JIM Kwok Lung - PolyU Electronic Theses

The Hong Kong Polytechnic University

Department of Applied Physics

BARIUM STRONTIUM TITANATE FERROELECTRIC

TUNABLE PHOTONIC AND PHONONIC CRYSTALS

JIM Kwok Lung

A thesis submitted in partial fulfillment of the requirements

for the degree of Doctor of Philosophy

January 2009

ABSTRACT

JIM Kwok Lung i

ABSTRACT

This thesis presents the results of theoretical simulations and experimental

investigations on developing electro-optically tunable photonic crystals and thermally

tunable phononic crystals based on the ferroelectric materials, Barium Strontium

Titanate (Ba0.7Sr0.3TiO3, BST).

One-dimensional photonic crystal (PC) consists of five periods, with each period

consisting of a 90 nm thick Ba0.7Sr0.3TiO3 layer and a 10 nm thick MgO layer, was

fabricated using pulsed laser deposition. X-ray diffraction study confirmed the epitaxial

growth of the Ba0.7Sr0.3TiO3 layers with good crystalline quality. A photonic bandgap

(transmission dip) with a centre wavelength at ~464 nm has been observed in the

transmission measurement which is consistent with simulation using the plane wave

expansion (PWE) method and the transfer matrix method. A 2-nm shift towards the

longer wavelength is observed when a dc voltage of 240 V (corresponding to an electric

field of about 12 MV/m) has been applied across the coplanar electrodes on the film

surface. The experimental result suggests that the electric field induced change in the

refractive index of Ba0.7Sr0.3TiO3 is about 0.5 %.

Photonic bandstructures and photonic bandgap maps of two-dimensional (2D)

Ba0.7Sr0.3TiO3-based photonic crystals with different cavity geometries (square or

ABSTRACT

JIM Kwok Lung ii

circular air rods) in square lattice were calculated using the PWE method. Simulation

results suggested comparable bandstructures and bandgap maps for square or circular air

rod photonic crystals, if (1) the dimension of the air rod is small compared to the

electromagnetic wavelengths inside the PC being considered, or (2) the frequencies of

the electromagnetic waves are less than 0.35(2c/a). The photonic bandgap maps of two

types of 2D Ba0.7Sr0.3TiO3-based PC, namely, the air-hole-in-BST type and the BST-rod-

in-air type both in square lattice and in triangular lattice were calculated. It is found that

PCs in triangular lattice contain richer bandgap feature in general. Bandgap features

along different symmetry directions have also been compared.

The refractive indices of the Ba0.7Sr0.3TiO3 thin film on a MgO (001) substrate

was measured using the prism coupling technique. The appropriate geometry of a single-

mode rib waveguide based on Ba0.7Sr0.3TiO3 thin film was determined by applying the

effective index method. A photonic crystal cavity embedded Ba0.7Sr0.3TiO3 rib

waveguide which functions as a tunable filter (on-off switch) for λ = 1550 nm was

designed with the help of the finite-difference time-domain (FDTD) simulation. The

required PC cavity is composed of two 5-row-4 PC mirrors, which is formed by air holes

arranged in triangular lattice in the Ba0.7Sr0.3TiO3 matrix, with cavity length of 800 nm.

The radius of the air holes is 250 nm and the periodicity is 625 nm. A 6-nm shift in the

resonant peak for a 0.5% change in the refractive index of the Ba0.7Sr0.3TiO3 was

illustrated in the simulation. Photonic crystal cavities were fabricated on a Ba0.7Sr0.3TiO3

rib waveguide by focused ion beam etching with satisfactory results.

ABSTRACT

JIM Kwok Lung iii

The shear and transverse wave velocities of Ba0.7Sr0.3TiO3 ceramic as a function

of temperature were determined using the ultrasonic through-transmission technique. A

drastic variation in the wave velocities was observed across the Curie temperature of

Ba0.7Sr0.3TiO3. Phononic crystal composed of Ba0.7Sr0.3TiO3 square rods in a matrix of

epoxy were fabricated using the dice-and-fill method. The width of the Ba0.7Sr0.3TiO3

rods is 200 µm with periodicity of 265 µm. The temperature dependence of the phononic

bandgaps of the phononic crystal was characterized by the reflection spectra which were

obtained using the ultrasonic pulse-echo technique. Thermal tuning of the phononic

bandgap was observed and the results were in good agreement with the phononic

bandstructure calculation by the plane wave expansion method.

LIST OF PUBLICATIONS

JIM Kwok Lung iv


1. K. L. Jim, D. Y. Wang, C. W. Leung, C. L. Choy, and H. L. W. Chan, “One-

dimensional tunable ferroelectric photonic crystals based on

Ba0.7Sr0.3TiO3/MgO multilayer thin films,” Journal of Applied Physics 103,

083107-6 (2008).

2. K. L. Jim, C. W. Leung, S. T. Lau, S. H. Choy, and H. L. W. Chan, “Thermal

tuning of phononic bandstructure in ferroelectric ceramic/epoxy phononic

crystal,” Applied Physics Letters 94, 193501-3 (2009).

3. K. L. Jim, F. K. Lee, J. Z. Xin, C. W. Leung, H. L. W. Chan, and Y. Chen,

“Fabrication of two-dimensional nanoscaled patterns on ceramic thin films by

soft ultraviolet nanoimprint lithography,” submitted to Microelectronic

Engineering, (2009).

4. K. L. Jim, C. W. Leung, and H. L. W. Chan, “Photonic crystal cavity

embedded barium strontium titanate thin-film rib waveguide prepared by

focused ion beam etching,” submitted to Thin Solid Films (2009).


JIM Kwok Lung v

5. K. L. Jim, D. Y. Wang, C. W. Leung, C. L. Choy, and H. L. W. Chan,

“Theoretical study of ferroelectric barium-strontium-titanate-based one-

dimensional tunable photonic crystals,” Proceedings of the Society of Photo-

Optical Instrumentation Engineers (SPIE) 6556, R5560 (2007).

6. D. Y. Wang, K. L. Jim, C. W. Leung, H. L. W. Chan, and C. L. Choy,

“Tunable ferroelectric photonic crystals based on Ba0.7Sr0.3TiO3/MgO

multilayer thin films,” Proceedings of the Society of Photo-Optical

Instrumentation Engineers (SPIE) 6556, Q5560 (2007).

7. K. L. Jim, D. Y. Wang, C. W. Leung, C. L. Choy, and H. L. W. Chan, “(Ba,

Sr)TiO3-based planar photonic bandgap crystal,” Proceedings of the Society

of Photo-Optical Instrumentation Engineers (SPIE) 6640, P6400 (2007).

8. K. L. Jim, C. W. Leung, C. L. Choy, and H. L. W. Chan, “Thermal shifting of

phononic bandgaps in barium strontium titanate-based structures,”

Acoustics’08, 1945 (2008).

ACKNOWLEDGEMENTS

JIM Kwok Lung vi

ACKNOWLEDGEMENTS

During my three years of Ph.D. study at the Hong Kong Polytechnic University, I

enjoyed my research and personal life everyday because of all the people who gave

endless support to me. First of all, I sincerely would like to express my gratitude to my

supervisors, Prof. H. L. W. Chan and Dr. C. W. Leung, for their invaluable guidance,

support, encouragement and patience throughout the whole period of my research study.

I would also like to thank Prof. K. H. Wong, Prof. J. Zhou (Tsinghua University)

and Prof. W. Y. Tam (HKUST), being the Board of Examiners of my thesis

examination, for their valuable comments which add advantages to this thesis.

I wish to show my appreciation to Prof. Y. Chen for his cordial welcome during

my exchange to the École Normale Supérieure in Paris. My thanks also go to friends I

made there including Ms. X. F. Ni, Mr. X. T. Zhou, Mr. L. Lei, Ms. Li Wang, Ms. Lu

Wang and Mr. Q. Zeng.

I would like to thank Prof. K. S. Chiang from the Optoelectronic Research Centre

in the Department of Electronic Engineering at the City University of Hong Kong who

has contributed resources to the success of this research work. My thanks are also

ACKNOWLEDGEMENTS

JIM Kwok Lung vii

extended to the members of his research team including Mr. K. P. Lor for his helpful

assistance in waveguide characterization.

I wish to thank Mr. Y. F. Chan from the Electron Microscope Unit at the

University of Hong Kong for his support in FIB etching.

My genuine thanks go to Dr. D. Y. Wang for his helpful advices and kind

assistance in pulsed laser deposition and prism coupling measurement, Dr. F. K. Lee for

his thoughtful assistance in waveguide fabrication and AFM investigations, Dr. K. C.

Cheng for his kind assistance in waveguide characterization, Dr. S. H. Choy for his

gentle help in fabrication of phononic crystals, and Dr. S. T. Lau for her helpful

assistance in reflection spectrum measurement of phononic crystals.

I wish to acknowledge the academic members in our department: Prof. C. L.

Choy and Dr. J. Y. Dai for their insightful suggestions during the group meeting

discussions. Thanks are due to Dr. P. F. Lee for his helpful advices in laser operation and

Mr. M. N. Yeung of the Materials Research Centre for his assistance in XRD

investigations.

I would like to thank my former and present colleagues in our department of their

assistance and discussions in the research work. They include (in no particular order):

Mr. Y. M. Tang, Ms. M. K. Li, Mr. C. K. Chow, Mr. S. M. Mok, and Dr. C. Y. Lam.

ACKNOWLEDGEMENTS

JIM Kwok Lung viii

I gratefully acknowledge the financial support from the Department of Applied

Physics and the Centre for Smart Materials of the Hong Kong Polytechnic University.

Most of all, I would like to express my deepest appreciations to my family

members for their continuous love, support, understanding and patience that inspired me

to continue and to mature both as a person and as a professional.

TABLE OF CONTENTS

JIM Kwok Lung ix

TABLE OF CONTENTS

ABSTRACT i

LIST OF PUBLICATIONS iv

ACKNOWLEDGEMENTS vi

TABLE OF CONTENTS ix

LIST OF FIGURE CAPTIONS xiv

LIST OF TABLE CAPTIONS xxii

LIST OF SYMBOLS xxiii

LIST OF ACRONYMS xxv

CHAPTER 1 INTRODUCTION 1

1.1 Background 1

1.2 Electro-optic effect in crystals 4

1.2.1 Refractive index 4

1.2.2 Optical birefringence 5

1.2.3 Electro-optic effect and electro-optic coefficients 7

1.3 Elastic properties during phase transition in ferroelectric materials 12

1.4 Barium strontium titanate in bulk and thin film forms 13

1.4.1 Bulk barium strontium titanate 13

1.4.2 Barium strontium titanate thin films 15

1.5 Photonic crystals 17

1.5.1 Theoretical tools for analyzing photonic crystals 19

1.5.1.1 Plane wave expansion method 20

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1.5.1.2 Scattering matrix method 22

1.5.1.3 Finite-difference time-domain method 23

1.5.1.4 Transfer matrix method 25

1.5.1.5 Comparison of the four methods 26

1.5.2 Tunable photonic crystals 28

1.5.2.1 Ferroelectric photonic crystals 29

1.6 Phononic crystals 30

1.6.1 Tunable phononic crystals 31

1.7 Scope of the present study 32

1.8 Statement of original contributions 34

CHAPTER 2 ONE-DIMENSIONAL TUNABLE PHOTONIC CRYSTAL

BASED ON BARIUM STRONTIUM TITANATE &

MAGNESIUM OXIDE MULTILAYERED THIN FILMS

35

2.1 Introduction 35

2.2 Simulation methods for one-dimensional photonic crystals 37

2.2.1 Computation of photonic bandstructure of one-dimensional

photonic crystal by the plane wave expansion method

37

2.2.2 Computation of transmission spectrum of one-dimensional

photonic crystal by the transfer matrix method

38

2.3 Theoretical study of Ba0.7Sr0.3TiO3/MgO one-dimensional photonic

crystals

40

2.4 Fabrication of Ba0.7Sr0.3TiO3/MgO one-dimensional photonic crystals 50

2.5 Characterization of one-dimensional Ba0.7Sr0.3TiO3/MgO photonic crystals 52

2.5.1 Structural characterization of the one-dimensional photonic

crystals

52

2.5.2 Optical characterization of the one-dimensional photonic crystals 53

2.6 Summary 59

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CHAPTER 3 PHOTONIC BANDSTRUCTURE STDUY ON TWO-

DIMENSIONAL BARIUM STRONTIUM TITANATE-

BASED PHOTONIC CRYSTALS

60

3.1 Introduction 60

3.2 Plane wave expansion method for two-dimensional photonic bandstructure

calculation

61

3.3 Photonic bandgap maps of two-dimensional Ba0.7Sr0.3TiO3-based photonic

crystals with different configurations

65

3.4 Photonic bandstructures and photonic bandgap maps of two-dimensional

Ba0.7Sr0.3TiO3-based photonic crystals with different shapes of air rods

70

3.5 Summary 76

CHAPTER 4 PHOTONIC CRYSTAL CAVITY EMBEDDED BARIUM

STRONTIUM TITANATE RIB WAVEGUIDES

77

4.1 Introduction 77

4.2 Characterization of Ba0.7Sr0.3TiO3 thin films 78

4.2.1 Structural characterization of Ba0.7Sr0.3TiO3 thin films 78

4.2.2 Optical characterization of Ba0.7Sr0.3TiO3 thin films 79

4.2.2.1 Basic principles of prism coupling technique 81

4.2.2.2 Guided modes in Ba0.7Sr0.3TiO3 thin films 85

4.3 Ba0.7Sr0.3TiO3 thin film optical rib waveguides 87

4.3.1 Light propagation in slab waveguides 87

4.3.1.1 Basic concepts of slab waveguides 87

4.3.1.2 Guided modes of slab waveguides 89

4.3.1.3 Cutoffs of asymmetric slab waveguides 93

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4.3.2 Design of Ba0.7Sr0.3TiO3 rib waveguides using the effective index

method

96

4.3.3 Fabrication of Ba0.7Sr0.3TiO3 rib waveguides 101

4.3.4 Optical characterization of Ba0.7Sr0.3TiO3 rib waveguides 104

4.4 Photonic crystal cavity embedded rib waveguides 106

4.4.1 Fabry-Perot cavity 106

4.4.2 Photonic crystal cavity 109

4.4.3 Design of photonic crystal cavity embedded Ba0.7Sr0.3TiO3 rib

waveguides

111

4.4.3.1 Appropriate geometry of the photonic crystal pattern 112

4.4.3.2 Effect of number of row 115

4.4.3.3 Effect of row configuration 118

4.4.3.4 Determination of cavity length 121

4.4.3.5 Electro-optic tuning of the resonant peak 122

4.4.4 Photonic crystal cavity preparation by focused ion beam etching 124

4.5 Summary 128

CHAPTER 5 THERMAL TUNING OF PHONONIC BANDGAPS

IN BARIUM STRONTIUM TITANATE/EPOXY

PHONONIC CRYSTALS

129

5.1 Introduction 129

5.2 Computation of phononic bandstructure by plane wave expansion method 130

5.2.1 Wave equation for an inhomogeneous elastic medium 131

5.2.2 Plane wave expansion for periodic elastic structure 132

5.2.3 Binary composite 134

5.3 Determination of temperature dependence of longitudinal and shear wave

velocities by ultrasonic through-transmission technique

138

5.3.1 Ultrasonic through-transmission technique 138

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5.3.2 Determination of longitudinal wave velocity 139

5.3.3 Determination of shear wave velocity 140

5.3.4 Temperature dependence of longitudinal and shear wave

velocities in Ba0.7Sr0.3TiO3 bulk sample

142

5.3.5 Temperature dependence of longitudinal and shear wave

velocities in epoxy sample

145

5.4 Barium strontium titanate/epoxy phononic crystals 146

5.4.1 Fabrication of the Ba0.7Sr0.3TiO3/epoxy phononic crystals 146

5.4.2 Reflection spectrum measurement by ultrasonic pulse-echo

technique

148

5.4.3 Phononic bandstructure computation of the Ba0.7Sr0.3TiO3/epoxy

phononic crystals

152

5.5 Summary 155

CHAPTER 6 CONCLUSIONS 156

6.1 Conclusions 156

REFERENCES 159

LIST OF FIGURE CAPTIONS

JIM Kwok Lung xiv


Figure 1.1 (a) Conceptual diagram of a photonic crystal circuit [Noda,

1999]. (b) Schematic diagram of add-drop wavelength-division

multiplexed (WDM) circuit composed of photonic crystals

[Kosaka, 1999b]. 2

Figure 1.2 Refractive index ellipsoid (optical indicatrix) for a positive

uniaxial crystal, 321 nnn . The optical axis is parallel to the

z-axis [Syms, 1992]. 7

Figure 1.3 Ultrasonic attenuations and velocities of a Ba0.7Sr0.3TiO3

ceramic for (a) longitudinal waves and (b) transverse waves

[Moreno-Gobbi, 2006]. 13

Figure 1.4 Lattice structure of Ba1-xSrxTiO3. Ba and Sr atoms occupy the

corner positions with Ti atom at the body center, surrounded by

the oxygen octahedron [Wang,2006a]. 14

Figure 1.5 (a) Room temperature lattice parameters [McQuarrie, 1955]

and (b) Relative permittivity versus temperature for Ba1-

xSrxTiO3 with different SrTiO3 contents [Smolenskii, 1954]. 15

Figure 1.6 Electro-optic coefficients as a function of x for Ba1-xSrxTiO3

thin films deposited on MgO (001) substrates at 650 ˚C [Wang,

2006a]. 17

Figure 1.7 Schematic diagrams of 1D, 2D, and 3D PCs. 18


JIM Kwok Lung xv

Figure 1.8 Transverse electric (TE) polarization photonic bandstructure of

a 2D PC in triangular lattice computed by the plane wave

expansion method. 21

Figure 1.9 Calculation model of the scattering matrix method for a 2D PC. 23

Figure 1.10 E

and H

field components in a Yee cell of dimensions x ,

y , and z [Yee, 1966]. 24

Figure 2.1 Schematic diagram of a one-dimensional photonic crystal

composed of alternative layers of dielectric material 1 and

material 2. 40

Figure 2.2 (a) Photonic bandstructure of a Ba0.7Sr0.3TiO3/MgO 1D PC

with f = 0.1. The shaded areas represent the photonic bandgaps.

(b) Transmission spectrum of a Ba0.7Sr0.3TiO3/MgO 1D PC (f =

0.1) with 1000 periods. 42

Figure 2.3 Photonic bandgap map of the first bandgap of the

Ba0.7Sr0.3TiO3/MgO 1D PC with different changes in the

refractive index of Ba0.7Sr0.3TiO3. 44

Figure 2.4 Changes of photonic bandgap edges of a Ba0.7Sr0.3TiO3/MgO

1D PC (f = 0.1) with different changes in the refractive index of

Ba0.7Sr0.3TiO3. 44

Figure 2.5 Photonic bandgap shift of a Ba0.7Sr0.3TiO3/MgO 1D PC (period

= 356 nm) with 1% change in refractive index of Ba0.7Sr0.3TiO3

(from n = 2.200 to n = 2.222). A small circle is placed for

illustration purpose. 47


JIM Kwok Lung xvi

Figure 2.6 Transmission spectrum of a Ba0.7Sr0.3TiO3/MgO 1D PC with

different numbers of periods (N). 49

Figure 2.7 Transmission spectrum of a 5-period Ba0.7Sr0.3TiO3/MgO 1D

PC for different angles of incidence. 49

Figure 2.8 Schematic diagram showing the experiment for observing the

photonic bandgap shift of a Ba0.7Sr0.3TiO3/MgO multilayered

thin films. 51

Figure 2.9 XRD θ/2θ scan of a Ba0.7Sr0.3TiO3/MgO multilayered thin

films. The inset shows the φ scan of the Ba0.7Sr0.3TiO3 (202)

and substrate (202) reflections. 53

Figure 2.10 (a) Observed transmission spectrum of the Ba0.7Sr0.3TiO3/MgO

multilayered thin films. The inset shows the transmissions

spectrum of a Ba0.7Sr0.3TiO3 thin film grown on MgO (001)

substrate; (b) Calculated transmission spectra of the

Ba0.7Sr0.3TiO3/MgO multilayered thin films and the

corresponding effective single film by TMM. The inset shows

the complex refractive index profile of the Ba0.7Sr0.3TiO3 used

for the calculation. 57

Figure 2.11 Measured transmission spectrum of the Ba0.7Sr0.3TiO3/MgO

multilayered thin films with (dotted curve) and without (solid

curve) the application of a dc voltage of 240 V. 58

Figure 3.1 High symmetry points in the first Brillouin zones of the (a)

square lattice and the (b) triangular lattice. 66


JIM Kwok Lung xvii

Figure 3.2 Photonic bandgap maps of 2D air-hole-in-BST type PC

(Ba0.7Sr0.3TiO3 matrix with circular air rods) in square lattice

along (a) Γ-X-M-Γ, (b) Γ-M, (c) Γ-X; in triangular lattice along

(d) Γ-M-K-Γ, (e) Γ-K, (f) Γ-M. The insets show the

configurations of the PCs and their corresponding Brillouin

zones. 68

Figure 3.3 Photonic bandgap maps of 2D BST-rod-in-air type PC (circular

Ba0.7Sr0.3TiO3 rods in air) in square lattice along (a) Γ-X-M-Γ,

(b) Γ-M, (c) Γ-X; in triangular lattice along (d) Γ-M-K-Γ, (e) Γ-

K, (f) Γ-M. The insets show the configurations of the PCs and

their corresponding Brillouin zones. 69

Figure 3.4 Photonic bandstructures of square lattice type 2D

Ba0.7Sr0.3TiO3-based PC with circular air rods, (a) 2r = 0.2a, (b)

2r = 0.5a, (c) 2r = 0.8a; with square air rods, (d) w = 0.2a, (e)

w = 0.5a, (f) w = 0.8a. 74

Figure 3.5 Photonic bandgap maps along the high symmetry direction of

square lattice type 2D Ba0.7Sr0.3TiO3-based PC with circular air

rods, (a) Γ-X-M-Γ, (b) Γ-M, (c) Γ-X; with square air rods, (d)

Γ-X-M-Γ, (e) Γ-M, (f) Γ-X. The dark green lines indicate the

position of w = 0.89a for visualization. 75

Figure 4.1 XRD θ/2θ scan of a Ba0.7Sr0.3TiO3 thin film grown on a MgO

(001) substrate. The inset shows the φ scan of the

Ba0.7Sr0.3TiO3 (202) and substrate (202) reflections. 79

Figure 4.2 Schematic diagram of the Metricon model 2010 prism coupler

system. 81


JIM Kwok Lung xviii

Figure 4.3 Intensity of reflected light against internal incident angle for

transverse electric (TE) modes. The knee corresponds to the

substrate mode. The inset shows the vibration directions of the

electric field for the TE and the TM modes. 83

Figure 4.4 Guided mode spectra excited at (a) λ = 632.8 nm and (b) λ =

1550 nm in the TE polarization for a ~550 nm thick

Ba0.7Sr0.3TiO3 film deposited on MgO (001) substrate. 86

Figure 4.5 A step-index slab waveguide of thickness d with a guided

mode. 88

Figure 4.6 Graphical solution of TE mode eigenvalue equation (Eq.

(4.24)). The intersections of the solid and dashed lines

correspond to the solutions. 94

Figure 4.7 Field profiles associated with the first four TE modes of an

asymmetric slab waveguide [Hunsperger, 2002]. 96

Figure 4.8 The effective index method on a rib waveguide. (a) The

original rib waveguide. (b) Transform the original rib

waveguide into three slab waveguides with effective index Neff1

and Neff2. (c) Transform the system into a symmetric slab

waveguide to determine the effective index Neff of the whole

structure. 98

Figure 4.9 Effective index Nm as function of thickness for a

Ba0.7Sr0.3TiO3/MgO slab waveguide at a wavelength of λ =

1550 nm. The refractive index of the film is 2.20. 99


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Figure 4.10 Effective index Nm versus rib width W for various rib height h

of a Ba0.7Sr0.3TiO3/MgO rib waveguide at a wavelength of λ =

1550 nm. Thickness of the film d is 550 nm. 100

Figure 4.11 Fabrication flow chart of thin film rib waveguides. 102

Figure 4.12 Atomic force micrograph of the Ba0.7Sr0.3TiO3 rib waveguide. 103

Figure 4.13 Schematic diagram of the transmission spectrum measurement

by end-fire coupling technique. 105

Figure 4.14 CCD image showing the output light spot from the rib

waveguide. The arrow indicates the position of the rib

waveguide. 105

Figure 4.15 Transmission spectrum of the Ba0.7Sr0.3TiO3 rib waveguide. 106

Figure 4.16 (a) Schematic diagram of a Fabry-Perot cavity. (b) Allowed

field distributions of different modes inside the cavity. (c)

Transmission spectrum of a Fabry-Perot resonator [Kasap,

2001]. 108

Figure 4.17 (a) Schematic diagram of a PC cavity formed by a pair of 3-

row PC mirrors. (b) Typical transmission spectrum of a PC

cavity. 110

Figure 4.18 PC pattern of triangular lattice type in a rib waveguide with the

Γ-M and Γ-K directions emphasized. 113


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Figure 4.19 (a) Simulation model and (b) transmission spectrum of the

PCCRWG in the FDTD calculation. 114

Figure 4.20 Transmission spectra for the PCCRWG with (a) 3-row, (b) 5-

row, (c) 7-row, (d) 9-row PC mirrors. 117

Figure 4.21 Two possible configurations of the 3-row PC mirrors: (a) 3-

row-3 PC mirrors, (b) 3-row-4 PC mirrors. 118

Figure 4.22 Transmission spectra of PCCRWG with different n-row-m PC

mirrors: (a) 3-row-3, (b) 3-row-4, (c) 4-row-3, (d) 4-row-4, (e)

5-row-3, (f) 5-row-4. 120

Figure 4.23 Schematic design of the PC cavity embedded Ba0.7Sr0.3TiO3 rib

waveguide with 5-row-4 PC mirrors. 123

Figure 4.24 Shifting of transmission resonant peaks for the PCCRWG

shown in Figure 4.23 when the refractive index of

Ba0.7Sr0.3TiO3 (nBST) is changed by +0.5% (red line) and -0.5%

(blue line). The green line corresponds to the effective

refractive index (nBST = 2.005). 124

Figure 4.25 Scanning electron micrograph of the Ba0.7Sr0.3TiO3 rib

waveguide with PC cavity prepared by FIB etching. 125

Figure 4.26 (a) Measured transmission spectra of the Ba0.7Sr0.3TiO3 rib

waveguide with and without PC cavity and (b) simulated

transmission spectrum of the Ba0.7Sr0.3TiO3 rib waveguide with

PC cavity. 127


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Figure 5.1 Schematic diagram of the ultrasonic through-transmission

technique. 139

Figure 5.2 Geometry of the propagation path of the ultrasonic wave. 141

Figure 5.3 Velocity of sound in silicone oil voil as a function of

temperature. 143

Figure 5.4 (a) Longitudinal and (b) Shear wave velocities as a function of

temperature for a Ba0.7Sr0.3TiO3 ceramic disc. Dotted lines are

added as visual aid. 144

Figure 5.5 Longitudinal and shear wave velocities as a function of

temperature for the epoxy (Epotek 301). 145

Figure 5.6 Fabrication flow chart of Ba0.7Sr0.3TiO3/epoxy phononic

crystals using a dice-and-fill technique. 147

Figure 5.7 Reflection spectra of the Ba0.7Sr0.3TiO3/epoxy phononic crystal

at different temperatures range from (a) 32 ˚C to 35 ˚C, (b) 35

˚C to 45 ˚C, and (c) 45 ˚C to 50 ˚C. The inset in (b) shows the

reflection spectra of the Ba0.7Sr0.3TiO3 ceramics at temperatures

range from 32 ˚C to 50 ˚C. 151

Figure 5.8 Simulation model of the Ba0.7Sr0.3TiO3/epoxy phononic crystal. 153

Figure 5.9 Computed phononic bandstructures for mixed mode along the

Γ-X direction of the phononic crystal with configuration shown

in Figure 5.8 at (a) 35 ˚C, and (b) 45 ˚C. The shaded areas

indicate the position of the phononic bandgaps. 154

LIST OF TABLE CAPTIONS

JIM Kwok Lung xxii

LIST OF TABLE CAPTIONS

Table 2.1 Photonic bandgap shift in wavelength at λ = 1550 nm for 0.3%

and 1.0% changes in the refractive index of Ba0.7Sr0.3TiO3. 46

Table 2.2 PLD conditions for Ba0.7Sr0.3TiO3 and MgO multilayered thin

films. 51

Table 4.1 Resonant peak values and locations of a 5-row-4 PCCRWG with

different cavity lengths. 121

Table 5.1 Material parameters of Ba0.7Sr0.3TiO3 & epoxy (Epotek 301) used

for the phononic crystal bandstructure computation. 146

LIST OF SYMBOLS

JIM Kwok Lung xxiii

LIST OF SYMBOLS

Symbol Description SI unit

a Period m

B Magnetic flux density T

c Speed of light ms-1

cl Longitudinal wave speed ms-1

ct Transverse wave speed ms-1

E Electric field strength Vm-1

f Filling fraction -

g

Reciprocal lattice vector m-1

H Magnetic field strength Am-1

k

Wave vector m-1

n Refractive index -

nBST Refractive index of Ba0.7Sr0.3TiO3 -

ne Extraordinary refractive index -

no Ordinary refractive index -

N Number of period -

Nm Effective index -

r Radius m

rc Effective linear electro-optic coefficient mV-1

rijk Linear electro-optic coefficient mV-1

Rc Effective quadratic electro-optic coefficient m2V-2

Tc Curie temperature K

u Displacement m

w Width m

ε Relative permittivity -

ηij Impermittivity -

LIST OF SYMBOLS

JIM Kwok Lung xxiv

λ Wavelength m

μ Relative permeability -

ρ Density kgm-3

ω Angular frequency s-1

Ω Dimensionless frequency -

LIST OF ACRONYMS

JIM Kwok Lung xxv

LIST OF ACRONYMS

Acronyms Description

1D One-dimensional

2D Two-dimensional

3D Three-dimensional

AFM Atomic force microscopy

BST Barium strontium titanate, Ba1-xSrxTiO3

BTO Barium titanate, BaTiO3

CCD Charge-coupled device

EM Electromagnetic

E-O Electro-optic

FDTD Finite-difference time-domain

FIB Focused ion beam

FP Fabry-Perot

FWHM Full width at half maximum

ICP Inductive coupled plasma

IR Infra-red

MST Multiple scattering theory

PBG Photonic bandgap

PBS Photonic bandstructure

PC Photonic crystal

PCCRWG Photonic crystal cavity embedded rib waveguide

PLZT Lanthanum modified lead zirconate titanate, Pb1-xLax(ZryTi1-y)1-0.25xO3

PWE Plane wave expansion

RF Radio frequency

RIE Reactive ion etching

SEM Scanning electron microscopy

LIST OF ACRONYMS

JIM Kwok Lung xxvi

STO Strontium titanate, SrTiO3

TE Transverse electric

TM Transverse magnetic

TMM Transfer matrix method

XRD X-ray diffraction

WDM Wavelength-division multiplexed

CHAPTER 1 INTRODUCTION

JIM Kwok Lung 1

CHAPTER 1

INTRODUCTION

1.1 Background

Over the past decade, there has been great research interest in the physical

properties of artificial structures comprised of two or more materials which differ in

certain properties. In the 1980’s, extensive efforts has been put on microstructures of

reduced dimensionality, such as quantum wells, quantum wires, and quantum dots. More

recently there has been increasing interest in macrostructures known as photonic crystals

(PCs) since Yablonovitch [Yablonovitch, 1987] and John [John, 1987] initiated the idea

of manipulating the photonic density of states by suitable geometric design of ‘photonic’

materials in a similar way of the well established quantum heterostructures studies on the

electronic bandstructure engineering in 1987. One of the distinctive features of PCs is

the creation of photonic bandgaps in which optical modes are forbidden in those regions.

Light with frequencies within the photonic bandgaps cannot propagate inside the PC

which can lead to interesting physical phenomena. For instance, atoms embedded inside

a PC can be locked in an excited state if the energy of this state falls within the photonic

bandgaps [Kurizki, 1988]. Anomalous Lamb shift is also suggested in the PC [John,

1990]. Moreover, observation of Anderson localization of light could be facilitated by

weak disordering of the perfect periodicity of a PC [John, 1987].


JIM Kwok Lung 2

In addition to the research interest on the more fundamental physical phenomena,

a new concept of photonic integrated circuits (Figure 1.1) based on the PCs has been

developed rapidly in the recent decade [Joannopoulos, 1997; Noda, 1999; Kosaka,

1999b; McGurn, 2000; McGurn, 2002]. The goal is to build ultrasmall and high-density

photonic integrated circuits that combine light emitters, waveguides and functional

devices all based on the PC elements.

(a) (b)

Figure 1.1 (a) Conceptual diagram of a photonic crystal circuit [Noda, 1999]. (b)

Schematic diagram of add-drop wavelength-division multiplexed (WDM)

circuit composed of photonic crystals [Kosaka, 1999b].

The light sources of photonic integrated circuits can be lasers. There are two

types of photonic crystal-based lasers. One of them is utilizing defect levels inside the

photonic bandgaps [Yablonovitch, 1991; Baba, 1996; Hirayama, 1996; Yoshie, 2001].

Painter et al. fabricated a 2D PC defect mode laser in InGaAsP which shows pulsed


JIM Kwok Lung 3

lasing oscillation by photopumping at room temperature [Painter, 1999]. Hwang et al.

demonstrated a photonic band gap laser, composed of GaInAsP/InP film on alumina,

operating at 1.54 μm at room temperature [Hwang, 2000]. The other type of photonic

crystal-based lasers is utilizing the photonic band-edge effect [Imada, 1999; Evans,

1997]. Notomi et al. illustrated directional lasing oscillation in a 2D organic PC lasers

[Notomi, 2001a]. Imada et al. showed the coherent lasing action in a surface-emitting

laser with PC structure by current injection.

A characteristic advantage of PC waveguides is the feasibility of ultrasmall bend

radii which leads to the possibility of very sharp bend waveguide and hence the

realization of a very compact photonic circuit. After theoretical prediction of sharp bend

PC waveguide by Mekis et al. [Mekis, 1996] in 1996, it was first achieved in microwave

regime [Lin, 1998], followed by the demonstration in optical regime [Baba, 1999]. PC

waveguides fabricated on silicon-on-insulator [Notomi, 2001b], AlGaAs/GaAs

[Yamada, 2001], and polymer [Liguda, 2001] are also investigated.

Many other PC elements such as channel drop filters [Fan, 1999; Noda, 2000;

Chutinan, 2001], superprisms [Kosaka, 1998; Kosaka, 1999a; Halevi, 1999; Notomi,

2000; Ochiai, 2001], and polarization splitters [Ohtera, 1999] are also studied

extensively.

An essential component of a photonic integrated circuit is the optical switch.

Without it, the photonic integrated circuit is not controllable. To realize PC-based optical


JIM Kwok Lung 4

switches, it is necessary for the PC to be tunable. Therefore, many schemes have been

proposed to realize tunable PCs [Villeneuve, 1996; Tran, 1996; Figotin, 1998; Leonard,

2000; de Lustrac, 1999; Halevi, 2000; Kee, 2001; Kang, 2001, Kim, 2001; Jia, 2003; Xu,

2003; Liu, 2005]. Among different tuning mechanisms, the scheme utilizing the electro-

optic (E-O) effect [Schmidt, 2005; Roussey, 2006] has an outstanding advantage because

the modulation frequency of the E-O effect can reach up to GHz [Taylor, 1999; Turner,

1966].

1.2 Electro-optic effect in crystals

1.2.1 Refractive index

The refractive index of a linear homogeneous medium is defined as:

v

cn , (1.1)

where c is the speed of light in vacuum and v is the speed of light inside the medium

[Griffiths, 1989]. According to Maxwell’s equations, v is given by:

c

v , (1.2)

where is the relative permittivity and is the relative magnetic permeability.

Combining Eq. (1.1) and Eq. (1.2) and setting to be 1 for nonmagnetic medium, the

refractive index can be expressed as:

n . (1.3)


JIM Kwok Lung 5

1.2.2 Optical birefringence

Crystals, in general, are anisotropic. That is, many of their properties depend on

the crystal direction. The relative permittivity which is related to the electronic

polarization depends on the crystal direction inasmuch as it is easier to displace electrons

along certain crystal directions. This means that the refractive index n of a crystal

depends on the direction of the electric field in the propagating light beam, i.e. the

directions of the wave vector and the polarization vectors. Maxwell’s equations allow

two waves with different polarizations to be propagated through the medium with

different velocities for a given wave vector. The refractive indices of the two waves as

functions of the wave vector k

can be obtained from the refractive index ellipsoid,

called the optical indicatrix [Syms, 1992] (Figure 1.2). If x, y, and z are the principal

axes of the relative permittivity tensor, the index ellipsoid is defined by the equation:

12

3

2

22

2

21

2

n

z

n

y

n

x, (1.4)

where xn 1 , yn 2 , zn 3 , and x , y , z are the principal relative

permittivities.

To determine the refractive indices of the medium for a light wave propagating

with wave vector k

, it is convenient to establish the central section which is formed by

the intersection of the index ellipsoid and a plane through the origin and normal to k

.

Generally, the central section is an ellipse. The major (OB) and minor (OA) axes of the

ellipse correspond to the two refractive indices of the medium for the two polarizations


JIM Kwok Lung 6

of the light wave. As a special case, the two possible waves with wave vector xkk ˆ

have refractive indices n2 and n3; and the displacement vectors D

for the two waves are

parallel to y and z , respectively. Similar arguments apply on ykk ˆ

and zkk ˆ

cases.

For this reason, n1, n2 and n3 are called the principal refractive indices.

The optical properties of a crystal are closely related to its crystal symmetry. For

a cubic crystal, the indicatrix is a sphere and all central sections are just circles with

radius 321 nnn .

For hexagonal, tetragonal and trigonal crystals, the indicatrix is an ellipsoid of

revolution about the principal symmetry axis z , as shown in Figure 1.2. The principal

axis is called the optic axis and all waves traveling along the optic axis have the same

phase velocity (refractive index) irrespective of their polarization. This group of crystals

is categorized as uniaxial with onnn 21 as the ordinary refractive index and enn 3

as the extraordinary refractive index. The difference between the two indices,

oe nnn , is called the birefringence. If the wave vector k

makes an angle with the

principal axis, the ordinary index (minor axis of the ellipse) is independent of while

the extraordinary index en varies with the angle as:

2

2

2

2

2

sincos1

eoe nnn

. (1.5)


JIM Kwok Lung 7

For the three remaining crystal systems with crystal symmetry of orthorhombic,

monoclinic and triclinic, the indicatrix is a triaxial ellipsoid. There are two circular

central sections and thus two optic axes. This group of crystals is categorized as biaxial.

Figure 1.2 Refractive index ellipsoid (optical indicatrix) for a positive uniaxial crystal,

321 nnn . The optical axis is parallel to the z-axis [Syms, 1992].

1.2.3 Electro-optic effect and electro-optic coefficients

Electro-optic (E-O) effects refer to changes in the refractive index of a material

induced by the application of an external electric field. When an electric field is applied,

both the size and the orientation of the indicatrix change. The modified equation of the

indicatrix under an applied field is given by [Syms, 1992]:

z

x

y k

A

B

O n1

n2

n3

ne()

no


JIM Kwok Lung 8

11323122

332

222

11 xzyzxyzyx , (1.6)

where ij is the impermittivity tensor which is related to the relative permittivity tensor

by:

ijij1 . (1.7)

A redistribution of the charges in the crystal, due to the presence of an external electric

field E

, leads to a change of the impermittivity tensor which can be expressed as:

lk

lkijklk

kijkij EEREr,

, (1.8)

where the first term and the second term represent the linear E-O effect (Pockels effect)

and quadratic E-O effect (Kerr effect), respectively. The coefficients ijkr and ijklR are the

corresponding linear and quadratic E-O coefficients. Since ij is a symmetric tensor, the

number of independent elements is limited and the subscript indices in ijkr and ijklR can

be reduced. For the third-rank tensor ijkr , the first and second indices are replaced by a

single index running from 1 to 6 in the following rule: 11 1, 22 2, 33 3, 23 or 32

4, 13 or 31 5, 12 or 21 6.

The linear E-O effect usually outweighs the quadratic E-O effect in magnitude.

The quadratic effect is, however, present in all materials [American national & IEEE

standard, 2003]. For materials with centrosymmetric structure such as ferroelectric

crystals in the paraelectric state, the linear component vanishes according to symmetry

consideration. The change in the impermittivity tensor ij is dominated by the


JIM Kwok Lung 9

quadratic term. In practice, the E-O effect is therefore either predominately linear or

quadratic with respect to E

and is thus characterized by either ijkr or ijklR only.

If the linear E-O effect is predominant, the change in the impermittivity tensor

ij , induced by an electric field ,,, zyx EEEE

, can be expressed as:

z

y

x

E

E

E

rrr

rrr

rrr

rrr

rrr

rrr

636261

535251

434241

333231

232221

131211

12

13

23

33

22

11

, (1.9)

where the 6 3 matrix ijr is called the electro-optic tensor. Taking into account the

crystal symmetry, some of the coefficients in the E-O tensor are zero and some of them

are equal in value or opposite in sign [Yariv, 1984]. For example, the E-O tensor of a

uniaxial crystal such as BaTiO3 which belongs to the point group 4mm has the form:

000

00

00

00

00

00

51

51

33

13

13

r

r

r

r

r

r . (1.10)

The complication of the cross-terms can often be avoided by applying the external field

parallel to one of the principal axes of the crystal. If the electric field is applied along the

z-axis of a uniaxial crystal EE ,0,0

, Eq. (1.6) together with Eq. (1.9) can be

reduced to:


JIM Kwok Lung 10

111 2

33222

132

zEr

nyxEr

n eo

. (1.11)

In this case, the principal axes of the indicatrix change their lengths without rotation.

Therefore, no cross terms are included. The modified indicatrix gives Eno and Ene

as:

ErnEn oo

1322

11 , and (1.12)

ErnEn ee

3322

11 . (1.13)

Applying the relation:

23 1

2

1

ndndn , (1.14)

Eq. (1.12) and Eq. (1.13) can be rewritten as:

ErnnEn ooo 133

2

1 , and (1.15)

ErnnEn eee 333

2

1 . (1.16)

The electric field induced birefringence can then be obtained by:

Ernrn

nrEnEnEn ce

e

oeeo

313

3

333

2

1

2

1

, (1.17)

where rc is the effective linear electro-optic coefficient (or linear electro-optic coefficient

for brevity) and is given by:

13

3

33 rn

nrr

e

oc

. (1.18)


JIM Kwok Lung 11

An effective quadratic E-O coefficient cR can be derived in a similar manner. For

materials exhibiting linear and quadratic E-O effect, the change of birefringence n

induced by an electric field with field strength E is then given by:

Ernn c3

2

1 , and (1.19(a))

23

2

1ERnn c , (1.19(b))

respectively and n is the relevant refractive index. crn3 and cRn3 are called the electro-

optic figures of merit. If crn3 or cRn3 is large, a significant change in refractive index

can be obtained even the applied electric field is moderate.

Typically, the linear E-O coefficients are in the order of 10-12 to 10-10 m/V. The

corresponding refractive index changes induced by a field of 106 V/m are in the range of

10-6 to 10-4. The typical quadratic E-O coefficients range from 10-18 to 10-14 m2/V2 in

crystals and 10-22 to 10-19 m2/V2 in liquids. For a field of 106 V/m, the corresponding

refractive index changes are 10-6 to 10-2 in crystals and 10-10 to 10-7 in liquids [Saleh,

1991]. In practice, both the linear and quadratic E-O effects are utilized to modulate

lightwave in telecommunication applications.

One class of the promising candidates for E-O applications comes from the

ferroelectric materials such as lanthanum modified lead zirconium titanate (PLZT)

[Haertling, 1971], lithium niobate (LNO) [de Toro, 1998], and barium strontium titanate

(BST) [Wang, 2006b].


JIM Kwok Lung 12

1.3 Elastic properties during phase transition in ferroelectric materials

Ferroelectric ceramics undergo a phase transition from tetragonal (ferroelectric)

phase to cubic (paraelectric) phase across the Curie temperature (Tc). In addition to the

lattice change, an elastic modulus anomaly, which results in drastic variations in the

longitudinal and transverse sound velocities and attenuations, is induced during the

phase transition [Huibregtse, 1959; Moreno-Gobbi, 2006; Frayssignes, 2005]. Figure 1.3

shows the sound velocities and attenuations of a Ba0.7Sr0.3TiO3 ceramic sample as a

function of temperature. It can be observed that there are three anomalies occur at

around 300, 220, and 130 K which correspond to the three phase transitions: cubic to

tetragonal, tetragonal to orthorhombic and orthorhombic to rhombohedral, respectively.

Since the Curie temperature of the Ba0.7Sr0.3TiO3 ceramic is at ~ 300 K, it is expected

that a device composed of Ba0.7Sr0.3TiO3 is thermally tunable at room temperature if the

characteristics of the device is related to the sound velocities of the constituting materials.


JIM Kwok Lung 13

Figure 1.3 Ultrasonic attenuations and velocities of a Ba0.7Sr0.3TiO3 ceramic for (a)

longitudinal waves and (b) transverse waves [Moreno-Gobbi, 2006].

1.4 Barium strontium titanate in bulk and thin film forms

1.4.1 Bulk barium strontium titanate

Barium strontium titanate (Ba1-xSrxTiO3 or BST), a solid solution system between

barium titanate (BaTiO3 or BTO) and strontium titanate (SrTiO3 or STO), is a

ferroelectric material. It has a perovskite (ABO3) structure (Figure 1.4) which may be

regarded as being formed by the substitution of Sr atoms for the Ba atoms in the barium

titanate lattice over the entire range of concentration x [Baumert, 1998]. The introduction

of Sr atoms into the barium titanate lattice influences both the crystalline structure and

the properties. The structure and properties of bulk Ba1-xSrxTiO3 have been reported in

the literature [Baumert, 1998; Landolt, 2002]. The ferroelectric-to-paraelectric phase


JIM Kwok Lung 14

transition (Curie transition) occurs at about 120 ˚C for BaTiO3 and -233 ˚C for SrTiO3.

Figure 1.5 shows the room-temperature lattice parameters and the relative permittivity

versus temperature for the Ba1-xSrxTiO3 as a function of the SrTiO3 content x. Taken as

the temperature location of the peak in the relative permittivity, the Curie temperature Tc

of Ba1-xSrxTiO3 exhibits an approximately linear relationship with the content of SrTiO3.

As can be seen from Figure 1.5(a), Ba1-xSrxTiO3 with x < 0.3 has a tetragonal (non-

centrosymmetric) structure at room temperature (~ 25 ˚C) and is thus in the ferroelectric

state. By contrast, Ba1-xSrxTiO3 with x > 0.3 has a cubic (centrosymmetric) structure and

is in the paraelectric state. Nevertheless, Figure 1.5(b) shows that the peak in the

permittivity for Ba0.7Sr0.3TiO3 occurs at about 35 ˚C, although the ferroelectric-to-

paraelectric transition has already started at room temperature. Therefore Ba0.7Sr0.3TiO3

should still exhibit ferroelectric behavior at room temperature.

Figure 1.4 Lattice structure of Ba1-xSrxTiO3. Ba and Sr atoms occupy the corner

positions with Ti atom at the body center, surrounded by the oxygen

octahedron [Wang,2006a].


JIM Kwok Lung 15

Figure 1.5 (a) Room temperature lattice parameters [McQuarrie, 1955] and (b)

Relative permittivity versus temperature for Ba1-xSrxTiO3 with different

SrTiO3 contents [Smolenskii, 1954].

1.4.2 Barium strontium titanate thin films

The research interests in Ba1-xSrxTiO3 thin films have originated from their

potential applications in integrated devices [Damjanovic, 1998]. For use in microwave

devices, Ba1-xSrxTiO3 thin films are more attractive than bulk materials due to the lower

operation voltage, smaller size and higher level of integration [Chang, 1999; Tseng,

1999]. Many thin film deposition techniques have been developed in the fabrication of

Ba1-xSrxTiO3 thin films and the properties of the films are strongly dependent on the

processing techniques.


JIM Kwok Lung 16

Early reports on Ba1-xSrxTiO3 thin films have focused on the dielectric properties.

The excellent dielectric properties of Ba1-xSrxTiO3 thin films motivated some researchers

to investigate their optical properties [Panda, 1998; Tcheliebou, 1997; Wang, 2004] and

electro-optic characteristics. Li et al. [Li, 2000] studied the E-O effects in Ba1-xSrxTiO3

thin films deposited on LAO (001) substrates. Both the Pockels and Kerr effects were

observed and the E-O coefficients were quite high. From then on, Ba1-xSrxTiO3 thin

films have been considered as promising candidates not only in microwave but also in

electro-optic applications.

The properties of Ba1-xSrxTiO3 thin films also depend on the composition. Wang

[Wang, 2006a] investigated the compositional dependence of the E-O properties of Ba1-

xSrxTiO3 thin films. The Ba1-xSrxTiO3 thin films exhibited a predominantly quadratic E-

O behavior, for which the quadratic E-O coefficients of Ba1-xSrxTiO3 thin films are in the

order of 10-17 to 10-18 m2/V2, in the entire composition range and the highest E-O

coefficient was found at x = 0.3 (Figure 1.6).

Owing to its relatively high E-O coefficient and room-temperature range Curie

temperature, Ba0.7Sr0.3TiO3 is chosen for device development in electro-optically tunable

photonic crystal applications and thermally tunable phononic crystal applications in this

study.


JIM Kwok Lung 17

Figure 1.6 Electro-optic coefficients as a function of x for Ba1-xSrxTiO3 thin films

deposited on MgO (001) substrates at 650 ˚C [Wang, 2006a].

1.5 Photonic crystals

Since the pioneering work of Yablonovitch [Yablonovitch, 1987] and John

[John, 1987], intensive research efforts have focused on the realization of photonic

crystals (PCs). Photonic crystals represent a special class of structured materials in

which the dielectric constant exhibits spatial periodic modulation with periodicity

comparable to the wavelength of electromagnetic (EM) waves (Figure 1.7). This type of

spatially modulated materials have been receiving particular attention in recent decades

due to their very promising applications in a number of fields, where EM wave is

employed as an information and energy carrier, such as optical communications, lasing ,

data storage and processing etc. These applications may find their motivations as an

analogy to the electronic industry, and are thus categorized as photonics in compliment

to electronics.


JIM Kwok Lung 18

Figure 1.7 Schematic diagrams of 1D, 2D, and 3D PCs.

The transmission of EM waves in PCs is conceptually the same as that of

electrons in atomic lattices. While the EM wave propagation obeys the Maxwell’s

equations, the electronic wave follows the Schrödinger equation in the framework of

quantum mechanics [Angelakis, 2004]. Therefore, a series of physical phenomena as

observed for electron propagation in atomic lattices, such as diffraction, dispersion,

localization, band and bandgap as well as excitations, can be one to one mapped onto

EM wave propagation in PCs. The solid state physics as the basis of modern electronics

thus provides a broad analogy platform on which photons and PCs may find their

functions for photonic industry and are also of some fundamental research interest.

Consequently, it is optimistically expected that the technology of photonics stemming

from PCs may find applications as extensive as modern electronics. Along this line, it

should be emphasized that the predominant advantages of photonics for information

processing and carrying are high speed, high reliability and huge capacity, which are

unrivaled by electronics.

One-dimensional (1D) Two-dimensional (2D) Three-dimensional (3D)


JIM Kwok Lung 19

Furthermore, there is no absolute length scale or dielectric function scale in PCs

such that the rescaled photonic energy dispersion spectrum becomes size dependent.

Therefore, rescaled spectra for all PCs of the same lattice alignment remain the same, no

matter how the PC lattice size changes [Joannopoulos, 1995]. These advantages allow

one to predict accurately the EM propagation in PCs without losing any physical

essence. This makes the structure design of PCs realistic and reliable.

By calculating the photonic bandstructures, Ho et al. [Ho, 1990] showed that

dielectric spheres arranged in a diamond structure possess a full photonic bandgap.

Utilizing this insulating property of PCs which arise from the existence of photonic

bandgaps, various applications, including large angle bending waveguides and resonant

cavities, are proposed.

1.5.1 Theoretical tools for analyzing photonic crystals

It is worth noted that theoretically the Maxwell equation description of EM wave

propagation in PCs is much more reliable than the issue of electrons in ionic crystals,

because electrons show high-order many-body interactions which do not exist for EM

radiations in PCs. Various theoretical approaches to analyze the properties of photonic

crystals through computation of photonic bandstructures and EM wave propagation in

PCs have been well developed. The three most commonly used techniques are the plane

wave expansion (PWE) method [Cassagne, 1996; Meade, 1992], the scattering matrix

method [Yonekura, 1999], and the finite-difference time-domain (FDTD) method


JIM Kwok Lung 20

[Tavlove, 1995; Yee, 1966]. Besides, transfer matrix method (TMM) is often used to

calculate the transmission and reflection properties of one-dimensional PCs [Li, 2007;

Wang, 2008] due to its simple algorithm and fast computation time.

1.5.1.1 Plane wave expansion method

The Maxwell’s equations can be rearranged to yield an eigenvalue equation:

rHc

rHr

2

21

. (1.40)

where r is the position-dependent dielectric function, H

the magnetic field, the

frequency, and c is the speed of light in vacuum. If the system is spatially periodic, the

magnetic field function rH

should be a Bloch function just like the electron wave

function in a lattice. Thus, the magnetic field function rH

together with the inverse

dielectric function r1 can be expressed by the Fourier expansion (plane wave

expansion) as:

g

rgig e

r

1

)(

1

, (1.41a)

g

rgkig eHrH

)()( , (1.41b)

where k

is the wave vector and g

the reciprocal lattice vector. Combining Eq. (1.41)

and Eq. (1.40) to obtain an eigenvalue equation:

gg

ggg Hc

aHgkgk

2

1

2

, (1.42)


JIM Kwok Lung 21

which can then be solved to yield the normal mode coefficients and frequencies of the

electromagnetic modes for each wave vector k

where a is the period of the photonic

crystal. The dispersion relation between frequency and wave vector k

(often called

the photonic bandstructure) with transverse electric (TE) polarization for a two-

dimensional (2D) PC in triangular lattice is shown in Figure 1.8. The shaded area

highlights the photonic bandgaps where EM waves with frequencies within the gaps

cannot propagate inside the photonic crystal. More elaborated discussions on the

photonic bandstructure calculation of 1D and 2D PCs are given in Chapter 2 and Chapter

3, respectively.

0

0.2

0.4

0.6

0.8

1Wave vector

Fre

que

ncy

( wa

/2pc

)

G M K G

TE Photonic Bandgap

Figure 1.8 Transverse electric (TE) polarization photonic bandstructure of a 2D PC in

triangular lattice computed by the plane wave expansion method.

Γ

MΓ

K


JIM Kwok Lung 22

1.5.1.2 Scattering matrix method

The general calculation model of a 2D PC for scattering matrix method is shown

in Figure 1.9. The refractive index of the columns is nc. Fields are calculated by solving

the Helmholtz equation using the Fourier-Bessel expansion of scattered fields from all

atoms. When P is located outside the columns, the field P is generally expressed as:

N

v mvvmmvinc PjmPrkHbPP

10

)1(, exp , (1.43)

where Pinc is the field of an incident wave from excitation points. v and N are the

index and total number of columns, respectively. )1(mH is the mth-order Hankel function

of the first kind, corresponding to the time-dependent function tje . 0k is the wave

number in vacuum, which is related to the vacuum wavelength , the angular frequency

, and the vacuum velocity of light c as ck //20 . Prv is the distance from

column v to point P, and Pv is the angle of line Pv against the x-axis. The

scattering matrix S can then be derived and its matrix elements mvS , are given by

[Yonekura, 1999]:

PrkJPrkHPrkJPrkHn

PrkJPrkJPrkJPrkJnS

vmvmvmvmc

vmvmvmvmcmv

0000

0000,

, (1.44)

where 1 for TE polarization and 2cn for TM polarization. The field distribution

and also the transmission spectrum can be obtained by the manipulation of the scattering

matrix S.


JIM Kwok Lung 23

Figure 1.9 Calculation model of the scattering matrix method for a 2D PC.

1.5.1.3 Finite-difference time-domain method

Maxwell’s equations describe the temporal change in the electric field E

upon

the spatial variation of the magnetic field H

, and vice versa. The finite-difference time-

domain (FDTD) is a method to solve the Maxwell’s equations by first discretizing the

equations via central differences in time and space and then numerically evaluating the

equations to obtain the temporal and spatial evolution of the electric field and magnetic

field. Taking the Maxwell’s curl equations as an example, they can be written in

Cartesian coordinates as six scalar equations. Two examples are:

y

E

z

E

t

H zyx

1

, and (1.45a)

x

H

z

H

t

Ezxy

1

. (1.45b)

P.

. xz

y

(P)

inc

r(P)


JIM Kwok Lung 24

The other four are symmetric equivalents of the above and are obtained by cyclic

permutation of x, y, and z.

The standard FDTD method is to solve Eq. (1.45) based on Yee’s mesh and

calculates the E

and H

field components at points on a grid with grid points spaced x ,

y , and z apart. The E

and H

field components are then interlaced in all three

spatial dimensions as shown in Figure 1.10. Furthermore, time is discretized in steps of

t . The E

field components are then evaluated at time tnt and the H

field

components at time tnt 2/1 , where n is an integer representing the computing

step.

Figure 1.10 E

and H

field components in a Yee cell of dimensions x , y , and z

[Yee, 1966].

(i, j, k)

Ex

Ez

Hx

Ey Ex

Ez

Ez

Hy

Hz Ey

EyEx

y

z

x

z

y

x


JIM Kwok Lung 25

Eq. (1.45) is then transformed to:

nkjiz

nkjiz

nkjiy

nkjiy

nkjix

nkjix EE

y

tEE

z

tHH ,1,,,1,,,,

2/1,,

2/1,,

, and(1.46a)

2/1,,

2/11,,

2/1,,

2/1,1,,,

1,,

nkjiy

nkjiy

nkjiz

nkjiz

nkjix

nkjix HH

z

tHH

y

tEE

. (1.46b)

The fields at a given mesh point, denoted by integers i, j, k at subsequent time interval

2/t can be solved iteratively in a leapfrog manner.

1.5.1.4 Transfer matrix method

The transfer matrix of a homogenous film is given by:

)coscos()cossin(cos

)cossin(cos

)coscos(

00

00

ndkndkin

ndkn

indk

M TE (1.47)

for TE polarization, where k0 is the wave vector of the EM wave, ψ the angle between k0

and the normal of the film, n and d the complex refractive index and the thickness of the

film, respectively. The transfer matrix of a multilayer film (Mmultilayer) consisting of N

layers is obtained by multiplication of the transfer matrices of the layers:

N

jjjjmultilayer dnM

mm

mmM

12221

1211 ),( . (1.48)

The transmittance and reflectance of the 1D PC (alternating multilayered films) can then

be calculated from the matrix elements of the matrix Mmultilayer. A more detailed

description on the computation of transmission spectrum of 1D PCs by the transfer

matrix method can be found in Section 2.2.2 in Chapter 2.


JIM Kwok Lung 26

1.5.1.5 Comparison of the four methods

Plane wave expansion (PWE) method is the most popular method for photonic

bandstructure computation where Maxwell’s equations are transformed to a set of

eigenvalue equations which are readily solved by many existing free software routines

such as LAPACK. This makes it extremely easy to implement computationally. PWE

has the distinctive advantages that it is easy to automate the identification of photonic

bands and bandgaps, and obtain the mode profiles with no additional efforts. The major

drawback of the PWE method is that the computational cost (storage and computation

time) increases parabolically with the number of plane waves. To achieve convergence,

more plane waves are required especially when the dielectric contrast is large.

Finite-difference time-domain (FDTD) method is well-suited for computing

properties that involve evolution of the fields, such as transmission and resonance decay-

time calculations. It also has the capacity for calculating photonic bandstructures and

finding resonant modes, by mapping the peaks in the Fourier transform of the time

response to some input. The main advantage of this is that all the frequencies (peaks) are

obtained at only one calculation. Furthermore, the FDTD method is capable of including

nonlinear effect and dispersion effect in the calculation. There are, however, several

disadvantages for the FDTD method. Some modes may be missed, which is especially

problematic in higher-order resonant cavity and waveguide calculations, due to weak

coupling or degeneracy of states. Although it is possible to obtain photonic bandstructure

information via Fourier transform of the time response, the frequency resolution is


JIM Kwok Lung 27

inversely related to the simulation time. Furthermore, single wavelength simulation is

needed for each mode in order to obtain the mode profile. For numerical stability reason,

the time-step size must be proportional to the spatial-grid size.

Scattering matrix method is usually employed to obtain the transmission

spectrum and the static electromagnetic field distribution of a PC. A distinctive

advantage of the scattering matrix method is its low computation cost. Usually, the

calculation converges rapidly for a round object with only -2 to +2 degree values

[Yonekura, 1999]. The calculation volume is proportional to the square of the number of

objects, so it is more advantageous for system with fewer numbers of period. However,

there are several limitations to the method. The major one is that it can only be applied

to 2D cases. Also, the scattering objects must be isolated from each other and must be

homogeneous. The obtained field distributions are static without time evolution.

The transfer matrix method (TMM) is mostly suitable for the calculation of the

transmission and reflection spectra of the one dimensional photonic crystals. The

algorithm is simple and the computation cost is extremely low. The TMM is also capable

of computing the transmission property in the case of off-axis incidence and PC with

finite numbers of period. However, photonic bandstructures and the EM field

distributions are not attainable using TMM.

Among different numerical methods, PWE method and the FDTD method are the

two most commonly adopted computational approaches to study PCs.


JIM Kwok Lung 28

1.5.2 Tunable photonic crystals

In the early studies of PCs, investigations are mainly focused on the passive

device applications such as sharply bent PC waveguides [Lin, 1998; Mekis, 1996], and

channel drop filters [Noda, 2000; Qiu, 2003] etc. If the photonic bandstructures of the

PCs can be modulated externally by external fields, more functional active PC devices

can be realized. In fact, different schemes have been proposed to realize the tunability of

PCs by external parameters. Schuller et al. [Schuller, 2003] fabricated a PC microcavity

structure in semiconductor slab waveguides infiltrated with liquid. A 9-nm shift in the

resonant peak is achieved when the temperature is risen from 20 ˚C to 70 ˚C. Utilizing

the rotating directors of the liquid crystals, Liu et al. [Liu, 2005] showed that the

photonic bandgaps of a PC structure infiltrated nematic liquid crystals are tunable under

an applied electric field. Yoshino et al. [Yoshino, 1999] demonstrated a nearly 20% shift

of wavelength in the reflection peak for an elastic polymer opal under a strain of 0.38.

Magnetically tunable PCs based on the Faraday and Voigt effects are also investigated

[Jia, 2003; Xu, 2003]. Different tuning mechanisms have their own benefits in various

applications. However, if high modulation speed is required, it is advantageous to

achieve the tunability through the E-O effect, for which the intrinsic response speed is

known to be in the gigahertz range [Taylor, 1999; Turner, 1966].


JIM Kwok Lung 29

1.5.2.1 Ferroelectric photonic crystals

Ferroelectric materials usually exhibit remarkable E-O effect and offer high

refractive indices and low photonic energy loss, which are desirable properties in PC

applications. It is therefore expected that high speed index-tunable PCs can be realized if

one fabricates PCs using ferroelectric materials.

Increasing research efforts have been put on the realization of ferroelectric PCs in

recent years. For example, controllable superprism effect [Scrymegeour, 2003; Xiong,

2003] and temporal modulation of the photonic bandgaps in 2D PCs [Takeda, 2004]

based on the E-O effect are studied theoretically. Schmidt et al. [Schmidt, 2005]

demonstrated an electro-optically tunable PC with a sub-1-V sensitivity using a polymer-

based PC slab waveguide resonator structure. Roussey et al. [Roussey, 2006] illustrated

the enhanced E-O effect due to slow light effect in lithium niobate PCs. Due to its high

E-O coefficient, lanthanum modified lead zirconium titanate (PLZT) is commonly used

for fabricating ferroelectric tunable PCs. Okamura et al. [Okamura, 2005] and Li et al.

[Li, 2003] demonstrated the fabrication of PLZT-based 2D and 3D ferroelectric PCs,

respectively. However, lead-containing materials are environmentally hazardous.

Therefore, lead-free material barium strontium titanate (BST), which have comparable

E-O properties with PLZT [Kim, 2003; Li, 2000], is one of the promising candidates for

building tunable ferroelectric PCs.


JIM Kwok Lung 30

1.6 Phononic crystals

Phononic crystals are periodic composite materials which have different acoustic

velocities and densities, with lattice constants on the scale of the acoustic wavelength.

They are the acoustic or elastic analogue of photonic crystals for light. As in the case of

photonic crystals, interest in phononic crystals, especially in two and three dimensions,

has grown especially rapidly during the last decade. This may be attributed to the distinct

advantages of using phononic crystals for studying the effects of lattice structure on the

propagation of classical waves. Moreover, the existence of both longitudinal and

transverse bulk waves, as well as surface waves, enriches the underlying physics. From

the experimental point of view, since both velocity and density differences contribute to

the acoustic contrast, phononic crystals with a wide range of acoustic properties can be

fabricated experimentally. In addition, well established pulsed ultrasonic and acoustic

techniques for the extraction of both amplitude and phase information make the

characterization of phononic crystals relatively easy and reliable to perform.

Much of the initial research in phononic crystals focused on phononic bandgaps,

which have their origin in the destructive interference of multiple scattered waves in

periodic structures. As a result, many phononic crystals with complete bandgaps can

now be fabricated, allowing wave transport in this frequency range to be investigated

and novel acoustic waveguides to be constructed [Torres, 2001; Vasseur, 2001].

Interesting ideas on how to construct compact phononic crystal sound insulators have

also been proposed and demonstrated [Liu, 2000]. When the periodicity of the phononic


JIM Kwok Lung 31

crystals is in the range of micron, the corresponding frequency range is in the MHz

regime, which is useful in the applications of medical imaging.

1.6.1 Tunable phononic crystals

In order to enhance functionality, it is desirable to tune the phononic bandgaps

actively. Several schemes have been proposed. The suggested bandgap tuning

mechanism is mainly achieved by changing the dimensions of the constituent materials

of the phononic crystals, such as by rotating the constituent steel rods placed in air [Feng,

2006], or through mechanical deformation of the elastomer/air phononic crystal

[Bertoldi, 2008], while Huang et al. [Huang, 2005] utilized the thermal expansion of the

quartz cylinders in the quartz/air system.

As discussed in Section 1.3, the sound velocities of ferroelectric materials exhibit

strong changes during phase transition across the Curie temperature. The sound

velocities of Ba0.7Sr0.3TiO3 ceramic undergo a drastic variation (>10%) across the Curie

temperature at around 30 ˚C. It is, therefore, expected that Ba0.7Sr0.3TiO3-based phononic

crystals are promising candidates for realizing thermally tunable phononic crystals.


JIM Kwok Lung 32

1.7 Scope of the present study

The main objective of the present research is to develop electro-optically tunable

photonic crystals and thermally tunable phononic crystals based on the ferroelectric

material, barium strontium titanate (Ba0.7Sr0.3TiO3).

This thesis consists of six Chapters. Following the introduction given in this

Chapter, the fabrication and characterization of one-dimensional photonic crystals

composed of alternating layers of Ba0.7Sr0.3TiO3 and magnesium oxide (MgO) grown on

MgO (001) substrates are discussed in Chapter 2. The tunability of the 1D PCs is

measured and the results are compared to the theoretical simulations by the plane wave

expansion method and the transfer matrix method.

In Chapter 3, the photonic bandstructures and photonic bandgap maps of two-

dimensional Ba0.7Sr0.3TiO3-based photonic crystals are studied. The effects on the

photonic bandstructures of different shapes of air rods arranged in different

configurations are studied.

Chapter 4 describes the investigation on the photonic crystal cavity embedded rib

waveguides. Photonic crystal cavity is embedded in the Ba0.7Sr0.3TiO3 rib waveguides.

The behaviour of the resonant peak in the transmission spectrum is simulated using the

finite-difference time-domain method. The fabrication and the characterization of the

Ba0.7Sr0.3TiO3-based photonic crystal cavity embedded rib waveguides are discussed.


JIM Kwok Lung 33

In Chapter 5, the thermal tuning of phononic bandstructure in phononic crystals

realized as Ba0.7Sr0.3TiO3 ceramic rods in an epoxy matrix is examined. The phononic

bandstructures are calculated and the shifting of the reflection dips in the reflection

spectrum of the phononic crystal at different temperatures measured by the ultrasonic

pulse-echo technique is demonstrated.

Conclusions are then given in Chapter 6.


JIM Kwok Lung 34

1.8 Statement of original contributions

To the best of my knowledge, the present work has made the following original

contributions:

1. The electro-optic tunability of one-dimensional photonic crystals composed of

multilayers of alternating Ba0.7Sr0.3TiO3 and MgO thin films was demonstrated.

A photonic bandgap shift of 2 nm, corresponding to a 0.5 % change in the

refractive index of Ba0.7Sr0.3TiO3 which is comparable to that of PLZT, was

measured under an applied dc voltage of 240 V (E ~ 12 MV/m).

2. Photonic crystal cavity structure was etched on a Ba0.7Sr0.3TiO3 rib waveguide by

a focused ion beam technique. A resonant peak located at ~1550nm in the

transmission spectrum was observed.

3. Two-dimensional phononic crystal realized as Ba0.7Sr0.3TiO3 ceramic rods in an

epoxy matrix was fabricated. The reflection dip in the reflection spectrum shifted

from 12 MHz to 14 MHz when the temperature increased from 35 ˚C to 45 ˚C.

ONE-DIMENSIOANL TUNABLE PHOTONIC CRYSTAL BASED ON BARIUM

CHAPTER 2 STRONTIUM TITANATE & MAGNESIUM OXIDE MULTILAYERED THIN FILMS

JIM Kwok Lung 35

CHAPTER 2

ONE-DIMENSIONAL TUNABLE PHOTONIC CRYSTAL

BASED ON BARIUM STRONTIUM TITANATE &

MAGNESIUM OXIDE MULTILAYERED THIN FILMS

2.1 Introduction

Over the past decade, there has been great interest in the study of photonic

crystals (PCs) due to their ability in manipulating photons and potential applications in

photonics information technology [Chow, 2000; Erchak, 2001; Mekis, 1996; Noda,

2000; Ren, 2006]. The properties of PCs depend on the configuration of the constituent

materials, which cannot be modified after fabrication. On the other hand, the photonic

bandstructures are also dependent on the refractive indices of the constituent materials. If

the photonic bandstructures of the PCs can be modulated externally by some other

means, the PCs may be applicable as active optical devices. Indeed, many schemes have

been proposed to realize the tunability of PCs by external parameters such as electric

field, magnetic field, temperature and strain [Jia, 2003; Leonard, 2000; Schuller, 2003;

Yoshino, 1999]. Different tuning mechanisms have their own benefits in various

applications. Recently, fast-response tunable photonic crystal filters [Alagappan, 2006;

Ha, 2001; Villar, 2003] have attracted much attention due to their important applications

in chip-to-chip and on-chip optical communications. For such applications, it is



JIM Kwok Lung 36

advantageous to achieve tunability through the electro-optic effect, for which the

intrinsic response speed is known to be in the gigahertz range [Taylor, 1999; Turner,

1966].

The electro-optic (E-O) effect is one of the interesting characteristics of

ferroelectric materials. The refractive index of a ferroelectric material can be modulated

by applying an external electric field. Therefore, it is expected that index-tunable PCs

can be realized if one fabricates PCs using ferroelectric materials. Barium strontium

titanate (BST) is considered to be a promising E-O material since the discovery of high

E-O coefficient in BST thin films [Li, 2000]. In some previous studies, Ba0.7Sr0.3TiO3

thin films epitaxially grown on single-crystal substrates showed low optical losses and

good E-O performances, indicating their potential uses in active optical devices [Wang,

2006b; Wang, 2007]. Many recent work have focused on self-assembled PC structures,

such as colloidal crystals and inverse opal structures [Bormashenko, 2005; Li, 2003].

Both of these structures have limited tunability and the fabrication processes are

relatively complicated. In comparison, periodic multilayered structures (1D PC) have

many advantages in terms of material properties, processability [Hong, 2006; Hu, 2005],

as well as cost [Urbas, 2000]. Moreover, it is possible to fabricate large-area PCs by

using multilayer structures. Therefore, one-dimensional photonic crystal is a good

starting point to investigate the tunability of the ferroelectric Ba0.7Sr0.3TiO3-based

photonic crystals.



JIM Kwok Lung 37

In this Chapter, we first briefly present the plane wave expansion (PWE) method

and the transfer matrix method (TMM) for calculations of the photonic bandstructure

and transmission spectrum of a one-dimensional photonic crystal (1D PC) structure,

respectively. The shift of photonic bandgaps with the change of the refractive index of

Ba0.7Sr0.3TiO3 is then discussed. The influence of number of period and the angle of

incidence on the transmission spectrum are also examined. Finally, the fabrication

details of a 1D PC composed of a Ba0.7Sr0.3TiO3/MgO multilayered thin films and its

characterization will be given.

2.2 Simulation methods for one-dimensional photonic crystals

2.2.1 Computation of photonic bandstructure of one-dimensional photonic crystal

by the plane wave expansion method

The photonic bandstructure of PCs can be calculated by using the plane wave

expansion (PWE) method [Ho, 1990; Plihal, 1991]. The Maxwell’s equations for

electromagnetic (EM) waves in a system with periodic distribution of dielectric

constants can be simplified to:

Hc

Hr

2

21

. (2.1)

Here, r is the position-dependent dielectric function, H

the magnetic field, the

frequency, and c is the speed of light in vacuum. As the system is periodic, the dielectric

function and the H

field can be expanded in terms of plane waves:



JIM Kwok Lung 38

g

rgig e

r

1

)(

1

, (2.2a)

g

rgkig eHrH

)()( , (2.2b)

where k

is the wave vector and g

is the reciprocal lattice vector. Both vectors are in

units of 2/a, where a is the period of the PC. For the propagation of light along the

normal (z) direction of a 1D PC composed of alternating layers of dielectric materials (ε1

and ε2), the transverse electric (TE) mode and the transverse magnetic (TM) mode are

degenerate. Combining Eq. (2.1) and Eq. (2.2), we obtain the following matrix equation:

gg

ggg HHgkgk

21

, (2.3)

with

,0,)sin(11

0),1(11

21

211

gforfg

fgf

gforff

g

(2.4)

where Ω=a/2c is the normalized frequency and f is the filling (volume) fraction of

material 1. The photonic bandstructure of the 1D PC (Ω against k) is then obtained by

solving the eigenvalue equation (Eq. (2.3)).

2.2.2 Computation of transmission spectrum of one-dimensional photonic crystal

by the transfer matrix method

The transfer matrix method (TMM) [Stenzel, 2005] utilizes the transfer

(characteristic) matrix to describe the propagation of electromagnetic (EM) waves inside



JIM Kwok Lung 39

a single homogenous film including the interfacial effects. Knowing the propagation

details of the EM waves inside a system, it is straightforward to obtain the transmission

spectrum of that system. The transfer matrix of a homogenous film is given by:

)coscos()cossin(cos

)cossin(cos

)coscos(

00

00

ndkndkin

ndkn

indk

M TE (2.5)

for TE-polarization, where k0 is the wave vector of the EM wave, ψ the angle between k0

and the normal of the film, n and d the complex refractive index and the thickness of the

film, respectively. The transfer matrix of a stack of multilayered films (Mmultilayer)

consisting of N layers is obtained by multiplication of the transfer matrices of the layers:

N

jjjjmultilayer dnM

mm

mmM

12221

1211 ),( . (2.6)

The transmission coefficient (t) and the transmittance (T) are then given by:

SSiiSS

iiTE nmmnnmm

nt

coscos)cos(

cos2

22211211 , and (2.7)

2

)cosRe(

)cosRe(TE

ii

SSTE t

n

nT

, (2.8)

where i and S denote the first (incident) layer and last (substrate) layer, respectively. The

transfer matrix and the expressions of transmission and transmittance for TM-

polarization are given by:

)coscos()cossin(cos

)cossin(cos

)coscos(

00

00

ndkndkn

i

ndkin

ndkM TM , (2.9)



JIM Kwok Lung 40

S

S

i

i

S

S

i

i

TM

nmm

nnmm

nt

coscos)

cos(

cos2

22211211 , and (2.10)

2

)cos

Re(

)cos

Re(

TM

i

i

S

S

TM t

n

nT

. (2.11)

2.3 Theoretical study of Ba0.7Sr0.3TiO3/MgO one-dimensional photonic crystals

A schematic diagram of a typical 1D PC is shown in Figure 2.1. One-

dimensional PCs are characterized by the refractive index contrast and the filling

fraction. The refractive index contrast is the ratio of the higher refractive index to the

lower refractive index (n2/n1) in the multilayered system. The filling fraction, f, is the

ratio between the thickness of the lower refractive index layer (the MgO layer in the

present case) and the period of the PC, i.e. f = d1/(d1+d2).

Figure 2.1 Schematic diagram of a 1D PC composed of alternating layers of dielectric

material 1 and material 2.

d1 d2 a=d1+d2

ε1 ε 2



JIM Kwok Lung 41

The photonic bandstructure of a Ba0.7Sr0.3TiO3/MgO 1D PC with f = 0.1

calculated by the plane wave expansion (PWE) method is given in Figure 2.2(a) as an

example. The refractive indices of Ba0.7Sr0.3TiO3 and MgO are taken to be 2.20 and 1.73,

respectively. The vertical axis represents the normalized frequency, Ω = a/2πc = a/λ,

where a is the period of the 1D PC and λ is the wavelength. The horizontal axis

represents the k-point along the symmetry direction of the first Brillouin zone. The

shaded areas illustrate the photonic bandgaps. Normally, the photonic bandgaps are

wider if the refractive index contrast is larger. One important aspect to be considered

when using the PWE method is the convergence of the normalized frequencies. As can

be seen from Figure 2.2(a), the photonic band edges appear at the Brillouin zone center

(k = 0) and the Brillouin zone edge (k = π/a) only for a 1D PC. Therefore, the

convergence of the normalized frequencies at the Brillouin zone center and the Brillouin

zone edge was tested by using 21 to 71 plane waves increasing in steps of 10. It was

found that the normalized frequencies converged to within 10-4 for 41 and more plane

waves. Thus, the photonic bandstructures of the 1D PC in this Chapter were calculated

using 71 plane waves to ensure convergence.

Figure 2.2(b) shows the transmission spectrum of the Ba0.7Sr0.3TiO3/MgO 1D PC

(f = 0.1) with 100 periods calculated by the transfer matrix method (TMM) for

comparison with Figure 2.2(a). It is clear that the two Figures are in excellent agreement

with each other, as the positions of the photonic bandgaps in Figure 2.2(a) correspond to

the frequencies at which there is no transmission in Figure 2.2(b).



JIM Kwok Lung 42

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5

Normalized wave vector, kz (2π /a )

No

rmal

ized

fre

qu

en

cy

(ωa

/2πc

)

(a)

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized frequency (Ω)

Tra

nsm

itta

nce

(b)

Figure 2.2 (a) Photonic bandstructure of a Ba0.7Sr0.3TiO3/MgO 1D PC with f = 0.1.

The shaded areas represent the photonic bandgaps. (b) Transmission

spectrum of a Ba0.7Sr0.3TiO3/MgO 1D PC (f = 0.1) with 1000 periods.



JIM Kwok Lung 43

When an external voltage is applied across the 1D PC, the refractive index of

Ba0.7Sr0.3TiO3 (nBST) would change due to the electro-optic effect. It will be shown later

that even a small variation (0.3%) in nBST would give rise to a bandgap shift in

wavelength of ~5 nm at λ = 1550 nm. Such a bandgap shift is sufficient for tunable

optical filter applications in wavelength division multiplexing technology [Limberger,

1998]. Ba0.7Sr0.3TiO3 thin film grown on MgO substrate exhibits a quadratic electro-

optic behavior [Wang 2006b; Wang 2007].

In order to examine how a change in the refractive index of Ba0.7Sr0.3TiO3 (nBST)

affects the photonic bandgap of the 1D PC, photonic bandgap maps of the first bandgap

in the Ba0.7Sr0.3TiO3/MgO 1D PC with different percentage changes in the refractive

index of Ba0.7Sr0.3TiO3 were calculated. The results are shown in Figure 2.3. It can be

seen from Figure 2.3 that when the refractive index of Ba0.7Sr0.3TiO3 increases, the

bandgap shifts to lower frequency. Moreover, the magnitude of the shift depends weakly

on the filling fraction, f, for f = 0-0.2. In the fabrication of Ba0.7Sr0.3TiO3/MgO 1D PC by

the pulsed laser deposition, it is more convenient to prepare samples with lower filling

fraction because the deposition rate of MgO is much lower than that of Ba0.7Sr0.3TiO3.

Therefore we concentrate our discussion on the case of f = 0.1. The changes in the

positions for the first and second bandgaps of a Ba0.7Sr0.3TiO3/MgO 1D PC with changes

in nBST is shown in Figure 2.4. It can be seen that the frequencies of the bandgaps

decrease linearly with increasing refractive index of Ba0.7Sr0.3TiO3, which is consistent

with the result in Figure 2.3.



JIM Kwok Lung 44

Figure 2.3 Photonic bandgap map of the first bandgap of the Ba0.7Sr0.3TiO3/MgO 1D

PC with different changes in the refractive index of Ba0.7Sr0.3TiO3.

y = -0.0041x + 0.4726

y = -0.0045x + 0.4552

y = -0.0021x + 0.2364

y = -0.0023x + 0.2274

0

0.1

0.2

0.3

0.4

0.5

0.6

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Percentage change of refractive index of Ba0.7Sr0.3TiO3 (%)

Ban

dg

ap e

dg

e (Ω

)

1st bandgap

2nd bandgap

Figure 2.4 Changes of photonic bandgap edges of a Ba0.7Sr0.3TiO3/MgO 1D PC (f =

0.1) with different changes in the refractive index of Ba0.7Sr0.3TiO3.



JIM Kwok Lung 45

Usually, the bandgap ratio, defined as the ratio of the bandgap width to the mid-

gap value, is the crucial quantity to be considered. It is often chosen to be as large as

possible in applications such as perfect mirrors and optical filters. However, the amount

of bandgap shift may be a more critical quantity in tunable optical filter application.

Therefore, we define the bandgap shift ratio, Rshift, as the ratio of the shift in the bottom

edge of the bandgap, shiftd , to the average of the shifted and unshifted bandgap bottom

edges, . Then, the bandgap shift in wavelength, dλshift, can be calculated as:

shiftshift

shift Rd

d

(2.12)

where λ is the fundamental wavelength to be filtered. The bandgap shift, dΩshift, is related

to the slope in Figure 2.4. As seen from the Figure, both the slope and the mid-gap value

of the second bandgap are approximately twice those of the first bandgap. These imply

that the bandgap shift ratios, Rshift, of the two gaps are almost the same. However, if a

specific wavelength is to be filtered, the periods of the 1D PC is doubled if the second

bandgap, instead of the first bandgap, is utilized.

As an example, we will illustrate a design of a 1D tunable PC. The mid-gap value

of the first bandgap is around 0.23. Therefore, the period of the 1D PC should be about

356 nm for a filtering-wavelength of 1550 nm. From Figure 2.4, the bandgap shift ratio,

Rshift, is approximately equal to the negative fractional change of the refractive index of

Ba0.7Sr0.3TiO3, ΔnBST/nBST. Replacing Rshift, by ΔnBST/nBST in Eq. (2.12), the bandgap shift

in wavelength, dλshift, can be obtained by:



JIM Kwok Lung 46

BST

BSTshift n

nd

. (2.13)

The bandgap shifts in wavelength, dλshift, for a filtering wavelength of 1550 nm for

different percentage change in nBST are given in Table 2.1.

Table 2.1 Photonic bandgap shift in wavelength at λ = 1550 nm for 0.3% and 1.0%

changes in the refractive index of Ba0.7Sr0.3TiO3.

Figure 2.5 shows the photonic bandgap shift of a Ba0.7Sr0.3TiO3/MgO 1D PC

with 1% change in refractive index of Ba0.7Sr0.3TiO3 (from n = 2.200 to n = 2.222). The

period of the 1D PC is 356 nm. A small circle is placed at the wavelength of 1550 nm

and the filling fraction of 0.1 for illustration purpose. When no voltage is applied across

the 1D PC, the refractive index of Ba0.7Sr0.3TiO3 is 2.200. The photonic bandgap of the

1D PC is between 1480 nm and 1540 nm. A light wave of wavelength 1550 nm is thus

able to propagate through the 1D PC. On the other hand, when an external voltage,

applied across the 1D PC, causes a 1% change in the refractive index of Ba0.7Sr0.3TiO3

due to the electro-optic effect, the bandgap of the 1D PC will shift about 15 nm towards

the longer wavelength. The light wave of wavelength 1550 nm then cannot propagate

inside the 1D PC in this situation. Thus the 1D PC acts as an optical filter with a tunable

Fractional change of the refractive

index of Ba0.7Sr0.3TiO3, ΔnBST/nBST (%)

Photonic bandgap shift in wavelength,

dλshift, at 1550 nm

0.3% 4.6 nm

1.0% 15.5 nm



JIM Kwok Lung 47

filtering window of width 15 nm. As can be seen from Figure 2.5, the band edge around

1550 nm is nearly horizontal in the filling fraction range between 0 and 0.2. Therefore

the 1D PC will work well as a tunable optical filter even if there is a deviation in the

filling fraction during the fabrication process.

1400

1450

1500

1550

1600

0 0.1 0.2Filling fraction

Wav

ele

ng

th (

nm

)

nBST=2.2

nBST=2.222(1% increase)

Band gap

Figure 2.5 Photonic bandgap shift of a Ba0.7Sr0.3TiO3/MgO 1D PC (period = 356 nm)

with 1% change in refractive index of Ba0.7Sr0.3TiO3 (from n = 2.200 to n

= 2.222). A small circle is placed for illustration purpose.

One of the drawbacks of the PWE method is the assumption of infinite extension

of the 1D PC. In practice, all 1D PCs are finite. In order to examine the effect of the

number of periods on the performance of the 1D PC filter, the transmission spectrum of

the 1D PC is calculated using the transfer matrix method (TMM). Figure 2.6 shows the

transmission spectrum of a Ba0.7Sr0.3TiO3/MgO 1D PC with a filling fraction of 0.1 for



JIM Kwok Lung 48

different numbers of periods. The 1D PC is assumed to be grown on a MgO substrate.

As can be seen from Figure 2.6, there are two dips at around Ω = 0.22 and Ω = 0.45,

which correspond to the two photonic bandgaps of the 1D PC (see Figure 2.2(a)). As the

number of periods of the 1D PC increases, the dips become deeper and the positions of

the dips shift toward the bandgap positions obtained by using the PWE method. The

smallest number of periods for achieving a transmittance of 0.5, which corresponds to -3

dB, is 5 if the second bandgap is considered. This means that the Ba0.7Sr0.3TiO3/MgO 1D

PC should have at least 5 periods in order to exhibit a significant filtering function.

Figure 2.7 shows the transmission spectrum of a 5-period Ba0.7Sr0.3TiO3/MgO 1D

PC for different angles of incidence. It can be observed that a small deviation (<5˚) from

normal incidence causes no significant difference in the transmission spectrum. When

the angle of incidence increases, the transmission dip (photonic bandgap) shifts to higher

normalized frequency (Ω = a/λ). An increase in angle of incidence is equivalent to a

longer period of the 1D PC experienced by the incident light wave since it will be

refracted inside the 1D PC. As Ω = a/λ, an increase of period corresponds to a higher

normalized frequency. Hence, the photonic bandgap shifts to higher normalized

frequency when the angle of incidence increases.



JIM Kwok Lung 49

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5Normalized frequency (Ω )

Tra

nsm

itta

nce

N = 3

N = 5

N = 7

N = 9

Figure 2.6 Transmission spectrum of a Ba0.7Sr0.3TiO3/MgO 1D PC with different

numbers of periods (N).

0

0.2

0.4

0.6

0.8

1

0.3 0.4 0.5 0.6

Normalized frequency (Ω )

Tra

nsm

itta

nce

0˚

5˚

10˚

15˚

20˚

Figure 2.7 Transmission spectrum of a 5-period Ba0.7Sr0.3TiO3/MgO 1D PC for

different angles of incidence.



JIM Kwok Lung 50

2.4 Fabrication of Ba0.7Sr0.3TiO3/MgO one-dimensional photonic crystals

The pulsed laser deposition (PLD) technique was employed to prepare a one-

dimensional photonic crystal, which consist of epitaxial Ba0.7Sr0.3TiO3/MgO

multilayered thin films. Alternating layers of Ba0.7Sr0.3TiO3 and MgO were deposited on

MgO (001) single crystal substrates by irradiating stoichiometric targets with a laser

beam of 248 nm wavelength and 25 ns pulse duration from a KrF excimer laser (Lambda

Physik COMPex 205). The pulse energy of the laser beam was 250 mJ and 350 mJ for

the deposition of the Ba0.7Sr0.3TiO3 and MgO layers, respectively, and the repetition rate

was 10 Hz. The distance between the target and the substrate was fixed at 50 mm. The

substrate temperature was maintained at 750 oC. The oxygen partial pressure was kept at

27 Pa during the laser ablation process. The deposition rate for Ba0.7Sr0.3TiO3 and MgO

was found to be about 20 nm/min and 5 nm/min, respectively. Table 2.2 summarizes the

PLD conditions for the fabrication of the Ba0.7Sr0.3TiO3 and MgO multilayered thin

films. Considering the increased difficulty in depositing thicker films by the PLD

technique, the fabricated 1D PC has five periods, with each period consisting of a ~90

nm thick Ba0.7Sr0.3TiO3 layer and a ~10 nm thick MgO layer. After deposition, the PC

was post-annealed at 1000 oC in a tube furnace for 3 h under oxygen atmosphere. The

crystal structure of the Ba0.7Sr0.3TiO3/MgO multilayered thin films was examined using

an X-ray diffractometer (Bruker D8 Discover) equipped with Cu Kα radiation.

The optical transmission spectrum of the Ba0.7Sr0.3TiO3/MgO multilayered thin

films was measured using a Perkin Elmer Lambda 18 UV-visible spectrometer. The



JIM Kwok Lung 51

electrode configuration used for applying an electric field consisted of two coplanar

electrodes of dimensions 1.0 × 8.0 mm2 separated by a 20 µm wide gap (see Figure 2.8).

Table 2.2 PLD conditions for Ba0.7Sr0.3TiO3 and MgO multilayered thin films.

Ba0.7Sr0.3TiO3 MgO

Target- substrate distance 50 mmLaser energy 250 mJ 350 mJ

Repetition rate of pulsed laser 10 HzAmbient gas O2

Total pressure of ambient gas 27 PaSubstrate temperature 750 oC

Growth rate ~ 20 nm/min ~ 5 nm/min

Figure 2.8 Schematic diagram showing the experiment for observing the photonic

bandgap shift of a Ba0.7Sr0.3TiO3/MgO multilayered thin films.



JIM Kwok Lung 52

2.5 Characterization of one-dimensional Ba0.7Sr0.3TiO3/MgO photonic crystals

2.5.1 Structural characterization of the one-dimensional photonic crystals

Figure 2.9 shows the θ/2θ x-ray diffraction (XRD) pattern of the

Ba0.7Sr0.3TiO3/MgO multilayered thin films. The fact that only (00l) peaks of the

Ba0.7Sr0.3TiO3 layers appear in the XRD patterns suggests a pure perovskite phase in the

Ba0.7Sr0.3TiO3 layers. It is believed that the diffraction peaks of the MgO layers are

submerged in the peaks of the MgO single crystal substrate, so no peaks of the MgO

layers can be observed. Rocking curve measurements of the Ba0.7Sr0.3TiO3 (002)

reflections revealed that the full width at half maximum (FWHM) is about 0.53o, which

reveals that the crystallites are of high quality. The in-plane alignment of the

Ba0.7Sr0.3TiO3 thin films with respect to the major axes of the (001) substrates was

confirmed by the XRD off-axis φ scan of the Ba0.7Sr0.3TiO3 (202) and MgO (202)

reflections, as shown in the inset of Figure 2.9, indicating epitaxial growth of the

Ba0.7Sr0.3TiO3 layers. Losses in optical devices usually originate from various structural

defects, such as point defects, grain boundaries, misorientation, and surface roughness.

Hence, fabrication of defect-free films, such as epitaxial single crystalline thin films, is

required for optical device applications.



JIM Kwok Lung 53

Figure 2.9 XRD θ/2θ scan of a Ba0.7Sr0.3TiO3/MgO multilayered thin films. The inset

shows the φ scan of the Ba0.7Sr0.3TiO3 (202) and substrate (202) reflections.

2.5.2 Optical characterization of the one-dimensional photonic crystals

The measured transmission spectrum of the Ba0.7Sr0.3TiO3/MgO multilayered

thin films is shown in Figure 2.10(a). The transparency of the multilayered films drops

sharply in the UV region and the absorption edge (threshold wavelength) is located at

318 nm, which is quite close to that of a single Ba0.7Sr0.3TiO3 thin film grown on a MgO

(001) substrate [Wang, 2006b]. There are four dips appeared in the transmission

spectrum at wavelengths around 830 nm, 610 nm, 460nm and 390 nm. They are

numbered as dips (1), (2), (3), and (4), respectively, in the figure. In order to confirm that

the transmission dips are not caused by the intrinsic absorptions of the perovskite phase

of Ba0.7Sr0.3TiO3, a Ba0.7Sr0.3TiO3 thin film of ~300 nm thickness was deposited on a

MgO (001) substrate under the same conditions. The transmission spectrum of the



JIM Kwok Lung 54

Ba0.7Sr0.3TiO3 thin film was measured and the result is shown in the inset of Figure

2.10(a). The film is highly transparent and does not show any absorption in the visible

region.

According to the PWE calculation (see Figure 2.2(a)), the fabricated 1D PC with

period of 100 nm should produce one photonic bandgap at around 440 nm (dip (3) in

Figure 2.10(a)), which corresponds to ~ 0.23, in the wavelength ranged from 300 nm

to 1100 nm. Other higher order PBGs (e.g. 220 nm which corresponds to ~ 0.46)

would fall below the absorption edge (318 nm). In fact, the four transmission dips

appeared in the measured transmission spectrum should be attributed to the Fabry-Perot

oscillation arisen from a stack of multilayered films of 500 nm thick in addition to the

photonic bandgap located at around 440nm. To verify this idea, a theoretical calculation

of the transmission spectra of the Ba0.7Sr0.3TiO3/MgO multilayered thin films and its

corresponding effective single film were conducted using the transfer matrix method

(TMM). Both the multilayered films and the effective single film are assumed to be

grown on a MgO substrate. The modeled multilayered films is composed of 5 periods of

alternating Ba0.7Sr0.3TiO3 and MgO layers with thicknesses of 90 nm and 10 nm,

respectively. The effective single film is modeled by a 500 nm thick film with refractive

index, )(ireffn , which is given by:

)()()( 1.09.0 irMgO

irBST

ireff nnn , (2.14)

where BSTn and MgOn are the refractive indices of Ba0.7Sr0.3TiO3 and MgO, respectively.

)(irn represents the real (imaginary) part of refractive index. The calculated spectra of



JIM Kwok Lung 55

the multilayered films (solid line) and the effective single film (dashed line) are shown in

Figure 2.10(b). The complex refractive index profile of Ba0.7Sr0.3TiO3 used for the TMM

calculation, which is shown in the inset of Figure 2.10(b), is obtained by the single Tauc-

Lorentz dispersion formula [Gao, 2007]. The refractive index of MgO is assumed to be a

constant of 1.73 with zero extinction coefficient in the TMM calculation.

The Fabry-Perot oscillation can be clearly noticed in the transmission spectrum

of the effective single film (dashed line) while a similar oscillation structure is also

observable in the transmission spectrum of the Ba0.7Sr0.3TiO3/MgO multilayered films

(solid line) shown in Figure 2.10(b). However, a distinct feature between the two spectra

is that the transmission level of every transmission dip in the transmission spectrum of

the effective single film is about the same, while each transmission dip of the

multilayered films is deeper than its smaller wavelength neighbour except for dip (4)

compared to dip (3). The occurrence of a photonic bandgap makes dip (3) extremely

deep.

The measured transmission spectrum, which is shown in Figure 2.10(a), exhibits

very similar features with the calculated transmission spectrum of the multilayered films.

Although there may be a trend of lower transmission for the smaller wavelength in the

measured transmission spectrum, the higher transmission level in dip (4) than that in dip

(3) strongly evidences the existence of a photonic bandgap at wavelength around 460

nm.



JIM Kwok Lung 56

Figure 2.11 shows the shift in the transmission spectrum when a dc voltage of

240 V (corresponding to an electric field of about 12 MV/m) is applied. With the applied

voltage, the photonic bandgap (transmission dip) at ~464 nm shifts slightly (~2 nm) to

longer wavelength. The electro-optic effect could be the origin of the bandgap tunable

phenomenon, as the refractive index of a ferroelectric material changes when it is

subjected to an external electric field. Since Ba0.7Sr0.3TiO3 thin films have good E-O

properties [Wang 2006b; Wang 2007], it is reasonable that the photonic bandgap is

shifted when an electric field is applied to the Ba0.7Sr0.3TiO3/MgO multilayered thin

films. Calculated from Eq. (2.13), a 2 nm shift of the gap-center corresponds to a 0.43%

change in nBST. Therefore, the experimental result suggests that the electric field induced

change in the refractive index of Ba0.7Sr0.3TiO3 is about 0.5 %, which is comparable to

that of (Pb, La)TiO3 thin films [Boudrioua, 1999].



JIM Kwok Lung 57

200 300 400 500 600 700 800 900 1000 11000

10

20

30

40

50

60

70

80

90

100

110

300 400 500 600 700 8000

20

40

60

80

100

Tra

nsm

ittan

ce (%

)

Wavelength ( nm )

Tra

nsm

issi

on

(%)

W avelength (nm)

200 300 400 500 600 700 800 900 1000 11000

10

20

30

40

50

60

70

80

90

100

110

BST/MgO multilayered films

Effective single film

1.6

1.8

2.0

2.2

2.4

2.6

200 400 600 800 1000 1200

0.0

0.2

0.4

0.6

0.8

1.0

Ref

ract

ive

inde

x n

Ext

inct

ion

coef

ficie

nt k

Wavelength (nm)

Tra

nsm

issi

on (

%)

Wavelength (nm)

Figure 2.10 (a) Observed transmission spectrum of the Ba0.7Sr0.3TiO3/MgO

multilayered thin films. The inset shows the transmissions spectrum of a

Ba0.7Sr0.3TiO3 thin film grown on MgO (001) substrate; (b) Calculated

transmission spectra of the Ba0.7Sr0.3TiO3/MgO multilayered thin films and

the corresponding effective single film by TMM. The inset shows the

complex refractive index profile of the Ba0.7Sr0.3TiO3 used for the

calculation.

(a)

(b)

(1)(2)(3)(4)

(1)(2)(3)(4)(5)



JIM Kwok Lung 58

0

20

40

60

80

100

200 300 400 500 600 700 800 900

Wavelength (nm)

Tra

ns

mis

sio

n (

%)

0V

240V

Figure 2.11 Measured transmission spectrum of the Ba0.7Sr0.3TiO3/MgO multilayered

thin films with (dotted curve) and without (solid curve) the application of a

dc voltage of 240 V.



JIM Kwok Lung 59

2.6 Summary

In this Chapter, we investigated the tunability of the photonic bandgap of a

Ba0.7Sr0.3TiO3/MgO 1D PC arising from the electro-optic effect. The relation between

the shift of the photonic bandgap and the change of the refractive index of Ba0.7Sr0.3TiO3

was examined using the plane wave expansion method. It is found that the gap-center

shifts linearly with the change of the refractive index of Ba0.7Sr0.3TiO3. A ferroelectric

one-dimensional photonic crystal consisting of five layers of Ba0.7Sr0.3TiO3 and five

layers of MgO alternately deposited on a MgO substrate was successfully fabricated by

the pulsed laser deposition technique. X-ray diffraction reveals that all the layers were

epitaxially grown on the MgO (001) single crystal substrate. A photonic bandgap at

~464 nm was observed in the transmission spectrum, which agrees well with the

theoretical calculation by the transfer matrix method. It is observed that the photonic

bandgap of the multilayered films could be tuned by an external electric field E. The

photonic bandgap shifts by about 2 nm when the PC is subjected to a dc voltage of 240

V (E ~ 12 MV/m). This shift corresponds to a 0.5% change in the refractive index of

Ba0.7Sr0.3TiO3.

PHOTONIC BANDSTRUCTURE STUDY ON TWO-DIMENSIONAL

CHAPTER 3 BARIUM STRONTIUM TITANATE-BASED PHOTONIC CRYSTALS

JIM Kwok Lung 60

CHAPTER 3

PHOTONIC BANDSTRUCTURE STDUY ON

TWO-DIMENSIONAL BARIUM STRONTIUM

TITANATE-BASED PHOTONIC CRYSTALS

3.1 Introduction

In order to investigate a photonic crystal (PC), it is always effective to first study

its photonic bandstructure (PBS) since the effect of the PC on the electromagnetic (EM)

waves is entirely characterized by the PBS. Consider a PC composed of a Ba0.7Sr0.3TiO3

matrix with air rods, two commonly prepared cross sections of the air rods are square

and circular. In order to decide which cross section has better performance in certain

applications, it is worth to examine how the shape of the air rods would affect the PBSs.

Many compact optical devices such as filters and waveguides have been

proposed in utilizing PCs [Mekis, 1996; Lončar, 2000; Qiu, 2003; Ren, 2006]. In such

applications, it is advantageous for the PC to possess large photonic bandgaps (PBGs).

To optimize the design in such applications, it is efficient to calculate the PBG maps in

order to obtain the configuration for the largest PBGs.



JIM Kwok Lung 61

In this Chapter, the plane wave expansion (PWE) for two-dimensional (2D)

photonic bandstructure calculation is briefly reviewed. Then, we will study the effects of

different shapes of air rods on the PBSs and the PBG maps of the Ba0.7Sr0.3TiO3-based

PCs. After that, the PBG maps of the Ba0.7Sr0.3TiO3-based PCs with different

configurations are discussed.

3.2 Plane wave expansion method for two-dimensional photonic bandstructure

calculation

The PBSs of PCs can be calculated by using the PWE method. The Maxwell’s

equations for EM waves in a system with periodic distribution of dielectric constants can

be simplified as:

Hc

Hr

2

21

, and (3.1a)

Ec

Er

2

21

. (3.1b)

Here, r is the position-dependent dielectric function, H

the magnetic field, E

the

electric field, the angular frequency, and c is the speed of light in vacuum. As the

system is periodic, the dielectric function, the magnetic field and the electric field can be

expanded in terms of plane waves:

g

rgig e

r

1

)(

1

, (3.2a)

g

rgkig eHrH

)()( , (3.2b)



JIM Kwok Lung 62

g

rgkig eErE

)()( , (3.2c)

where k

is the wave vector, g

the reciprocal lattice vector and r

is the position vector.

Throughout this Chapter, the wave vectors and the reciprocal lattice vectors are

measured in unit of 2/a where a is the period of the PC in consideration, and the lengths

(position vectors) are measured in unit of a. In other words, all k-space vectors and

lengths are dimensionless.

Combining Eq. (3.1) and Eq. (3.2), we obtain the following matrix equations:

gg

ggg HHgkgk

21

, and (3.3a)

gg

ggg EEgkgk

21

, (3.3b)

where the Fourier coefficient of the inverse dielectric function can be calculated by the

integral:

Ag rgir

drA

exp

111

, (3.4)

and A is the area of the unit cell. The normalized frequency is give by:

a

c

a

2, (3.5)

where is the wavelength in vacuum. The PBS of the 2D PC (Ω against k) is then

obtained by solving Eq. (3.3) after evaluating the Fourier coefficient of the inverse

dielectric function given in Eq. (3.4). In general, the integral in Eq. (3.4) needs to be

calculated numerically. However, if the shapes of the dielectric components in the unit



JIM Kwok Lung 63

cell are simple enough, we can calculate the integral analytically. We will show two

cases of different air rod shapes for a 2D PC with square lattice in the following. The

first one contains circular air rods, and the second one contains square air rods in the unit

cells.

Circular air rod

We denote the radius and the dielectric constant of the circular air rod by ra and

a , respectively, and the dielectric constant of the background material by b . The

inverse dielectric function is thus given by:

rSr bab

1111

, (3.6)

where rS

is the structure factor which is defined such that:

.0

,1

a

a

rrfor

rrforrS

(3.7)

Substituting Eq. (3.6) and Eq. (3.7) into Eq. (3.4), the Fourier coefficient of the inverse

dielectric function for the circular rod, gc1 , has the form of:

,011

2

,01

1

1,

gforrg

rgJf

gforff

a

a

bac

b

c

a

c

cg

(3.8)

where J1 is the Bessel function of the first kind, and:

2

2

a

rf a

c

, (3.9)



JIM Kwok Lung 64

is the volume fraction of the circular air rod.

Square air rod

The calculation of the Fourier coefficient for the case of square air rod is similar

to that of circular one. In this case, we denote the width and the dielectric constant of the

square air rod by wa and a , respectively, and the dielectric constant of the background

material is denoted by b . The inverse dielectric function is again given by Eq. (3.6)

while rS

is defined in this case as:

.2

0

,2

1

a

a

wrfor

wrfor

rS

(3.10)

Substituting Eq. (3.6) and Eq. (3.10) into Eq. (3.4), the Fourier coefficient of the inverse

dielectric function for the square rod, 1,

sg , has the form of:

,0sinsin11

,01

1,

gforwg

wg

wg

wgf

gforff

ay

ay

ax

ax

bas

b

s

a

s

sg

(3.11)

where gx and gy are the x- and y-components of the reciprocal lattice vector, respectively,

and:

2

2

a

wf a

s , (3.12)

is the volume fraction of the square air rod.



JIM Kwok Lung 65

3.3 Photonic bandgap maps of two-dimensional Ba0.7Sr0.3TiO3-based photonic

crystals with different configurations

In many applications such as filters and waveguides, it is advantageous for the

photonic crystal to possess large photonic bandgaps. It is thus worth to investigate the

photonic bandgap features of a PC in different configurations. Two most commonly

chosen configurations in 2D PC are square lattice and triangular lattice [Villeneuve,

1992; Plihal, 1991]. The high symmetry points in the first Brillouin zone for both lattices

are shown in Figure 3.1. The computation of the photonic bandstructures (PBSs) and

photonic bandgap (PBG) maps were performed using the software BandSOLVETM 3.0

from RSoft Design Group, Inc. which utilizes PWE method for the calculation. The

dielectric constants of the air and the Ba0.7Sr0.3TiO3 at optical frequency were assumed to

be 1.00 and 4.84 ( 2BSTn ), respectively. All PBSs were obtained using 64 x 64 plane

waves. The eigenvalue tolerance was set to be 10-8. The first 60 bands were calculated

for each of the transverse electric (TE) mode and transverse magnetic (TM) mode.

Figure 3.2 shows the PBG maps of 2D air-hole-in-BST type PC (Ba0.7Sr0.3TiO3 matrix

with circular air rods) in square and triangular lattices while Figure 3.3 shows the PBG

maps of 2D BST-rod-in-air type PC (circular Ba0.7Sr0.3TiO3 rods in air) in square and

triangular lattices.

To illustrate how the PBG maps can facilitate the design of photonic crystals,

consider Figure 3.2(d) which shows the PBG map of the 2D air-hole-in-BST type PC



JIM Kwok Lung 66

(Ba0.7Sr0.3TiO3 matrix with circular air rods) in triangular lattice along all the high

symmetry directions Γ-M-K-Γ (Figure 3.1). It can be seen from Figure 3.2(d) that there

(a) (b)

Figure 3.1 High symmetry points in the first Brillouin zones of the (a) square lattice

and the (b) triangular lattice.

is a large TE photonic bandgap at r = 0.4 a. The TE PBG extends from Ω = 0.35 to Ω =

0.50 with the mid-gap as Ωmid-gap = 0.425. If we aim to design a PC pattern which

possesses a TE PBG around λ = 1550 nm, we can calculate the required period of PC by

Ω = a/λ by setting Ω = Ωmid-gap. The required period of PC is thus given by a = Ωmid-gap ·

λ = 0.425 · 1550 nm ≈ 660 nm. The radius of the air hole is given by r = 0.4 a = 264 nm.

The corresponding TE PBG, which can be calculated by λ = a/Ω, extends from λ = 1320

nm (Ω = 0.50) to λ = 1886 nm (Ω = 0.35).

In literature, usually only the PBG maps along all the high symmetry directions

(Γ-X-M-Γ in square lattice and Γ-M-K-Γ in triangular lattice) are considered. However,

we can control the direction of the incident light in some applications. It may be,

therefore, reasonable to relax the all-direction PBG requirement, but focus only on a

Γ

X M

Γ

M K



JIM Kwok Lung 67

certain direction each time. The Γ-X and Γ-M PBG maps for square lattice and the Γ-M

and Γ-K PBG maps for triangular lattice are thus calculated for comparison. It can be

observed from Figure 3.2 and Figure 3.3 that 2D PCs in triangular lattice contain richer

photonic bandgap features in general. This is consistent with previous studies which

show that PCs with a more circular-like Brillouin zone usually contain richer PBG

features [Cassagne, 1996; Wang, 2001]. Also, BST-rod-in-air type PCs contain much

richer PBG features than the air-hole-in-BST type PCs. Complete photonic bandgaps

(Simultaneous TE and TM PBGs along all high symmetry directions in the first

Brillouin zone) are absent in the considered configurations. For square lattice PCs, there

are much more PBGs along the Γ-X direction than the Γ-M direction and significant

simultaneous PBGs can merely be found in the Γ-X direction. For triangular lattice PCs,

more PBGs can be found along the Γ-M direction than the Γ-K direction and significant

simultaneous PBGs can be found in the Γ-M direction only.



JIM Kwok Lung 68

(a) (d)

(b) (e)

(c) (f)

Figure 3.2 Photonic bandgap maps of 2D air-hole-in-BST type PC (Ba0.7Sr0.3TiO3

matrix with circular air rods) in square lattice along (a) Γ-X-M-Γ, (b) Γ-M,

(c) Γ-X; in triangular lattice along (d) Γ-M-K-Γ, (e) Γ-K, (f) Γ-M. The

insets show the configurations of the PCs and their corresponding

Brillouin zones.



JIM Kwok Lung 69

(a) (d)

(b) (e)

(c) (f)

Figure 3.3 Photonic bandgap maps of 2D BST-rod-in-air type PC (circular

Ba0.7Sr0.3TiO3 rods in air) in square lattice along (a) Γ-X-M-Γ, (b) Γ-M, (c)

Γ-X; in triangular lattice along (d) Γ-M-K-Γ, (e) Γ-K, (f) Γ-M. The insets

show the configurations of the PCs and their corresponding Brillouin

zones.



JIM Kwok Lung 70

3.4 Photonic bandstructures and photonic bandgap maps of two-dimensional

Ba0.7Sr0.3TiO3-based photonic crystals with different shapes of air rods

Beside different lattice types, photonic bandstructures and photonic bandgap

maps of 2D air-hole-in-BST type PCs with different shapes (circular and square) of air

rods are also investigated. In order to facilitate the discussion, we define the diameter of

the circular air rod (2ra) and the width of the square air rod (wa) as the characteristic

lengths (l). They will be expressed in unit of the period of the PC, a.

Figure 3.4 shows the photonic bandstructures of the 2D Ba0.7Sr0.3TiO3-based PCs

containing circular and square air rods with characteristic lengths of 0.2a, 0.5a and 0.8a.

It can be seen that the corresponding PBSs of the circular-air-rod PC and the square-air-

rod PC with the same characteristic length are very similar. For l = 0.2a (Figure 3.4(a)

and Figure 3.4(d)), the two corresponding PBSs are almost identical in the whole range

of frequency from Ω = 0.0 to Ω = 1.0. If we focus only on the small frequency range (Ω

< 0.35), the corresponding PBSs are nearly the same even for different characteristic

lengths. By contrast, the discrepancies between the PBSs become increasingly apparent

with larger characteristic lengths (Figure 3.4(c) and Figure 3.4(f)).

The results may not be unexpected, if we consider the wavelengths of the EM

waves inside the 2D PC ( ). For frequencies small than 1.0 (Ω < 1.0), the

corresponding obtained from Eq. (5) are greater than the period of the PC, a. As the

refractive index (n) of Ba0.7Sr0.3TiO3 is about 2.20, the wavelengths of the EM waves



JIM Kwok Lung 71

inside the 2D PC ( n/ ) are still larger than 0.45a. It could be imagined that the EM

waves with Ω < 1.0 ( > 0.45a) are not sensitive to the slight difference in shapes of

the air rods, if the characteristic length of the air rods is much less than . It is therefore

not surprising that the PBSs between the circular- and square-air-rod PCs would appear

similar when the characteristic length of the air rods is 0.2a only. As the characteristic

length becomes comparable (l = 0.5a) or even larger (l = 0.8a) than the wavelengths of

EM waves inside the 2D PC ( ), the discrepancies between the photonic bandstructures

of the circular- and square-air-rod PCs become apparent.

A more concrete interpretation of the results can be made by examining the

Fourier coefficients of the inverse dielectric function for the two cases of circular air

rods and the square air rods, which are given by Eq. (3.8) and Eq. (3.11) respectively.

The two equations, containing all the geometric information of the PCs, are the only

differences in the photonic bandstructure calculation for the two cases. If the reciprocal

lattice vectors, g

, or the characteristic length of the air rod, l, are small ( 1lg

), the

sinc function (sin x/x) for the square-air-rod case and the Bessel function of the first kind

for the circular-air-rod case can be approximated as the following:

21

xxJ , for small x, (3.13)

1sin

x

x, for small x. (3.14)

Putting Eq. (3.13) into Eq. (3.8) and Eq. (3.14) into Eq. (3.11), the Fourier coefficients

of the inverse dielectric function for the circular-air-rod and the square-air-rod have the

forms of:



JIM Kwok Lung 72

,011

,01

1

gforf

gforff

g

bac

b

c

a

c

c

(3.15)

,011

,01

1

gforf

gforff

g

bas

b

s

a

s

s

(3.16)

in cases of small g

vectors or small characteristic length l of air rods. Under such

approximation, the Fourier coefficients of the inverse dielectric function for the circular-

air-rod and the square-air-rod attain identical forms. The different definitions of fs and fc,

however, should be carefully noted, as will be discussed later. Since the photonic

bandstructure depends highly on the Fourier coefficients of the inverse dielectric

function, the PBSs for the circular-air-rod PC and the square-air-rod PC would therefore

be very similar in situations whenever the 1lg

approximation is valid. It is known

that small g

vectors correspond to low frequencies. Thus, for cases of low frequencies

or small characteristic length of air rods, the photonic bandstructures of the circular-air-

rod PC and the square-air-rod PC would be very similar.

Nevertheless, the consistency of the photonic bandstructures for the two different

geometries at large characteristic lengths of air rods is not as good as that for the small

ones. We have calculated the photonic bandgap maps of the 2D Ba0.7Sr0.3TiO3-based

PCs with circular air rods and square air rods along the symmetry directions Γ-X-M-Γ,

Γ-M and Γ-X to further illustrate this point. As shown in Figure 3.5, there are significant



JIM Kwok Lung 73

discrepancies in the corresponding PBG maps for large characteristic lengths of air rod

even for small frequencies. If we simply compare the PBG maps at la =1.0 [i.e. ra = 0.5

(2ra = 1.0) and wa = 1.0], the PBG features are actually quite different. It seems to

contradict to the previous conclusion that the PBSs should be the same if the Fourier

coefficients of the inverse dielectric function are the same. In fact, Eq. (3.15) and Eq.

(3.16) are not exactly the same because the volume fractions for the circular air rod [Eq.

(3.9)] and the square air rod [Eq. (3.12)] are not identical indeed. In order to acquire

equal volume fractions, the diameter of the circular air rod should correlate with 0.89

times of the width of the square air rod [e.g. ra = 0.5 (2ra = 1) should correlate with wa =

0.89]. We have highlighted the positions of wa = 0.89 with dark green lines in the

bandgap maps for visualization. It can be observed that the corresponding PBG maps

show better consistency when we take into account the volume fraction correction.



JIM Kwok Lung 74

(a) (d)

(b) (e)

(c) (f)

Figure 3.4 Photonic bandstructures of square lattice type 2D Ba0.7Sr0.3TiO3-based PC

with circular air rods, (a) 2r = 0.2a, (b) 2r = 0.5a, (c) 2r = 0.8a; with

square air rods, (d) w = 0.2a, (e) w = 0.5a, (f) w = 0.8a.



JIM Kwok Lung 75

(a) (d)

(b) (e)

(c) (f)

Figure 3.5 Photonic bandgap maps along the high symmetry direction of square

lattice type 2D Ba0.7Sr0.3TiO3-based PC with circular air rods, (a) Γ-X-M-Γ,

(b) Γ-M, (c) Γ-X; with square air rods, (d) Γ-X-M-Γ, (e) Γ-M, (f) Γ-X. The

dark green lines indicate the position of w = 0.89a for visualization.



JIM Kwok Lung 76

3.5 Summary

In this Chapter, we have studied the photonic bandgap maps of two types of 2D

Ba0.7Sr0.3TiO3-based PC, namely, the air-hole-in-BST type and the BST-rod-in-air type

both in square lattice and in triangular lattice. The photonic bandstructures and photonic

bandgap maps of the PC were computed using the plane wave expansion method. It is

found that PCs in triangular lattice contain richer PBG feature in general. Photonic

bandgap features along different symmetry directions have also been compared.

We have also compared the photonic bandstructures and the photonic bandgap

maps of the square lattice type 2D Ba0.7Sr0.3TiO3-based PC with circular air rods and

square air rods. It was found that there are two situations at which the photonic

bandstructures and the photonic bandgap maps of the circular air rod PCs and square air

rod PCs would be nearly identical: (1) if the characteristic length of the air rod is small (l

< 0.45a), and (2) if the frequency range is low enough (Ω < 0.35). Furthermore, a better

correlation in the photonic bandstructures and the photonic bandgap maps between the

circular and square air rod 2D PC can be obtained if the volume fraction correction, i.e.

fc = fs wa = 0.89 x (2ra), has been taken into account.

PHOTONIC CRYSTAL CAVITY EMBEDDED

CHAPTER 4 BARIUM STRONTIUM TITANATE RIB WAVEGUIDES

JIM Kwok Lung 77

CHAPTER 4


BARIUM STRONTIUM TITANATE

RIB WAVEGUIDES

4.1 Introduction

Photonic crystal (PC) devices have received much attention over the past decade

for their great potentials of creating miniaturised photonic components essential for

compact photonic integrated circuits. One of the crucial PC devices is the photonic

crystal filter which is very promising in wavelength division multiplexing (WDM)

applications [Hu, 2007; Jugessur, 2004; Pustai, 2002]. The realization of good

nanometric optical structures on ceramic materials continues to be a difficult task due to

its well-known resistance towards standard machining techniques like wet etching or

even dry etching by reactive ion etching (RIE). Focused ion beam (FIB) etching is an

efficient technique to fabricate nanosized holes with high aspect ratio [Roussey, 2005].

Since the etching time for FIB greatly depends on the area of the PC pattern, it is

advantageous for the required area of PC patterns to be as small as possible. Patterning

PC cavities on rib waveguides may alleviate the problem. Therefore, this Chapter aims

to demonstrate the feasibility of developing PC cavity embedded Ba0.7Sr0.3TiO3 rib

waveguides to function as a tunable optical filter at the telecommunication wavelength,

1550 nm.



JIM Kwok Lung 78

In this Chapter, we first discuss the characterization of BST thin films both

structurally by XRD and optically by prism coupling technique. Then, the required

dimension of Ba0.7Sr0.3TiO3 rib waveguides supporting single mode wave propagation is

examined after the introduction of dielectric waveguide theory. Next, fabrication and

characterization of Ba0.7Sr0.3TiO3 rib waveguides are discussed. Finally, the design and

fabrication of photonic crystal cavity embedded Ba0.7Sr0.3TiO3 rib waveguides are

illustrated.

4.2 Characterization of Ba0.7Sr0.3TiO3 thin films

4.2.1 Structural characterization of Ba0.7Sr0.3TiO3 thin films

The θ/2θ x-ray diffraction (XRD) pattern of the Ba0.7Sr0.3TiO3 thin film grown on

a MgO (001) substrate is shown in Figure 4.1. Since only (00l) peaks of the

Ba0.7Sr0.3TiO3 appear in the XRD patterns, it indicates a pure perovskite phase in the

Ba0.7Sr0.3TiO3 thin film. The in-plane alignment of the Ba0.7Sr0.3TiO3 thin films with

respect to the major axes of the (001) substrates was confirmed by the XRD off-axis φ

scan of the Ba0.7Sr0.3TiO3 (202) and MgO (202) reflections, as shown in the inset of

Figure 4.1, confirming an epitaxial growth of the Ba0.7Sr0.3TiO3 film.



JIM Kwok Lung 79

Figure 4.1 XRD θ/2θ scan of a Ba0.7Sr0.3TiO3 thin film grown on a MgO (001)

substrate. The inset shows the φ scan of the Ba0.7Sr0.3TiO3 (202) and

substrate (202) reflections.

4.2.2 Optical characterization of Ba0.7Sr0.3TiO3 thin films

The photonic bandstructures, and thus the optical properties, of photonic crystals

(PCs) depend merely on the geometric arrangement and the refractive indices of the

constituent materials. It is therefore very important to first obtain the refractive index

information of the constituent materials before we can design the appropriate

configuration of the PCs.

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E+06

20 25 30 35 40 45 50

2 Theta (degree)

Inte

ns

ity

(a.

u.)

BST (001)

MgO (002)

BST (002)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

-120 -90 -60 -30 0 30 60 90 120 150 180 210 240

Phi (degree)

Inte

ns

ity

(a.

u.)



JIM Kwok Lung 80

In order to measure the refractive indices of the Ba0.7Sr0.3TiO3 thin films grown

on MgO (001) substrates, the prism coupling technique was employed. A prominent

advantage of this method is that no further processing on the thin film is needed since

the film acts as a planar waveguide during the measurement.

Prism coupling is an advanced optical waveguiding technique for measuring both

the thickness and the refractive index/birefringence of dielectric films rapidly and

accurately [Tien, 1969; Ulrich, 1973]. For many thin film and optical waveguide

applications, the prism coupling technique offers unique advantages over conventional

measurements based on ellipsometry or spectrophotometry. One prominent advantage of

the prism coupling measurement is that the thickness of the films needs not be known in

advance. Moreover, a resolution in refractive index of ±0.0005 can be achieved, which is

an order or magnitude better than other techniques. For simultaneous measurements of

the thickness and the refractive index using the prism coupling method, the film should

be sufficiently thick to support two or more guided modes.

The thickness of the film to be studied is the single most important factor

determining whether ellipsometry or prism coupling is the preferred technique. In

general, ellipsometry is the clear choice for investigating thin films with thicknesses

smaller than a few hundred nanometers. For studying moderate-to-thick films, prism

coupling technique is usually superior. Consequently, ellipsometry and prism coupling

can be viewed as complementary techniques.



JIM Kwok Lung 81

4.2.2.1 Basic principles of prism coupling technique

The prism coupling measurements were carried out using a Metricon Model 2010

prism coupler equipped with a 45˚-45˚-90˚ rutile (TiO2) prism (Metricon corporation,

Pennington, USA). The schematic diagram of the instrument is shown in Figure 4.2.

Figure 4.2 Schematic diagram of the Metricon model 2010 prism coupler system.

Two laser systems are installed to generate the light beam. The first one is a He-

Ne laser with a wavelength of 632.8 nm, which is the commonly used wavelength for

optical characterizations of ferroelectric thin films. The other laser system is a laser

diode with a wavelength of 1550 nm, since the optical waveguide may have potential use

in the near infrared (IR) region for telecommunication applications. To conduct a

α

αp

σ

np

Waveguide, n1

Substrate, n2

Air gap, n3

Laser beam

Rutile prism

Photodector



JIM Kwok Lung 82

measurement, the film was brought into contact with the base of the rutile prism by

means of a pneumatically-operated coupling head, creating a small air gap between the

film and the prism. The laser beam struck the base of the prism and was totally reflected

at the prism base onto a photodetector. At certain discrete values of incident angle,

known as mode angles (m), photons can tunnel through the air gap into a guided optical

propagation mode of the film, causing a sharp dip in the intensity of light reaching the

detector as illustrated in Figure 4.3. The mode angle m is given by:

pp

pm nn

n sinsinsinsinsin 1

1

, (4.1)

where np is the refractive index of the prism, n1 the refractive index of the film, σ the

incident angle of the laser beam at the prism-film interface (internal incident angle), α

the incident angle of the laser beam at the air-prism interface (external incident angle)

and αp is the base angle of the prism (αp = 45˚ in this case). Figure 4.3 shows the

transverse electric (TE) mode measurement only. The Metricon 2010 prism coupler

system actually supports two polarizations, namely the transverse electric (TE)

polarization, and the transverse magnetic (TM) polarization (see inset of Figure 4.3). The

electric field of the laser beam is vibrating parallel to the film (in-plane direction) for TE

polarization, while the electric field of the laser beam is vibrating perpendicular to the

film (out-of-plane direction) for TM polarization. Therefore, a TE mode measurement is

used for measuring the refractive index along the in-plane direction, and a TM mode

measurement is used for measuring the refractive index along the out-of-plane direction.



JIM Kwok Lung 83

3436384042444648505254Internal angle of incidence (degree)

Ref

lect

ed in

ten

sity

(a

. u.)

TE0

knee

TE1

Figure 4.3 Intensity of reflected light against internal incident angle for transverse

electric (TE) modes. The knee corresponds to the substrate mode. The

inset shows the vibration directions of the electric field for the TE and the

TM modes.

Refractive index and thickness of the film are the two parameters that can be

measured using the prism coupling technique. For a given substrate type, the mode angle

of a given order of mode is related to the refractive index and thickness of the film by the

“transverse resonance condition” [Syms, 1992] which is expressed as:

mdkn m 222cos2 13122

1 , (4.2)

TETM



JIM Kwok Lung 84

where d is the thinkness of the film, n1 the refractive index of of the film, 2

k the

free-space wavenumber with λ the free-space wavelength of the incident light, m = 0, 1,

2, 3 …, the order of the mode, and 12 and 13 are the Fresnel phase shift at the film-

substrate and film-air interfaces, respectively. The phase shift terms 12 and 13 can be

evaluated by considering the total internal reflections at the film-substrate and film-air

interfaces and are given by:

m

m

i

n

nn

n

n

cos

sinarctan

1

22

21

2

2

112

, and (4.3a)

m

m

i

n

nn

n

n

cos

sinarctan

1

23

21

2

3

113

, (4.3b)

where n1, n2, n3 are refractive indices of the film, the substrate and the air gap,

respectively, i = 0 for TE modes and i = 1 for TM modes. Eq. (4.2) is transcendental

which needs to be solved numerically. Two measured mode angles, which give two Eq.

(4.2) corresponding to two orders of mode, are sufficient to determine simultaneously

the refractive index n1 and thickness d of the film. If more than two modes are observed,

the problem is overspecified. It is possible to calculate the independent refractive index

and thickness of the film from each pair of modes and compare them to obtain the mean

and standard deviation of the refractive index and thickness from all the pairs of modes

for consistency check.

After the reflected intensity spectrum is measured, the mode angles can be

located using the software incorporated in the prism coupler system which then



JIM Kwok Lung 85

calculates the refractive index and thickness (with mean and standard deviation if more

than two mode angles are observed) of the film.

4.2.2.2 Guided modes in Ba0.7Sr0.3TiO3 thin films

Figure 4.4 shows the guided mode spectra of a 550 nm thick Ba0.7Sr0.3TiO3 thin

film grown on MgO (001) substrate in TE polarization. There are three TE modes for λ =

632.8 nm, while there is only one for λ = 1550 nm. The reflectivity dips of each guided

mode are very sharp, indicating that good confinement of lightwave is achieved and the

film is potentially useful for waveguide device applications. At the end of the spectra, a

sharp fall of the reflected intensity (knee) is observed, which is associated with the

substrate mode.

Using the mode equations (Eq. (4.2) and Eq. (4.3)) and the observed angular

positions of the TE modes at λ = 632.8 nm (Figure 4.4(a)), the refractive indices and the

thickness can be computed. The calculated film thickness and ordinary refractive index

(n0) are 547 ± 4 nm and 2.3107 ± 0.0006. Using the thickness obtained, the refractive

index at λ = 1550 nm can also be calculated even though only one guided TE mode is

measured. The calculated refractive index at at λ = 1550 nm is 2.2209. Since there is

only one mode being observed, no standard deviation can be obtained.



JIM Kwok Lung 86

20

30

40

50

60

70

80

40 42 44 46 48 50 52 54 56

Internal angle of incidence (degree)

Ref

lect

ed in

ten

sity

(a

. u.)

(a)

10

20

30

40

50

60

40 42 44 46 48 50 52 54 56

Internal angle of incidence (degree)

Ref

lect

ed in

ten

sity

(a

. u.)

(b)

Figure 4.4 Guided mode spectra excited at (a) λ = 632.8 nm and (b) λ = 1550 nm in

the TE polarization for a ~550 nm thick Ba0.7Sr0.3TiO3 film deposited on

MgO (001) substrate.



JIM Kwok Lung 87

4.3 Ba0.7Sr0.3TiO3 thin film optical rib waveguides

4.3.1 Light propagation in slab waveguides

4.3.1.1 Basic concepts of slab waveguides

Dielectric slabs are the simplest optical waveguides. By studying the properties

of the slab waveguides, an insight into the waveguide properties of more complicated

dielectric waveguides can be gained. Consider a step-index waveguide (Figure 4.5), in

which the refractive index changes abruptly along the thickness direction at the surface

of the core, it has finite dimension in the thickness direction but infinite dimension in the

length and width directions. The core region of the waveguide, with a refractive index

n1, is deposited on a substrate with refractive index n2. The refractive index of the

medium above the core (cladding) is n3. The refractive index n3 may be unity if the

region above the core is air, or it may have some other value if the core is surrounded by

dielectric materials on both sides. For true mode guidance to occur, it is necessary that n1

is larger than n2 and n3.

Without loss of generality, we assume that n1 > n2 ≥ n3. A slab waveguide is

called symmetric if n2 = n3 and asymmetric if n2 ≠ n3. The modes of symmetric slab

waveguides are simpler than those of the asymmetric ones because they can be

expressed either as even or odd field distributions. Moreover, the lowest-order mode of a

symmetric slab waveguide does not have a cutoff frequency, which implies such mode



JIM Kwok Lung 88

can, in principle, propagate at arbitrarily low frequencies while all modes of an

asymmetric slab waveguide become cutoff if the frequency of operation is sufficiently

low. The guided lightwaves are confined in the core region under total internal reflection

at the dielectric interfaces. According to the Snell’s law, the critical angles, 13,C and

12,C , for total internal reflection at the upper and the lower interfaces are given by:

1

313,sin

n

nC , and (4.4a)

1

212,sin

n

nC . (4.4b)

Figure 4.5 A step-index slab waveguide of thickness d with a guided mode.

Most commonly the cladding layer is air, which leads to n2 > n3 and 12,C > 13,C .

When 12,C < < 90˚, the lightwave is confined in the core layer by total internal

n1kn1kx

n1kz = β

d

n3

n1

n2

Cladding

Substrate

Core

x

y

z

x = 0

x = -d



JIM Kwok Lung 89

reflection at both interfaces and it propagates along a zig-zag path as shown in Figure

4.5. This is called a guided mode. For a slab waveguide, the plane wave propagation

constant in the wave-normal direction is kn1, where 2

k and is the wavelength of

light in free space. The propagation constant along the z-direction is obtained by:

sin11 knnkz . (4.5)

The speed of the lightwave parallel to the waveguide is given by:

/0 kc , (4.6)

where 0c is the speed of light in free space. Another frequently used quantity is the

effective index of refraction of a slab waveguide Nm, which is defined as the ratio of the

speed of light in free space to the speed of the lightwave parallel to the waveguide:

k

cNm

0 . (4.7)

4.3.1.2 Guided modes of slab waveguides

An optical mode is a discrete solution of Maxwell’s equations which satisfies all

the boundary conditions and remains the same spatial distribution during propagation.

Maxwell’s equations for plane waves in an isotropic, lossless dielectric medium is given

by [Agrawal, 2004]:

HiE

0 , and (4.8)

EniH

20 , (4.9)



JIM Kwok Lung 90

where H

and E

are the magnetic and electric field vectors. 0 and 0 are the dielectric

permittivity and magnetic permeability of free space, respectively. 02 c

is the

angular frequency and 00

0

1

c is the light speed in free space. We now use the

geometry of slab waveguides to simplify the two curl equations. Assuming that the x-

axis is normal to the waveguide plane and that the waveguide is infinitely wide along the

y-axis, we note that both E

and H

are independent of y. Moreover, both of them vary

with z as ziexp . Then the electromagnetic fields vary as:

ztieyxEE ,

, and (4.10)

ztieyxHH ,

, (4.11)

where E and H are electric and magnetic fields in the plane perpendicular to the

propagation direction, respectively. We can thus remove the y and z derivatives from Eq.

(4.8) and Eq. (4.9) using:

0

y

E, (4.12a)

0

y

H, (4.12b)

Eiz

E

, and (4.12c)

Hiz

H

. (4.12d)



JIM Kwok Lung 91

The resulting set of six equations, when written in Cartesian coordinates, has two distinct

sets of linearly polarized solutions, known as the transverse electric (TE) and transverse

magnetic (TM) modes, depending on whether we choose Ez = 0 or Hz = 0.

In the case of TE modes, Ex = Ez = 0, while Ey satisfies:

02222

2

y

y Enkx

E . (4.13)

The magnetic field H

is related to yE as:

x

EiEHHHH y

yzyx00

,0,,,

. (4.14)

In the case of TM modes, 0 zx HH , while yH satisfies:

02222

2

y

y Hnkx

H . (4.15)

The electric field E

is related to yH as:

x

H

n

iH

nEEEE y

yzyx 20

20

,0,,,

. (4.16)

The two orthogonal TE and TM modes are distinguished in order to discuss the

dispersion characteristics of the guided modes. Here we consider the TE mode first. The

same analysis can be made for TM mode. Assuming that the core layer has a thickness d,

Eq. (4.13) can be solved within each layer using n = n1, n2, or n3 for the three layers

shown in Figure 4.5. Since the refractive index is constant in each layer, the general

solution can be written in terms of sinusoidal and exponential functions as:

dx

x

y

edBdA

xBxA

Ae

E

sincos

sincos

dxfor

dxfor

xfor

0

0

),(

),(

),(

substrate

core

cladding

(4.17)



JIM Kwok Lung 92

with the abbreviations:

11222

1 cos knkn , (4.18)

2222

21

222

2 knnkn , and (4.19)

2223

21

223

2 knnkn (4.20)

The constants A and B can be determined from the boundary conditions at the two

interfaces which require the tangential components of E

and H

being continuous

across them. In the case of TE modes, these boundary conditions are satisfied if yE and

zH are continuous at x = 0 and x = -d, provided:

0 BA , and (4.21)

0sincoscossin BddAdd . (4.22)

This homogeneous system of equations has a solution only if its determinant vanishes,

i.e.:

0cossinsincos dddd . (4.23)

This eigenvalue equation for TE modes can be rewritten in a simpler form as:

2

tan d . (4.24)

Similarly, the eigenvalue equation for TM modes can be derived as:

4

122

32

2

22

23

21tan

nnn

nnnd

. (4.25)



JIM Kwok Lung 93

4.3.1.3 Cutoffs of asymmetric slab waveguides

In the case of an asymmetric slab with 32 nn , cutoff frequencies exist in the

waveguide. The eigenvalue equations, Eq. (4.24) and Eq. (4.25), are solved in order to

determine the cutoff frequencies for the TE and TM modes supported by the waveguide.

Some information can be obtained about the solutions of the TE mode eigenvalue

equation, Eq. (4.24), by considering Figure 4.6. The solid lines in the figure are the

branches of the tangent function of d . The dashed lines show the function dF that

represents the right hand side of Eq. (4.25). From Eq. (4.18) to Eq. (4.20), we obtain:

ddd

ddkddF

2

2223

21

2222

21

2

2223

21

2222

21

dkdnndkdnnd

dkdnndkdnnd

. (4.26)

Figure 4.6 was drawn for 1122

21 kdnn , and 242

32

1 kdnn . The pole in the

dF curve occurs at the point where the denominator of Eq. (4.26) vanishes. The

dF curve ends at the point:

dkdnn 22

21 , (4.27)

since one of the square root expression in Eq. (4.26) becomes imaginary as d exceeds

the value given by Eq. (4.27). The d coordinates of intersections of the solid and

dashed lines give the solutions of the eigenvalue Eq. (4.24). Each solution gives one TE

mode of the slab waveguide. As shown in Figure 4.6, four guided modes were obtained

for the conditions that were used to draw the figure.



JIM Kwok Lung 94

Figure 4.6 Graphical solution of TE mode eigenvalue equation (Eq. (4.24)). The

intersections of the solid and dashed lines correspond to the solutions.

We define a parameter V that combines the difference of the squares of the

refractive indices of the core and substrate with information about the operating

wavelength and the thickness of the core as:

kdnnV 22

21 . (4.28)

As the value of V decreases, the endpoint of the dashed lines moves to the left, so that it

crosses fewer braches of the tangent function. If V decreases, the number of guided

modes is thus reduced. Hence, the total number of modes supported by a waveguide

depends on the thickness d, the refractive indices (n1, n2, and n3) and wavelength of



JIM Kwok Lung 95

the propagation light. In case of sufficiently thin core, low frequency, or small refractive

index difference, no guided mode can exist.

The cutoff value of V, Vc, for each guided TE mode can be calculated from Eq.

(4.24) where the cutoff points occur when 0 . We thus can obtain Vc from Eq. (4.18),

Eq. (4.20) and Eq. (4.24):

mnn

nnTEVc

22

21

23

22arctan , (4.29)

where the integer m is the mode number with m = 0, 1, 2, 3…. Similarly, the cutoff value

of cV for the TM mode is,

mnn

nn

n

nTMVc

22

21

23

22

23

21arctan . (4.30)

A waveguide will support a single TE and a single TM mode when its thickness

is chosen such that10

mcmc VVV . This is the single-mode condition for slab

waveguides. A single propagation mode is generally required in a practical waveguide

device since this mode possesses the lowest optical loss. The field distribution for each

TE mode is completely specified in terms of xEy as given in Eq. (4.17). Figure 4.7

shows, as an example, the mode profiles for the first four TEm modes (m = 0, 1, 2 and 3)

for an asymmetric slab waveguide. The value of m denotes the number of nodes within

the core layer at which the field amplitude vanishes. For the fundamental modes, TE0

and TM0, there are no nodes within the core.



JIM Kwok Lung 96

Figure 4.7 Field profiles associated with the first four TE modes of an asymmetric

slab waveguide [Hunsperger, 2002].

4.3.2 Design of Ba0.7Sr0.3TiO3 rib waveguides using the effective index method

The effective index method (EIM), proposed by Knox and Toulios [Knox, 1970],

is one of the most popular methods for the analysis of rib waveguides in millimeter-

wave and visible lightwave integrated circuits. It is, therefore, employed to obtain the

appropriate geometry of the Ba0.7Sr0.3TiO3 rib waveguides in this study.

In the EIM, the “transverse resonance condition”, Eq. (4.2), is modified by using

the parameters defined in Eq. (4.18) to Eq. (4.20) and is given by:

md 1312 . (4.31)

The phase shift terms in Eq. (4.3) are modified to:

i

n

n2

2

112 arctan , and (4.32a)



JIM Kwok Lung 97

i

n

n2

3

113 arctan , (4.32b)

where m = 0, 1, 2 , 3 …, i = 0 for TE mode and i = 1 for TM mode. By substituting Eq.

(4.7) into Eq. (4.18) to Eq. (4.20), , , and take the forms as:

kNn m 221 , (4.33)

knN m 22

2 , and (4.34)

knN m 23

2 . (4.35)

Combining Eq. (4.31) to Eq. (4.35), one can obtain the “transverse resonance condition”

in terms of the effective refractive index Nm:

imdnnnNN m ,,,,,, 321 . (4.36)

Knox and Toulios [Knox, 1970] proposed to consider a rib waveguide, which is shown

in Figure 4.8(a), as consisting of three regions: the rib region in the center, and the two

regions on the left and the right of the rib. The rib region is then regarded as a slab

waveguide of thickness d while the two regions on the left and the right as a slab

waveguide of thickness d-h (Figure 4.8(b)). Applying Eq. (4.31) to Eq. (4.35), the

effective index of the rib region, Neff1, and the effective index of the region on the left

and the right hand sides, Neff2 can be evaluated. Finally, the whole system can be

considered as a symmetric waveguide with a “core” layer having an effective index Neff1

and thickness of W, and “substrate” and “cladding” layers with effective index Neff2

(Figure 4.8(c)). The effective index of the whole system can then be calculated.



JIM Kwok Lung 98

Figure 4.8 The effective index method on a rib waveguide. (a) The original rib

waveguide. (b) Transform the original rib waveguide into three slab

waveguides with effective index Neff1 and Neff2. (c) Transform the system

into a symmetric slab waveguide to determine the effective index Neff of

the whole structure.

W

h

d

n3

n1

n2

W

Neff1 Neff2Neff2

Neff

(a)

(b)

(c)

n1

n2

Neff2 Neff1 Neff2

d

n3

d-h d-h



JIM Kwok Lung 99

In order to determine the appropriate height of the rib for the Ba0.7Sr0.3TiO3 rib

waveguides to maintain single mode wave propagation, the effective index Nm as a

function of d at λ = 1550 nm is plotted and the result is shown in Figure 4.9. It can be

seen from Figure 4.9 that only the first TE mode (TE0) is supported if the waveguide has

a thickness d between 150 nm and 700 nm. To develop the Ba0.7Sr0.3TiO3 rib waveguides

used at λ = 1550 nm, the thickness d of the Ba0.7Sr0.3TiO3 films is chosen to be 550 nm.

The corresponding effective index Nm of this chosen d value, which is in fact Neff1, can

be obtained from Figure 4.9 to be 2.015.

1.65

1.70

1.75

1.80

1.85

1.90

1.95

2.00

2.05

2.10

2.15

2.20

2.25

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Thickness (mm)

Eff

ecti

ve in

de

x N

m

TE0

TE1

nBST

nMgO

Figure 4.9 Effective index Nm as function of thickness for a Ba0.7Sr0.3TiO3/MgO slab

waveguide at a wavelength of λ = 1550 nm. The refractive index of the

film is 2.20.



JIM Kwok Lung 100

To determine the appropriate width of rib for maintaining single mode wave

propagation, we need to obtain Neff1 and Neff2, which can actually be read from Figure

4.9 directly. As an example, take h = 50 nm (with the chosen d = 550 nm), d-h = 500 nm.

Neff2 is the Nm value corresponds to the thickness of 500 nm which gives 1.99. With the

values of Neff1 and Neff2 known, the effective index of the rib waveguide as function of

rib width W can be obtained by applying Eq. (4.31) to Eq. (4.35) again. Similar

calculations can be performed for other values of h and the results are shown in Figure

4.10.

1.96

1.97

1.98

1.99

2.00

2.01

2.02

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Rib width W (mm)

Eff

ecti

ve In

dex

Nm

a

bc

a

b

c

TE00TE01

a: h = 50 nmb: h = 60 nmc: h = 70 nmd: h = 90 nm

dd

Figure 4.10 Effective index Nm versus rib width W for various rib height h of a

Ba0.7Sr0.3TiO3/MgO rib waveguide at a wavelength of λ = 1550 nm.

Thickness of the film d is 550 nm.



JIM Kwok Lung 101

It can be observed from Figure 4.10 that, for h = 60 nm, single mode propagation

can be maintained if W is smaller than 2.6 μm. Practically speaking, it is preferable to

have a larger h for the rib waveguide since it is very difficult to distinguish whether the

light is guided inside the rib waveguide or just in the planar waveguide next to it during

a waveguide transmission measurement via end-fire coupling (will be discussed in

Section 4.3.4) if h is of several tenths nanometers only. However, the required rib width

for single mode propagation becomes smaller as h increases and needs to be smaller than

2.0 μm, which is approaching the ordinary photolithographic limit of 1.0 μm, for h = 90

nm. Therefore, considering the difficulties may be encountered during the fabrication

process and the measurement stage, h and W are chosen to be 60 nm and 2.5 μm,

respectively. The corresponding effective index Nm of the rib waveguide, which can be

read from Figure 4.10, is 2.005.

4.3.3 Fabrication of Ba0.7Sr0.3TiO3 rib waveguides

Photolithographic patterning followed by dry etching was employed for the

fabrication of Ba0.7Sr0.3TiO3 rib waveguides from the Ba0.7Sr0.3TiO3 film on MgO

substrate structure. The fabrication flow chart is shown in Figure 4.11. To remove the

particulates on the film surface, the Ba0.7Sr0.3TiO3 thin film was immersed in an

ultrasonic bath of acetone and ethanol for 15 min consecutively and then dried in a

stream of compressed air. After cleaning, a 100-nm thick chromium (Cr) film was

deposited on the Ba0.7Sr0.3TiO3 film by RF magnetron sputtering to serve as the

sacrificial etching mask layer. A positive photoresist (AZ 5214) was then deposited on



JIM Kwok Lung 102

top of the Cr layer by spin coating at a spinning speed of 4000 rpm for 40 s, resulting in

a photoresist layer of 1.5 μm thick. After baking at 100 ˚C for 1 min, the photoresist

layer was exposed under a mask to high intensity ultraviolet light for 15 s in a Model

800 MBA double side mask aligner (OAI). The exposed photoresist was then immersed

in a developer (AZ300 Mif) for 30 s. Using an etching solution of Ce(NO3)4 in aceric

acid, the Cr layer without the protection of photoresist was removed. An O2 plasma

etching was then carried out for 10 min in order to strip off the photoresist residue,

leaving the bare waveguide patterns of the Cr layer on the film surface.

Figure 4.11 Fabrication flow chart of thin film rib waveguides.

BST

MgO

CrBST

MgO

Cr PR

BST

MgO

Mask

UV light

Cr PR

BST

MgO

CrPR

BST

MgO

CrBST

MgO

Cr

MgO

BST

MgO

BSTFilm deposition (PLD)

Sputtering ofCr layer

Spin coating of photoresist (PR)

UV exposure of photoresist (PR)

Developing of photoresist (PR)

Wet etching Cr followed byO2 plasma

ICP-RIE of BST

Removal of Cr



JIM Kwok Lung 103

In order to achieve a smooth etched surface, reactive ion etching (RIE) process

with inductive coupled plasma (ICP) source was carried out in a ICP-98 system

(Microelectronics R&D Center, Chinese Academy of Sciences) to etch away the

unprotected region of the film for the fabrication of the desired rib pattern. Due to its

high degree of anisotropy, RIE is a better choice for patterning features in the submicron

range compared to the isotropic wet chemical etching process [Schneider, 1998]. In the

dry etching process, a gas mixture of CF4 and Ar in the ratio of 1:4 was used as the

etchant gas. A self bias power of 150 W and a RF power of 400 W were used. The gas

pressure was kept at 30 mTorr. An etching rate of 6 nm/min was achieved using the

above conditions. The 550-nm thick Ba0.7Sr0.3TiO3 film was etched for 10 min which

resulted in a 60 nm etching depth with a rib width of ~2.5 μm as confirmed by the

atomic force microscopic measurement (Figure 4.12). Finally, the input and output

endfaces of the waveguide were cleaved for light coupling measurement.

Figure 4.12 Atomic force micrograph of the Ba0.7Sr0.3TiO3 rib waveguide.



JIM Kwok Lung 104

4.3.4 Optical characterization of Ba0.7Sr0.3TiO3 rib waveguides

The transmission spectrum of the Ba0.7Sr0.3TiO3 film/MgO rib waveguide was

measured (at the Optoelectronic Research Centre, Department of Electronic Engineering

of the City University of Hong Kong) using the end-fire coupling technique. The

schematic diagram of the end-fire coupling method is shown in Figure 4.13. The light

source used was a TE-polarized laser diode (Amonic ALS-15CL, C+L band ASE

broadband source) which can produce a light spectrum of wavelength from 1500 nm to

1620 nm. The laser beam was butt-coupled to the cleaved input endface of the

waveguide by a polarization-maintaining single mode optical fiber (diameter ~ 8 μm).

The transmitted light at the output endface of the waveguide was measured using an

infrared charge-coupled device (CCD) camera after the light beam had passed through a

micro-objective lens. Since the size of the waveguide is very small, the positions of the

fibers, the micro-objective lens and the waveguide were carefully adjusted using three

xyz micron-positioners mounted on an optical vibration isolator. The optimum alignment

between the input fiber and the waveguide was accomplished by maximizing the

intensity of the output light spot (Figure 4.14). After the alignment between the input

fiber and the waveguide, the CCD camera was replaced by another optical fiber which

was connected to an optical spectrum analyzer (Agilent 86140) for signal processing.

The output fiber was butt-coupled to the cleaved output endface in order to collect output

signal. The transmission spectrum (with wavelength from 1500 nm to 1620 nm) of the

Ba0.7Sr0.3TiO3 rib waveguide is shown in Figure 4.15. It can be seen that the waveguide

can guide light of wavelength from 1530 nm to 1600 nm reliably.



JIM Kwok Lung 105

Figure 4.13 Schematic diagram of the transmission spectrum measurement by end-fire

coupling technique.

Figure 4.14 CCD image showing the output light spot from the rib waveguide. The

arrow indicates the position of the rib waveguide.

Computer-connected CCD camera

Micro-objective lens

BST rib waveguide

Single mode optical fiber

Broadband laser source

xyz micro-positioners



JIM Kwok Lung 106

-80

-70

-60

-50

-40

-30

1500 1520 1540 1560 1580 1600 1620

Wavelength (nm)

Inte

nsi

ty (

dB

m)

Figure 4.15 Transmission spectrum of the Ba0.7Sr0.3TiO3 rib waveguide.

4.4 Photonic crystal cavity embedded rib waveguides

4.4.1 Fabry-Perot cavity

Figure 4.16(a) shows a Fabry-Perot (FP) cavity consisting of two parallel perfect

plane mirrors, M1 and M2, separated by a distance L [Kasap, 2001]. Assume that the

cavity is filled with a dielectric of refractive index n. A plane wave with free-space

wavenumber /20 k travelling inside the cavity will reflect back and forth, and the

total field travelling in both directions must everywhere be given by the sum of one

combined field travelling in the +z direction and a similar field travelling in the –z

direction as:

nzikEnzikEzE 00 expexp , (4.37)



JIM Kwok Lung 107

The boundary conditions require that E = 0 at z = 0 and z = L. The former condition

gives E- = -E+, which allows us to write Eq. (4.37) as:

nzkEzE 00 sin . (4.38)

The latter condition is satisfied when:

mnLk 0 (4.39a)

Ln

m

2

, (4.39b)

where m = 1, 2, 3 …, is the mode number. Eq. (4.39) is often called the longitudinal

resonance condition. The field distributions for different mode number are shown in

Figure 4.16(b). If the two mirrors are not perfect but partially reflecting with reflectance

R, light outside the cavity can couple into the cavity and light being resonant inside the

cavity can transmit to the outside (Figure 4.16(c)). The output intensity will be a fraction

of the input intensity. The amount of transmitted light is determined by the reflectance R.

A fraction of the incident light (1-R) will enter the cavity as Iincident, of which part (again

1-R) will leave the cavity as Itransmitted. Thus the ratio of transmitted to incident light is

given by:

LnkRR

R

I

I

incident

dtransmitte

022

2

sin41

1

. (4.40)

It can be observed, from Eq. (4.40), that resonant peaks will occur in the transmission

spectrum of a Fabry-Perot resonator when the longitudinal resonance condition, Eq.

(4.39), is fulfilled.



JIM Kwok Lung 108

(a) (b)

(c)

Figure 4.16 (a) Schematic diagram of a Fabry-Perot cavity. (b) Allowed field

distributions of different modes inside the cavity. (c) Transmission

spectrum of a Fabry-Perot resonator. [Kasap, 2001]

Partially reflecting mirrorsTransmission



JIM Kwok Lung 109

4.4.2 Photonic crystal cavity

If a light beam with frequencies within the photonic bandgap of a PC strikes on

that PC which is semi-infinite extended, it will be totally reflected. The PC acts like a

perfect mirror. If the PC is, however, finite, the reflection will not be perfect and there is

chance for the light to tunnel through the PC. In this way, the PC acts like a partially

reflecting mirror and is denoted as a PC mirror. A photonic crystal cavity, which is

similar to a Fabry-Perot (FP) cavity, is formed by separating two sets of PC mirror

(finite PC) by a distance. The cavity length is defined by the center-to-center distance

between the nearest rows of the two PC mirrors. Figure 4.17(a) shows the schematic

diagram of a photonic crystal cavity. Similar to the situation in FP cavity, a fraction of

light, although with frequencies within the photonic bandgap, can tunnel through the PC

mirror and enter the cavity. Being reflected back and forth inside the cavity, most light

interferes destructively. Only light with certain wavelengths satisfying the longitudinal

resonance condition (Eq. (4.39)) is allowed inside the cavity. Therefore, the transmission

spectrum of a PC cavity resembles that of a FB cavity in such a way that transmission

peaks appear at certain resonant wavelengths inside the region of the photonic bandgap.

A typical transmission spectrum of a PC cavity is shown in Figure 4.17(b).



JIM Kwok Lung 110

(a)

(b)

Figure 4.17 (a) Schematic diagram of a PC cavity formed by a pair of 3-row PC

mirrors. (b) Typical transmission spectrum of a PC cavity.

Photonic

bandgap

Resonant peak

PC mirrorsPC cavity



JIM Kwok Lung 111

4.4.3 Design of photonic crystal cavity embedded Ba0.7Sr0.3TiO3 rib waveguides

The photonic crystal cavity embedded Ba0.7Sr0.3TiO3 rib waveguide (PCCRWG)

is planned to function as a tunable filter (on-off switch) for λ = 1550 nm. Therefore, the

main purpose of the design is to determine the appropriate geometry of the PC cavity so

that it possesses a photonic bandgap (approximate zero transmission) at the wavelengths

around but contains a resonant peak at λ = 1550 nm, which corresponds to an “on” state

of the device. By changing the refractive index of Ba0.7Sr0.3TiO3 via the electro-optic

effect, the resonant peak should be shifted away from λ = 1550 nm, which corresponds

to an “off” state, since the longitudinal resonance condition for the cavity is altered.

The design flow is thus set as the following: First, the appropriate geometry of

the PC pattern for a larger photonic bandgap at wavelength around 1550 nm was

determined. Then, the effects on the resonant peak for different numbers of row and row

configurations in the PC mirror were examined. After that, the required cavity length for

the 1550-nm resonant peak was studied. Finally, the electro-optic tuning of the resonant

peak was illustrated. The transmission spectra of the photonic crystal cavity embedded

Ba0.7Sr0.3TiO3 rib waveguide were simulated by the software FullWAVETM 6.0 (RSoft

Design Group, Inc.) which utilizes the finite-difference time-domain (FDTD) algorithm.

Two-dimensional FDTD simulations were used in order to keep the computation time in

a reasonable range. An effective index Nm of 2.005 was assumed for the Ba0.7Sr0.3TiO3

rib waveguide.



JIM Kwok Lung 112

4.4.3.1 Appropriate geometry of the photonic crystal pattern

As the PC pattern is constructed by focused ion beam technique, it is obvious that

hole-type PC pattern is a better choice than rod-type PC in view of fabrication process.

Therefore, only hole-type PC pattern is considered. Since the Ba0.7Sr0.3TiO3 rib

waveguide is designed for light propagation of TE fundamental mode, larger TE

bandgaps in the PC pattern is preferred. Refer to Figure 3.2, Ba0.7Sr0.3TiO3 PCs with air

holes arranged in triangular lattice possess larger TE gaps than those in square lattice.

Hence, Ba0.7Sr0.3TiO3 PCs with air holes arranged in triangular lattice is chosen as the

required photonic crystal pattern.

If we arrange the PC pattern such that the Γ-M direction of the triangular lattice

is along the waveguide direction (Figure 4.18), the light guided in the waveguide will

then strike the PC pattern mostly along the Γ-M direction. Therefore we can consider the

photonic bandgap map for the Γ-M direction (Figure 3.2(f)) only. It can be observed

from Figure 3.2(f) that there is a large TE gap (including the joint gap) at r = 0.4 a (a is

the period of the PC). The bandgap extends from Ω = 0.30 to Ω = 0.45. Choosing a =

625 nm (then r = 0.4a = 250 nm), the corresponding bandgap in term of wavelength,

which can be calculated from Ω = a/λ, extends from λ = 1390 nm to λ = 2080 nm.

The simulation model and the transmission spectrum of the PC cavity embedded

Ba0.7Sr0.3TiO3 rib waveguide (PCCRWG) with the above-mentioned PC pattern (a = 625

nm and r = 250 nm) are shown in Figure 4.19. The cavity length was set to be 770 nm



JIM Kwok Lung 113

(which is estimated according to Eq. (4.39) for λ = 1550 nm, n = 2.005, and m = 2). It

can be seen from Figure 4.19 that the transmission is approximate zero from 1300 nm to

1800 nm, and a transmission peak occurs at λ = 1550 nm. This confirms the proof-of-

concept of the photonic crystal cavity embedded Ba0.7Sr0.3TiO3 rib waveguide.

Figure 4.18 PC pattern of triangular lattice type in a rib waveguide with the Γ-M and

Γ-K directions emphasized.

Γ

M K

Waveguide direction



JIM Kwok Lung 114

(a)

(b)

Figure 4.19 (a) Simulation model and (b) transmission spectrum of the PCCRWG in

the FDTD calculation.

625 nm

250 nm

770 nm

Monitor

Incident light



JIM Kwok Lung 115

4.4.3.2 Effect of number of row

Figure 4.20 shows the transmission spectra of the photonic crystal cavity

embedded Ba0.7Sr0.3TiO3 rib waveguide with different numbers of row in the PC mirrors.

The cavity length is kept constant (L = 770 nm) for each case. As expected, the resonant

peak value is lower for the case of PC mirrors with more rows which should have a more

well-defined photonic bandgap and thus reflect light more efficiently. On the other hand,

the peak is less sharp for PC mirrors with fewer rows although they have a higher

transmission peak value. There is nearly 100% transmission for the resonant peak in the

case of 3-row PC mirrors. The transmission decreases to ~0.5 for the 5-row PC mirrors.

Surprisingly, the transmission drops rapidly to <3% and <1% for the 7-row and 9-row

PC mirrors, respectively. This indicates that the bandgap is already quite well-defined

for a 7-row PC mirror so that much less light can tunnel through the PC mirror even only

2 more rows are added. Therefore, PC mirror with 5 rows or fewer should be used for

satisfactory amount of transmission.

It should be noted that the position of the resonant peak shifts toward the smaller

wavelength for PC mirrors with more rows even the cavity length is kept unchanged.



JIM Kwok Lung 116

(a)

(b)



JIM Kwok Lung 117

(c)

(d)

Figure 4.20 Transmission spectra for the PCCRWG with (a) 3-row, (b) 5-row, (c) 7-

row, (d) 9-row PC mirrors.



JIM Kwok Lung 118

4.4.3.3 Effect of row configuration

In the previous Section, an example of a PC cavity embedded Ba0.7Sr0.3TiO3 rib

waveguide with 3-row PC mirrors is illustrated. In fact, there are two possible

configurations to set up the 3-row PC mirrors which are shown together in Figure 4.21.

The one in Figure 4.21(a) has a pair of 3-row PC mirrors with the first row containing 3

air holes, while the one in Figure 4.21(b) has a pair of 3-row PC mirrors with the first

row containing 4 air holes. We denote the former one as 3-row-3 PC mirror and the latter

one as 3-row-4 PC mirror. In other words, a n-row-m PC mirror represents a n-row PC

mirror with m air holes in the first row.

(a) (b)

Figure 4.21 Two possible configurations of the 3-row PC mirrors: (a) 3-row-3 PC

mirrors, (b) 3-row-4 PC mirrors.

The transmission spectra for PC cavity embedded Ba0.7Sr0.3TiO3 rib waveguide

with different n-row-m PC mirrors (n = 3, 4, 5 and m = 3, 4) are calculated. The results

are shown in Figure 4.22. The distinctness of the photonic bandgaps and the sharpness of

3-row-43-row-3



JIM Kwok Lung 119

the resonant peaks are similar for the corresponding pairs having the same n (number of

row in the PC mirror) but different m (number of air hole in the first row). However,

their resonant peak values are usually of great difference. It is interesting to note that the

resonant peak value is always higher when (n+m) is odd for the same n. Taking into

consideration the tradeoff between the resonant peak value and the sharpness of the

resonant peak, the 5-row-4 configuration is chosen for its relatively sharp resonant peak

which still have a transmission peak value ~ 0.5.



JIM Kwok Lung 120

Figure 4.22 Transmission spectra of PCCRWG with different n-row-m PC mirrors: (a)

3-row-3, (b) 3-row-4, (c) 4-row-3, (d) 4-row-4, (e) 5-row-3, (f) 5-row-4.

3row3 3row4

4row44row3

5row3 5row4



JIM Kwok Lung 121

4.4.3.4 Determination of cavity length

After the configuration of the PC pattern is settled, the cavity length needs to be

fine tuned so that the resonant peak is situated at λ = 1550 nm. The transmission spectra

of the 5-row-4 PCCRWG with different cavity lengths were simulated. The peak values

and the locations of the resonant peaks with different cavity lengths are summarized in

Table 4.1. It is found that, for resonant peak situated at λ = 1550 nm, the required cavity

length should be 800 nm. The peak value is coincidently the highest among the

simulated cases.

Table 4.1 Resonant peak values and locations of a 5-row-4 PCCRWG with different

cavity lengths.

Cavity length

(nm)

Location of the

resonant peak (nm)

Transmission value

of the peak

770 1520 0.43

780 1530 0.29

790 1540 0.30

800 1550 0.52

810 1560 0.38



JIM Kwok Lung 122

4.4.3.5 Electro-optic tuning of the resonant peak

The schematic design of the PC cavity embedded Ba0.7Sr0.3TiO3 rib waveguide

with 5-row-4 PC mirrors is shown in Figure 4.23. Two coplanar electrodes are placed

parallel to the rib waveguide. When an external voltage is applied across the electrodes,

the refractive index of the Ba0.7Sr0.3TiO3 (nBST) will change due to the electro-optic

effect. The transmission resonant peak should be shifted accordingly. Figure 4.24

illustrates such shift in the resonant peak assuming the refractive index of Ba0.7Sr0.3TiO3

is changed by ±0.5% for an applied electric field of about 12 MV/m (suggested by the

1D PC results discussed in Section 2.5.2). It can be seen from Figure 4.24 that the

resonant peak is situated at λ = 1551 nm for nBST = 2.005. If the refractive index of the

Ba0.7Sr0.3TiO3 varies through the electro-optic effect, the resonant peak shifts 6 nm

toward the smaller wavelength (λ = 1545 nm) if nBST is decreased by 0.5%, while it

shifts 6 nm toward the larger wavelength (λ = 1557 nm) if nBST is increased by 0.5%. For

nBST = 2.005, the transmission value at λ = 1551 nm is 0.53 while the transmission values

at λ = 1545 and 1557 nm are both 0.26 which is half of the transmission value

(corresponds to -3 dB) at λ = 1551 nm. It should, thus, provide sufficient difference in

the transmission value to distinguish the “on” and “off” states for the PC cavity

embedded Ba0.7Sr0.3TiO3 rib waveguide to be functioned as a tunable filter for λ = 1550

nm.



JIM Kwok Lung 123

Figure 4.23 Schematic design of the PC cavity embedded Ba0.7Sr0.3TiO3 rib waveguide

with 5-row-4 PC mirrors.

W 2.5 μm

L 800 nm

a 625 nm

r 250 nm

L

W

a

r

Electrodes

Ba0.7Sr0.3TiO3

Rib waveguide



JIM Kwok Lung 124

Figure 4.24 Shifting of transmission resonant peaks for the PCCRWG shown in Figure

4.23 when the refractive index of Ba0.7Sr0.3TiO3 (nBST) is changed by

+0.5% (red line) and -0.5% (blue line). The green line corresponds to the

effective refractive index (nBST = 2.005).

4.4.4 Photonic crystal cavity preparation by focused ion beam etching

The photonic crystal cavity pattern is prepared on the Ba0.7Sr0.3TiO3 rib

waveguide using focused ion beam (FIB) etching. The etching process is carried out in a

Quanta 200 3D DualBeam (FIB/SEM) system (at the Electron Microscope Unit, The

University of Hong Kong). Figure 4.25 shows the scanning electron micrograph of the

Ba0.7Sr0.3TiO3 rib waveguide with PC cavity patterns prepared by FIB etching. It can be



JIM Kwok Lung 125

seen from Figure 4.25 that holes with diameter of 490±20 nm with period of 630±20 nm

arranged in triangular lattice have been produced nicely. The cavity length L is 800±20

nm. The bias voltage and the beam current used during operation are 30 kV and 30 pA,

respectively. The etching time for each hole is about 1 min. The etching depth of the

holes is ~ 440 nm. It is, thus, demonstrated that FIB is an efficient way to prepare

photonic crystal cavity on the Ba0.7Sr0.3TiO3 rib waveguide.

Figure 4.25 Scanning electron micrograph of the Ba0.7Sr0.3TiO3 rib waveguide with PC

cavity prepared by FIB etching.

Figure 4.26 shows the measured and simulated transmission spectra of the

Ba0.7Sr0.3TiO3 rib waveguide with PC cavity. It can be seen that the measured spectrum

L



JIM Kwok Lung 126

agrees well with the simulated one. A resonant peak is situated at λ ~ 1560 nm in the

measured transmission. The small discrepancy in the position of the resonant peak

between the measured spectrum and the simulated spectrum may be due to the

fabrication deviation of the cavity length from 800 nm. Also, the intensity of the

resonant peak is ~ 20 dBm larger than those of other wavelengths for both measured and

simulated spectra. In addition, the bandgap effect of the PC pattern can be evaluated

when compared the transmission spectrum of the Ba0.7Sr0.3TiO3 waveguide with PC

cavity to that of the Ba0.7Sr0.3TiO3 waveguide without PC cavity. The transmission

intensity level of the Ba0.7Sr0.3TiO3 waveguide with PC cavity drops by ~ 30 dBm

relative to that of the Ba0.7Sr0.3TiO3 waveguide without PC cavity. Such a great decrease

in the transmission intensity should be attributed to the photonic bandgap effect due to

the PC pattern on the Ba0.7Sr0.3TiO3 waveguide.



JIM Kwok Lung 127

-80

-70

-60

-50

-40

-30

1.50 1.52 1.54 1.56 1.58 1.60 1.62Wavelength (mm)

Inte

nsity

(dB

m)

(a)

(b)

Figure 4.26 (a) Measured transmission spectra of the Ba0.7Sr0.3TiO3 rib waveguide with

and without PC cavity and (b) simulated transmission spectrum of the

Ba0.7Sr0.3TiO3 rib waveguide with PC cavity.

With PC cavity

Without PC cavity



JIM Kwok Lung 128

4.5 Summary

The refractive indices of the Ba0.7Sr0.3TiO3 thin film on a MgO (001) substrate is

measured using the prism coupling technique. The rib waveguide dimensions for single

mode TE wave propagation is determined by applying the effective index method. A

photonic crystal cavity embedded Ba0.7Sr0.3TiO3 rib waveguide which functions as a

tunable filter (on-off switch) for λ = 1550 nm is designed with the help of FDTD

simulation. The photonic crystal cavity is composed of two 5-row-4 PC mirrors, which is

formed by air holes arranged in triangular lattice in the Ba0.7Sr0.3TiO3 matrix. The radius

of the air holes is 250 nm and the periodicity is 625 nm. The required cavity length for

the resonant peak to be situated at λ = 1550 nm is 800 nm. Photonic crystal cavities are

fabricated on a Ba0.7Sr0.3TiO3 rib waveguide by focused ion beam etching with

satisfactory results. The transmission properties of the Ba0.7Sr0.3TiO3 rib waveguide with

PC cavity have been measured. The results agree well with the FDTD simulation.

THERMAL TUNING OF PHONONIC

CHAPTER 5 BANDGAPS IN BST/EPOXY PHONONIC CRYSTALS

JIM Kwok Lung 129

CHAPTER 5

THERMAL TUNING OF PHONONIC BANDGAPS

IN BARIUM STRONTIUM TITANATE/EPOXY

PHONONIC CRYSTALS

5.1 Introduction

Phononic crystals, which are the elastic and acoustic analogs of photonic crystals,

are periodic composite materials having different acoustic velocities and densities with

lattice constants on the scale of the wavelength of sound or ultrasound [Page, 2004].

Extending the concept of photonic bandgaps to acoustic waves, phononic crystals

possesses phononic bandgaps for which the acoustic waves at these frequencies cannot

propagate and lead to a total reflection regime in such frequency range. This makes

phononic crystal very attractive for the design of acoustic band-pass filters [Wu, 2009].

If frequency and the width of the phononic bandgaps can be tuned, promising

applications such as tunable filters can be realized [Khelif, 2003; Pennec, 2004]. Since

the phononic bandstructures of a phononic crystal depend on the acoustic velocities and

densities of the constituent materials, changes in such properties would cause

modification in the phononic bandgaps. Ferroelectric materials will undergo phase

transition, at which the longitudinal and shear acoustic wave velocities change



JIM Kwok Lung 130

significantly, across the Curie temperature [Wu, 1992; Cheng, 1994; Frayssignes, 2005].

Since Ba0.7Sr0.3TiO3 has its ferroelectric-to-paraelectric phase transition at ~35 ˚C

[Zhang, 1998; Wang, 2005], it becomes a promising candidate as a building block of

thermally tunable phononic crystal near room temperature.

In this Chapter, we first present the calculation of phononic bandstructure using

the plane wave expansion method, and then we describe the measurement of the

longitudinal and shear wave velocities of Ba0.7Sr0.3TiO3 bulk ceramics by the ultrasonic

through-transmission technique. The fabrication of Ba0.7Sr0.3TiO3/epoxy phononic

crystals with the corresponding phononic bandstructures and the characterization of the

phononic bandgaps by ultrasonic pulse-echo technique are then discussed.

5.2 Computation of phononic bandstructure by plane wave expansion method

Similar to the case in photonic crystals, several theoretical methods have been

developed to study phononic crystals. They include the plane wave expansion (PWE)

method [Kushwaha, 1994], the multiple scattering theory (MST) method [Kafesaki,

1999], and the finite-difference time-domain (FDTD) method [Cao, 2004]. Among these

methods, the PWE method is extensively used to calculate phononic bandstructures

because of its convenience and easiness to apply. Therefore, we also employed the PWE

method for the calculation of phononic bandstructures. In this Section, we describe

briefly the computation of phononic bandstructures of two-dimensional phononic

crystals by the PWE method.



JIM Kwok Lung 131

5.2.1 Wave equation for an inhomogeneous elastic medium

Consider an inhomogeneous but isotropic elastic medium which is characterized by

three material parameters: the mass density r , the longitudinal speed of sound rcl

,

and the transverse speed of sound rct

, the equation of motion governing the lattice

displacement tru ,

is given by:

2

2

t

ui

j

ij

x

)2()(2)2(2 222222tl

jijkkt

jijij

j

kktl

j

ijt cc

xuc

xu

x

ucc

x

uc

, (5.1)

where σij is the stress tensor which is given by:

ijkktlijtij uccuc )2(2 222 . (5.2)

j

k

i

k

i

j

j

iij x

u

x

u

x

u

x

uu

2

1(5.3)

is the deformation tensor, and 3,2,1iui being the components of the displacement

vector tru ,

. The convention of summation over equal indices applies.

Assume that the medium is linear, the quadratic term in uij in Eq. (5.3) can

thereby be omitted. Then Eq. (5.1) can be simplified to a more compact form as:

2

2

t

ui

iti

tlit ucux

ccuc

)()2( 22222

uccxx

uc tl

iit

)2()( 222



JIM Kwok Lung 132

uccxx

ucuc tl

iitit

)2( 2222 . (5.4)

It should be noted that this equation is valid for arbitrary inhomogeneity. Since r ,

rcl

, and rct

are, however, position dependent, Eq. (5.4) cannot be separated into

two equations, one for the longitudinal displacement which satisfies 0 u

, and the

other for the transverse displacement which satisfies 0 u

, as in the case for

homogeneous medium [Kushwaha, 1994].

5.2.2 Plane wave expansion for periodic elastic structure

If the inhomogeneous elastic medium exhibits spatial periodicity, the three material

parameters, r , rcl

, and rct

, can then be expanded in the Fourier series. In fact,

the algebra would be less complicated if we expand 2lc and 2

tc , rather than lc and tc ,

as the speeds of sound always appear in Eq. (5.4) in the forms of 2lc and 2

tc .

Therefore the three material parameters are expanded as

G

rGieGr

, (5.5a)

G

rGil eGrcr

2 , and (5.5b)

G

rGit eGrcr

2 . (5.5c)

where G

represent the reciprocal lattice vectors with corresponding periodicity of the

medium which may be one, two, or three dimensional. The summation in Eq. (5.5)



JIM Kwok Lung 133

covers all reciprocal lattice vectors which correspond to the Bravais lattice vectors in the

real space. The Fourier coefficients G

, G

, and G

, are given by:

rGi

C

errdV

G

31, (5.6a)

rGil

C

ercrrdV

G

231 , and (5.6b)

rGit

C

ercrrdV

G

231 , (5.6c)

where the integration is over the unit cell and Vc is its volume. Since the medium is

periodic, the displacement tru ,

must satisfy the Bloch theorem which states as:

G

rGiK

trKi eGuetru

, , (5.7)

where K

is the Bloch wave vector, and is the angular frequency of the wave.

Substituting Eq. (5.5) and Eq. (5.7) into Eq. (5.4), followed by multiplication by

rGiexp , and integration over the unit cell, Eq. (5.4) transforms to an infinite set of

linear equations as:

G

K GKGKGuGG

GKGKGuGG K

02 2 GuGGGKGKGuGGGG KK

, (5.8)

for the eigenvectors GuK

. For each given Bloch vector K

, Eq. (5.8) has solutions for

some eigenvalues Kn

, where n = 1, 2, … is the first, second, etc. vibrational band.



JIM Kwok Lung 134

5.2.3 Binary composite

Let us consider phononic crystal composed of binary composite for which there are

only two materials, labeled as a and b, in every unit cell. Material a (b) is characterized

by the three parameters a ( b ), lac ( lbc ), and tac ( tbc ). The filling ratio, f, is defined as

the occupancy ratio for material a in the unit cell. Thus, the occupancy ratio for material

b is (1-f). The Fourier coefficients in Eq. (5.6) can be calculated. To illustrate, we will

take G

as an example. For 0G

,

ffG ba 10

, (5.9)

which is actually the average density. If 0G

,

b

rGib

ca

rGia

c

erdV

erdV

G

33 11

a

rGiba

cba

rGib

c

erdV

erdV

33 11 . (5.10)

The first term in the right hand side is zero. The integral in the second term defines a

structure function:

a

rGi

c

erdV

GF

31. (5.11)

Therefore we have,

,

,1

GFGF

ffG

ba

ba

.0

,0

Gfor

Gfor

(5.12)

Similarly, G

and G

can be expressed by:



JIM Kwok Lung 135

,

1222

222

GFGFcGFcc

cfcfcG

llbblaa

llbblaa

.0

,0

Gfor

Gfor

(5.13)

,

1222

222

GFGFcGFcc

cfcfcG

llbblaa

llbblaa

.0

,0

Gfor

Gfor

(5.14)

Notice that the structure function GF

, given in Eq. (5.11), contains all the required

geometrical information of the binary system for the computation of phononic

bandstructure. Moreover, the form of GF

depends only on the geometry of material a

but not on the type of the Bravais lattice considered since the vector G

in evaluating

GF

is arbitrary.

To proceed further, we first single out the term GG

in the summation in Eq.

(5.8) as:

GuGKGKGuGK KK

22

GG

K GuGKGKGGF

GuGKGK K

GuGKGK K

2

02 GuGuGKGK KK

. (5.15)

Eq. (5.15) can be further rewritten in dimensionless quantities if we introduce a lattice

parameter a for normalization purpose and define the following quantities:

2/aKk

, (5.16a)



JIM Kwok Lung 136

2/aGg , (5.16b)

tCa 2/ , (5.16c)

2/122/1 // tt cC , (5.16d)

2/122/1

// tl cC , (5.16e)

tb

lb

tbbtaa

lbblaa

c

c

fccf

fccf2/1

22

222/1

1/

1/

, (5.16f)

ff ba

ba

1/

1/

, (5.16g)

fccf

cc

lbblaa

lbblaa

1/

1/22

22

, and (5.16h)

fccf

cc

tbbtaa

tbbtaa

1/

1/22

22

. (5.16i)

The dimensionless version of Eq. (5.15) is then given by:

gugkgkgugkkk

1222

gg

kgugkgkggF

gugkgkk

gugkgkk

2

022 gugugkgkkk

. (5.17)

In principle the phononic bandstructure (normalized frequency, , versus normalized

wave vector, k

) can be obtained by solving Eq. (5.17). It is, however, computationally

inconvenient to handle Eq. (5.17) because the matrix operating on k

u

itself depends on



JIM Kwok Lung 137

the normalized frequency which is exactly what we propose to obtain. Therefore it is

advantageous to transform Eq. (5.17) into the form:

g

kkgg gugukA

2 , (5.18)

where the dyadic matrix kA gg

is independent of . After some algebra, Eq. (5.17) can

be expressed in the form of Eq. (5.18) with:

gggggg MNkA

1 , (5.19a)

gggg gkgkIgkMgg

122

gkgkggF gg

21

gkgkIgkgk

gkgk 2 , (5.19b)

gggggg ggFN

1 . (5.19c)

With specified crystalline symmetry and the structure of the unit cell, the corresponding

phononic bandstructure can be obtained by solving Eq. (5.18), with quantities defined in

Eq. (5.19).

A home written computer programme, based on the plane wave expansion described

in this section, has been developed for the computation of phononic bandstructure of a

phononic crystal composed of binary composites with square or circular rods arranged in

square lattice.



JIM Kwok Lung 138

5.3 Determination of temperature dependence of longitudinal and shear wave

velocities by ultrasonic through-transmission technique

5.3.1 Ultrasonic through-transmission technique

The longitudinal and shear wave velocities of a solid can be measured by the

ultrasonic through-transmission technique [Freemantle, 1998]. The experimental setup

for the ultrasonic through-transmission technique is shown in Figure 5.1. The sample,

mounted in a sample holder, is placed in between two ultrasonic transducers. The sample

holder and the transducers are supported by a rigid stainless steel frame. The vertical

position of the sample holder can be adjusted and the angle of incidence of the ultrasonic

beam on the sample can be varied. The whole setup is placed into a tank filled with

silicone oil which is used to improve coupling ultrasound to the sample. The silicone oil

has been supplied by Phone-Poulenc in France and has a known sound velocity of 998.2

m/s at 24˚C. The ultrasonic transducers used are a pair of plane immersion transducers

with diameter of 8 mm and have a center frequency of 10 MHz. The maximum operating

temperature of the transducers is about 100˚C. One transducer functions as the

transmitter and the other as the receiver. An ultrasonic analyzer (Panametrics 5900PR) is

employed to generate a sharp pulse to provide a broadband excitation to the transmitting

transducer. The ultrasonic beam, transmitted through the sample, is then picked up by

the receiving transducer. The received signal is amplified by the analyzer (voltage gain =

40 dB) before displaying on a digitizing oscilloscope (HP Infinium). In order to adjust

the temperature of the silicone oil bath, a temperature controlling system, which consists



JIM Kwok Lung 139

of a temperature controller (Eurotherm 2408) with accuracy of ±0.1˚C, a 100W heater

and a stirrer, has been installed.

Figure 5.1 Schematic diagram of the ultrasonic through-transmission technique.

5.3.2 Determination of longitudinal wave velocity

If the sample has a larger longitudinal wave velocity than the velocity of sound in

silicone oil, the ultrasonic pulse, with the sample inserted, will take a shorter time to

reach the receiving transducer and will shift forward in time. This negative time shift

l is related to the longitudinal wave velocity in the sample lv by:

oilll vv

t11 , (5.20)

where t is the sample thickness and oilv is the velocity of sound in the silicone oil.

(a)

(b)

(c)(d)

(f)(h)

(e)

(i)(j)

(a). Ultrasonic analyzer

(b). Digital oscilloscope

(c). Temperature control system

(d). Tank

(e). Sample holder

(f). 10MHz Immersion

Transducers

(g). Sample

(h). K-type thermocouple

(i). Stirrer

(j). Heater

(g)(f)



JIM Kwok Lung 140

Starting with no sample placed in the path of the ultrasonic beam, the received

signal was displayed on the oscilloscope while the position of the first pulse peak was

noted. The sample was then inserted in the path, and the negative time shift l in the

first pulse peak was recorded. The longitudinal wave velocity in the sample Lv was

readily calculated using Eq. (5.20).

5.3.3 Determination of shear wave velocity

Both the shear and longitudinal waves are excited in the sample if the ultrasonic

beam impinges on the sample at an off-normal angle. The longitudinal and the shear

waves in the sample will be refracted since their velocities are different from the sound

wave velocity in the silicone oil. The angle of refraction )(, lsr corresponds to shear

(longitudinal) wave is given by the Snell’s law:

)()(,sin

sin

ls

oil

lsr

i

v

v

. (5.21)

where i is the angle of incidence, oilv is the velocity of sound in silicon oil, and )(lsv is

the shear (longitudinal) wave velocity in the sample. If the velocity of sound in the

silicone oil is less than the wave velocity in the sample, there exists a critical angle

beyond which a total internal reflection of the wave occurs. Since the longitudinal wave

velocity is always larger than the shear wave velocity in solid, the critical angle for

longitudinal wave is smaller than that of shear wave. This opens a window of angles

where only the shear wave can propagate through the sample while the longitudinal

wave is totally reflected.



JIM Kwok Lung 141

Starting with normal incidence of the ultrasonic beam on the sample as described

in the previous Section, the received signal due to the longitudinal wave will decrease in

amplitude when the sample is rotated off-normal and finally disappear when the incident

angle is beyond the critical angle for the longitudinal wave in the sample. The incident

angle of the ultrasonic beam is varied by rotating the sample through the sample holder.

Figure 5.2 Geometry of the propagation path of the ultrasonic wave.

Refer to Figure 5.2, the path length x through the sample is given by:

2

sin1

oil

si v

v

tx

. (5.22)

The difference in transit time s with and without the sample in place can be expressed

as:

oil

isr

ss vv

x

,cos1. (5.23)

Substituting Eq. (5.21) and Eq. (5.22) into Eq. (5.23), the shear wave velocity sv is

given by:

t

x

θr

θi



JIM Kwok Lung 142

oilsi

i

i

oils vt

tvv

cos

sinarctansin

sin. (5.24)

5.3.4 Temperature dependence of longitudinal and shear wave velocities in

Ba0.7Sr0.3TiO3 bulk sample

The longitudinal and shear wave velocities of a Ba0.7Sr0.3TiO3 bulk sample as a

function of temperature were measured using the ultrasonic through-transmission

technique. The sample used was a Ba0.7Sr0.3TiO3 ceramic disc with a diameter of 20 mm

and thickness of 2.38 mm. Since the velocity of sound in silicone oil is also a function of

temperature, such temperature dependence was measured first and was taken into

account when using Eq. (5.20) later. The velocity of sound in the silicone oil as a

function of temperature is shown in Figure 5.3. The Ba0.7Sr0.3TiO3 sample was then

inserted in between the transducers for the longitudinal wave velocity measurement. It

should be noted that in order to accurately measure the longitudinal wave velocity in the

sample, the incident ultrasonic beam should be normal to the sample face. This could be

confirmed by maximizing the amplitude of the received signal. Figure 5.4(a) shows the

longitudinal wave velocity of the Ba0.7Sr0.3TiO3 ceramic disc as a function of

temperature. It can be observed from Figure 5.4(a) that the Ba0.7Sr0.3TiO3 ceramic disc

has a relatively steady longitudinal velocity vl of ~5100 m/s below 35 ˚C which is the

Curie temperature of Ba0.7Sr0.3TiO3. When the temperature rises from 35 ˚C, vl increases

gradually until reaches a saturation of ~6300 m/s at 45 ˚C (see Table 5.1).



JIM Kwok Lung 143

In order to perform the shear wave velocity measurement, the angle of incidence

should be larger than the critical angle for the longitudinal wave. The critical angle for

the longitudinal wave lC , , which can be calculate by

l

oillC v

varcsin, , (5.25)

is 11.5˚ for vl = 5000 m/s. To ensure that all longitudinal waves were totally reflected,

the angle of incidence is set to be 15˚, which is larger than the critical angle corresponds

to vl = 4000 m/s. The shear wave velocity as a function of temperature is shown in

Figure 5.4(b). Similar to the trend of the longitudinal wave velocity shown in Figure

5.4(a), the shear wave velocity has a steady value of ~3000 m/s below the Curie

temperature (35 ˚C) and increases with temperature until saturates at ~4000 m/s at 45 ˚C

(see Table 5.1).

850

900

950

1000

1050

20 25 30 35 40 45 50 55

Temperature (˚C)

v oil (

m/s

)

Figure 5.3 Velocity of sound in silicone oil voil as a function of temperature.



JIM Kwok Lung 144

4000

4500

5000

5500

6000

6500

7000

20 25 30 35 40 45 50 55Temperature (˚C)

Vel

oci

ty (

m/s

)

(a)

2000

2500

3000

3500

4000

4500

20 25 30 35 40 45 50 55Temperature (˚C)

Vel

ocity

(m

/s)

(b)

Figure 5.4 (a) Longitudinal and (b) Shear wave velocities as a function of temperature

for a Ba0.7Sr0.3TiO3 ceramic disc. Dotted lines are added as visual aid.



JIM Kwok Lung 145

5.3.5 Temperature dependence of longitudinal and shear wave velocities in epoxy

sample

The longitudinal and shear wave velocities of the epoxy (Epotek 301) were

measured using the ultrasonic through-transmission technique as described in the

previous Section. The results are shown in Figure 5.5. It can be seen that the longitudinal

and the shear wave velocities of the epoxy decrease slightly as temperature increases

(see Table 5.1).

500

1000

1500

2000

2500

3000

3500

30 32 34 36 38 40 42 44 46 48 50 52

Temperature (˚C)

Vel

oci

ty (

m/s

)

Longitudinal waveShear wave

Figure 5.5 Longitudinal and shear wave velocities as a function of temperature for the

epoxy (Epotek 301).



JIM Kwok Lung 146

Table 5.1 Material parameters of Ba0.7Sr0.3TiO3 & epoxy (Epotek 301) used for the

phononic crystal bandstructure computation.

Ba0.7Sr0.3TiO3

= 6050 kgm-3

Epoxy (Epotek 301)

= 1130 kgm-3

Temperature vs (m/s) vl (m/s) vs (m/s) vl (m/s)

35 ˚C 3008 5233 1180 2580

45 ˚C 3892 (29.4%) 6317 (20.7%) 1150 (2.5%) 2530 (2.0%)

5.4 Barium strontium titanate/epoxy phononic crystals

5.4.1 Fabrication of the Ba0.7Sr0.3TiO3/epoxy phononic crystals

Ba0.7Sr0.3TiO3/epoxy phononic crystals were fabricated using the dice-and-fill

technique [Taunaumang, 1994]. The fabrication flow chart is shown in Figure 5.6. A

Ba0.7Sr0.3TiO3 ceramic disc with diameter of 20 mm and thickness of 2.38 mm was diced

using an automatic wafer dicing saw (Disco DAD 321) equipped with a diamond blade

(Disco NBC-Z 2050 55 x 0.06 x 40) of width 60 µm giving a groove width of 65 µm due

to blade vibration. The periodicity of subsequent cuttings was set to be 265 µm. After a

set of parallel cuttings, epoxy (Epotek 301 supplied by Epoxy Technology) was filled

into the grooves. The sample was, subsequently, degassed in a vacuum chamber for 30

min and then cured at 40 ˚C for 30 min. A second set of identical cutting was then

performed in a direction perpendicular to the first set of cutting. After filling epoxy into

the grooves again, the sample was degassed in a vacuum chamber for 30 min and cured



JIM Kwok Lung 147

at 40 ˚C for 12 hours. Grinding was performed on both sides of the sample to remove

excess Ba0.7Sr0.3TiO3 and epoxy.

Figure 5.6 Fabrication flow chart of Ba0.7Sr0.3TiO3/epoxy phononic crystals using a

dice-and-fill technique.

Dicing

Dicing after 90˚ rotation

Filling epoxy

Grinding

Filling epoxy

BST



JIM Kwok Lung 148

5.4.2 Reflection spectrum measurement by ultrasonic pulse-echo technique

The reflection spectrum of a phononic crystal can be obtained by the ultrasonic

pulse-echo technique. Compared to the through-transmission technique, ultrasonic pulse-

echo technique is more suitable for the characterization of phononic bandgaps in

phononic crystals for frequency in the MHz range where the attenuation loss is very high.

Consider a phononic crystal composed of lossless materials, ultrasonic beam with

frequencies outside the phononic bandgaps will transmit through it while beam with

frequencies inside the phononic bandgaps will be reflected entirely. Therefore, a peak in

the reflection spectrum, which implies a phononic bandgap, corresponds to a dip in the

transmission spectrum for an ideal phononic crystal. However, the composite materials

in real situation are not lossless, especially at high frequency. The ultrasonic beam will

suffer attenuation inside the phononic crystal. This diminishes the transmission signal

significantly which makes the detection much more difficult. Also, a dip in the

transmission spectrum can now be attributed to phononic bandgap reflection or/and

material absorption. This causes the interpretation of a transmission dip ambiguous.

Therefore the fabricated Ba0.7Sr0.3TiO3/epoxy phononic crystal was characterized by the

ultrasonic pulse-echo reflection technique.

The experimental setup for the ultrasonic pulse-echo technique is identical to the

one used for the through-transmission technique shown in Figure 5.1 except that only

one transducer is used for both transmitting and receiving purposes. Since the thickness

of our samples was about 1 mm only and the ultrasonic beam was intended to impinge



JIM Kwok Lung 149

the sample in the in-plane direction so that the beam can experience the periodicity of

the phononic crystal, a transducer with diameter of 1 mm was used in order to provide a

more confined ultrasonic beam. The center frequency of the transducer was 10 MHz. A

phononic crystal with 10 periods of Ba0.7Sr0.3TiO3/epoxy was placed in front of the

transducer at a distance of about 10 mm in water. Similar to the case in through-

transmission technique, an ultrasonic analyzer (Panametrics 5900PR) was employed to

generate a sharp pulse to provide a broadband excitation to the transducer. The ultrasonic

beam, reflected from the sample, was then picked up by the same transducer. The

received signal was amplified by the analyzer (voltage gain = 40 dB) before displaying

on a digitizing oscilloscope (HP Infinium). A Fast Fourier Transform was then

performed on the received signal (time response) to obtain the reflection spectrum

(frequency response). The measurement was repeated at different temperatures range

from 32 ˚C to 50 ˚C. The reflection spectra of the Ba0.7Sr0.3TiO3/epoxy phononic crystal

at different temperatures are shown in Figure 5.7. As can be seen from Figure 5.7(a), a

reflection peak is situated at ~9.5 MHz. Away from the reflection peak, two dips can be

observed at around 7.5 MHz and 12 MHz. The positions of the reflection peak and the

two dips remain nearly unchanged for the temperature from 32 ˚C to 35 ˚C. When the

temperature rises from 35 ˚C to 45 ˚C, it can been observed from Figure 5.7(b) that the

high frequency dip is moving from 12 MHz to 14 MHz gradually, while the low

frequency dip and the reflection peak remain approximately the same positions at 7.5

MHz and 9.5 MHz, respectively. When the temperature further increases from 45 ˚C to

50 ˚C, the reflection peak and the two dips are again remain almost unmoved as can be

noted in Figure 5.7(c).



JIM Kwok Lung 150

As mentioned previously in this Section, ultrasonic beam with frequencies inside

the phononic bandgaps would reflect totally if impinged on the phononic crystal.

Therefore, a peak in the reflection spectrum should be an indication of a phononic

bandgap. Moreover, the positions of the dips around the peak should be related to the

positions of the phononic bandgap edges, although they should not simply be considered

as the exact locations of the bandgap edges. The low frequency bandgap edge does not

shift while the high frequency bandgap edge shifts towards higher frequencies when the

temperature rises from 35 ˚C to 45 ˚C.

(a)

-60

-50

-40

-30

-20

-10

0

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Frequency (MHz)

Inte

nsi

ty (

dB

m)



JIM Kwok Lung 151

(b)

(c)

Figure 5.7 Reflection spectra of the Ba0.7Sr0.3TiO3/epoxy phononic crystal at different

temperatures range from (a) 32 ˚C to 35 ˚C, (b) 35 ˚C to 45 ˚C, and (c) 45

˚C to 50 ˚C. The inset in (b) shows the reflection spectra of the

Ba0.7Sr0.3TiO3 ceramics at temperatures range from 32 ˚C to 50 ˚C.

-60

-50

-40

-30

-20

-10

0

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Frequency (MHz)

Inte

nsi

ty (

dB

m)

35˚C

45˚C

Increasing temperature

-60

-50

-40

-30

-20

-10

0

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Frequency (MHz)

Inte

ns

ity

(d

Bm

)-60

-50

-40

-30

-20

-10

0

5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Frequency (MHz)

Inte

ns

ity

(d

Bm

)



JIM Kwok Lung 152

5.4.3 Phononic bandstructure computation of the Ba0.7Sr0.3TiO3/epoxy phononic

crystals

In order to interpret the measured reflection spectra of the phononic crystal as

discussed in the previous Section, the phononic bandstructures of a Ba0.7Sr0.3TiO3/epoxy

phononic crystal were calculated using the home written computer programme. The

simulation model is illustrated in Figure 5.8. The phononic crystal was composed of

square Ba0.7Sr0.3TiO3 rods with widths of 200 µm arranged in a square lattice with a

period of 265 µm in a matrix of epoxy. The material parameters used for the simulation

are summarized in Table 5.1. As the experiment was performed with ultrasonic pulses

directed to the sample along the dicing direction through water, it is expected that mixed

mode waves, with wave vectors along the Γ-X direction, are predominately excited

along the Γ-X direction during measurement. Thus, the phononic bandstructures for

mixed mode along the Γ-X direction of the phononic crystal at 35 ˚C and 45 ˚C in the

frequency range of interest (0-18MHz) are calculated and the results are shown in Figure

5.9. As can be seen from Figure 5.9, the phononic crystal possesses many phononic

bandgaps. The lowest bandgap starts at 3 MHz for both temperature while the highest

bandgap positioned at ~12 MHz and ~16 MHz for T = 35 ˚C and 45 ˚C, respectively.

Since the longitudinal and transverse wave velocities in Ba0.7Sr0.3TiO3 change greatly

when the temperature rises from 35 ˚C to 45 ˚C, it is not surprised that the phononic

bandgaps have been modified accordingly. One interesting point to note is that if the

phononic bands in between the bandgaps are ignored for which all bandgaps are viewed

as combined to form one larger bandgap, the lower edge of the combined phononic



JIM Kwok Lung 153

bandgap does not shift at all while the upper edge of it shifts from 12 MHz to 16 MHz

when the temperature changes from 35 ˚C to 45 ˚C. This agrees very well with the

observations from the reflection spectra of the phononic crystal discussed in the previous

Section.

Figure 5.8 Simulation model of the Ba0.7Sr0.3TiO3/epoxy phononic crystal.

Configurationd

a

Ba0.7Sr0.3TiO3

Epoxy

d = 200 μm

a = 265 μm

f = (d/a)2 = 0.57



JIM Kwok Lung 154

0

2

4

6

8

10

12

14

16

18

0

Fre

quen

cy (

MH

z)

(a)

0

2

4

6

8

10

12

14

16

18

0

Fre

quen

cy (

MH

z)

(b)

Figure 5.9 Computed phononic bandstructures for mixed mode along the Γ-X

direction of the phononic crystal with configuration shown in Figure 5.8 at

(a) 35 ˚C, and (b) 45 ˚C. The shaded areas indicate the position of the

phononic bandgaps.

X

X



JIM Kwok Lung 155

5.5 Summary

The shear and transverse wave velocities of Ba0.7Sr0.3TiO3 ceramic as a function

of temperature were determined using the ultrasonic through-transmission technique. A


Ba0.7Sr0.3TiO3.

Phononic crystals composed of Ba0.7Sr0.3TiO3 rods in a matrix of epoxy were

fabricated using the dice-and-fill method. The width of the Ba0.7Sr0.3TiO3 rods was 200

µm with periodicity of 265 µm. The temperature dependence of the phononic bandgaps

of the phononic crystal was characterized by the reflection spectra which were obtained

using the ultrasonic pulse-echo technique. Thermal tuning of the phononic bandgap was

observed and the results were in good agreement with the phononic bandstructure

calculation by the plane wave expansion method.

CHAPTER 6 CONCLUSIONS

JIM Kwok Lung 156

CHAPTER 6

CONCLUSIONS

6.1 Conclusions

The present research focused on the feasibility of developing electro-optically

tunable photonic crystals and thermally tunable phononic crystals based on the

ferroelectric materials Ba0.7Sr0.3TiO3.

The tunability of the photonic bandgap of a Ba0.7Sr0.3TiO3/MgO one-dimensional

(1D) photonic crystal (PC) arising from the electro-optic effect was investigated. The

relation between the shift of the photonic bandgap and the change of the refractive index

of Ba0.7Sr0.3TiO3 was examined using the plane wave expansion (PWE) method. It is

found that the gap-center shifts linearly with the change of the refractive index of

Ba0.7Sr0.3TiO3. One-dimensional PC consisting of five periods, with each period having a

90 nm thick Ba0.7Sr0.3TiO3 layer and a 10 nm thick MgO layer, was fabricated using

pulsed laser deposition. X-ray diffraction revealed that all the layers were epitaxially

grown on the MgO (001) single crystal substrate. A photonic bandgap at ~464 nm was

observed in the transmission spectrum, which agreed well with the theoretical

calculation by the transfer matrix method. A 2-nm shift towards the longer wavelength

was shown when a dc voltage of 240 V (corresponding to an electric field of about 12


JIM Kwok Lung 157

MV/m) is applied across the coplanar electrodes on the film surface. This shift

corresponds to a 0.5% change in the refractive index of Ba0.7Sr0.3TiO3.

The photonic bandstructures and the photonic bandgap maps of the square lattice

type two-dimensional Ba0.7Sr0.3TiO3-based PC with circular air rods and square air rods.

The photonic bandstructures of the PC were computed using the PWE method. It was

found that there were two situations at which the photonic bandstructures and the

photonic bandgap maps of the circular air rod PCs and square air rod PCs would be

nearly identical: (1) if the characteristic length of the air rod is small (l < 0.45a), and (2)

if the frequency range is low enough (Ω<0.35). Furthermore, a better correlation in the

photonic bandstructures and the photonic bandgap maps between the circular and square

air rod 2D PC was obtained if the volume fraction correction, i.e. fc = fs wa = 0.89 x

(2ra), was considered. The photonic bandgap maps of two types of 2D Ba0.7Sr0.3TiO3-

based PC, namely, the air-hole-in-BST type and the BST-rod-in-air type both in square

lattice and in triangular lattice were studied. The result suggested that PCs in triangular

lattice contain richer bandgap feature in general. Bandgap features along different

symmetry directions have also been compared.

The refractive indices of the Ba0.7Sr0.3TiO3 thin film on a MgO (001) substrate

were measured using the prism coupling technique. The rib waveguide dimensions for

single mode TE wave propagation was determined by applying the effective index

method. A photonic crystal cavity embedded Ba0.7Sr0.3TiO3 rib waveguide which

functioned as a tunable filter (on-off switch) for λ = 1550 nm was designed using the


JIM Kwok Lung 158

finite-difference time-domain (FDTD) simulation. The required PC cavity was

composed of two 5-row-4 PC mirrors, which was formed by air holes arranged in a

triangular lattice in the Ba0.7Sr0.3TiO3 matrix, with cavity length of 800 nm. The radius

of the air holes was 250 nm and the periodicity was 625 nm. A 6-nm shift in the resonant

peak for a 0.5% change in the refractive index of the Ba0.7Sr0.3TiO3 was demonstrated in

the simulation. Photonic crystal cavities were fabricated on a Ba0.7Sr0.3TiO3 rib

waveguide by focused ion beam etching with satisfactory results.

The shear and transverse wave velocities of Ba0.7Sr0.3TiO3 ceramics as a function

of temperature were measured using the ultrasonic through-transmission technique. A


Ba0.7Sr0.3TiO3. Phononic crystal composed of Ba0.7Sr0.3TiO3 rods in a matrix of epoxy

was fabricated using the dice-and-fill method. The width of the Ba0.7Sr0.3TiO3 rods was

200 µm with a periodicity of 265 µm. The temperature dependence of the phononic

bandgaps of the phononic crystal was characterized by the reflection spectra which were

obtained using the ultrasonic pulse-echo technique. The reflection dip shifted from 12

MHz to 14 MHz when the temperature increased from 35 ˚C to 45 ˚C. Thermal tuning of

the phononic bandgap was observed and the results were in good agreement with the

phononic bandstructure calculation by the plane wave expansion method.

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JIM Kwok Lung 159

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