Xiaoning Jia

Liquid crystal based optical phased-array beam steering

Academic year 2014-2015Faculty of Engineering and ArchitectureChairman: Prof. dr. ir. Rik Van de WalleDepartment of Electronics and Information Systems

Master of Science in Photonics EngineeringMaster's dissertation submitted in order to obtain the academic degree of

Counsellor: Ir. Oliver WillekensSupervisors: Prof. dr. ir. Jeroen Beeckman, Prof. dr. ir. Kristiaan Neyts

Xiaoning Jia

Liquid crystal based optical phased-array beam steering

Academic year 2014-2015Faculty of Engineering and ArchitectureChairman: Prof. dr. ir. Rik Van de WalleDepartment of Electronics and Information Systems

Master of Science in Photonics EngineeringMaster's dissertation submitted in order to obtain the academic degree of

Counsellor: Ir. Oliver WillekensSupervisors: Prof. dr. ir. Jeroen Beeckman, Prof. dr. ir. Kristiaan Neyts

iii

Acknowledgement

First of all, I am very grateful for professor Jeroen Beeckman and professor Kristiaan Neyts for

giving me this opportunity to do my thesis at the Liquid Crystals and Photonics group, and

thank you for your kind guidance and consistent solicitude in this academic year. Moreover,

I would like to give my deepest gratitude to my thesis counsellor Oliver, thank you for your

continuously guidance and enormous help during the time span of this thesis, I really enjoyed the

informal discussion with you about the practical issues regarding this thesis, and the knowledge

and ideas I learned from you benefited me and will be benefit me during my life, and thank

you for reading my thesis and providing valuable suggestions, this thesis would not have been

possible without your guidance and help.

I would like to thank my parents for raising me up, for believing in me unconditionally, for

loving me. I wouldn’t have been here without your support and love, thank you for making me

who am I today and I hope I can make you proud.

Finally I would like to thank my friends who have been by my side during the two years in Gent,

Xiaomin, Ali, Boyang, Yuting and all of the others. Thank you for backing me up when I am

down, for listening to my complaints, for sharing my happiness and sorrows.

Xiaoning JIA,June 2015

iv

Permission of use on loan

The author gives permission to make this master dissertation available for consultation and to

copy parts of this master dissertation for personal use. In all cases of other use, the copyright

terms have to be respected, in particular with regard to the obligation to state explicitly the

source when quoting results from this master dissertation.

09-June-2015

Liquid Crystal Based Optical Phased-Array BeamSteering

Xiaoning Jia

Supervisor(s): Ir. Oliver Willekens, Prof. dr. ir. Jeroen Beeckman, Prof. dr. ir. Kristiaan Neyts.

Abstract— In this article, study of a beam steering device based on ne-matic liquid crystals was performed. A blazed phase grating approximationhas been made for the analysis and simulations for the devices with variousconfigurations have been performed. Characterizations of the fabricatedcells were carried out and beam steering was observed to various extent.

Keywords—Liquid crystals, beam steering, PZT, blazed grating, diffrac-tion.

I. INTRODUCTION

IN our modern society, beam steering becomes more and morecritical to many optical systems such as fiber-optical switches

and free space communications. Conventional beam steeringdevices are implemented with mechanically controlled mirrors,prisms and lenses. These technologies suffer from disadvan-tages such as mechanical complexity, high cost, and limitedsteering speed. However, the study of beam steering modulesbased on liquid crystals (LCs) have been proved feasible andvarious configurations of such devices have been reported. Ingeneral, the working principle of these devices relies on theelectro-optically controlled refractive index of liquid crystals,writing a linearly changed refractive index profile in LC leads toa linear phase retardation of light such that incident light can bediffracted into discrete angles.

II. THEORETICAL ANALYSIS

It is well known that light is an electromagnetic wave de-scribed by Maxwell’s equations, and it can be diffracted whenincident on optical elements with dimensions comparable to thewavelength being used. A blazed grating can diffract light intoone single order by inducing a linearly changing optical pathlength (OPL). The beam steering device based on LC can ap-proximate the behaviour of such a blazed grating if the refrac-tive index profile can be well controlled, which is in this thesisimplemented by spin coating a layer of lead zirconate titanate(PZT) on top of widely spaced electrodes. Thus by selectivelyswitching the LC molecules at different positions the resultingrefractive index profile will have a blazed grating shape. Thediffraction angles of a blazed grating are given by the gratingequation:

sin(θm) = ±mλ

Λ(1)

in which λ is the wavelength, Λ is the grating period and m isthe diffraction order. To which order the light can be diffracteddepends on the maximum phase delay1 caused by the grating.Since the refractive index profile of LCs can be electrically con-trolled, the resulting output angle can thus be tuned by changingthe applied voltage over the LC device.

1The phase delay depends on the refractive index and the thickness.

III. SIMULATION

The simulation program is based on Q-tensor theory, whichcan calculate the director distribution on each point of the mesh,and then the director distribution is transformed into optical pathlength (OPL) using Matlab. The 10µm thick cell with symmet-ric finger electrodes was first simulated to study the function-ality of the PZT layer. By applying a square voltage with fre-quency of 1kHz and with amplitude V1=V2=+5V on the twofinger electrodes, the rubbing direction and the polarization ofthe incident light are both along x-direction, the OPL of such acase is shown in Figure 1.

Fig. 1. Simulated OPL profile showing the nearly flat phase profile of lightafter propagating through a PZT-enhanced finger-pattern. Its performanceis clearly much better than that of the reference cell, where in between twoelectrodes, the mesogens hardly switch, thus resulting in a much larger OPLthere.

It can be seen that the OPL of the cell with PZT is almostconstant while in the cell without PZT the OPL varies in a rangeabout one wavelength. Inspection of the electric field showsthat the reason for this is that PZT extends the electric field intothe region between the electrodes and thus the mesogens in thisregion will experience the electric field more clearly and thustend to switch.

Figure 2 shows the OPL of the 10µm thick cell for beamsteering usage with asymmetric finger electrodes and with alayer of PZT on top of electrodes. Again both the rubbing andthe polarization of the light are along the x-direction thus thelight will see an extraordinary refractive index ne at the voltage-off state, but once a sufficiently strong voltage is applied on theelectrode the LC molecules above it will tilt in vertical direc-tion and light will feel an ordinary refractive index no. One cansee that as the applied voltage increases on one electrode com-pared to the other one and the common grounded electrode, theOPL changes from a constant to an almost linearly increasing

Fig. 2. Optical phase profile of a vertical field switching cell with asymmetricfinger pattern where the finger electrodes are at different potentials.

shape, which is due to the functionality of the PZT layer. Fur-ther increase of the voltage leads to uniform switching of the LCmolecules in the whole cell, thus the OPL in the gap region andabove the electrode applied 0V decrease.

IV. EXPERIMENT

To understand the functionality of the PZT layer, the switch-ing behavior of the two cells2 with symmetric finger electrodeswere observed under polarizing microscope, with the appliedvoltages of V1=V2=+5V, shown in Figure 3 and Figure 4. It canbe seen that for the cell without PZT a clear periodic structureis visible, due to the fact that the LC molecules switch differ-ently above the finger electrodes and in between them, while forthe PZT cell the periodic pattern is less clear, this is becausethe LC molecules in the whole cell are more or less switchinguniformly due to the presence of the PZT layer, leading to analmost constant phase retardation.

Fig. 3. Switching behavior of the LC molecules when the two finger electrodesare applied V1=V2=+5V (reference cell).

With a good understanding of the PZT layer, beam steer-ing cells with asymmetric finger electrodes were fabricated andcharacterized. To increase fabrication yield, six modules wereassembled on one single substrate. They differ in the electrodeperiod, i.e. the largest gap between two neighboring electrodes,ranging from 30µm to 80µm, in each of them two wires weresoldered to the bondpads connecting the finger electrodes. Thecommon electrode was shared by all the six modules and sol-dered with one wire. Cells with thickness of 10µm and 20µmwere fabricated, one of the fabricated cells is shown in Figure 5.

2All the LC cells studied in this thesis are used in transmission, not reflection.

Fig. 4. Switching behavior of the LC molecules when the two finger electrodesare applied V1=V2=+5V (PZT cell).

Fig. 5. Fabricated cell with six modules connected to 2× 4 + 1 = 9 wires.

A simple setup was built to characterize the finished cell.The setup consists of a horizontally aligned standard HeNelaser(632.8nm), a combination of polarizer-λ4 plate-polarizer forpolarization control, a diaphragm for cleaning the beam, an anal-yser and the projection screen. The LC cell was placed betweenthe diaphragm and the analyser, a linear CCD camera was usedto capture the diffraction pattern on the screen.

All the fabricated cells exhibit beam steering ability to variousextent. Figure 6 shows the diffraction pattern of the modulewith an electrode period of 70µm and cell thickness of 20µm,a layer of PZT was spin coated on top of the electrodes. In thisexperiment, both the initial rubbing direction of the LC and thepolarization direction of incident light are perpendicular to thelength direction of finger electrodes, the two finger electrodesare applied voltages V1=0 and V2=+V, respectively, with V theroot mean square value of the square wave at 1kHz, changingfrom 0 to 10Volt.

Fig. 6. Beam steering for the 20µm thick PZT cell with electrode period 70µm.

One can see that as the voltage increases, the beam is gradu-ally shifted from the 0th order to the 1st order, 2nd order and soon, until it reaches the maximum angle at 4.65◦. The efficiencyat this angle is about 74%. If the voltage is further increasedthe beam starts gradually shifting back towards the 0th order.The reason for this is that if the voltage is so high (10V) themolecules in the whole cell are switched, as can be seen fromthe OPL in Figure 2, resulting in a reduced phase retardation,thus the steering angle is smaller.

The reference cell was fabricated to compare the beam steer-ing capability with the PZT cell, and the only difference fromthe PZT cell is the absence of PZT layer, and also the refer-ence cell is 10µm thick. The same setup and method were usedto characterize the reference cell, and the diffraction pattern isshown in Figure 7.

Fig. 7. Beam steering for the reference cell with electrode period 70µm.

As it can be seen, instead of diffracted into single orders, thelight is more spread out in several orders at smaller angles, inother words, the beam steering ability is rather limited, whichis mainly due to the orientations of LC molecules in the gap re-gion not being properly controlled, leading to an irregular phaseprofile.

V. CONCLUSION

The most important results from this thesis work have beenpresented. The functionality of the PZT layer was proved tohave the ability of extending the electric field into the gap regionbetween electrodes, leading to a well defined phase profile ofthe incident light. With the benefit from PZT, the beam steeringdevice based on liquid crystals was shown to steer the beam intosingle diffraction orders, the steering angle is tunable simply bychanging the applied voltage, while the reference cell withoutthe PZT layer exhibits very limited beam steering capability.

ACKNOWLEDGEMENT

The author would like to acknowledge professor JeroenBeeckman and professor Kristiaan Neyts for offering this re-search topic and their concern and guidance throughout the aca-demic year, and the deepest gratitude goes to Oliver who wasalways there to help and whose guidance and help have madethis thesis possible.

REFERENCES

[1] Matic, R. M. ”Blazed phase liquid crystal beam steering”.OE/LASE’94, 1994, 194-205

[2] Resler, D.; Hobbs, D.; Sharp, R.; Friedman, L.; Dorschner, T.”High-efficiency liquid-crystal optical phased-array beam steer-

ing”. Optics letters, Optical Society of America, 1977-, 1996, 21,689-691

[3] Stockley, J. ; Serati, S. ”Advances in liquid crystal beam steeringOptical Science and Technology”. the SPIE 49th Annual Meeting,2004, 32-39

[4] Wang, X.; Wilson, D.; Muller, R.; Maker, P. ; Psaltis, D. ”Liquid-crystal blazed-grating beam deflector”. Applied Optics, OpticalSociety of America, 2000, 39, 6545-6555

CONTENTS viii

Contents

List of Figures xi

List of Tables xii

1 Liquid Crystals 1

1.1 Overview of this thesis work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Order Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 Liquid Crystal Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.3 Anisotropy in Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Q-tensor Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Uniaxial and Biaxial Nematics . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.2 Q Tensor Order Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Beam Steering 9

2.1 Mechanical Beam Steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Non-mechanical Beam Steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.1 Liquid Crystal Prisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Liquid Crystal Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.3 Liquid Crystal Phased Arrays . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.4 Liquid Crystal Polarization Gratings . . . . . . . . . . . . . . . . . . . . . 17

3 Theoretical Analysis 21

3.1 Diffraction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1 Thin Surface Grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.2 Blazed Grating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Cell Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 Blazed Grating Based on Liquid Crystals . . . . . . . . . . . . . . . . . . 24

3.2.2 PZT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.3 Symmetric Finger Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.4 Asymmetric Finger Electrodes . . . . . . . . . . . . . . . . . . . . . . . . 26

CONTENTS ix

4 Simulations 32

4.1 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Finite-Difference Time-Domain method . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3.1 Symmetric Finger Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3.2 Asymmetric Finger Electrodes . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Experiments 41

5.1 Fabrication Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.2 Fabricated Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.3 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.3.1 Measurement Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.4 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.4.1 Symmetric Finger Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.4.2 Asymmetric Finger Electrodes . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Conclusions 55

LIST OF FIGURES x

List of Figures

1.1 In the isotropic phase (left), there is no long range ordering of the molecules,

whereas for molecules in the nematic phase, one of the many liquid crystalline

phases, there is. This long-range order is often denoted by the pseudovector ~L,

the director [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Liquid crystal phases[2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Liquid crystal chiral phases [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Blue phases, left: BP I, right: BP II.[5] . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Uniaxial structure of liquid crystals.[6] . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6 Biaxial structure of liquid crystals.[6] . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.7 Two directors and Euler angles.[6] . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Nonmechanical beam steering configurations.[14] . . . . . . . . . . . . . . . . . . 10

2.2 Output light path of two identical rotating prims.[10] . . . . . . . . . . . . . . . . 11

2.3 Basic structure of the LC prism proposed by Lin.[25] . . . . . . . . . . . . . . . . 13

2.4 Beam deflection angle versus driving voltage.[25] . . . . . . . . . . . . . . . . . . 13

2.5 Schematic build-up of a proposed LC-based lens[28]. . . . . . . . . . . . . . . . . 15

2.6 Focal length as function of voltage.[28] . . . . . . . . . . . . . . . . . . . . . . . . 15

2.7 Blazed phase liquid crystal beam steering device.[20] . . . . . . . . . . . . . . . . 16

2.8 Setup of Pancharatam QHQ: (a) and (e) polarizer; (b) and (d) quarter-wave plate;

and (c) half-wave plate.[39] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.9 Director profile in LCPG cell[34]: (a) Top-view; (b) Side-view (Off-state); (c) Side-

view (On-state); Diffraction property (d) RCP and (e) LCP polarized incident

light (Off-state; (f) Transmission of light (Off-state . . . . . . . . . . . . . . . . . 18

2.10 Ternary PG beam steering design: (a) ternary steerer design, (b) number of

steering angles and calculated transmittance versus number of stages.[35] . . . . 19

3.1 Plane wave incident on window with slits.[45] . . . . . . . . . . . . . . . . . . . . 21

3.2 Blazed grating and its transmission[45] . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Refractive index as function of applied voltage for the nematic LC mixure E7 .[46] 25

3.4 Cell structure with symmetric finger electrodes . . . . . . . . . . . . . . . . . . . 27

3.5 Cell structure with asymmetric finger electrodes . . . . . . . . . . . . . . . . . . . 28

3.6 Finger electrode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

LIST OF FIGURES xi

4.1 One unit of the mesh for cell with 2 electrodes. . . . . . . . . . . . . . . . . . . . 33

4.2 Yee grid for FDTD method.[48] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Director profile of the two cells when +7V and -7V are applied to the two finger

electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4 Optical path length of the two cells with different voltage applied on the finger

electrodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.5 Angular spectrum of the cells when applied +7V and -7V. . . . . . . . . . . . . . 36

4.6 Potential distribution when the two finger electrodes are both applied +5V voltage. 37

4.7 OPL when two consecutive electrodes are applied +5V and +5V. . . . . . . . . . 38

4.8 Angular spectrum of the cells when applied +5V and +5V. . . . . . . . . . . . . 39

4.9 OPL after passing through the cells with asymmetric finger electrodes. . . . . . . 39

4.10 Angular spectrum when the cells are applied +3V and 0V. . . . . . . . . . . . . . 40

5.1 Mask of the asymmetric finger patterns. . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 Lift-off process steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.3 Glue pattern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.4 One of the fabricated cells, observed using two polarizers, with parallel transmis-

sion axes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.5 Image of the break and short circuit of the cell under polarization microscope . 45

5.6 Transmission spectrum of one cell. . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.7 Schematic of the set up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.8 Sets of voltage profiles applied to the finger electrodes of the symmetric finger

patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.9 Diffraction pattern when the finger electrodes are in applied voltage V1=+V and

V2=-V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.10 Defects causing scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.11 Diffraction pattern when the finger electrodes are applied voltage V1=+V and

V2=0V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.12 Diffraction pattern when the finger electrodes are applied voltage V1=+V and

V2=+V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.13 Cells under polarizing microscope. . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.14 Diffraction pattern of the electrodes at the voltage-off state. . . . . . . . . . . . . 51

5.15 Diffraction pattern for the cell No.3 with P = 70µm. . . . . . . . . . . . . . . . . 52

5.16 Diffraction pattern of cell No. 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.17 Diffraction pattern of the reference cell with P = 70µm. . . . . . . . . . . . . . . 53

5.18 Diffraction pattern using white light, from a Xenon lamp. . . . . . . . . . . . . . 54

LIST OF TABLES xii

List of Tables

3.1 Parameters of the cell with symmetric finger electrode . . . . . . . . . . . . . . . 29

3.2 Parameters of the cell with asymmetric finger electrode . . . . . . . . . . . . . . 29

3.3 Diffraction angle of the cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.1 Thickness of the cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Cells tested in experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Acronyms

LC Liquid Crystal

ITO Indium Tin Oxide

PZT Lead Zirconate Titanate

OPD Optical Path Difference

OPL Optical Path Length

FDTD Finite Difference in Time Domain

FSM Fast Steering Mirror

MEMS Microelectromechanical systems

LCPG Liquid Crystal Polarization Grating

RMS Root Mean Square

BPLC Blue Phase Liquid Crystal

AC Alternating Current

LIQUID CRYSTALS 1

Chapter 1

Liquid Crystals

One of the key aspects to this dissertation is the switching behaviour of liquid crystals. Therefore,

in this chapter, the relevant background knowledge about this material and its electro-optic

properties will be given. At the end of this chapter, the reader will have a feel for how liquid

crystals can be used to achieve beam steering, a topic for which several examples will be shown

in the next chapter.

1.1 Overview of this thesis work

This dissertation is organized as follows. In the first chapter, the theoretical background on liquid

crystals will be given, sufficient to ensure the reader has the basics to understand the thesis.

This background information touches the topics of types and properties of liquid crystals, and

the Q-tensor theory which is used in the simulation work in this thesis.

In the second chapter, both mechanical and non-mechanical beam steering will be reviewed.

However, due to their many advantages, which will become apparent, this thesis will focus on

the non-mechanical beam steering that can be obtained by using a special electrically tunable

material, known as liquid crystals.

In chapter three, the theoretical analysis regarding the liquid crystal cells1 fabricated and char-

acterized in this thesis work will be given, as well as some preliminary knowledge on diffraction

theory.

In chapter four, simulations results will be discussed, first the simulation of the cell with sym-

metric finger electrodes were performed to examine the functionality of the PZT layer, then the

cells with asymmetric finger electrodes were investigated via simulation.

Chapter five presents the experiment results, both for the two configuration of cell, and analysis

based on the results will be given.

Chapter six presents the conclusion of the thesis.

1Units of functional liquid crystal device.

1.2 Introduction 2

1.2 Introduction

Liquid crystals is a broad identifier used to denote a special state of matter that exists in

between the solid and liquid phases. It is sometimes also called a mesophase, as it exhibits

physical properties from both solids and liquids. LC molecules have some translational freedom

in the material just like fluid in a liquid, and also their orientation exhibits long-range ordering

just like in a solid where molecules are usually spaced in a periodic fashion, thereby creating a

crystal lattice. This unique combination of liquid and solid phases give liquid crystals interesting

properties like high anisotropy and low viscosity. Generally the liquid crystals consist of organic

molecules that have disc or rod-like shape, and so they can be represented by rigid rods or

ellipsoids, and they can be classified into two types: thermotropic and lyotropic liquid crystals.

The thermotropic liquid crystals are formed as a result of thermal equilibrium of molecule

interactions. When the thermal equilibrium is broken by increasing the energy of the system,

e.g. heating the material, the material will experience a transition from the solid crystalline phase

into an isotropic liquid state, but in between it will have the liquid crystal phase. Lyotropic

liquid crystals however, get the long-range ordering when the concentration of the mesogen in

a solvent reaches a certain critical density. In the discussion that follows, when we refer to a

liquid crystal it will always be of the thermotropic kind, unless explicitly mentioned otherwise.

1.2.1 Order Parameters

Since liquid crystal molecules can be represented by rods or ellipsoids and they have long-range

ordering, it would be convenient to model the long-range ordering by a vector called director~L, which represents the average orientation direction of the long axis of the molecules. The

molecules have no preference to align along the long axis or the inverse because of their central

symmetry however, which means there is no difference if we turn the molecules up side down,

i.e. −~L is equivalent to ~L. The representation for the nematic phase is shown in Figure 1.1.

Figure 1.1: In the isotropic phase (left), there is no long range ordering of the molecules, whereas

for molecules in the nematic phase, one of the many liquid crystalline phases, there is. This

long-range order is often denoted by the pseudovector ~L, the director [1]

.

When there is thermal motion in the material, the orientation of the individual liquid crystal

molecules will change as well as the long-range ordering. The overall degree of how well the

molecules are aligned to each other can be represented by the scalar order parameter S:

1.2 Introduction 3

(a) Isotropic (b) Nematic (c) Smectic

Figure 1.2: Liquid crystal phases[2]

S =1

2

⟨3(~k · L)2 − 1

⟩=

1

2

⟨3 cos2 θ − 1

⟩(1.1)

where ~L is the director, ~k is the direction of the long axis of each molecule, and θ is the deviation

angle between ~L and ~k. In the isotropic state the long axis of the molecules are oriented randomly

(Figure 1.1) and thus the order parameter for the isotropic liquid is zero. When the temperature

decreases, the molecules are aligned in the same direction and the order parameter is 1.

1.2.2 Liquid Crystal Phases

Liquid crystals can be classified into different phases depending on molecule properties and the

types of ordering, including orientational ordering, which means the orientation of the overall

director of the molecules, and positional ordering, which means how the molecules are locally

structured. When the temperature is high, the molecules in thermotropic liquid crystals are

randomly oriented, leading to an isotropic phase, the liquid crystals will just behave like an

ordinary isotropic liquid, and when decreasing the temperature, different phases can be distin-

guished. Generally speaking, the thermotropic liquid crystals will experience the following phase

transitions upon cooling:

isotropic - nematic - smectic A - smectic C - crystalline

Nematic Phase

The nematic phase is typically the first mesophase to be observed when decreasing the temper-

ature, which means it has higher thermal energy than any other phase, resulting in the least

ordered long-range orientation of the molecules: the molecules do not (yet) have any positional

order, however the long-axis of the molecules aligns approximately in the same direction. Figure

1.2a shows the molecule structure of the nematic phase.

1.2 Introduction 4

Smectic Phase

If the temperature decreases further, the thermal energy will also decrease, and the smectic phase

appears, the molecules in smectic phase will have a preference to align into planes, thereby giving

it additional positional order, shown in Figure 1.2c. The smectic phase can be further divided

into smectic A, smectic B, and smectic C phases, according to the degree of order and the

orientation of the director within a plane. Different from the nematic phase, the molecules in

the smectic A phase tend to form layers, and the director within such a layer is perpendicular

to the molecule layers, so it is parallel to the layer normal. Also the molecules have rotational

freedom around their long axes. In the smectic C phase however, the director forms a small

angle with respect to the layer normal, and the rotational freedom along the long axis is lost.

The smectic B phase is more like the crystalline solid, in this type of phase the molecules still

arrange in layers, while in the perpendicular direction they are packed hexagonally. Note that

these phase transitions are reversible, simply by changing the temperature[1].

Cholesteric Phase

The nematic phase is formed when the building blocks of the liquid crystal are achiral molecules,

which is the type of molecules that when interchanging the functional groups the resulting

molecule is identical to itself, meaning the molecules have a mirror plane. When the molecules

are chiral, i.e. the molecules do not have such a mirror plane, the energy in this liquid crystal

phase will be minimum when the directors in each slab have a slight twist with respect to the

previous slab. So rather than having the same direction in the achiral case, the chiral molecules

will form a helical structure in one direction. The period for which the director of one slab

returns to its starting direction after some twist is called the pitch. Figure 1.3a shows such

a cholesteric phase, generally the chiral nematic phase can be generated by introducing chiral

molecular dopants into the nematic mesophase.

Blue Phase

The chiral molecules can also form a double-twist structure, called the blue phase (BP). It is a

mesophase in between the isotropic phase and the chiral nematic phase, the configuration of the

molecules is shown in Figure 1.3b. The existence of the blue phase could be explained, because

the free energy is lower when the molecules also twist in the second direction, however this kind

of structure can only exist in a very small temperature interval (only a few degrees). Recently

researchers have managed to extend the temperature range of the blue phase to more than 60

degrees at room temperature by adding monomers to the blue phase liquid crystal mixture and

polymerizing it [4].

The double-twist structure of liquid crystal molecules cannot extend infinitely due to the unstable

nature of this structure, thus the molecules will stack together in cylinders that then form cubic

structures with dimensions of hundreds of nanometers. These cubic structures behave like Bragg

reflectors that reflect light of particular wavelengths. Because the lattice constant of this cubic

1.2 Introduction 5

(a) Chiral nematic phase (b) Double-twist structure

Figure 1.3: Liquid crystal chiral phases [3]

Figure 1.4: Blue phases, left: BP I, right: BP II.[5]

arrangement is on the order of several hundreds of nanometers, the Bragg reflections from a blue

phase liquid crystal are ususally in the visible wavelength range (350-700 nm). The name stems

from this: the first mixture of this type was found to have a bright blue color.

The molecules of a BPLC can be packed into cubes in different ways, and according to this

different types of blue phase can be distinguished, for example the blue phase I has a face-

centred cubic structure and the blue phase II shows a simple cubic structure, as illustrated in

Figure 1.4.

Due to the blue phase liquid crystal’s unique structure, the blue phase liquid crystal is optically

isotropic in the absence of an external electric field, but it becomes anisotropic when an electric

1.3 Q-tensor Theory 6

field is present, thus it is possible to make polarization-independent devices with blue phase

liquid crystals.

1.2.3 Anisotropy in Liquid Crystals

The anisotropy in liquid crystals originates from the shape and long-range ordering of the

molecules, for example the optical properties parallel and perpendicular to the director are

different. Depending on how many optical axes (the macroscopic rotational axis) the liquid

crystal has, it can be categorized as either uniaxial (one optical axis) or biaxial (two optical

axes) materials.

In uniaxial materials, when the incident light is polarized along the optical axis, it will experience

the extraordinary refractive index (ne), and when light is polarized perpendicular to the optical

axis it will experience the ordinary refractive index no. This phenomenon in which the refractive

index depends on the polarization direction is called birefringence:

∆n = ne − no (1.2)

When ne > no, we call the material optically positive, while when no > ne it is called optically

negative. These are the two cases when the polarization of the light is either parallel or perpen-

dicular to the optical axis. When the direction of propagation of the light makes an angle θ with

respect to the director, then it can be decomposed into two perpendicular polarization compo-

nents, one of which will experience no, and the other polarization will experience a combination

of refractive index of no and ne depending on the angle θ, which is expressed by:

neff =

(cos2 θ

n2o

+sin2 θ

n2e

)− 12

(1.3)

1.3 Q-tensor Theory

1.3.1 Uniaxial and Biaxial Nematics

As explained previously, the degree of long-range ordering can be represented by a scalar order

parameter with the expression shown in formula (1.1). Assuming that the director is along the

z’ direction and we only consider a small amount of molecules enclosed by a sphere, shown in

Figure 1.5, then the director distribution viewed from y’-z’ plane is identical to that viewed from

x’-z’ plane, with an angle of θmaxm representing the maximum deviation angle between individual

molecules and the z’ direction, and the molecules are randomly distributed when viewed from

the x’-y’ plane, i.e. the molecules have rotational symmetry around z’ axis.

However in biaxial liquid crystal materials the molecules do not have an axis of rotational sym-

metry as in the uniaxial case, as shown in Figure 1.6, the director distribution is different when

1.3 Q-tensor Theory 7

Figure 1.5: Uniaxial structure of liquid crystals.[6]

Figure 1.6: Biaxial structure of liquid crystals.[6]

viewing from y’-z’, x’-z’, and x’-y’ plane, thus the light will experience three different refrac-

tive indices when propagating along three orthogonal axes, leading to trirefringence, therefore

instead of one scalar order parameter or one director, two perpendicular directors need to be

defined (the third one is orthogonal to them) as well as the two time-and-space dependent scalar

order parameter S1 and S2 along the two directors.

1.3 Q-tensor Theory 8

Figure 1.7: Two directors and Euler angles.[6]

In summary, for biaxial materials two directors ~n(r, t), ~m(r, t) and two scalar order parameters

S1(r, t), S2(r, t) need to be defined, all of them are dependant on time and space. Furthermore,

if we assume |~n| = |~m| = 1, the two directors can be represented by Euler angels θ, φ and ψ, as

shown in Figure 1.7, therefore the biaxial material can be fully described by five variables: θ, φ,

ψ, S1 and S2.

1.3.2 Q Tensor Order Parameter

The idea of Q tensor theory is to use a 3× 3 matrix to describe the orientation of molecules in

liquid crystal:

Q =( q1 q2 q3q2 q4 q5q3 q5 −q1−q4

)(1.4)

q1∼q5 are five independent variables whose values are functions of θ, φ, ψ, S1 and S2. The degree

of ordering in three orthogonal directions can be determined by the three eigenvalues of the Q

tensor λ1, λ2 and λ3, when the three eigenvalues are identical, the material is isotropic, for a

uniaxial material only two of the three eigenvalues are identical, and when the three eigenvalues

are different from each other, the material is biaxial.

BEAM STEERING 9

Chapter 2

Beam Steering

In our modern society, beam steering is an essential technique for many optical systems: fiber-

optical switches or connectors, projection displays, and free space communications.[7, 8, 9, 10]

Steering the beam into large angles without influencing other functionalities plays a vital role

in the developments of these systems.

Generally speaking, beam steering can be achieved by introducing a linear phase retardation to

the wavefront of any lightwave, in particular a laser beam, the capability of how large angles the

beam can be steered into is determined by the slope of the phase retardation profile, it is easy

to imagine that larger retardation slope leads to larger steering angles. To generate the linear

phase retardation profile a linear change of optical path difference (OPD) is needed, which tilts

the incident phase front and thereby steers the light beam. This steering can be achieved either

mechanically or non-mechanically.

2.1 Mechanical Beam Steering

The traditional approach to redirect a beam of light is to use macroscopically sized mechanically

controlled devices. A series of applications of rotating mirrors [11, 12], rotating prisms [10, 13]

and decentered lenses [13, 14] have been reported will be discussed in this section. Examples of

these configurations can be found in Figure 2.1.

Rotating Mirrors

Fast steering mirrors (FSM) are typically lightweight mirrors mounted to a flexure support

system controlled by electromechanical actuators, such as e.g. a simple stepper motor. The

motion of the mirrors are driven by actuators such as voice coils and piezoelectric devices. One

typical configuration of such a FSM is shown in Figure 2.1a.

In this configuration, a single axis hinge acts as a flexure that tilts the mirror in one direction,

an actuator is attached to the base to drive the movement of the mirror. Different types of

actuators can be used depending on the requirement of the movements speed of the mirror: a

2.1 Mechanical Beam Steering 10

(a) Rotating mirror (b) Achromatic doublet prism (c) Decentered lens

Figure 2.1: Nonmechanical beam steering configurations.[14]

high speed (2kHz) with small tilts can be achieved by a piezoelectric driver, and medium speed

for large tilt angles requires a voice coil. Note that this device can only provide beam steering

in one direction, and two-orthogonal-direction beam steering can be implemented by using two

such devices in series.

One obvious drawback of such a device is that the errors in the steering angle are doubled with

respect to the errors in the actuator since it works in reflective mode. Therefore a sensor and

feedback system is needed to correct the errors, making the system more complex and thus

increasing cost. Also the speed and accuracy is limited by the frequency response and resolution

of the actuators.

Rotating Prisms

We know that can be deflected when incident on a prism, and the output light will follow a

circular path when the prism is rotated around the incident direction of the light. However, one

major problem of prisms is dispersion, which is why white light breaks up into its components,

a phenomenon which we often observe in our daily life when light propagates through a glass

window: rainbow edges can be observed at the border between light and shadow. The origin

of dispersion traces back to the wavelength-dependent refractive index of any optical material.

Because of this dependency of the refractive index on the actual wavelength in any optical

material, no single prism can be used for beam steering for a wide range of wavelengths. The

solution to this issue is by using doublet prisms where dispersion is corrected by cementing two

prisms made of different optical materials together, such as flint and crown glass. The steering

ability of the prisms can be extended by cascading two identical prisms, as shown in Figure

2.2, the circle with radius δ at the center is the trajectory of the light by rotating one single

prism, after cascading the second prism the output light will follow another circle centered at

an arbitrary point of the first circle, thus the incident light can be steered at a maximum angle

of 2δ, denoted by the big circle with dashed lines. Figure 2.1b shows two possible configurations

of two cascaded rotating prisms.

2.1 Mechanical Beam Steering 11

Figure 2.2: Output light path of two identical rotating prims.[10]

Note that the first configuration in Figure 2.1b has a disadvantage that it will induce a blind spot

at the horizontal axis, which is unwanted in broadband spectrum beam steering applications, this

can be solved by using the second configuration in Figure 2.1b, the configuration is identical to

itself when rotated 180◦ of one prism, which means it is possible to deflect the light into 0◦ for all

considered wavelengths, thus no blind spot on the axis is generated, however this configuration

will introduce a bit more dispersion than the first one, but in most cases it is acceptable.

Decentered Lenses

By cascading two single lenses in such a way that the front focal point of the second lens coincides

with the back focal point of the first lens, the incident plane-wave light source can be steered

into a non-zero angle when moving the second lens in a direction perpendicular to the optical

axis. Figure 2.1c shows an example of such a decentered lens set.

The maximum angle in which the light can be deflected can be calculated by:

θmax = arctan

(d

2f

)= arctan

(1

2F

)(2.1)

in which f and F are the focal length and f-number of the lens, respectively, d is the diameter

of the lens. So we can see from formula (2.1) that a smaller f-number of the lens gives larger

steering angles, but a smaller f-number will introduce a lot of fabrication problems such as more

spherical aberrations, and also we can see that when the second lens is moved too much in the

vertical axis vignetting happens and part of the incident light cannot reach the second lens,

shown in the shadow region in Figure 2.1c. However to solve this issue more optical elements,

like field lenses, need to be added to the system, which gives other problems such as complexity,

poor liability and steering speed, and difficult to control.

2.2 Non-mechanical Beam Steering 12

2.2 Non-mechanical Beam Steering

As explained in the previous section, macro-mechanical beam steering suffers from disadvan-

tages such as mechanical complexity, high cost of maintenance, limited steering speed and low

inherent reliability. Therefore, it is essential to induce non-mechanical beam steering, which al-

lows random access pointing, and it can also decrease complexity, increase reliability, and reduce

costs. Reports have been made on using multiplexed volume holography [15, 16], microelectro-

mechanical systems (MEMS)[17, 18] and liquid crystals[19, 20, 21] to achieve nonmechanical

beam steering.

Liquid crystal (LC) based beam steering has valuable merits such as large optical aperture,

needing only a low driving voltage, exhibiting high birefringence and the devices can be fabri-

cated with mature available technologies. Because of these advantages, we will review some of

the state-of-art liquid crystal based beam steering technologies in this section.

LC based Beam Steering Devices

Liquid crystal based beam steering devices have been developed for many decades[22]. Due

to the the large birefringence1 of LC, light can be deflected to large angles, even with small

thicknesses of the LC devices, and with low driving voltages. The basic idea of these devices

is to generate an optical path difference (OPD) at different positions within the cell so that

the resulting phase profile will have a saw-tooth shape, leading to the deflection of the incident

light. A number of different optical devices have been investigated based on this idea: liquid

crystal prisms, liquid crystal lenses, liquid crystal phased arrays and liquid crystal polarization

gratings. The working principle of each is explained in the next few paragraphs.

2.2.1 Liquid Crystal Prisms

The concept of LC prisms was first introduced by U. Schmidt and W. Thust[23], and then

developed by several research groups[24, 25]. Based on the electrode pattern the LC prisms can

be divided into stair-case and continuous-ramp phase modulations, the stair-case LC prisms use

discrete electrodes, while the latter one uses uniform electrodes which gives it the advantage

of being more easily fabricated and controlled. One recent study presented a novel simple-

structured easily controllable LC prism[25], shown in Figure 2.3. The LC prism consists of a

wedge-like layer of liquid crystals sandwiched between two indium tin oxide (ITO) electrodes,

the thickness of the liquid crystal layer and the ramp of the prism is determined by the diameters

of the spherical spacers located on two opposite sides of the cell, the liquid crystal molecules

in the cell are aligned parallelly by rubbing the polyimide (PI) coated on the ITO electrodes.

Because of the non-uniform thickness of the LC layer, light passing through will experience a

different optical path length at the two sides of the cell, so that the incident light is finally

deflected into another angle. Furthermore, the slope of the optical path length can be tuned by

1Optical properties depend on the polarization and propagation direction of incident light.

2.2 Non-mechanical Beam Steering 13

Figure 2.3: Basic structure of the LC prism proposed by Lin.[25]

(a) One dimension (b) Two dimension

Figure 2.4: Beam deflection angle versus driving voltage.[25]

applying different voltages to the two ITO electrodes, which means that the deflection angle can

be controlled purely electronically, by applying differing voltages.

Both the deflection angles in the experiment and in simulation at various voltages are shown

in Figure 2.4a. It can be seen that the measured deflection angles are in good agreement with

the simulation results, and the deflection angle is approximately 19.48◦ at the voltage-off state,

which is due to the refraction occurring at the two glass substrate interfaces. When a voltage is

applied, the beam can be deflected in a range from about 13.75◦to 19.48◦. Another important

parameter for beam steering is the angular resolution, which defines how precise one can deflect

the beam. In this research the angle resolution is better than 0.0573◦ and it is directly related

to the resolution of the driving voltage: if the resolution of the driving voltage is improved, so

will the angular resolution also be improved.

One-dimensional beam steering and scanning can be achieved by using one such LC prism,

while by cascading two LC prisms with slope gradient directions perpendicular to each other,

two-dimensional beam steering is feasible, Figure 2.4b shows two graphs of voltage-dependent

deflection angles in the x and y directions, the difference in the two curves is caused by the

difference in the spacer distance in the two LC prisms.

Although the LC prisms have advantages like being easy to fabricate, easy to control and also

having a low manufacturing cost, the efficiency of this kind of device is low due to the losses

of the prism arrays, and further investigation and improvement of this technique need to be

performed.

2.2 Non-mechanical Beam Steering 14

2.2.2 Liquid Crystal Lens

Lenses typically work in transmission and they can change (converge or diverge) the path of

the incident light due to refraction. The phase front of the incident plane wave will experience

a different optical path length (OPL) at different positions within the lens, measured from the

center of it (it is assumed lenses are centro-symmetric), thus the beam converges or diverges

depending on the shape of the lens: if the optical path length experienced at the edges of the

lens is bigger than that at the center, light will traverse the lens near its center more quickly

than at the edge and thus will act to diverge a collimated beam of light. This can be easily

shown by applying Huygens’s principle to the exit plane of the lens. A simple conventional lens

is made of a piece of transmissive material, such as glass, and the optical power of the lens

is defined by its shape, which imposes a major impact on the behavior of the incident light,

thus the main drawback of a conventional lens is that the optical power is fixed and cannot be

tuned. Therefore, an interesting research field is the tunable lens. The idea of a normal lens is

to introduce a space dependent OPL, which is given by:

OPL =

∫ L

0n(r)dr (2.2)

where n(r) is the space dependent refractive index and L is the physically measurable length

that the light traverses, so an intuitive solution is to engineer the refractive index profile n(r)

to have a parabolic shape, instead of the physical shape of the lens. This is where the concept

of the liquid crystal-based lenses comes in, since the refractive index of the LC depends on the

voltage applied over the LC, which means that the optical power of the lens can be changed,

by tuning the control voltage. Researchers reported different configurations of such LC-based

lenses[26, 27, 28]: one approach to achieve LC-based lenses is by using parabolic electrodes[28],

Figure 2.5 shows the fabrication process of such a device. The transparent metal electrode is

deposited on the parabolic surface of the bottom glass substrate, then a polymer whose refractive

index matches with the glass substrate is injected in the sag area, and then an LC cell with

only planar top electrode is laid on top of the polymer. The alignment layer of the LC cell is

antiparallelly rubbed and has a pretilt angle of 3◦.

When a voltage is applied, the planar top and parabolic bottom electrode will generate an

inhomogeneous electric field, thus creating a gradient refractive index profile in the LC layer,

which will induce a focusing effect to the incident light, and the focal length can be approximated

by Fresnel’s formula (2.3):

f =r2

2δndLC(2.3)

where r is the radius of the lens aperture, dLC is the thickness of the LC layer, and δn is the

index difference between the center of the lens and the border, which depends on the electric

field difference, therefore the focal length is continuously controlled by adjusting the applied

voltage.

2.2 Non-mechanical Beam Steering 15

Figure 2.5: Schematic build-up of a proposed LC-based lens[28].

Figure 2.6: Focal length as function of voltage.[28]

Figure 2.6 shows the focal length of the lens as the applied voltage changes. At 0V , the focal

length is infinity due to the match of refractive index of the polymer and the glass substrate,

and the director of the LC molecules is homogeneously aligned. When the voltage is increased,

the LC molecules at the center and at the border will switch differently due to the gradient

electric field induced by the parabolic electrode, so the light will experience a parabolic refractive

index profile and lensing effect occurs, which means the focal length decreases. When the LC

molecules at the border are fully switched but the molecules at the center have no switching,

the focal length will reach its minimum value. To put it differently: when the refractive index

difference between the edge of the lens and its center reaches the maximum, the lens will reach

its maximal optical power. If the voltage continues to increase, the LC molecules at the center

will also reorient and thus the focal length will again increase, but at a slower rate.

This kind of LC lens possesses several merits, such as the fabrication process, which is relatively

simple, and having a simple electrode and a uniformly thick LC layer, which results in a uniform

response over the whole device.

2.2 Non-mechanical Beam Steering 16

Figure 2.7: Blazed phase liquid crystal beam steering device.[20]

2.2.3 Liquid Crystal Phased Arrays

The origin of the liquid crystal phased-array (LCPA) beam steering concept can date back to

the 1970s, since then numerous LC based beam steering studies were performed and different

device configurations are reported [20, 29, 30, 31], and also LC materials other than nematic

liquid crystals are used in beam steering devices, e.g. blue phase LC [32], and cholesteric LC [33].

The working principle of these beam steering devices is essentially the same however: the LC

material is sandwiched between one top planar electrode and multiple stripe electrodes at the

bottom, and by applying a different potential difference relative to the common top electrode for

each of the stripe electrodes, a sloped refractive index profile like a blazed grating is generated,

causing the light to be deflected. Figure 2.7 shows the configuration of such a beam steering

device.

The device works in reflective mode, an anti-reflection layer is coated on top of the upper

substrate to decrease the loss due to the reflection at the first interface (air-glass), the LC

layer is sandwiched between two glass substrates, the material used for the optional dielectric

mirror is a high resistivity oxide, the upper common electrode is formed by coating a layer of

ITO, the bottom electrodes are made of aluminum, under which a thin layer of transparent

resistive metallic oxide stripes are used to create a gradient electric field between the two stripe

electrodes, so that the LC molecules will switch gradually, leading to a linear gradient refractive

index profile. To give an idea on the order of magnitude for the typical dimensions of these

devices, the proposed modulator period is 5µm, the cell thickness is 15.7µm, the width of the

bottom electrodes are 1µm with 2µm gap (active region) between the two electrodes on top of

the same resistive stripe.

In the voltage-off state, the LC molecules are oriented along the bottom electrodes, but when

proper voltages are applied, due to the thickness of the cell being much larger than the width

of each active region, the electric field in the direction parallel to the bottom substrate is much

larger than the electric field between the upper and bottom electrodes, so that the LC molecules

in the active region will rotate in the plane parallel to the bottom substrate. That is, they twist in

the plane rather than tilting in the vertical direction, therefore when the horizontally polarized

2.2 Non-mechanical Beam Steering 17

Figure 2.8: Setup of Pancharatam QHQ: (a) and (e) polarizer; (b) and (d) quarter-wave plate;

and (c) half-wave plate.[39]

light incident on the cell, it will experience a change in the refractive index and resulting in

a different phase delay at different positions above each active region, which can lead to the

deflection of the light.

Note that the device can also work in transmission, simply by removing the dielectric mirror

at the bottom, but working in transmissive mode would require twice the LC layer thickness to

obtain the same phase shift as in the reflection mode. That is usually not desirable, furthermore,

the response time of the LC layer is proportional to the square of its thickness [20], so the response

time of the device working in reflective mode is one fourth of that working in transmissive mode

with similar deflection characteristics, thus making a reflective device much faster.

The experimental results shows that the device can deflect the incident light up to 5◦, however

the efficiency is very low, less than 9%, which is mainly due to the interference between the light

reflected back by the mirror under the bottom electrodes and the light incident on the mirror

in the active region of the device. The efficiency can be increased by embedding the bottom

electrodes and the resistive layer underneath the mirror, but in this approach two difficulties

need to be solved. Firstly burying the electrodes and the resistive layer may induce deformations

to the surface of the mirror which will make the optical quality of the mirror worse. Second,

there is uncertainty that the electrodes and the resistive layer can create a linear electric field

profile at the surface of the mirror, which is critical to generate a linear gradient refractive index

profile.

2.2.4 Liquid Crystal Polarization Gratings

Another approach to achieve beam steering with high efficiency is by using the liquid crystal

polarization grating (LCPG). This type of devices steer beams by modulating the polarization

of the light, rather than amplitude or phase. A few reports have been made on this technique

[21, 34, 35, 36, 37] In order to understand how this LCPG works, let’s first consider a simple

setup called the Pancharatam QHQ stack [38] shown in Figure 2.8.

The light becomes circularly polarized after passing through the first linear polariser and the

first quarter wave plate. For simplicity, we assume it is right-handed circular polarization, which

can be denoted by the Jones vector [39]:

2.2 Non-mechanical Beam Steering 18

Figure 2.9: Director profile in LCPG cell[34]: (a) Top-view; (b) Side-view (Off-state); (c) Side-

view (On-state); Diffraction property (d) RCP and (e) LCP polarized incident light (Off-state;

(f) Transmission of light (Off-state

Ein =[ExinEyin

]=[Exini·Exin

](2.4)

According to Jones calculus, the light after passing through the half wave plate is the product

of the Jones vector of the input light and the Jones matrix of the half wave plate as well as the

transformation matrices:

Eout =[

cosβ − sinβsinβ cosβ

]·[

1 00 −1

]·[

cosβ sinβ− sinβ cosβ

]·[Exini·Exin

]=[

Exinei·2β

−i·Exinei·2β]

(2.5)

So in formula (2.5) we can see that the right-hand polarized input light is converted to left-hand

polarized output light, but with a phase delay of ei·2β, when we change β (the angle between the

slow axis of the λ2 plate and the x-axis) linearly from 0 to π, the phase delay and the resulting

phase profile will change linearly from 0 to 2π. The λ2 plate can be also implemented by using

liquid crystals, simply by rotating the azimuth angle of the LC directors from 0 to 2π, this can

be achieved for example by using the polarization-sensitive alignment material and interfering

two orthogonally circular-polarized UV laser beams onto the material such that the periodic LC

alignment structure is formed, shown in left figure in Figure 2.9

At the voltage-off state, the incident light can be deflected into the +1 order or -1 order depending

on whether it is left-hand or right-hand circular polarization, the diffraction angle is simply given

by

θ = arcsinλ

Λ+ θin (2.6)

Where θin is the incident angle, λ is the wavelength of the light and Λ is the period of the

polarization grating. When a sufficiently high voltage is applied (voltage-on state), the LC

molecules will switch vertically and therefore the polarization grating vanishes, the incident

2.2 Non-mechanical Beam Steering 19

Figure 2.10: Ternary PG beam steering design: (a) ternary steerer design, (b) number of steering

angles and calculated transmittance versus number of stages.[35]

light will just pass through the cell without changing its directions. Studies show that these

kinds of LCPGs can have a steering efficiency as high as 99% [36, 40, 41, 42] and the efficiency

remains high as long as the incident angle is below 20◦[43].

Note that the LCPGs can only steer the incident light to three discrete angles, the 0th order and

the ±1 orders, the angle at +(-)1 order is determined by the period of the polarization grating,

for example, if the optical communication wavelength at 1550nm is used, the grating period

needs to be 8.9µm to achieve a deflection angle of 10◦ according to formula (2.6). For practical

use, the device needs to be combined with a fine angle steering stage to precisely control the

deflection angle.

To achieve even larger steering angles, multiple stages of the LCPGs can be cascaded [35].

Figure2.10 shows such a design, which shows a wide angle beam steering as large as 40◦, the

steering angle can be doubled by each PG stage without a significant decrease in efficiency. The

number of steering angles M can be calculated by

M = 3N (2.7)

where N is the number of stages of PGs, and the transmittance T is given by

T = (η+1)N (1−D)N (1−R)2N (1−A)2N (2.8)

where η+1 is the diffraction efficiency of each PG, D is the diffuse scattering of each PG, and R

and A are the Fresnel reflectance and absorption losses, respectively.

2.2 Non-mechanical Beam Steering 20

The PGs can also be made into passive devices, reports have been made in which they fabricated

the PGs using polymerizable liquid crystals, which is called reactive mesogen [44]. This technique

allows reduction of scattering losses and smaller grating periods [42] , which means the steering

angle for each single PG can become larger, but then the gratings are not erasable with voltage

and thus the device will always steer the beam to the intrinsic angle, which is dependent on the

wavelength of the incident light. The study in paper [42] shows that the passive LCPG can also

have a diffraction efficiency larger than 99%.

The major advantages of the LCPGs are the lack of flyback2 and reset effect in these cells,

which are the main reasons for low diffraction efficiency, and also the cell only has a thickness of

half-wave optical path, which diminishes the scattering and absorption due to the liquid crystal

itself, and the thickness does not depend on the steering angle nor the aperture size.

2The width for phase difference changes from 2π to 0.

THEORETICAL ANALYSIS 21

Chapter 3

Theoretical Analysis

3.1 Diffraction Theory

It is well know that light is an electromagnetic wave described by Maxwell’s equations, however

in general, light can be considered as a bundle of light rays, travelling along straight lines, and it

can be reflected or refracted by optical elements. This research field is called geometric optics,

but the assumptions made in geometric optics only hold true when the dimensions of the optical

elements are much larger than the wavelength of light. When light is interacting with optical

elements with dimensions that are comparable to the wavelength of light, it must be regarded as

waves, and one intrinsic property of the nature of waves is diffraction, thus it is very necessary

to have a thorough understanding of the diffraction theory.

3.1.1 Thin Surface Grating

Figure 3.1: Plane wave incident on window with slits.[45]

3.1 Diffraction Theory 22

Consider the configuration shown in Figure 3.1, where light passes through a periodic structure

consisting of windows of slits. The transmitted field through such a structure can be described

by the product of the incident field and the transmission function of the periodic structure

as predicted by Fourier Optics. Fourier analysis teaches us that the Fourier transform of the

product of functions is the convolution of the Fourier transforms of the two individual functions.

The transmission of the finite width structure shown in Figure 3.1 can be written as [45]:

t(x) =N∑n=1

t1(x− xn) (3.1)

with xn = (n− 1)Λ, and its Fourier transform:

T (fx) = T1(fx)× ej(N−1)δ/2 × sinNδ/2

sin δ/2(3.2)

where N is the number of periods (or slits), Λ is the period of the structure, fx = sin θ/λ is

the spatial frequency and δ = −2πfxΛ is the phase delay. We can see the overall transmission

function consists of two contributions : T1(fx) is the transmission function of one single period,

and sinNδ/2sin δ/2 is an amplitude modulating factor, originating from the periodic nature of the

structure. When a plane wave propagating perpendicularly towards the structure is transmitted

by the structure, the transmitted field can be expressed by:

F (φ(x, 0+)) = T (fx) (3.3)

because the Fourier transform of a plane wave is 1. 0+ in this context means it is the transmitted

field, and the field has peaks at:

fx = ±mΛ

(3.4)

which is equivalent to

sin(θm) = ±λΛ

(3.5)

where θm is the diffraction angle of m-th order, this is the case when the light is normally

incident on the structure. When the k-vector of the incident light makes an angle θinwith the

surface normal, then comes the famous grating equation:

sin(θm)− sin(θin) = ±mλ

Λ(3.6)

3.1.2 Blazed Grating

The blazed grating is a periodic structure where in one single period the thickness is linearly

increasing to a maximum, predefined value, shown in Figure 3.2a, such a structure can diffract

light into different output angles, which are determined by the difference in optical path length

(OPL) at different position of the grating. The OPL is defined as the product of the physical

3.1 Diffraction Theory 23

(a) Blazed grating (b) Transmission of blazed grating

Figure 3.2: Blazed grating and its transmission[45]

length a beam of light traverses and the refractive index the light experiences. In this case, the

grating is made of a homogeneous material with refractive index n1, and the light is diffracted

by the grating to the medium with refractive index of n2, the period of the grating is Λ and the

thickness profile of the grating can be written as d(x) = xΛd, where d is the total thickness, so

the OPL after transmission of a distance d can be described by:

OPL = d(x)n1 + (d− d(x))n2 = dn2 + dx

Λ(n1 − n2) (3.7)

It can be seen that at different locations in the grating (different x) the OPL will be different

and larger thickness of the grating corresponds to larger OPL. Assuming an incident plane

wave, then the phase after transmission through the one period of the grating, which is also the

transmission function of such a phase grating, can be described by:

t1(x) = e+j 2πλ

(n2−n1)xd/Λe−j2πdλn2 (3.8)

So the plane wave will experience different phase delay at different position within one period

of the grating and will be diffracted towards the path that is perpendicular to the phase front,

the output angle of the diffracted light can be determined by working out the Fourier transform

of the transmission function of the periodic grating, and since we are only interested in the

intensity of the transmitted light, so we take the modulus square of the Fourier transform, from

the previous analysis(equation (3.2)) this can be written as:

|T (fx)|2 =sin2(Nδ/2)

sin2(δ/2)

sin2(π(fx − f0)Λ)

(π(fx − f0)Λ)2(3.9)

3.2 Cell Structure 24

with f0 = (n2 − n1) dλΛ , the plot of this function is shown in Figure 3.2b, the first factor in

equation (3.9) has peaks at fx = ±mΛ , and is only related to the number of periods N and the

period Λ; if the number of periods increases the peaks will become sharper and larger grating

periods lead to smaller distances between the peaks. The second factor in the transmission

function is a sinc function and it acts like an envelop. It is known that the sinc function sinc(x)

has its maximum at x = 0, in this case π(fx − f0)Λ = 0, thus fx − f0 = 0, so the maximum of

the transmission function is dependent on the value of f0, which means light can be diffracted

to the direction with spatial frequency fx = ±mΛ , which is equivalent to say that light can be

diffracted to the direction where the optical path difference (OPD) is an integer times of 2π:

δ = m2π, with m the diffraction order.

3.2 Cell Structure

The goal of this thesis work is to achieve beam steering utilising liquid crystals based on the

diffraction theory as described in last section. To achieve this goal, a certain device configuration

has to be implemented. In this work we fabricated and characterized the liquid crystal cells with

two different configurations, the difference lies in the electrode patterns, one configuration uses

interdigitated electrodes (a ”finger pattern”) with a symmetric gap between the fingers on each

side, while the other cell is fabricated with asymmetric finger electrodes In each of the two

configurations, two types of cells can be distinguished, with the difference of whether there is

a thin layer of lead zirconate titanate (PZT) coated on top of the electrodes. Furthermore,

in the asymmetric finger electrodes case, there are two rubbing directions of the liquid crystal

molecules with respect to the direction of the stripe finger electrodes, more information about

these configurations of the cell will be provided in a next chapter.

3.2.1 Blazed Grating Based on Liquid Crystals

The blazed grating discussed previously is based on the variation of the thickness of medium

within one single period, so that the resulting optical path difference (which results in a phase

difference for an incident plane wave) will have a saw-tooth profile. Note that this phase profile

can also by generated by using liquid crystals, as liquid crystal molecules can respond to an

external electric field by twisting or tilting. This change of orientation of LC molecules will lead

to the change in refractive index experienced by the incident light. As explained in chapter two,

suppose the incident light is propagating along the z direction and with polarization along the x

direction, rubbing direction of the liquid crystal is also along the x direction. When there is no

external electric field applied the light will experience a refractive index of ne, when the electric

field is applied along the z direction, the molecules will tilt and light will experience an effective

refractive index described by equation (1.3), and the voltage-dependent refractive index has the

shape shown in Figure 3.3. We can recognise three different regions in this curve, at low voltage

the refractive index remains at ne because the voltage is not enough to switch the molecules.

When the voltage reaches a threshold the molecules start switching and the refractive index

decreases, the dependence is roughly linear. At high voltages the curve saturates which means

3.2 Cell Structure 25

Figure 3.3: Refractive index as function of applied voltage for the nematic LC mixure E7 .[46]

the liquid crystal is fully switched: the molecules are all oriented along the electric field, they

cannot tilt further.

Imagine the electric field is distributed in the cell such that the resulting refractive index profile

of the liquid crystal is like a blazed grating, and it can be expressed as:

n(x) = n2 −x

Λ(n2 − n1) (3.10)

which leads to the same transmission function described in equation (3.8). The difference be-

tween this kind of blazed grating and the normal grating is that the thickness of the liquid

crystal cell is uniform, while the OPL profile still resembles that of the normal blazed grating.

3.2.2 PZT

The liquid crystal based blazed gratings have advantages such as tunable diffraction angle,

however, to implement a blazed grating with liquid crystals is not easy: the phase profile of

the transmitted light has to be perfectly linear, this means a homogeneous electric field has

to be induced over one period of the cell. In reality, for obvious reasons, electrodes need to

be separated by gaps to allow appropriate voltage combinations to be applied. The existance

of gaps between the electrodes will lead to big variations in the electric field profile, thus the

molecule orientations in this gap region will deviate from the desired pattern, therefore the

resulting refractive index profile is not smooth.

One innovative solution to this problem of fringe fields is to reduce the gap between electrodes,

one can imagine that when the gap is small compared to the thickness of the cell, a better,

smoother refractive index profile can be obtained, however this approach is prone to fabrication

process errors: when the dimension of the gaps reaches the fabrication resolution limitations,

there will be a higher chance that two adjacent electrodes can be shorted, which should definitely

be avoided.

3.2 Cell Structure 26

An interesting alternative approach is to linearize the electric field by smoothening the fringe

fields. Efforts have been made by using a resistive layer in contact with the electrodes[20]. As

discussed in chapter 2 (shown in Figure 2.7), the two electrodes with different voltages are seated

on top of the resistive stripes, so the linear, gradient electric field is created above the resistive

stripes.

This thesis is based on a similar idea, where instead of using resistive material a material with

high dielectric constant was used. TO be specific, a layer of lead zirconate titanate (PZT) is

coated on top of the electrodes. Although PZT is well known for its piezoelectric effect, the

most important and the only property that matters in this thesis work is that it has a very high

dielectric constant, up to 500. As will become clear, the electric field induced by the electrodes

will be extended into the gap region leading to a smoother field distribution and thus refractive

index profile of the liquid crystal.

3.2.3 Symmetric Finger Electrodes

To understand the functionality of the PZT layer, cells with different electrode configurations

were designed and fabricated. In the first configuration, symmetric finger patterned electrodes

were used, where one cell has a thin layer of PZT on top of the electrodes, and the other one

without PZT layer was used as reference. Then, after experimentally verifiying the performance

improvement of the PZT cell, a set of cells with asymmetric finger electrodes were fabricated

for the purpose of beam steering.

Figure 3.4 shows one unit of the cell structure with symmetric finger electrodes. The liquid

crystal layer is sandwiched between two glass substrates, a common electrode made of a thin

layer of ITO is deposited on the top substrate. On the bottom substrate, a symmetric finger

electrode pattern, also made of ITO, is used (illustrated in Figure 3.6), electrodes 1 and 3 are

connected together to one bondpad, and electrode 2 and 4 are connected to the other bondpad.

Due to the transparency of the ITO, there should no diffraction pattern at the voltage-off state.

At the voltage-on state, due to the periodic nature of the structure, light will be diffracted,

with the diffraction angles at different orders determined by the grating equation. However,

analytical calculation of how light is distributed at these orders is not possible, thus we will

characterize this cell by numerical simulation and experiments, and the results will be presented

in chapters 4 and 5. Table 3.1 lists the most important parameters of this cell.

3.2.4 Asymmetric Finger Electrodes

The main difference between the cells with symmetric finger electrodes and the ones under

discussion here, the asymmetric finger patterned cells, lies in the electrode pattern, as can be

seen in Figure 3.6a. Also instead of ITO, the bottom electrodes are made of a non-transparent

metal. Similar to the cells with symmetric finger electrodes, electrode 1 and electrode 3 are

connected to one bonding pad while electrode 2 and electrode 4 are connected to the other

3.2 Cell Structure 27

(a) Cell configuration without PZT layer

(b) Cell configuration with PZT layer

Figure 3.4: Cell structure with symmetric finger electrodes

3.2 Cell Structure 28

(a) Rubbing direction perpendicular to electrodes

(b) Rubbing direction parallel to electrodes

Figure 3.5: Cell structure with asymmetric finger electrodes

3.2 Cell Structure 29

Table 3.1: Parameters of the cell with symmetric finger electrode

wavelength used 632.8nm

LC E7

no 1.5189

ne 1.7304

thickness 10µm

electrode width 6µm

electrode gap(G) 6µm

grating period(P) 18µm

PZT thickness 1µm

bonding pad, so that by applying two different voltages at the two bonding pads a gradient

electric field in the region between electrode 2 and 3 is created. The period P (distance from

electrode 1 to electrode 3) depends on the specific region of the cell under test and varies from

30µm to 80µm. A PZT layer is coated on top of the finger electrode layer to smooth out the

electric variations, thus a linear refractive index profile is formed. The rubbing direction of the

liquid crystal is either parallel or perpendicular with respect to the finger electrodes. A reference

cell with the same configuration but with no PZT layer is also fabricated and characterized, with

the rubbing direction perpendicular to the electrodes. The fabrication process of the cell will be

discussed in chapter 5. Here we list the most important parameters of the cell, summarized in

table 3.2.

Table 3.2: Parameters of the cell with asymmetric finger electrode

wavelength 632.8nm

LC E7

no 1.5189

ne 1.7304

thickness 10µm or 20µm

electrode width 6µm

electrode gap(G) 6µm

grating period(P) 30µm− 80µm

PZT thickness 800nm

Diffraction Angle Calculation

Since the finger electrode is made of metal that is not transparent, when the light is incident on

the cell the electrode itself will cause diffraction, and the diffraction angle of the electrode θe in

the voltage-off state (the liquid crystal layer acts as an isotropic medium to the light) can be

easily calculated using the grating equation (3.6), assuming normal incidence:

3.2 Cell Structure 30

(a) Asymmetric finger electrode (b) Symmetric finger electrode

Figure 3.6: Finger electrode

θe = ± arcsinmλ

Λ(3.11)

where λ is the wavelength of the light and Λ is the period of the grating. (30µm− 80µm)

In order to analytically calculate the diffraction angle of the device in the voltage-on state, a

few assumptions have to be made. First we assume the PZT layer can extend the electric field

into the gap leading to a linear gradient field, furthermore we assume the voltage dependent

birefringence ∆n(V ) of the liquid crystal is linear, so we work in the linear region indicated in

Figure 3.3. Another assumption we make is that, since the gap between electrode 1 and 2 is

much smaller than that between electrode 2 and 3, we can ignore what happens above electrodes

1 and 2, i.e. we assume there is no flyback region at the resets of the refractive index profile.

With these assumptions the diffraction angle of the cell can be calculated again by the grating

equation:

θcell = ± arcsinmλ

Λ(3.12)

Note that different from the case with electrodes, where light is diffracted into several orders,

light can be diffracted to a single order, as long as the phase difference:

∆φ =(n2 − n1)d

λ2π (3.13)

is a multiple of 2π. Then the cell works like a higher order blazed grating. The maximum

diffraction angle occurs when the difference of refractive indices is maximum: ∆nmax = ne−no,then the maximum phase difference is:

∆φmax =(ne − no)d

λ2π =

(1.7304− 1.5189)× 10µm

0.633µm2π = 3.34π (3.14)

3.2 Cell Structure 31

When the phase difference is not a multiple of 2π, light will be diffracted into the (m− 1)th and

(m− 1)th order, in this case between 3rd and 4th order. Table 3.3 list the first order diffraction

angle and the maximum diffraction angle for the grating period 30µm to 80µm.

Table 3.3: Diffraction angle of the cell

grating period 30µm 40µm 50µm 60µm 70µm 80µm

1st order angle(◦) 1.21 0.907 0.725 0.604 0.52 0.453

maximum angle(3rd order)(◦) 3.63 2.72 2.18 1.81 1.55 1.36

SIMULATIONS 32

Chapter 4

Simulations

In this chapter we will give an introduction to the simulations done in this thesis work, in order

to verify the theory and the experiments, and get more understanding of the behavior of the

liquid crystal based beam steering device. The simulation was done by first using the Q-tensor

Matlab program[47] to calculate the director distribution of the liquid crystal when an external

voltage was applied, and then the diffraction pattern and angular spectrum was simulated by

using commercial software FDTD solutions. Both of the simulation results of the cell with

symmetric finger electrodes and asymmetric finger electrodes will be presented in this chapter.

4.1 Mesh Generation

Usually in two-dimensional numerical simulations, two different meshes with different cell shapes

are used: triangles and quadrilaterals. In this thesis we choose a triangular mesh for its advantage

of being able to mesh complex geometries and it is also fast and easy to generate. The mesh

was created by the free software called gmsh. A well-written mesh is built parametrically:

by defining a few variables upfront, the mesh can be altered easily, by changing only those

parameters, without having to change the location of multiple points defining the geometry. For

the topic of interest, these parameters include the thickness of the LC layer and PZT layer,

the width of the electrodes and spacing between them. Figure 4.1 shows one example of the

generated mesh, the two electrodes are colored light blue, above which there is the PZT layer,

and on top of the that layer, there is the LC layer. Note that a periodic boundary is defined on

the left and right edges of this mesh in order to simulate the real device.

The size of the mesh has to be controlled properly: a denser mesh will give a more accurate

simulation result and a more detailed LC director distribution, but it requires more calculation

time and more computer memory. A good solution to this is by using variable mesh size at

different positions, e.g. in the PZT layer and near the electrodes, the electric potential changes

rapidly leading to abrupt changes of the LC directors near the PZT-LC interface, thus the mesh

size in the PZT layer was chosen very small, and above the PZT-LC interface a linearly increasing

mesh size is used. Once the mesh is defined, the orientation of the LC molecule directors will

be calculated by using the provided Matlab program based on Q-tensor theory.

4.2 Finite-Difference Time-Domain method 33

Figure 4.1: One unit of the mesh for cell with 2 electrodes.

4.2 Finite-Difference Time-Domain method

Light propagation in liquid crystals can be calculated using Jones Calculus, which divides the

liquid crystal layer into thin sublayers, and each layer can be modelled by a Jones matrix, by

taking the product of the Jones matrices, the polarisation and relative phase of the light can be

calculated after passing through the liquid crystal layer. However, in this thesis work we used

FDTD solutions instead of Jones Calculus, because of its simplicity. Here we briefly give an

introduction to the finite-difference time-domain (FDTD) method.

FDTD is a numerical method to calculate the electromagnetic fields based on finite differences,

in which time and space are divided into small intervals called the Yee grid, shown in Figure

4.2, so that the time-dependent Maxwell’s equations can be solved by iterating over the electric

field and the magnetic field in the time domain.

Figure 4.2: Yee grid for FDTD method.[48]

The Maxwell equations in a linear, isotropic non-dispersive materials can be written as:

∂E

∂t= −1

ε∇×H− σ

εE (4.1)

∂H

∂t= − 1

µ∇×E (4.2)

4.3 Simulation Results 34

We can see that the time derivative of the E-field depends on the E-field itself and the curl of

the H-field, so that the E-field after a small time interval E(t0 + δt) depends on the E-field at

time t0 (E(t0)), which has been stored, and the spatial distribution of the H-field. The similar

dependence can be found in the H-field, so that by iterating over the E-field and H-field in time

we can calculate all the values of the E and H-field at any time and space.

4.3 Simulation Results

The simulation result of the Q tensor program contains five independent variables which describe

the director orientations of the liquid crystal, to visualize the director distribution, a free software

called LCview [49] developed by the Computer Modelling Group at the University College

London is used and it can also provide information on the potential distribution and the tilt and

twist angles of the molecules at different locations. As we mentioned previously, to calculate the

light propagation in the liquid crystal we used FDTD solutions, which requires a rectangular

mesh instead of a triangular one, thus the Q tensor variable on the triangular mesh has to

be interpolated into a rectangular mesh, which is done with an in-house developed MATLAB

function.

4.3.1 Symmetric Finger Electrodes

In order to understand the functionality of the PZT layer, we did simulations on the two cells

with symmetric finger electrodes, by applying different combinations of voltages on the two finger

electrodes. Figure 4.3 plots the director distribution and potential profile of the two cells when

potentials of V1=+7V and V2=-7V were applied 1 on the two finger electrodes, with the common

electrode grounded, the rubbing direction is along the x-direction. We see the molecules above

the two finger electrodes are twisted, this is because above the electrodes the electric potential

is so large that the molecules there have to induce some twist to obtain a minimum energy

state. The difference in director distributions between the two cells is not obvious, however

the twist in the PZT cell seems to be smaller than that in the reference cell, and the twist is

believed to be responsible for the phase variations when light passes through the liquid crystal

layer. To further investigate the difference between the two cells, we plot the optical path length

(OPL = neffd/λ) after light with polarization along the x-axis 2 passing through the LC layer,

shown in Figure 4.4a, the red short lines denote the position of the two electrodes. We see

that due to the fact that molecules above the two finger electrodes switch (tilt) in the vertical

direction under influence of the electric field, the light will see a small refractive index no in

that region, in the region between the finger electrodes, the electric field is smaller and thus LC

molecules are less tilted, so light will experience a larger refractive index, and in the middle the

molecules are horizontally aligned thus light will see the largest refractive index ne. Important

to note is that in the region above the finger electrodes (denoted by the circles in Figure 4.4a) the

phase variation in the PZT cell is smaller than that in the reference cell, which is an advantage

1The applied voltage in the simulation is DC voltage.2Thus polarization is along the rubbing direction

4.3 Simulation Results 35

because rapidly varying phase variations will cause scattering when light passes through, leading

to losses. The reason behind this phenomenon is that the electric field is extended into the gap

between electrodes by the PZT layer thus the phase variation caused by the twist is reduced.

(a) Reference cell.

(b) PZT cell.

Figure 4.3: Director profile of the two cells when +7V and -7V are applied to the two finger

electrodes.

As we discussed in chapter 3, the PZT layer is assumed to linearize the phase profile between the

two finger electrodes, to verify this assumption, the OPL of the light (polarized along x-direction)

after passing through the cell with applied voltage V1=7V and V2=0V is plotted in Figure 4.4b.

If one of the finger electrodes is held at 0V, then the molecules above these electrodes will not

switch due to the common electrode also being grounded, so the light will see a refractive index

of ne and have a large optical path length. In between the finger electrodes the molecules will

switch more and more when approaching the other electrode which is held at 7V, leading to a

gradually decreasing optical path length. In this region, it can be clearly seen that the phase

changing in the PZT cell is much more linear than that in the reference cell, thus one can expect

the PZT cell will give a better diffraction pattern in the experiment.

The angular spectrum can be obtained by doing simulations with FDTD progams. The Q tensor

program output is first interpolated to a rectangular mesh, then the files containing the x, y

and z components of the refractive indices at each points of the mesh are imported into FDTD

solutions, after which the optical simulations can be performed. The light source used in the

simulation is a plane wave with wavelength λ at 632.8nm, and the polarization is along the

x-direction. Figure 4.5 shows the angular spectrum of the two cells when the applied voltage is

4.3 Simulation Results 36

(a) OPL when applied +7V and -7V. (b) OPL when applied +7V and 0V.

Figure 4.4: Optical path length of the two cells with different voltage applied on the finger

electrodes.

V1=+7V and V2=-7V3. Note that the grating period Λ in this case is 18µm, so according to the

grating equation(3.6), the diffraction orders should appear at 4

θm = arcsinmλ

Λ= arcsinm

0.6328µm

18µm≈ m2.015◦ (4.3)

(a) Reference cell. (b) PZT cell.

Figure 4.5: Angular spectrum of the cells when applied +7V and -7V.

We can see from Figure 4.5a, in the angular spectrum of the reference cell, the diffraction orders

appear at the expected angles, which are multiples of 2◦5, and the maximum intensity appears

at 8◦, furthermore, the angular spectrum is not symmetric. While in the PZT cell, the angular

spectrum is more symmetric, and the maximum intensity also appears at 2◦ and −8◦, note that

in the PZT cell diffraction orders also appear at multiple of 1◦, which corresponds to a grating

period of 36µm. The reason for this stays unknown. One proper guess might be that due to

the presence of the PZT layer, the liquid crystals respond to positive and negative voltages in

different ways, leading to a doubled grating period.

3The signs indicate that the AC signals are in anti-phase, i.e. 180◦ out of phase, but having equal amplitude4The light is incident along the substrate normal in all of the simulations.5We assume the diffraction angle is small.

4.3 Simulation Results 37

(a) Reference cell.

(b) PZT cell with εPZT = 10.

(c) PZT cell with εPZT = 50.

Figure 4.6: Potential distribution when the two finger electrodes are both applied +5V voltage.

4.3 Simulation Results 38

Another approach to verify that PZT can extend the electric field into the region between

electrodes is by doing a simulation where both of the finger electrodes are applied positive

voltages. Figure 4.6 shows such a case and we see that in the reference cell the electric field

is concentrated in the region near the electrodes. When we add a layer of PZT and adjust its

dielectric constant to 10, the electric field starts to extend into the gap region. Upon increasing

the dielectric constant to 50, the electric field is almost extended to the whole region of the

cell. In reality, the PZT has a dielectric constant of 400 − 500, thus one can imagine that the

molecules in between the finger electrodes will also switch because of the PZT layer.

Figure 4.7 shows the OPL of the two cells when V1=+5V and V2 =+5V, with the common

electrode grounded. One can see that in the PZT cell the OPL is more or less constant, which

is what we expected, while in the reference cell it resembles the blazed grating, and the optical

path difference between the maximum and minimum is approximately one wavelength, so that

we can expect the cell can diffract incident light into the +(-)1 order.

Figure 4.7: OPL when two consecutive electrodes are applied +5V and +5V.

The angular spectrum for this case is calculated in the same way, shown in Figure 4.8. We see

that in the reference cell most light is diffracted into the -1 order, at an angle of 2◦, which agrees

with the phase profile, and at the +2 order and -2 order moderate peaks can be observed. This

is because the phase profile is not a perfect blazed grating and also the optical path difference

is not exactly one wavelength. In the PZT cell light stays at 0◦, due to the fact that all the

molecules in the cell are switching because of the presence of PZT.

4.3.2 Asymmetric Finger Electrodes

A similar simulation was performed for the cells with asymmetric finger electrodes. In this case

both the rubbing direction and the polarization of the incident light are along the x-direction.

Figure 4.9 shows the OPL for the cells with an electrode period of 60µm. It can be seen that at

4.3 Simulation Results 39

(a) Reference cell. (b) PZT cell.

Figure 4.8: Angular spectrum of the cells when applied +5V and +5V.

(a) PZT cell. (b) Reference cell.

Figure 4.9: OPL after passing through the cells with asymmetric finger electrodes.

4.3 Simulation Results 40

the voltage-off state or when only a small voltage (eg. 1V) is applied, the OPL stays constant

because the incident light wave experiences the extraordinary refractive index, ne. As the voltage

increases, the LC molecules start to switch and the refractive index changes. In the PZT cell,

the OPL becomes fairly linear at modest voltages (eg. 3V). However, as the voltage is further

increased the OPL over the electrode gap region continues to decrease and finally becomes flat at

10V, this is again due to the PZT extending the electric field into the gap region and triggering

the switch there. In the reference cell, similar behavior can be found at the region above the

electrode, while at the gap region the OPL stays almost unchanged due to lack of electric field.

The angular spectrum of the two cells when the two finger electrodes are supplied with 3V and

0V relative to the planar top electrode is shown in Figure 4.10.

(a) PZT cell. (b) Reference cell.

Figure 4.10: Angular spectrum when the cells are applied +3V and 0V.

It can be seen that for the PZT cell the incident light is diffracted into one single order at 1.2◦,

which corresponds to the +2nd order for the cell with the electrode period of 60µm (see Table

3.3). This is because at this voltage combination the maximum OPL is about 2λ, as shown in

Figure 4.9a. While for the reference cell another strong peak appears at −1.2◦, which is due to

the phase profile not being very much approximating the ideal blazed grating situation, as can

be seen in Figure 4.9b.

EXPERIMENTS 41

Chapter 5

Experiments

5.1 Fabrication Process

A general introduction to the fabrication process of liquid crystal cells is given in this chapter.

Due to safety concerns and timing considerations, the author was not present for all steps of

the fabrication process, but participated in those that he could. The fabrication process of the

beam steering cells is involves mask design, lift-off, surface anchoring, assembly, liquid crystal

filling and finally soldering of conducting wires to the on-glass electrodes.

Mask design. The electrode patterns were designed with the IPKISS software. Masks that

are made with it can be easily viewed with Klayout[50]. To increase yield of the fabrication,

six finger electrode patterns are combined on a single 1 inch substrate. These 6 regions will

be called the ”active regions” for convenience. The geometry of the mask is shown in Figure

5.1a. The letters a-f indicate these active regions, with an electrode period ranging from 30µm

to 80µm. Figure 5.1b shows part of the zoomed electrode pattern in active region f, with the

typical dimensions annotated: the width of electrodes is 6µm with a fixed gap of 6µm between

them1. The electrode period is 80µm, and thus the largest non-trivial spacing between two

finger electrodes is 80µm− 2× 6µm− 6µm = 62µm. Note that this value varies for each active

region with different electrode periods, but this will be mentioned in the text.

Lift-off process. For the common electrode, ITO was used and requires a single step of

deposition. For the bottom electrode the material we used was Platinum (Pt), which is the

least reactive metal, but it resists etching rather well, thus it is more easy to make the electrode

pattern in an indirect way, rather than by etching. In this thesis we used a process called lift-off,

which is shown schematically in Figure 5.2.

The following steps are used in the lift-off process: (I) A clean glass substrate is readied. (II)

Photoresist is deposited on top of the glass substrate by means of spincoating. (III) The photore-

sist is exposed to UV light and subsequently developed such that the region where the electrodes

1The width of electrodes and fixed gap are identical in every action region and every cell.

5.1 Fabrication Process 42

(a) Mask of finger electrodes consists of six active

regions. From a to f, the pitch increases linearly

from 30µm to 80µm.

(b) Zoomed pattern of active region f.

Figure 5.1: Mask of the asymmetric finger patterns.

Figure 5.2: Lift-off process steps.

should be on the substrate are left open in the resist, while the other regions remain covered with

5.1 Fabrication Process 43

Figure 5.3: Glue pattern.

photoresist. (IV) Platinum is deposited on the whole surface, thus at the electrode region it is

attached to the substrate, while at other regions it stays on the photoresist. (V) The photoresist

is stripped away by a solution. Because the photoresist is organic, acetone is typically used to

do this. The result is that the Platinum that was on top of resist patches will also be removed,

while the Platinum attached to the surface of the substrate will stay. (VI) A final patterned

electrode is created.

Surface anchoring. Before filling the cell with liquid crystals, the alignment direction has

to be predefined. In this process a layer of nylon mixture was spin coated on top of the two

substrates that would be used to make a cell. These two substrates, after deposition of the nylon-

based alignment layer, were then cured at 180◦ for 4 hours so that the nylon was cross-linked.

Finally the substrates were mechanically rubbed, using a soft piece of cloth. The direction in

which the cloth is pulled over the substrate defines the alignment direction for the rod-shaped

liquid crystal molecules.

Cell assembly and LC filling. After rubbing, the two substrates are glued together. This

is done by using a syringe containing a mix of UV curable glue and spherical spacers, whose

diameter would determine the final cell thickness. The syringe movement was controlled by an

automated translation stage. The glue pattern is a rectangle with two openings at opposing

sides, shown in Figure 5.3. This pattern allows us to fill the cell with a liquid crystal mixture

through these openings by means of capillary forces. The glue containing spacers was deposited

on the side with nylon of the top substrate , and then the bottom substrate was attached to the

top substrate by applying pressure, which was very important to ensure uniform cell thickness.

If the cell thickness is not uniform then it would act like an LC prism and induce a phase shift

at the voltage-off state, as we discussed in Chapter 2. After this the glue was cured by UV light,

which gives a firm attachment of the two substrates. Note that the glue pattern should be large

enough to cover all the six active regions, otherwise the active region outside the glue pattern

would not be covered by liquid crystals, which is obviously unwanted.

When the cell was assembled, the nematic liquid crystal mixture known as E7 was injected in

the cell cavity. Since the cell is very thin, only a small drop of liquid crystals was needed. The

liquid crystal was filled in via one of the two openings of the glue pattern, and it would then

reach the whole region of the cell because of the capillary effect.

Connecting to bondpads. The last step of making the cells is connecting wires to the

bondpads of the electrodes so that a driving voltage can be applied. In this case we use ultrasonic

5.2 Fabricated Cells 44

Figure 5.4: One of the fabricated cells, observed using two polarizers, with parallel transmission

axes.

soldering, which could allow the solder to penetrate the PZT layer and connect more easily with

the electrodes on the glass substrate.

5.2 Fabricated Cells

In total four of these asymmetric cells were fabricated, including three cells with PZT layer

and one reference cell without PZT, each of them has six active regions with bottom electrode

period from 30µm to 80µm, Figure 5.4 shows one of these cells. However, due to fabrication

errors, some of these active regions are not functioning. After fabrication we checked these

active regions under crossed polarizers and with the microscope. It was shown that due to

improper filling, some of them are not covered fully by liquid crystals, so these regions could not

be used in experiments, and in some of the active regions there were disconnected electrodes and

short circuits between the two bottom finger electrodes, shown in Figure 5.5a. Disconnected

electrodes would not be a big problem since only the LC molecule near the broken electrode

would not be switching due to lack of electric field, while the problem of short circuits is in

theory more severe, because it would induce a current (and thus generate heat) in the liquid

crystal and the cell might be burned. To avoid this, in experiment we should apply voltages

only in a short time, but long enough for the LC molecules to switch. It was observed under

polarizing microscope that the short circuit only has local influence on the behavior of liquid

crystals. Figure 5.5b shows the switching of the liquid crystals under polarizing microscope,

electrode 1 and the common electrode were grounded, a 10V potential difference was applied to

electrode 2. As a result of this, different colors appeared in different regions. This is because

the locally switched molecules will affect the anisotrosopy of the liquid crystals thus change the

polarization of the light, and the degree of the polarization change is wavelength dependent.

We see that near the short circuit location the color is the same which means the LC molecules

above it are uniformly switched, i.e, the potential on the two electrodes is the same, however,

the behavior of the liquid crystals right of electrode 2 are not affected by the short circuit, as

can be seen from the color fringes.

Furthermore, since the steering angle is related to the thickness of the cell, it is important to

characterize the thickness of the cell after fabrication. The thickness of the cells was measured

5.2 Fabricated Cells 45

(a) Electrodes short circuit and break. (b) Switching of LC near the short circuit

Figure 5.5: Image of the break and short circuit of the cell under polarization microscope

Figure 5.6: Transmission spectrum of one cell.

before filling them with liquid crystal, by using a spectrometer: the gap between the common

electrode and the bottom finger electrodes acts like a FabryPerot cavity and the transmission

spectrum of one of these cells is shown in Figure 5.6.

By counting how many transmission peaks are there within a certain wavelength range, the

thickness of the cell can be easily calculated using the formula:

∆λ =λ2

2naird(5.1)

in which ∆λ is the free spectral range, in this case it equals the interested wavelength range

divided by how many peaks inside this range, λ is the center wavelength, nair is the refractive

index of air, and d is the thickness of the cell. Table 5.1 lists the expected and measured thickness

of the four cells.

5.3 Measurements 46

Table 5.1: Thickness of the cells

Cell No. 1 2 3 Reference

Expected d (µm) 10 10 20 10

Measured d (µm) 12.5 13.5 23.4 13.6

Figure 5.7: Schematic of the set up.

5.3 Measurements

5.3.1 Measurement Setup

A simple experiment setup was built to perform the measurement, and the schematic of the

experiment setup is shown in Figure 5.7.

- LP1, LP2, LP3: linear polarizer.

- QWP: quarter wave plate.

- PH: Pinhole.

The light source used in this experiment is a HeNe laser, emitting light at 623.8nm. The laser

is aligned by using a small diaphragm which is mounted on a translation stage with moving

direction along the light path, if the light can pass through the diaphragm when it is moved

along the light path, then the laser beam is well aligned. A linear polarizer (LP1) is placed in

front of the laser, making sure the light after it is linearly polarized along a direction that we

can easily determine, the quarter wave plate converts the linear polarized light into circular po-

larization, and thus another linear polarizer (LP2) can select any linear polarization by rotating

the polarizer, but in this setup it won’t change the transmitted intensity. A diaphragm is placed

before the liquid crystal cell to clean the beam, the cell is mounted on a translation stage which

can move perpendicularly to the propagation direction of the laser, an additional linear polarizer

is used after the cell because the liquid crystal will induce some twist when applying a voltage,

as discussed in chapter 4, causing the polarization of the light to change, which is unwanted

in practice. The voltage applied to the cell is generated by a function generator. An 8 bits

CCD camera is placed next to the laser (now shown in the schematic) to image the diffraction

5.4 Results and Analysis 47

pattern at the screen. A sheet of millimeter paper is fixed on the screen to measure the distance

d between the steered light spot and the light spot of the 0th order, the distance between the

cell and the screen is denoted D, so the measured steering angle can be found by:

θ = arctand

D(5.2)

5.4 Results and Analysis

All the fabricated cells are characterized to evaluate their performance, the diffraction patterns

are captured by the CCD camera. First the cells with symmetric finger electrodes are tested to

verify the functionality of the PZT, and then experiments are done with the cells with asymmetric

finger electrodes to check their beam steering abilities. In all of the measurements, the cells are

placed in such a way that the strips of the electrodes are along vertical direction, so a diffraction

pattern should be observed in the horizontal direction because of the periodicity in this direction.

5.4.1 Symmetric Finger Electrodes

Voltage generation. For the cells with symmetric finger electrodes, two different sets of

square wave voltages are applied, with frequency at 1kHz. First, the two finger electrodes are

applied +V and -V, by using two channels of the wave function generator and setting them 180◦

out of phase, and then the two finger electrodes are applied voltages of +V and 0V, in both

cases V is the root mean square (RMS) value and changes from 0 to 10, with a step of 0.5V.

The voltage profile is shown in Figure 5.10.

(a) +V and -V. (b) +V and 0

Figure 5.8: Sets of voltage profiles applied to the finger electrodes of the symmetric finger

patterns

Diffraction patterns. Figure 5.9 shows the diffraction pattern when the two cells are applied

voltages of +V and -V on the finger electrodes, with the polarization of the incident light along

the rubbing direction. It can be seen that when V=0V the light stays at the 0th order because the

light beam experiences a uniform refractive index of ne, when the potential difference increases,

the diffraction pattern appears due to different switching behavior of the LC molecules above the

finger electrodes and in the gap between them. For the reference cell the diffraction angles match

5.4 Results and Analysis 48

reasonably well with the calculation and simulation, which are approximately at multiples of 2◦.

While in the PZT cell besides the diffraction orders at multiple of 2◦, additional orders appear at

integer times of 0.99◦, which matches with our simulation as presented in chapter 4. This means

the grating period is 36µm instead of 18µm, and only when V=6V these orders disappear, in

fact when V is further increased to V=8.5, 9.5 and 10V, these orders also disappear, at other

voltages the orders at multiple of 0.99◦ stay there, while in the simulations, no matter what

value V is, these orders always stay there and the reason behind this remains unclear.

(a) Reference cell.

(b) PZT cell

Figure 5.9: Diffraction pattern when the finger electrodes are in applied voltage V1=+V and

V2=-V.

Note that in both cells the scattering is strong, the reasons for this scattering are three folded.

First, as we discussed in Chapter 4, the molecules above the electrodes will induce some twist

to obtain minimum energy, leading to phase variations at that region and cause scattering for

the incident light. Second, defects in the LC are formed once an external filed is applied,

as illustrated in Figure 5.10a: the LC molecules in between the electrodes which are applied

+5V and -5V will switch horizontally, and the LC molecules above the electrodes will switch

vertically, in between these regions the LC molecules don’t have a preferable direction to switch

thus resulting in defects, as denoted by the red dot. The breaks in the finger electrodes are the

third reason causing scattering, when voltages are applied on the electrodes, the LC molecules

near the breaks will not switch, leading to phase variations and scatting arises. It can be observed

in Figure 5.9 that the scattering in the PZT cell is less than that in the reference cell, the reason

for this is that the PZT layer can smoothen the phase variations of the light in the region above

the electrodes, as described in Chapter 4, thus the scattering is less in the PZT cell.

When one of the two finger electrodes is held at +V and the other one is grounded, the diffraction

grating period should be doubled (Λ = 18µm × 2 = 36µm) compared to the case where the

finger electrodes are held at +V and -V, as can be seen in Figure 4.4. The diffraction angles

can be calculated by equation (4.3) which are approximately multiples of 1.01◦, the diffraction

5.4 Results and Analysis 49

(a) Defect when applied voltage.[51] (b) Breaks in finger electrodes under polarizing mi-

croscope.

Figure 5.10: Defects causing scattering

patterns shown in Figure 5.11 agree with the calculation, in which the diffraction orders appear

at multiples of 1.01◦. We also see that the scattering in the PZT cell is again less than that in

the reference cell and for the same reason as mentioned before.

(a) Reference cell.

(b) PZT cell.

Figure 5.11: Diffraction pattern when the finger electrodes are applied voltage V1=+V and

V2=0V.

When the two voltages applied on the finger electrodes are in phase, i.e. V1=V2=+V, the

diffraction patterns of the two cells are shown in Figure 5.12, we can see that for the PZT cell

only a small part of the light is diffracted into the ±1 order, most of the light remains at the

0th order, however in the reference cell the diffraction pattern resembles that when V1=+V

and V2=-V, and three strong diffraction orders appear at −1.98◦ and ±4.02◦, corresponding

to -1st and ±2nd order, respectively. This matches with the simulation shown in Figure 4.7,

however the light spots at ±4.02◦ seem as strong as the spot at −1.98◦, this may be because

the phase profile in this case is not a perfect blazed grating shape, and the OPL is not exactly

one wavelength.

5.4 Results and Analysis 50

(a) Reference cell.

(b) PZT cell.

Figure 5.12: Diffraction pattern when the finger electrodes are applied voltage V1=+V and

V2=+V.

Figure 5.13 shows the switching behavior of the two cells under polarizing microscope when

V1=V2=+5. We see that for the reference cell a clear periodic structure is visible, due to

the fact that the LC molecules switch differently above the finger electrodes and in between

them, while for the PZT cell the periodic pattern is less pronounced. This is because the LC

molecules in the whole cell are more or less switching uniformly due to the presence of the PZT

layer, leading to an almost constant phase retardation and thus the light experiences almost no

diffraction.

(a) Reference cell. (b) PZT cell.

Figure 5.13: Cells under polarizing microscope.

5.4.2 Asymmetric Finger Electrodes

The cells fabricated for beam steering usage are tested in the setup. As we previously mentioned,

due to imperfect fabrication some of the active regions are not functioning, here we list the

parameters of the cells tested in the experiment, shown in Table 5.2.

In cells No. 1,2 and 3 there is a layer of PZT on top of the finger electrodes, whereas in the

reference cell the PZT layer is absent. The thickness indicated in this table is the thickness

as obtained from measurements with the spectrometer. ⊥ and ‖ refer to the initial rubbing

direction, which is perpendicular or parallel with respect to the length direction of the finger

electrodes. All other designs parameters for these cells are identical.

5.4 Results and Analysis 51

Table 5.2: Cells tested in experiment

Cell No. 1 2 3 Reference

Thickness(µm) 12.5 13.5 23.4 13.6

Rubbing direction ⊥ ‖ ⊥ ⊥Electrode period P(µm) 70,80 30∼80 30∼70 30∼80

At the voltage-off state a diffraction pattern is expected due to the existence of the periodic

arrangement of non-transparent metal electrodes, and the angles at which the diffraction orders

should appear are independent of the thickness or rubbing direction of the cells. During the

experiment the diffraction pattern at the voltage-off state for all cells tested was observed and

Figure 5.14 shows this pattern for the reference cell with electrode period of 70µm, for this cell

the calculated angles at which diffraction orders should appear are ±0.52◦, ±1.03◦, etc. We can

see in Figure 5.14 the measured diffraction angles are at ±0.51◦, ±1.05◦, which agree very well

with the calculations.

Figure 5.14: Diffraction pattern of the electrodes at the voltage-off state.

Diffraction patterns. The cells tested in experiment show beam steering capacity to various

extents. Among these cells, the active region with electrode period P = 70µm in cell No. 3

shows the most satisfactory result, the diffraction pattern for this active region is shown in

Figure 5.15.

In this case the polarization of the incident light is along the rubbing direction, so at voltage-off

state the light will experience a refractive index of ne, when a positive voltage (V1) is applied on

one of the finger electrodes (E1), and the other one (E2) and the common electrode are grounded,

the LC molecules above E1 will tilt in vertical direction so light will see a smaller refractive index

neff2 while molecules above E2 remains unswitched thus light will see refractive index of ne,

in the region between the two electrodes the refractive index will change gradually from neff

to ne, leading to a periodic phase grating. Figure 5.15 shows a smooth shift of the main peak

from 0◦ to a maximum angle of 4.65◦ as the voltage V1 increased. The maximum angle of 4.65◦

corresponds to the +9 order, in theory the maximum steering angle can be calculated by:

OPL =(ne − no)d

λ=

(1.7304− 1.5188)× 23.4µm

0.633µm= 7.8 ≈ 8 (5.3)

2When the molecules are fully switched neff = no

5.4 Results and Analysis 52

Figure 5.15: Diffraction pattern for the cell No.3 with P = 70µm.

so the calculated maximum steering angle is:

θmax = arcsin8λ

Λ=

8× 0.633µm

70µm= 4.15◦ (5.4)

The match is satisfactory, and the error is most likely due to the measured thickness of the

cell not being accurate. Interesting to note that as we further increase voltage V1 above 3.5V,

the main peak of the diffracted light beam shifts back toward 0◦, denoted by the blue arrow in

Figure 5.15, this can be explained by the fact that when the voltage is high, the LC molecules

in between the electrodes gap are switched because the PZT layer extends the electric field into

the whole region of the cell, as described in Figure 4.9a in Chapter 4. The diffraction efficiency

for the maximum diffraction angle at 4.65◦ is 74%.

For the cell with initial alignment direction along the length direction of finger electrodes (Cell

No. 2 in Table 5.2), in this case the polarization direction of the incident light is also along

the length direction of finger electrodes, an analyser with polarization direction also along the

length direction of finger electrodes is placed to filter out the undesired orthogonal polarization.

The voltage applied in this case is identical to the above one, the diffraction pattern for grating

period P equals 40µm and 70µm are shown in Figure 5.16.

We can see that the the cell with grating period of 40µm works better than the cell with grating

period of 70µm. This is because if the gap between finger electrodes is too large, then the PZT

layer cannot extend the electric field into the region that is far from the electrode, thus the

director profile in this region is not as well controlled, leading to a distortion from the linear

refractive index profile, in other words, the PZT can only work in a limited distance.

For the reference cell, light can also be deflected from 0◦, however the beam steering ability is

quite poor, Figure 5.17 shows the diffraction pattern for the grating period of P = 70µm for

5.4 Results and Analysis 53

(a) P=40µm.

(b) P=70µm.

Figure 5.16: Diffraction pattern of cell No. 2.

Figure 5.17: Diffraction pattern of the reference cell with P = 70µm.

the reference cell, the polarization direction of the incident light is perpendicular to the length

direction of finger electrodes.

It can be seen that as we increase the voltage applied on one of the finger electrodes, light

can be diffracted into higher orders, however quite an amount of light stays at the 0th order,

furthermore, comparing to the diffraction pattern for the PZT cell No. 3 shown in Figure 5.15,

light is more uniformly distributed at higher orders, instead of concentrating at one single order

as in the case of PZT cells, this is because without the PZT layer the director orientations in

between the finger electrodes are not well-defined, thus the resulting phase profile when light

passes through is deviated from the linear shape, causing the light to be to diffracted into

multiple orders.

The experiment is also performed with a white light source. It will be diffracted because the

dimension of the finger electrodes is comparable with the wavelengths under consideration,

however, since the white light source is incoherent, the interference from different diffraction

5.4 Results and Analysis 54

contributions would not happen, thus a slowly shifting white spot is expected. Figure 5.18

shows the diffraction pattern using a Xenon lamp.

Figure 5.18: Diffraction pattern using white light, from a Xenon lamp.

We see that different from what we expected, white light is split into colored strips due to

dispersion, i.e. different light components with different wavelengths will experience different

phase retardation for the same cell, leading to different diffraction angles. Furthermore, when

the voltage exceeds 4.5V, the diffracted light shifts back towards 0◦, due to the molecules in the

whole cell being fully switched.

CONCLUSIONS 55

Chapter 6

Conclusions

Beam steering cells based on liquid crystals were studied, both theoretically and experimentally.

First, cells with symmetric finger electrodes were characterized to understand the functionality

of a PZT layer, a layer made of a material with a high dielectric constant. Both the simulation

and experiment showed that the PZT layer can reduce rapid phase variations and thus decrease

scattering, by extending the electric field into the region between finger electrodes where the

electric field is usually no longer strong enough to lead to the switching of the mesogens. This

leads to a nicely defined orientation profile of liquid crystal directors, thus the incident light will

experience a (more or less) linear phase retardation.

Then the cells with asymmetric finger electrodes are investigated for beam steering usage, all

these cells exhibit beam steering ability to various extent. An improvement of the beam steering

ability for the PZT cell compared to the reference cell was observed, due to the well controlled

phase profile of the incident light in the gap between the finger electrodes. The PZT cell with

thickness of 20µm and with grating period of 70µm gave a maximum steering angle of 4.65◦.

Comparing the PZT cells with grating periods of 40µm and 70µm led to the conclusion that

the PZT layer only has a limited working distance and thus the design parameters, such as PZT

layer thickness and LC layer thickness, need to be considered on a per application basis.

In conclusion, beam steering based on liquid crystals has been demonstrated and it is shown

that simply by spin coating a layer of PZT on top of the finger electrodes the beam steering

capability of the cells can be strongly improved, and the cell configuration investigated in this

thesis work only requires two finger electrodes, which is an advantage in both the fabrication

process as well as during operation.

BIBLIOGRAPHY 56

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