sideband shortened to decrease the duration of the radar pulse. Fig.

6.7a-b show the resulting zero locations and the corresponding spectral amplitude. Fig. 6.7c shows a

period of the temporal waveform, with superoscillations occurring in the design region. A period of this

wave, padded by a suitabe sequence of zero voltage, forms the voltage input sequence to the AWG, the

output of which is measured and shown in Fig. 6.7d. Fig. 6.7e shows a closeup on the design region

and compares the calculated and measured waveforms to a sinc function of the same bandwidth. The

temporal waveform has a calculated 3dB width of 0.57 ns, which is 35% narrower than the sinc pulse,

for which the 3dB width measures 0.88 ns. This pulse width can be further narrowed through tradeoffs

with the duration of the design interval and the sideband amplitude. As we shall observe, this narrowed

pulse leads to a direct improvement in the range resolution in a radar imaging scheme.


Figure 6.7: Design for a sharp temporal superoscillatory radar pulse. (a) Resulting zerolocations, with open circles denoting zeros inside the design region. (b) The spectralamplitude distribution. (c) The calculated temporal waveform over a period. (d) Themeasured temporal waveform, when one period of the waveform is sent to the AWG. (e)Comparison between the calculated (blue) waveform, the measured (green, with dots)waveform and the Fourier transform-limited sinc (red, dashed) over the design region.


Figure 6.8: A schematic of the test radar system. Solid arrows denote the signal pathof the radar pulse V (t); the dashed arrow denots the signal path for the impulse I(t)

6.4.3 Experimental Apparatus

Fig. 6.8 shows a schematic of the radar test system. First, a test pulse V (t) is generated by an AWG

bandlimited to 500 MHz. This pulse modulates a 4.2 GHz carrier supplied by the pulse signal generator

(PSG), then gets amplified and launched through a horn antenna. The pulse is reflected off a metallic

plate — representative of a point scatterer — placed a set distance away, recollected by the horn,

demodulated and observed with an oscilloscope. An additional impulse signal I(t), connected directly

from the AWG to the oscilloscope, serves as the trigger signal. To observe the radar return signals which

are inherently faint in energy, one needs to take care to remove or reduce noise from all parts of the radar

system. In the present radar test system, in addition to using test components which are impedance

matched to 50 Ω within the bandwidth of operation, systematic noise caused by faint back reflections are

removed by capturing two traces at each distance: first a trace is taken with the scatterer present; then

a background trace is taken with the scatterer absent, and subtracted from the former trace to remove

back reflection components within the system. Calibration with one scatterer distance establishes a time

of zero delay; thereafter, for a scatterer placed at an arbitrary distance away, the delay T attained by

the superoscillatory peak determines the scatterer’s distance through the simple relation

d =c0T

2± R3dB

2. (6.12)


Figure 6.9: Reflection traces of the sinc radar pulse at four image distances: 2.50± 0.02m, 3.00± 0.02 m, 3.50± 0.02 m and 4.00± 0.02 m. (top to bottom)

Figure 6.10: Reflection traces of the superoscillatory radar pulse at four image distances:2.50± 0.02 m, 3.00± 0.02 m, 3.50± 0.02 m and 4.00± 0.02 m. (top to bottom)


Figure 6.11: Radar range resolution of the superoscillatory pulse (blue) and the sincpulse (green, dashed). The scatterer is placed at 3.00 m away from the horn. The3dB resolutions are 101 mm and 164 mm respectively. Hence the superoscillatory pulseachieves a range resolution 38%-improved from that of the sinc pulse.

6.4.4 Single Scatterer Experiment

When a sinc pulse is used as the radar pulse, Fig. 6.9 shows reflection traces at four image distances:

2.50 ± 0.02 m, 3.00 ± 0.02 m, 3.50 ± 0.02 m, and 4.00 ± 0.02 m. When a superoscillatory pulse is

used instead, the corresponding traces are shown in Fig. 6.10. The range measurements, with precision

quoted from the 3dB half width of the radar peak, are shown in Table 6.4. The superoscillatory reflection

traces yield range measurements which agree with physical measurements, and hence demonstrate the

system’s accuracy for single target detection. Notwithstanding slight distortions, the overall shapes of

the superoscillatory pulses, including the superoscillatory peaks, are preserved.

Fig. 6.11 compares the superoscillatory pulse with a 500 MHz sinc pulse as they reflect from a

scatterer placed at 3.00±0.02 m away from the horn. The superoscillation peak can be clearly observed,

with a 3dB resolution of 101 mm, which is 38% improved over the 500 MHz sinc radar pulse, which

has a 3dB resolution of 164 mm. This obtained percentage improvement agrees well with theoretical

calculation of 35%, presented earlier in this section, and thus shows that the reduced temporal width of

a radar pulse directly translates into a improvement in the range precision of a radar system.


Physical Measurement Sinc Pulse Superoscillatory Pulse

2.50 ± 0.02 m 2.500 ± 0.077 m 2.500 ± 0.045 m

3.00 ± 0.02 m 3.055 ± 0.082 m 3.013 ± 0.051 m

3.50 ± 0.02 m 3.571 ± 0.074 m 3.539 ± 0.048 m

4.00 ± 0.02 m 4.068 ± 0.078 m 4.049 ± 0.051 m

Table 6.4: Single object range measurement using a superoscillatory radar pulse and asinc radar pulse

Figure 6.12: Simulated radar signals from a pair of object closely spaced in range. (a)Simulated radar signatures from a sinc radar pulse. (b) Simulated radar signatures froma superoscillatory radar pulse. In both plots, the black lines denote object locations.

6.4.5 Resolving Two Scatterers in Close Range

Beyond improving the range precision of the radar system, a superoscillatory radar pulse extends a

radar system’s functionality by resolving two closely-spaced point objects. When radar signals from two

closely-spaced point objects superimpose upon one another, the resultant signal can be mathematically

described as

s(t) = A1(t) cos(ωct) +A2(t−∆t) cos(ωc(t−∆t)), (6.13)

where A1 and A2 are the reflected signal amplitudes from the two objects, ωc describes the carrier

frequency, and ∆t describes the time delay between the returning signals. Upon in-phase-quadrature

(IQ) demodulation, and filtering out higher-order harmonics, the observed signal becomes

sIQ(t) =1

2A1(t) +

1

2A2(t−∆t) exp(jωc∆t)), (6.14)

and its amplitude becomes


Figure 6.13: Simulated radar signatures from a pair of objects spaced 140 mm apart.The top panel shows the reflection trace for the superoscillatory radar pulse; the bottompanel shows the reflection for the sinc radar pulse.

‖sIQ(t)‖ =1

2A2

1(t) +1

2A2

2(t−∆t) +A1(t)A2(t) cos(jωc∆t)). (6.15)

This reflection trace is converted into distance by a simple application of (6.10).

The last term in (6.15) varies with ωc∆t. This fact makes complicated the determination of a

minimum resolvable distance between two objects, and disallows the establishment of a limit analogous

to Rayleigh’s diffraction limit or its coherent equivalent. Generally speaking, (6.15) forbids the resolving

of the separate objects unless the separation distance is wider than the width of the individual radar

signals. This fact is shown in Fig. 6.12a for the system launching a sinc radar pulse, which plots the

simulated reflection signal of two objects spaced distance s apart, and reflecting signals of identical

strength. While for some separation distances, reflection traces seem to resolve the two scatterers, the

radar system does not consistently conclude the existence of two scatterers until their range separation

goes beyond 230 mm. However, when the radar system launches instead a superoscillatory radar pulse,

the objects can be resolved at a closer separation, as long as they stay within the confines of the design

region. This fact is shown in Fig. 6.12b, where it is evident that the two objects are reliably resolve for

the range of distance from 120 mm to 200 mm.

Fig. 6.13 shows the radar reflection signals for the superoscillatory and sinc radar pulses when the two

objects are separated by a distance of 140 mm. Despite the existence of large sidebands, the two objects


Figure 6.14: Experimental radar signatures from a pair of objects spaced 140 mm apart.The top panel shows the reflection trace for the superoscillatory radar pulse; the bottompanel shows the reflection for the sinc radar pulse.

are clearly distinguishable in the superoscillatory reflection signal; however they are not distinguishable

in the sinc reflection signal. Experimental reflection signals are shown in Fig. 6.14. While the sideband of

the superoscillatory reflection signal undergoes visible distortions due to system noise and imperfections,

a reasonable agreement has been achieved between the calculated and measured reflection signals. In

particular the object are successfully resolved in the measured superoscillatory reflection signal, but are

unsuccessfully resolved in the sinc counterpart. From the measured radar signal, the separation between

the two closely-spaced objects is deduced to be 173 mm — which is accurate to within the precision of

the system.

6.4.6 Discussion

The experiment in this section has shown the capability of a temporal superoscillatory pulse to increase

the range resolution of a radar system. It has also shown that such a pulse can resolve two closely ranged

objects whereas a conventional radar pulse fails to do so. Admittedly, the achievement of increased

range resolution comes with an important drawback — that the design region must be large enough to

encompass all objects, such that return signals from all scatterers superimpose without any overlap with

signal components from the high amplitude sidebands. This precludes the imaging of electrically large

objects as well as objects with large separations in range. Improvements to the superoscillatory waveform

can yield a larger design region and further improve the range resolution, but it will likely increase the


extent of the sideband region, and therefore might not be desirable for radar imaging applications. Hence

at present, a superoscillatory radar system would be most suitable in niche applications for imaging single

objects, such as the detection of missiles (or other potentially dangerous objects) incoming from space.

Additionally, it can be effectively image a small cluster of closely-ranged objects. In similar spirit, it can

also provide superior performance in sensors which detect small deviations across two or more surfaces.

The demonstration of superoscillatory radar imaging motivates a few directions of future investi-

gation. Firstly, it can be conceived that the temporal truncation operation theoretically explored in

Section 6.3 should yield a bandwidth-extended pulse, practical for radar imaging and more generally

in applications which require sharp probe or signal pulses. Such a pulse can be synthesized through

mixing a superoscillatory signal with a conventional windowing waveform, which should fit within the

operation bandwidth without difficulty. The resulting pulse can come of use for systems which benefit

from having the sharpest waveform variation available, but need to accommodate a strict bandwidth

requirement. Secondly, the application of superoscillation theory to the detection filter of a radar imag-

ing system should also warrant interest — as in some cases, it may be more practical to implement a

superoscillatory filter than it is to construct, then launch, a superoscillatory radar pulse. Finally, the

prospect of numerical superoscillatory post-processing, and how it compares to or corroborates with ex-

isting post-processing algorithms for radar detection, is a worthy topic of investigation, and may result

in improvements in radar detection algorithms.

6.5 Summarizing Conclusion

This chapter has reported an investigation on temporal superoscillations. Leveraging work presented in

previous chapters, two superoscillation waveforms are designed and synthesized — a waveform which

oscillates 30% faster than the waveform bandwidth, as well as a sharp pulse which is 47%-compressed

from the transform-limited sinc waveform. These represent the first ever reported demonstration of tem-

poral superoscillatory electromagnetic waveforms. Using as a radar pulse one period of a superoscillatory

sharp pulse, this work has also demonstrated that superoscillatory radar imaging can improve a radar

system’s range precision by 38% over a sinc pulse of the same bandwidth, and resolve two scatterers at

a close range of 140 mm, which has been unresolvable for the conventional sinc radar pulse. Advantages

and drawbacks to superoscillatory pulse synthesis and radar imaging have been discussed, and future

directions of investigations have been suggested which will form worthy extensions to this thesis.

Chapter 7

Conclusion

7.1 Summary

This thesis has investigated the properties of superoscillatory electromagnetic waves, and demonstrated

their design, synthesis and application towards two kinds of super-resolution imaging: sub-diffraction

imaging and radar range imaging. All objectives listed in the introductory chapter have been achieved.

Through viewing superoscillations in the perspective of antenna array theory, this work has presented a

new way to understand superoscillations and a new method to design superoscillatory electromagnetic

waveforms. This method provides handles to control both the superoscillatory and non-superoscillatory

region of the waveform, and hence leads to the design of waveforms practical for sub-diffraction imaging

and related applications. Imaging devices based on these waveforms have subsequently been implemented

in the spatial domain. Two microwave focusing devices — the sub-diffraction focusing screen and the

sub-diffraction focusing waveguide — have been designed. Simulation results from the former device

and simulation and experimental results from the latter demonstrate successful sub-diffraction focusing

at five wavelengths from the screen plane where the excitations are generated. This image distance

represents a tenfold improvement in working distance of evanescent-wave-based sub-wavelength focusing

devices, of comparable focal quality. Subsequently, the Optical Super-Microscope has been designed

and demonstrated, which successfully performed scan-less sub-diffraction far-field optical imaging, with

resolution improved to about 70% that of the diffraction limit for both the Abbe and the Rayleigh cases.

An investigation has also been pursued on time-domain superoscillatory electromagnetic waveforms,

which resulted in the first demonstration of temporal superoscillatory electromagnetic waveforms, as

well as a sharp superoscillatory pulse which has improved the range resolution of a radar system by 38%

119

Chapter 7. Conclusion 120

over the transform-limited sinc radar pulse.

7.2 Future Directions

As often happens in the pursuit of knowledge, the answering of one question raises multiple further

questions; resolving a mystery opens doors to many others. Hence unsatisfied is the curious mind, and

the learned humbly realizes how much he does not yet know. In this spirit, this thesis report concludes

with a summary on promising directions of future research identified as a fallout of this thesis.

Design of Superoscillatory Waveforms

This thesis has developed a method whereby superoscillatory waveforms, like antenna patterns, are

designed by judicious placement of zeros. While this thesis has mostly used the Tschebyscheff zero

placement, or some modification thereof, to design superoscillatory sharp waveforms, an exploration on

placing zeros using optimization algorithms might lead to solutions which better satisfy unique sets of

application-specific waveform requirements. Similarly, it is worthwhile to investigate how a superoscilla-

tory waveform improves with increasing degrees of freedom — which can be made available by increasing

the number of zeros in the method.

Superoscillatory Focusing devices

In terms of future directions on superoscillatory focusing devices, a demonstration of a free-space su-

peroscillatory focusing screen on existing technology platforms, such as the transmitarray or the meta-

surface, would serve as a solid step towards demonstrating practical applications for superoscillatory

waveform focusing. Thereafter, a multitude of applications could be pursued, which involve scan-based

superoscillatory imaging and sensing, and superoscillatory waveform synthesis.

Optical Super-Microscopy

In regards to the Optical Super-Microscope, while experimental findings presented in this work have ef-

fectively proven the principle of superoscillatory far-field sub-diffraction imaging, further developments,

which include the demonstration of ‘stitched’ wide-field imaging and sub-wavelength optical imaging, will

represent significant steps towards the practical deployment of the OSM. As discussed in Chapter 5, the

OSM can in principle perform wide-field imaging by selectively illuminating, and taking images of, por-

tions on the object plane. These partial images can then be stitched together by simple post-processing


to provide a combined FOV which is much larger than that for a single shot imaging OSM. Since the

required selective illumination lies well within the diffraction limit, a custom-fabricated diffractive opti-

cal element will suffice to produce such an illumination; thereafter, coordinating object movement and

image acquisition, as well as image stitching, should be easily handled by software.

As for performing sub-wavelength imaging with the OSM, three major improvements to the OSM

should sufficiently develop it for such an application. Firstly, the numerical aperture of the of the optical

system should be increased. Secondly, a magnification stage needs to be added, either before or after

the superoscillatory stage, to magnify the sub-wavelength image. Finally, for improved superoscillation

filtering quality, the SLM filter should ideally be replaced, possibly by a custom-fabricated filter. As all

aforementioned improvements entail components and operations well within fabrication and alignment

capabilities of current technology, no foreseeable roadblock exist, it is very reasonable to expect the

successful demonstration of sub-wavelength optical imaging with an OSM upon further research and

development. A sub-wavelength OSM will undoubtedly ready the device for optical microscopy in

cutting-edge biological applications.

Temporal Superoscillation

On the topic of temporal superoscillation, one worthwhile future direction research is an investigation on

bandwidth expansion through the judicious truncation of superoscillatory waves. Chapter 6 has shown

that bandwidth expansion occurs if one truncates the waveform to exclude the non-superoscillatory

sideband. If such judicious truncation can be optimized to maximize the generation of high frequency

components beyond the bandwidth of the original waveform, and if a system can be devised which effec-

tively controls and generates such waveforms, this investigation will prove useful for many applications in

RF electronics. With regards to radar imaging, investigations on superoscillatory detection filtering, and

a more in-depth study relating the SNR to resolution improvements, would prove practical and worth-

while. Thirdly, an exploration on further applications for temporal superoscillatory electromagnetic

waves may prove fruitful as well.

Other Areas

Finally it is worthwhile to mention a few areas which lie beyond, but are closely-related, to the scope of

the present thesis. Firstly, research on numerical superoscillation — where superoscillatory modulation

or filtering is virtually applied in a computational process — may lead to diverse applications in imaging

and in signal processing in general. Numerical superoscillation is attractive in that it removes the need


for the high-precision hardware which will be required to (a) generate an aggressive superoscillatory

wave (one with a high superoscillatory ratio S >> 1) or (b) implement an aggressive superoscillatory

filter. However, precision is required in detecting the image, and sometimes it is non-trivial to collect

information on both the amplitude and the phase of an image or signal — especially at optical frequencies.

Nonetheless, whereas earlier works on superoscillation regarded such schemes as impractically sensitive

[90], recent simulations [88] and experiments [87] show such schemes are possible. Notwithstanding,

further understanding is needed to clarify the strengths and weaknesses to numerical superoscillation,

and to identify sensible areas of application for such an approach.

Secondly, an investigation on the nature of sensitivity of superoscillatory waveforms may lead to

breakthroughs in superoscillation engineering. A better understanding on the nature of sensitivity of

a superoscillatory waveform will greatly help one determine its applicability and reliability for specific

applications. Moreover, while the sensitivity of superoscillation functions have heretofore been viewed

as a barrier to their synthesis and application, perhaps it is also possible to utilize this sensitivity as an

advantage, for example in a sensing application. These two reasons form strong motivations for a deeper

investigation into the sensitive nature of superoscillations.

Finally, it is certainly worthwhile to explore the application of superoscillatory wave theory in other

wave-based imaging modalities, which may include telescopy, ground (or object) penetration imaging,

and medical imaging. It is very plausible that such explorations bring practical improvements to these

existing imaging systems.

7.3 Contributions

The work described in this thesis has led to the following academic contributions

Journal Papers

[J6] A.M.H. Wong and G.V. Eleftheriades, “An optical super-microscope for far-field, real-time

imaging beyond the diffraction limit”, Scientific Reports, vol. 3, 1715, Apr. 2013.

[J5] A.M.H. Wong and G.V. Eleftheriades, “Advances in imaging beyond the diffraction limit”,

IEEE Photonics Journal, vol. 4, no. 2, pp. 586–589, Apr. 2012. (Invited)


[J4] A.M.H. Wong and G.V. Eleftheriades, “Superoscillatory radar imaging: improving radar

range resolution beyond fundamental bandwidth limitations”, IEEE Microwave and Wireless

Component Letters, vol. 22, no. 3, pp. 147–149, Mar. 2012.

[J3] A.M.H. Wong and G.V. Eleftheriades, “Sub-wavelength focusing at the multi-wavelength

range using superoscillations: an experimental demonstration”, IEEE Transactions on An-

tennas and Propagation, vol. 59, no. 12, pp. 4766–4776, Dec. 2011.

[J2] A.M.H. Wong and G.V. Eleftheriades, “Temporal pulse compression beyond the Fourier

transform limit”, IEEE Transactions on Microwave Theory and Techniques, vol. 59, no. 9,

pp. 2173–2179, Sept. 2011.

[J1] A.M.H. Wong and G.V. Eleftheriades, “Adaptation of Schelkunoff’s superdirective an-

tenna theory or the realization of superoscillatory antenna arrays”, IEEE Antennas and

Wireless Propagation Letters, vol. 9, pp. 315–318, Apr. 2010.

Conference Papers

[C3] A.M.H. Wong and G.V. Eleftheriades, “Superdirectivity-based superoscillatory waveform

design: a practical path to far-field sub-diffraction imaging”, The 8th European Conference

on Antennas and Propagation (EuCAP), Apr. 2014. (accepted).

[C2] G.V. Eleftheriades, L. Markley and A.M.H. Wong, “Sub-wavelength focusing and imaging

using shifted-beam and super-oscillation antenna arrays”, IEEE International Symposium

of Antenna Technology and applied Electromagnetics (ANTEM), Jun. 2012. (invited).

[C1] A.M.H. Wong and G.V. Eleftheriades, “Superoscillatory antenna arrays for sub-diffraction

focusing at the multi-wavelength range in a waveguide environment”, IEEE International

Symposium on Antennas and Propagation (AP-S), Paper 230.5. Jul. 2010. (Student Com-

petition Finalist)

Appendix A

The Heisenberg Uncertainty

Principle

This appendix provides a derivation for the Heisenberg uncertainty principle, which is mainly adapted

from [117]. Other treatments include [118] and [91].

A.1 Defining Conditions

First let the forward and inverse Fourier transforms be defined in conformity to elsewhere in the thesis:

F (kx) =

∫ ∞−∞

f(x)e−jkxxdx;

f(x) =1

2π

∫ ∞−∞

F (kx)ejkxxdx.

(A.1)

Further, let a 1D function f(x) be defined with unit energy and a sufficiently fast decay towards infinity.

Mathematically, f(x) satisfies the following:

∫ ∞−∞|f(x)|2dx = 1;

limx→∞

√xf(x) = 0;

(A.2)

Although the variables x and kx directly imply applicability towards spatial resolution considerations

addresses in this thesis, these two variables give general representation to any reciprocal domain. For

this unit energy function, Parseval’s theorem gives

124

Appendix A. The Heisenberg Uncertainty Principle 125

∫ ∞−∞|f(x)|2dx =

1

2π

∫ ∞−∞|F (kx)|2dx = 1. (A.3)

The 2nd moment widths of such a in x and kx are given as:

∆x =

∫ ∞−∞

x2|f(x)|2dx; ∆kx =

∫ ∞−∞

kx2|F (kx)|2dkx. (A.4)

A.2 Deriving the Uncertainty Principle

The Cauchy-Schwarz’s Inequality states that

∣∣∣∣∣∫ b

a

g1g2 dx

∣∣∣∣∣2

≤∫ b

a

|g1|2dx∫ b

a

|g2|2dx. (A.5)

Let g1 (x) = xf ; g2 (x) = dfdx . Applying these to (A.5), with integration across all of x, gives

∣∣∣∣∫ ∞−∞

xf

(df

dx

)dx

∣∣∣∣2 ≤ ∫ ∞−∞|xf |2 dx

∫ ∞−∞

∣∣∣∣ dfdx∣∣∣∣2 dx . (A.6)

Solving the LHS of (A.6) by integration by parts yields

∫ ∞−∞

xf

(df

dx

)dx =

[xf2(x)

]−∞∞ −

∫ ∞−∞

f2dx−∫ ∞−∞

xf

(df

dx

)dx = −1

2

∫ ∞−∞

f2dx

⇒∣∣∣∣∫ ∞−∞

xf

(df

dx

)dx

∣∣∣∣2 =1

4

∣∣∣∣∫ ∞−∞

f2dx

∣∣∣∣2 ≤ 1

4

(∫ ∞−∞|f |2 dx

)2

=1

4,

(A.7)

where the last inequality was written by applying (A.5) again, with g1 = g2 = f . Rearranging RHS of

(A.6) gives

∫ ∞−∞|xf |2dx

∫ ∞−∞

∣∣∣∣ dfdx∣∣∣∣2dx =

∫ ∞−∞

x2 |f |2dx(

1

2π

∫ ∞−∞|jkxF (kx)|2dkx

)=

1

2π

∫ ∞−∞

x2 |f |2dx∫ ∞−∞

kx2 |F |2dkx

= ∆x∆kx.

(A.8)

Substituting (A.7) and (A.8) into (A.6) gives

∆x∆kx ≥1

4. (A.9)

Equation (A.9) is known as the Heisenberg uncertainty principle.

Appendix B

The Design of Dolph-Tschebyscheff

Superdirective Antenna Arrays

B.1 Introduction

The Dolph-Tschebyscheff method is first proposed by Dolph [3]. It uses an expansion of Tschebyscheff

polynomials to design the narrowest antenna beam with a prescribed, uniform sidelobe level. Particu-

larly, when the elemental separation distance is smaller than half wavelength, the Dolph-Tschebyscheff

method results in superdirective excitation currents. Balanis [119] highlights pioneering works on

Dolph-Tschebyscheff superdirective antennas and provides an in-depth tutorial on designing Dolph-

Tschebyscheff antenna arrays through the aforementioned polynomial expansion approach. However, for

theoretical investigations it might be more preferable to employ a method based on the calculation of

zero locations, because as noted in another early work on superdirective antennas [103], calculating zero

locations requires less precision than calculating current excitations for the same superdirective antenna.

Hence, many antenna and superoscillation designs within this thesis have followed a novel method which

computes zero locations of a superdirective antenna pattern based on simple mappings from a single

Tschebyscheff polynomial. This method represents a slight modification to the traditional, but leads to

an increased precision needed for theoretically investigations in strongly superdirective arrays.

Fig. B.1 shows a diagram of the antenna array which denotes the geometry of variables used. Let

us first assume a case where the number of antenna elements N is odd, hence the number of zeros

M = N − 1 is even (the case of even N , odd M will be discussed toward the end of this appendix).

Further, let us require that the waveform be symmetric about a broadside peak at θ = 90. In this case

126

Appendix B. The Design of Dolph-Tschebyscheff Superdirective Antenna Arrays 127

Figure B.1: A diagram of the antenna which denotes the co-ordinates and defines theangle θ. Each antenna is represented by a black node, and is separated by distance dfrom its neighbors.

the array factor can be expressed as

AF (θ) =

bN/2c∑n=−bN/2c

cne−jnkxd = c0 +

bN/2c∑n=1

2cn cos(nkxd),

where kx = k cos θ

(B.1)

Defining

z = e−jkxd;u = <z = cos(kxd) (B.2)

allows one to rewrite (B.1) as

AF (u) = c0 + 2

bN/2c∑n=1


= c0 + 2

bN/2c∑n=1

cnTn(u),

(B.3)

where Tn(u) is the n’th order Tschebyscheff polynomial of the first kind. (The interested reader is

referred to section 4.1 of this thesis for an introduction to Tschebyscheff polynomials.) Since (B.3)

shows that AF (u) is a sum of bN/2c Tschebyscheff polynomials, which collectively span the space of the

bN/2c’th degree polynomial, AF (u) can in general be any bN/2c’th degree polynomial in the space of

u, when one chooses an appropriate coefficient set cn.


This work aims to design a waveform AF (kx) with the narrowest possible peak at broadside, and a

constant sidelobe level of Asl for θ ∈ [0, 180] ⇒ kx ∈ [−k, k]. In u-space this region of interest (ROI)

is mapped to u ∈ [cos(kd), 1], where the desired waveform AF (u) attains the peak amplitude at u = 1.

From the definition of Tschebyscheff polynomials, it can be seen that the desired waveform would be

a Tschebyscheff polynomial of order bN/2c, but in a linearly-shifted space v such that the ROI will be

shifted to v ∈ [−1, v0], where v0 is the point for which TbN/2c(v0) = 1/Asl. In mathematical terms,

AF (u) = TbN/2c(v), (B.4)

where

v = αu+ β =1 + v0

1− cos(kd)u− 1 + v0 cos(kd)

1− cos(kd)

⇒ u =v − βα

=(1− cos(∆kL))vzq + (1 + v0 cos(∆kL))

1 + v0

(B.5)

The above mappings allow us to find the null locations in the array factor AF (θ). The zero locations

of a Tschebyscheff polynomial of degree bN/2c can be analytically derived as

vzq = cos

(π

2

2q − 1

bN/2c

), for q = 1, ..., bN/2c. (B.6)

Applying (B.6) into (B.5) allows one to find the set uzq of null locations in u-space:

uzq =(1− cos(kd))vzq + (1 + v0 cos(kd))

1 + v0

for q = 1, ..., bN/2c.(B.7)

Thereafter, noting the fact that all nulls of AF (θ) must be located on the unit circle in z-space, one

finds the set of zero locations zn for EbN/2c(z) as


where n = ±1, ...,±bN/2c.(B.8)

Having obtained zero locations in z-space, one can find the corresponding nulls in AF (θ) as

θn = cos−1 arg(zn)

kd, (B.9)

and the array factor coefficients cn by expanding


Figure B.2: Superdirective antenna arrays and their antenna patterns. (Reproducedfrom Fig. 2.3.) (a) Current Excitations (array factors) for 3 antennas arrays of length2λ, with 11, 21 and 31 elements respectively. (b) The corresponding far-field angulardistributions, compared to that of a uniform array.

AF (θ) = cbN/2c

bN/2c∏n=−bN/2c

z − wn (B.10)

into the polynomial form of (B.1).

Finally, for an array with an even number of elements (hence an odd number of nulls), the extra null

needs to be placed at u = −1 to retain a symmetric antenna pattern with a peak at broadside.

In chapter two of this thesis, a plot has shown array excitations and antenna patterns of three antenna

arrays designed using this method. Each array has an overall size of 2λ, but have differing number of

elements (namely 11, 21 and 31 elements), which leads to respective elemental separations of λ/5, λ/10

and λ/15. The plot is hereby reproduced in Fig. B.2; the corresponding array excitation coefficients and

zero locations (in z-space) are given in Tables B.1 to B.3.


Zero Location Excitation Weight

1 0.3251− 0.9457i 0.01285759576

2 0.4442− 0.8959i -0.08189899567

3 0.6370− 0.7709i 0.2660423446

4 0.8297− 0.5582i -0.5670018719

5 0.9489− 0.3157i 0.8702448128

6 0.9489 + 0.3157i -1.0

7 0.8297 + 0.5582i 0.8702448128

8 0.6370 + 0.7709i -0.5670018719

9 0.4442 + 0.8959i 0.2660423446

10 0.3251 + 0.9457i -0.08189899567

11 N/A 0.01285759576

Table B.1: Zero locations and excitation weights for the 11-element superdirective an-tenna


1 0.8102− 0.5862i 9.183186356× 10−6

2 0.8193− 0.5734i −1.658969847× 10−4

3 0.8366− 0.5478i 1.439656353× 10−3

4 0.8605− 0.5095i −7.978240055× 10−3

5 0.8885− 0.4588i 0.03166097707

6 0.9180− 0.3966i -0.09562600986

7 0.9461− 0.3240i 0.2280571477

8 0.9699− 0.2435i -0.4397331499

9 0.9872− 0.1592i 0.6961726066

10 0.9964− 0.0853i -0.913836274

11 0.9964 + 0.0853i 1.0

12 0.9872 + 0.1592i -0.913836274

13 0.9699 + 0.2435i 0.6961726066

14 0.9461 + 0.3240i -0.4397331499

15 0.9180 + 0.3966i 0.2280571477

16 0.8885 + 0.4588i -0.09562600986

17 0.8605 + 0.5095i 0.03166097707

18 0.8366 + 0.5478i −7.978240055× 10−3

19 0.8193 + 0.5734i 1.439656353× 10−3

20 0.8102 + 0.5862i −1.658969847× 10−4

21 N/A 9.183186356× 10−6




1 0.9138− 0.4062i 9.084451052× 10−9

2 0.9156− 0.4020i −2.606841272× 10−7

3 0.9193− 0.3936i 3.626913964× 10−6

4 0.9246− 0.3810i −3.258235112× 10−5

5 0.9313− 0.3644i 2.123452457× 10−4

6 0.9390− 0.3438i −1.069411727× 10−3

7 0.9476− 0.3195i 4.328831788× 10−3

8 0.9565− 0.2917i -0.01446287415

9 0.9655− 0.2606i 0.04064368218

10 0.9740− 0.2265i -0.09740929043

11 0.9818− 0.1900i 0.2011733939

12 0.9885− 0.1515i -0.3608112044

13 0.9937− 0.1117i 0.5652542808

14 0.9974− 0.0722i -0.7767518668

15 0.9993− 0.0384i 0.9389213206

16 0.9993 + 0.0384i -1.0

17 0.9974 + 0.0722i 0.9389213206

18 0.9937 + 0.1117i -0.7767518668

19 0.9885 + 0.1515i 0.5652542808

20 0.9818 + 0.1900i -0.3608112044

21 0.9740 + 0.2265i 0.2011733939

22 0.9655 + 0.2606i -0.09740929043

23 0.9565 + 0.2917i 0.04064368218

24 0.9476 + 0.3195i -0.01446287415

25 0.9390 + 0.3438i 4.328831788× 10−3

26 0.9313 + 0.3644i −1.069411727× 10−3

27 0.9246 + 0.3810i 2.123452457× 10−4

28 0.9193 + 0.3936i −3.258235112× 10−5

29 0.9156 + 0.4020i 3.626913964× 10−6

30 0.9138 + 0.4062i −2.606841272× 10−7

31 N/A 9.084451052× 10−9


Appendix C

Calibrating the Spatial Light

Modulator

The SLM used in this thesis is the SDE1024 SLM kit from Cambridge Correlators, which is a reflective

SLM with a 1028 × 724 pixel array, of pitch size 9 µm. Each pixel within the array can receive an 8-bit

signal which controls the signal reflection level for that pixel. This signal level will be represented with

the variable l ∈ [0, 255], which alters the pixel birefringence to varying degrees, which in turn causes

an effective change in the SLM’s reflection coefficient, when suitable polarization optics is included to

select an incoming and an outgoing polarization. Characterization is needed, however, to obtain more

information on the complex reflection coefficient r(l), which is in general a complex quantity, for which

both its amplitude and phase vary with l in a manner which depends on polarization components within

the optical system. This appendix presents a calibration method which is adapted from [120] to find the

SLM’s complex reflection coefficient corresponding to each input signal level.

C.1 Visibility for Grating Diffraction Orders

This section considers the diffraction of a two level grating structure, the diffraction orders of which will

help with SLM calibration. The analysis begins by approximating the SLM panel as an infinite binary

grating along one of its principle directions, where the reflection amplitude is r(l1) for a half-period, and

r(l2) for another half period. The following will show that, by measuring the intensity of the zeroth

and first diffraction orders from this grating, one can determine, relative to a proportional constant, the

complex reflection coefficient r(l) = |r(l)|ejφ(l) ∀ l.

132

Appendix C. Calibrating the Spatial Light Modulator 133

The derivation begins by simplifying terminology in the following manner:

1. Let l1 = 0;

2. Let r (l1) = r (0) = r0; and

3. Rename l2 to l.

Hereafter, The grating reflection function can be written as:

f(x) = [p(x)] ∗

[ ∞∑n=−∞

δ (x− nW )

], where p(x) =

r(l) 0 ≤ x < W/2

r0 W/2 ≤ x < 1

0 otherwise

. (C.1)

The the following visibility equation describes how well one can distinguish zeroth and first diffraction

orders:

V =I0 − I1I0 + I1

, where I0 = Iconst|F (kx = 0)|2 and I1 = Iconst|F (kx = π/W )|2. (C.2)

Here Iconst is a proportionality constant incorporating the object light intensity and the electromagnetic

properties of free space, while F (kx) denotes the Fourier transform of f(x). Now, intensity of the zeroth

and first diffraction orders are respectively

|F (kx = 0)|2 = Iconst

∣∣∣∣∣∫ W

0

f(x)dx

∣∣∣∣∣2

= Iconst

∣∣∣∣∣∫ W/2

0

r(l)dx+

∫ W

W/2

r0dx

∣∣∣∣∣2

= Iconst

∣∣∣∣W2 (r(l) + r0)

∣∣∣∣2=W 2Iconst

4|r(l) + r0|2 ,

(C.3)

and

∣∣∣∣F (kx =2π

W

)∣∣∣∣2 = Iconst

∣∣∣∣∣∫ W

0

f(x)e−j2πxW dx

∣∣∣∣∣2

= Iconst

∣∣∣∣∣r(l)∫ W

2

0

e−j2πxW dx+ r0

∫ W

W/2

e−j2πxW dx

∣∣∣∣∣2

= Iconst

∣∣∣∣jW2π [r(l) (e−π − 1)

+ r0(l)(1− e−π

)]∣∣∣∣2=W 2Iconst

π2|(r(l)− r0)|2.

(C.4)


Substituting (C.3) and (C.4) into (C.2), one can express the visibility as

V (l) =π2|r(l) + r0|2 − 4 |r(l)− r0|2

π2 |r(l) + r0|2 + 4 |r(l)− r0|2

=(R(l) +R0)

(π2 − 4

)+ 2 |r(l)r0| cos (φ(l)− φ0)

(π2 + 4

)(R(l) +R0) (π2 + 4) + 2 |r(l)r0| cos (φ(l)− φ0) (π2 − 4)

=c+ U(l)cos (φ(l)− φ0)

1 + cU(l)cos (φ(l)− φ0),

(C.5)

where the last line uses the substitutions

R(l) = |r(l)|2; R0 = |r0|2; φ0 = φ(0); c =π2 − 4

π2 + 4; and U(l) =

2 |r0r(l)|R0 +R(l)

. (C.6)

C.2 Performing SLM Calibration

Equations (C.5) and (C.6) imply that when the reflectivityR(l) is known for all values of l, one can deduce

the reflection phase φ(l) by measuring the visibility V (l) of the grating diffraction pattern considered

in this section. The following first describes the calibration setup, then summarizes the calibration

procedure.

Calibration Setup

The calibration was performed with the collection components in place for the Optical Super-Microscope

(OSM). The object and Lens 1 from the OSM, as shown from the schematic in Fig. 5.9, are removed, and

replaced with a beam expander which ensures the incident illumination impinges onto the SLM sensor

array with uniform amplitude and phase. Moreover, the beam splitter, which remains in the calibration

setup, provides the same polarization filter to light beams incident to and reflected from the SLM, in

the same way it would within the assembled OSM. In this configuration, a matrix of signal values loaded

the SLM will generate a spatially dependant reflection coefficient profile r(x, y), which will become the

field profile of the wave reflected from the SLM. Lens 2 Fourier transforms this reflected wave so that

its far-field diffraction pattern is mapped onto the image plane.

Calibration Procedure

First, the reflected power from the SLM is recorded when the input signal l is repeated across the device.

This measurement allows one to find R(l) (with respect to R0), which square-rooted gives |r(l)|. Second,

a grating structure is displayed on the SLM, implementing the function described by (C.1). The grating


period W is chosen such that both the zeroth and first diffraction orders are both located within the field

of view of a CCD camera placed on the image plane, but well separated from one another to minimize

overlap between the orders. For each value of l (which is swept from 1 to 255), the camera captures

an image at the image plane, and the power in each the zeroth and first diffraction orders are obtained

by summing intensity readings within in two square regions centering upon the respective diffraction

peaks. The visibility is hence calculated using (C.2). Finally, knowing V (l), R0 and R(l), substitution

into (C.5) and (C.6) determines r(l).

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by Alex Man Hon Wong - University of Toronto T-Space · Alex Man Hon Wong Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto 2014