Beam Propagation Modelling of Whispering Gallery Microcavities by Mohammad Amin Cheraghi ...

84 Pages·2015·3.08 MB·English

Beam Propagation Modelling of Whispering Gallery Microcavities
by
Mohammad Amin Cheraghi Shirazi
B.Sc., University of Tehran, 2011
A Thesis Submitted in Partial Ful(cid:12)llment of the
Requirements for the Degree of
MASTER OF APPLIED SCIENCE
in the Department of Electrical and Computer Engineering
' Mohammad Amin Cheraghi Shirazi, 2015
University of Victoria
All rights reserved. This thesis may not be reproduced in whole or in part, by
photocopying or other means, without the permission of the author.
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Beam Propagation Modelling of Whispering Gallery Microcavities
by
Mohammad Amin Cheraghi Shirazi
B.Sc., University of Tehran, 2011
Supervisory Committee
Dr. Tao Lu, Supervisor
(Department of Electrical and Computer Engineering)
Dr. T. Aaron Gulliver, Departmental Member
(Department of Electrical and Computer Engineering)
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Supervisory Committee
Dr. Tao Lu, Supervisor
(Department of Electrical and Computer Engineering)
Dr. T. Aaron Gulliver, Departmental Member
(Department of Electrical and Computer Engineering)
ABSTRACT
Whispering Gallery Mode (WGM) microcavities have a wide range of applications
from fundamental physics researches to engineering applications due to their ultra
high quality factor (Q). For example, an ultra-high Q WGM cavity can be used as an
bio/nanosensor since a nano particle bound to the surface of the cavity will result in
a resonance wavelength shift. In the last decade lots of research have been conducted
on this topic, as a result, WGM biosensors are emerging as one of the mainstream
senors.
This thesis presents an efficient beam propagation method (BPM) simulation tool
to study the light propagation behaviour in WGM cavities. Using this tool, the
perturbation of the cavity properties caused by a polystyrene nano bead attached to
the surface of a WGM silica microcavity is investigated. Furthermore, we numerically
verify a three times sensitivity enhancement by fabricating a nanohole at the surface
of the WGM cavity sensor.
In addition, we study the open cavity structures, cavity-waveguide coupling, huge
WGM cavities, and deformed microcavities radiation. Finally, the impact of fabri-
cation inaccuracy on asymmetric WGM cavities is investigated in terms of quality
factor degradation.
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Contents
Supervisory Committee ii
Abstract iii
Table of Contents iv
List of Figures vi
Acknowledgements x
Dedication xi
1 Introduction 1
1.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Background 4
2.1 Quality factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Waveguide-Cavity Coupling . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Buildup Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Finite Difference Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4.1 Uniform Finite Difference Method . . . . . . . . . . . . . . . . 10
2.4.2 Non Uniform Finite Difference Method . . . . . . . . . . . . . 11
3 Finite Difference Mode Solver 13
3.1 Cartesian Mode Solver . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Cylindrical Mode Solver . . . . . . . . . . . . . . . . . . . . . . . . . 17
4 Finite Difference Beam Propagation Method 21
4.1 Two Dimensional Cartesian FD-BPM . . . . . . . . . . . . . . . . . . 21
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4.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.1.2 Transparent Boundary Condition . . . . . . . . . . . . . . . . 26
4.2 Cylindrical FD-BPM . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2.1 2D FD-BPM . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.2 Perfectly Matched Layer . . . . . . . . . . . . . . . . . . . . . 31
4.2.3 3D FD-BPM . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2.4 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 35
5 Applications 48
5.1 Nano Particle Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2 Enhanced Sensor Structure . . . . . . . . . . . . . . . . . . . . . . . . 51
5.3 Waveguide-Cavity Coupling . . . . . . . . . . . . . . . . . . . . . . . 57
5.4 Huge Cavity Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.5 Asymmetric Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6 Summary 64
Bibliography 66
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List of Figures
Figure 2.1 An illustration of taper-cavity coupling. The dashed red box
shows the coupling area as a 4-port transfer function. The 4
ports are the taper and cavity inputs and outputs. . . . . . . . 6
Figure 2.2 The taper output transmission vs. the normalized coupling pa-
rameter K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Figure 2.3 One dimensional (cid:12)nite difference scheme. . . . . . . . . . . . . . 11
Figure 3.1 Illustration of the cartesian and cylindrical coordinate systems. 13
Figure 3.2 The (cid:12)rst (red) and third (blue) order mode of a 8-(cid:22)m silica slab
waveguide in the air (left axis). The dotted black line indicates
the refractive index pro(cid:12)le of the waveguide on the right axis. . 17
Figure 3.3 The (cid:12)rst (blue), second (red), and third (green) order whispering
gallery modes of a silica ring resonator in the air (left axis). The
major radius is 45(cid:22)m and the width of the waveguide is 10(cid:22)m.
The dotted black line shows the refractive index pro(cid:12)le of the
waveguide on the right axis. . . . . . . . . . . . . . . . . . . . . 19
Figure 3.4 Four different whispering gallery modes of a silica micro toroid
in water in the central wavelength of 970nm. The one on the top
left is the fundamental mode. The major radius of the toroid is
45(cid:22)m and the minor diameter is 10(cid:22)m. The black lines show the
boundaries of the cavities. . . . . . . . . . . . . . . . . . . . . . 20
Figure 4.1 Field intensity inside a 4-mm long, 8-(cid:22)m core diameter bent (cid:12)ber. 29
Figure 4.2 Re(cid:13)ectivity in dB vs. different (cid:27) values. (cid:27) = 2:5(cid:2)1016 is the
0 0
optimum value. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Figure 4.3 The (cid:12)eld intensity of a straight waveguide, launching toward the
computation window edge, for three different (cid:27) values: 0 (top
0
left) , 2:5(cid:2)1016 (bottom), and 1020 (top right). The white lines
indicate the inner 2-(cid:22)m PML edges. . . . . . . . . . . . . . . . 33
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Figure 4.4 Discretization scheme in r-z plain. . . . . . . . . . . . . . . . . 34
Figure 4.5 Field intensity of a 45(cid:22)m major radius and 10(cid:22)m width silica
ring, in logarithm scale. The radiations is observable. The solid
white lines are illustrating the ring borders and the dashed black
lines are showing the PML layer boundaries. . . . . . . . . . . . 36
Figure 4.6 Intensity distribution of a huge microdisk with the radius of 1mm. 37
Figure 4.7 (a) Quality factor vs. grid spacings in the r and ϕ direction. (b)
cross section view of (a) at ∆r = 3:1 nm and it’s relative error.
(c) variations for the cross section at ∆ϕ = 0:0015 radians with
the corresponding relative error. . . . . . . . . . . . . . . . . . 39
Figure 4.8 Resonance wavelength of the ring resonator and its relevant error
vs. grid size in the radial direction. . . . . . . . . . . . . . . . . 40
Figure 4.9 The gaussian (cid:12)eld pattern (red) after (a) 0, (b) 25, (c) 125 round
of propagation inside a 45-(cid:22)m major radius and 10-(cid:22)m channel
width silica ring resonator. The dashed blue curve is the funda-
mental mode pro(cid:12)le. . . . . . . . . . . . . . . . . . . . . . . . . 42
Figure 4.10Magnitude of the overlap factor between the output pro(cid:12)le and
mode pro(cid:12)le vs. round trip number. The red curve illustrates
the overlap factor departure from unity. . . . . . . . . . . . . . 43
Figure 4.11Power at the output of a 45(cid:22)m silica ring resonator after each
round of propagation normalized to the (cid:12)rst round power for
resonance (red) and for (cid:21) (1+ 1 ) (green). . . . . . . . . . . . 44
res 2Q
Figure 4.12Buildup factor and its relative error vs. grid spacing in the radial
direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Figure 4.13Quality factor of a micro toroid and its relative error vs. grid
spacing in the ϕ direction. R is the outer radius of the toroid
(R = 50(cid:22)m). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Figure 4.14Resonancewavelength(a)andqualityfactor(b)ofamicrotoroid
and its relative error vs. the transverse plane grid spacings. . . 47
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Figure 5.1 r-ϕ plain cross section view of micro toroid electrical (cid:12)eld pro-
(cid:12)le (a) and (cid:12)eld intensity in logarithm scale (b), perturbed by
a 400-nm radius polystyrene bead. The black lines depict the
cavity outer edge and the bead boundaries. (c) The resonance
wavelength shift vs. the bead radius using 2D BPM (green), 3D
BPM (red), and perturbation method (black). . . . . . . . . . . 49
Figure 5.2 Resonance wavelength shift arising from a 400-nm radius PS
bead and its relative error vs. grid spacing in the ϕ direction. . 50
Figure 5.3 Resonance wavelength shift arising from a 400-nm radius PS
bead and its relative error vs. grid spacing in the r and z direction. 51
Figure 5.4 (a) Field intensity (in logarithm scale) of a 1:15 mm radius disk
at r-ϕ cross section. The black line is the cavity boundaries. The
r-z cross section view of the (cid:12)eld intensity in logarithm scale (b)
after the notch (ϕ = 5◦), (c) inside the notch (ϕ = 3◦), and (d)
before the notch (ϕ = 1◦). . . . . . . . . . . . . . . . . . . . . . 52
Figure 5.5 (a) Wavelength shift vs. the hole depth for a 150-nm radius
hole-bead. The dashed red curve is the cavity mode pattern. (b)
Enhancement vs. hole-bead radius (blue) for a 475-nm depth
hole. Black line is the ultimate enhancement for this cavity (i.e.
2:23). Red curve is the quality factor vs. the hole-bead radius
(right axis) and the green line is the bare cavity Q. (c) Field
intensity in a logarithm scale for a silica microtoroid with a 475-
nm depth, 150-nm radius hole on it. The same radius PS bead
is placed inside the hole. . . . . . . . . . . . . . . . . . . . . . . 54
Figure 5.6 Theratioofresonancewavelengthshift(∆(cid:21))andcavitylinewidth
((cid:14)(cid:21)) vs. the ratio of the hole depth to the bead-hole radius. . . 56
Figure 5.7 The (cid:12)eld intensity in logarithm scale for (a) 300 nm (b) 625 nm
full hole depth. The hole-bead radius is 150 nm. . . . . . . . . 57
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Figure 5.8 (a) The 1:2-(cid:22)m diameter taper output transmission vs. the
microsphere-taper gap, modeled by the FD-BPM (red) and mea-
sured experimentally (black). (b) The normalized coupling pa-
rameter K vs. the gap. The FD-BPM and experimental results
are presented in black and red curves respectively. (c) The r-ϕ
plain cross section view of the (cid:12)eld intensity plot in logarithm
scale. The black lines indicates the borders of the 67-(cid:22)m diam-
eter microsphere and 1:2-(cid:22)m diameter taper. The experimental
results are extracted from [1]. . . . . . . . . . . . . . . . . . . . 59
Figure 5.9 r-z plain view of the (cid:12)eld intensity in a (a) 90-(cid:22)m and a (b)
1-mm diameter toroid. The black line is the resonator edge. r-ϕ
plain view of the (cid:12)eld intensity in logarithm scale in a (c) 90-(cid:22)m
and a (d) 1-mm diameter toroid. (c) The purple line is showing
the resonator outer edge and the black lines are showing the 2-
(cid:22)m PML. (d) The black line is showing the resonator outer edge
and the white lines are showing the 2-(cid:22)m PML. . . . . . . . . . 61
Figure 5.10The (cid:12)eld intensities at the second round of propagation for (a)
R = 45 (cid:22)m and R = 37:5 (cid:22)m as well as (b) R = 45 (cid:22)m
0◦ 90◦ 0◦
and R = 32:5 (cid:22)m. (c) Quality factor vs. ellipticity. . . . . . . 63
90◦
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ACKNOWLEDGEMENTS
I would like to express my gratitude to my supervisor Dr. Tao Lu for his insightful
guidance and helpful suggestions through my graduate studies. I would also like to
thank my colleagues Niloofar Sadeghi, WenYan Yu, Serge Vincent, Xuan Du, and
LeYuan Pan for their helpful discussion and assistance.