characterisation of acoustic waves in optical waveguides

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by  Neil Stewart

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Citation: Sriratanavaree, S. (2014). The characterisation of acoustic waves in optical
waveguides. (Unpublished Doctoral thesis, City University London)
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Sriratanavaree, Suchara (2014). The characterisation of acoustic waves in optical waveguides.
(Unpublished Doctoral thesis, City University London)
City Research Online
Original citation: Sriratanavaree, Suchara (2014). The characterisation of acoustic waves in optical
waveguides. (Unpublished Doctoral thesis, City University London)
Permanent City Research Online URL: http://openaccess.city.ac.uk/3483/

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CHARACTERISATION
OF
ACOUSTIC WAVES
IN OPTICAL WAVEGUIDES

BY
Suchara SRIRATANAVAREE

A THESIS SUBMITTED FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
AT

CITY UNIVERSITY LONDON

PHOTONICS MODELLING GROUP
SCHOOL OF ENGINEERING AND MATHEMATICAL SCIENCES
NORTHAMPTON SQUARE, LONDON EC1V 0HB
UNITED KINGDOM

JANUARY 2014
TABLE OF CONTENTS

List of Tables............................................................................................................... vi
List of Figures ............................................................................................................ vii
Symbols and Abbreviations ...................................................................................... xiv
Acknowledgements .................................................................................................. xvii
Declaration… .......................................................................................................... xviii
Abstract……… ......................................................................................................... xix

1 INTRODUCTION ............................................................... 1

1.1 Introduction to Acoustic Waves.................................................................... 2
1.1.1 Properties of Acoustic plane wave - Wavelength, Frequency
and Velocity and their interaction with optical wave ......................... 2
1.2 Nonlinear Scattering Effects ......................................................................... 3
1.3 Stimulated Brillouin Scattering (SBS) and basic theory ............................... 4
1.3.1 Basic Theory ....................................................................................... 5
1.3.2 Physical process .................................................................................. 5
1.3.3 Applications of SBS Phenomenon ...................................................... 7
1.4 Numerical methods ....................................................................................... 8
1.5 Photonics ..................................................................................................... 10
1.6 Aims and Objectives ................................................................................... 10
1.7 Structure of the thesis .................................................................................. 12
1.8 Summary ..................................................................................................... 14

2 ACOUSTIC THEORY ...................................................... 15

2.1 Introduction ................................................................................................ 15
2.2 Sound Wave in Nonpiezoelectric Materials: One Dimensional Theory ..... 16
2.2.1 Stress ................................................................................................. 17
i
Table of contents

2.2.2 Displacement and Strain in One Dimensional .................................. 18
2.2.3 Equation of Motion ........................................................................... 19
2.3 Tensor Notation and Constitutive Relations Materials ............................... 19
2.3.1 Displacement and Strain ................................................................... 20
2.3.2 Stress in a Rectangular Cube ............................................................ 20
2.3.3 Hooke’s Law and Elasticity .............................................................. 23
2.3.4 Tensor Notation ................................................................................ 24
2.3.5 Reduced Subscript Notation ............................................................. 25
2.4 Waves in Isotropic Media .......................................................................... 27
2.4.1 Lamb Constants ............................................................................... 27
2.4.2 Shear and Longitudinal Waves ........................................................ 28
2.5 Refraction and Snell's Law ........................................................................ 29
2.5.1 Plane Wave Reflection and Refraction ............................................. 30
2.6 Modes of Sound Wave Propagation ........................................................... 33
2.7 Surface Waves ............................................................................................ 35
2.7.1 Rayleigh Wave in an Isotropic Medium ........................................... 36
2.8 2D Acoustic Waveguide ............................................................................. 38
2.9 Properties of Waves in Waveguide ............................................................. 41
2.10 Mode Classification .................................................................................... 43
2.10.1 Pure Longitudinal Mode ................................................................. 43
2.10.2 Pure Transverse Modes ................................................................... 45
Torsional Mode .......................................................................... 47
Radial Mode ............................................................................... 48
Torsional-Radial Mode .............................................................. 50
2.10.3 Quasi-acoustic Modes ..................................................................... 52
Quasi-Longitudinal ..................................................................... 52
Quasi-Transverse ........................................................................ 53
2.11 Summary ............................................................................................................ 55

3 FINITE ELEMENT METHOD ...................................... 56

3.1 Introduction ................................................................................................ 56
ii
Table of contents

3.2 Variational Formulations ............................................................................ 58
3.2.1 Generalization of an Element............................................................ 59
3.3 Discretization of the Problem ..................................................................... 59
3.4 Interpolation Functions and the Shape Function......................................... 60
3.5 Triangle Coordinates in Two dimensions ................................................... 62
3.6 Implementation of the Acoustic Waveguide Code with FEM .................... 64
3.6.1 Finite Element Formulation .............................................................. 66
3.6.2 Generalised Coordinate and the Order of the Polynomials .............. 66
3.6.3 Effect of Crystal Symmetry on Elastic Constants............................. 69
3.7 Benchmarking - Steel Waveguide .............................................................. 73
3.7.1 Comparison of the FEM Numerical Results ..................................... 74
3.7.2 Bending Mode ................................................................................... 78
3.7.3 Torsional Mode ................................................................................. 82
3.7.4 Hybridness ........................................................................................ 85
3.8 Summary ..................................................................................................... 86

4 SILICA FIBRES ................................................................ 87

4.1 Introduction ................................................................................................ 87
4.2 Weakly Guiding Silica Fibres ..................................................................... 88
4.2.1 Case 1: Shear Modes ......................................................................... 88
4.2.2 Case 2: Longitudinal Modes ............................................................. 95
4.2.3 Case 3: ∆V /V ≪1 and ∆V /V ≪1 ................................................ 96
L L S S
4.3 Silica Nanowires ....................................................................................... 100
4.3.1 Silica Nanowire with an Acoustic Low Index Cladding ................ 100
4.3.2 Studied of the Optical Waveguide for
the Designs of Acousto-optic Interaction ....................................... 101
4.3.3 The Analysis of the Acousto-optic Interaction with
Waveguide Mode at Radius, R=0.6 μm .......................................... 102
4.3.4 Stimulated Brillouin Scattering (SBS) Generation ......................... 105
4.4 Summary .................................................................................................. 107

iii
Table of contents

5 SILICA RECTANGULAR WAVEGUIDE .................. 108

5.1 Introduction ............................................................................................... 108
5.2 Weakly Guiding SiO Rectangular Waveguide ........................................ 109
2
5.2.1 The Comparison of Two Different Dimensions of Waveguides .... 110
5.2.2 Effect of the Width Variation ......................................................... 118
5.2.3 Variation of the Waveguide Widths with Different Frequencies ... 119
5.2.4 Results of the Overlap Between Optical and Acoustic Modes ....... 121
5.3 Strongly Guiding Silica Rectangular Waveguide ..................................... 128
5.3.1 Waveguide Structure ....................................................................... 128
5.4 Acousto-optic Interaction .......................................................................... 146
5.4.1 Study of the Waveguide when H is Varied ..................................... 148
5.4.2 Study of the Waveguide when W is Varied .................................... 151
5.5 Strip waveguide ........................................................................................ 154
5.5.1 Study of Longitudinal modes in Porous Silicon Waveguide .......... 154
5.5.2 Study of Overlap Between Optical and Acoustic Modes ............... 157
5.5.3 Study of Shear modes in Silicon Strip Waveguide ......................... 161
5.5.4 Study of the PS Waveguide when Width (W) is Varied ................. 164
5.6 Summary ................................................................................................... 166

6 SLOT WAVEGUIDE ...................................................... 167

6.1 Introduction ............................................................................................... 167
6.2 Analysis of the Slot Waveguide ................................................................ 168
6.2.1 Waveguide Structure ....................................................................... 168
6.2.2 Study of Slot Waveguide Acoustic Modes ..................................... 170
6.3 Acousto-optic Interaction in Slot Waveguide ........................................... 176
6.3.1 Effect of Slot Width (W ) ............................................................... 178
S
6.3.2 The SBS Frequency ........................................................................ 178
6.3.3 Varied Height (H) of the Slot.......................................................... 181
6.3.4 Varied Width W (Si) on the Side of Slot ........................................ 182
6.4 Summary ................................................................................................... 184
iv
Table of contents

7 CONCLUSION AND SUGGESTIONS FOR
FURTHER WORK ......................................................... 185

7.1 Conclusion .................................................................................................. 185
7.2 Suggestions for future work ........................................................................ 192

Appendix A: Stress, Strain and Reduced Notation ............................................... 193

Appendix B: References ....................................................................................... 206
v
LIST OF TABLES

Chapter 2
Table 2.1 Tensor and vector components .................................................................................. 25
Table 2.2 Example of the reduced tensor notation ..................................................................... 25
Table 2.3 Relations between the various elastic constants and Poisson’s ratio σ
for an isotropic solid .................................................................................................. 28
Table 2.4 Summarized possible wave modes in solid ................................................................ 34

Chapter 3
Table 3.1 Material properties of steel and xenon gas used in this work .................................... 74
Table 3.2 The eigenfrequency of the first two Steel waveguide modes with
Atkin's extrapolation calculation and comparison with references papers ................. 74

Chapter 4
Table 4.1 Material properties of a silica waveguide used in this work ...................................... 88
Table 4.2 Material properties of silica and xenon waveguide.................................................. 100

Chapter 5
Table 5.1 Material properties of silica with 3% Ge doped silica ............................................. 109
Table 5.2 Material properties and elastic stiffness constants for the strip waveguide ............. 154
Table 5.3 The material properties of the waveguide ................................................................ 161

Chapter 6
Table 6.1 Material properties and elastic stiffness constants for the slot waveguide ............... 169

vi
LIST OF FIGURES

Chapter 1
Figure 1.1 Spontaneous Brillouin scattering top
and Stimulated Brillouin scattering bottom phenomenon .............................................. 5

Chapter 2
Figure 2.1 (a) Longitudinal wave propagation,
(b) shear wave propagation (full cube not shown) ....................................................... 16
Figure 2.2 (a) Stress in the longitudinal direction for a slab of length l,
(b) stress in the shear direction .................................................................................... 17
Figure 2.3 Application of general stress components ................................................................... 20
Figure 2.4 The diagram of reflected and refracted longitudinal (V ) wave .................................. 29
L
Figure 2.5 Longitudinal wave incident on a free surface. There are two reflected waves:
"longitudinal wave" and "shear wave" ........................................................................ 30
Figure 2.6 Shear wave incident on a free surface .......................................................................... 32
Figure 2.7 Longitudinal and shear wave direction of propagation ................................................ 33
Figure 2.8 Configuration for acoustic surface wave analysis ........................................................ 36
Figure 2.9 Isotropic Rayleigh wave velocity V as a function of the bulk shear wave
R
velocity V and the bulk longitudinal wave velocity V .............................................. 37
s L
Figure 2.10 The 2D acoustic waveguide with the propagation direction (a) rectangular
and (b) circular ............................................................................................................. 38
Figure 2.11 Wave motion of guided waves in plates for (a) symmetric mode
and (b) antisymmetric mode ........................................................................................ 42
Figure 2.12 Wave motion of guided waves in bars or pipes for (a) longitudinal mode,
(b) flexural mode and (c) torsional mode ..................................................................... 42
Figure 2.13 U profile of the longitudinal L mode or U mode .................................................... 43
z 01 z
Figure 2.14 Higher order longitudinal (a) L (b) L of the U modes ........................................... 44
11 21 z
Figure 2.15 Fundamental shear S along y-axis for the U displacement (a) Uy mode,
01 y
(b) U displacement (c) propagation in the y direction in fibre .................................... 45
y
Figure 2.16 Fundamental shear S for the U displacement (a) Ux mode,
01 x
(b) the displacement in the x direction in fibre along x-axis ........................................ 46
Figure 2.17 Fundamental shear S for the vector plot U U U ......................................... 46
01 t x y
Figure 2.18 Shear S for the Torsional mode (T) (a) U (b) U displacement in fibre ................... 47
11 x y
Figure 2.19 Shear S for the Torsional mode (T) (a) vector plot U U U ,
11 t x y
(b) the rotation of the twist in the fibre ........................................................................ 48
Figure 2.20 S for the Radial mode (R) (a) U (b) U displacement in fibre .................................. 48
11 x y
vii