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Lehigh UniversityLehigh Preserve
Theses and Dissertations
2011
Novel Waveguide Architectures for Light Sourcesin Silicon PhotonicsRavi S. TummidiLehigh University
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Recommended CitationTummidi, Ravi S., "Novel Waveguide Architectures for Light Sources in Silicon Photonics" (2011). Theses and Dissertations. Paper1268.
Novel Waveguide Architectures for Light Sources
in Silicon Photonics
by
Ravi Sekhar Tummidi
Presented to the Graduate and Research Committee
of Lehigh University
in Candidacy for the Degree of
Doctor of Philosophy
in
Electrical and Computer Engineering
Lehigh University
September 2011
ii
Copyright © by Ravi Sekhar Tummidi
September 2011
iii
Approved and recommended for acceptance as a dissertation in partial fulfillment for the degree of Doctor of Philosophy.
_____________________ Date __________________________________________ Prof. Thomas L. Koch (Dissertation Director)
_____________________ Accepted Date
Committee Members:
__________________________________________ Prof. Thomas L. Koch (Chairman and Advisor)
__________________________________________ Prof. Volkmar R. Dierolf __________________________________________ Prof. Michael J. Stavola __________________________________________ Prof. Nelson Tansu __________________________________________ Dr. Mark A. Webster
iv
Acknowledgement
I first of all would like to thank my advisor, Prof. Thomas L. Koch. One of the
proudest moments of my life was when Prof. Koch offered me to be his student. I hope I
have done justice to the confidence he bestowed on me. I have learnt a great deal from
him both as a scientist and as a person and will keep making my best effort to put it in to
practice.
The work detailed in this dissertation wouldn’t have been possible with out the
support, encouragement and collaboration of Dr. Robert M. Pafchek, Dr. Thach G.
Nguyen, Dr. Mark A. Webster, Prof. Arnan Mitchell and Kangbaek Kim. It was a very
fulfilling scientific endeavor because of their company and I hope it continues into the
future.
I would also like to thank my Ph.D. committee Prof. Volkmar R. Dierolf, Prof.
Michael J. Stavola, Prof. Nelson Tansu and Dr. Mark A.Webster for their insightful
queries and suggestions over the course of my graduate work at Lehigh. I would
especially like to thank Prof. Nelson Tansu under whom I did my master’s for getting me
started in the photonics field.
Over the course of my graduate work at Lehigh I had the incredible privilege to be a
part of the Silicon Laser Multi University Research Initiative (MURI) project. As a result
v
I got the unique opportunity to interact and collaborate with some of the brightest
scientists in my field for which I am very thankful.
Thanks to all my colleagues at the Center for Optical Technologies (COT) for their
support and encouragement over the years. A special thanks to Anne Nierer. Life as a
graduate student became much simpler with her around at COT.
Growing up I was privileged to be taught by some wonderful teachers who always
encouraged and prodded me to do better. The list is just too long but I have to mention
Alamelu madam, Shyamala madam and Sesha madam by name. This work is my “Guru
Dakshina” to all my teachers. I hope I have lived up to your expectations.
I want to thank my friends and relatives for their good wishes. I have to single out
Kalyan, my cousin, with out whose encouragement I wouldn’t have come to Lehigh or to
the United States for that matter. Thank you bhaiyya. Last but not the least, I want to
thank my mom, Mrs. T. Jhansi and my sister Sree Lekha, for their love and support with
out which I couldn’t have persevered through this challenging period. Thank you for your
patience through all these years.
I also gratefully acknowledge financial support from Army Research Laboratory
(ARL), Pennsylvania Ben Franklin Technology Development Authority (BFTDA) and
Air Force Office of Scientific Research (AFOSR) Silicon Laser MURI under Dr. Gernot
Pomrenke for the work carried out and presented in this dissertation.
vi
to my father
T.N.S. Satya Narayana
vii
Table of Contents
Acknowledgement ........................................................................................................... IV
List of Figures ................................................................................................................... X
Abstract .............................................................................................................................. 1
Chapter 1. Introduction ........................................................................................... 3
1.1 The Problem and Our Approach ................................................................ 6
1.2 Thesis Organization ................................................................................. 11
Chapter 2. Ultra Low Loss Quasi Planar Ridge Waveguides ............................ 13
2.1 Theory of Lateral Leakage Loss in Shallow Ridge Waveguides ............ 15
2.2 Experimental Evidence of “Magic Widths” ............................................ 19
2.3 Numerical Analysis of Lateral Leakage Loss in Shallow Ridge Straight
Waveguides .............................................................................................................. 22
2.3.1 TM-TE Mode Coupling at a Step Discontinuity............................... 22
2.3.2 Width Dependent Propagation Loss ................................................. 25
2.4 Waveguide Fabrication ............................................................................ 28
2.5 Waveguide Loss Measurement Technique .............................................. 30
2.6 Results and Discussion ............................................................................ 32
2.7 An Ultra Compact Waveguide polarizer based on “Anti-Magic Widths”
…………………………………………………………………………..35
viii
2.7.1 Polarizer Design Principle and Analysis........................................... 36
Chapter 3. Anomalous Losses in Curved Waveguides and Directional
Couplers at “Magic Widths”.......................................................................................... 43
3.1 Lateral Leakage of TM-like Mode in Thin Ridge SOI Ring ................... 44
3.2 Rigorous Simulation Approaches ............................................................ 46
3.3 Numerical Results and Discussion .......................................................... 47
3.3.1 The Impact of Waveguide Curvature on the Leakage Cancellation
Without Re-Entry ............................................................................................. 47
3.3.2 Leakage Cancellation in Waveguide Rings with Re-Entry: “Magic
Radius” Phenomenon ....................................................................................... 50
3.3.3 Mode Field Distribution .................................................................... 56
3.3.4 Wavelength Dependence of Lateral Leakage Loss ........................... 58
3.4 Directional Coupler ................................................................................. 60
Chapter 4. Low Loss Si-SiO2-Si TM Slot Waveguides ....................................... 63
4.1 Waveguide Design and Fabrication ......................................................... 64
4.2 Experimental Evaluation ......................................................................... 69
Chapter 5. Erbium Doped 8 nm Horizontal Slot Waveguides ........................... 72
5.1 Erbium Doped Slot Waveguide Fabrication ............................................ 74
5.2 Photoluminescence Measurements .......................................................... 75
5.3 Time Resolved Photoluminescence Measurements ................................. 76
5.4 Spontaneous Emission in Waveguide Media .......................................... 80
5.4.1 Spontaneous Emission into a Particular Waveguide Mode .............. 81
5.4.2 Total Spontaneous Emission Rate Including All Modes .................. 86
ix
5.4.3 Relation to Purcell Factor ................................................................. 92
5.5 Analysis of Experimental Results and Discussion .................................. 98
Chapter 6. Future Work and Outlook ............................................................... 101
6.1 Mask Overview ...................................................................................... 102
6.2 Passive Ridge Waveguides .................................................................... 103
6.3 Passive Horizontal Si-SiO2-Si Slot Waveguides ................................... 107
6.4 Ring Resonators for Critical Coupling in Horizontal Si-SiO2-Si Slot
Waveguides ............................................................................................................ 115
6.5 Active Horizontal Si-Er doped SiO2-Si Slot Waveguides ..................... 119
6.6 Summary and Future Work ................................................................... 123
Appendix A: Spontaneous Emission Rate in a Slot Waveguide ............................... 127
Bibliography .................................................................................................................. 151
x
List of Figures
Figure 1.1. Normalized profile of the dominant electric field component of the
fundamental TE and TM mode for the 8 nm horizontal slot waveguide shown in the inset
............................................................................................................................................. 9
Figure 1.2. A shallow ridge horizontal slot waveguide structure ................................... 10
Figure 2.1. (a) Ridge waveguide geometry (b) Slab mode indexes used in effective index
calculations. The modal indexes for the guided TE-like and TM-like modes, respectively,
lie intermediate between their slab values for slab thicknesses t1 and t2 .We use index
values of nSi = 3.475 and nSiO2 = 1.444 at 1.55 μm wavelength. ...................................... 15
Figure 2.2. (a) Phase-matching diagram showing TM-like waveguide mode phase
matched to a propagating TE slab mode in the lateral cladding (b) Ray picture of the TM
mode lateral leakage ......................................................................................................... 17
Figure 2.3. (a) AFM image of a waveguide formed by wet-etching (b) Experimentally
measured optical transmission (log scale) for the fundamental TE-like and TM-like
modes for the waveguides ................................................................................................. 20
Figure 2.4. (a) AFM image of a waveguide formed using thermal oxidation (b)
Experimentally measured optical transmission (log scale) for the fundamental TE-like
and TM-like modes for the waveguides............................................................................ 21
xi
Figure 2.5. Mode coupling diagram showing the TM-TE mode coupling at the ridge
boundary ........................................................................................................................... 23
Figure 2.6. (Color) Magnitude of the reflected TM mode, reflected and transmitted TE
modes as a function of the incident TM mode propagation angle .................................... 23
Figure 2.7. (Color) Relative phase of the reflected and transmitted TE modes as a
function of the incident TM mode propagation angle ...................................................... 24
Figure 2.8. (Color) Simulated loss of TM-like mode in shallow ridge waveguide shown
in inset which shows strong width dependence ................................................................ 26
Figure 2.9. Electric field distributions of the fundamental TM-like mode for (a) 1.2 μm
and (b) 1.429 μm waveguide widths ................................................................................. 27
Figure 2.10. Processing steps for shallow ridge waveguide fabrication ......................... 28
Figure 2.11. (a) AFM trace of a 1.44 μm ridge waveguide with <0.2 nm surface
roughness (b) Quasi-planar ridge SOI waveguide geometry. BOX thickness is 2μm ...... 29
Figure 2.12. Setup to measure the resonance characteristics of (a) a weakly coupled ring
resonator with separate thru and drop ports and (b) an exemplary drop-port response .... 30
Figure 2.13. Drop-port responses for the TE and TM modes for a 400 μm radius ring
resonator with “magic width” waveguide of 1.43 μm ...................................................... 33
Figure 2.14. (a) XY view and (b) XZ view of the 3D plot of loss of the fundamental TM
mode of a SOI ridge waveguide with 220 nm c-Si in the core (inset) .............................. 38
Figure 2.15. Loss and effective Index nTM-WG of the fundamental TM-like mode of a
SOI ridge waveguide with 220 nm c-Si in the core for (a) 90 nm (b) 148 nm and (c) 150
nm ridge heights ................................................................................................................ 40
xii
Figure 3.1. (a) Cross section, and (b) plan view and mode coupling diagram of a SOI
thin-ridge ring resonator. Waveguide dimensions are shown ........................................... 44
Figure 3.2. (Color) Simulated propagation loss of the TM-like guided mode of a thin-
ridge SOI bent waveguide in the absence of secant TE waves inside the ring (a) as a
function of waveguide width for different bend radii and (b) as a function of the bend
radius when the waveguide width is fixed at 1.43 μm. Conventional bending losses are
also shown. ........................................................................................................................ 49
Figure 3.3. Simulation domain used to model rings with R0 = 100 μm in FEM ............ 51
Figure 3.4. (Color) Propagation loss of the fundamental TM-like mode of a ring as a
function of ring radius. Waveguide width is W = 1.43 μm ............................................... 52
Figure 3.5. (Color) Propagation loss of the fundamental TM-like mode of a ring
resonator as a function of the waveguide width for different ring radii ........................... 54
Figure 3.6. Propagation loss of the fundamental TM-like mode of a ring as a function of
the ring waveguide width and ring radius. Radius around (a) 200 μm and (b) 400 μm ... 55
Figure 3.7. Electric field distributions of the fundamental TM-like mode of a ring with a
waveguide width of 1.35 μm and radius at (a) 199.62 μm (high loss) and (b) 198.95 μm
(low loss) ........................................................................................................................... 57
Figure 3.8. Electric field distributions of the fundamental TM-like mode of a ring with a
radius of (a) 199.84 μm (high loss) and (b) 200.46 μm (low loss). Waveguide width of
1.43 μm ............................................................................................................................. 58
Figure 3.9. Wavelength dependent loss of the TM-like mode in thin-ridge SOI rings
with a waveguide width of (a) 1.43 μm and (b) 1.35 μm and ring radii as shown ........... 60
xiii
Figure 3.10. (Color) Inset: Cross sectional view of ridge waveguides in a directional
coupler configuration with waveguide centre to centre separation S and a symmetric
waveguide width of W. Graphs: Loss of the fundamental and first order TM super mode
of the directional coupler as a function of waveguide width (W) for a waveguide centre to
centre of (a) S=3 μm and (b) S=1 μm ............................................................................... 62
Figure 4.1. Processing steps for shallow ridge horizontal slot waveguide. Shallow ridge
waveguide in step 1 formed using the process described in Fig. 2.10 .............................. 65
Figure 4.2. Scanning electron micrograph of a similar planar slot structure with 8.3 nm
slot visible using ESED mode imaging ............................................................................ 66
Figure 4.3. Weighting function for index changes for TE and TM modes, illustrating the
14X enhancement of sensitivity to index change in the SiO2 slot for the TM mode over
TE mode for the slot waveguide shown in the inset. The integrated weighting function
across the thickness of the SiO2 slot gives an effective confinement of 23.5%. .............. 68
Figure 4.4. Scan of ring resonance illustrating ΔνFWHM =625 MHz with a corresponding
Q~3 X 105 ......................................................................................................................... 70
Figure 5.1. (a) Implant and anneal conditions and (b) cross sectional view of Erbium
doped slot waveguide ........................................................................................................ 74
Figure 5.2. Setup for measurement of Erbium doped slot waveguide luminescence ...... 75
Figure 5.3. Erbium doped slot waveguide photoluminescence at different waveguide
coupled pump powers, expressed as power spectral density (PSD) ................................. 76
Figure 5.4. (a) Pump pulse and (b) Luminescence signal using gated integrator and
BOXCAR averager with system preamplifiers. Observed peak luminescence matches
calculated value ................................................................................................................. 78
xiv
Figure 5.5. (a) Photoluminescence measured with the system pre-amplifiers replaced by
a low noise high gain TIA in conjunction with gated integrator and BOXCAR averager
(b) Luminescence decay signal on a semi-log plot with a lifetime of ~ 152 μsec ........... 79
Figure 5.6. (Inset) Ultra-thin 8.3 nm slot slab waveguide structure. Plot of effective
inverse modal thickness in the vertical direction as a function of position for both the TE
and TM modes .................................................................................................................. 85
Figure 5.7. Plot of F(z)/(n(z)Ffree) as defined in Eq. 5.30 as function of position in the 8.3
nm slot structure shown in the inset. ................................................................................. 90
Figure 5.8. Plot of total spontaneous emission rate enhancement, averaged over
polarizations, as a function of slot thickness. Also shown are all the contributions,
dominated by the TM slot guided mode, along with the efficiency of emission into just
the TM slot mode. Values indicated for the 8.3 nm slot case ........................................... 91
Figure 5.9. A Fabry-perot resonator of length Lc formed by inserting mirrors of
reflectivity R in an arbitrarily long waveguide of length L .............................................. 94
Figure 6.1. Image showing the mask set consisting of 54 blocks arranged in a 9 by 6 grid.
An exemplary block is marked ....................................................................................... 102
Figure 6.2. Image of a typical individual mask block as marked in Fig 6.1 ................. 103
Figure 6.3. Transverse TM mode count for a 220 nm fixed Si core ridge waveguide
shown in the inset as a function of waveguide width and ridge height. TM mode count is
indicated .......................................................................................................................... 104
Figure 6.4. Fundamental TM mode loss for the shallow 220 nm fixed Si core ridge
waveguide shown in the inset of Fig 6.3 as a function of waveguide width for ridge
heights at and in the vicinity of 7 nm .............................................................................. 105
xv
Figure 6.5. (Color) Fundamental TM mode loss for the shallow 220 nm fixed Si core
ridge waveguide shown in the inset of Fig 6.3 in ring resonator configuration as a
function of ring radius. The width of the waveguide in the ring (WR) is 1.458 μm and
1.35 μm. The ridge height is (a) 6 nm (b) 7 nm and (c) 8 nm ........................................ 106
Figure 6.6. Weighting function confinement % at 1550 nm in (a) a-Si and (b) 10 nm
SiO2 slot as a function of c-Si and a-Si thickness ........................................................... 110
Figure 6.7. Analytical Bending loss at 1480 nm for a 10 nm ridge slot waveguide
structure with a radius of (a) 400 μm and (b) 200 μm as a function of the c-Si thickness
in the core and amorphous silicon thickness with the waveguide width equaling the
“magic width” at 1550 nm .............................................................................................. 111
Figure 6.8. (a) Schamatic of a 10 nm ridge, 100 nm a-Si – 15 nm SiO2 – 213.25 nm c-Si
slot waveguide structure (b) Fundamental TM mode loss of this straight waveguide at
1530 nm and 1480 nm as a function of the width W of the c-Si ridge ........................... 113
Figure 6.9. (Color) Fundamental TM mode loss for the 15 nm slot, waveguide shown in
Fig. 6.8 (a) in ring resonator configuration as a function of ring radius around (a) 400 μm
and (b) 450 μm. The width of the c-Si ridge waveguide in the ring is 1.605 μm. Traces
shown for ridge heights of 9 nm, 10 nm and 11 nm ....................................................... 114
Figure 6.10. (a) Schematic of TWR ring resonator coupled to a bus waveguide (b) Power
transmission through the bus waveguide as a function of the normalized loaded quality
factor (QL) ....................................................................................................................... 116
Figure 6.11. Coupling Quality factor for the 15 nm slot, 10 nm ridge structure as a
function of the minimum gap between the tangential bus waveguide and the ring as
shown in Fig 6.12 (a) ...................................................................................................... 117
xvi
Figure 6.12. Section of a bus waveguide-resonator system showing the coupler
configuration (a) Regular coupler - Bus waveguide running tangentially to the ring with
gap of G μm at minimum separation between the two (b) Curved coupler - Bus
waveguide wrapped around the ring to increase the interaction length and hence to reduce
the QC at a gap of G μm. The curved interaction length is parameterized by the arc angle
θ ....................................................................................................................................... 117
Figure 6.13. (Color) Coupling Quality factor for the 15 nm slot, 10 nm ridge waveguide-
resonator system with a curved coupler as a shown in Fig 6.12 (b). Arc angle θ is defined
in Fig. 6.12(b). Gap is fixed at 1.152 μm ........................................................................ 118
Figure 6.14. (a) Exemplary erbium doped slot waveguide ring resonator coupled to a bus
waveguide to aid pumping of erbium in the slot (b) Cross section of the slot waveguide
constituting the ring ........................................................................................................ 120
Figure 6.15. (Color) Minimum erbium concentration required in the slot for the structure
shown in Fig. 6.14 to reach threshold as a function of the coupling gap G for different
intrinsic resonator Qs ...................................................................................................... 120
Figure 6.16. Threshold pump power as a function of coupling gap for the ring resonator
configuration shown in Fig. 6.14 with Q = 4X105 and slot erbium concentration of
7.7X1020 cm-3 .................................................................................................................. 121
Figure 6.17. Output lasing power at 1530 nm as a function of pump power and coupling
gap for the configuration shown in Fig. 6.14 with Q = 4X105 and slot erbium
concentration of 7.7X1020 cm-3 ...................................................................................... 122
Figure A.1. A five layer dielectric slab structure representative of the slot waveguide 130
xvii
Figure A.2. Illustration of the definition of the reflection and transmission coefficients r12
and t12 .............................................................................................................................. 131
1
Abstract
Of the many challenges which are threatening to derail the success trend set by
Moore’s Law, perhaps the most prominent one is the “Interconnect Bottleneck”. The
metallic interconnections which carry inter-chip and intra-chip signals are increasingly
proving to be inadequate to carry the enormous amount of data due to band-width
limitations, cross talk and increased latency. A silicon based optical interconnect is
showing enormous promise to address this issue in a cost effective manner by leveraging
the extremely matured CMOS fabrication infrastructure. An optical interconnect system
consists of a low loss waveguide, modulator, photo detector and a light source. Of these
the only component yet to be demonstrated in silicon is a CMOS compatible electrically
pumped silicon based laser.
The present work is our endeavor towards the goal of a practical light source in
silicon. To this end we have focused our efforts on horizontal slot waveguide which
consists of a nm thin low index silica layer sandwiched between two high index silicon
layers. Such a structure provides an exceptionally high confinement for the TM-like
mode in the thin silica slot. The shallow ridge profile of the waveguide allows in
principle for lateral electrical access to the core of the waveguide for excitation of the slot
embedded gain material like erbium or nano-crystal sensitized erbium using tunneling,
polarization transfer or transport.
2
Low losses in the proposed structure are paramount due to the low gain expectation
(~1dB/cm) from CMOS compatible gain media. This dissertation details the novel
techniques conceived to mitigate the severe lateral radiation leakage loss of the TM-like
mode in these waveguides and resonators using “Magic Widths” and “Magic Radii”
designs. New fabrication techniques are discussed which were developed to achieve
ultra-smooth waveguide surfaces to substantially reduce the scattering induced losses in
the Silicon-on-Insulator (SOI) high index contrast system. This enabled us to achieve
resonators with Qs of 1.6X106 for the TE-like mode in non-slot configurations and 3X105
for the TM-like mode in full slot configuration, the highest yet reported for this type of
structure and close to our design requirements for a laser.
Erbium was incorporated into the silica slot just 8.3 nm thick and photoluminescence
was observed in full waveguide configuration. A simple phenomenological model based
on spontaneous emission into a waveguide mode was developed, which predicted >10X
Purcell enhancement of the luminescence decay in these slot waveguides even in the
absence of a resonator, a result also yielded by a rigorous quantum electrodynamic
analysis. These enhanced spontaneous emission rates were experimentally verified using
time resolved photoluminescence decay and luminescence power measurements.
The results so far indicate that these slot structures could be the enablers for very
efficient LEDs due to the highly preferential characteristic of the spontaneous emission to
go into the single guided mode. The future goal will be to harness this behavior for novel
silicon photonic light sources.
3
Chapter 1. Introduction
It would be apt to say that for photonics to become as pervasive and ubiquitous as
electronics, it would have to go the “integration” way. The ability to monolithically
integrate transistors into a single silicon chip lent integrated circuits to economies of
scale. This has driven their exponentially increasing processing power, reliability and
functionality but at much reduced space, power requirements and cost, as was famously
predicted by “Moore’s Law”[1].
Indeed the same incentives exist for photonic integration and the idea of a photonic
“super chip” containing integrated optical components for generating , modulating,
switching , guiding, detecting and amplifying light was recognized as early as 1969 [2].
The ensuing two decades saw a significant research effort [3, 4] towards this goal. The
early work in the field utilized materials which were already well known for their relative
ease of manipulating and generating light such as Lithium Niobate (LiNbO3) and III-V
semiconductors such as Gallium Arsenide (GaAs) and Indium Phosphide (InP)
respectively.
However, mid 1980s saw silicon arising as a contender for photonic integration due
its transparency and hence capability to guide light at telecommunication wavelengths of
1.3 and 1.55 μm [5] and the possibility offered by it of placing electronic and photonic
4
components side by side on a single silicon chip. This was despite the fact that silicon is
sub optimal for both modulation and generation of light. Being a centro-symmetric
crystal silicon lacks the linear electro-optic (Pockels) effect, the traditional means for
modulating light and being an indirect band gap material means light generation in
silicon is a very inefficient process. However, the possibility of utilizing the already well
understood technical know how on silicon due to its prominence in the electronics
industry and leveraging the extremely matured multi-billion dollar complementary-
metal-oxide-semiconductor (CMOS) fabrication infrastructure drove the research to
overcome these deficiencies of silicon by ingenious and often un-conventional means.
An immediate pressing problem for which silicon integrated photonics is ideally
suited to provide solution is the “Interconnect Bottleneck”[6]. The insatiable desire to
pack ever more number of transistors per unit area on the silicon real estate to keep up
with the predictions of “Moore’s Law” has required a commensurate down scaling in the
interconnect cross sectional dimensions and also increase in interconnect lengths.
Decrease in interconnect cross sectional dimensions results in increased resistance
leading to higher signal latencies. This has been countered in the past by replacing
aluminum with copper as the medium for interconnect to reduce the resistivity, using low
k dielectric materials to reduce the capacitance and by increasing the number of metal
layers on a silicon chip. However these are one time improvements and it’s a matter of
time before the problem resurfaces. This coupled with the increased power requirements
and electro-magnetic interference (EMI) associated with dense packing of metallic
interconnects have the potential to derail the success trend set by “Moore’s Law”.
5
Herein perhaps lies an opportunity for photonics to come onboard silicon and
intimately integrate with the electronics in the form of an optical interconnect to provide
signaling and clock distribution. An optical interconnect in silicon offers the promise of
decreasing the interconnect delays and providing higher band width (BW) to keep pace
with transistor speed improvements, while potentially lowering power consumption and
being resistant to EMI [7]. A hypothetical optical interconnect will consist of a photon
source, preferably a laser whose light is coupled to a waveguide matrix acting as
interconnects between different on chip components. Light is converted into data using a
modulator driven directly by a standard CMOS driver. Light is then routed to the other
end of interconnect to a photo detector which converts the light into photo current. The
current is then transformed into a conventional digital voltage signal using a CMOS
trans-impedance amplifier (TIA). Such an intimate integration of electronics and
photonics together on chip would relax the impedance matching conditions while
providing voltage scaling and superior performance at much lower power levels.
Tremendous progress has been made over the years to realize all the components
which will constitute a photonic interconnect in silicon, the so-called silicon photonics
toolkit. Silicon photonics relies heavily on Silicon-On-Insulator (SOI) technology [8]. It
was only after SOI was adopted by mainstream CMOS that a more comprehensive
development of silicon photonics began. The large index contrast between silicon and the
buried oxide (BOX) layer allows definition of waveguides with very tight bend radii
which is necessary for efficient packing of waveguides on chip and miniaturization of
various other optical components. On the active components side, modulators based on
free carrier plasma dispersion effect [9] in silicon with 40 GHz 3dB bandwidth [10, 11]
6
have been realized in fully CMOS compliant fabrication processes. The flexibility
provided by monolithic electronic and photonic integration on silicon has been utilized to
improve the performance of germanium detectors on silicon. The ability to connect the
transistors of the TIA placed in close proximity to the germanium photo detectors on
silicon using CMOS metallization results in extremely low capacitances and hence higher
detector speeds. As a testament of the strength of silicon photonics integration with
CMOS technology, a 20 Gbps dual XFP transceiver [10] and 40 Gbps WDM transceiver
based on the various components discussed above have already been realized.
Arguably the most critical component of an optical interconnect system on silicon is
the “photon source”. The most ideal avatar of such a photon source would be an on chip
monolithically integrated CMOS compatible electrically pumped silicon based laser.
Various approaches[12] have been taken to realize the same, however, to date a
continuous wave electrically pumped silicon based laser operating at room temperature is
yet to be demonstrated.
1.1 The Problem and Our Approach
The present work is our endeavor towards achieving a CMOS compatible electrically
driven light source in silicon. With the aim of utilizing erbium as the extrinsic gain
medium, we focused our efforts on horizontal slot waveguides. Here I enumerate the key
design challenges associated with an extrinsic light source in silicon which will put the
approach we have taken into perspective.
7
1. Erbium (Er): Erbium doped materials are of great interest in thin film integrated
optoelectronic integration technology due to the Er3+ intra-4f emission at 1.54 μm, a
standard telecommunication wavelength at which silicon is transparent. The radiative
lifetime corresponding to the first excited state (~1.54μm) is usually in millisecond
range, depending on the host material. As a result the emission and absorption cross
sections are relatively small, typically 10-21-10-20 cm2. Therefore reasonable optical
gain values (~3 dB) can only be achieved when the signal beam encounters a large
amount of excited Er (1021-1020 Er/cm2). At chip scale integration dimensions, this
would require a high density of Er, typically in the range 0.1 to 1 at. %. At such high
Er concentrations, the distance between Er ions is very small and electric dipole-
dipole interactions between Er ions can reduce the gain performance of the device. So
we can’t expect a lot of gain from Er doped silica, therefore our designed structure
should have very low losses for it to lase.
2. Silica (SiO2) host: Er will be hosted in SiO2 which has a lower index (1.45)
compared to silicon which has an index of 3.47 at 1.54μm. As a result it is
challenging to achieve high modal confinement factor (Γ) in the material (SiO2)
containing the gain medium (Er) in conventional waveguide architectures. Therefore
new waveguide configurations have to be explored which can provide high modal
confinement in a low index material.
3. Tunnel injection: The structure we design should also provide means of electrical
excitation of the gain medium in a non-invasive manner i.e. which doesn’t depend on
hot electron injection for excitation of the gain medium as it causes degradation of the
silica layer embedding Er. This suggests that the excitation be based on some
8
tunneling or polarization transfer or transport mechanism. Though the mechanism is
not clear yet, such non-invasive mechanisms are usually operative only at nano-meter
dimension scales. So a plausible non-destructive electrical excitation of the Er would
require that the silica layer embedding the gain medium be very thin i.e. couple of
nano-meters thick.
4. Doping and charge accumulation: Electrical drive for excitation of the gain media
requires doping to increase the conductivity. In addition, the envisioned tunneling
mechanism would require charge accumulation at the interfaces adjacent to the
dielectric containing Er. Optical modal overlap with these free carrier regions would
increase the modal loss and hence the lasing threshold. Therefore the designed
structure should minimize these losses.
5. Scattering and material losses: Due to the high index contrast between silicon and
silica, scattering induced losses can be predominant in this material system. In
addition to this, the losses could also increase if the device design requires utilization
of other materials such as amorphous or poly silicon which have losses higher than
their crystalline counterpart. With very small gain expectation from Er, these losses
have to be minimized to achieve lasing operation.
In light of these design challenges, we concentrated our research efforts on horizontal slot
waveguides as shown in the inset of Fig. 1.1. The structure consists of a low index
material, in our case silica, sandwiched between two high index silicon layers. This
structure when operated in TM mode provides huge optical confinement in the low index
silica slot layer compared to the TE mode as can be seen from Fig. 1.1, which shows the
normalized profile of the dominant electric field component of the TE and TM modes for
9
the horizontal slot waveguide shown in the inset. The electrical field confinement in the
slot will increase as the slot thickness is increased, but for the structure to be suitable for
our purpose we require the slots to be ultra thin i.e. couple of nano-meters thick so as to
enable excitation of the slot embedded gain media by tunneling, polarization transfer or
transport. But even such ultra thin slot dimensions will not seriously compromise the
modal gain as we will show later in Sec. 4.1. In fact for this particular structure the TM
mode will see fourteen times higher gain enhancement due to a gain medium like Er
embedded in the slot compared to the TE mode or more precisely 23.5% of the local gain
in the 8 nm slot will be seen by the TM mode. This is quite phenomenal considering the
fact the slot is just 8 nm thick and is of lower index.
Figure 1.1. Normalized profile of the dominant electric field component of the fundamental TE and
TM mode for the 8 nm horizontal slot waveguide shown in the inset
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 0.2 0.4 0.6 0.8 1Normalized Electric Field
Wav
egui
de P
ositi
on (µ
m)
TE TM
150 nm Si
100 nm Si
8 nm SiO2
BOX
10
The structure we want to realize is shown in Fig. 1.2. The structure will consist of an
ultra thin silica slot sandwiched between an upper amorphous and a lower crystalline
silicon layer to provide large TM mode confinement. The slot will be a thermally grown
oxide which will allow precise control over its thickness. The horizontal slot design
allows for planar fabrication technology which eliminates the etching induced roughness
of a vertical slot design thereby lowering the scattering losses. We also utilize a novel
fabrication technique which is detailed later in this thesis which results in ultra smooth
waveguide surfaces. It is also a shallow ridge design to keep the waveguide sidewall
roughness at the minimum. A shallow ridge structure is also critical for allowing
electrical drive access to the waveguide core with minimal loss impact as we intend to
use amorphous silicon as the top cladding layer which suffers from low mobility. The
waveguides are also of specific widths referred to as the “magic widths” (MW) to
eliminate the TM mode radiation leakage loss which can be prevalent in these shallow
ridge designs and can render the device useless for TM mode operation. This aspect is
central to this thesis and is discussed in detail. Finally, the upper cladding layer is a
superior quality deposited amorphous silicon film.
Figure 1.2. A shallow ridge horizontal slot waveguide structure
a-SiEr-SiO2
c-Si
11
1.2 Thesis Organization
The remaining part of this thesis is organized to mirror the step-step sequential process
which was undertaken to achieve low loss horizontal slot waveguides as lasing element
for a CMOS compatible light source in silicon. Chapter two details the realization of the
ultra low loss quasi planar ridge waveguide which is a critical primitive to the horizontal
slot waveguide. Here new fabrication techniques are discussed which were developed to
achieve ultra-smooth waveguide surfaces to substantially reduce the scattering induced
losses in SOI high index contrast system. Resonator Qs of 1.6X106 for the TE mode were
achieved as a result. These fabrication methods were then combined with the novel
phenomenon of “magic widths” conceived to mitigate the severe lateral leakage losses for
the application critical TM mode resulting in Qs of 6.8X105, the highest yet reported for
such shallow ridge structures. Further more, we discuss the design of an ultra compact
waveguide polarizer with significant improvements over any current designs as an
application of this phenomenon. In chapter three we take a keener look at the “magic
width” phenomenon and show how it is impacted in curved waveguides and directional
couplers. In doing so we discovered new anomalous losses over and above any
conventional reported losses in these structures which led to the framing of new design
rules to achieve highest performance devices. In chapter four we apply “magic widths”
and the ultra smooth fabrication techniques to design and realize a very low loss TM like
mode quasi planar horizontal slot waveguide. A key factor contributing to the low loss in
the these structures is a superior quality deposited PECVD amorphous silicon film
developed in house with a bulk loss of ~3 dB/cm which is in the league of best results
reported by other groups. Having achieved low loss slot waveguides, the next obvious
12
step was to incorporate Erbium into the slot. Chapter five discusses ultra thin 8 nm
Erbium doped slot waveguides. Room temperature photo luminescence results were
obtained in full waveguide configuration and time resolved spectroscopy showed
evidence of enhanced spontaneous emission rates or “Purcell Enhancement” in these slot
waveguides. This was quite surprising as this phenomenon is usually associated with
resonators and is typically explained using quantum electrodynamics. We however show
that the simple phenomenological model based on conventional expressions for
spontaneous emission into a waveguide mode precisely captures the “Purcell
enhancement” in slot waveguides. The results are also verified against a full quantum
electrodynamic analysis of the structure. Finally chapter six sets a detailed and definitive
direction for future work based on the findings of this thesis.
13
Chapter 2. Ultra Low Loss Quasi Planar Ridge Waveguides
A low loss shallow ridge waveguide is a vital precursor to the low loss shallow ridge
slot waveguide. Devoid of the slot layer and the upper amorphous silicon cladding, it is a
much simpler structure to work with. Information gained by analyzing various loss
mechanisms in this basic structure proved vital in the design of low loss slot waveguides.
Quasi planar or shallow ridge waveguides by themselves are of strong interest for active
SOI devices such as modulators, tunable filters and sensor applications. Optimized
voltage metrics for field-induced effects demand tight confinement primarily only in one
dimension, and the quasi-planar ridge affords the highest possible confinement in the
vertical direction while simultaneously providing a near-ideal geometry for control of
applied fields and conductive lateral access.
Also, more recently Densmore et. al. [13] have shown that high-index-contrast
systems operating in the TM mode are ideally suited for interferometric sensors that rely
on perturbations in the modal effective index caused by the interaction of the evanescent
tail of the mode with solutions or material adsorbed on the waveguide surface. In
particular, silicon-on-insulator (SOI) was shown to provide an order of magnitude
increase in sensitivity compared to other material systems. Sensitivity of evanescent-
wave surface sensors depends critically on maximizing field overlap with the adsorbed
species, but precision in phase measurement also requires long interaction lengths in
14
either a long interferometric configuration or a high-Q resonator. So waveguide
propagation losses will also impose a limit on the sensitivity. Thin-core shallow ridge
waveguides can potentially optimize both these attributes. However, it is well known that
waveguide scattering losses, which can be shown to scale as (Δn)2 through analytical
means [14], are exacerbated in tight vertical confinement SOI guides both due to the high
index contrast and the higher modal amplitude at the interfaces with the low-index
cladding layers. Reactive ion etching (RIE) is typically used to pattern Si-core SOI
waveguides and significant (multi-nanometer scale) sidewall roughness is commonly
encountered from standard lithographic and etching processes. This suggests alternative
fabrication techniques for highly smooth waveguides that minimize exposure of the mode
to etched interfaces. Also, more importantly, the desired TM-mode operation in shallow
ridge waveguides can also lead to inherent severe lateral radiation leakage loss in high
index-contrast systems like SOI. These losses which are over and above the scattering
losses can be just too high to render these waveguides practically useless for TM mode
operation.
Firstly the mechanism of lateral leakage loss for the TM-like mode in shallow ridge
waveguide is elucidated using a simple phenomenological model. This is followed by
experimental verification of the TM mode lateral leakage phenomenon and a way to curb
it [15, 16]. Rigorous full vector models are then discussed which were developed to
analyze and aid the accurate design of these structures[17]. The insights gained from
these were then combined with novel fabrication techniques yielding ultra smooth
waveguide surfaces to design and fabricate very low loss shallow ridge waveguides,
which are detailed next[18, 19]. Finally, an interesting application in the form of an ultra
15
compact TE pass polarizer utilizing the lateral leakage phenomenon in shallow ridge
waveguides is proposed and analyzed using full vector numerical techniques[20].
2.1 Theory of Lateral Leakage Loss in Shallow Ridge Waveguides
Under an equivalent slab model for the ridge waveguide geometry (Fig. 2.1 (a)), the
lateral leakage loss for the TM-like mode is due to TM/TE mode conversion at the ridge
Figure 2.1. (a) Ridge waveguide geometry (b) Slab mode indexes used in effective index calculations.
The modal indexes for the guided TE-like and TM-like modes, respectively, lie intermediate between
their slab values for slab thicknesses t1 and t2 .We use index values of nSi = 3.475 and nSiO2 = 1.444 at
1.55 μm wavelength.
Slab Thickness (nm)
Slab
Mod
al E
ffect
ive
inde
x
TE-like mode Neff
TM-like mode Neff
t1 t2
W
AirSi
SiO2
TE
TM
t1
t2
(a)
(b)
16
boundary [15, 16, 21-23]. It should be emphasized that this effect is not caused by any
surface or side-wall roughness. With reference to slab waveguide dispersion curves in
Fig. 2.1 (b), for the case of TE-like modes, this mode conversion at the boundary cannot
lead to any propagating field or leakage loss in the lateral cladding since longitudinal
phase-matching requires that any fields generated at the ridge boundaries be laterally
evanescent in the slab lateral cladding for both the TE and TM generated fields.
However, in the case of the TM-like mode, while phase-matching requires that the lateral
TM slab mode be evanescent in the lateral cladding, any TE slab field component
generated at the boundary is phase-matched to a laterally propagating TE slab mode at
some angle θ in the lateral slab cladding region.
This phase-matching is illustrated in Fig. 2.2 (a). Here TEβ and TMβ represent the
propagation constants of the TE-like and TM-like modes of the ridge waveguide,
respectively. This diagram illustrates that the propagation constant of the TM-like mode
is much less than that of the TE-like mode, and for many practical rib guides may also lie
below the propagation constant of the unguided TE slab mode cladTEk . Since this slab mode
is unguided by the rib, it may propagate at any angle and if cladTETM k<β , it is possible to
rotate cladTEk by an angle θ such that the TE-slab and TM-guided mode are phase-matched
in the z direction. Since the guided mode is then phase-matched to a radiation mode, it is
possible that leakage may occur if there is some means for mode conversion.
For a thin enough lateral slab thickness , and certainly in the case of a strip or “wire”
waveguide where , the TE slab effective index can lie below the TM-like mode effective
index, and hence, avoid this phase-matched leakage. However, for electrical access, these
17
Figure 2.2. (a) Phase-matching diagram showing TM-like waveguide mode phase matched to a
propagating TE slab mode in the lateral cladding (b) Ray picture of the TM mode lateral leakage
designs may not be practical, and thus, this leakage loss must be well understood.
The leakage process is illustrated in Fig. 2.2 (b). Since the z components of all
propagation constants are conserved, all waves develop the same relative phase along the
length of guide and we can discuss phase in the lateral direction only.
Starting at the bottom, a TM mode is guided by the rib and is represented as the solid
ray incident on the right wall, where the angle of incidence is such that TM total internal
reflection occurs. However, due to the step discontinuity at the rib wall, mode conversion
from TM to TE can occur, and it can be shown from mode-matching calculations [23]
(Sec 2.3.1) that TE transmitted and reflected propagating waves are produced that are
approximately equal in magnitude, but are ~π radians out of phase. The reflected TE
radiation mode traverses across the core as shown by a dashed line. Upon total internal
TMβ
TEβ
)(cladTEk
)(,cladTEzk
x
z
TETM
W
TE
)(,
coreTEeffn
)(,
coreTMeffn
)(,
cladTEeffn
)(,
cladTMeffn
)(,
cladTEeffn
)(,
cladTMeffn
For TE:
For TM:
θ
(a)
(b)
18
reflection, the TM mode experiences a negative phase shift TIRφ and the combination of
TIRφ with the phase from a single traverse of the guide to the left is zero for the
fundamental mode.
At the left rib wall, the TM mode generates additional small reflected and transmitted
TE propagating waves. The new transmitted TE wave, with a relative phase of π radians
as noted above, combines with the previous reflected TE wave that has traversed the
guide with a phase shift of Wk coreTEx ⋅, . Thus, if this phase shift across a single traverse for
the TE in the core is a multiple of π2 , the TE waves will interfere destructively. This
leads to a width dependence for the leakage minima that satisfies a resonance-like
condition [22] of π2, ⋅=⋅ mWk coreTEx , or alternatively stated[15, 16],
( )…321
22,,,
,,
=−
⋅= mNn
mWTMeff
coreTEeff
λ (2.1)
where coreTEeffn ,
is the two-dimensional slab effective index of the TE mode in the core while
TMeffN , is the full three-dimensional effective index of the TM-like mode for the
structure. TMeffN , has a weak W dependence, so some care must be used if precision is
required. From this argument, we would expect to see significant leakage loss for TM
propagation, except at precise, specific waveguide widths --- ‘‘magic widths’’ satisfying
the resonance condition in Eq. (2.1), where the leakage loss would be greatly reduced due
to destructive interference of radiating TE waves. It is also interesting to note that the
next higher order mode will be lossy at these widths, since the radiated fields add in
phase for this mode. This may allow relatively wider guides with effectively single-mode
behavior.
19
2.2 Experimental Evidence of “Magic Widths”
A series of ridge waveguides with widths varying from 0.5 to 1.8 μm in 50 nm
increments with slab thickness t1=205 nm and t2=190 nm (i.e. a 15 nm ridge height) were
fabricated on an SOI wafer with a 2 μm buried oxide (BOX) layer thickness [15, 16].
These waveguides were formed by wet-etching, a process using which low propagation
losses of 0.7 dB/cm for the TE mode at 1.55 μm have already been reported [24].
Waveguides were also fabricated using a thermal oxidation process (discussed in Sec.
2.4) which results in smoother and rounded sidewalls. Fig. 2.3 (a) shows the atomic force
microscope (AFM) profile of the waveguide surface as obtained using the wet etch
process and Fig 2.3 (b) shows the total relative transmitted fiber-coupled power measured
as a function of waveguide width for both the TE and TM input polarizations. Fig. 2.4
shows the corresponding data for the waveguides formed using thermal oxidation.
Here each data point corresponds to the measured power averaged over ten nominally
identical waveguides with a length of 1.5 cm. It can be readily observed that the TE-like
mode has a net low loss weakly dependent upon waveguide width, whereas the TM-like
mode has in general high loss except at some specific values of waveguide widths. For
both the wet etched samples and the thermal oxide case, these low loss waveguide widths
are 0.72 and 1.44 μm.
A simple effective index model was used to calculate the TE and TM modal effective
indices. These were then substituted into Eq. (2.1) to obtain widths of 0.72 and 1.44 μm
for the first two resonances which are in excellent agreement with the measured results,
20
Figure 2.3. (a) AFM image of a waveguide formed by wet-etching (b) Experimentally measured
optical transmission (log scale) for the fundamental TE-like and TM-like modes for the waveguides
as shown in Figs. 2.3 (b) and 2.4 (b). The data of Fig. 2.4 (b) show that the waveguides
with a smooth rounded sidewall as obtained using the thermal oxidation process still
display the TM-like mode lateral leakage loss, but show a broader width dependence at
the loss minima. This is believed to be due to a weaker TM/TE conversion at these
slightly rounded ridge boundaries.
W
15 nm
TE
TM
Waveguide width, W (µm)
Rel
ativ
e Tr
ansm
issi
on (d
B)
W=0.72 µm W=1.44 µm
(a)
(b)
21
Figure 2.4. (a) AFM image of a waveguide formed using thermal oxidation (b) Experimentally
measured optical transmission (log scale) for the fundamental TE-like and TM-like modes for the
waveguides
The data presented in Figs. 2.3 (b) and 2.4 (b) exhibit some subtle characteristics
which warrant further investigation. To gain comprehensive insight into the behavior of
the TM-like mode in shallow ridge waveguides, a full vector numerical analysis of these
structures was carried out.
W
15 nm
TE
TM
Waveguide width, W (µm)
Rel
ativ
e Tr
ansm
issi
on (d
B)
W=0.72 µm W=1.44 µm
(a)
(b)
22
2.3 Numerical Analysis of Lateral Leakage Loss in Shallow Ridge Straight
Waveguides
To numerically evaluate the leakage loss of the shallow ridge SOI waveguides, full
vectorial mode matching (MM) technique [23, 25] and finite element method (FEM) with
perfectly matched layer (PML) boundary conditions [26, 27] were employed. For the
mode matching simulations the waveguide cross-section was divided into a number of
uniform multilayer sections. In each section, the waveguide field was expressed as a
superposition of normal modes including guided and radiation modes for both TE and
TM polarizations of the corresponding slab. By matching the fields of these sections at
the ridge boundary, the mode of the ridge waveguide can be determined. The propagation
loss of the mode is then calculated from the imaginary part of the mode effective index.
The ridge waveguide is placed vertically between two perfectly conducting planes to
discretize the radiation spectrum but are kept far from the ridge so as to have minimal
impact on the leakage properties. The accuracy of a mode matching simulation depends
strongly on the number of normal modes used in the field expansion. Here, 50 pairs of
TM and TE modes are used in each waveguide section. We now discuss the results
obtained from the numerical simulations[17].
2.3.1 TM-TE Mode Coupling at a Step Discontinuity
Firstly, the mode coupling at a single ridge boundary when the TM fundamental
mode of the waveguide is obliquely incident on the ridge interface from the waveguide
core as illustrated in Fig. 2.5 is considered. The mode matching formulation is applied
23
only on the interface plane at the ridge wall. Fig. 2.6 shows the magnitude of the
reflection/transmission coefficients of the reflected fundamental TM mode and the
reflected/transmitted TE modes as a function of propagation angle (shown in Fig. 2.5) of
the incident TM mode. It can be seen from these plots that there is significant TM-TE
mode coupling at the ridge boundary. Under total internal reflection, not all the power of
Figure 2.5. Mode coupling diagram showing the TM-TE mode coupling at the ridge boundary
Figure 2.6. (Color) Magnitude of the reflected TM mode, reflected and transmitted TE modes as a
function of the incident TM mode propagation angle
205 nm
Air
Si 190 nm
Si02
TM
TETE
TM
x
yz
Core Cladding
°θ
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25
0
4
8
0 20
TM Reflected, ГTMTE Reflected, ГTETE Transmitted, TTE
Propagation Angle, θº
TM R
efle
ctio
n C
oeffi
cien
t
TM→
TE Reflection/Transm
issionC
oefficient
(TTE -Г
TE ) X 1e4
θº
24
the incident TM wave is transferred into the reflected TM wave. A small amount of
power is coupled into TE transmitted and reflected propagating waves.
The transmitted and reflected TE waves are similar, but not identical in magnitude.
The inset of Fig. 2.6 shows the difference in the magnitude between the transmitted and
reflected TE waves. This difference increases as the propagation angle of the incident TM
wave increases. In addition, it can be seen that the coupling coefficients between the TM
and TE waves increase almost linearly with the propagation angle of the incident TM
wave. Fig. 2.7 shows the relative phase of the reflected and transmitted TE waves. We
see that there is a small difference in phase between the reflected and transmitted TE
waves.
Figure 2.7. (Color) Relative phase of the reflected and transmitted TE modes as a function of the
incident TM mode propagation angle
0
20
40
60
80
100
120
140
160
180
200
0 5 10 15 20 25 30
TE ReflectedTE Transmitted
Propagation Angle, θº
Phas
e A
ngle
(º)
25
2.3.2 Width Dependent Propagation Loss
If a waveguide is formed by combining two such ridge interfaces as shown in the
inset of Fig. 2.8, significant loss due to TM-TE mode coupling at the ridge boundaries
can be expected. Based on the above numerical analysis we can now make a small
correction to Eq. (2.1) for the waveguide widths at which TE radiation from the two
interfaces coherently cancels and we have
( )…3,2,1,2
2,
2,
=−
⋅⎟⎠⎞
⎜⎝⎛ Δ+
= mNn
mW
TMeffcore
TEeff
λπφ
(2.2)
where φΔ is the phase difference between the reflected and transmitted TE waves while
other parameters are the same as defined w.r.t. Eq. (2.1). The resonant “magic widths”
calculated from Eq. (2.2) are about 10 nm higher than those obtained from Eq. (2.1).
We simulated the leakage loss due to mode coupling of the TM-like mode in SOI
shallow ridge waveguides using the mode matching formulation and FEM with PML.
The simulated leakage loss of the TM-like mode as a function of the waveguide width of
a straight waveguide is presented in Fig. 2.8 with both methods yielding similar results.
It is evident that the leakage loss depends strongly on the waveguide width. There
exist “magic widths” at which the leakage loss is minimal. The first two magic widths
obtained from the simulations are 0.71 μm and 1.429 μm. These magic widths are in
excellent agreement with the experimental results presented in Sec. 2.2.
At magic widths, perfect leakage cancellation does not occur due to the imperfect
balance of the reflected and transmitted TE waves as shown in the inset of Fig. 2.6. The
26
Figure 2.8. (Color) Simulated loss of TM-like mode in shallow ridge waveguide shown in inset which
shows strong width dependence
leakage loss at the first resonance is higher than that at the second resonance. This trend
agrees well with the experimental results in Figs. 2.3 (b) and 2.4 (b). When the
waveguide width increases, the TM wave inside the ridge region approaches glancing
incidence to satisfy the phase matching condition for the guided mode. This small
propagation angle makes the TM-TE coupling decrease as shown in Fig. 2.6.
Figure 2.9 (a) and (b) show the electrical field distributions of all three vectorial
components of the fundamental TM-like guided mode for waveguides with widths 1.2
and 1.429 μm respectively. The hybrid nature of the mode can be inferred by the presence
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Mode MatchingFEM+PML
Waveguide width, W (µm)
Loss
(dB
/cm
)
AirSiSiO2
W
205 nm 190 nm
27
of the small Ey field component which is ~10X smaller than the dominant Ex field. The
mode also shows the presence of a strong longitudinal field component Ez. Importantly,
the Ey component exhibits strong radiation in the 1.2 μm waveguide case which is
missing at 1.429 μm which is a “magic width”. This correlates with their propagation loss
values shown in Fig. 2.8. Radiation can also be seen for the longitudinal field component
Ez as the TE-like radiation is propagating in the yz plane.
Figure 2.9. Electric field distributions of the fundamental TM-like mode for (a) 1.2 μm and (b) 1.429
μm waveguide widths
(a) (b)
Ex Ex
Ey Ey
Ez Ez
28
2.4 Waveguide Fabrication
Six inch Soitec SOI substrates were used to fabricate the ridge waveguides. The
silicon device layer in the SOI wafers is 205 nm thick with a 2 μm BOX layer
underneath. To minimize any sidewall roughness, the ridges were formed using an ultra
smooth thermal oxidation process [18, 19]. Fig. 2.10 gives a process flow illustrating the
fabrication sequence we employed in this study. To form a waveguide structure with this
technique an appropriate masking material, such as Si3N4, is advantageous because
oxygen diffusion through the nitride layer is negligible, thereby protecting the silicon
underneath. A patterned nitride layer can thus serve as a mask during a high-temperature
oxygen annealing process. This technique not only minimizes side-wall roughness, but
also has the advantage of providing high uniformity across the wafer. Furthermore, if an
oxide film or some other semi permeable layer is incorporated between the Si and Si3N4,
the resulting waveguide profile can be optimized for a given application by adjusting the
oxide layer thickness in order to control the oxygen diffusion through this layer near the
ridge sidewalls. Optical losses are also minimized in this design by the reduced etch
depth into the silicon device layer associated with the ridge waveguide as compared to
wire waveguides, where the silicon device layer is fully etched. This is accomplished
Figure 2.10. Processing steps for shallow ridge waveguide fabrication
PECVD SiO2 and Si3N4 Mask25 nm Si3N4
100 nm SiO2 c-Si 205 nm
Thermal Oxidation HF Strip
29
simply by controlling the oxidation time, which gives a precise control on the height of
the waveguide ridges compared to traditional etching processes. It should be noted that
reducing the waveguide optical loss of the ridge waveguide is the first step in achieving
low loss slot structures.
The detailed fabrication process for the ridge waveguide formation started with the
deposition of a 100 nm plasma-enhanced chemical vapor deposited (PECVD) SiO2
followed by a 25 nm thick PECVD Si3N4 layer. A 1 μm thick AZ703 resist was spin
coated onto the wafer and exposed with a 1X projection printer. A 6:1 ammonium
fluoride: hydrofluoric acid (NH3F:HF) solution was used to transfer the pattern into the
nitride/oxide layers. Once the pattern was formed, the resist was stripped and the
resulting structure was then oxidized at 975 °C in O2 for 17 min to form 13 nm ridges.
The residual mask and the thermally formed SiO2 were then stripped with HF, resulting
in a waveguide structure with very smooth silicon surface and side wall, as shown in the
atomic force microscope (AFM) trace in Fig. 2.11 (a). The RMS surface roughness was
measured to be less than 0.2 nm, which is an order of magnitude less than that typically
Figure 2.11. (a) AFM trace of a 1.44 μm ridge waveguide with <0.2 nm surface roughness (b) Quasi-
planar ridge SOI waveguide geometry. BOX thickness is 2μm
(a) (b)
BOX
WSi 192 nm 13 nm Ridge
Si Substrate
30
obtained with conventional etching processes. After the ridge waveguides were formed
with the process described above, a layer of resist was applied to protect the surface
during dicing and end facet polishing of the resulting chips. Finally, the protective resist
layer was removed from the finished chips using acetone and alcohol. Fig. 2.11 (b) shows
the sectional view of the final fabricated structure.
2.5 Waveguide Loss Measurement Technique
To accurately quantify the losses of our waveguides, we use the resonance
characteristics of under coupled ring resonators [24, 28]. Fig. 2.12 shows the schematic
of the ring resonator circuit formed using the ridge waveguide described above and the
full width at half-maximum (FWHM) of the drop port intensity response νπω Δ=Δ 2
Figure 2.12. Setup to measure the resonance characteristics of (a) a weakly coupled ring resonator
with separate thru and drop ports and (b) an exemplary drop-port response
(a)
(b)
Tunable Laser1460-1580 nm
Polarization controller
Six axis Piezo Stage
Six axis Piezo Stage
Ge Detector900-1700 nm
Optical Power meter
Couplers
0
0.2
0.4
0.6
0.8
1
-2 -1 0 1 2
Drop Port Response
Frequency Offset (GHz)
Dro
p P
ort R
espo
nse
(arb
.)
νΔ
31
as measured by scanning the input excitation frequency across the resonance.
The ring configuration in Fig. 2.12 (a) has separate through and drop ports and
consists of two directional couplers that are designed to be under coupled so that the ring
losses are dominated by the waveguide losses rather than the input/output coupling of the
ring to the thru-port and drop-port waveguides. Using the nomenclature of Yariv[29], the
expression for line width for our structure is
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −=Δ
21
2112
tt
ttLrt
g
ττν
ω (2.3)
where gν is the group velocity, rtL is the round-trip length, τ is the field transmission of
the waveguide for one round trip of the ring, and 21 tt , are the field transmission
coefficients of the directional couplers. The couplers are assumed to be lossless, giving
122 =+ κt where κ is the field coupling coefficient. To simplify the expression in Eq.
2.3, we can write ( )2/rtLexp ατ −= with α being the waveguide intensity attenuation
coefficient and if we let ( )211 /rtLexpt α−= and ( )222 /rtLexpt α−= , then for the case
of low loss ( )1≈τ and weak coupling ( )121 ≈≈ tt , we obtain the simple expression
grt ναω =Δ (2.4)
The round-trip loss rtα is the sum of the waveguide propagation losses α and the
coupling losses from the two directional couplers ( 1α and 2α ), but, if the ring is designed
to be weakly coupled, then gναω =Δ . Since precision frequency scans allow for an
accurate determination of resonance width, this provides an unambiguous and
conservative figure for waveguide transmission losses. If desired, the weak contributions
from coupler losses can be removed for a correction.
32
Loss measurements were conducted using an external cavity laser diode with a line
width of <10MHz with continuous single-longitudinal mode and very fine-tuning
capability. The test setup is shown in Fig.2.12. The laser output was fed through a
polarization controller and tapered fibers controlled by six-axis piezo translation stages
were used to couple the signal into and out of the waveguide ports. The output from the
waveguide was detected by a power meter, followed by analog to digital conversion
allowing computerized data collection. The shape of the resonance profile was recorded
by scanning the laser across the resonance peaks, as shown in Fig. 2.12 (b), allowing for
the determination of the FWHM line widths and corresponding resonator losses.
2.6 Results and Discussion
Fig. 2.13 shows the average TE and TM drop-port response for a 400 μm radius ring
resonator with a waveguide “magic width” of 1.43 μm in the ring as well as in the
straight and coupler sections. Both of these responses were obtained on the same device.
While the waveguides have very strong birefringence, with calculated phase index values
of nTE = 2.768 and nTM =1.740, the group index values are nearly identical due almost
entirely to the strong structural waveguide dispersion. The measured group index values
were ng-TE =3.721 and ng-TM = 3.847 for the TE and TM modes, respectively, as
determined from the free spectral range of the resonators, and agree fairly well with
calculated values including several percent corrections from silicon material dispersion.
Using these measured group index values, the measured line widths of ΔνFWHM-TE =
124.37 MHz and ΔνFWHM-TM = 286.33 MHz correspond to losses of 0.42 dB/cm and 1.00
33
Figure 2.13. Drop-port responses for the TE and TM modes for a 400 μm radius ring resonator with
“magic width” waveguide of 1.43 μm
dB/cm for the TE and the TM modes, respectively. As noted in Sec 2.5, this provides a
conservative figure because all the loss is ascribed to propagation. Conventional bending
losses are readily estimated analytically and for the waveguide design used in this
experiment, are calculated to be <0.03 dB/cm for TE modes and orders of magnitude
smaller for TM modes and have been, therefore, neglected. The couplers in the device are
fabricated with a straight waveguide running tangentially to the ring with the gap
between the edges of the ridge at the closest point being 1.87 μm. The coupling
coefficient for this design was evaluated using the two dimensional RSOFT BeamPROP
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1537.15 1537.16 1537.17 1537.18 1537.19 1537.2Wavelength (nm)
Dro
p po
rt re
spon
se (a
rb.)
QTE=1.57x106
ΔνFWHM,TE=124.37 MHzαTE=0.42 dB/cm
QTM=6.81x105
ΔνFWHM,TM=286.33 MHzαTM=1.0 dB/cm
400 µm Ring, 1.43 µm “Magic width” Ridge Waveguide
34
program[30], which yielded a coupling coefficient of 0.18% for the TE mode and 0.19%
for the TM mode per coupler. Using these computed values, a correction can be included
for the two couplers combined, with a contributed loss of 0.062 dB/cm for the TE mode
and 0.066 dB/cm for the TM mode. This results in waveguide propagation losses of 0.36
dB/cm and 0.94 dB/cm and for the TE and TM modes, respectively[19].
A variation in the loss values with wavelength for the TM mode was observed as
expected due to the “magic width” effect. Since the actual fabricated width may not
precisely correspond to the “magic width”, the wavelength was tuned in these
measurements in order to minimize the FWHM of the drop port responses and the
minima were obtained at 1537 nm. We also saw small variations of successive
resonances exhibiting ±10% differences in the peak widths about the average loss values
quoted above, which might be defect related and are under investigation.
Additionally, while the TM loss is quite low, it is not as low as the TE loss even at the
“magic width” values. This excess loss may be due to additional anomalous radiation
losses resulting form imperfect radiation cancellation in curved waveguides even at
“magic widths” and is discussed in chapter 3.
The corresponding Q values for these ring resonators are 1.6X106 and 6.8X105 and
for TE and TM, respectively[19]. To our knowledge, these are the lowest published loss
values for TE and TM mode SOI waveguides with ultrahigh vertical confinement designs
suitable for active device configurations and optimized sensors.
35
2.7 An Ultra Compact Waveguide polarizer based on “Anti-Magic Widths”
A polarizer is a critical component in photonic integrated circuits (PICs) for coherent
communication, optical signal processing, switching networks and sensor applications.
With the goal of achieving high modal extinction ratios in minimal device sizes, various
polarizer designs based on birefringent crystals, metal clad waveguides, anisotropic
overlays, proton exchanged waveguide sections, photonic crystal and liquid crystal have
been proposed and demonstrated. These designs rely on utilization of more complex
materials and fabrication techniques, and in some cases have very stringent fabrication
tolerances. In the silicon-on-insulator (SOI) platform, for example, it is desirable to have
a polarizer design based on the existing standard fabrication sequences that is capable of
providing high polarization-dependent loss for μm-scale devices. Such a component may
be important, for example, in providing very high extinction of undesired residual TM
light in PIC’s using polarization diversity where both incoming polarizations are mapped
into TE modes on the chip.
Here we propose a very simple yet extremely efficient waveguide polarizer design
based on the lateral leakage phenomenon in shallow ridge SOI waveguides discussed
above[20]. It is commonly understood that TM modes in such structures are often leaky,
and it is therefore often asserted that PIC’s designed with ridge guides are “TE only”.
However, when etched deeply enough, ridge guides can carry TM light as a bound mode
without radiation loss. Furthermore, as we illustrated above, even for shallow ridge
designs, TM mode leakage can be mitigated at waveguide widths where the radiative
leakage destructively interferes [16, 17] (so-called “magic widths”), leading to TM losses
below 1 dB/cm. We now discuss in more depth how this loss can be engineered with
36
width control to provide advantageous device characteristics. In particular, we illustrate
that waveguide widths selected to provide constructive interference of radiation can
provide extremely large TM radiation loss, thereby providing an ideal polarizer fabricated
using simple, and non-critical, dimension control of the ridge. Rigorous analysis shows
that the structure[20] is capable of achieving significant improvements in polarizer
performance over prior designs.
2.7.1 Polarizer Design Principle and Analysis
Referring back to Fig. 2.8, illustrating the fundamental TM mode loss in shallow
ridge waveguides, we see that barring a few precise waveguide widths-“magic widths”,
the TM mode loss is in general quite high at most waveguide widths. This leakage loss
occurs because the effective index of the TM mode of the waveguide (nTM-WG) is less than
that of the TE slab mode in the lateral cladding (nTE-CLAD). The TM mode can therefore
phase match to the TE slab mode propagating at a particular angle relative to the
waveguide axis, causing the waveguide TM mode to leak in to the TE slab mode in the
cladding. At the “magic width”, there is destructive interference of radiation wave fronts
resulting in low loss for the TM mode [16, 17]. It is also critical to note that the TE mode
of the waveguide in these structures is a perfectly guided mode due to its higher index
value and always has low loss.
This waveguide width dependent loss mechanism of the TM mode in SOI ridge
waveguides is the basis of our efficient TE-pass polarizer. While the destructive
interference at the “magic width” inhibits radiation loss, widths where the radiation
37
components constructively interfere, referred to as “anti-magic widths,” will yield very
high TM loss as illustrated in Fig. 2.8 while leaving the TE mode unaffected.
With the goal of maximizing the TM mode loss, we performed a rigorous full vector
analysis of the SOI ridge waveguide similar to the one shown in the inset of Fig. 2.8
using analytical mode matching and finite element techniques. The analyzed structure
now has 220 nm of crystalline silicon (c-Si) in the core, with R nm as the ridge height
and is shown in the inset of Fig. 2.14 (a). Therefore thickness of c-Si in the cladding is
(220-R) nm and W μm is the waveguide width.
Fig. 2.14 (a) shows the XY view of a 3D plot of loss of the fundamental TM mode of
this ridge waveguide as a function of waveguide width and ridge height as obtained using
mode matching method. The white portion on the top left corner in the plot corresponds
to the region where the TM mode is at cut off due to the cessation of total internal
reflection within the waveguide. The cyclic behavior of the TM mode loss as a function
of waveguide width at a particular ridge height is clearly apparent. The simulation results
indicate that the fundamental TM mode suffers a loss as high as 34,000 dB/cm for
waveguide widths in the vicinity of 0.4 μm and 20,000 dB/cm for waveguides around 1.1
μm for a ridge height of 150 nm as can be clearly seen from Fig. 2.14 (b), which is the
XZ view of the 3D loss plot. This indicates that with a 10 μm long waveguide, a 34 dB
extinction ratio is achievable which is 30X improvement over the theoretical design for a
SOI TE-pass waveguide polarizer presented in[31]. The present result is also in the
league of an ultra-compact TM-pass silicon nanophotonic waveguide polarizer analyzed
and proposed in [32] with 26 dB extinction ratio over a 10 μm device length.
38
Figure 2.14. (a) XY view and (b) XZ view of the 3D plot of loss of the fundamental TM mode of a SOI
ridge waveguide with 220 nm c-Si in the core (inset)
(a)
(b)
Waveguide width, W (µm)
Rid
ge H
eigh
t, R
(nm
)Loss dB/cm
x104
220 nmRidge, R nm
WAirSi
SiO2
Waveguide width, W (µm)
Loss
(dB
/cm
)
x104
39
As discussed above, the high loss for the TM mode in these SOI shallow ridge
waveguides occurs because nTM-WG < nTE-CLAD. It is therefore anticipated that as the ridge
height is increased, a point will be reached where nTM-WG > nTE-CLAD hence the TM-like
mode will be no longer leaky and becomes purely guided. To show that this is indeed the
case, Fig. 2.15 depicts the loss and effective index of the TM mode as a function of
waveguide width for three ridge heights, (a) 90 nm (b) 148 nm and (c) 150 nm calculated
using FEM-PML and mode matching technique. It is evident that both models give
identical results. For the 90 nm ridge height, the entire waveguide width span of 0.1 to 3
μm exhibits lateral leakage loss as nTM-WG < nTE-CLAD (= 2.375) for this width range.
However, as we increase the ridge height to 148 nm for which nTE-CLAD =1.85 (indicated
in the plot), there are regions where nTM-WG > nTE-CLAD causing TM mode to become a
purely guided with zero lateral leakage loss in these width spans. Similar observations are
made for the 150 nm ridge which has nTE-CLAD =1.829. As nTM-WG of the waveguide
(a)
1.4
1.5
1.6
1.7
1.8
1.9
0.0E+00
2.0E+03
4.0E+03
6.0E+03
8.0E+03
1.0E+04
1.2E+04
1.4E+04
0.1 0.6 1.1 1.6 2.1 2.6
MM LossFEM LossMM IndexFEM Index
Loss
(dB
/cm
)
Waveguide width, W (µm)
TM0 Index
90 nm Ridge
40
Figure 2.15. Loss and effective Index nTM-WG of the fundamental TM-like mode of a SOI ridge
waveguide with 220 nm c-Si in the core for (a) 90 nm (b) 148 nm and (c) 150 nm ridge heights
(b)
(c)
1.4
1.5
1.6
1.7
1.8
1.9
0.0E+00
5.0E+03
1.0E+04
1.5E+04
2.0E+04
2.5E+04
3.0E+04
3.5E+04
0.1 0.6 1.1 1.6 2.1 2.6
Loss
(dB
/cm
)
Waveguide width, W (µm)
TM0 Index
148 nm Ridge
nTE-CLAD
1.4
1.5
1.6
1.7
1.8
1.9
0.0E+00
5.0E+03
1.0E+04
1.5E+04
2.0E+04
2.5E+04
3.0E+04
3.5E+04
4.0E+04
0.1 0.6 1.1 1.6 2.1 2.6
Loss
(dB
/cm
)
Waveguide width, W (µm)
TM0 Index
150 nm Ridge
nTE-CLAD
41
increases with waveguide width for a given ridge height, it is clear that the condition nTM-
WG < nTE-CLAD will start failing at larger waveguide widths first. As the ridge height is
further increased incrementally beyond 150 nm, loss maximas for the TM mode will first
vanish at 1.1 μm and then at 0.4 μm.
The TE mode in our structure is a perfectly guided mode, unlike the design in [31]
where the TE mode is also a leaky mode and can compromise the extinction ratio. In the
present design the extinction ratio also scales with the waveguide length as the TM mode
leaking into the TE slab mode propagates laterally away from the waveguide. This
considerably eases the design of the polarizer compared to the one in [32] where the TE
mode leaks into the substrate and gets reflected back into the waveguide leading to
particular lengths that maximize the extinction ratio. It is also clear from the Fig. 2.14(b)
that a high loss for the TM mode can be obtained over reasonably wide range of
waveguide widths in the vicinity of the “anti-magic width,” thereby easing fabrication
related tolerances significantly compared to other polarizer designs. The optimum
polarizer design should target narrower waveguide widths (~ 0.4 μm) and a deeper ridge
height that is slightly less than the ridge height beyond which TM mode is no longer
leaky. This regime maintains single mode operation for both TE and TM modes and also
achieves the highest loss for the TM mode.
The fact that at any given ridge height both “magic widths” and “anti-magic widths”
exist with an extraordinary level of TM mode loss disparity between the two provides a
high degree of design flexibility without any associated fabrication complexity. This
polarizer can conceivably follow an imperfect TM to TE polarization rotator, with an
42
adiabatic waveguide taper connecting the two sections, causing a very strong reduction in
polarization cross talk levels.
In conclusion, we have proposed and analyzed an ultra-compact SOI based TE-pass
polarizer utilizing the programmable loss characteristics of the leaky TM mode in ridge
waveguides. Analysis shows that the presented structure, which is both simple and
fabrication tolerant, can achieve a multifold improvement in polarizer performance over
prior designs. The ultimate performance of such a polarizer will of course depend on the
proper engineering of follow-on waveguides or other PIC components to avoid re-capture
of the radiated light, a topic which is under analysis.
43
Chapter 3. Anomalous Losses in Curved Waveguides and Directional
Couplers at “Magic Widths”
We have seen that shallow ridge waveguides operating in TM mode, which is critical
for our application, experience severe inherent lateral leakage loss. This was overcome by
controlling the width of the waveguides to “magic widths” at which the radiation loss
components experience interferometric cancellation [15, 16, 19]. However, it is possible
that the rings, bends and directional couplers, which are part of the ring resonator
architecture we use to measure the waveguide losses, might be significantly impacting
the effectiveness of the “magic widths” as it relies on precision destructive interference of
the radiation wave fronts. Understanding their effects on “magic width” is important also
to facilitate the incorporation of these waveguides into practical functional subsystems
like resonators, add-drop filters and active components like modulators, switches and
emitters. Here we investigate the impact of waveguide curvature[33] and directional
coupling [17, 34]on the “magic width” phenomenon using full vector analytical mode-
matching and finite element modeling techniques. We show here new anomalous losses
over and above any previously investigated or published results, and thus show that
additional design constraints have to be imposed to achieve highest performance devices.
44
3.1 Lateral Leakage of TM-like Mode in Thin Ridge SOI Ring
Figure 3.1 shows the cross-sectional (Fig. 3.1 (a)) and plan view (Fig. 3.1 (b)) of a
section of a ring resonator formed by a thin-ridge SOI waveguide. Consider a simplified
ray-tracing view of wave propagation. The rays of the guided TM-like mode of the ring
are shown in Fig. 3.1 (b). In general, when the rays of a guided mode are incident on the
waveguide boundary, they are totally internally reflected. However, as has been shown
previously [16, 17] in Sec. 2.3.1, when a TM ray of a thin-ridge SOI waveguide is
incident on a waveguide boundary, transmitted and reflected TE rays are generated in
addition to the reflected TM ray due to strong TM-TE mode coupling at the waveguide
ridge boundary. On the outer waveguide boundary, the transmitted TE ray (TTE1)
propagates away from the ring, while the reflected TE ray (RTE1) propagates across the
Figure 3.1. (a) Cross section, and (b) plan view and mode coupling diagram of a SOI thin-ridge ring
resonator. Waveguide dimensions are shown
190 nm Si
15 nm Ridge
BOX
xφ
r
RW
+
R
W
TETM
TTE2
RTE1
RTE2TTE1
(a)
(b)
45
waveguide, intersecting the inner waveguide boundary where it is largely transmitted.
The reflected TE ray (RTE1) then traverses a secant across the ring, intersecting with the
inner and outer waveguide boundaries again, and then propagates away from the
waveguide mostly unaltered. On the inner waveguide boundary, the incident TM ray
generates reflected TE (RTE2) and transmitted TE (TTE2) rays. The reflected TE ray (RTE2)
propagates across the waveguide to radiate away from the ring. The transmitted TE ray
(TTE2) traverses a secant across the ring, intersecting the inner and outer waveguide
boundaries and propagates away unaltered. The TM guided mode suffers from high
leakage loss due to power coupling to TE radiation at the two waveguide boundaries.
Based on the above ray model, at any point outside the ring, there are four different
TE rays: transmitted (TTE1) and reflected (RTE1) TE rays generated from the TM ray
incident on the outer waveguide boundary, and the transmitted (TTE2) and reflected (RTE2)
TE rays generated from TM-TE mode coupling on the inner waveguide boundary. As the
relative phases of all these TE rays depend on both the ring radius and waveguide width,
it is possible that there might exist some combinations of waveguide width and ring
radius for which all four of these TE waves interfere destructively outside the ring,
resulting in cancellation of the lateral leakage radiation.
Inside the ring, there are reflected (RTE1) TE rays generated from the outer waveguide
boundary and transmitted (TTE2) TE rays generated from the inner waveguide boundary.
The relative phase between these TE rays depends primarily on the waveguide width. For
some waveguide widths, these TE rays can interfere constructively to generate a strong
TE field inside the ring. Such widths will be called “anti-magic widths”. On the other
46
hand, if the waveguide is at a “magic width”, the TE rays inside the ring will be out of
phase, resulting in minimization of the TE field inside the ring.
3.2 Rigorous Simulation Approaches
To rigorously model the TM-like modes of thin-ridge bent waveguides, a full
vectorial mode matching technique [25] in cylindrical coordinates [35] was employed. In
the mode matching technique, the waveguide cross-section is divided into a number of
radially uniform sections. Each section corresponds to a multi-layer slab. In each section,
the waveguide mode field was expanded into a superposition of the TE and TM normal
modes of the corresponding slab waveguide. The amplitudes of slab normal modes in
each section are the solutions of Bessel equations. By matching the fields of these
sections at the vertical interfaces between two adjacent regions, the modes of the
waveguide can be determined. The propagation loss of each mode was then calculated
from the imaginary part of the complex azimuthal propagation constant. The calculated
loss includes both the lateral leakage and conventional bending loss.
In the mode matching implementation, to avoid treatment of the continuum of the
radiation modes of each section, two perfectly conducting planes were introduced above
and below the waveguide [23, 35]. The positions of these conducting planes were chosen
to be sufficiently far away from the Si core so that they do not affect the waveguide loss.
In order to achieve adequate accuracy, a sufficiently large number of normal modes
which include guided and radiation modes in both the silica and air claddings must be
included in the field expansion. Here 50 pairs of TM and TE normal modes were used in
47
each waveguide section. There is no boundary imposed on the lateral direction and
therefore, the mode matching simulation can accurately model the lateral leakage in thin-
ridge SOI waveguides.
To validate the mode matching results, the structure was also simulated using a finite
element model (FEM)[26] with cylindrical perfectly matched layer (C-PML) boundary
conditions; developed in house for full vectorial analysis of an axi-symmetric structure.
The FEM model utilized the COMSOL[27] FEM engine along with its geometry
definition, meshing and post-processing features.
3.3 Numerical Results and Discussion
This section presents the numerical results of the analysis of the lateral leakage of the
TM-like mode in thin-ridge SOI bent waveguides and ring resonators. The waveguide
dimensions are shown in Fig. 3.1 (a).
3.3.1 The Impact of Waveguide Curvature on the Leakage Cancellation Without
Re-Entry
The effectiveness of the leakage cancellation at “magic widths” is limited by the
imperfect balance of the generated reflected and transmitted TE waves at the waveguide
boundaries [17] (Sec. 2.3.1). It is expected that the imbalance in the amplitude of the
generated TE waves is further enhanced by the waveguide curvature of bent waveguides.
48
The impact of the waveguide curvature on the leakage cancellation at the “magic widths”
is investigated here.
To isolate the effect of the waveguide width, the secant TE waves inside the ring
were absorbed so that they do not re-enter the waveguide region. For the mode matching
simulation, this was done by forcing the amplitudes of the TE waves inside the ring to
take the form of Hankel functions instead of Bessel function of first kind as in [35]. In the
FEM simulation, a C-PML layer was placed inside the ring to absorb the secant TE
waves. It should be noted that by preventing the TE waves from re-entering the
waveguide region, we are effectively simulating the loss characteristics of a section of a
ring resonator i.e. a bent waveguide structure. Fig. 3.2 (a) shows the simulated
propagation loss of a thin-ridge bent waveguide as a function of waveguide width for
different bend radii without considering re-entrant TE waves. The results for a straight
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.3 1.35 1.4 1.45 1.5 1.55 1.6
100200400600Straight
Waveguide width, W (µm)
Loss
(dB
/cm
)
Ring Radius, R (µm)
(a)
49
Figure 3.2. (Color) Simulated propagation loss of the TM-like guided mode of a thin-ridge SOI bent
waveguide in the absence of secant TE waves inside the ring (a) as a function of waveguide width for
different bend radii and (b) as a function of the bend radius when the waveguide width is fixed at
1.43 μm. Conventional bending losses are also shown.
waveguide [17] are also shown in Fig. 3.2 (a) which has a “magic width” of 1.43 μm. The
results obtained from the mode matching and FEM simulations are in good
agreement[33]. The two simulation approaches make significantly different
approximations. The FEM simulation assumes a limited simulation domain and a finite,
numerically implemented PML. The mode matching simulation considered only a limited
number of normal modes of the slab regions. Both simulation approaches produced the
same solution, proving that the solution is not a result of either of these approximations
and hence is likely to be a valid representation of the actual waveguide behavior. It is
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
100 200 300 400 500 600 700 800 900 1000Ring Radius, R (µm)
Loss
(dB
/cm
)
Total Loss
Conventional Bending Loss
(b)
50
clear that all waveguides with different bend radii show the same “magic width” which is
identical to that of a straight waveguide at 1.43 μm. However, the propagation loss
increases as the bend radius is decreased. Fig. 3.2 (b) shows the propagation loss
calculated for different bend radii with the waveguide width fixed at the “magic width” of
1.43 μm. Also shown is the conventional bending loss for these curved waveguide
sections. It is evident that the conventional bending losses are too small to account for the
predicted propagation losses in these curved waveguides. Even at a bend radius of 100
μm the conventional bending loss of a bent waveguide with equivalent refractive index
contrast is only 0.13 dB/cm, which is much smaller than the loss of ~2 dB/cm obtained
from the simulations. It is thus evident that the radius of a bent waveguide has a
significant impact on the effectiveness of the leakage cancellation at the “magic width”.
As shown in Sec. 2.3.1, the leakage cancellation is affected by the imbalance of the
reflected and transmitted TE waves generated from TM-TE mode coupling. For a bent
waveguide, the inner and outer waveguide boundaries will have different radii and thus
the reflections from these boundaries will differ significantly. This will exacerbate the
imperfect cancellation at the “magic width”. The amplitude difference between the
radiated TE waves increases as the bend radius reduces. Therefore, the leakage
cancellation is less effective with smaller radius as seen in Fig. 3.2.
3.3.2 Leakage Cancellation in Waveguide Rings with Re-Entry: “Magic Radius”
Phenomenon
Referring back to Fig. 3.1 (b), when the reflected TE wave from the outer boundary
(RTE1) and transmitted TE wave (TTE2) from the inner boundary traverse a secant across
51
the ring and intersect with the waveguide again, it is possible for these TE waves to
interfere destructively with the transmitted TE wave (TTE1) from outer boundary and
reflected TE wave (RTE2) from inner boundary resulting in low loss propagation for the
TM-like mode. The effect of the secant TE waves inside the ring was simulated by
allowing them to re-enter the waveguide region. For the mode matching simulation, this
was done by setting the form of the TE slab modes of the inner section to be Bessel
functions of the first kind. For the FEM simulation, the C-PML inside the ring was
removed and a PEC boundary was placed significantly far away from the inner boundary
of the waveguide so that this PEC boundary does not interfere with the TE waves inside
the ring as illustrated in Fig. 3.3. The waveguide width was fixed at the “magic width” of
1.43 μm. Simulations were conducted for rings with radii in a range of ±3 μm around
nominal radii R0 = 100, 200, and 400 μm. In the FEM, due to the necessity of a large
simulation domain and hence proportionately high computational resources and time
requirements for accurate modeling of the loss characteristics, simulations were
performed only for the R0 = 100 μm case.
Figure 3.3. Simulation domain used to model rings with R0 = 100 μm in FEM
SiO2
Si
Radius, R=100 µmW
C-PML
C-PML
Air
89.5 µm
14 µm
70 µm
1.5 µm
C-PML
52
Figure 3.4 shows the propagation loss of the TM-like mode as a function of the ring
radius. Both mode matching and FEM simulations produced very similar results. It can
be seen that although the waveguide width is at the “magic width” of the straight
waveguide, perfect leakage cancellation and hence low loss does not occur at all ring
radii. Complete leakage cancellation only occurs at specific “magic radii” for which the
secant TE waves interfere destructively with TE waves outside the ring. The loss of the
TM-like mode shows a cyclic dependence on the ring radius.
The propagation loss shown in Fig. 3.4 includes both lateral leakage loss due to TM-
TE coupling and the conventional bending loss. For large rings, at “magic radii”, the
Figure 3.4. (Color) Propagation loss of the fundamental TM-like mode of a ring as a function of ring
radius. Waveguide width is W = 1.43 μm
1.E-02
1.E-01
1.E+00
1.E+01
-3 -2 -1 0 1 2 3R-R0 (µm)
Loss
(dB
/cm
)
R0200 µm
R0400 µm
R0100 µm
R0100 µm
FEM
53
propagation loss approaches zero due to near-perfect leakage cancellation and negligible
bending loss. However, for smaller rings e.g. R0 = 100 μm, the losses at the “magic radii”
are limited by conventional bending losses. This effect can be seen clearly in Fig. 3.4 by
comparing the propagation losses at the “magic radii” for the R0 = 100 μm ring case with
the conventional bending loss for R0 = 100μm. The small difference between the bending
loss and the total loss at “magic radii” can be attributed to the imperfect leakage
cancellation resulting from imbalance in the amplitudes of the reflected and transmitted
TE waves for small rings as discussed above.
We now consider the dependence of the lateral leakage loss on both the ring radius
and waveguide widths other than the “magic width” of the straight waveguide. Fig. 3.5
shows the variation of the propagation loss with waveguide width for a number of ring
radii R. Also shown is the loss for a straight waveguide. Again, good agreement is
achieved between the results obtained from the mode matching and FEM simulations. It
can be seen that the loss depends strongly on both the waveguide width and ring radius.
Considering the loss at a waveguide width of 1.43 μm, which is the optimum width to
achieve lowest loss for a straight waveguide. For bent waveguides, the loss has a local
minimum at 1.43 μm, but significantly lower losses can now be achieved at other
waveguide widths.
It is instructive to combine the results of Figs. 3.4 and 3.5 into a single three
dimensional plot showing the loss as a function of both ring radius and waveguide width
as shown in Figs. 3.6 (a) and (b) for rings with radius around 200 μm and 400 μm,
respectively. The cyclic dependence of propagation loss with ring radii at a particular
waveguide width is clearly visible and so is the presence of multiple low loss waveguide
54
Figure 3.5. (Color) Propagation loss of the fundamental TM-like mode of a ring resonator as a
function of the waveguide width for different ring radii
widths for a ring with particular radius. For any waveguide width, there exist multiple
“magic radii” at which the propagation losses are low. It can be seen from Fig. 3.6 that it
is best to design a ring with waveguide width equal to the “magic width” of a straight
waveguide and then select the ring radius to be one of the “magic radii” at this waveguide
width. The sensitivity of the propagation loss to waveguide width and radius variations at
these “magic radius” - “magic width” combinations is lowest.
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.3 1.35 1.4 1.45 1.5 1.55 1.6
100 100 (FEM)200 300400 Straight
Waveguide Width (µm)
Loss
(dB
/cm
)Radius, R (µm)
55
Figure 3.6. Propagation loss of the fundamental TM-like mode of a ring as a function of the ring
waveguide width and ring radius. Radius around (a) 200 μm and (b) 400 μm
(a)
(b)
Waveguide Width (µm)
Rin
g R
adiu
s, R
(µm
) Loss, Log10(dB/cm
)
Waveguide Width (µm)
Rin
g R
adiu
s, R
(µm
) Loss, Log10(dB/cm
)
56
3.3.3 Mode Field Distribution
Using the mode matching simulation, the mode profile of the TM-like guided mode
was also calculated. Fig.3.7 (a) and (b) show the vectorial components of the electric
field distributions of the TM-like mode of a ring with a waveguide width fixed at 1.35
μm and radius of 199.62 μm (high loss) and 198.95 μm (low loss), respectively. Similar
field distributions were also obtained from FEM simulations. Comparing the vertical
component (Ex), it can be seen that the TM-like component is almost identical for both
ring radii. Similar to straight waveguides, there is a significant radial component (Er) of
TE-like mode despite the mode being TM-like. Inside the ring, the radial field
components are very similar for both ring radii. At an “anti-magic width” (w = 1.35μm),
the generated TE-waves inside the ring interfere constructively to give strong radial field
inside the ring regardless of whether the mode is high or low loss. Outside the ring,
however, there is a significant radial field component at the “anti-magic radius” of 199.62
μm. For the ring radius of 198.95 μm which is a “magic radius” for 1.35 μm waveguide
width, destructive interference of TE waves outside the ring result in the cancellation of
the radial field component outside the ring. The azimuthal field component (Eφ) also
shows similar radiation behavior as the radial field component.
Next, we consider the field distribution when the waveguide width is at the “magic
width” of the straight waveguide i.e.1.43 μm. Fig. 3.8 (a) and (b) show the field
distributions of the TM-like modes with radius of 199.84 μm (“anti-magic radius”) and
200.46 μm (“magic radius”) respectively. Comparing the field amplitudes of the radial
and azimuthal components inside the rings in Fig. 3.7 and Fig. 3.8, it is clear that at the
57
“magic width”, these field components inside the ring are significantly reduced due to the
destructive interference of TE waves generated from two waveguide boundaries.
However, unlike the case of a straight waveguide, the cancellation is not perfect due to
the impact of the waveguide curvature as discussed in Sec. 3.3.2. As a result there is still
a small TE component inside the ring. At the “magic radius”, the radiation field outside
the ring is further suppressed resulting in low loss propagation.
Figure 3.7. Electric field distributions of the fundamental TM-like mode of a ring with a waveguide
width of 1.35 μm and radius at (a) 199.62 μm (high loss) and (b) 198.95 μm (low loss)
(b) (a)
Ex Ex
Er Er
Eφ Eφ
r (µm) r (µm)
58
Figure 3.8. Electric field distributions of the fundamental TM-like mode of a ring with a radius of (a)
199.84 μm (high loss) and (b) 200.46 μm (low loss). Waveguide width of 1.43 μm
3.3.4 Wavelength Dependence of Lateral Leakage Loss
The “magic width” in straight waveguides has a strong dependence on the
wavelength. Similar dependency is also expected from the lateral leakage loss of the TM-
like mode of a thin-ridge SOI bent waveguide. Using the mode matching simulator, we
calculated the wavelength dependence of the propagation loss for rings with a waveguide
width of 1.43 μm (“magic width”) and radii of 400 μm and 200.5 μm, and a waveguide
(a) (b)
Ex Ex
Er Er
Eφ Eφ
r (µm) r (µm)
59
width of 1.35 μm (“anti-magic width”) and radii of 399.02 μm and 198.95 μm. The
chosen radii are the “magic radii” for the corresponding waveguide widths at λ = 1.55
μm. Fig. 3.9 (a) and (b) shows the propagation loss as a function of wavelength for
waveguides with widths of 1.43 μm and 1.35 μm, respectively. It is evident that the
propagation loss depends strongly on the wavelength. The “magic width/magic radius” at
a given wavelength may become “anti-magic width/anti-magic radius” when the
operating wavelength changes. Thus, when designing a ring resonator, care should be
taken to engineer a ring which exhibits low loss at the required resonant wavelength. The
wavelength dependent behavior of the propagation loss also presents an opportunity to
eliminate unwanted out of band resonances.
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.545 1.550 1.555 1.560 1.565 1.570 1.575 1.580 1.585
R=400 µm
Wavelength (µm)
Loss
(dB
/cm
)
R=200.5 µm
(a)
60
Figure 3.9. Wavelength dependent loss of the TM-like mode in thin-ridge SOI rings with a
waveguide width of (a) 1.43 μm and (b) 1.35 μm and ring radii as shown
3.4 Directional Coupler
We also investigated the impact of “magic width” radiation cancellation in directional
coupler geometries. A directional coupler can be formed by two parallel straight
waveguides, a cross sectional view of which is shown in the inset of Fig. 3.10 (a). The
behavior of a directional coupler can be rigorously represented as a superposition of the
fundamental and first order supermodes of the structure. Low coupler loss therefore
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.545 1.550 1.555 1.560 1.565 1.570 1.575 1.580 1.585
Wavelength (µm)
Loss
(dB
/cm
)
R=399.02 µm
R=198.95 µm
(b)
61
requires low losses from both the even and odd supermodes. Fig. 3.10 illustrates the loss
behavior of the two TM supermodes calculated using FEM-PML and mode matching
technique for the directional coupler operating in two coupling regimes. When the
waveguide center to center to separation is large (S=3 μm, Lcouple=361 μm, Fig. 3.10(a)),
both modes have a loss minima at 0.71 μm, which is also the first order magic width of
the single isolated waveguide. However, reducing the waveguide separation to increase
coupling (S=1 μm, Lcouple=16 μm, Fig. 3.10(b)) results in a loss increase for the two
modes at the original magic waveguide widths. The new “magic widths” are 0.88 μm and
0.73 μm for the fundamental and first order supermodes respectively. To balance the two
supermode losses for good coupler operation, for example, the coupler can be designed at
a waveguide width of 0.775 μm, but the loss value itself is then rather high at 10 dB/cm.
While this excess loss only occurs for the designed coupler interaction length, it could
(a)
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
0.55 0.6 0.65 0.7 0.75 0.8 0.85
TM_0 FEM+PML
TM_0 Mode Matching
TM_1 FEM+PML
TM_1 Mode Matching
Waveguide width, W (µm)
Loss
(dB
/cm
)
S
BOX
Si
15 nm Ridge
190 nm
Separation, S = 3 µm
W
62
Figure 3.10. (Color) Inset: Cross sectional view of ridge waveguides in a directional coupler
configuration with waveguide centre to centre separation S and a symmetric waveguide width of W.
Graphs: Loss of the fundamental and first order TM super mode of the directional coupler as a
function of waveguide width (W) for a waveguide centre to centre of (a) S=3 μm and (b) S=1 μm
impact resonator or other device functionality and must be carefully contemplated when
designing high-Q components.
(b)
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95
TM_0 FEM+PML
TM_0 Mode Matching
TM_1 FEM+PML
TM_1 Mode Matching
Waveguide width, W (µm)
Loss
(dB
/cm
)
Separation, S = 1 µm
63
Chapter 4. Low Loss Si-SiO2-Si TM Slot Waveguides
Slot dielectric optical waveguides[36] have recently generated considerable interest
due to their ability to strongly concentrate light into nano meter scale low index
dielectrics, thereby providing enhanced nonlinearities, modulation, gain or possible
sensor applications. In the high index contrast Si–SiO2–Si system, for example, standard
waveguide calculations reveal that 20% of the optical power can be confined in SiO2 slots
as narrow as 10 nm. To realize active functions, such as lasing using active dielectric gain
media, it is critical to achieve nm-scale slot waveguides with very low optical losses.
While relatively wider etched vertical slots have been demonstrated with losses of 8
dB/cm[37], this method poses difficulties in scaling to nm slot dimensions, and scattering
from etched slot surfaces contributes undesirable losses.
Horizontal slot waveguides can be realized with smoother wafer layer interfaces
using standard planar processing and transverse-magnetic (TM) mode operation, but also
present challenges in achieving low loss. The most flexible and expedient means for
creating the horizontal slot structure utilizes a thin dielectric slot layer on a single crystal
SOI device layer, followed by deposition of amorphous silicon (a-Si) or poly-silicon,
both of which contribute undesirable propagation loss. Recently, propagation loss of 1.5
dB/cm has been demonstrated for buried a-Si waveguide structures[38], and since only
the upper Si layer in the horizontal slot guide is comprised of a-Si, this suggests that
64
carefully designed slot waveguides could achieve even lower losses if scattering losses
could be eliminated. In Sec. 2.4 we discussed a novel technique for waveguide
fabrication resulting in ultra smooth waveguide surfaces[18]. This allowed us to achieve
a loss as low as 0.36 dB/cm for the TE mode[19] in very tight vertical confinement, 0.2
μm core SOI waveguide with shallow ridge. This was essentially the result of minimizing
the surface scattering losses. These guides additionally offer the important benefit of
electrical access to the waveguide core as required for providing active functionalities.
We then showed experimentally that realizing low loss TM mode propagation in these
shallow ridge structures requires precise fabrication of waveguides at “magic widths” to
effectively negate the inherent lateral leakage of TM quasi-modes[15-17, 19]. Here, we
demonstrate TM mode propagation losses of 1.83 dB/cm in 8.3 nm horizontal slot guides
designed at the “magic width” and fabricated with a deposited a-Si layer as the top
cladding layer[39, 40], which we believe is both the lowest-loss and narrowest slot
waveguide demonstrated to date.
4.1 Waveguide Design and Fabrication
The horizontal Si---SiO2---Si waveguide is grown on commercial SOI wafers with a
buried oxide (BOX) layer of thickness 2 µm. Figure 4.1 shows the schematic of the
process flow for forming the slot waveguides. The 205 nm crystalline silicon device layer
(c-Si) has a 7.5 nm step ridge fabricated using the ultra-smooth waveguide core local
thermal oxidation process[18] to avoid etching-induced surface roughness as detailed in
Sec. 2.4, and is then thinned with blanket oxidation to a core thickness of 150 nm, leaving
65
a lateral field c-Si layer of 143-nm thickness. The 8.3 nm SiO2 slot layer between the c-Si
and yet to be deposited a-Si layer is formed using blanket thermal oxidation. The 100 nm
thick a-Si layer is then deposited using a plasma-enhanced chemical vapor deposition
system at 180 °C, at a plasma power of 40 W, and deposition rate of ~1 Å/sec. The
precise thickness of the SiO2 layer for this study was determined using ellipsometry,
while the a-Si layer thickness was determined by step profiling. For low loss TM
performance, minimizing lateral radiation leakage also requires that the waveguide width
Figure 4.1. Processing steps for shallow ridge horizontal slot waveguide. Shallow ridge
waveguide in step 1 formed using the process described in Fig. 2.10
c-Si 205 nm
BOX
7.5 nm Ridge
1
Blanket Thermal Oxidation
BOX
c-Si 150 nm 2
BOX
c-Si 150 nm
HF Strip
3 8 nm SiO2
BOX
c-Si 150 nm
Slot Thermal Oxidation
4
PECVD Amorphous Silicon Deposition
BOX
c-Si 150 nm8 nm SiO2
100 nm a-Si
W=1.51 µm ( Magic width)7.5 nm Ridge
5
66
be precisely controlled to a ‘‘magic width’’ as given by Eq. 2.1. In our case the result is
1.51 µm which is achieved by properly tailoring the original mask so that the correct
value is achieved after all process steps. Fig. 4.2 provides a scanning electron
micrograph of a similar structure, taken using the environmental secondary electron
detection (ESED) mode to best image the SiO2 slot. Further more the fabricated structure
provides 16.3 % field intensity confinement in the 8.3 nm silica slot (Fig. 1.1) but from
optical gain or loss perspective the relevant quantity is the confinement of the weighting
function as we show in the analysis below:
The change in the propagation constant (Δβ) due to a perturbation Δε(x,y) of the dielectric
constant of the waveguide is given by [41]
P
dydxEE∫ ∫∞
∞−
∞
∞−
⋅Δ=Δ
*εωβ
(4.1)
Figure 4.2. Scanning electron micrograph of a similar planar slot structure with 8.3 nm slot
visible using ESED mode imaging
67
If we assume for simplicity that we have perturbation only in the slot, then
⎩⎨⎧ Δ
=Δslotoutside
slotinsidenn slotslot
02 0ε
ε (4.2)
Gain or loss in a homogeneous medium is
imagnc
Δ=Δ ωα 2 (4.3)
but for the slot waveguide (assuming for simplicity that nreal-slot >> nimag-slot) the modal
gain or loss is given using Eq. (4.1) by
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ ⋅⋅⎥⎦⎤
⎢⎣⎡ Δ=Δ
∫∫− P
dydxEEncn
csectioncrossslot
slotreal
slotimagmode
*0_2
2ε
ωα (4.4)
The first factor in the Eq. (4.3) is the bulk gain or loss due to the homogeneous slot
medium and the second factor is the correction term which arises due to waveguide
confinement. We notice that this term is not the usual intensity confinement.
Now considering the case of a 1D slab waveguide structure, for small changes in material
index we have
)()())(( xnxnxn Δ≅Δ 22 (4.5)
The modal effective index change which is proportional to the gain or loss experienced
by the mode of the slab waveguide will be
∫∞
∞−
⋅Δ=Δ dxxfxnneff )()( (4.6)
where f(x) is a weighting function and is given by
( )∫∞
∞−
⋅×+×
⋅=
dxzHEHE
EExncxf
ˆ
)()(
**
*02 ε
(4.7)
68
Now a plot (Fig. 4.3) of this weighting function f(x) for the TE and TM modes for the 8.3
nm horizontal slot slab waveguide structure shows a huge confinement for the TM mode
in the slot. For this particular structure this means that the TM mode will see 14X higher
modal gain or loss relative to the TE mode for any gain or loss achieved in the SiO2 slot
dielectric. To be more specific, a full 23.5% of the local gain in the 8.3 nm slot will be
seen by the TM mode. This is quite phenomenal considering the fact the slot is just 8.3
nm thick and is of lower index. So for example, if the dielectric slot were appropriately
doped with rare earths to achieve a peak gain of 13.7 dB/cm in the 8.3 nm dielectric slot
Figure 4.3. Weighting function for index changes for TE and TM modes, illustrating the 14X
enhancement of sensitivity to index change in the SiO2 slot for the TM mode over TE mode for the
slot waveguide shown in the inset. The integrated weighting function across the thickness of the SiO2
slot gives an effective confinement of 23.5%.
-0.5
-0.4
-0.3
-0.2
-0.10
0.1
0.2
0.3
0.4
0.5
0 10 20 30Weighting function (µm-1)
Wav
egui
de P
ositi
on (µ
m)
TE TM
150 nm Si
100 nm Si
8 nm SiO2
BOX
69
[42], the modal gain would be 3.2 dB/cm. This is attractive as it could provide sufficient
gain for lasing while potentially allowing for electrical excitation of the nm scale slot
medium. Another critical thing to notice in the plot are the spikes of the TM weighting
function at the interfaces between different layers which make it more susceptible to
surface and interface roughness scattering losses. This makes achieving ultra smooth
waveguide surfaces all that more critical.
4.2 Experimental Evaluation
We evaluated the waveguide propagation losses of the horizontal slot waveguides
using the weakly coupled ring resonator geometry as discussed Sec. 2.5. Compared to the
cut-back or Fabry---Pérot fringe methods, this provides a more reliable loss measurement
due to unambiguous and precise resonance characteristic measurement of a single device
made using a scanning single-frequency laser. With negligible contribution from
coupling loss, the loss would be given by
( )g
FWHMelogcmdB
ννπα Δ⋅⋅⋅=⎟
⎠⎞
⎜⎝⎛ 210 10 (4.8)
where ΔνFWHM is the measured drop port full-width at half maximum line width and νg
is
the waveguide group velocity. The experiment is conducted using an external cavity laser
diode with a continuous single longitudinal mode fine wavelength scanning feature. Fig.
4.4 shows the smallest observed linewidth of Δν=625MHz, yielding a resonator of
Q~3X105.
The measured group index of ng=4.1, obtained from the free-spectral range of the
70
Figure 4.4. Scan of ring resonance illustrating ΔνFWHM =625 MHz with a corresponding Q~3 X 105
resonator is quite close to the numerically modeled value of 4.0, which is significantly
larger than the calculated phase index of nphase=2.1 for the TM mode of this slot structure.
Using the experimental group index, the corresponding loss is 2.33 dB/cm if all the loss
is ascribed to propagation. However, the calculated coupling coefficient for the
intentionally weak directional couplers in this resonator is 2.15% per coupler, and the two
couplers combined thus contribute a loss of 0.50 dB/cm for our 600 µm radius ring,
leading to a waveguide transmission loss of 1.83 dB/cm [39, 40].
In addition to loss variation with wavelength that is expected due to the ‘‘magic
width’’ effect, we also see small variations in successive resonances. We believe these
may be due to instrumentation and facet reflectivity effects and are under investigation.
But even if averaged, these yield a loss increase of only ~25% above the minimum value.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
1566.15 1566.2 1566.25 1566.3 1566.35 1566.4 1566.45 1566.5
QTM=3.06 x 105
ΔνFWHM,TM=625 MHzαTM=2.33 dB/cm
ΔνFSR,TM= 19.40 GHzD
rop
port
resp
onse
(arb
.)
Wavelength (nm)
600 µm Ring, 1.51 µm “Magic width” Slot Waveguide
71
An estimate of loss of our amorphous silicon film was also made. For this we refer
back to our result for the TM mode loss of 0.94 dB/cm in shallow ridge waveguides[19]
(Sec. 2.6). If we now attribute all the excess loss of 0.89 dB/cm in going from ridge
waveguide to the slot waveguide to a-Si and consider the 32.5 % weighting function
confinement in the 100 nm a-Si film in the slot waveguide, we get an upper limit of 2.8
dB/cm for the a-Si loss. This loss value for a-Si puts it in the class of some of the best
results reported[38] so far for low loss or good optical quality amorphous silicon films.
TM slot mode losses from conventional bending radiation are vanishingly small at
our large ring radius. Calculations reveal that a 0.1 dB/cm excess loss would be incurred
for a reduced radius of ~320 µm. However, with a modest increases in ridge height to
only 15 or 30 nm, which is not expected to increase loss significantly, the calculated 0.1
dB/cm conventional bending loss radius reduces to 82 and 24 µm respectively due to the
excellent lateral bend tolerance of TM modes. TE mode bending losses at these shallow
ridge heights are exorbitantly high and hence TE mode drop ports couldn’t be measured.
In conclusion, we have demonstrated a-Si---SiO2---c-Si 8.3-nm slot waveguides with
quasi-TM-mode loss of 1.83 dB/cm using an ultra smooth, low-loss shallow ridge
fabrication technique. This result illustrates that low-loss slot waveguides can be
achieved with nm-scale slots using deposited amorphous silicon for the upper layer of the
slot structure, providing flexibility in designing high-index-contrast, high-confinement
passive and active horizontal slot waveguide devices.
72
Chapter 5. Erbium Doped 8 nm Horizontal Slot Waveguides
Erbium (Er) has played a key role in advancing the long haul telecommunication
optical networks by providing means for repeaterless amplification of optical signals at
the telecommunication wavelength of 1.55 μm. Trivalent Erbium Er3+ (the preferred
bonding state of Er) has an incomplete 4f electronic shell that is shielded from the outside
world by the closed 5s and 5p shells. As a result, rather sharp optical intra-4f transitions
can be achieved from erbium doped materials. The transition from the first excited state
to the ground state in Er3+ occurs at an energy of 0.8 eV, corresponding to a wavelength
of 1.54 μm, a standard telecommunication wavelength at which the standard silica based
optical fibers have maximum transparency. This precise attribute of Erbium is utilized in
Erbium doped fiber amplifiers (EDFAs)[43] to amplify optical signals at 1.54 μm. A
continuous pump laser is used to create population inversion between the ground state
and the first excited state in Er3+, whereupon stimulate emission serves to amplify a high
frequency telecommunication signal at 1.54 μm. The availability of miniature
semiconductor diodes to pump the EDFAs, coupled with its exquisite gain characteristics,
polarization insensitivity, temperature stability, quantum limited noise figure and
immunity to interchannel crosstalk meant EDFAs could be deployed widely in a cost
effective manner while at the same time setting a high standard for network performance.
73
It is perhaps safe to argue that it is mostly because of the availability of the EDFAs, that
the optical telecommunication networks are as well developed as they are today.
With the long haul conquered the next logical frontier for Erbium is chip based
applications where it is being investigated as a source of optical gain. However, the
challenges now are even more formidable than before as has been indicated in Sec. 1.1.
In spite of these, considerable progress has been made in devising erbium doped
waveguide amplifiers (EDWAs) [44, 45] and more recently chip based erbium doped
lasers [46-48]. However these structures are designed to operate under optical pumping
conditions and provide no clear direction for extension to electrical drivability in a non-
invasive manner. As a result they have not been widely employed in chip scale
applications.
We believe horizontal slot waveguides with ultra narrow slots in principle provide
means of electrical excitation of the slot embedded gain media like Erbium through non-
invasive mechanisms like tunneling, polarization transfer and transport. However, as a
first step towards evaluating the suitability of such a device to provide optical gain under
electrical excitation it is instructive to characterize the behavior of the device under
optical pumping scheme. Having achieved encouraging loss values for the horizontal slot
waveguide structures for TM mode operation, erbium implantation of the slot was
undertaken. The samples were characterized for photoluminescence and erbium lifetime.
In the process, an unexpected physical phenomenon - Purcell enhancement[49] of
luminescence decay associated with erbium luminescence centers in ultra thin slots came
to forefront. The same was thoroughly analyzed theoretically and experimentally and
74
could have significant positive implications on the realization of novel light sources in
silicon[50, 51].
5.1 Erbium Doped Slot Waveguide Fabrication
For Erbium doped slot waveguides the challenge was incorporation of erbium into the
ultra thin slot with minimal exposure of the underlying crystalline silicon layer to
Erbium. This entailed utilization of extremely low implantation energy of just 2 keV and
a 45° implantation angle. A relatively higher dosage of erbium of 6X1020 cm-3 was used
to compensate for the low implantation energies. The Erbium doped slot samples
followed the same fabrication sequence as was utilized for the formation of the passive
slot structures described in Sec. 4.1 (Fig. 4.1) except for the implantation of Erbium in
the ultra thin slot followed by Erbium activation anneal at 700 °C in nitrogen ambient for
15 minutes prior to amorphous silicon film deposition. Fig. 5.1 shows the implantation
conditions and the final Erbium doped slot structure obtained after all the processing
steps, with structural attributes same as the final passive slot structure shown in Fig. 4.1.
Figure 5.1. (a) Implant and anneal conditions and (b) cross sectional view of Erbium doped slot
waveguide
(a) (b)
100 nm a-Si
Si Substrate
BOX 143 nm c-Si
8 nm Er/SiO2
7.5 nm Ridge
W=1.51 µm (magic width)
Amorphous Silicon deposition
Erbium Implantation2 KeV 6E20 cm-3
Activation anneal700°C, N2 ambient, 15 min
75
5.2 Photoluminescence Measurements
Photoluminescence measurements were done in full waveguide configuration by end
pumping straight slot waveguides using a high power 1480 nm pump laser. Fig. 5.2
shows a schematic of the experimental setup used to measure the photoluminescence.
Light was coupled in and out of the waveguides using lens tipped fibers precision aligned
to the waveguide using six axis piezo stages. The pump was TM polarization to utilize
the higher confinement characteristic of the TM mode in the slot. Cascaded band pass
filters with centre wavelength of 1480 nm and 1550 nm with 30 nm bandwidth were used
on the input and output respectively to provide more than 60 dB of out of band isolation.
The luminescence spectrum was measured on an ANDO AQ6317 optical spectrum
analyzer with a resolution bandwidth (RBW) of 0.5 nm. Fig. 5.3 shows the measured
luminescence spectra at different waveguide coupled pump powers which shows clear
Erbium characteristics. This we believe is the first observation of Erbium luminescence
in ultrathin slot waveguides in full 3D waveguide configuration primarily made possible
due to the low losses of these waveguides. The waveguide transmission loss is ~3 dB/cm
for the TM mode as determined from the ring resonator measurements.
Figure 5.2. Setup for measurement of Erbium doped slot waveguide luminescence
Pump1480 nm
Polarization controller
CascadedFilters OSA
1550 nm Band pass
Six Axis Piezo stages
Er doped Slot waveguides
CascadedFilters
1480 nm Band pass
76
Figure 5.3. Erbium doped slot waveguide photoluminescence at different waveguide coupled pump
powers, expressed as power spectral density (PSD)
5.3 Time Resolved Photoluminescence Measurements
Considerable effort was put into accurately measuring the Erbium spontaneous
emission lifetime to quantify the radiative and non-radiative processes associated with
Erbium in the ultra narrow 8 nm slot. The measurement is made complex by the fact that
only 2.6 nW of integrated luminescence power could be collected from these waveguides
even under saturation conditions. In the absence of a sensitive detector like photo
multiplier tube (PMT), a setup had to be designed to increase the signal to noise ratio of
the luminescence.
-85
-83
-81
-79
-77
-75
1525 1535 1545 1555 1565
~ 8mW
~ 40mW
Wavelength (nm)
PSD
(dB
m)
RBW=0.5 nm
Waveguide Coupled Pump Power
77
The setup for lifetime measurement was similar to that showed in Fig 5.2, except that
the luminescence was now detected using a Thorlabs InGaAs PIN fiber coupled detector
instead of OSA. The detector has a responsivity of 0.83 at 1550 nm, a capacitance of 0.7
pf and a noise equivalent power of 10-15 W/(Hz)0.5. Even with a load resistance of 10 kΩ
which corresponds to a bandwidth of 22.7 MHz, we can expect to see only 21.25 μV at a
signal to noise ratio (SNR) of 0.12 for 2.6 nW of optical power coupled to the detector,
which would be impossible to see on a digitizing oscilloscope. To improve the signal to
noise ratio, a scheme using a lock-in amplifier was initially implemented to quantify the
Erbium lifetimes using the frequency domain technique. The method yielded very fast
lifetimes in the range of ~150 μsec which seemed unusually fast compared to the
expected millisecond range lifetimes for Erbium in silica matrix. The results were also
convoluted by the presence of multiple life times for Erbium making it challenging to
interpret the frequency domain data. To corroborate the fast lifetimes, a more direct time
domain technique was implemented using a gated integrator and BOXCAR averager
system with preamplifiers. The system pre-amplifiers supplied with the BOXCAR have a
gain of 5 and a noise of 6.4 nV/(Hz)0.5. The system allows cascading of five such
amplifiers together. The best expected signal output with 2.6 nW of coupled optical
power is 0.013 V with a SNR of 12.7 by using 15 μsec gate width (GW) and ten thousand
sample averaging. Fig. 5.4 shows the pump pulse and the luminescence signal obtained
under these conditions. Although the luminescence can be seen, the decay is not smooth.
This meant a further improvement in SNR was required to get a quantitative
measurement of the Erbium lifetime. This was only possible if we improved on our pre-
amplifiers as we were already operating at the limit of the BOXCAR at ten thousand
78
Figure 5.4. (a) Pump pulse and (b) Luminescence signal using gated integrator and BOXCAR
averager with system preamplifiers. Observed peak luminescence matches calculated value
samples averaging. The system pre-amplifiers were replaced by a ultra low noise
(3pA/(Hz)0.5) trans- impedance amplifier (TIA) with a bandwidth from DC to 2 MHz and
a gain of 10 MV/A. With this TIA, the expected signal was 4.25 V with a SNR of 543.
Fig. 5.5 (a) shows the measured luminescence signal under these new conditions and the
signal is finally clean enough to measure the Erbium lifetime. Fig. 5.5 (b) shows the
luminescence decay on a semi-log scale under saturation conditions. The measured
Erbium lifetime is 152 μsec, similar to what we had obtained using the frequency domain
method using a lock-in amplifier.
A fast lifetime of ~152 μsec suggests a strong non-radiative decay channel for
erbium. However it is also possible that other factors might me contributing to the fast
lifetime including the waveguide structural impact on the radiative spontaneous emission
rates. This will be the topic of the next section.
(a) (b)
-5.0E-03
-1.0E-17
5.0E-03
1.0E-02
1.5E-02
0 1 2 3 4 5 6 7-0.20.00.20.40.60.81.01.2
0 1 2 3 4 5 6 7Time (x100 μsec)
Box
car o
/p (V
olts
) Er Luminescence
Nor
mal
ized
Box
car o
/p (V
olts
) Pump pulse @1480 nm
Time (x100 μsec)
79
Figure 5.5. (a) Photoluminescence measured with the system pre-amplifiers replaced by a low noise
high gain TIA in conjunction with gated integrator and BOXCAR averager (b) Luminescence decay
signal on a semi-log plot with a lifetime of ~ 152 μsec
0.00.51.01.52.02.53.03.54.04.55.0
0 2 4 6 8 10Time (x100 μsec)
Box
car o
/p (V
olts
)
1 µsec GW,1K Sample Avg.
15 µsec GW,10K Sample Avg.
Er Luminescence
(b)
(a)
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
3.445 5.445 7.445 9.445Time (x100 μsec)
152 μsec
Nor
mal
ized
Lum
ines
cenc
e D
ecay
Sig
nal
80
5.4 Spontaneous Emission in Waveguide Media
The atomic spontaneous emission rate can be expressed in terms of the zero-point
fluctuations of the electromagnetic field at the position of the atom. The local zero-point
field fluctuations depend on the photon density of states and on the electromagnetic field
strengths of the modes. As the electromagnetic fields of the modes depend strongly on
the configuration and on the electromagnetic properties of the materials, the spontaneous
emission rate can be either increased or decreased, depending on the electromagnetic
properties of the atom’s environment. The possibility of modifying the spontaneous
emission rate by changing the electromagnetic environment was first pointed out by E.M.
Purcell in 1946 [49], where it was implicitly assumed that a sizable modification in
spontaneous emission rates could only be brought about by putting the atom in a high Q
resonator. However very recently pronounced broadband enhancements of spontaneous
emission rates has been modeled and observed in Er-doped Si/SiO2 slot waveguides [52,
53] i.e. even in a non-resonator configuration. Due to the high index contrast at the
Si/SiO2 interfaces, the electromagnetic field strongly concentrates in the narrow slot
region (for the TM mode), leading to an increase in the zero-point fluctuations. The zero-
point field fluctuations can be calculated by applying quantum electrodynamic formalism
which requires determination of the complete set of electromagnetic modes which can
exist in the configuration so as to quantize the electromagnetic field [54, 55]. The
modification of the spontaneous emission rate can also be obtained from a semi classical
treatment by calculating the emitted power from a classical dipole source which is also
equivalent to the work done on a dipole by its own reflected field [56, 57].
81
However, here we show that Purcell enhancement in waveguides is already correctly
captured in conventional spontaneous emission expressions[58]. A quantum
electrodynamic derivation of enhanced spontaneous emission rate in waveguides is also
shown to prove the equivalence of the two approaches[58]. We also show how this result
for a waveguide naturally maps into the traditional expression for Purcell factor which is
usually associated with a resonator[58]. The results are then compared with those
obtained using a full numerical QED calculation which includes the contribution from the
radiation modes. Finally the experimental results presented above are analyzed in light of
this new information[50, 51].
5.4.1 Spontaneous Emission into a Particular Waveguide Mode
This sub-section closely follows materials from unpublished notes used in ECE 450,
Optoelectronics Physics and Lightwave Technology II, taught by Prof. Thomas L. Koch
[58].
Elementary treatments of spontaneous emission into waveguides often invoke the
fraction of spontaneous emission emitted into a waveguide mode by dividing the solid
angle ΔΩ mode of the mode by the total 4π emission. From simple diffraction theory
modemode An2
2λ=ΔΩ (5.1)
where Amode is the mode effective area, which is assigned the value
Γ= active
modeA
A (5.2)
82
by considering the confinement factor Г of the mode within the active emitting area
Aactive. This leads to a fraction of spontaneous emission emitted into both directions of
one waveguide polarization mode to be given by
activeactivesp AnAn
F Γ=Γ⋅⋅= 2
2
2
2
441
212
πλλ
π (5.3)
where the confinement factor is given by the classical expression based upon mode field
amplitudes E and H ,
( )∫ ∫
∫∞
∞−
∞
∞−
∗
∗
⋅×+×
⋅≡Γ
dxdy
dxdyncactiveA
activeo
zHEHE
EE2
ˆ*
ε (5.4)
The spontaneous emission rate (SER) into one waveguide polarization mode is then
30
30
221
2
2
3411
cn
AnF
activemediumhomospsp
WGntoisp επωμ
πλ
ττ⋅Γ=⋅= (5.5)
which includes the standard “free space” SER for a medium of index n for a transition
with a dipole moment µ21. It is remarkable, however, that when applied to slot
waveguides we find that the “fraction of spontaneous emission” Fsp into the waveguide
mode given above can easily exceed unity. While this is seemingly a nonsensical result,
it is actually rigorously correct and the factor Fsp above is exactly the Purcell
enhancement factor for emission into a waveguide mode. This follows rigorously from a
quantum electrodynamic treatment of inhomogeneous dielectrics as derived for example
by Dalton, et al. [59]. The quantized electric field operator in an inhomogeneous
dielectric medium )x(rε is
( )∑ +− −⋅=n
nti†
nnti
nn eaieait )()(),( * xAxA
2xE ωωω (5.6)
83
where the vector basis An(x) set satisfies the self-adjoint operator eigen value wave
equation
( ) nn
nnr c
AAAx
12
22 ω
ε=⋅∇∇−∇
)( (5.7)
along with the modified radiation gauge criterion 0Ax =⋅∇ nr )(ε .These vectors,
proportional to the vector potential, are thus complete under the orthogonalization
condition
mnV
nmrnm xd ,*)( δεε =⋅≡ ∫ 3
0 AAxAA (5.8)
Assuming a transition dominated by dipole emission, the spontaneous transition rate from
level b to level a into mode An (x) is given by Fermi’s Golden rule as
( ) )()(ˆ
)(,),(,
* ωδμπω
ωδπ
−−⋅⋅+=
−−⋅+=−
abnbann
abnnspontba
EEN
EEbNetaNW
22
2
xAρ1
xxE12
(5.9)
where Nn is the photon occupancy of the mode and ρ is the unit vector in the dipole
direction. For a channel waveguide of length L and propagation constant βn we have
Lyxe nzi
nn
0
φxAε
β ),()( = (5.10)
where φn (x, y) is scaled so that An (x) satisfies the normalization condition above. The
density of states for forward and backward emission is given by
( ) cnLE geff −⋅⋅=
πρ
22)( (5.11)
So the spontaneous emission rate averaged over dipole orientations becomes
84
⎟⎟⎠
⎞⎜⎜⎝
⎛×=
= −−
),(
),(),(
yxAn
yxc
nyxW
modemediumhomosp
nbabageff
spontba
1411
φ3
2
2
2
0
2
λπτ
εμω
(5.12)
where the effective area of the mode for an emitter at (x, y) is
2φ1
),(),(
yxnnyxA
ngeffmode
−
≡ (5.13)
The first term in Eq. 5.12 is the standard homogeneous medium spontaneous emission
rate and the second term (enclosed in brackets) which accounts for the impact on the
emission rate due to the waveguide configuration is referred to as the Purcell factor.
Using the equivalence between the Poynting vector and the group velocity times the
energy density; it can be shown that the confinement factor defined earlier is also equal to
∫ −=ΓactiveA
ngeff dxdyyxnn 2φ ),( (5.14)
If we then average the SER across the active emitting region we find that indeed
Γ= active
modeA
yxA ),( (5.15)
and hence we obtain the same expression for the Purcell factor as the one we wrote for
Fsp in Eq. 5.3
activesp An
FFactorPurcell Γ== 2
2
4πλ (5.16)
We thus see that the conventional expressions for emission into waveguide modes still
apply, and indeed predict a SER into the waveguide mode alone that exceeds the total
homogenous medium result whenever we satisfy
85
( )
22
zHEHE
EE21⎟⎠⎞
⎜⎝⎛⋅>
⋅×+×
⋅=
∫ ∫∞
∞−
∞
∞−
∗
∗
λπε n
dxdy
yxncyxA
o
mode ˆ
),(),( *
(5.17)
for an emitter at the position (x, y)[58]. The equivalent 1-D slab waveguide analysis can
be carried out similarly and yields a spontaneous enhancement factor
)(1
221
1 xtnnn
Fmode
effDsp ⋅⎟
⎠⎞
⎜⎝⎛=−
λ (5.18)
where [tmode (x)]-1 i.e. the inverse modal thickness is the function above in Eq. 5.17 with
the denominator integrated over just the vertical dimension x.
So for the ultra-thin 8.3 nm slot waveguide shown in the inset of Fig. 5.6, Fig. 5.6
illustrates a dramatic enhancement of emission expected into the TM slot mode for
Figure 5.6. (Inset) Ultra-thin 8.3 nm slot slab waveguide structure. Plot of effective inverse modal
thickness in the vertical direction as a function of position for both the TE and TM modes
-0.5
-0.4
-0.3
-0.2
-0.10
0.1
0.2
0.3
0.4
0.5
0 10 20 30Inverse Modal Thickness (µm-1)
Wav
egui
de P
ositi
on (
µm)
TE TM
150 nm Si
100 nm Si
8.3 nm SiO2
BOX
86
emitters placed in the slot. The average effective modal thickness across the active slot
layer in the vertical direction is calculated to be nmtt activemode 35/ =Γ= which is 1/30th
of a wavelength in the medium, yielding a ~11X emission rate enhancement into just the
slot TM mode compared to homogeneous silica.
5.4.2 Total Spontaneous Emission Rate Including All Modes
To obtain the total SER enhancement, the calculation must also include emission into
radiation modes that may also get modified by the structure. To do this rigorously, a
quantum electrodynamic approach was used to calculate the zero-point fluctuations of the
electromagnetic field and the spontaneous emission rate from a non-absorbing dielectric
film. Our formulation closely follows the one given in [54] where the authors analyze the
spontaneous emission rate from a lossless dielectric film bounded by two non-absorbing
dielectric half spaces of arbitrary refractive indices. We extended this methodology to the
five layer slot structure i.e. Si-SiO2-Si slabs bounded by air and SiO2 dielectric half
spaces.
The process requires the quantization of the electromagnetic field which in turn
requires that the complete set of spatial electromagnetic modes that exist in the
configuration be determined. These modes must be orthonormalized with respect to the
scalar product corresponding to the electromagnetic energy density. The set consists of
radiation modes pertaining to plane waves that are incident from either dielectric half
space for both orthogonal polarizations. In addition, guided modes of either polarization
may exist. These modes are evanescent in both dielectric half spaces. Spontaneous
87
emission rate and zero-point field fluctuations are then calculated by incorporating the
quantized field in the Fermi’s Golden Rule. Here we summarize the key results of the
formulation. For a complete derivation, the reader is referred to Appendix A –
Spontaneous Emission Rate in a Slot Waveguide.
Suppose that an atom makes a spontaneous dipole transition from a state 2 to a
state 1 thereby emitting a photon of energy 0ω . The spontaneous emission can occur
in any mode of the electromagnetic field of frequency 0ω , or equivalently of wave
number ck /00 ω= in vacuum. The transition rate into a particular electromagnetic mode
with parameters μλκ ,, say, and with wave number k in vacuum, is given by [60]
)(ˆ kkifc ED −Η 0
2
2
2 δπ (5.19)
where i and f are the initial and final states of the combined atom-radiation system
and EDΗ is the electric-dipole interaction part of the complete Hamiltonian of the atom -
radiation system. In the initial state of the electromagnetic field, no photons are present.
In the final state there is one photon in the mode with parameters ( )μλκ ,, .
According to Fermi’s Golden Rule, the total spontaneous emission rate is obtained by
integrating Eq. 5.19 over all the final states. Suppose that the dipole moment of the atom
is parallel to one of the axes of the Cartesian coordinate system and let j be x, y, or z. Let
jE be the jth component of the transverse electric field operator. Then the total
spontaneous emission rate of an atom at position r whose dipole moment is parallel to
the jth axis is [61]
88
)(zFDc
e j2
122
221 πτ
= (5.20)
where e is the electron charge, 12D is the dipole matrix element of the atomic transition,
and
( )
∑ ∑ ∫∫
∑ ∑ ∫∫∫
=
∞
= >
= =<+
−+
−
=
PSyx
PSnkkk
yxj
j
min
yx
dkdkkkrE
dkdkdkkkrE
zF
,
,
,
))(()(
)()(
)(
λ ν ββ
λννλκ
λ μμλκ
λν
μ
βδεω
δεω
10
2
0
5
40
2
0
2
221
22
(5.21)
is the contribution (per wave number) of all modes of wave number 0k to the jth
component of the zero-point field fluctuations.
When polar coordinates are introduced in the ( )yx kk , plane:
)(insk),(cosk yx ϕβϕβ == (5.22)
Eq. 5.21 becomes
∑ ∑ ∫ ∫
∑ ∑ ∫ ∫ ∫
=
∞
=
∞
=
= =
∞
=
−+
−
=
PS sincos
PS
nk
ksincos
j
j
min
ddkkrE
ddkdkkrE
zF
, ))(),((
, )),(),((
,
))(()(
)()(
)(
λ ν
π
β
λν
ϕβϕβκνλκ
λ μ
π
ϕβϕβκμλκ
λν
μ
ϕβββδεω
ϕββδεω
1
2
00
2
0
5
4
2
0 0 00
2
0
2
2
(5.23)
∑ ∑ ∫
∑ ∑ ∫ ∫
=<
≥ =
= = =
+
=
PSkk
sinkcosk
PS
nk
ksincos
j
j
min
dkdk
dkrE
ddrE
zF
, ))()(),()((
, )),(),((
,
)()()(
)(
)(
λ ν
π λνλ
νϕβϕβκ
νλκ
λ μ
π
ϕβϕβκμλκ
λν
λν
λν
μ
ϕββεω
ϕββεω
000
0
1
2
000
2
0
0
5
4
2
0 0
2
0
0
2
2
(5.24)
89
The integral over ϕ for the three components j=x, y, z in Eq. 5.24 can be computed
explicitly. For the mean of the three components,
[ ])()()()( zFzFzFzF zyx ++=31 (5.25)
the result of the integral over ϕ yields a rather concise formula. The quantity )(zF is the
zero-point field fluctuation of a randomly oriented atom at depth z and is therefore of
special interest. After further simplifications it is given by
∑ ∑
∑ ∑ ∫
=<
≥ =
= = =
+
=
PSkk
k
PS
nk
k
j
min
kdk
dkrE
drE
zF
, )),((
, ),,(
,
)()()(
)(
)(
λ ν
λνλ
νβκ
νλκ
λ μ βκμλκ
λν
λν
μ
ββεωπ
ββεωπ
00
0
100
0
2
0
0
5
4 0 0
2
0
0
232
232
(5.26)
The remaining integrals with respect to β have to be computed numerically.
Consider the zero-point field fluctuations in a homogeneous dielectric with a
refractive index n1. The components of the zero point fluctuations in this homogeneous
case are given by
freezyx FnFFF 1=== (5.27)
where freeF denotes the vacuum field fluctuations (in free space):
20
2
30
6 cF free
επω= (5.28)
For an atom at depth z inside the slab of index n1 in a multi-dielectric slab structure,
having a dipole moment parallel to the jth coordinate axis, the spontaneous emission rate
relative to the value in a bulk n1 index medium will be
free
j
FnzF
1
)( (5.29)
while for a randomly oriented atom the relative transition rate is
90
freeFnzF
1
)( (5.30)
with )(zF being defined by Eq. 5.25.
Finally, the mean spontaneous emission rate for a film of thickness d relative to bulk
film medium is obtained by computing the mean of Eq. 5.30 over the thickness of the
film.
Figure 5.7 shows a plot of the ( )freeFznzF )()( for the 8.3 nm slot structure shown in
the inset. It is clear that an emitter placed in the silica slot will see large enhancement in
the SER relative to that in a bulk silica medium. Furthermore, Fig 5.8 shows the mean
relative SER enhancement for an emitter placed in a T nm thick silica slot. The thickness
Figure 5.7. Plot of F(z)/(n(z)Ffree) as defined in Eq. 5.30 as function of position in the 8.3 nm slot
structure shown in the inset.
-250
-200
-150
-100
-50
0
50
100
150
200
250
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
8.3 nm SiO2
Slab
Pos
ition
(nm
)
( )freeFznzF )()(
BOX
150 nm c-Si
100 nm a-Si
Air
91
of the a-Si and c-Si have been fixed at 100 nm and 150 nm respectively. In addition to the
total enhancement, the contributions from TM guided, TE guided and combined radiation
modes have been plotted separately to elucidate the contributions of each to the total SER
enhancement. Based upon these results, for T=8.3 nm we can expect a 13.1X
enhancement in the SER for an emitter placed in the silica slot relative to that in a bulk
silica medium. This means our analytical result of above of ~11.06X is quite accurate
even though we ignored the contributions of the radiation and TE modes to it. This is
Figure 5.8. Plot of total spontaneous emission rate enhancement, averaged over polarizations, as a
function of slot thickness. Also shown are all the contributions, dominated by the TM slot guided
mode, along with the efficiency of emission into just the TM slot mode. Values indicated for the 8.3
nm slot case
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0
2
4
6
8
10
12
14
16
18
20
0 10 20 30 40 50 60 70 80 90 100
TM G
uided Efficiency
Slot Thickness, T(nm)
SER
Enh
ance
men
t in
Slot
Total
TE
TM
Radiation
13.07
0.86
92
because in the slot structure the dominant SER is into the TM polarized guided mode.
The efficiency into the TM guided mode is plotted on the right vertical axis in Fig. 5.8
and is as high as ~ 90% in this structure for the thinnest of slots.
5.4.3 Relation to Purcell Factor
The traditional expression for the Purcell factor is different from the one derived
above, which raises the question about the relationship between the two. This sub-section
also closely parallels the notes from Optoelectronics Physics and Lightwave Technology
II [58].
The Purcell enhancement factor F is usually associated with resonators and is written
in terms of resonator Q and mode volume V as
VQ
nF
3
2 26
⎟⎠⎞
⎜⎝⎛= λ
π (5.31)
In this resonator picture the spontaneous emission rate enhancement is usually cast
in-terms of very high density of states at the resonance as captured by the resonator Q
value. However, the actual basis for this effect is due to the anomalously large amplitude
of the modal field in such a resonator, and correspondingly, the anomalously large
amplitude of the vacuum field fluctuations that can be viewed as being responsible for the
spontaneous emission. The spontaneous emission changes we have seen in the
waveguide are due to this exact same phenomenon. To see this, we can very easily
extend the waveguide treatment to include a resonator, and get the Purcell ratio
above[58].
93
To do this, rather than viewing the Purcell effect as arising from an increase in the
density of states, one can embed the resonator in a volume large enough such that the
phase variance upon transmitting through the resonator does not significantly affect the
density of states of the large volume, even at the resonant frequency. We will show here
that this same requirement on the size of the volume being large is also sufficient to
guarantee that the energy inside the resonator, while possibly large at resonance, is still
vanishingly small compared to the energy outside the resonator in the large volume. This
means that the large volume modes can still be used without changing their normalization
to the large volume, and the density of states is also unchanged for these modes.
As an application of this, we can consider the two-dimensional channel waveguide.
We already derived the spontaneous emission rate into such a waveguide which is given
by
2
0
2
ρ ),(ˆ * yxc
nW n
babageffspontba ϕ
εμω
⋅= −− (5.32)
where we have not averaged over all angles of the atomic dipole moment. If we place two
reflectors of power reflectivity R in this waveguide separated by a length Lc as shown in
Fig. 5.9, this makes a small Fabry-Perot resonator of length Lc with the transmission
characteristic given by
c
c
Li
Li
FP eReTt β
β
ω 21 ⋅−=)( (5.33)
and if we let R = 1−δ where δ is understood to be small, we obtain the power
transmission of the mirror as T = δ and the Q of the resonator is given by
δω
cnL
Q geffc −= (5.34)
94
Figure 5.9. A Fabry-perot resonator of length Lc formed by inserting mirrors of reflectivity R in
an arbitrarily long waveguide of length L
We now want to stipulate that the total external length of the waveguide L is so large
that the number of external modes that “sample” the spectral width of the resonance for a
mode in question is large. If we consider propagation along the z-axis as illustrated in the
Fig. 5.9, the usual periodic boundary condition, or phase condition, for the modes of the
huge cavity in the z direction would be
MLL resonatorc πϕβ 2=+− )( (5.35)
and the number of modes in an interval dβ is then
πβ
βϕ
2d
dd
LLdM resonatorcz ⎟⎟
⎠
⎞⎜⎜⎝
⎛ +−= )( (5.36)
We know that the resonator phase will vary rapidly in the region of the resonance, so
to significantly oversample the resonance, we must require that, even in the vicinity of
the resonance,
)( cresonator LLd
d −<<β
ϕ (5.37)
At resonance, the phase derivative above is given by
ωβϕ
geff
resonator
ncQ
dd
−
= 2 (5.38)
Eout EoutEin
Lc
R R
L
z
95
So to have a high density of external modes to smoothly and effectively sample through
the resonance we need
cLLn
cnL
Q cgeffgeffc
2)( −
<<= −− ωδ
ω (5.39)
which says that we just need to ensure that δ
cLL
2>> , which is easy to do
mathematically, recalling that L is just the large, otherwise arbitrary dimension of our
waveguide.
We also want to stipulate that the normalization of the long waveguide modes is not
impacted by the presence of the resonator. Note that this normalization is given by
1A22 =⋅∫∫ dxdyndz
An
L
(5.40)
where A covers the area of the lateral mode and L is the very long external length of the
waveguide. Because22
AE nn ∝ , clearly if the overall normalization is not impacted we
would require that the integrated value of2
En inside the resonator (i.e., the energy) is
negligibly small compared to the total integrated value of 2
E n along the entire length of
the waveguide. To compute this, we need to evaluate the field enhancement of an
external mode that is coupled into the resonator. The ratio of the field outside to the field
inside the resonator is simply given by
δ== TEE
in
out (5.41)
96
Thus the energy density outside compared to the energy density inside will be
proportional tocQnL geffc ω
δ −= . Then if the energy density inside is U, the energy density
outside is cQnL
U geffc ω− , and we require for negligible energy in the resonator
LcQnL
ULU geffcc
ω−<<⋅ (5.42)
which thus requires that
ωgeffncQL
−
>> or δ
cLL >> (5.43)
and which we notice is identical to the condition we needed to make sure the phase
distortion of the resonator transmission was not strong enough to impact the sampling of
the resonance feature with many external long waveguide modes.
From this point, the enhancement inside the resonator is obvious because the long
waveguide modes normalization is unaffected, and the density of states is identical to
what we have already used to calculate spontaneous emission into the waveguide modes.
The only difference is the fact that the amplitude of the electric field locally inside the
mode is now larger by a factor as noted above,
δ1=
out
in
EE
(5.44)
and since the spontaneous rate scaled as 2
E ),( yxn , the spontaneous rate is just increased
by exactly a factor of
ωδ geffcguidechannelsp
resonatorsp
nLcQ
WW
−−−
− == 1 (5.45)
97
Referring to our previous calculation of spontaneous emission rate into the channel
waveguide mode, we then have our result for the enhancement of spontaneous emission
inside the resonator, as compared to free space, as being
( )tEnhancemenResonatorWaveguideW
W
guidechannelspontba
resspontba
×= −−−
−−
or
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅⋅=
−
−−− ω
ϕε
μω
geffcn
babageffresspontba nL
cQyxc
nW
2
0
2
ρ ),(ˆ *
(5.46)
and the ratio to the free space spontaneous emission rate is then
1
30
322
0
2
3ρ
−−
−
−−−
−−
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅⋅=
cnyx
cn
nLcQ
WW
baban
babageff
geffc
spacefreespontba
resspontba
επωμϕ
εμω
ω),(ˆ *
or
23
2 ρ12
3 ),(ˆ~ * yxnnL
Qnn
nF
WW
ngeffcgeff
spacefreespontba
resspontba
ϕλπ
⋅⎟⎠⎞
⎜⎝⎛== −
−
−−−
−−
(5.47)
which we recognize as essentially the Purcell factor F quoted earlier when we equate
[ ] 12 −
−≡ ),( yxnnA ngeffeff ϕ (5.48)
for an atom located inside the resonator at a position (x, y) to get
effgeff VnnQ
nF 1
26~ 3
2−
⋅⎟⎠⎞
⎜⎝⎛= λ
π (5.49)
where 2
ceffeff
LAV = . The additional factor of ½ arises if we consider the position of the
atom to be at the peak of a longitudinal standing wave inside the cavity rather than its
average value along the cavity, as would be clear if the cavity was only half a wavelength
98
long and we used the same method to define the cavity length as we have for the cavity
area, and the factor of geffn
n
−
can be seen as a correction for dispersive effects.
5.5 Analysis of Experimental Results and Discussion
The measured lifetime of 152 μsec for Erbium in the slot waveguide structures
suggests a strong non-radiative decay. However in spite of this we could still a see an
integrated collected luminescence power of 2.6 nW from these waveguides under
saturation conditions at room temperature. We now ask the question: How much power
do we expect to see coming out of these waveguides under the Purcell enhancement
condition?
Using the rigorous calculation of the emission rate into the forward direction of the
waveguide mode, including the Purcell factor, we have
active
eff
bulkradmode-waveguide-spont An
WΓ
⎟⎠⎞
⎜⎝⎛⋅⋅=
−
2
41
211 λ
πτ (5.50)
The total emitted power into the waveguide mode is
active
effact
bulkrademitted An
ALhfN
PΓ
⎟⎠⎞
⎜⎝⎛⋅⋅⋅⋅⋅
⋅=
−
22
41
21 λ
πν
τ (5.51)
where f is the fraction of erbium that is optically active.
To determine N2, the excited state erbium population, we make use of the fact that we
are pumping erbium to saturation. Using the experimentally determined value of nsp~1.6
at 1.48 μm which is known from erbium amplifiers we have
99
2
22
2
12
2
2
161
NNNNN
NNN
Nntottot
sp
−=
−−=
−≈=
)(.
totNN 72702 .≈⇒
(5.52)
The collected power would then be
collection
active
effactive
bulkrad
tot
expected
AnALhfN
P
ηλπ
ντ
⋅⋅Γ
⎟⎠⎞
⎜⎝⎛⋅⋅⋅⋅⋅⋅=
−
64041
217270 2
.. (5.53)
where we have included a factor of 0.64 to account for the integrated waveguide
transmission loss of 3 dB/cm and a ηcollection term which accounts for 10-13 dB of
waveguide to output lens tipped fiber coupling losses.
So the power which we can expect to collect under the Purcell enhancement condition
would be
[ ] fnW2613Pexpected ⋅−~ (5.54)
This figure seems quite reasonable and in comparing with the experimentally
measured value of ~2.6 nW suggests that only ~10-19% of the high density implanted Er
is optically active. This is highly likely as the erbium concentration in the slot
waveguides is quite high at 6X1020 cm-3. So our experimental results are consistent with
the calculated Purcell enhancement in the slot channel waveguides.
These results are quite interesting because if we remind ourselves of the reasons for
the desirability of lasers at the first place like their high modulation bandwidths, high
optical efficiency into the guided mode and for some applications their exteremely
narrow spectrum, we have seen that these conditions are met in slot waveguides even
without a resonator or threshold behavior. Perhaps the only quality missing is the
coherence. This implies that erbium doped slot waveguides can find application in very
100
efficient LED sources where all the emission is into a single preferred waveguide mode,
with improved efficiency resulting from the increased dominance of radiative decay over
any nonradiative decay channels, and in some cases increased modulation bandwidth
from the faster decay times.
101
Chapter 6. Future Work and Outlook
The purpose here is to set a definitive direction for future work which can be carried
out based on the theoretical and experimental findings of this thesis. A new
photolithography mask set was designed to further improve on the already verified
experimental results and to also put the new theoretical conclusions on sound
experimental footing. The new mask was designed with an aim to corroborate the “magic
radius” phenomenon in shallow ridge waveguides, seek optical gain in erbium doped
silica slot waveguides and to comprehensively verify Purcell enhancement in slot
waveguides so as to pave way for an electrically driven light source in Silicon. Here I
discuss in the design of the new mask.
We first start with a brief overview of the mask and then go into the details of its
various parts. The designs are aided by comprehensive simulation results which are
presented in detail. Finally the key findings of this thesis are summarized and outlook of
their role in silicon photonics is provided.
102
6.1 Mask Overview
The mask primarily consists of designs for (a) Passive ridge waveguides (b) Passive
horizontal Si---SiO2---Si slot waveguides (c) Critical coupling designs in ring resonators and
(d) Active horizontal Si---Er doped SiO2---Si slot waveguide devices
The mask was laid using Tanner EDA L-edit software using its programming feature
to automate the mask layout. Fig 6.1 shows the image of the mask with dimensions of 9
cm by 10 cm which is used to expose a 6’’ SOI wafer using a 1X projection photolitho-
Figure 6.1. Image showing the mask set consisting of 54 blocks arranged in a 9 by 6 grid. An
exemplary block is marked
9 cm
10 cm
103
Figure 6.2. Image of a typical individual mask block as marked in Fig 6.1
graphy system. The mask consists of 54 blocks which are arranged in a 9 by 6 grid. An
exemplary block is marked in Fig 6.1. Each block which in turn translates into a 1 cm by
1.6 cm SOI chip consists typically of 60 to 80 unique devices. Figure 6.2 shows the
image of a typical block.
6.2 Passive Ridge Waveguides
Passive ridge waveguides in the form of ring resonators were designed to verify the
“magic width” and “magic radius” phenomenon in shallow ridge waveguides, the
theoretical aspects of which were detailed in chapter 2 and 3. The base SOI to work with
1.6 cm
1.0
cm
104
now has a crystalline silicon thickness of 220 nm and a BOX thickness of 3 μm. Fig 6.3
shows the transverse TM mode count as a function of waveguide width W and ridge
height for the ridge waveguide shown in the inset with a fixed silicon core thickness of
220 nm. At a waveguide width around 1.45 μm where we expect the second order “magic
width” to be, we see that we require a ridge height of < 7.5 nm for single TM-like mode
operation. A single mode operation is preferable to minimize noise in the drop port
responses, resulting in a cleaner signal to infer the waveguide losses from. To ascertain
the sensitivity of the “magic width” to nm level ridge height variations, Fig. 6.4 shows
the loss of the fundamental TM-like mode for a straight waveguide as a function of
Figure 6.3. Transverse TM mode count for a 220 nm fixed Si core ridge waveguide shown in the
inset as a function of waveguide width and ridge height. TM mode count is indicated
10
20
30
40
50
60
70
80
0.5 1.0 1.5 2.0 2.5 3.0
Rid
ge h
eigh
t (nm
)
Waveguide Width W (µm)
Si SubstrateBOX
220 nmc-SiRidge W
1 2 3
< 7.5 nm
105
Figure 6.4. Fundamental TM mode loss for the shallow 220 nm fixed Si core ridge waveguide shown
in the inset of Fig 6.3 as a function of waveguide width for ridge heights at and in the vicinity of 7 nm
waveguide width for ridge heights at and in the vicinity of 7 nm. We see that the second
order magic width for this straight 220 nm Si core structure is 1.458 μm and shows little
variation with ridge height fluctuations.
Figure 6.5 shows the TM mode loss as a function of ring radius around 400 μm.
Traces are shown for two waveguide widths in the ring. One is for a ring waveguide
width of 1.458 μm, which is also the magic width for a straight waveguide and second
one is for 1.35 μm, which is an anti-magic width for the straight waveguide. We also
show these traces for three ridge heights of (a) 6 nm (b) 7 nm and (c) 8 nm. We see that
the TM loss is a strong function of both the ring radius and ring waveguide width. Low
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.3 1.35 1.4 1.45 1.5 1.55 1.6
6 nm7 nm9 nm
TM m
ode
Loss
(db/
cm)
Waveguide Width W (µm)
2nd order MW1.458 µm
Ridge Height
106
Figure 6.5. (Color) Fundamental TM mode loss for the shallow 220 nm fixed Si core ridge waveguide
shown in the inset of Fig 6.3 in ring resonator configuration as a function of ring radius. The width of
the waveguide in the ring (WR) is 1.458 μm and 1.35 μm. The ridge height is (a) 6 nm (b) 7 nm and
(c) 8 nm
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
399 399.143 399.286 399.429 399.572 399.715 399.858 400.001 400.144 400.287 400.43 400.573 400.716 400.859
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
399 399.143 399.286 399.429 399.572 399.715 399.858 400.001 400.144 400.287 400.43 400.573 400.716 400.859
1.E-02
1.E-01
1.E+00
1.E+01
399 399.143 399.286 399.429 399.572 399.715 399.858 400.001 400.144 400.287 400.43 400.573 400.716 400.859
7 nm
8 nm
Ridge Height 6 nm
Ring Radius (µm)
Loss
(dB
/cm
)Lo
ss (d
B/c
m)
WR 1.35 µm
WR 1.458 µm
Loss
(dB
/cm
)
(a)
(b)
(c)
107
losses can only be achieved at particular ring radii –“magic radii”, the positions of which
also depend on the width of the waveguide in the ring. These results also indicate that the
“magic radii” are sensitive to ridge height variations. Considering the case of 1.35 μm
width, at 6 nm the “magic radius” for the structure is 399.858 μm with loss <0.01 dB/cm,
however for a ridge height of 7 nm the TM mode will incur a loss of ~10 dB/cm for the
same radius.
With the goal of tracing out the curves shown in Fig. 6.5 experimentally thereby
verifying the “magic radius” phenomenon, weakly coupled ring resonators were laid out
as shown in Fig 6.2. The ring radius was varied from 399 μm to 401 μm in 0.143 μm
steps. The width of the through and drop port bus waveguides was always fixed at 1.458
μm while separate blocks were laid out for 1.458 μm and 1.35 μm ring waveguide width
case respectively. Each block also consists of different coupling gaps to vary the coupling
strengths from weak (0.06%) to strong (5.29%). This would allow us to remove the
contribution of the coupling loss to the overall loss as ascertained by measuring the drop
port responses. In addition to this, separate blocks were also laid out to account for over
and under exposure during the photolithography process as over or under exposure results
in a change in the waveguide widths from their nominal values.
6.3 Passive Horizontal Si-SiO2-Si Slot Waveguides
Ring resonator structures were laid out to determine the fundamental TM mode loss
in passive horizontal Si – SiO2 – Si slot waveguides. The intrinsic resonator quality factor
determined from these passive structures is a key input to the active slot structure design
108
as discussed later in this chapter. The passive structure was also designed with the
intention of using it as a test vehicle for the active devices. Again, starting with a base
SOI with 220 nm of crystalline silicon thickness and 3 μm BOX, the two design
parameters are thickness of various layers i.e. the c-Si layer, the SiO2 slot layer and the
amorphous silicon (a-Si) layer and the ridge height. The thicknesses of different layers
should be selected to achieve preferably single TM mode operation. The field and
weighting function confinement should be maximized in the ultra thin (5 to 15 nm) slot
layer to maximize gain; on the other hand the same should be reduced in the amorphous
silicon layer to minimize the absorption losses. From the electrical design point of view,
it is preferable to have thicker c-Si and a-Si layers. So the a-Si thickness has to be
selected to balance the electrical conductivity and absorption losses. It was also agreed
upon to avoid the long blanket thermal oxidation step utilized for thinning out the c-Si
layer thickness before the slot oxidation as it was noticed to be causing pin holes in the c-
Si layer. The designed structure should also minimize substrate leakage losses. A shallow
ridge height should be selected to minimize the side wall scattering losses and also to aid
conductive lateral electrical access to the waveguide core. The ridge height will also be a
criterion in setting the minimum ring radius that can be utilized along with the waveguide
width. It must also be pointed out that the above characteristics of the structure are
desired at both the signal wavelength (~1530 nm) and the pump wavelength (~1480 nm).
In circumstances where it is difficult or cumbersome to satisfy the conditions for both the
wavelengths, the signal wavelength design takes precedence over the pump as the signal
emission from Er is anticipated to be much weaker compared to the excitation pump.
109
Figure 6.6 (a) and (b) show the weighting function confinement in the a-Si and 10 nm
SiO2 slot layer for a 5 layer (Air – a-Si – SiO2 – c-Si – BOX) slab structure as a function
of c-Si thickness and a-Si thickness at a wavelength of 1550 nm. The darker blue region
in the top right corner of the plots corresponds to the design space where there are >1 TM
guided modes. Based on various criterion mentioned above, it is preferable to work in the
top right corner of this c-Si and a-Si layer thickness design space. However due to
practical limitations on the quality control of the a-Si at thicknesses >100 nm, an
amorphous silicon thickness of 100 nm was selected for the design. A 100 nm of a-Si and
216 nm c-Si (thickness of c-Si after 10 nm slot oxidation and avoiding any blanket
oxidation) provides 17% weighting function confinement in the 10 nm slot and 15%
confinement in the a-Si layer. The corresponding values for the 100 nm a-Si –10 nm SiO2
– 150 nm c-Si structure which closely mimics the old design are 17% and 19 %
respectively. Therefore the new design will minimize the confinement in the amorphous
silicon layer with no penalty on the confinement in the SiO2 slot. Calculations were also
performed for a 15 nm SiO2 slot structure which yielded weighting function confinement
values of 22% in SiO2 slot and 13% in a-Si for a 100 nm a-Si – 15 nm SiO2 – 213 nm c-Si
structure. Similar calculations were also performed at a wavelength of 1480 nm. The
confinement values at 1480 nm are very close to that at 1550 nm wavelength.
With the layer thicknesses fixed, a ridge height of 10 nm was chosen. Figure 6.7 (a)
and (b) show the analytical bending loss for the 3D slot waveguide structure for a ring
radius of 400 μm and 200 μm respectively. The bending loss is shown as a function of
thickness of c-Si in the core and the amorphous silicon which is uniform in both the core
as well as the cladding. Therefore, with the 10 nm ridge height, the thickness of the c-Si
110
Figure 6.6. Weighting function confinement % at 1550 nm in (a) a-Si and (b) 10 nm SiO2 slot as a
function of c-Si and a-Si thickness
WF Confinement in a Si
cSi 216nmaSi 100nm
15%
cSi 150nmaSi 100nm
19%
WF Confinement in slot
cSi 216nmaSi 100nm
17%
cSi 150nmaSi 100nm
17%
(a)
(b)
111
Figure 6.7. Analytical Bending loss at 1480 nm for a 10 nm ridge slot waveguide structure with a
radius of (a) 400 μm and (b) 200 μm as a function of the c-Si thickness in the core and amorphous
silicon thickness with the waveguide width equaling the “magic width” at 1550 nm
Log10 of Bending Loss in dB/cm
cSi 216nmaSi 100nm
1.04
cSi 150nmaSi 100nm
-1.51
Log10 of Bending Loss in dB/cm
cSi 216nmaSi 100nm
-2.33
cSi 150nmaSi 100nm
-7.53
Radius 400 µm
Radius 200 µm
(a)
(b)
112
layer in the cladding is 10 nm less than that in the core. The figures show the bending
loss at 1480 nm wavelength with the waveguide width at the corresponding 1550 nm
wavelength “magic width” of the structure as obtained using the magic width equation.
For the selected layer thicknesses of 100 nm a-Si – 10 nm SiO2 – 216 nm c-Si and a 10
nm ridge, the bending loss at 400 μm is 4.67X10-3 dB/cm and for 200 μm is 10.96 dB/cm.
The corresponding values at 1550 nm are slightly less severe at 3.09X10-3 dB/cm and
8.71 dB/cm respectively and the values for the 15 nm slot layer structure (100 nm a-Si –
15 nm SiO2 – 213 nm c-Si and a 10 nm ridge) are slightly better than the 10 nm slot case.
We also notice that the old slot structure has much lower bending loss values compared
to the new design as indicated in the figure. For the 10 and 15 nm slot structures with 10
nm ridge height, for the bending loss to be <1X10-3 dB/cm requires that the ring radius be
>410 μm. With this in mind, radii of 400 μm and 450 μm were selected for the slot
waveguide structures.
Figure 6.8 (b) shows the fundamental TM mode loss of the 15 nm slot, 10 nm ridge
waveguide (Fig 6.8 (a)) as determined using COMSOL FEM. The actual slanted profile
of the waveguide side walls as determined using AFM was utilized in the simulation. The
loss is plotted as a function of the width of the c-Si ridge waveguide, as this can be
directly related to the width on the mask. The loss is shown at both the signal (1530 nm)
and the pump (1480 nm) wavelength. Material dispersion was taken into account in doing
these calculations. As we can see the TM mode loss minima for the two wavelengths are
at slightly different waveguide widths. Choosing to design at the “magic width” of 1.605
μm at the signal wavelength, the loss at 1480 nm at this waveguide width is ~0.1 dB/cm.
113
Figure 6.8. (a) Schamatic of a 10 nm ridge, 100 nm a-Si – 15 nm SiO2 – 213.25 nm c-Si slot waveguide
structure (b) Fundamental TM mode loss of this straight waveguide at 1530 nm and 1480 nm as a
function of the width W of the c-Si ridge
Figure 6.9 (a) and (b) show the TM mode loss for the 15 nm slot, 10 nm ridge
structure shown in Fig 6.8 (a) as a function of ring radius around 400 μm and 450 μm
respectively at a wavelength of 1530 nm. The width of the c-Si ridge waveguide in the
ring is 1.605 μm (“magic width” for the straight waveguide at 1530 nm). Traces are
100 nm a-Si
Si Substrate
BOX 203.25 nm c-Si
15 nm SiO2
10 nm Ridge
W
Waveguide Width W, C-Si Ridge (µm)
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8
1.48 mic 1.53 mic
Loss
(dB
/cm
)
1.605 µm @1530 nm1.585 µm @1480 nm
15 nm Slot10 nm Ridge
(a)
(b)
114
shown for 9 nm, 10 nm and 11 nm ridge heights. Similar to the passive ridge waveguides,
we see that the positions of the “magic radii” are sensitive to the ridge height variations.
To verify the positions of the “magic radii”, weakly coupled ring resonators were laid out
using the same layout principle as detailed in Sec. 6.2 for passive ridge waveguides.
Structures were also designed for the 10 nm slot case.
Figure 6.9. (Color) Fundamental TM mode loss for the 15 nm slot, waveguide shown in Fig. 6.8 (a) in
ring resonator configuration as a function of ring radius around (a) 400 μm and (b) 450 μm. The
width of the c-Si ridge waveguide in the ring is 1.605 μm. Traces shown for ridge heights of 9 nm, 10
nm and 11 nm
(a)
(b) 1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
451 451.273 451.546 451.819 452.092 452.365 452.638 452.911 453.184 453.457 453.73
9 nm 10 nm 11 nm
Ring Radius (µm)
Loss
(dB
/cm
)
Ridge Height
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
400 400.231 400.462 400.693 400.924 401.155 401.386 401.617 401.848 402.079 402.31 402.541 402.772
9 nm 10 nm 11 nm
Loss
(dB
/cm
)
Ring Radius (µm)
Ridge Height
115
6.4 Ring Resonators for Critical Coupling in Horizontal Si-SiO2-Si Slot
Waveguides
A ring resonator is a travelling-wave-resonator (TWR). Therefore by appropriate
design of the waveguide-resonator coupling, a complete 100% transmission of energy
from a bus waveguide to the resonator is possible[29]. From the active device
perspective, critical coupling at the pump wavelength can be beneficial for the judicious
utilization of the pump power for pumping erbium in the slot in a ring resonator
configuration. Engineering of the waveguide-resonator coupling structure to achieve
critical coupling can be quite challenging. Here we discuss our attempts to achieve
critical coupling in the shallow ridge slot waveguides.
Fig 6.10 (a) shows the TWR ring resonator coupled to a bus waveguide. In such
TWR-waveguide structures, in the weak coupling regime, the power transmission and the
total loaded quality factor (QL) at resonance are expressed as
2
0 11
CO
CO
in
out
QQQQ
PP
T//
)(+−
==ω (6.1)
111 −−− += COL QQQ (6.2)
where QO and QC are the intrinsic and coupling quality factors respectively. The coupling
quality factor is used as a measure of the rate of energy decay from the cavity to the
waveguide. Hence, stronger the waveguide-resonator coupling, smaller is the QC. As can
be inferred from Eq. 6.1, in order to completely transfer energy from the waveguide to
the resonator at resonance, requires QO to be equal to QC and as a result the loaded
quality factor QL becomes half of QO. This condition is referred to as the critical
116
coupling. Therefore from Eq. 6.1 and 6.2, by changing QC, both QL and the power
transmission through the bus waveguide can be controlled as shown in Fig. 6.10 (b).
Figure 6.11 shows the coupling quality factor for the 15 nm slot, 10 nm ridge
waveguide structure as a function of the minimum gap between the straight bus
waveguide and the ring as obtained using RSOFT 2D simulations. The bus waveguide is
Figure 6.10. (a) Schematic of TWR ring resonator coupled to a bus waveguide (b) Power
transmission through the bus waveguide as a function of the normalized loaded quality factor (QL)
Pin Pout
ωOQO
-50
-40
-30
-20
-10
0
0.1 0.3 0.5 0.7 0.9
QC <QO QC >QO
Over Coupled
UnderCoupled
Pow
er T
rans
mis
sion
(dB
)
QL/ QO
(a)
(b)
117
Figure 6.11. Coupling Quality factor for the 15 nm slot, 10 nm ridge structure as a function of the
minimum gap between the tangential bus waveguide and the ring as shown in Fig 6.12 (a)
Figure 6.12. Section of a bus waveguide-resonator system showing the coupler configuration (a)
Regular coupler - Bus waveguide running tangentially to the ring with gap of G μm at minimum
separation between the two (b) Curved coupler - Bus waveguide wrapped around the ring to increase
the interaction length and hence to reduce the QC at a gap of G μm. The curved interaction length is
parameterized by the arc angle θ
Regular Coupler
Ring
BusCurved Coupler
Ring
Bus
θ
(a) (b)
1.E+05
1.E+06
1.E+07
1.E+08
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
RR400,Lam1.53RR400,Lam1.48RR450,Lam1.53RR450,Lam1.48
Cou
plin
g Q
, QC
Gap, G (µm)
4.E+05
118
running tangentially to the ring as shown in Fig 6.12 (a). The QC is shown for ring radii
of 400 μm and 450 μm at both the pump (1480 nm) and signal (1530 nm) wavelengths.
The width of the waveguide in both the ring and the straight sections is at the “magic
width” at 1530 nm. Assuming an intrinsic quality factor of 4X105, we see that the
coupling gap needs to be ~0.8 μm to achieve the critical coupling condition i.e. QO= QC.
But, our photolithography system limits the minimum achievable gap for this 15 nm slot
structure to 1.152 μm at which QC > QO and hence the ring will be under coupled at this
gap. So in order to satisfy the critical coupling condition at the larger gap of 1.152 μm
requires increased coupling strength at this gap. This can be achieved by increasing the
interaction length between the bus waveguide and the ring using the configuration shown
in Fig 6.12(b). Figure 6.13 now shows the QC for the same structure evaluated in Fig 6.11
Figure 6.13. (Color) Coupling Quality factor for the 15 nm slot, 10 nm ridge waveguide-resonator
system with a curved coupler as a shown in Fig 6.12 (b). Arc angle θ is defined in Fig. 6.12(b). Gap is
fixed at 1.152 μm
1.00E+05
1.00E+06
0 1 2 3 4 5 6 7
RR400,Lam1.53RR400,Lam1.48RR450,Lam1.53RR450,Lam1.48
Arc Angle (º)
4.E+05
Cou
plin
g Q
, QC
119
as a function of the arc angle θ as defined in Fig 6.12 (b). The gap is now fixed at 1.152
μm. We notice that for arc angles > 4.2° and > 5.8°, a QC of < 4X105 can be achieved at
1530 nm and 1480 nm respectively for both 400 and 450 μm radius rings for this 15 nm
slot structure.
Similar calculations were also performed for the 10 nm slot, 10 nm ridge structure.
Structures to verify the critical coupling were laid out consisting of ring resonators with
curved coupler configurations with the arc angle θ increased incrementally from 1° to 7°.
6.5 Active Horizontal Si-Er doped SiO2-Si Slot Waveguides
Designs for erbium doped silica slot waveguides were incorporated with the aim of
optically pumping these structures at 1480 nm to achieve optical gain in a ring resonator
configuration. The key design parameter here is the intrinsic quality factor Q of the
resonator which is obtained by doing loss measurements on a passive horizontal slot ring
resonator. Assuming we can achieve an intrinsic resonator Q of 4X105, we summarize
here results predicting the behavior of our erbium doped silica slot waveguide ring laser.
The analysis is based on coupled mode formalism for a resonator coupled to an
external waveguide[47]. As an exemplary case, we consider 100 nm a-Si – 15 nm Er
doped SiO2 – 213 nm c-Si, 10 nm ridge horizontal slot structure configured as a 450 μm
ring resonator coupled to a straight bus waveguide as shown in Fig 6.14. A pump at 1480
nm is coupled to the ring through the bus waveguide to excite the erbium in the slot so as
to collect erbium signal emission at the output of the bus waveguide. Figure 6.15 shows
120
Figure 6.14. (a) Exemplary erbium doped slot waveguide ring resonator coupled to a bus waveguide
to aid pumping of erbium in the slot (b) Cross section of the slot waveguide constituting the ring
Figure 6.15. (Color) Minimum erbium concentration required in the slot for the structure shown in
Fig. 6.14 to reach threshold as a function of the coupling gap G for different intrinsic resonator Qs
the minimum erbium concentration required in the slot to reach threshold at a particular
coupling gap G for different intrinsic resonator Qs. The minimum erbium concentration
required to reach threshold increases as the gap decreases for a given intrinsic Q.
(a) (b)
100 nm a-Si
Si Substrate
BOX 203.25 nm c-Si
15 nm Er/SiO2
10 nm Ridge
1.605 µm
PPin PPout
Q
450 µm
SPout
λ1480 nm
Gap, G (µm)
Erbi
um C
once
ntra
tion
(m-3
)
7.6763e26 5.6642e26
121
Assuming an intrinsic Q of 4X105, which is in the range of best results we have achieved
so far for the TM-like mode in horizontal slot waveguides, we see that we require the
minimum erbium concentration in the slot to be 7.7X1020 cm-3 to reach threshold for all
gaps >1.1 μm. As the achievable resonator Q improves, threshold can be reached for
lower erbium concentrations in the slot. Figure 6.16 shows the pump threshold power as a
function of the coupling gap with the erbium concentration fixed at 7.7X1020 cm-3 and
resonator Q of 4X105. Threshold powers < 10 mw can be achieved for gaps in the range
of 1.2 μm to 1.9 μm. It must however be pointed out that in these calculations we have
neglected co-operative upconversion, excited state absorption and pair induced quenching
effects which can be prevalent at such high erbium concentrations. Fig. 6.17 shows the
expected lasing power at 1530 nm from this 450 μm ring resonator with a Q of 4X105 and
7.7X1020 cm-3 slot erbium concentration as a function of coupling gap and pump power.
Figure 6.16. Threshold pump power as a function of coupling gap for the ring resonator
configuration shown in Fig. 6.14 with Q = 4X105 and slot erbium concentration of 7.7X1020 cm-3
Gap, G (µm)
Thre
shol
d Po
wer
(mW
) >
122
Figure 6.17. Output lasing power at 1530 nm as a function of pump power and coupling gap for the
configuration shown in Fig. 6.14 with Q = 4X105 and slot erbium concentration of 7.7X1020 cm-3
The white portion of the plot corresponds to the region where the structure doesn’t reach
threshold for the given parameters. We see that we can expect lasing power of ~ 90 μW
at a gap of 1.3 μm when pumped at 40 mW.
Ring resonator structures at “magic radii” and “magic width” with single coupler and
coupling gaps in the range of 1.1 to 1.7 μm were laid out in a similar scheme as shown in
Fig 6.2. Calculations were also done for the 10 nm slot case where minimum erbium
concentration to reach threshold is higher due smaller pump/signal confinement in the
thinner slot compared to the 15 nm case.
Pum
p Po
wer
(m
W)
Gap, G (µm)
µW
123
6.6 Summary and Future Work
A horizontal slot waveguide operating in TM mode is arguably the most practical
architecture for achieving an electrically pumped silicon based light source. The design
allows for fabrication using standard CMOS planar processing and provides the
necessary means for electrical excitation of the slot embedded gain media. This compared
to other architectures like micro-toroids or disks on pedestals which require elaborate
fabrication and testing mechanisms and provide absolutely no means for electrical
excitation of the gain media. Usually such complex fabrication techniques are resorted to
for essentially one simple reason – to reduce the loss i.e. to achieve ultra high Q
resonators. However doing so makes them unsuitable for wide deployment.
Minimizing the losses in waveguide architectures suitable for light sources in silicon
photonics i.e. a horizontal slot waveguide, has been the overarching theme of this thesis.
This has required overcoming some stringent conceptual barriers, devising novel
fabrication techniques and improvements in material quality.
Before our work, it was usually argued that the TM like mode in shallow ridge
waveguides is always leaky and hence high loss. However, TM mode operation is
fundamentally critical to our application due to its high confinement in the slot. A
comprehensive analysis of this problem was presented and we showed that this TM mode
leakage loss can be mitigated by simple width control of the waveguide to “magic
widths”.
Achieving smooth waveguide surfaces is absolutely critical for achieving low loss in
a high index contrast system like SOI. However the challenge here is to do so using
techniques which still fit into the frame work of standard CMOS planar processing.
124
Novel fabrication techniques based on thermal oxidation were devised which yielded
ultra-smooth waveguide surfaces. Shallow ridge waveguides fabricated using this method
were demonstrated with a Q of 1.6X106 for the TE-like mode and when combined with
the “magic width” designs yielded a Q of 6.8X105 for the TM-like mode, the highest yet
reported for such shallow ridge structures with tight vertical confinement.
Comprehensive full vector numerical models were also developed for rigorous
analysis and quick design of these structures. These techniques, namely mode matching
(MM) and finite element method (FEM) were then utilized to analyze the lateral leakage
phenomenon in structures which usually don’t lend themselves to simple
phenomenological methods such as ring resonators and directional couplers, which are
key components in integrated photonics. In analyzing these, new anomalous losses, over
and above any reported values came forth. To counteract these losses, new design
principles like the “magic radii” were conceived and elucidated to achieve highest
performance in such devices.
These novel design concepts and fabrication techniques were then applied to the
fabrication of horizontal slot waveguides. As the design required utilization of amorphous
silicon as the top cladding layer which can be high loss, considerable improvement in the
optical quality of the PECVD deposited amorphous silicon had to be achieved. With ~3
dB/cm loss for our amorphous silicon, puts it in the league of the best results reported by
other groups. This combined with a carefully engineered design at “magic widths”
resulted in a Q of 3X105 for the TM-like mode in a ultra thin 8 nm horizontal slot
waveguide.
125
Erbium was then implanted into a silica slot just 8 nm thick. A relatively higher
dosage of erbium at 6X1020 cm-3 was utilized as a safety net against the very low
implantation energies of just 2 keV. Nevertheless, erbium photoluminescence was
observed in full waveguide configuration primarily made possible due to the low TM-like
mode losses of these waveguides. A noise analysis based test setup was designed to
extract the temporal characteristics from the very weak erbium luminescence signal
which indicated some unusually fast erbium lifetime which didn’t seem to fit the
conventional wisdom. Other groups working on similar structures also reported seeing
similar results around the same time and attributed these fast lifetimes in slot waveguides
to enhanced spontaneous emission rate or Purcell enhancement, a phenomenon usually
associated with high Q resonators and explained it using a quantum electrodynamic
formalism. We however showed that the conventional simple expressions for
spontaneous emission into a waveguide mode correctly capture the Purcell enhancement
in slot waveguides and showed our experimental results in slot waveguides to be
consistent with this phenomenon. A full numerical quantum electrodynamic analysis of
these slot structures was also carried out to further buttress the simple “fraction of
spontaneous emission into waveguide mode” model. This enhanced spontaneous
emission rate characteristic of the slot waveguides make them even more interesting as it
means that laser like characteristics – preferential emission into single waveguide mode,
high modulation bandwidth can now be achieved in slot waveguides even with out a
resonator and threshold behavior. This means slot waveguides can be the basis for very
efficient LEDs, without the very stringent constraint of achieving gain.
126
The novel waveguide architectures detailed in this thesis can play a crucial role for
photonic integration in silicon. The ultra low loss shallow ridge waveguides are ideal
candidates for high performance sensors, tunable filters and ultra small foot print
polarizers. Comprehensive simulation techniques which predict their behavior with great
accuracy have been put in place to allow their comprehensive analysis and seamless
integration.
In line with the findings of this thesis, the future work will be geared towards putting
not yet experimentally proven theoretical aspects like the “magic radius” phenomenon on
sound experimental footing. It is also anticipated that by using a much lower erbium
concentration in the slot compared to the present case of 6X1020 cm-3 would allow a more
robust verification of Purcell enhancement in slot waveguides and evaluating its
consequences on light sources in silicon. To study these and to further improve on the
already established results, an improved mask was designed and laid out.
127
Appendix A
Spontaneous Emission Rate in a Slot Waveguide
The formulation presented here closely follows the one given in [54] where the
authors apply quantum electrodynamics to calculate the zero-point fluctuations of the
electromagnetic field and the spontaneous emission rate from a non-absorbing dielectric
film bounded by two non-absorbing dielectric half spaces of arbitrary refractive indices.
Here we extend their methodology to the five layer slot structure i.e. Si-SiO2-Si slabs
bounded by air and SiO2 dielectric half spaces. The entire derivation is given here for
completeness.
The quantization of the electromagnetic field requires that the complete set of spatial
electromagnetic modes that exist in the configuration be determined. These modes must
be orthonormalized with respect to the scalar product corresponding to the
electromagnetic energy density. For the dielectric film the set of modes is well known. It
consists of radiation modes pertaining to plane waves that are incident from either
dielectric half space for both orthogonal polarizations. In addition, guided modes of either
polarization may exist. These modes are evanescent in both dielectric half spaces.
128
We will list the radiation and guided modes in Sec. A.2 and A.3 respectively. The
electromagnetic field will be quantized in Sec. A.4. In Sec. A.5 formulas for the
spontaneous emission rate and the zero-point field fluctuations will be derived. It should
be mentioned that the local-field effects on a molecular scale may be very important in
determining the spontaneous emission rate [61]. However as we are only interested in the
emission rate of the film relative to the bulk value, the spatial electromagnetic modes do
not have to be corrected for local field effects.
A.1 Notations
Considering an electromagnetic plane wave in a medium that is translation invariant
in the x and y directions of a cartesian co-ordinate system (x, y, z). Let the medium have a
refractive index n. The time dependence of time harmonic fields will be given by the
factor exp(-iωt), where ω>0 is the frequency. Let k =(kx , ky , kz) be the wave vector of the
plane wave so that
22222 nkkkk zyx =++ (A.1)
where k= ω(ε0 μ0)1/2 is the wave number in vacuum. The components kx and ky are
assumed real but kz = (k2 n2 - kx2
– ky2)1/2 is purely imaginary when kx
2 + ky
2 > k2 n2. We
choose the branch of the complex square root such that the cut is along the negative real
axis so that when a is positive, a½ is positive and (-a) ½ = + i a½. Hence kz is positive
imaginary when kx2
+ ky2 > k2 n2 and the wave is then evanescent in the z direction. The
plane wave can be written in a unique way as a linear combination of two orthogonally
polarized plane waves, namely TE or S polarized wave whose electric field vector is
129
parallel to the (x, y) plane and a TM or P polarized plane wave for which the magnetic
field vector is parallel to the (x, y) plane. We introduce the following two unit vectors
corresponding to the electric field vector of the two polarizations:
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛−
+=
0
1
21
22x
y
yx
kk
kkSk
)(),(ι (A.2)
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
+−+++=
)()()(),(ˆ
2221
2221
222
1
yx
zy
zx
yxzyx kkkkkk
kkkkkPkι (A.3)
Then the S- polarized plane wave is given by
),(ˆ)( Sk)rkexp(iArE ι⋅= (A.4)
),(ˆ)()(
)( Pkrkiexpk
kkkArH yxz ι
με
⋅++
⎟⎟⎠
⎞⎜⎜⎝
⎛=
21
22221
0
0 (A.5)
where A is the amplitude of the electric field strength. Similarly for the TM or P-
polarized waves we have
),(ˆ)( Pk)rkexp(iArE ι⋅= (A.6)
),(ˆ)()(
)( Skrkiexpkkk
nkArH
yxz
ιμε
⋅++
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
21
222
221
0
0 (A.7)
Figure A.1 shows the configuration we will study here. The Cartesian coordinate system
(x, y, z) is chosen with its origin in the middle of the layer with index n1 which in our case
will be the slot layer. The z axis is perpendicular to the interfaces. For given real kx and ky
we will use the notation:
( ) 54321212222 ,,,,
/=−−≡ jforkknkk yxjjz (A.8)
130
Figure A.1. A five layer dielectric slab structure representative of the slot waveguide
where the branch of the square root was described above. In addition +jk and −
jk are the
wave vectors given by
),,(,),,( jzyxjjzyxj kkkkkkkk −== −+ (A.9)
As the time dependence of the fields is assumed to be exp(-iωt), it implies that when kjz is
real, +jk and −
jk are the wave vectors of plane waves that propagate in the positive and
the negative z directions respectively. When kjz is imaginary, these plane waves decay
exponentially in the positive and the negative z directions respectively.
n5
n3
n1
n2
n4
z=d/2xy
z
z=-d/2
z=-(d/2)-b
z=(d/2)+aAir
Si
SiO2
Si
SiO2
131
Finally, we recall that for a plane wave impinging on the interface between two
dielectrics with refractive indices n1 and n2 incident from dielectric 1 as shown in the Fig.
A.2, the reflection and transmission coefficients are given by
onpolarizatiSforkk
krt
kkkkr
zz
z
zz
zz
21
11212
21
2112
21+
=+=
+−=
(A.10)
and
.onpolarizatiPfor
nk
nk
nk
rt
nk
nk
nk
nk
r
zz
z
zz
zz
22
22
1
1
21
1
1212
22
22
1
1
22
22
1
1
12
21
+=+=
+
−=
(A.11)
Figure A.2. Illustration of the definition of the reflection and transmission coefficients r12 and t12
n1
n2
z
t12
r12
132
A.2 Radiation Modes
We first start by describing the radiation modes that can exist in the configuration of
Fig. A.1. We distinguish between modes that are incident from medium 4 and modes that
are incident from medium 5 and between S and P polarizations.
Mode incident from medium 4
For every frequency ω and every kx and ky, satisfying
24
222 nkkk yx <+ (A.12)
where k= ω(ε0 μ0)1/2, there are S and P polarized electromagnetic fields in the structure
consisting of plane waves and standing waves of which kx and ky are the components of
the wave vectors in the x and y directions. The electric field of the S polarized mode is
given by
[ ]
[ ]
[ ][ ]
[ ]⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪
⎨
⎧
⎟⎠⎞
⎜⎝⎛ +−≤⋅+⋅
⎟⎠⎞
⎜⎝⎛−≤≤⎟
⎠⎞
⎜⎝⎛ +−⋅+⋅
⎟⎠⎞
⎜⎝⎛≤≤⎟
⎠⎞
⎜⎝⎛−⋅+⋅
⎟⎠⎞
⎜⎝⎛ +≤≤⎟
⎠⎞
⎜⎝⎛⋅+⋅
⎟⎠⎞
⎜⎝⎛ +≥⋅
=
−−++
−−++
−−++
−−++
++
bdzforSkrkiexpBSkrkiexpA
dzbdforSkrkiexpDSkrkiexpCA
dzdforSkrkiexpFSkrkiexpEA
adzdforSkrkiexpHSkrkiexpGA
adzforSkrkiexpIA
rE
SS
SSS
SSS
SSS
SS
2
22
22
22
2
444444
2242244
1141144
3343344
5544
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(
)(
ιι
ιι
ιι
ιι
ι
(A.13)
where
133
[ ] [ ] [ ]
[ ] [ ]
[ ] [ ] [ ]
122
2222
222222
222222
22
4212
42113
424
41241312
432135
432351312
431352412
43352413
4 Functionkbdbkiexpr
kbdbkdkiexprkbdiexpr
kbddkiexprrrkbdakbkdkiexpr
kbdakbkiexprrrkbdakdkiexprrr
kbdakiexprrr
B zz
zzz
z
zz
zzzz
zzz
zzz
zz
S )()(
)()(
)()()(
)(
+−−+−++
+−−+−+
+−++++−+−+−++
+−−
=
(A.14)
122
222
22
22
22
2
4242
421131242
4321351242
432351342
4 Function
kbdkbdiexpt
kbdkbddkiexprrt
kbdakkbddkiexprrt
kbdakkbdiexprrt
Czz
zzz
zzzz
zzz
S⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ ++
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ ++−
⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−+⎟
⎠⎞
⎜⎝⎛ ++−
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−+⎟
⎠⎞
⎜⎝⎛ +
=
(A.15)
122
222
22
22
22
2
421242
4211342
43213542
43235131242
4 Function
kbdkbdiexprt
kbdkbddkiexprt
kbdakkbddkiexprt
kbdakkbdiexprrrt
Dzz
zzz
zzzz
zzz
S⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ −−−
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ −−+
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−+⎟
⎠⎞
⎜⎝⎛ −−+
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−+⎟
⎠⎞
⎜⎝⎛ −−−
=
(A.16)
12
22
22
432135134221
4214221
4 Function
kbdakbkkdiexprrtt
kbdbkkdiexptt
Ezzzz
zzz
S⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−++⎟
⎠⎞
⎜⎝⎛+
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−+⎟
⎠⎞
⎜⎝⎛
=
(A.17)
134
12
22
3
223
4321354221
421134221
4 Function
kbdakbkkdiexprtt
kbdbkkdiexprtt
Fzzzz
zzz
S⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−++⎟
⎠⎞
⎜⎝⎛+
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−+⎟
⎠⎞
⎜⎝⎛
=
(A.18)
122 4321422113
4 Function
kbdkdbkdkiexptttG
zzzzS
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛−+
= (A.19)
12
22 432135132142
4 Function
kbdkadbkdkiexprtttH
zzzzS
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ +++
=
(A.20)
122 432135132142
4 Function
kadkbdakbkdkiexpttttI
zzzzS
⎢⎣
⎡
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ +−++
=
(A.21)
and
[ ] [ ] [ ] [ ] [ ] [ ] [ ]z
zzz
zzzz
zzzz
bkiexprrakdkiexprrakiexprr
akbkdkiexprrdkiexprrbkdkiexprrakbkiexprrrr
Function
22412
31351233513
321352411312
2124133235241312
22222
2211
++−+
++−−+−++
=
(A.22)
with the reflection and transmission coefficients defined by Eq. A.10. When the factor
SA4 is chosen such that
( )21
4234 2
1⎟⎟⎠
⎞⎜⎜⎝
⎛=
z
PorS
kkA /π
(A.23)
it follows that for every k~ , xk~ and yk~ satisfying Eq. A.12, with E~
being the electric field
of the corresponding S polarized mode, we have
135
∫∫∫ −−−=⋅ )~()~()~(~
* kkkkkkrdEE yyxx δδδεε
0
(A.24)
where ε denotes the piecewise constant function, which is equal to εj in medium j and
where δ is the Dirac’s delta function. All radiation modes are parameterized by the triplet
kx , ky , k.
The electric field of the P-polarized mode corresponding to kx, ky, k and satisfying Eq.
A.12 is given by
[ ]
[ ]
[ ]
[ ]
[ ]⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪
⎨
⎧
⎟⎠⎞
⎜⎝⎛ +−≤⋅+⋅
⎟⎠⎞
⎜⎝⎛−≤≤⎟
⎠⎞
⎜⎝⎛ +−⋅+⋅
⎟⎠⎞
⎜⎝⎛≤≤⎟
⎠⎞
⎜⎝⎛−⋅+⋅
⎟⎠⎞
⎜⎝⎛ +≤≤⎟
⎠⎞
⎜⎝⎛⋅+⋅
⎟⎠⎞
⎜⎝⎛ +≥⋅
=
−−++
−−++
−−++
−−++
++
bdzforPkrkiexpBPkrkiexpA
dzbdforPkrkiexpDPkrkiexpCnnA
dzdforPkrkiexpFPkrkiexpEnnA
adzdforPkrkiexpHPkrkiexpGnnA
adzforPkrkiexpInnA
rE
PP
PPP
PPP
PPP
PP
2
22
22
22
2
444444
2242242
44
1141141
44
3343343
44
5545
44
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(
)(
ιι
ιι
ιι
ιι
ι
(A.25)
where PPPPPPPP IandHGFEDCB 44444444 ,,,,,, are given by Eqs.A.14 through A.22 but now
the reflection and transmission coefficients are given by Eq. A.11. The choice of PA4 as
given by Eq. A.23 again leads to the satisfaction of the orthonormality relation given by
Eq. A.24 for the electric fields of the two P-polarized modes.
Mode incident from medium 5
For all values of k, kx and ky satisfying
25
222 nkkk yx <+ (A.26)
there are S and P polarized electromagnetic fields that are incident from medium 5 and
consist of plane waves and standing waves of which kx and ky are the components of the
136
wave vectors in the x and y directions. The electric field of the S polarized mode is given
by
[ ]
[ ]
[ ][ ]
[ ]⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪
⎨
⎧
⎟⎠⎞
⎜⎝⎛ +−≤⋅
⎟⎠⎞
⎜⎝⎛−≤≤⎟
⎠⎞
⎜⎝⎛ +−⋅+⋅
⎟⎠⎞
⎜⎝⎛≤≤⎟
⎠⎞
⎜⎝⎛−⋅+⋅
⎟⎠⎞
⎜⎝⎛ +≤≤⎟
⎠⎞
⎜⎝⎛⋅+⋅
⎟⎠⎞
⎜⎝⎛ +≥⋅+⋅
=
−−
++−−
++−−
++−−
++−−
bdzforSkrkiexpIA
dzbdforSkrkiexpHSkrkiexpGA
dzdforSkrkiexpFSkrkiexpEA
adzdforSkrkiexpDSkrkiexpCA
adzforSkrkiexpBSkrkiexpA
rE
SS
SSS
SSS
SSS
SS
2
22
22
22
2
4455
2252255
1151155
3353355
555555
),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
)(
ι
ιι
ιι
ιι
ιι
(A.27)
where
[ ] [ ] [ ]
[ ] [ ]
[ ] [ ] [ ]
1222
222222
2222
2222
222
53112
5313
532124
532241312
535
51351312
52352412
521352413
5 Functionkadakdkiexpr
kadakiexprkadakbkdkiexpr
kadakbkiexprrrkadiexpr
kaddkiexprrrkadbkiexprrr
kadbkdkiexprrr
B zzz
zz
zzzz
zzz
z
zz
zz
zzz
S )()(
)()(
)()()(
)(
+−+++−−
+−++++−+−
+−−+−++−−
+−+
=
(A.28)
137
122
222
222
2222
5353
531131253
532241253
5321241353
5 Function
kadkadiexpt
kadkaddkiexprrt
kadkadbkiexprrt
kadkadbkdkiexprrt
Czz
zzz
zzz
zzzz
S⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ ++
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ ++−
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ +++
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ +++−
=
(A.29)
122
2
22
2222
222
5311253
531353
53212453
53224131253
5 Function
kadkaddkiexprt
kadkadiexprt
kadkadbkdkiexprt
kadkadbkiexprrrt
Dzzz
zz
zzzz
zzz
S⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ −−+
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ −−−
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ −−++
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ −−−
=
(A.30)
12
22
22
532124125331
5315331
5 Function
kadakbkkdiexprrtt
kadakkdiexptt
Ezzzz
zzz
S⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−++⎟
⎠⎞
⎜⎝⎛+
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−+⎟
⎠⎞
⎜⎝⎛
=
(A.31)
122
3
22
23
531125331
5321245331
5 Function
kadakkdiexprtt
kadakbkkdiexprtt
Fzzz
zzzz
S⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−+⎟
⎠⎞
⎜⎝⎛+
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−++⎟
⎠⎞
⎜⎝⎛
=
(A.32)
122 5321533112
5 Function
kadakkddkiexptttG
zzzzS
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−+⎟
⎠⎞
⎜⎝⎛−
= (A.33)
138
12
22 532124533112
5 Function
kadakkbddkiexprtttH
zzzzS
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−+⎟
⎠⎞
⎜⎝⎛ ++
= (A.34)
122 5432124123153
5 Function
kadkbdakbkdkiexpttttI
zzzzzS
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛ +−++
=
(A.35)
and Function1 is same as defined in Eq. A.22. When the factor SA5 is chosen such that
( )21
5235 2
1⎟⎟⎠
⎞⎜⎜⎝
⎛=
z
PorS
kkA /π
(A.36)
the electric field is orthonormal in the sense of Eq. A.24. Similarly, for the P-polarized
modes that are incident from medium 5, the electric field is
[ ]
[ ]
[ ]
[ ]
[ ]⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪
⎨
⎧
⎟⎠⎞
⎜⎝⎛ +−≤⋅
⎟⎠⎞
⎜⎝⎛−≤≤⎟
⎠⎞
⎜⎝⎛ +−⋅+⋅
⎟⎠⎞
⎜⎝⎛≤≤⎟
⎠⎞
⎜⎝⎛−⋅+⋅
⎟⎠⎞
⎜⎝⎛ +≤≤⎟
⎠⎞
⎜⎝⎛⋅+⋅
⎟⎠⎞
⎜⎝⎛ +≥⋅+⋅
=
−−
++−−
++−−
++−−
++−−
bdzforPkrkiexpInnA
dzbdforPkrkiexpHPkrkiexpGnnA
dzdforPkrkiexpFPkrkiexpEnnA
adzdforPkrkiexpDPkrkiexpCnnA
adzforPkrkiexpBPkrkiexpA
rE
PP
PPP
PPP
PPP
PP
2
22
22
22
2
4454
55
2252252
55
1151151
55
3353353
55
555555
),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
)(
ι
ιι
ιι
ιι
ιι
(A.37)
where PPPPPPPP IandHGFEDCB 55555555 ,,,,,, are given by Eqs.A.28 through A.35 but now
the reflection and transmission coefficients are given by Eq. A.11. The choice of PA5 as
given by Eq. A.36 again leads to the satisfaction of the orthonormality relation given by
Eq. A.24 for the electric fields of the two P-polarized modes.
139
A.3 Guided Modes
As we are specifically interested in slot waveguides – we have a specific layered
configuration. We have n4 (Silica -BOX) =1.444, n2 (Crystalline silicon) =3.4797, n1 (Slot
oxide) =1.444, n3 (Amorphous silicon) =3.4797 and n5 (Air) =1 at 1550 nm wavelength.
The guided modes may exist when
( ) ( ) ( )
( ) ( ) ( )4797.3444.1
..2/122
23
2/12241
kkkk
einornkkknornk
yx
yx
<+<
<+<
(A.38)
The guided modes are evanescent in n5(Air) and n4(BOX-SiO2). Hence k5z and k4z must
be purely imaginary. Also in our case of the slot structure, k1z will also be purely
imaginary. The length of the projection of the wave vector on the (x, y) plane,
( ) 2/122yx kk +≡β , is called the propagation constant of the guided mode. For a given
value of the wave number k in vacuum, or equivalently for a given frequency ω, there are
at most a finite number of propagation constants in the interval defined by Eq. A.38. The
set of βs are different for the two polarizations. We will consider S and P polarization
separately.
S Polarized Guided Modes
The relation to be satisfied by the propagation constant β of S-polarized guided mode
is
[ ] [ ] [ ] [ ] [ ] 12
2222
22
22412
31351233513
321352411312
2124133235241312
=−++−
+++++++−
z
zzz
zzzz
zzzz
bkiexprrakdkiexprrakiexprr
akbkdkiexprrdkiexprrbkdkiexprrakbkiexprrrr ][][
(A.39)
where the reflection coefficients are given by Eq. A.10.
140
The electric field of an S-polarized guided mode is given by
[ ]
[ ]
[ ][ ]
[ ]⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪
⎨
⎧
⎟⎠⎞
⎜⎝⎛ +−≤⋅
⎟⎠⎞
⎜⎝⎛−≤≤⎟
⎠⎞
⎜⎝⎛ +−⋅+⋅
⎟⎠⎞
⎜⎝⎛≤≤⎟
⎠⎞
⎜⎝⎛−⋅+⋅
⎟⎠⎞
⎜⎝⎛ +≤≤⎟
⎠⎞
⎜⎝⎛⋅+⋅
⎟⎠⎞
⎜⎝⎛ +≥⋅
=
−−
−−++
−−++
−−++
++
bdzforSkrkiexpJE
dzbdforSkrkiexpHSkrkiexpGE
dzdforSkrkiexpFSkrkiexpE
adzdforSkrkiexpDSkrkiexpCE
adzforSkrkiexpAE
rE
Sg
Sg
Sg
Sg
Sg
Sg
Sg
Sg
Sg
Sg
Sg
Sg
2
22
22
22
2
44
2222
1111
3333
55
),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(
)(
ι
ιι
ιι
ιι
ι
(A.40)
where
[ ]z
zzzSg akiexprr
kadakkdiexpttA
33513
5313513
2122
+
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−+⎟
⎠⎞
⎜⎝⎛
= (A.41)
[ ]z
zzSg akiexprr
kdkdiexptC
33513
3113
2122
+
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛
= (A.42)
[ ]z
zzSg akiexprr
kadkdiexprtD
33513
313513
21
222
+
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ ++⎟
⎠⎞
⎜⎝⎛
= (A.43)
[ ] [ ] [ ]z
zzzSg akiexprr
dkiexprakdkiexprF33513
1133135
212
+++= (A.44)
[ ]z
zzSg
bkiexprr
kdkdiexptG
224
12
2112
21
22
−+
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛−
= (A.45)
141
[ ]z
zzSg
bkiexprr
kbdkdiexprt
H2
24
12
2124
12
21
222
−+
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−⎟
⎠⎞
⎜⎝⎛−
= (A.46)
[ ]z
zzzSg
bkiexprr
kbdbkkdiexprtt
J2
24
12
42124
2412
21
22
−+
⎥⎦
⎤⎢⎣
⎡
⎭⎬⎫
⎩⎨⎧
⎟⎠⎞
⎜⎝⎛ +−−⎟
⎠⎞
⎜⎝⎛−
= (A.47)
When
( )( )
( ) ( ) ( )[ ]( ) ( )[ ]
( ) ( ) ( )
( ) ( ) ( )[ ]( ) ( )[ ]
( )( )
21
4
424
22
22
2
22
11
21
33
33
3
23
5
525
22
21
21
21
212
1
12
12
21
22
21
−
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
+−+
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−+
−−−−+++
⎥⎥⎦
⎤
⎢⎢⎣
⎡++++
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−−+
−−+++
+−
=
z
zSg
Sg
zzSg
Sg
zzSg
Sg
z
Sg
Sg
Sg
Sg
Sg
Sgz
Sg
Sg
z
zzSg
Sg
zzSg
Sg
z
Sg
Sg
Sg
Sg
z
zSg
Sg
Sg
kbdkexp
JJn
bkiexpdkiexpHGi
bkiexpdkiexpHGi
kbHHGGn
dFFdkSinhFFk
n
akiexpdkiexpDCi
akiexpdkiexpDCi
kaDDCCn
kadkexp
AAn
E
*
*
***
**
*
***
*
π
(A.48)
is chosen, the electric fields become normalized in the sense that for two S-polarized
guided modes corresponding to xk , yk ,ν and xk~ , yk~ ,ν~ with electric fields E and
E~
respectively, we have
∫∫∫ −−=⋅ ννδδδεε
~* )~()~(
~yyxx kkkkdrEE
0
(A.49)
where 1=ννδ ~ if νν ~= and = 0 otherwise. ν is the mode label.
142
It is also important to note that in deriving SgE above we have assumed a particular
index structure. In particular we have assumed an oxide slot. As a result k1z is taken to be
imaginary.
P Polarized Guided Modes
The relation to be satisfied by the propagation constant β of P-polarized guided
modes is same as that for S-polarized mode i.e. Eq. A.39, but with the reflection and
transmission coefficients given by Eq. A.11.
The electric field of a P-polarized guided mode is given by
[ ]
[ ]
[ ]
[ ]
[ ]⎪⎪⎪⎪⎪⎪⎪
⎩
⎪⎪⎪⎪⎪⎪⎪
⎨
⎧
⎟⎠⎞
⎜⎝⎛ +−≤⋅
⎟⎠⎞
⎜⎝⎛−≤≤⎟
⎠⎞
⎜⎝⎛ +−⋅+⋅
⎟⎠⎞
⎜⎝⎛≤≤⎟
⎠⎞
⎜⎝⎛−⋅+⋅
⎟⎠⎞
⎜⎝⎛ +≤≤⎟
⎠⎞
⎜⎝⎛⋅+⋅
⎟⎠⎞
⎜⎝⎛ +≥⋅
=
−−
−−++
−−++
−−++
++
bdzforPkrkiexpJnnE
dzbdforPkrkiexpHPkrkiexpGnnE
dzdforPkrkiexpFPkrkiexpE
adzdforPkrkiexpDPkrkiexpCnnE
adzforPkrkiexpAnnE
rE
Pg
Pg
Pg
Pg
Pg
Pg
Pg
Pg
Pg
Pg
Pg
Pg
2
22
22
22
2
444
1
22222
1
1111
33333
1
555
1
),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(),(ˆ)(
),(ˆ)(
)(
ι
ιι
ιι
ιι
ι
(A.50)
where Pg
Pg
Pg
Pg
Pg
Pg JandGFDCA ,,,, are given by Eqs.A.41 through A.47 but now the
reflection and transmission coefficients are given by Eq. A.11.
When
143
( )( )
( ) ( ) ( )[ ]( ) ( )[ ]
( ) ( ) ( )
( ) ( ) ( )[ ]( ) ( )[ ]
( )( )
21
4
4
22
22
2
11
33
33
3
5
5
1
22
21
21
21
212
1
12
12
21
22
21
−
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
+−+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−+
−−−−+++
++++
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
−−−+
−−+++
+−
=
z
zPg
Pg
zzPg
Pg
zzPg
Pg
z
Pg
Pg
Pg
Pg
Pg
Pgz
Pg
Pg
z
zzPg
Pg
zzPg
Pg
z
Pg
Pg
Pg
Pg
z
zPg
Pg
kbdkexp
JJ
bkiexpdkiexpHGi
bkiexpdkiexpHGi
kbHHGG
dFFdkSinhFFk
akiexpdkiexpDCi
akiexpdkiexpDCi
kaDDCC
kadkexp
AA
nE
*
*
***
**
*
***
*
π
(A.51)
is chosen, orthonormality condition given by Eq. A.49 is satisfied. Again, in deriving PgE
above we have assumed the slot slab structure, where k1z is purely imaginary.
A.4 Field Quantization
The radiation modes are labeled by the triplet ),,( kkk yx=κ where ck /ω= is the
wave number in vacuum, by the index λ, which specifies the polarization (λ = S or P) and
by the index μ which specifies the medium from which the radiation mode is incident (i.e.
μ = 4 or 5). The electric fields of the radiation modes are thus denoted by μλκE .
The guided modes are labeled by ),( yx kk=κ , λ and ν, where ν is the index of
propagation constant of the guided mode. The electric field of the guided mode is
denoted by νλκE .
144
The orthogonality relations, Eqs. A.24 and A.49 for the modes can be summarized as
follows:
∫∫∫ −=⋅2121222111 21
0μμλλμλκ δδκκδ
εε
μλκ)(* drEE (A.52a)
∫∫∫ −=⋅2121222111 21
0ννλλνλκ δδκκδ
εε
νλκ)(* drEE (A.52b)
∫∫∫ =⋅ 0222111
0
drEE *μλκνλκε
ε (A.52c)
The set of electromagnetic fields of the radiation and guided modes is a complete
orthonormal set of eigen functions of Maxwell’s equations in the space of all pairs
),( HE of transverse square integrable vector fields. By definition transverse fields satisfy
00 =⋅∇=⋅∇ HE ,)(ε (A.53)
Let μλκa and *ˆ μλκa be the destruction and creation operators corresponding to the
radiation modes and let νλκa and *ˆ νλκa be the corresponding operators for the guided
modes. These operators satisfy the commutation relations
2121222111 21 μμλλμλκ δδκκδμλκ
)(]ˆ,ˆ[ * −=aa (A.54a)
2121222111 21 ννλλνλκ δδκκδνλκ
)(]ˆ,ˆ[ * −=aa (A.54b)
0222111222111
== ]ˆ,ˆ[]ˆ,ˆ[ **μλκμλκ μλκμλκ aaaa (A.54c)
0222111222111
== ]ˆ,ˆ[]ˆ,ˆ[ **νλκνλκνλκνλκ aaaa (A.54d)
0222111222111222111
=== ]ˆ,ˆ[]ˆ,ˆ[]ˆ,ˆ[ ***μλκνλκμλκνλκμλκνλκ aaaaaa (A.54e)
The transverse electric field operator is
),(ˆ),(ˆ),(ˆ trEtrEtrE +− += (A.55)
where
145
( )
∑ ∑ ∫∫
∑ ∑ ∫∫∫
=
∞
= >=
= =<+ =
+
−⎟⎟⎠
⎞⎜⎜⎝
⎛+
−⎟⎟⎠
⎞⎜⎜⎝
⎛
=
PSyxkc
PSnkkk
yx
kc
min
yx
dkdkatiexprEi
dkdkdkatiexprEi
trE
, )(
,
,
ˆ)()(
ˆ)()(
),(ˆ
λ ν ββνλκβωνλκ
λ μμλκ
ω
μλκ
λν
λν
μ
ωεω
ωεω
1
21
0
5
4
21
0
2
221
22
(A.56)
and ),(ˆ trE − is the adjoint of ),(ˆ trE + .
When Eqs. A.52 and A.54 are used it follows that the contribution of the transverse
electric field operator to the Hamiltonian for the total electromagnetic field is given by
( )
∑ ∑ ∫∫
∑ ∑ ∫∫∫
∫∫∫
=
∞
= >=
= =<+
=
⎥⎦⎤
⎢⎣⎡ ++
⎥⎦⎤
⎢⎣⎡ +
=⋅
PSyxkc
PSnkkk
yxkc
min
yx
dkdkaa
dkdkdkaa
rdtrEtrE
,
*)(
,
*
*
,
ˆˆ
ˆˆ
),(ˆ),(ˆ
λ ν ββνλκνλκβω
λ μμλκμλκω
λν
λν
μ
ω
ω
ε
1
5
4
21
21
21
2121
21
22
(A.57)
The contribution to the Hamiltonian of the magnetic field operator is identical to that of
the electric field operator. Hence the total Hamiltonian of the electromagnetic field is
given by the well known superposition of the number operators for all electromagnetic
modes.
A.5 Spontaneous Emission and Zero Point Field Fluctuations
Suppose that an atom makes a spontaneous dipole transition from a state 2 to a
state 1 thereby emitting a photon of energy 0ω . The spontaneous emission can occur
in any mode of the electromagnetic field of frequency 0ω , or equivalently of wave
146
number ck /00 ω= in vacuum. The transition rate into a particular electromagnetic mode
with parameters μλκ ,, say, and with wave number k in vacuum, is given by[60]
)(ˆ kkifc ED −Η 0
2
2
2 δπ (A.58)
where i and f are the initial and final states of the combined atom-radiation system
and EDΗ is the electric-dipole interaction part of the complete Hamiltonian of the atom -
radiation system. In the initial state of the electromagnetic field, no photons are present.
In the final state there is one photon in the mode with parameters μλκ ,, . Because the
transition rate is much smaller than the transition frequency, the Lorentzian line is
extremely peaked. Hence, in a good approximation only modes of energy identical to the
transition energy contribute to the transition.
According to Fermi’s “golden rule”, the total spontaneous emission rate is obtained
by integrating Eq. A.58 over all the final states. Suppose that the dipole moment of the
atom is parallel to one of the axes of the Cartesian coordinate system and let j be x, y, or
z. Let jE be the jth component of the transverse electric field operator. Then the total
spontaneous emission rate of an atom at position r whose dipole moment is parallel to
the j axis is [61]
)(zFDc
e j2
122
221 πτ
= (A.59)
where e is the electron charge, 12D is the dipole matrix element of the atomic transition,
and
147
( )
∑ ∑ ∫∫
∑ ∑ ∫∫∫
=
∞
= >
= =<+
−+
−
=
PSyx
PSnkkk
yxj
j
min
yx
dkdkkkrE
dkdkdkkkrE
zF
,
,
,
))(()(
)()(
)(
λ ν ββ
λννλκ
λ μμλκ
λν
μ
βδεω
δεω
10
2
0
5
40
2
0
2
221
22
(A.60)
where in the integrand of the first term to the right, kc=ω whereas in the integrand of
the second term )(βω λνkc= . The function jF clearly depends only on the coordinate z
and is independent of x and y. Note that without the delta function in the integrands, Eq.
A.60 is the jth component of the zero-point electromagnetic field
fluctuations 00 2)(ˆ rE j . Hence )(zF j is the contribution (per wave number) of all
modes of wave number 0k to the jth component of the zero-point field fluctuations. We
will refer to )(zF j as the jth component of the zero-point field fluctuations in the
understanding that only the contributions of the modes of wave number 0k are meant.
When polar coordinates are introduced in the ( )yx kk , plane:
)(insk),(cosk yx ϕβϕβ == (A.61)
Eq. A.60 becomes
∑ ∑ ∫ ∫
∑ ∑ ∫ ∫ ∫
=
∞
=
∞
=
= =
∞
=
−+
−
=
PS sincos
PS
nk
ksincos
j
j
min
ddkkrE
ddkdkkrE
zF
, ))(),((
, )),(),((
,
))(()(
)()(
)(
λ ν
π
β
λν
ϕβϕβκνλκ
λ μ
π
ϕβϕβκμλκ
λν
μ
ϕβββδεω
ϕββδεω
1
2
00
2
0
5
4
2
0 0 00
2
0
2
2
(A.62)
148
∑ ∑ ∫
∑ ∑ ∫ ∫
=<
≥ =
= = =
+
=
PSkk
sinkcosk
PS
nk
ksincos
j
j
min
dkdk
dkrE
ddrE
zF
, ))()(),()((
, )),(),((
,
)()()(
)(
)(
λ ν
π λνλ
νϕβϕβκ
νλκ
λ μ
π
ϕβϕβκμλκ
λν
λν
λν
μ
ϕββεω
ϕββεω
000
0
1
2
000
2
0
0
5
4
2
0 0
2
0
0
2
2
(A.63)
The integral over ϕ for the three components j=x, y, z in Eq. A.63 can be computed
explicitly. The results for the x and y components are identical as they should be.
For the mean of the three components,
[ ])()()()( zFzFzFzF zyx ++=31 (A.64)
the result of the integral over ϕ yields a rather concise formula. The quantity )(zF is the
zero-point field fluctuation of a randomly oriented atom at depth z and is therefore of
special interest. It is easy to infer from the formulas for the electric fields of the modes in
Sec. A.2 and Sec. A.3 that for )),(),(( 0ksincos ϕβϕβκ = and
))()(),()(( ϕβϕβκ λν
λν sinkcosk 00= the squared amplitudes
22∑=
j
j rErE )()(μλκμλκ (A.65)
and
22∑=
j
j rErE )()(νλκνλκ (A.66)
are independent of ϕ . Hence we may take 0=ϕ in these sums. It then follows from Eq.
A.63 that
149
∑ ∑
∑ ∑ ∫
=<
≥ =
= = =
+
=
PSkk
k
PS
nk
k
j
min
kdk
dkrE
drE
zF
, )),((
, ),,(
,
)()()(
)(
)(
λ ν
λνλ
νβκ
νλκ
λ μ βκμλκ
λν
λν
μ
ββεωπ
ββεωπ
00
0
100
0
2
0
0
5
4 0 0
2
0
0
232
232
(A.67)
The remaining integrals with respect to β have to be computed numerically.
Consider now the zero-point field fluctuations in a homogeneous dielectric with a
refractive index n1. By substituting n5=n3=n2=n4=n1 into the expressions for the radiation
modes listed in Sec. A.2, we find
⎪⎩
⎪⎨⎧
===⋅
===⋅=
−−
++
),,(,,,),(ˆ)(
),,(,,,),(ˆ)()(
kkkkPSforkirkiexpA
kkkkPSforkirkiexpArE
yx
yx
λμλ
λμλμλκ
5
4
11
11
(A.68)
where 12122 nkkk yx <+ /)( and
( )21
1232
1⎟⎟⎠
⎞⎜⎜⎝
⎛=
zkkA /π
(A.69)
to ensure the modes are orthogonal. Hence the radiation modes reduce to plane waves. In
a homogeneous space the guided modes obviously don’t exist. By substituting Eq. A.68
and A.69 into Eq. A.63, we find for the components of the zero-point fluctuations in a
homogeneous dielectric of refractive index n1:
freezyx FnFFF 1=== (A.70)
where freeF denotes the vacuum field fluctuations (in free space):
20
2
30
6 cF free
επω= (A.71)
150
For an atom at depth z inside the slab of index n1 in a multi-dielectric slab structure,
having a dipole moment parallel to the jth coordinate axis, the spontaneous emission rate
relative to the value in bulk n1 index medium will be
free
j
FnzF
1
)( (A.72)
while for a randomly oriented atom the relative transition rate will be
freeFnzF
1
)( (A.73)
with )(zF being defined by Eq. A.64.
Finally, the mean spontaneous emission rate for the film of thickness d relative to
bulk film medium is obtained by computing the mean of Eq. A.73 over the thickness of
the film.
Results for the slot waveguide are shown in Fig. 5.7, which shows the mean relative
spontaneous emission rate for an emitter embedded in a silica slot relative to that in a
bulk silica medium.
The structure was also analyzed using a formulation presented in [55] where a transfer
matrix approach is utilized to determine the fields in the dielectric layers. Here the
authors also utilized a modified set of radiative modes characterized by a single outgoing
component as it is claimed that using the conventional set of radiative modes (used
above) gives rise to quantum interference between different outgoing modes. For the slot
structure under analysis, both the methods however, yielded exactly the same results.
151
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155
RAVI S. TUMMIDI, Ph.D.
Room 206 D, Sinclair laboratory, Center for Optical Technologies, Lehigh University, Bethlehem, PA 18015Ph: (484) 767 5920 [email protected] http://www.lehigh.edu/~rst205/ravi.html
EDUCATION
Lehigh University, Bethlehem, PA Candidate for Doctor of Philosophy in Electrical Engineering June 2007 to August 2011 Thesis Advisor: Prof. Thomas L. Koch • Cumulative GPA: 3.92/4.00 • Air Force office of Scientific Research (AFOSR), MURI Silicon Laser Project Graduate Research
Assistantship at Lehigh University from January 2007 till present • 6 Journal and 10 Conference publications, 1 Patent Filed, >25 Citations
Lehigh University, Bethlehem, PA Master of Science in Electrical Engineering August 2005 to May 2007 Advisor: Prof. Nelson Tansu • Cumulative GPA: 3.97/4.00 • Phi Beta Delta honors (Inducted in 2006), Lehigh University • Graduate Teaching Assistantship at Lehigh University from January 2006 till December 2006 • 3 Conference publications
North Maharashtra University, India Bachelor of Engineering in Electronics and Telecommunication August 1999 to June 2003 • First Class with Distinction (Top 2% of 180 students) • Implemented an automated telephone call routing circuitry. Routes calls based on a person’s
location identification in a premises using a transceiver chip
RESEARCH AND PROFESSIONAL EXPERIENCE
Lightwire, Inc., Allentown, PA June 2011 till Present Photonic Device Engineer • Developing components for high speed, low power CMOS photonic interconnect technology
LehighUniversity, Center for Optical Technologies Jan 2007 to August 2011 Ph.D. Candidate and Research Assistant • Principal design responsibilities in an AFOSR funded Silicon Laser MURI led by MIT, where
eight universities are targeting a CMOS compatible electrically driven silicon laser technology for nano-photonic interconnect applications
• Proposed, analyzed and demonstrated ultra low loss (very high Q) shallow ridge horizontal slot waveguides at “Magic Widths” operating in TM mode which is a critical ingredient in our overall design for an electrically driven photon source in Silicon
• Developed new fabrication techniques to achieve ultra smooth waveguide surfaces to substantially reduce waveguide scattering losses in high index contrast systems like SOI
• Combined the new fabrication techniques and novel resonator designs to achieve resonators with Qs of 1.6×106 for the TE-like mode in non-slot configurations and 3×105 for the TM-like mode in full slot configuration, the highest yet reported for this type of structure and close to our design requirements for a laser
• Incorporated Erbium into a slot just 8.3 nm thick and observed photoluminescence in full waveguide configuration. Performed Quantum electro dynamic (QED) numerical analysis which predicted >10 times Purcell enhancement in these non-resonator structures. Verified the same experimentally
• Collaborated with a research group at RMIT, Australia on low loss resonator designs which resulted in >8 joint publications
• Presented bimonthly project progress reports and annual reviews totaling >20 since January 2007 to MURI team comprising of research groups at MIT, Caltech, Stanford, Cornell, Boston U., U. Delaware & U. Rochester. Only student working on the project at Lehigh University
156
Lehigh University, Center for Optical Technologies Aug 2005 to Jan 2007 Master’s Student and Research Assistant • Performed extensive numerical modeling of VCSELs and quantum well structures • Executed design and MOCVD epitaxy of III-Nitride semiconductor optoelectronic devices for
mid-IR applications • Strong knowledge in material characterization using photoluminescence, atomic force microscopy
(AFM), scanning electron microcopy (SEM) and Hall measurement • Liaised with external vendors for the installation and commissioning of the MOCVD system at
Lehigh University and gained experience in system and support equipment maintenance
Videsh Sanchar Nigam Limited (VSNL), India Nov 2003 to July 2005 Transmission Engineer • Maintained international telecommunication links, fiber optic communication equipment –
MUX/DEMUX, digital cross connects, SDH and PDH equipment • Gained familiarity with usage of various transmission test equipment – Advanced network tester,
PFA 30/35, spectrum analyzer, HP communication analyzer • Selection for the post was on all India basis. Was one among fifty who got selected from the initial
34,000 candidates who applied for the positions
ACTIVITIES
• Reviewer for IEEE Photonics, OSA Optics Express, OSA Optical Materials Express journals • Founding member of the IEEE Photonics Society student chapter at Lehigh University in 2008
and Vice President for the chapter • Student member of IEEE Photonics Society, Optical Society of America (OSA), International
Society for Optics and Photonics (SPIE) • Captain of the University Badminton and Volleyball teams during undergraduate studies
REFEREED JOURNAL PUBLICATIONS
1. Naser Dalvand, Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch, and Arnan Mitchell, “Thin-ridge Silicon-on-Insulator waveguides with directional control of lateral leakage radiation”, Optics Express, 19, (6), pp. 5635-5643, March 2011
2. Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch and Arnan Mitchell, “Lateral leakage in TM-like mode Thin-ridge Silicon-on-Insulator Bent waveguides and Ring Resonators”, Optics Express 18, (7), pp. 7243-7252, March 2010
3. Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch and Arnan Mitchell, “Lateral leakage in TM-like whispering gallery mode of thin-ridge silicon-on-insulator disk resonators”, Optics Letters, 2009, 34, (7), pp. 980-982
4. Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch and Arnan Mitchell, “Rigorous Modeling of Lateral Leakage Loss in SOI Thin-Ridge Waveguides and Couplers”, Photonics Technology Letters, IEEE, 2009, 21, (7), pp. 486-488
5. Robert M. Pafchek, Jinjin Li, Ravi S. Tummidi and Thomas L. Koch, “Low Loss Si – SiO2 – Si 8 nm Slot Waveguides”, Photonics Technology Letters, IEEE, 2009, 21, (6), pp. 353-355
6. Robert M. Pafchek, Ravi S. Tummidi, Jinjin Li, Mark A. Webster, E. Chen and Thomas L. Koch, “Low-loss SOI Shallow-Ridge TE and TM Waveguides Formed Using Thermal Oxidation,” Applied Optics, 2009,48,(5), pp. 958-963
CONFERENCE PUBLICATIONS
1. Ravi S. Tummidi, Thach G. Nguyen, Arnan Mitchell and Thomas L. Koch, “An Ultra-compact waveguide polarizer based on “Anti-Magic-Widths””, IEEE Group Four Photonics (GFP), London, UK, September, 2011 (Accepted)
2. Naser Dalvand, Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch, and Arnan Mitchell, “Directional control of lateral leakage loss in Thin-ridge Silicon-on-Insulator waveguides”, IEEE/OSA Conference on Lasers and Electro-Optics (CLEO) 2011, Baltimore MD, May, 2011
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3. Ravi S. Tummidi, Robert M. Pafchek, Kangbaek Kim and Thomas L. Koch, “Purcell-enhanced Erbium Emission in Low-loss Horizontal nm-Scale Slot Silicon Photonic Waveguides”, Invited Talk V1.2, 2011 Materials Research Society Spring Meeting and Exhibit, San Francisco, April 25 - 29, 2011
4. Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch and Arnan Mitchell, “”Magic Radius” Phenomenon in Thin-ridge SOI Ring Resonators: Theory and Preliminary Observations”, IEEE/OSA Conference on Lasers and Electro-Optics (CLEO) 2010, San Jose CA, May, 2010
5. Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch, and Arnan Mitchell, “Mode-Quenching in Thin Ridge Silicon on Insulator Disk Resonators with Lateral Leakage”, Australian Conference on Optical Fiber Technology (ACOFT), The University of Adelaide, Australia, Nov-Dec 2009
6. Ravi S. Tummidi*, Robert M. Pafchek, Kangbaek Kim and Thomas L. Koch, “Modification of spontaneous emission rates in shallow ridge 8.3 nm Erbium doped silica slot waveguides”, IEEE Group Four Photonics (GFP), San Francisco, CA, September, 2009
7. Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch and Arnan Mitchell, “Thin-Ridge SOI Disk and Ring Resonators with "Magic Radius" and "Magic Width" Phenomena ”, IEEE/OSA Conference on Lasers and Electro-Optics (CLEO) 2009, Baltimore MD, June, 2009
8. Ravi S. Tummidi*, Thach G. Nguyen, Arnan Mitchell and Thomas L. Koch, “Anomalous losses in curved waveguides and directional couplers at “magic widths” ”, in Proc. of the IEEE Lasers and electro-optics society (LEOS), Newport Beach, CA, November, 2008
9. Thach G. Nguyen, Ravi S. Tummidi, Mark A. Webster, Thomas L. Koch, and Arnan Mitchell, “Analysis of Lateral Leakage Loss in Silicon-On-Insulator Thin-Rib Waveguides”, Australian Conference on Optical Fiber Technology (ACOFT), Sydney, Australia, July 2008
10. Robert M. Pafchek, Jinjin Li, Ravi S. Tummidi* and Thomas L. Koch, “ Low Loss Si – SiO2 – Si 8 nm Slot Waveguides”, in Proc. of the IEEE/OSA Conference on Lasers and Electro-Optics (CLEO) 2008, San Jose CA, May 2008
11. R. A. Arif, Ravi S. Tummidi, Y. K. Ee, and N. Tansu, “Design Analysis of Lattice-Matched AlInGaN-GaN Quantum Wells for Optimized Intersubband Absorption in the Mid-IR Regime,” in Proc. of the SPIE Photonics West 2007, Physics and Simulation of Optoelectronics Devices XV, San Jose, CA, Jan 2007
12. H. Li, J. T. Perkins, H. M. Chan, R. P. Vinci, Y. K. Ee, R. A. Arif, Ravi S. Tummidi, J. Li, and N.Tansu, “Nanopatterning of Sapphire for GaN Heteroepitaxy by Metalorganic Chemical Vapor Deposition,” in Proc. of the MRS Fall Meeting 2006: Symposium I: Advances in III-V Nitride Semiconductor Materials and Devices, Boston, MA, USA, November-December 2006
13. Z. Jin, Ravi S. Tummidi*, Y. P. Gupta, D. M. Schindler, and N. Tansu, “Quasi-Guided-Optical-Waveguide VCSELs for Single-Mode High-Power Applications,” in Proc. of the IEEE/OSA Conference on Lasers and Electro-Optics (CLEO) 2006, Long Beach, CA, May 2006
* Oral Presenter
INVITED SEMINARS
1. Ravi S. Tummidi, “Novel waveguide architectures for light sources in Silicon Photonics”, Lightwire Inc., Allentown, PA, February 2011
2. Ravi S. Tummidi, “Novel waveguide architectures for light sources in Silicon Photonics”, GE Global Research, Niskayuna, NY, March 2011
PATENTS
1. “Chip-Based Slot Waveguide Spontaneous Emission Light Sources”, US Patent Application 12/799,171 filed April 20, 2010 claiming priority of provisional patent application 61/214,313 filed April 22, 2009. Inventors: Harry A. Atwater, Jr., Ryan M. Briggs, Mark L. Brongersma, Young Chul Jun, Thomas L. Koch and Ravi S. Tummidi