novel waveguide architectures for light sources in silicon photonics

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Theses and Dissertations

2011

Novel Waveguide Architectures for Light Sourcesin Silicon PhotonicsRavi S. TummidiLehigh University

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Novel Waveguide Architectures for Light Sources

in Silicon Photonics

by

Ravi Sekhar Tummidi

Presented to the Graduate and Research Committee

of Lehigh University

in Candidacy for the Degree of

Doctor of Philosophy

in

Electrical and Computer Engineering

Lehigh University

September 2011

ii

Copyright © by Ravi Sekhar Tummidi

September 2011

iii

Approved and recommended for acceptance as a dissertation in partial fulfillment for the degree of Doctor of Philosophy.

_____________________ Date __________________________________________ Prof. Thomas L. Koch (Dissertation Director)

_____________________ Accepted Date

Committee Members:

__________________________________________ Prof. Thomas L. Koch (Chairman and Advisor)

__________________________________________ Prof. Volkmar R. Dierolf __________________________________________ Prof. Michael J. Stavola __________________________________________ Prof. Nelson Tansu __________________________________________ Dr. Mark A. Webster

iv

Acknowledgement

I first of all would like to thank my advisor, Prof. Thomas L. Koch. One of the

proudest moments of my life was when Prof. Koch offered me to be his student. I hope I

have done justice to the confidence he bestowed on me. I have learnt a great deal from

him both as a scientist and as a person and will keep making my best effort to put it in to

practice.

The work detailed in this dissertation wouldn’t have been possible with out the

support, encouragement and collaboration of Dr. Robert M. Pafchek, Dr. Thach G.

Nguyen, Dr. Mark A. Webster, Prof. Arnan Mitchell and Kangbaek Kim. It was a very

fulfilling scientific endeavor because of their company and I hope it continues into the

future.

I would also like to thank my Ph.D. committee Prof. Volkmar R. Dierolf, Prof.

Michael J. Stavola, Prof. Nelson Tansu and Dr. Mark A.Webster for their insightful

queries and suggestions over the course of my graduate work at Lehigh. I would

especially like to thank Prof. Nelson Tansu under whom I did my master’s for getting me

started in the photonics field.

Over the course of my graduate work at Lehigh I had the incredible privilege to be a

part of the Silicon Laser Multi University Research Initiative (MURI) project. As a result

v

I got the unique opportunity to interact and collaborate with some of the brightest

scientists in my field for which I am very thankful.

Thanks to all my colleagues at the Center for Optical Technologies (COT) for their

support and encouragement over the years. A special thanks to Anne Nierer. Life as a

graduate student became much simpler with her around at COT.

Growing up I was privileged to be taught by some wonderful teachers who always

encouraged and prodded me to do better. The list is just too long but I have to mention

Alamelu madam, Shyamala madam and Sesha madam by name. This work is my “Guru

Dakshina” to all my teachers. I hope I have lived up to your expectations.

I want to thank my friends and relatives for their good wishes. I have to single out

Kalyan, my cousin, with out whose encouragement I wouldn’t have come to Lehigh or to

the United States for that matter. Thank you bhaiyya. Last but not the least, I want to

thank my mom, Mrs. T. Jhansi and my sister Sree Lekha, for their love and support with

out which I couldn’t have persevered through this challenging period. Thank you for your

patience through all these years.

I also gratefully acknowledge financial support from Army Research Laboratory

(ARL), Pennsylvania Ben Franklin Technology Development Authority (BFTDA) and

Air Force Office of Scientific Research (AFOSR) Silicon Laser MURI under Dr. Gernot

Pomrenke for the work carried out and presented in this dissertation.

vi

to my father

T.N.S. Satya Narayana

vii

Table of Contents

Acknowledgement ........................................................................................................... IV

List of Figures ................................................................................................................... X

Abstract .............................................................................................................................. 1

Chapter 1. Introduction ........................................................................................... 3

1.1 The Problem and Our Approach ................................................................ 6

1.2 Thesis Organization ................................................................................. 11

Chapter 2. Ultra Low Loss Quasi Planar Ridge Waveguides ............................ 13

2.1 Theory of Lateral Leakage Loss in Shallow Ridge Waveguides ............ 15

2.2 Experimental Evidence of “Magic Widths” ............................................ 19

2.3 Numerical Analysis of Lateral Leakage Loss in Shallow Ridge Straight

Waveguides .............................................................................................................. 22

2.3.1 TM-TE Mode Coupling at a Step Discontinuity............................... 22

2.3.2 Width Dependent Propagation Loss ................................................. 25

2.4 Waveguide Fabrication ............................................................................ 28

2.5 Waveguide Loss Measurement Technique .............................................. 30

2.6 Results and Discussion ............................................................................ 32

2.7 An Ultra Compact Waveguide polarizer based on “Anti-Magic Widths”

…………………………………………………………………………..35

viii

2.7.1 Polarizer Design Principle and Analysis........................................... 36

Chapter 3. Anomalous Losses in Curved Waveguides and Directional

Couplers at “Magic Widths”.......................................................................................... 43

3.1 Lateral Leakage of TM-like Mode in Thin Ridge SOI Ring ................... 44

3.2 Rigorous Simulation Approaches ............................................................ 46

3.3 Numerical Results and Discussion .......................................................... 47

3.3.1 The Impact of Waveguide Curvature on the Leakage Cancellation

Without Re-Entry ............................................................................................. 47

3.3.2 Leakage Cancellation in Waveguide Rings with Re-Entry: “Magic

Radius” Phenomenon ....................................................................................... 50

3.3.3 Mode Field Distribution .................................................................... 56

3.3.4 Wavelength Dependence of Lateral Leakage Loss ........................... 58

3.4 Directional Coupler ................................................................................. 60

Chapter 4. Low Loss Si-SiO2-Si TM Slot Waveguides ....................................... 63

4.1 Waveguide Design and Fabrication ......................................................... 64

4.2 Experimental Evaluation ......................................................................... 69

Chapter 5. Erbium Doped 8 nm Horizontal Slot Waveguides ........................... 72

5.1 Erbium Doped Slot Waveguide Fabrication ............................................ 74

5.2 Photoluminescence Measurements .......................................................... 75

5.3 Time Resolved Photoluminescence Measurements ................................. 76

5.4 Spontaneous Emission in Waveguide Media .......................................... 80

5.4.1 Spontaneous Emission into a Particular Waveguide Mode .............. 81

5.4.2 Total Spontaneous Emission Rate Including All Modes .................. 86

ix

5.4.3 Relation to Purcell Factor ................................................................. 92

5.5 Analysis of Experimental Results and Discussion .................................. 98

Chapter 6. Future Work and Outlook ............................................................... 101

6.1 Mask Overview ...................................................................................... 102

6.2 Passive Ridge Waveguides .................................................................... 103

6.3 Passive Horizontal Si-SiO2-Si Slot Waveguides ................................... 107

6.4 Ring Resonators for Critical Coupling in Horizontal Si-SiO2-Si Slot

Waveguides ............................................................................................................ 115

6.5 Active Horizontal Si-Er doped SiO2-Si Slot Waveguides ..................... 119

6.6 Summary and Future Work ................................................................... 123

Appendix A: Spontaneous Emission Rate in a Slot Waveguide ............................... 127

Bibliography .................................................................................................................. 151

x

List of Figures

Figure 1.1. Normalized profile of the dominant electric field component of the

fundamental TE and TM mode for the 8 nm horizontal slot waveguide shown in the inset

............................................................................................................................................. 9

Figure 1.2. A shallow ridge horizontal slot waveguide structure ................................... 10

Figure 2.1. (a) Ridge waveguide geometry (b) Slab mode indexes used in effective index

calculations. The modal indexes for the guided TE-like and TM-like modes, respectively,

lie intermediate between their slab values for slab thicknesses t1 and t2 .We use index

values of nSi = 3.475 and nSiO2 = 1.444 at 1.55 μm wavelength. ...................................... 15

Figure 2.2. (a) Phase-matching diagram showing TM-like waveguide mode phase

matched to a propagating TE slab mode in the lateral cladding (b) Ray picture of the TM

mode lateral leakage ......................................................................................................... 17

Figure 2.3. (a) AFM image of a waveguide formed by wet-etching (b) Experimentally

measured optical transmission (log scale) for the fundamental TE-like and TM-like

modes for the waveguides ................................................................................................. 20

Figure 2.4. (a) AFM image of a waveguide formed using thermal oxidation (b)

Experimentally measured optical transmission (log scale) for the fundamental TE-like

and TM-like modes for the waveguides............................................................................ 21

xi

Figure 2.5. Mode coupling diagram showing the TM-TE mode coupling at the ridge

boundary ........................................................................................................................... 23

Figure 2.6. (Color) Magnitude of the reflected TM mode, reflected and transmitted TE

modes as a function of the incident TM mode propagation angle .................................... 23

Figure 2.7. (Color) Relative phase of the reflected and transmitted TE modes as a

function of the incident TM mode propagation angle ...................................................... 24

Figure 2.8. (Color) Simulated loss of TM-like mode in shallow ridge waveguide shown

in inset which shows strong width dependence ................................................................ 26

Figure 2.9. Electric field distributions of the fundamental TM-like mode for (a) 1.2 μm

and (b) 1.429 μm waveguide widths ................................................................................. 27

Figure 2.10. Processing steps for shallow ridge waveguide fabrication ......................... 28

Figure 2.11. (a) AFM trace of a 1.44 μm ridge waveguide with <0.2 nm surface

roughness (b) Quasi-planar ridge SOI waveguide geometry. BOX thickness is 2μm ...... 29

Figure 2.12. Setup to measure the resonance characteristics of (a) a weakly coupled ring

resonator with separate thru and drop ports and (b) an exemplary drop-port response .... 30

Figure 2.13. Drop-port responses for the TE and TM modes for a 400 μm radius ring

resonator with “magic width” waveguide of 1.43 μm ...................................................... 33

Figure 2.14. (a) XY view and (b) XZ view of the 3D plot of loss of the fundamental TM

mode of a SOI ridge waveguide with 220 nm c-Si in the core (inset) .............................. 38

Figure 2.15. Loss and effective Index nTM-WG of the fundamental TM-like mode of a

SOI ridge waveguide with 220 nm c-Si in the core for (a) 90 nm (b) 148 nm and (c) 150

nm ridge heights ................................................................................................................ 40

xii

Figure 3.1. (a) Cross section, and (b) plan view and mode coupling diagram of a SOI

thin-ridge ring resonator. Waveguide dimensions are shown ........................................... 44

Figure 3.2. (Color) Simulated propagation loss of the TM-like guided mode of a thin-

ridge SOI bent waveguide in the absence of secant TE waves inside the ring (a) as a

function of waveguide width for different bend radii and (b) as a function of the bend

radius when the waveguide width is fixed at 1.43 μm. Conventional bending losses are

also shown. ........................................................................................................................ 49

Figure 3.3. Simulation domain used to model rings with R0 = 100 μm in FEM ............ 51

Figure 3.4. (Color) Propagation loss of the fundamental TM-like mode of a ring as a

function of ring radius. Waveguide width is W = 1.43 μm ............................................... 52

Figure 3.5. (Color) Propagation loss of the fundamental TM-like mode of a ring

resonator as a function of the waveguide width for different ring radii ........................... 54

Figure 3.6. Propagation loss of the fundamental TM-like mode of a ring as a function of

the ring waveguide width and ring radius. Radius around (a) 200 μm and (b) 400 μm ... 55

Figure 3.7. Electric field distributions of the fundamental TM-like mode of a ring with a

waveguide width of 1.35 μm and radius at (a) 199.62 μm (high loss) and (b) 198.95 μm

(low loss) ........................................................................................................................... 57

Figure 3.8. Electric field distributions of the fundamental TM-like mode of a ring with a

radius of (a) 199.84 μm (high loss) and (b) 200.46 μm (low loss). Waveguide width of

1.43 μm ............................................................................................................................. 58

Figure 3.9. Wavelength dependent loss of the TM-like mode in thin-ridge SOI rings

with a waveguide width of (a) 1.43 μm and (b) 1.35 μm and ring radii as shown ........... 60

xiii

Figure 3.10. (Color) Inset: Cross sectional view of ridge waveguides in a directional

coupler configuration with waveguide centre to centre separation S and a symmetric

waveguide width of W. Graphs: Loss of the fundamental and first order TM super mode

of the directional coupler as a function of waveguide width (W) for a waveguide centre to

centre of (a) S=3 μm and (b) S=1 μm ............................................................................... 62

Figure 4.1. Processing steps for shallow ridge horizontal slot waveguide. Shallow ridge

waveguide in step 1 formed using the process described in Fig. 2.10 .............................. 65

Figure 4.2. Scanning electron micrograph of a similar planar slot structure with 8.3 nm

slot visible using ESED mode imaging ............................................................................ 66

Figure 4.3. Weighting function for index changes for TE and TM modes, illustrating the

14X enhancement of sensitivity to index change in the SiO2 slot for the TM mode over

TE mode for the slot waveguide shown in the inset. The integrated weighting function

across the thickness of the SiO2 slot gives an effective confinement of 23.5%. .............. 68

Figure 4.4. Scan of ring resonance illustrating ΔνFWHM =625 MHz with a corresponding

Q~3 X 105 ......................................................................................................................... 70

Figure 5.1. (a) Implant and anneal conditions and (b) cross sectional view of Erbium

doped slot waveguide ........................................................................................................ 74

Figure 5.2. Setup for measurement of Erbium doped slot waveguide luminescence ...... 75

Figure 5.3. Erbium doped slot waveguide photoluminescence at different waveguide

coupled pump powers, expressed as power spectral density (PSD) ................................. 76

Figure 5.4. (a) Pump pulse and (b) Luminescence signal using gated integrator and

BOXCAR averager with system preamplifiers. Observed peak luminescence matches

calculated value ................................................................................................................. 78

xiv

Figure 5.5. (a) Photoluminescence measured with the system pre-amplifiers replaced by

a low noise high gain TIA in conjunction with gated integrator and BOXCAR averager

(b) Luminescence decay signal on a semi-log plot with a lifetime of ~ 152 μsec ........... 79

Figure 5.6. (Inset) Ultra-thin 8.3 nm slot slab waveguide structure. Plot of effective

inverse modal thickness in the vertical direction as a function of position for both the TE

and TM modes .................................................................................................................. 85

Figure 5.7. Plot of F(z)/(n(z)Ffree) as defined in Eq. 5.30 as function of position in the 8.3

nm slot structure shown in the inset. ................................................................................. 90

Figure 5.8. Plot of total spontaneous emission rate enhancement, averaged over

polarizations, as a function of slot thickness. Also shown are all the contributions,

dominated by the TM slot guided mode, along with the efficiency of emission into just

the TM slot mode. Values indicated for the 8.3 nm slot case ........................................... 91

Figure 5.9. A Fabry-perot resonator of length Lc formed by inserting mirrors of

reflectivity R in an arbitrarily long waveguide of length L .............................................. 94

Figure 6.1. Image showing the mask set consisting of 54 blocks arranged in a 9 by 6 grid.

An exemplary block is marked ....................................................................................... 102

Figure 6.2. Image of a typical individual mask block as marked in Fig 6.1 ................. 103

Figure 6.3. Transverse TM mode count for a 220 nm fixed Si core ridge waveguide

shown in the inset as a function of waveguide width and ridge height. TM mode count is

indicated .......................................................................................................................... 104

Figure 6.4. Fundamental TM mode loss for the shallow 220 nm fixed Si core ridge

waveguide shown in the inset of Fig 6.3 as a function of waveguide width for ridge

heights at and in the vicinity of 7 nm .............................................................................. 105

xv

Figure 6.5. (Color) Fundamental TM mode loss for the shallow 220 nm fixed Si core

ridge waveguide shown in the inset of Fig 6.3 in ring resonator configuration as a

function of ring radius. The width of the waveguide in the ring (WR) is 1.458 μm and

1.35 μm. The ridge height is (a) 6 nm (b) 7 nm and (c) 8 nm ........................................ 106

Figure 6.6. Weighting function confinement % at 1550 nm in (a) a-Si and (b) 10 nm

SiO2 slot as a function of c-Si and a-Si thickness ........................................................... 110

Figure 6.7. Analytical Bending loss at 1480 nm for a 10 nm ridge slot waveguide

structure with a radius of (a) 400 μm and (b) 200 μm as a function of the c-Si thickness

in the core and amorphous silicon thickness with the waveguide width equaling the

“magic width” at 1550 nm .............................................................................................. 111

Figure 6.8. (a) Schamatic of a 10 nm ridge, 100 nm a-Si – 15 nm SiO2 – 213.25 nm c-Si

slot waveguide structure (b) Fundamental TM mode loss of this straight waveguide at

1530 nm and 1480 nm as a function of the width W of the c-Si ridge ........................... 113

Figure 6.9. (Color) Fundamental TM mode loss for the 15 nm slot, waveguide shown in

Fig. 6.8 (a) in ring resonator configuration as a function of ring radius around (a) 400 μm

and (b) 450 μm. The width of the c-Si ridge waveguide in the ring is 1.605 μm. Traces

shown for ridge heights of 9 nm, 10 nm and 11 nm ....................................................... 114

Figure 6.10. (a) Schematic of TWR ring resonator coupled to a bus waveguide (b) Power

transmission through the bus waveguide as a function of the normalized loaded quality

factor (QL) ....................................................................................................................... 116

Figure 6.11. Coupling Quality factor for the 15 nm slot, 10 nm ridge structure as a

function of the minimum gap between the tangential bus waveguide and the ring as

shown in Fig 6.12 (a) ...................................................................................................... 117

xvi

Figure 6.12. Section of a bus waveguide-resonator system showing the coupler

configuration (a) Regular coupler - Bus waveguide running tangentially to the ring with

gap of G μm at minimum separation between the two (b) Curved coupler - Bus

waveguide wrapped around the ring to increase the interaction length and hence to reduce

the QC at a gap of G μm. The curved interaction length is parameterized by the arc angle

θ ....................................................................................................................................... 117

Figure 6.13. (Color) Coupling Quality factor for the 15 nm slot, 10 nm ridge waveguide-

resonator system with a curved coupler as a shown in Fig 6.12 (b). Arc angle θ is defined

in Fig. 6.12(b). Gap is fixed at 1.152 μm ........................................................................ 118

Figure 6.14. (a) Exemplary erbium doped slot waveguide ring resonator coupled to a bus

waveguide to aid pumping of erbium in the slot (b) Cross section of the slot waveguide

constituting the ring ........................................................................................................ 120

Figure 6.15. (Color) Minimum erbium concentration required in the slot for the structure

shown in Fig. 6.14 to reach threshold as a function of the coupling gap G for different

intrinsic resonator Qs ...................................................................................................... 120

Figure 6.16. Threshold pump power as a function of coupling gap for the ring resonator

configuration shown in Fig. 6.14 with Q = 4X105 and slot erbium concentration of

7.7X1020 cm-3 .................................................................................................................. 121

Figure 6.17. Output lasing power at 1530 nm as a function of pump power and coupling

gap for the configuration shown in Fig. 6.14 with Q = 4X105 and slot erbium

concentration of 7.7X1020 cm-3 ...................................................................................... 122

Figure A.1. A five layer dielectric slab structure representative of the slot waveguide 130

xvii

Figure A.2. Illustration of the definition of the reflection and transmission coefficients r12

and t12 .............................................................................................................................. 131

1

Abstract

Of the many challenges which are threatening to derail the success trend set by

Moore’s Law, perhaps the most prominent one is the “Interconnect Bottleneck”. The

metallic interconnections which carry inter-chip and intra-chip signals are increasingly

proving to be inadequate to carry the enormous amount of data due to band-width

limitations, cross talk and increased latency. A silicon based optical interconnect is

showing enormous promise to address this issue in a cost effective manner by leveraging

the extremely matured CMOS fabrication infrastructure. An optical interconnect system

consists of a low loss waveguide, modulator, photo detector and a light source. Of these

the only component yet to be demonstrated in silicon is a CMOS compatible electrically

pumped silicon based laser.

The present work is our endeavor towards the goal of a practical light source in

silicon. To this end we have focused our efforts on horizontal slot waveguide which

consists of a nm thin low index silica layer sandwiched between two high index silicon

layers. Such a structure provides an exceptionally high confinement for the TM-like

mode in the thin silica slot. The shallow ridge profile of the waveguide allows in

principle for lateral electrical access to the core of the waveguide for excitation of the slot

embedded gain material like erbium or nano-crystal sensitized erbium using tunneling,

polarization transfer or transport.

2

Low losses in the proposed structure are paramount due to the low gain expectation

(~1dB/cm) from CMOS compatible gain media. This dissertation details the novel

techniques conceived to mitigate the severe lateral radiation leakage loss of the TM-like

mode in these waveguides and resonators using “Magic Widths” and “Magic Radii”

designs. New fabrication techniques are discussed which were developed to achieve

ultra-smooth waveguide surfaces to substantially reduce the scattering induced losses in

the Silicon-on-Insulator (SOI) high index contrast system. This enabled us to achieve

resonators with Qs of 1.6X106 for the TE-like mode in non-slot configurations and 3X105

for the TM-like mode in full slot configuration, the highest yet reported for this type of

structure and close to our design requirements for a laser.

Erbium was incorporated into the silica slot just 8.3 nm thick and photoluminescence

was observed in full waveguide configuration. A simple phenomenological model based

on spontaneous emission into a waveguide mode was developed, which predicted >10X

Purcell enhancement of the luminescence decay in these slot waveguides even in the

absence of a resonator, a result also yielded by a rigorous quantum electrodynamic

analysis. These enhanced spontaneous emission rates were experimentally verified using

time resolved photoluminescence decay and luminescence power measurements.

The results so far indicate that these slot structures could be the enablers for very

efficient LEDs due to the highly preferential characteristic of the spontaneous emission to

go into the single guided mode. The future goal will be to harness this behavior for novel

silicon photonic light sources.

3

Chapter 1. Introduction

It would be apt to say that for photonics to become as pervasive and ubiquitous as

electronics, it would have to go the “integration” way. The ability to monolithically

integrate transistors into a single silicon chip lent integrated circuits to economies of

scale. This has driven their exponentially increasing processing power, reliability and

functionality but at much reduced space, power requirements and cost, as was famously

predicted by “Moore’s Law”[1].

Indeed the same incentives exist for photonic integration and the idea of a photonic

“super chip” containing integrated optical components for generating , modulating,

switching , guiding, detecting and amplifying light was recognized as early as 1969 [2].

The ensuing two decades saw a significant research effort [3, 4] towards this goal. The

early work in the field utilized materials which were already well known for their relative

ease of manipulating and generating light such as Lithium Niobate (LiNbO3) and III-V

semiconductors such as Gallium Arsenide (GaAs) and Indium Phosphide (InP)

respectively.

However, mid 1980s saw silicon arising as a contender for photonic integration due

its transparency and hence capability to guide light at telecommunication wavelengths of

1.3 and 1.55 μm [5] and the possibility offered by it of placing electronic and photonic

4

components side by side on a single silicon chip. This was despite the fact that silicon is

sub optimal for both modulation and generation of light. Being a centro-symmetric

crystal silicon lacks the linear electro-optic (Pockels) effect, the traditional means for

modulating light and being an indirect band gap material means light generation in

silicon is a very inefficient process. However, the possibility of utilizing the already well

understood technical know how on silicon due to its prominence in the electronics

industry and leveraging the extremely matured multi-billion dollar complementary-

metal-oxide-semiconductor (CMOS) fabrication infrastructure drove the research to

overcome these deficiencies of silicon by ingenious and often un-conventional means.

An immediate pressing problem for which silicon integrated photonics is ideally

suited to provide solution is the “Interconnect Bottleneck”[6]. The insatiable desire to

pack ever more number of transistors per unit area on the silicon real estate to keep up

with the predictions of “Moore’s Law” has required a commensurate down scaling in the

interconnect cross sectional dimensions and also increase in interconnect lengths.

Decrease in interconnect cross sectional dimensions results in increased resistance

leading to higher signal latencies. This has been countered in the past by replacing

aluminum with copper as the medium for interconnect to reduce the resistivity, using low

k dielectric materials to reduce the capacitance and by increasing the number of metal

layers on a silicon chip. However these are one time improvements and it’s a matter of

time before the problem resurfaces. This coupled with the increased power requirements

and electro-magnetic interference (EMI) associated with dense packing of metallic

interconnects have the potential to derail the success trend set by “Moore’s Law”.

5

Herein perhaps lies an opportunity for photonics to come onboard silicon and

intimately integrate with the electronics in the form of an optical interconnect to provide

signaling and clock distribution. An optical interconnect in silicon offers the promise of

decreasing the interconnect delays and providing higher band width (BW) to keep pace

with transistor speed improvements, while potentially lowering power consumption and

being resistant to EMI [7]. A hypothetical optical interconnect will consist of a photon

source, preferably a laser whose light is coupled to a waveguide matrix acting as

interconnects between different on chip components. Light is converted into data using a

modulator driven directly by a standard CMOS driver. Light is then routed to the other

end of interconnect to a photo detector which converts the light into photo current. The

current is then transformed into a conventional digital voltage signal using a CMOS

trans-impedance amplifier (TIA). Such an intimate integration of electronics and

photonics together on chip would relax the impedance matching conditions while

providing voltage scaling and superior performance at much lower power levels.

Tremendous progress has been made over the years to realize all the components

which will constitute a photonic interconnect in silicon, the so-called silicon photonics

toolkit. Silicon photonics relies heavily on Silicon-On-Insulator (SOI) technology [8]. It

was only after SOI was adopted by mainstream CMOS that a more comprehensive

development of silicon photonics began. The large index contrast between silicon and the

buried oxide (BOX) layer allows definition of waveguides with very tight bend radii

which is necessary for efficient packing of waveguides on chip and miniaturization of

various other optical components. On the active components side, modulators based on

free carrier plasma dispersion effect [9] in silicon with 40 GHz 3dB bandwidth [10, 11]

6

have been realized in fully CMOS compliant fabrication processes. The flexibility

provided by monolithic electronic and photonic integration on silicon has been utilized to

improve the performance of germanium detectors on silicon. The ability to connect the

transistors of the TIA placed in close proximity to the germanium photo detectors on

silicon using CMOS metallization results in extremely low capacitances and hence higher

detector speeds. As a testament of the strength of silicon photonics integration with

CMOS technology, a 20 Gbps dual XFP transceiver [10] and 40 Gbps WDM transceiver

based on the various components discussed above have already been realized.

Arguably the most critical component of an optical interconnect system on silicon is

the “photon source”. The most ideal avatar of such a photon source would be an on chip

monolithically integrated CMOS compatible electrically pumped silicon based laser.

Various approaches[12] have been taken to realize the same, however, to date a

continuous wave electrically pumped silicon based laser operating at room temperature is

yet to be demonstrated.

1.1 The Problem and Our Approach

The present work is our endeavor towards achieving a CMOS compatible electrically

driven light source in silicon. With the aim of utilizing erbium as the extrinsic gain

medium, we focused our efforts on horizontal slot waveguides. Here I enumerate the key

design challenges associated with an extrinsic light source in silicon which will put the

approach we have taken into perspective.

7

1. Erbium (Er): Erbium doped materials are of great interest in thin film integrated

optoelectronic integration technology due to the Er3+ intra-4f emission at 1.54 μm, a

standard telecommunication wavelength at which silicon is transparent. The radiative

lifetime corresponding to the first excited state (~1.54μm) is usually in millisecond

range, depending on the host material. As a result the emission and absorption cross

sections are relatively small, typically 10-21-10-20 cm2. Therefore reasonable optical

gain values (~3 dB) can only be achieved when the signal beam encounters a large

amount of excited Er (1021-1020 Er/cm2). At chip scale integration dimensions, this

would require a high density of Er, typically in the range 0.1 to 1 at. %. At such high

Er concentrations, the distance between Er ions is very small and electric dipole-

dipole interactions between Er ions can reduce the gain performance of the device. So

we can’t expect a lot of gain from Er doped silica, therefore our designed structure

should have very low losses for it to lase.

2. Silica (SiO2) host: Er will be hosted in SiO2 which has a lower index (1.45)

compared to silicon which has an index of 3.47 at 1.54μm. As a result it is

challenging to achieve high modal confinement factor (Γ) in the material (SiO2)

containing the gain medium (Er) in conventional waveguide architectures. Therefore

new waveguide configurations have to be explored which can provide high modal

confinement in a low index material.

3. Tunnel injection: The structure we design should also provide means of electrical

excitation of the gain medium in a non-invasive manner i.e. which doesn’t depend on

hot electron injection for excitation of the gain medium as it causes degradation of the

silica layer embedding Er. This suggests that the excitation be based on some

8

tunneling or polarization transfer or transport mechanism. Though the mechanism is

not clear yet, such non-invasive mechanisms are usually operative only at nano-meter

dimension scales. So a plausible non-destructive electrical excitation of the Er would

require that the silica layer embedding the gain medium be very thin i.e. couple of

nano-meters thick.

4. Doping and charge accumulation: Electrical drive for excitation of the gain media

requires doping to increase the conductivity. In addition, the envisioned tunneling

mechanism would require charge accumulation at the interfaces adjacent to the

dielectric containing Er. Optical modal overlap with these free carrier regions would

increase the modal loss and hence the lasing threshold. Therefore the designed

structure should minimize these losses.

5. Scattering and material losses: Due to the high index contrast between silicon and

silica, scattering induced losses can be predominant in this material system. In

addition to this, the losses could also increase if the device design requires utilization

of other materials such as amorphous or poly silicon which have losses higher than

their crystalline counterpart. With very small gain expectation from Er, these losses

have to be minimized to achieve lasing operation.

In light of these design challenges, we concentrated our research efforts on horizontal slot

waveguides as shown in the inset of Fig. 1.1. The structure consists of a low index

material, in our case silica, sandwiched between two high index silicon layers. This

structure when operated in TM mode provides huge optical confinement in the low index

silica slot layer compared to the TE mode as can be seen from Fig. 1.1, which shows the

normalized profile of the dominant electric field component of the TE and TM modes for

9

the horizontal slot waveguide shown in the inset. The electrical field confinement in the

slot will increase as the slot thickness is increased, but for the structure to be suitable for

our purpose we require the slots to be ultra thin i.e. couple of nano-meters thick so as to

enable excitation of the slot embedded gain media by tunneling, polarization transfer or

transport. But even such ultra thin slot dimensions will not seriously compromise the

modal gain as we will show later in Sec. 4.1. In fact for this particular structure the TM

mode will see fourteen times higher gain enhancement due to a gain medium like Er

embedded in the slot compared to the TE mode or more precisely 23.5% of the local gain

in the 8 nm slot will be seen by the TM mode. This is quite phenomenal considering the

fact the slot is just 8 nm thick and is of lower index.

Figure 1.1. Normalized profile of the dominant electric field component of the fundamental TE and

TM mode for the 8 nm horizontal slot waveguide shown in the inset

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1Normalized Electric Field

Wav

egui

de P

ositi

on (µ

m)

TE TM

150 nm Si

100 nm Si

8 nm SiO2

BOX

10

The structure we want to realize is shown in Fig. 1.2. The structure will consist of an

ultra thin silica slot sandwiched between an upper amorphous and a lower crystalline

silicon layer to provide large TM mode confinement. The slot will be a thermally grown

oxide which will allow precise control over its thickness. The horizontal slot design

allows for planar fabrication technology which eliminates the etching induced roughness

of a vertical slot design thereby lowering the scattering losses. We also utilize a novel

fabrication technique which is detailed later in this thesis which results in ultra smooth

waveguide surfaces. It is also a shallow ridge design to keep the waveguide sidewall

roughness at the minimum. A shallow ridge structure is also critical for allowing

electrical drive access to the waveguide core with minimal loss impact as we intend to

use amorphous silicon as the top cladding layer which suffers from low mobility. The

waveguides are also of specific widths referred to as the “magic widths” (MW) to

eliminate the TM mode radiation leakage loss which can be prevalent in these shallow

ridge designs and can render the device useless for TM mode operation. This aspect is

central to this thesis and is discussed in detail. Finally, the upper cladding layer is a

superior quality deposited amorphous silicon film.

Figure 1.2. A shallow ridge horizontal slot waveguide structure

a-SiEr-SiO2

c-Si

11

1.2 Thesis Organization

The remaining part of this thesis is organized to mirror the step-step sequential process

which was undertaken to achieve low loss horizontal slot waveguides as lasing element

for a CMOS compatible light source in silicon. Chapter two details the realization of the

ultra low loss quasi planar ridge waveguide which is a critical primitive to the horizontal

slot waveguide. Here new fabrication techniques are discussed which were developed to

achieve ultra-smooth waveguide surfaces to substantially reduce the scattering induced

losses in SOI high index contrast system. Resonator Qs of 1.6X106 for the TE mode were

achieved as a result. These fabrication methods were then combined with the novel

phenomenon of “magic widths” conceived to mitigate the severe lateral leakage losses for

the application critical TM mode resulting in Qs of 6.8X105, the highest yet reported for

such shallow ridge structures. Further more, we discuss the design of an ultra compact

waveguide polarizer with significant improvements over any current designs as an

application of this phenomenon. In chapter three we take a keener look at the “magic

width” phenomenon and show how it is impacted in curved waveguides and directional

couplers. In doing so we discovered new anomalous losses over and above any

conventional reported losses in these structures which led to the framing of new design

rules to achieve highest performance devices. In chapter four we apply “magic widths”

and the ultra smooth fabrication techniques to design and realize a very low loss TM like

mode quasi planar horizontal slot waveguide. A key factor contributing to the low loss in

the these structures is a superior quality deposited PECVD amorphous silicon film

developed in house with a bulk loss of ~3 dB/cm which is in the league of best results

reported by other groups. Having achieved low loss slot waveguides, the next obvious

12

step was to incorporate Erbium into the slot. Chapter five discusses ultra thin 8 nm

Erbium doped slot waveguides. Room temperature photo luminescence results were

obtained in full waveguide configuration and time resolved spectroscopy showed

evidence of enhanced spontaneous emission rates or “Purcell Enhancement” in these slot

waveguides. This was quite surprising as this phenomenon is usually associated with

resonators and is typically explained using quantum electrodynamics. We however show

that the simple phenomenological model based on conventional expressions for

spontaneous emission into a waveguide mode precisely captures the “Purcell

enhancement” in slot waveguides. The results are also verified against a full quantum

electrodynamic analysis of the structure. Finally chapter six sets a detailed and definitive

direction for future work based on the findings of this thesis.

13

Chapter 2. Ultra Low Loss Quasi Planar Ridge Waveguides

A low loss shallow ridge waveguide is a vital precursor to the low loss shallow ridge

slot waveguide. Devoid of the slot layer and the upper amorphous silicon cladding, it is a

much simpler structure to work with. Information gained by analyzing various loss

mechanisms in this basic structure proved vital in the design of low loss slot waveguides.

Quasi planar or shallow ridge waveguides by themselves are of strong interest for active

SOI devices such as modulators, tunable filters and sensor applications. Optimized

voltage metrics for field-induced effects demand tight confinement primarily only in one

dimension, and the quasi-planar ridge affords the highest possible confinement in the

vertical direction while simultaneously providing a near-ideal geometry for control of

applied fields and conductive lateral access.

Also, more recently Densmore et. al. [13] have shown that high-index-contrast

systems operating in the TM mode are ideally suited for interferometric sensors that rely

on perturbations in the modal effective index caused by the interaction of the evanescent

tail of the mode with solutions or material adsorbed on the waveguide surface. In

particular, silicon-on-insulator (SOI) was shown to provide an order of magnitude

increase in sensitivity compared to other material systems. Sensitivity of evanescent-

wave surface sensors depends critically on maximizing field overlap with the adsorbed

species, but precision in phase measurement also requires long interaction lengths in

14

either a long interferometric configuration or a high-Q resonator. So waveguide

propagation losses will also impose a limit on the sensitivity. Thin-core shallow ridge

waveguides can potentially optimize both these attributes. However, it is well known that

waveguide scattering losses, which can be shown to scale as (Δn)2 through analytical

means [14], are exacerbated in tight vertical confinement SOI guides both due to the high

index contrast and the higher modal amplitude at the interfaces with the low-index

cladding layers. Reactive ion etching (RIE) is typically used to pattern Si-core SOI

waveguides and significant (multi-nanometer scale) sidewall roughness is commonly

encountered from standard lithographic and etching processes. This suggests alternative

fabrication techniques for highly smooth waveguides that minimize exposure of the mode

to etched interfaces. Also, more importantly, the desired TM-mode operation in shallow

ridge waveguides can also lead to inherent severe lateral radiation leakage loss in high

index-contrast systems like SOI. These losses which are over and above the scattering

losses can be just too high to render these waveguides practically useless for TM mode

operation.

Firstly the mechanism of lateral leakage loss for the TM-like mode in shallow ridge

waveguide is elucidated using a simple phenomenological model. This is followed by

experimental verification of the TM mode lateral leakage phenomenon and a way to curb

it [15, 16]. Rigorous full vector models are then discussed which were developed to

analyze and aid the accurate design of these structures[17]. The insights gained from

these were then combined with novel fabrication techniques yielding ultra smooth

waveguide surfaces to design and fabricate very low loss shallow ridge waveguides,

which are detailed next[18, 19]. Finally, an interesting application in the form of an ultra

15

compact TE pass polarizer utilizing the lateral leakage phenomenon in shallow ridge

waveguides is proposed and analyzed using full vector numerical techniques[20].

2.1 Theory of Lateral Leakage Loss in Shallow Ridge Waveguides

Under an equivalent slab model for the ridge waveguide geometry (Fig. 2.1 (a)), the

lateral leakage loss for the TM-like mode is due to TM/TE mode conversion at the ridge

Figure 2.1. (a) Ridge waveguide geometry (b) Slab mode indexes used in effective index calculations.

The modal indexes for the guided TE-like and TM-like modes, respectively, lie intermediate between

their slab values for slab thicknesses t1 and t2 .We use index values of nSi = 3.475 and nSiO2 = 1.444 at

1.55 μm wavelength.

Slab Thickness (nm)

Slab

Mod

al E

ffect

ive

inde

x

TE-like mode Neff

TM-like mode Neff

t1 t2

W

AirSi

SiO2

TE

TM

t1

t2

(a)

(b)

16

boundary [15, 16, 21-23]. It should be emphasized that this effect is not caused by any

surface or side-wall roughness. With reference to slab waveguide dispersion curves in

Fig. 2.1 (b), for the case of TE-like modes, this mode conversion at the boundary cannot

lead to any propagating field or leakage loss in the lateral cladding since longitudinal

phase-matching requires that any fields generated at the ridge boundaries be laterally

evanescent in the slab lateral cladding for both the TE and TM generated fields.

However, in the case of the TM-like mode, while phase-matching requires that the lateral

TM slab mode be evanescent in the lateral cladding, any TE slab field component

generated at the boundary is phase-matched to a laterally propagating TE slab mode at

some angle θ in the lateral slab cladding region.

This phase-matching is illustrated in Fig. 2.2 (a). Here TEβ and TMβ represent the

propagation constants of the TE-like and TM-like modes of the ridge waveguide,

respectively. This diagram illustrates that the propagation constant of the TM-like mode

is much less than that of the TE-like mode, and for many practical rib guides may also lie

below the propagation constant of the unguided TE slab mode cladTEk . Since this slab mode

is unguided by the rib, it may propagate at any angle and if cladTETM k<β , it is possible to

rotate cladTEk by an angle θ such that the TE-slab and TM-guided mode are phase-matched

in the z direction. Since the guided mode is then phase-matched to a radiation mode, it is

possible that leakage may occur if there is some means for mode conversion.

For a thin enough lateral slab thickness , and certainly in the case of a strip or “wire”

waveguide where , the TE slab effective index can lie below the TM-like mode effective

index, and hence, avoid this phase-matched leakage. However, for electrical access, these

17

Figure 2.2. (a) Phase-matching diagram showing TM-like waveguide mode phase matched to a

propagating TE slab mode in the lateral cladding (b) Ray picture of the TM mode lateral leakage

designs may not be practical, and thus, this leakage loss must be well understood.

The leakage process is illustrated in Fig. 2.2 (b). Since the z components of all

propagation constants are conserved, all waves develop the same relative phase along the

length of guide and we can discuss phase in the lateral direction only.

Starting at the bottom, a TM mode is guided by the rib and is represented as the solid

ray incident on the right wall, where the angle of incidence is such that TM total internal

reflection occurs. However, due to the step discontinuity at the rib wall, mode conversion

from TM to TE can occur, and it can be shown from mode-matching calculations [23]

(Sec 2.3.1) that TE transmitted and reflected propagating waves are produced that are

approximately equal in magnitude, but are ~π radians out of phase. The reflected TE

radiation mode traverses across the core as shown by a dashed line. Upon total internal

TMβ

TEβ

)(cladTEk

)(,cladTEzk

x

z

TETM

W

TE

)(,

coreTEeffn

)(,

coreTMeffn

)(,

cladTEeffn

)(,

cladTMeffn

)(,

cladTEeffn

)(,

cladTMeffn

For TE:

For TM:

θ

(a)

(b)

18

reflection, the TM mode experiences a negative phase shift TIRφ and the combination of

TIRφ with the phase from a single traverse of the guide to the left is zero for the

fundamental mode.

At the left rib wall, the TM mode generates additional small reflected and transmitted

TE propagating waves. The new transmitted TE wave, with a relative phase of π radians

as noted above, combines with the previous reflected TE wave that has traversed the

guide with a phase shift of Wk coreTEx ⋅, . Thus, if this phase shift across a single traverse for

the TE in the core is a multiple of π2 , the TE waves will interfere destructively. This

leads to a width dependence for the leakage minima that satisfies a resonance-like

condition [22] of π2, ⋅=⋅ mWk coreTEx , or alternatively stated[15, 16],

( )…321

22,,,

,,

=−

⋅= mNn

mWTMeff

coreTEeff

λ (2.1)

where coreTEeffn ,

is the two-dimensional slab effective index of the TE mode in the core while

TMeffN , is the full three-dimensional effective index of the TM-like mode for the

structure. TMeffN , has a weak W dependence, so some care must be used if precision is

required. From this argument, we would expect to see significant leakage loss for TM

propagation, except at precise, specific waveguide widths --- ‘‘magic widths’’ satisfying

the resonance condition in Eq. (2.1), where the leakage loss would be greatly reduced due

to destructive interference of radiating TE waves. It is also interesting to note that the

next higher order mode will be lossy at these widths, since the radiated fields add in

phase for this mode. This may allow relatively wider guides with effectively single-mode

behavior.

19

2.2 Experimental Evidence of “Magic Widths”

A series of ridge waveguides with widths varying from 0.5 to 1.8 μm in 50 nm

increments with slab thickness t1=205 nm and t2=190 nm (i.e. a 15 nm ridge height) were

fabricated on an SOI wafer with a 2 μm buried oxide (BOX) layer thickness [15, 16].

These waveguides were formed by wet-etching, a process using which low propagation

losses of 0.7 dB/cm for the TE mode at 1.55 μm have already been reported [24].

Waveguides were also fabricated using a thermal oxidation process (discussed in Sec.

2.4) which results in smoother and rounded sidewalls. Fig. 2.3 (a) shows the atomic force

microscope (AFM) profile of the waveguide surface as obtained using the wet etch

process and Fig 2.3 (b) shows the total relative transmitted fiber-coupled power measured

as a function of waveguide width for both the TE and TM input polarizations. Fig. 2.4

shows the corresponding data for the waveguides formed using thermal oxidation.

Here each data point corresponds to the measured power averaged over ten nominally

identical waveguides with a length of 1.5 cm. It can be readily observed that the TE-like

mode has a net low loss weakly dependent upon waveguide width, whereas the TM-like

mode has in general high loss except at some specific values of waveguide widths. For

both the wet etched samples and the thermal oxide case, these low loss waveguide widths

are 0.72 and 1.44 μm.

A simple effective index model was used to calculate the TE and TM modal effective

indices. These were then substituted into Eq. (2.1) to obtain widths of 0.72 and 1.44 μm

for the first two resonances which are in excellent agreement with the measured results,

20

Figure 2.3. (a) AFM image of a waveguide formed by wet-etching (b) Experimentally measured

optical transmission (log scale) for the fundamental TE-like and TM-like modes for the waveguides

as shown in Figs. 2.3 (b) and 2.4 (b). The data of Fig. 2.4 (b) show that the waveguides

with a smooth rounded sidewall as obtained using the thermal oxidation process still

display the TM-like mode lateral leakage loss, but show a broader width dependence at

the loss minima. This is believed to be due to a weaker TM/TE conversion at these

slightly rounded ridge boundaries.

W

15 nm

TE

TM

Waveguide width, W (µm)

Rel

ativ

e Tr

ansm

issi

on (d

B)

W=0.72 µm W=1.44 µm

(a)

(b)

21

Figure 2.4. (a) AFM image of a waveguide formed using thermal oxidation (b) Experimentally

measured optical transmission (log scale) for the fundamental TE-like and TM-like modes for the

waveguides

The data presented in Figs. 2.3 (b) and 2.4 (b) exhibit some subtle characteristics

which warrant further investigation. To gain comprehensive insight into the behavior of

the TM-like mode in shallow ridge waveguides, a full vector numerical analysis of these

structures was carried out.

W

15 nm

TE

TM


Rel

ativ

e Tr

ansm

issi

on (d

B)

W=0.72 µm W=1.44 µm

(a)

(b)

22

2.3 Numerical Analysis of Lateral Leakage Loss in Shallow Ridge Straight

Waveguides

To numerically evaluate the leakage loss of the shallow ridge SOI waveguides, full

vectorial mode matching (MM) technique [23, 25] and finite element method (FEM) with

perfectly matched layer (PML) boundary conditions [26, 27] were employed. For the

mode matching simulations the waveguide cross-section was divided into a number of

uniform multilayer sections. In each section, the waveguide field was expressed as a

superposition of normal modes including guided and radiation modes for both TE and

TM polarizations of the corresponding slab. By matching the fields of these sections at

the ridge boundary, the mode of the ridge waveguide can be determined. The propagation

loss of the mode is then calculated from the imaginary part of the mode effective index.

The ridge waveguide is placed vertically between two perfectly conducting planes to

discretize the radiation spectrum but are kept far from the ridge so as to have minimal

impact on the leakage properties. The accuracy of a mode matching simulation depends

strongly on the number of normal modes used in the field expansion. Here, 50 pairs of

TM and TE modes are used in each waveguide section. We now discuss the results

obtained from the numerical simulations[17].

2.3.1 TM-TE Mode Coupling at a Step Discontinuity

Firstly, the mode coupling at a single ridge boundary when the TM fundamental

mode of the waveguide is obliquely incident on the ridge interface from the waveguide

core as illustrated in Fig. 2.5 is considered. The mode matching formulation is applied

23

only on the interface plane at the ridge wall. Fig. 2.6 shows the magnitude of the

reflection/transmission coefficients of the reflected fundamental TM mode and the

reflected/transmitted TE modes as a function of propagation angle (shown in Fig. 2.5) of

the incident TM mode. It can be seen from these plots that there is significant TM-TE

mode coupling at the ridge boundary. Under total internal reflection, not all the power of

Figure 2.5. Mode coupling diagram showing the TM-TE mode coupling at the ridge boundary

Figure 2.6. (Color) Magnitude of the reflected TM mode, reflected and transmitted TE modes as a

function of the incident TM mode propagation angle

205 nm

Air

Si 190 nm

Si02

TM

TETE

TM

x

yz

Core Cladding

°θ

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25

0

4

8

0 20

TM Reflected, ГTMTE Reflected, ГTETE Transmitted, TTE

Propagation Angle, θº

TM R

efle

ctio

n C

oeffi

cien

t

TM→

TE Reflection/Transm

issionC

oefficient

(TTE -Г

TE ) X 1e4

θº

24

the incident TM wave is transferred into the reflected TM wave. A small amount of

power is coupled into TE transmitted and reflected propagating waves.

The transmitted and reflected TE waves are similar, but not identical in magnitude.

The inset of Fig. 2.6 shows the difference in the magnitude between the transmitted and

reflected TE waves. This difference increases as the propagation angle of the incident TM

wave increases. In addition, it can be seen that the coupling coefficients between the TM

and TE waves increase almost linearly with the propagation angle of the incident TM

wave. Fig. 2.7 shows the relative phase of the reflected and transmitted TE waves. We

see that there is a small difference in phase between the reflected and transmitted TE

waves.

Figure 2.7. (Color) Relative phase of the reflected and transmitted TE modes as a function of the

incident TM mode propagation angle

0

20

40

60

80

100

120

140

160

180

200

0 5 10 15 20 25 30

TE ReflectedTE Transmitted

Propagation Angle, θº

Phas

e A

ngle

(º)

25

2.3.2 Width Dependent Propagation Loss

If a waveguide is formed by combining two such ridge interfaces as shown in the

inset of Fig. 2.8, significant loss due to TM-TE mode coupling at the ridge boundaries

can be expected. Based on the above numerical analysis we can now make a small

correction to Eq. (2.1) for the waveguide widths at which TE radiation from the two

interfaces coherently cancels and we have

( )…3,2,1,2

2,

2,

=−

⋅⎟⎠⎞

⎜⎝⎛ Δ+

= mNn

mW

TMeffcore

TEeff

λπφ

(2.2)

where φΔ is the phase difference between the reflected and transmitted TE waves while

other parameters are the same as defined w.r.t. Eq. (2.1). The resonant “magic widths”

calculated from Eq. (2.2) are about 10 nm higher than those obtained from Eq. (2.1).

We simulated the leakage loss due to mode coupling of the TM-like mode in SOI

shallow ridge waveguides using the mode matching formulation and FEM with PML.

The simulated leakage loss of the TM-like mode as a function of the waveguide width of

a straight waveguide is presented in Fig. 2.8 with both methods yielding similar results.

It is evident that the leakage loss depends strongly on the waveguide width. There

exist “magic widths” at which the leakage loss is minimal. The first two magic widths

obtained from the simulations are 0.71 μm and 1.429 μm. These magic widths are in

excellent agreement with the experimental results presented in Sec. 2.2.

At magic widths, perfect leakage cancellation does not occur due to the imperfect

balance of the reflected and transmitted TE waves as shown in the inset of Fig. 2.6. The

26

Figure 2.8. (Color) Simulated loss of TM-like mode in shallow ridge waveguide shown in inset which

shows strong width dependence

leakage loss at the first resonance is higher than that at the second resonance. This trend

agrees well with the experimental results in Figs. 2.3 (b) and 2.4 (b). When the

waveguide width increases, the TM wave inside the ridge region approaches glancing

incidence to satisfy the phase matching condition for the guided mode. This small

propagation angle makes the TM-TE coupling decrease as shown in Fig. 2.6.

Figure 2.9 (a) and (b) show the electrical field distributions of all three vectorial

components of the fundamental TM-like guided mode for waveguides with widths 1.2

and 1.429 μm respectively. The hybrid nature of the mode can be inferred by the presence

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.E+04

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Mode MatchingFEM+PML


Loss

(dB

/cm

)

AirSiSiO2

W

205 nm 190 nm

27

of the small Ey field component which is ~10X smaller than the dominant Ex field. The

mode also shows the presence of a strong longitudinal field component Ez. Importantly,

the Ey component exhibits strong radiation in the 1.2 μm waveguide case which is

missing at 1.429 μm which is a “magic width”. This correlates with their propagation loss

values shown in Fig. 2.8. Radiation can also be seen for the longitudinal field component

Ez as the TE-like radiation is propagating in the yz plane.

Figure 2.9. Electric field distributions of the fundamental TM-like mode for (a) 1.2 μm and (b) 1.429

μm waveguide widths

(a) (b)

Ex Ex

Ey Ey

Ez Ez

28

2.4 Waveguide Fabrication

Six inch Soitec SOI substrates were used to fabricate the ridge waveguides. The

silicon device layer in the SOI wafers is 205 nm thick with a 2 μm BOX layer

underneath. To minimize any sidewall roughness, the ridges were formed using an ultra

smooth thermal oxidation process [18, 19]. Fig. 2.10 gives a process flow illustrating the

fabrication sequence we employed in this study. To form a waveguide structure with this

technique an appropriate masking material, such as Si3N4, is advantageous because

oxygen diffusion through the nitride layer is negligible, thereby protecting the silicon

underneath. A patterned nitride layer can thus serve as a mask during a high-temperature

oxygen annealing process. This technique not only minimizes side-wall roughness, but

also has the advantage of providing high uniformity across the wafer. Furthermore, if an

oxide film or some other semi permeable layer is incorporated between the Si and Si3N4,

the resulting waveguide profile can be optimized for a given application by adjusting the

oxide layer thickness in order to control the oxygen diffusion through this layer near the

ridge sidewalls. Optical losses are also minimized in this design by the reduced etch

depth into the silicon device layer associated with the ridge waveguide as compared to

wire waveguides, where the silicon device layer is fully etched. This is accomplished

Figure 2.10. Processing steps for shallow ridge waveguide fabrication

PECVD SiO2 and Si3N4 Mask25 nm Si3N4

100 nm SiO2 c-Si 205 nm

Thermal Oxidation HF Strip

29

simply by controlling the oxidation time, which gives a precise control on the height of

the waveguide ridges compared to traditional etching processes. It should be noted that

reducing the waveguide optical loss of the ridge waveguide is the first step in achieving

low loss slot structures.

The detailed fabrication process for the ridge waveguide formation started with the

deposition of a 100 nm plasma-enhanced chemical vapor deposited (PECVD) SiO2

followed by a 25 nm thick PECVD Si3N4 layer. A 1 μm thick AZ703 resist was spin

coated onto the wafer and exposed with a 1X projection printer. A 6:1 ammonium

fluoride: hydrofluoric acid (NH3F:HF) solution was used to transfer the pattern into the

nitride/oxide layers. Once the pattern was formed, the resist was stripped and the

resulting structure was then oxidized at 975 °C in O2 for 17 min to form 13 nm ridges.

The residual mask and the thermally formed SiO2 were then stripped with HF, resulting

in a waveguide structure with very smooth silicon surface and side wall, as shown in the

atomic force microscope (AFM) trace in Fig. 2.11 (a). The RMS surface roughness was

measured to be less than 0.2 nm, which is an order of magnitude less than that typically

Figure 2.11. (a) AFM trace of a 1.44 μm ridge waveguide with <0.2 nm surface roughness (b) Quasi-

planar ridge SOI waveguide geometry. BOX thickness is 2μm

(a) (b)

BOX

WSi 192 nm 13 nm Ridge

Si Substrate

30

obtained with conventional etching processes. After the ridge waveguides were formed

with the process described above, a layer of resist was applied to protect the surface

during dicing and end facet polishing of the resulting chips. Finally, the protective resist

layer was removed from the finished chips using acetone and alcohol. Fig. 2.11 (b) shows

the sectional view of the final fabricated structure.

2.5 Waveguide Loss Measurement Technique

To accurately quantify the losses of our waveguides, we use the resonance

characteristics of under coupled ring resonators [24, 28]. Fig. 2.12 shows the schematic

of the ring resonator circuit formed using the ridge waveguide described above and the

full width at half-maximum (FWHM) of the drop port intensity response νπω Δ=Δ 2

Figure 2.12. Setup to measure the resonance characteristics of (a) a weakly coupled ring resonator

with separate thru and drop ports and (b) an exemplary drop-port response

(a)

(b)

Tunable Laser1460-1580 nm

Polarization controller

Six axis Piezo Stage

Six axis Piezo Stage

Ge Detector900-1700 nm

Optical Power meter

Couplers

0

0.2

0.4

0.6

0.8

1

-2 -1 0 1 2

Drop Port Response

Frequency Offset (GHz)

Dro

p P

ort R

espo

nse

(arb

.)

νΔ

31

as measured by scanning the input excitation frequency across the resonance.

The ring configuration in Fig. 2.12 (a) has separate through and drop ports and

consists of two directional couplers that are designed to be under coupled so that the ring

losses are dominated by the waveguide losses rather than the input/output coupling of the

ring to the thru-port and drop-port waveguides. Using the nomenclature of Yariv[29], the

expression for line width for our structure is

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ −=Δ

21

2112

tt

ttLrt

g

ττν

ω (2.3)

where gν is the group velocity, rtL is the round-trip length, τ is the field transmission of

the waveguide for one round trip of the ring, and 21 tt , are the field transmission

coefficients of the directional couplers. The couplers are assumed to be lossless, giving

122 =+ κt where κ is the field coupling coefficient. To simplify the expression in Eq.

2.3, we can write ( )2/rtLexp ατ −= with α being the waveguide intensity attenuation

coefficient and if we let ( )211 /rtLexpt α−= and ( )222 /rtLexpt α−= , then for the case

of low loss ( )1≈τ and weak coupling ( )121 ≈≈ tt , we obtain the simple expression

grt ναω =Δ (2.4)

The round-trip loss rtα is the sum of the waveguide propagation losses α and the

coupling losses from the two directional couplers ( 1α and 2α ), but, if the ring is designed

to be weakly coupled, then gναω =Δ . Since precision frequency scans allow for an

accurate determination of resonance width, this provides an unambiguous and

conservative figure for waveguide transmission losses. If desired, the weak contributions

from coupler losses can be removed for a correction.

32

Loss measurements were conducted using an external cavity laser diode with a line

width of <10MHz with continuous single-longitudinal mode and very fine-tuning

capability. The test setup is shown in Fig.2.12. The laser output was fed through a

polarization controller and tapered fibers controlled by six-axis piezo translation stages

were used to couple the signal into and out of the waveguide ports. The output from the

waveguide was detected by a power meter, followed by analog to digital conversion

allowing computerized data collection. The shape of the resonance profile was recorded

by scanning the laser across the resonance peaks, as shown in Fig. 2.12 (b), allowing for

the determination of the FWHM line widths and corresponding resonator losses.

2.6 Results and Discussion

Fig. 2.13 shows the average TE and TM drop-port response for a 400 μm radius ring

resonator with a waveguide “magic width” of 1.43 μm in the ring as well as in the

straight and coupler sections. Both of these responses were obtained on the same device.

While the waveguides have very strong birefringence, with calculated phase index values

of nTE = 2.768 and nTM =1.740, the group index values are nearly identical due almost

entirely to the strong structural waveguide dispersion. The measured group index values

were ng-TE =3.721 and ng-TM = 3.847 for the TE and TM modes, respectively, as

determined from the free spectral range of the resonators, and agree fairly well with

calculated values including several percent corrections from silicon material dispersion.

Using these measured group index values, the measured line widths of ΔνFWHM-TE =

124.37 MHz and ΔνFWHM-TM = 286.33 MHz correspond to losses of 0.42 dB/cm and 1.00

33

Figure 2.13. Drop-port responses for the TE and TM modes for a 400 μm radius ring resonator with

“magic width” waveguide of 1.43 μm

dB/cm for the TE and the TM modes, respectively. As noted in Sec 2.5, this provides a

conservative figure because all the loss is ascribed to propagation. Conventional bending

losses are readily estimated analytically and for the waveguide design used in this

experiment, are calculated to be <0.03 dB/cm for TE modes and orders of magnitude

smaller for TM modes and have been, therefore, neglected. The couplers in the device are

fabricated with a straight waveguide running tangentially to the ring with the gap

between the edges of the ridge at the closest point being 1.87 μm. The coupling

coefficient for this design was evaluated using the two dimensional RSOFT BeamPROP

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1537.15 1537.16 1537.17 1537.18 1537.19 1537.2Wavelength (nm)

Dro

p po

rt re

spon

se (a

rb.)

QTE=1.57x106

ΔνFWHM,TE=124.37 MHzαTE=0.42 dB/cm

QTM=6.81x105

ΔνFWHM,TM=286.33 MHzαTM=1.0 dB/cm

400 µm Ring, 1.43 µm “Magic width” Ridge Waveguide

34

program[30], which yielded a coupling coefficient of 0.18% for the TE mode and 0.19%

for the TM mode per coupler. Using these computed values, a correction can be included

for the two couplers combined, with a contributed loss of 0.062 dB/cm for the TE mode

and 0.066 dB/cm for the TM mode. This results in waveguide propagation losses of 0.36

dB/cm and 0.94 dB/cm and for the TE and TM modes, respectively[19].

A variation in the loss values with wavelength for the TM mode was observed as

expected due to the “magic width” effect. Since the actual fabricated width may not

precisely correspond to the “magic width”, the wavelength was tuned in these

measurements in order to minimize the FWHM of the drop port responses and the

minima were obtained at 1537 nm. We also saw small variations of successive

resonances exhibiting ±10% differences in the peak widths about the average loss values

quoted above, which might be defect related and are under investigation.

Additionally, while the TM loss is quite low, it is not as low as the TE loss even at the

“magic width” values. This excess loss may be due to additional anomalous radiation

losses resulting form imperfect radiation cancellation in curved waveguides even at

“magic widths” and is discussed in chapter 3.

The corresponding Q values for these ring resonators are 1.6X106 and 6.8X105 and

for TE and TM, respectively[19]. To our knowledge, these are the lowest published loss

values for TE and TM mode SOI waveguides with ultrahigh vertical confinement designs

suitable for active device configurations and optimized sensors.

35

2.7 An Ultra Compact Waveguide polarizer based on “Anti-Magic Widths”

A polarizer is a critical component in photonic integrated circuits (PICs) for coherent

communication, optical signal processing, switching networks and sensor applications.

With the goal of achieving high modal extinction ratios in minimal device sizes, various

polarizer designs based on birefringent crystals, metal clad waveguides, anisotropic

overlays, proton exchanged waveguide sections, photonic crystal and liquid crystal have

been proposed and demonstrated. These designs rely on utilization of more complex

materials and fabrication techniques, and in some cases have very stringent fabrication

tolerances. In the silicon-on-insulator (SOI) platform, for example, it is desirable to have

a polarizer design based on the existing standard fabrication sequences that is capable of

providing high polarization-dependent loss for μm-scale devices. Such a component may

be important, for example, in providing very high extinction of undesired residual TM

light in PIC’s using polarization diversity where both incoming polarizations are mapped

into TE modes on the chip.

Here we propose a very simple yet extremely efficient waveguide polarizer design

based on the lateral leakage phenomenon in shallow ridge SOI waveguides discussed

above[20]. It is commonly understood that TM modes in such structures are often leaky,

and it is therefore often asserted that PIC’s designed with ridge guides are “TE only”.

However, when etched deeply enough, ridge guides can carry TM light as a bound mode

without radiation loss. Furthermore, as we illustrated above, even for shallow ridge

designs, TM mode leakage can be mitigated at waveguide widths where the radiative

leakage destructively interferes [16, 17] (so-called “magic widths”), leading to TM losses

below 1 dB/cm. We now discuss in more depth how this loss can be engineered with

36

width control to provide advantageous device characteristics. In particular, we illustrate

that waveguide widths selected to provide constructive interference of radiation can

provide extremely large TM radiation loss, thereby providing an ideal polarizer fabricated

using simple, and non-critical, dimension control of the ridge. Rigorous analysis shows

that the structure[20] is capable of achieving significant improvements in polarizer

performance over prior designs.

2.7.1 Polarizer Design Principle and Analysis

Referring back to Fig. 2.8, illustrating the fundamental TM mode loss in shallow

ridge waveguides, we see that barring a few precise waveguide widths-“magic widths”,

the TM mode loss is in general quite high at most waveguide widths. This leakage loss

occurs because the effective index of the TM mode of the waveguide (nTM-WG) is less than

that of the TE slab mode in the lateral cladding (nTE-CLAD). The TM mode can therefore

phase match to the TE slab mode propagating at a particular angle relative to the

waveguide axis, causing the waveguide TM mode to leak in to the TE slab mode in the

cladding. At the “magic width”, there is destructive interference of radiation wave fronts

resulting in low loss for the TM mode [16, 17]. It is also critical to note that the TE mode

of the waveguide in these structures is a perfectly guided mode due to its higher index

value and always has low loss.

This waveguide width dependent loss mechanism of the TM mode in SOI ridge

waveguides is the basis of our efficient TE-pass polarizer. While the destructive

interference at the “magic width” inhibits radiation loss, widths where the radiation

37

components constructively interfere, referred to as “anti-magic widths,” will yield very

high TM loss as illustrated in Fig. 2.8 while leaving the TE mode unaffected.

With the goal of maximizing the TM mode loss, we performed a rigorous full vector

analysis of the SOI ridge waveguide similar to the one shown in the inset of Fig. 2.8

using analytical mode matching and finite element techniques. The analyzed structure

now has 220 nm of crystalline silicon (c-Si) in the core, with R nm as the ridge height

and is shown in the inset of Fig. 2.14 (a). Therefore thickness of c-Si in the cladding is

(220-R) nm and W μm is the waveguide width.

Fig. 2.14 (a) shows the XY view of a 3D plot of loss of the fundamental TM mode of

this ridge waveguide as a function of waveguide width and ridge height as obtained using

mode matching method. The white portion on the top left corner in the plot corresponds

to the region where the TM mode is at cut off due to the cessation of total internal

reflection within the waveguide. The cyclic behavior of the TM mode loss as a function

of waveguide width at a particular ridge height is clearly apparent. The simulation results

indicate that the fundamental TM mode suffers a loss as high as 34,000 dB/cm for

waveguide widths in the vicinity of 0.4 μm and 20,000 dB/cm for waveguides around 1.1

μm for a ridge height of 150 nm as can be clearly seen from Fig. 2.14 (b), which is the

XZ view of the 3D loss plot. This indicates that with a 10 μm long waveguide, a 34 dB

extinction ratio is achievable which is 30X improvement over the theoretical design for a

SOI TE-pass waveguide polarizer presented in[31]. The present result is also in the

league of an ultra-compact TM-pass silicon nanophotonic waveguide polarizer analyzed

and proposed in [32] with 26 dB extinction ratio over a 10 μm device length.

38

Figure 2.14. (a) XY view and (b) XZ view of the 3D plot of loss of the fundamental TM mode of a SOI

ridge waveguide with 220 nm c-Si in the core (inset)

(a)

(b)


Rid

ge H

eigh

t, R

(nm

)Loss dB/cm

x104

220 nmRidge, R nm

WAirSi

SiO2


Loss

(dB

/cm

)

x104

39

As discussed above, the high loss for the TM mode in these SOI shallow ridge

waveguides occurs because nTM-WG < nTE-CLAD. It is therefore anticipated that as the ridge

height is increased, a point will be reached where nTM-WG > nTE-CLAD hence the TM-like

mode will be no longer leaky and becomes purely guided. To show that this is indeed the

case, Fig. 2.15 depicts the loss and effective index of the TM mode as a function of

waveguide width for three ridge heights, (a) 90 nm (b) 148 nm and (c) 150 nm calculated

using FEM-PML and mode matching technique. It is evident that both models give

identical results. For the 90 nm ridge height, the entire waveguide width span of 0.1 to 3

μm exhibits lateral leakage loss as nTM-WG < nTE-CLAD (= 2.375) for this width range.

However, as we increase the ridge height to 148 nm for which nTE-CLAD =1.85 (indicated

in the plot), there are regions where nTM-WG > nTE-CLAD causing TM mode to become a

purely guided with zero lateral leakage loss in these width spans. Similar observations are

made for the 150 nm ridge which has nTE-CLAD =1.829. As nTM-WG of the waveguide

(a)

1.4

1.5

1.6

1.7

1.8

1.9

0.0E+00

2.0E+03

4.0E+03

6.0E+03

8.0E+03

1.0E+04

1.2E+04

1.4E+04

0.1 0.6 1.1 1.6 2.1 2.6

MM LossFEM LossMM IndexFEM Index

Loss

(dB

/cm

)


TM0 Index

90 nm Ridge

40

Figure 2.15. Loss and effective Index nTM-WG of the fundamental TM-like mode of a SOI ridge

waveguide with 220 nm c-Si in the core for (a) 90 nm (b) 148 nm and (c) 150 nm ridge heights

(b)

(c)

1.4

1.5

1.6

1.7

1.8

1.9

0.0E+00

5.0E+03

1.0E+04

1.5E+04

2.0E+04

2.5E+04

3.0E+04

3.5E+04

0.1 0.6 1.1 1.6 2.1 2.6

Loss

(dB

/cm

)


TM0 Index

148 nm Ridge

nTE-CLAD

1.4

1.5

1.6

1.7

1.8

1.9

0.0E+00

5.0E+03

1.0E+04

1.5E+04

2.0E+04

2.5E+04

3.0E+04

3.5E+04

4.0E+04

0.1 0.6 1.1 1.6 2.1 2.6

Loss

(dB

/cm

)


TM0 Index

150 nm Ridge

nTE-CLAD

41

increases with waveguide width for a given ridge height, it is clear that the condition nTM-

WG < nTE-CLAD will start failing at larger waveguide widths first. As the ridge height is

further increased incrementally beyond 150 nm, loss maximas for the TM mode will first

vanish at 1.1 μm and then at 0.4 μm.

The TE mode in our structure is a perfectly guided mode, unlike the design in [31]

where the TE mode is also a leaky mode and can compromise the extinction ratio. In the

present design the extinction ratio also scales with the waveguide length as the TM mode

leaking into the TE slab mode propagates laterally away from the waveguide. This

considerably eases the design of the polarizer compared to the one in [32] where the TE

mode leaks into the substrate and gets reflected back into the waveguide leading to

particular lengths that maximize the extinction ratio. It is also clear from the Fig. 2.14(b)

that a high loss for the TM mode can be obtained over reasonably wide range of

waveguide widths in the vicinity of the “anti-magic width,” thereby easing fabrication

related tolerances significantly compared to other polarizer designs. The optimum

polarizer design should target narrower waveguide widths (~ 0.4 μm) and a deeper ridge

height that is slightly less than the ridge height beyond which TM mode is no longer

leaky. This regime maintains single mode operation for both TE and TM modes and also

achieves the highest loss for the TM mode.

The fact that at any given ridge height both “magic widths” and “anti-magic widths”

exist with an extraordinary level of TM mode loss disparity between the two provides a

high degree of design flexibility without any associated fabrication complexity. This

polarizer can conceivably follow an imperfect TM to TE polarization rotator, with an

42

adiabatic waveguide taper connecting the two sections, causing a very strong reduction in

polarization cross talk levels.

In conclusion, we have proposed and analyzed an ultra-compact SOI based TE-pass

polarizer utilizing the programmable loss characteristics of the leaky TM mode in ridge

waveguides. Analysis shows that the presented structure, which is both simple and

fabrication tolerant, can achieve a multifold improvement in polarizer performance over

prior designs. The ultimate performance of such a polarizer will of course depend on the

proper engineering of follow-on waveguides or other PIC components to avoid re-capture

of the radiated light, a topic which is under analysis.

43

Chapter 3. Anomalous Losses in Curved Waveguides and Directional

Couplers at “Magic Widths”

We have seen that shallow ridge waveguides operating in TM mode, which is critical

for our application, experience severe inherent lateral leakage loss. This was overcome by

controlling the width of the waveguides to “magic widths” at which the radiation loss

components experience interferometric cancellation [15, 16, 19]. However, it is possible

that the rings, bends and directional couplers, which are part of the ring resonator

architecture we use to measure the waveguide losses, might be significantly impacting

the effectiveness of the “magic widths” as it relies on precision destructive interference of

the radiation wave fronts. Understanding their effects on “magic width” is important also

to facilitate the incorporation of these waveguides into practical functional subsystems

like resonators, add-drop filters and active components like modulators, switches and

emitters. Here we investigate the impact of waveguide curvature[33] and directional

coupling [17, 34]on the “magic width” phenomenon using full vector analytical mode-

matching and finite element modeling techniques. We show here new anomalous losses

over and above any previously investigated or published results, and thus show that

additional design constraints have to be imposed to achieve highest performance devices.

44

3.1 Lateral Leakage of TM-like Mode in Thin Ridge SOI Ring

Figure 3.1 shows the cross-sectional (Fig. 3.1 (a)) and plan view (Fig. 3.1 (b)) of a

section of a ring resonator formed by a thin-ridge SOI waveguide. Consider a simplified

ray-tracing view of wave propagation. The rays of the guided TM-like mode of the ring

are shown in Fig. 3.1 (b). In general, when the rays of a guided mode are incident on the

waveguide boundary, they are totally internally reflected. However, as has been shown

previously [16, 17] in Sec. 2.3.1, when a TM ray of a thin-ridge SOI waveguide is

incident on a waveguide boundary, transmitted and reflected TE rays are generated in

addition to the reflected TM ray due to strong TM-TE mode coupling at the waveguide

ridge boundary. On the outer waveguide boundary, the transmitted TE ray (TTE1)

propagates away from the ring, while the reflected TE ray (RTE1) propagates across the

Figure 3.1. (a) Cross section, and (b) plan view and mode coupling diagram of a SOI thin-ridge ring

resonator. Waveguide dimensions are shown

190 nm Si

15 nm Ridge

BOX

xφ

r

RW

+

R

W

TETM

TTE2

RTE1

RTE2TTE1

(a)

(b)

45

waveguide, intersecting the inner waveguide boundary where it is largely transmitted.

The reflected TE ray (RTE1) then traverses a secant across the ring, intersecting with the

inner and outer waveguide boundaries again, and then propagates away from the

waveguide mostly unaltered. On the inner waveguide boundary, the incident TM ray

generates reflected TE (RTE2) and transmitted TE (TTE2) rays. The reflected TE ray (RTE2)

propagates across the waveguide to radiate away from the ring. The transmitted TE ray

(TTE2) traverses a secant across the ring, intersecting the inner and outer waveguide

boundaries and propagates away unaltered. The TM guided mode suffers from high

leakage loss due to power coupling to TE radiation at the two waveguide boundaries.

Based on the above ray model, at any point outside the ring, there are four different

TE rays: transmitted (TTE1) and reflected (RTE1) TE rays generated from the TM ray

incident on the outer waveguide boundary, and the transmitted (TTE2) and reflected (RTE2)

TE rays generated from TM-TE mode coupling on the inner waveguide boundary. As the

relative phases of all these TE rays depend on both the ring radius and waveguide width,

it is possible that there might exist some combinations of waveguide width and ring

radius for which all four of these TE waves interfere destructively outside the ring,

resulting in cancellation of the lateral leakage radiation.

Inside the ring, there are reflected (RTE1) TE rays generated from the outer waveguide

boundary and transmitted (TTE2) TE rays generated from the inner waveguide boundary.

The relative phase between these TE rays depends primarily on the waveguide width. For

some waveguide widths, these TE rays can interfere constructively to generate a strong

TE field inside the ring. Such widths will be called “anti-magic widths”. On the other

46

hand, if the waveguide is at a “magic width”, the TE rays inside the ring will be out of

phase, resulting in minimization of the TE field inside the ring.

3.2 Rigorous Simulation Approaches

To rigorously model the TM-like modes of thin-ridge bent waveguides, a full

vectorial mode matching technique [25] in cylindrical coordinates [35] was employed. In

the mode matching technique, the waveguide cross-section is divided into a number of

radially uniform sections. Each section corresponds to a multi-layer slab. In each section,

the waveguide mode field was expanded into a superposition of the TE and TM normal

modes of the corresponding slab waveguide. The amplitudes of slab normal modes in

each section are the solutions of Bessel equations. By matching the fields of these

sections at the vertical interfaces between two adjacent regions, the modes of the

waveguide can be determined. The propagation loss of each mode was then calculated

from the imaginary part of the complex azimuthal propagation constant. The calculated

loss includes both the lateral leakage and conventional bending loss.

In the mode matching implementation, to avoid treatment of the continuum of the

radiation modes of each section, two perfectly conducting planes were introduced above

and below the waveguide [23, 35]. The positions of these conducting planes were chosen

to be sufficiently far away from the Si core so that they do not affect the waveguide loss.

In order to achieve adequate accuracy, a sufficiently large number of normal modes

which include guided and radiation modes in both the silica and air claddings must be

included in the field expansion. Here 50 pairs of TM and TE normal modes were used in

47

each waveguide section. There is no boundary imposed on the lateral direction and

therefore, the mode matching simulation can accurately model the lateral leakage in thin-

ridge SOI waveguides.

To validate the mode matching results, the structure was also simulated using a finite

element model (FEM)[26] with cylindrical perfectly matched layer (C-PML) boundary

conditions; developed in house for full vectorial analysis of an axi-symmetric structure.

The FEM model utilized the COMSOL[27] FEM engine along with its geometry

definition, meshing and post-processing features.

3.3 Numerical Results and Discussion

This section presents the numerical results of the analysis of the lateral leakage of the

TM-like mode in thin-ridge SOI bent waveguides and ring resonators. The waveguide

dimensions are shown in Fig. 3.1 (a).

3.3.1 The Impact of Waveguide Curvature on the Leakage Cancellation Without

Re-Entry

The effectiveness of the leakage cancellation at “magic widths” is limited by the

imperfect balance of the generated reflected and transmitted TE waves at the waveguide

boundaries [17] (Sec. 2.3.1). It is expected that the imbalance in the amplitude of the

generated TE waves is further enhanced by the waveguide curvature of bent waveguides.

48

The impact of the waveguide curvature on the leakage cancellation at the “magic widths”

is investigated here.

To isolate the effect of the waveguide width, the secant TE waves inside the ring

were absorbed so that they do not re-enter the waveguide region. For the mode matching

simulation, this was done by forcing the amplitudes of the TE waves inside the ring to

take the form of Hankel functions instead of Bessel function of first kind as in [35]. In the

FEM simulation, a C-PML layer was placed inside the ring to absorb the secant TE

waves. It should be noted that by preventing the TE waves from re-entering the

waveguide region, we are effectively simulating the loss characteristics of a section of a

ring resonator i.e. a bent waveguide structure. Fig. 3.2 (a) shows the simulated

propagation loss of a thin-ridge bent waveguide as a function of waveguide width for

different bend radii without considering re-entrant TE waves. The results for a straight

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.3 1.35 1.4 1.45 1.5 1.55 1.6

100200400600Straight


Loss

(dB

/cm

)

Ring Radius, R (µm)

(a)

49

Figure 3.2. (Color) Simulated propagation loss of the TM-like guided mode of a thin-ridge SOI bent

waveguide in the absence of secant TE waves inside the ring (a) as a function of waveguide width for

different bend radii and (b) as a function of the bend radius when the waveguide width is fixed at

1.43 μm. Conventional bending losses are also shown.

waveguide [17] are also shown in Fig. 3.2 (a) which has a “magic width” of 1.43 μm. The

results obtained from the mode matching and FEM simulations are in good

agreement[33]. The two simulation approaches make significantly different

approximations. The FEM simulation assumes a limited simulation domain and a finite,

numerically implemented PML. The mode matching simulation considered only a limited

number of normal modes of the slab regions. Both simulation approaches produced the

same solution, proving that the solution is not a result of either of these approximations

and hence is likely to be a valid representation of the actual waveguide behavior. It is

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

100 200 300 400 500 600 700 800 900 1000Ring Radius, R (µm)

Loss

(dB

/cm

)

Total Loss

Conventional Bending Loss

(b)

50

clear that all waveguides with different bend radii show the same “magic width” which is

identical to that of a straight waveguide at 1.43 μm. However, the propagation loss

increases as the bend radius is decreased. Fig. 3.2 (b) shows the propagation loss

calculated for different bend radii with the waveguide width fixed at the “magic width” of

1.43 μm. Also shown is the conventional bending loss for these curved waveguide

sections. It is evident that the conventional bending losses are too small to account for the

predicted propagation losses in these curved waveguides. Even at a bend radius of 100

μm the conventional bending loss of a bent waveguide with equivalent refractive index

contrast is only 0.13 dB/cm, which is much smaller than the loss of ~2 dB/cm obtained

from the simulations. It is thus evident that the radius of a bent waveguide has a

significant impact on the effectiveness of the leakage cancellation at the “magic width”.

As shown in Sec. 2.3.1, the leakage cancellation is affected by the imbalance of the

reflected and transmitted TE waves generated from TM-TE mode coupling. For a bent

waveguide, the inner and outer waveguide boundaries will have different radii and thus

the reflections from these boundaries will differ significantly. This will exacerbate the

imperfect cancellation at the “magic width”. The amplitude difference between the

radiated TE waves increases as the bend radius reduces. Therefore, the leakage

cancellation is less effective with smaller radius as seen in Fig. 3.2.

3.3.2 Leakage Cancellation in Waveguide Rings with Re-Entry: “Magic Radius”

Phenomenon

Referring back to Fig. 3.1 (b), when the reflected TE wave from the outer boundary

(RTE1) and transmitted TE wave (TTE2) from the inner boundary traverse a secant across

51

the ring and intersect with the waveguide again, it is possible for these TE waves to

interfere destructively with the transmitted TE wave (TTE1) from outer boundary and

reflected TE wave (RTE2) from inner boundary resulting in low loss propagation for the

TM-like mode. The effect of the secant TE waves inside the ring was simulated by

allowing them to re-enter the waveguide region. For the mode matching simulation, this

was done by setting the form of the TE slab modes of the inner section to be Bessel

functions of the first kind. For the FEM simulation, the C-PML inside the ring was

removed and a PEC boundary was placed significantly far away from the inner boundary

of the waveguide so that this PEC boundary does not interfere with the TE waves inside

the ring as illustrated in Fig. 3.3. The waveguide width was fixed at the “magic width” of

1.43 μm. Simulations were conducted for rings with radii in a range of ±3 μm around

nominal radii R0 = 100, 200, and 400 μm. In the FEM, due to the necessity of a large

simulation domain and hence proportionately high computational resources and time

requirements for accurate modeling of the loss characteristics, simulations were

performed only for the R0 = 100 μm case.

Figure 3.3. Simulation domain used to model rings with R0 = 100 μm in FEM

SiO2

Si

Radius, R=100 µmW

C-PML

C-PML

Air

89.5 µm

14 µm

70 µm

1.5 µm

C-PML

52

Figure 3.4 shows the propagation loss of the TM-like mode as a function of the ring

radius. Both mode matching and FEM simulations produced very similar results. It can

be seen that although the waveguide width is at the “magic width” of the straight

waveguide, perfect leakage cancellation and hence low loss does not occur at all ring

radii. Complete leakage cancellation only occurs at specific “magic radii” for which the

secant TE waves interfere destructively with TE waves outside the ring. The loss of the

TM-like mode shows a cyclic dependence on the ring radius.

The propagation loss shown in Fig. 3.4 includes both lateral leakage loss due to TM-

TE coupling and the conventional bending loss. For large rings, at “magic radii”, the

Figure 3.4. (Color) Propagation loss of the fundamental TM-like mode of a ring as a function of ring

radius. Waveguide width is W = 1.43 μm

1.E-02

1.E-01

1.E+00

1.E+01

-3 -2 -1 0 1 2 3R-R0 (µm)

Loss

(dB

/cm

)

R0200 µm

R0400 µm

R0100 µm

R0100 µm

FEM

53

propagation loss approaches zero due to near-perfect leakage cancellation and negligible

bending loss. However, for smaller rings e.g. R0 = 100 μm, the losses at the “magic radii”

are limited by conventional bending losses. This effect can be seen clearly in Fig. 3.4 by

comparing the propagation losses at the “magic radii” for the R0 = 100 μm ring case with

the conventional bending loss for R0 = 100μm. The small difference between the bending

loss and the total loss at “magic radii” can be attributed to the imperfect leakage

cancellation resulting from imbalance in the amplitudes of the reflected and transmitted

TE waves for small rings as discussed above.

We now consider the dependence of the lateral leakage loss on both the ring radius

and waveguide widths other than the “magic width” of the straight waveguide. Fig. 3.5

shows the variation of the propagation loss with waveguide width for a number of ring

radii R. Also shown is the loss for a straight waveguide. Again, good agreement is

achieved between the results obtained from the mode matching and FEM simulations. It

can be seen that the loss depends strongly on both the waveguide width and ring radius.

Considering the loss at a waveguide width of 1.43 μm, which is the optimum width to

achieve lowest loss for a straight waveguide. For bent waveguides, the loss has a local

minimum at 1.43 μm, but significantly lower losses can now be achieved at other

waveguide widths.

It is instructive to combine the results of Figs. 3.4 and 3.5 into a single three

dimensional plot showing the loss as a function of both ring radius and waveguide width

as shown in Figs. 3.6 (a) and (b) for rings with radius around 200 μm and 400 μm,

respectively. The cyclic dependence of propagation loss with ring radii at a particular

waveguide width is clearly visible and so is the presence of multiple low loss waveguide

54

Figure 3.5. (Color) Propagation loss of the fundamental TM-like mode of a ring resonator as a

function of the waveguide width for different ring radii

widths for a ring with particular radius. For any waveguide width, there exist multiple

“magic radii” at which the propagation losses are low. It can be seen from Fig. 3.6 that it

is best to design a ring with waveguide width equal to the “magic width” of a straight

waveguide and then select the ring radius to be one of the “magic radii” at this waveguide

width. The sensitivity of the propagation loss to waveguide width and radius variations at

these “magic radius” - “magic width” combinations is lowest.

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

1.3 1.35 1.4 1.45 1.5 1.55 1.6

100 100 (FEM)200 300400 Straight

Waveguide Width (µm)

Loss

(dB

/cm

)Radius, R (µm)

55

Figure 3.6. Propagation loss of the fundamental TM-like mode of a ring as a function of the ring

waveguide width and ring radius. Radius around (a) 200 μm and (b) 400 μm

(a)

(b)


Rin

g R

adiu

s, R

(µm

) Loss, Log10(dB/cm

)


Rin

g R

adiu

s, R

(µm

) Loss, Log10(dB/cm

)

56

3.3.3 Mode Field Distribution

Using the mode matching simulation, the mode profile of the TM-like guided mode

was also calculated. Fig.3.7 (a) and (b) show the vectorial components of the electric

field distributions of the TM-like mode of a ring with a waveguide width fixed at 1.35

μm and radius of 199.62 μm (high loss) and 198.95 μm (low loss), respectively. Similar

field distributions were also obtained from FEM simulations. Comparing the vertical

component (Ex), it can be seen that the TM-like component is almost identical for both

ring radii. Similar to straight waveguides, there is a significant radial component (Er) of

TE-like mode despite the mode being TM-like. Inside the ring, the radial field

components are very similar for both ring radii. At an “anti-magic width” (w = 1.35μm),

the generated TE-waves inside the ring interfere constructively to give strong radial field

inside the ring regardless of whether the mode is high or low loss. Outside the ring,

however, there is a significant radial field component at the “anti-magic radius” of 199.62

μm. For the ring radius of 198.95 μm which is a “magic radius” for 1.35 μm waveguide

width, destructive interference of TE waves outside the ring result in the cancellation of

the radial field component outside the ring. The azimuthal field component (Eφ) also

shows similar radiation behavior as the radial field component.

Next, we consider the field distribution when the waveguide width is at the “magic

width” of the straight waveguide i.e.1.43 μm. Fig. 3.8 (a) and (b) show the field

distributions of the TM-like modes with radius of 199.84 μm (“anti-magic radius”) and

200.46 μm (“magic radius”) respectively. Comparing the field amplitudes of the radial

and azimuthal components inside the rings in Fig. 3.7 and Fig. 3.8, it is clear that at the

57

“magic width”, these field components inside the ring are significantly reduced due to the

destructive interference of TE waves generated from two waveguide boundaries.

However, unlike the case of a straight waveguide, the cancellation is not perfect due to

the impact of the waveguide curvature as discussed in Sec. 3.3.2. As a result there is still

a small TE component inside the ring. At the “magic radius”, the radiation field outside

the ring is further suppressed resulting in low loss propagation.

Figure 3.7. Electric field distributions of the fundamental TM-like mode of a ring with a waveguide

width of 1.35 μm and radius at (a) 199.62 μm (high loss) and (b) 198.95 μm (low loss)

(b) (a)

Ex Ex

Er Er

Eφ Eφ

r (µm) r (µm)

58

Figure 3.8. Electric field distributions of the fundamental TM-like mode of a ring with a radius of (a)

199.84 μm (high loss) and (b) 200.46 μm (low loss). Waveguide width of 1.43 μm

3.3.4 Wavelength Dependence of Lateral Leakage Loss

The “magic width” in straight waveguides has a strong dependence on the

wavelength. Similar dependency is also expected from the lateral leakage loss of the TM-

like mode of a thin-ridge SOI bent waveguide. Using the mode matching simulator, we

calculated the wavelength dependence of the propagation loss for rings with a waveguide

width of 1.43 μm (“magic width”) and radii of 400 μm and 200.5 μm, and a waveguide

(a) (b)

Ex Ex

Er Er

Eφ Eφ

r (µm) r (µm)

59

width of 1.35 μm (“anti-magic width”) and radii of 399.02 μm and 198.95 μm. The

chosen radii are the “magic radii” for the corresponding waveguide widths at λ = 1.55

μm. Fig. 3.9 (a) and (b) shows the propagation loss as a function of wavelength for

waveguides with widths of 1.43 μm and 1.35 μm, respectively. It is evident that the

propagation loss depends strongly on the wavelength. The “magic width/magic radius” at

a given wavelength may become “anti-magic width/anti-magic radius” when the

operating wavelength changes. Thus, when designing a ring resonator, care should be

taken to engineer a ring which exhibits low loss at the required resonant wavelength. The

wavelength dependent behavior of the propagation loss also presents an opportunity to

eliminate unwanted out of band resonances.

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.545 1.550 1.555 1.560 1.565 1.570 1.575 1.580 1.585

R=400 µm

Wavelength (µm)

Loss

(dB

/cm

)

R=200.5 µm

(a)

60

Figure 3.9. Wavelength dependent loss of the TM-like mode in thin-ridge SOI rings with a

waveguide width of (a) 1.43 μm and (b) 1.35 μm and ring radii as shown

3.4 Directional Coupler

We also investigated the impact of “magic width” radiation cancellation in directional

coupler geometries. A directional coupler can be formed by two parallel straight

waveguides, a cross sectional view of which is shown in the inset of Fig. 3.10 (a). The

behavior of a directional coupler can be rigorously represented as a superposition of the

fundamental and first order supermodes of the structure. Low coupler loss therefore

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.545 1.550 1.555 1.560 1.565 1.570 1.575 1.580 1.585

Wavelength (µm)

Loss

(dB

/cm

)

R=399.02 µm

R=198.95 µm

(b)

61

requires low losses from both the even and odd supermodes. Fig. 3.10 illustrates the loss

behavior of the two TM supermodes calculated using FEM-PML and mode matching

technique for the directional coupler operating in two coupling regimes. When the

waveguide center to center to separation is large (S=3 μm, Lcouple=361 μm, Fig. 3.10(a)),

both modes have a loss minima at 0.71 μm, which is also the first order magic width of

the single isolated waveguide. However, reducing the waveguide separation to increase

coupling (S=1 μm, Lcouple=16 μm, Fig. 3.10(b)) results in a loss increase for the two

modes at the original magic waveguide widths. The new “magic widths” are 0.88 μm and

0.73 μm for the fundamental and first order supermodes respectively. To balance the two

supermode losses for good coupler operation, for example, the coupler can be designed at

a waveguide width of 0.775 μm, but the loss value itself is then rather high at 10 dB/cm.

While this excess loss only occurs for the designed coupler interaction length, it could

(a)

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

0.55 0.6 0.65 0.7 0.75 0.8 0.85

TM_0 FEM+PML

TM_0 Mode Matching

TM_1 FEM+PML

TM_1 Mode Matching


Loss

(dB

/cm

)

S

BOX

Si

15 nm Ridge

190 nm

Separation, S = 3 µm

W

62

Figure 3.10. (Color) Inset: Cross sectional view of ridge waveguides in a directional coupler

configuration with waveguide centre to centre separation S and a symmetric waveguide width of W.

Graphs: Loss of the fundamental and first order TM super mode of the directional coupler as a

function of waveguide width (W) for a waveguide centre to centre of (a) S=3 μm and (b) S=1 μm

impact resonator or other device functionality and must be carefully contemplated when

designing high-Q components.

(b)

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.E+03

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

TM_0 FEM+PML

TM_0 Mode Matching

TM_1 FEM+PML

TM_1 Mode Matching


Loss

(dB

/cm

)

Separation, S = 1 µm

63

Chapter 4. Low Loss Si-SiO2-Si TM Slot Waveguides

Slot dielectric optical waveguides[36] have recently generated considerable interest

due to their ability to strongly concentrate light into nano meter scale low index

dielectrics, thereby providing enhanced nonlinearities, modulation, gain or possible

sensor applications. In the high index contrast Si–SiO2–Si system, for example, standard

waveguide calculations reveal that 20% of the optical power can be confined in SiO2 slots

as narrow as 10 nm. To realize active functions, such as lasing using active dielectric gain

media, it is critical to achieve nm-scale slot waveguides with very low optical losses.

While relatively wider etched vertical slots have been demonstrated with losses of 8

dB/cm[37], this method poses difficulties in scaling to nm slot dimensions, and scattering

from etched slot surfaces contributes undesirable losses.

Horizontal slot waveguides can be realized with smoother wafer layer interfaces

using standard planar processing and transverse-magnetic (TM) mode operation, but also

present challenges in achieving low loss. The most flexible and expedient means for

creating the horizontal slot structure utilizes a thin dielectric slot layer on a single crystal

SOI device layer, followed by deposition of amorphous silicon (a-Si) or poly-silicon,

both of which contribute undesirable propagation loss. Recently, propagation loss of 1.5

dB/cm has been demonstrated for buried a-Si waveguide structures[38], and since only

the upper Si layer in the horizontal slot guide is comprised of a-Si, this suggests that

64

carefully designed slot waveguides could achieve even lower losses if scattering losses

could be eliminated. In Sec. 2.4 we discussed a novel technique for waveguide

fabrication resulting in ultra smooth waveguide surfaces[18]. This allowed us to achieve

a loss as low as 0.36 dB/cm for the TE mode[19] in very tight vertical confinement, 0.2

μm core SOI waveguide with shallow ridge. This was essentially the result of minimizing

the surface scattering losses. These guides additionally offer the important benefit of

electrical access to the waveguide core as required for providing active functionalities.

We then showed experimentally that realizing low loss TM mode propagation in these

shallow ridge structures requires precise fabrication of waveguides at “magic widths” to

effectively negate the inherent lateral leakage of TM quasi-modes[15-17, 19]. Here, we

demonstrate TM mode propagation losses of 1.83 dB/cm in 8.3 nm horizontal slot guides

designed at the “magic width” and fabricated with a deposited a-Si layer as the top

cladding layer[39, 40], which we believe is both the lowest-loss and narrowest slot

waveguide demonstrated to date.

4.1 Waveguide Design and Fabrication

The horizontal Si---SiO2---Si waveguide is grown on commercial SOI wafers with a

buried oxide (BOX) layer of thickness 2 µm. Figure 4.1 shows the schematic of the

process flow for forming the slot waveguides. The 205 nm crystalline silicon device layer

(c-Si) has a 7.5 nm step ridge fabricated using the ultra-smooth waveguide core local

thermal oxidation process[18] to avoid etching-induced surface roughness as detailed in

Sec. 2.4, and is then thinned with blanket oxidation to a core thickness of 150 nm, leaving

65

a lateral field c-Si layer of 143-nm thickness. The 8.3 nm SiO2 slot layer between the c-Si

and yet to be deposited a-Si layer is formed using blanket thermal oxidation. The 100 nm

thick a-Si layer is then deposited using a plasma-enhanced chemical vapor deposition

system at 180 °C, at a plasma power of 40 W, and deposition rate of ~1 Å/sec. The

precise thickness of the SiO2 layer for this study was determined using ellipsometry,

while the a-Si layer thickness was determined by step profiling. For low loss TM

performance, minimizing lateral radiation leakage also requires that the waveguide width

Figure 4.1. Processing steps for shallow ridge horizontal slot waveguide. Shallow ridge

waveguide in step 1 formed using the process described in Fig. 2.10

c-Si 205 nm

BOX

7.5 nm Ridge

1

Blanket Thermal Oxidation

BOX

c-Si 150 nm 2

BOX

c-Si 150 nm

HF Strip

3 8 nm SiO2

BOX

c-Si 150 nm

Slot Thermal Oxidation

4

PECVD Amorphous Silicon Deposition

BOX

c-Si 150 nm8 nm SiO2

100 nm a-Si

W=1.51 µm ( Magic width)7.5 nm Ridge

5

66

be precisely controlled to a ‘‘magic width’’ as given by Eq. 2.1. In our case the result is

1.51 µm which is achieved by properly tailoring the original mask so that the correct

value is achieved after all process steps. Fig. 4.2 provides a scanning electron

micrograph of a similar structure, taken using the environmental secondary electron

detection (ESED) mode to best image the SiO2 slot. Further more the fabricated structure

provides 16.3 % field intensity confinement in the 8.3 nm silica slot (Fig. 1.1) but from

optical gain or loss perspective the relevant quantity is the confinement of the weighting

function as we show in the analysis below:

The change in the propagation constant (Δβ) due to a perturbation Δε(x,y) of the dielectric

constant of the waveguide is given by [41]

P

dydxEE∫ ∫∞

∞−

∞

∞−

⋅Δ=Δ

*εωβ

(4.1)

Figure 4.2. Scanning electron micrograph of a similar planar slot structure with 8.3 nm slot

visible using ESED mode imaging

67

If we assume for simplicity that we have perturbation only in the slot, then

⎩⎨⎧ Δ

=Δslotoutside

slotinsidenn slotslot

02 0ε

ε (4.2)

Gain or loss in a homogeneous medium is

imagnc

Δ=Δ ωα 2 (4.3)

but for the slot waveguide (assuming for simplicity that nreal-slot >> nimag-slot) the modal

gain or loss is given using Eq. (4.1) by

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ ⋅⋅⎥⎦⎤

⎢⎣⎡ Δ=Δ

∫∫− P

dydxEEncn

csectioncrossslot

slotreal

slotimagmode

*0_2

2ε

ωα (4.4)

The first factor in the Eq. (4.3) is the bulk gain or loss due to the homogeneous slot

medium and the second factor is the correction term which arises due to waveguide

confinement. We notice that this term is not the usual intensity confinement.

Now considering the case of a 1D slab waveguide structure, for small changes in material

index we have

)()())(( xnxnxn Δ≅Δ 22 (4.5)

The modal effective index change which is proportional to the gain or loss experienced

by the mode of the slab waveguide will be

∫∞

∞−

⋅Δ=Δ dxxfxnneff )()( (4.6)

where f(x) is a weighting function and is given by

( )∫∞

∞−

⋅×+×

⋅=

dxzHEHE

EExncxf

ˆ

)()(

**

*02 ε

(4.7)

68

Now a plot (Fig. 4.3) of this weighting function f(x) for the TE and TM modes for the 8.3

nm horizontal slot slab waveguide structure shows a huge confinement for the TM mode

in the slot. For this particular structure this means that the TM mode will see 14X higher

modal gain or loss relative to the TE mode for any gain or loss achieved in the SiO2 slot

dielectric. To be more specific, a full 23.5% of the local gain in the 8.3 nm slot will be

seen by the TM mode. This is quite phenomenal considering the fact the slot is just 8.3

nm thick and is of lower index. So for example, if the dielectric slot were appropriately

doped with rare earths to achieve a peak gain of 13.7 dB/cm in the 8.3 nm dielectric slot

Figure 4.3. Weighting function for index changes for TE and TM modes, illustrating the 14X

enhancement of sensitivity to index change in the SiO2 slot for the TM mode over TE mode for the

slot waveguide shown in the inset. The integrated weighting function across the thickness of the SiO2

slot gives an effective confinement of 23.5%.

-0.5

-0.4

-0.3

-0.2

-0.10

0.1

0.2

0.3

0.4

0.5

0 10 20 30Weighting function (µm-1)

Wav

egui

de P

ositi

on (µ

m)

TE TM

150 nm Si

100 nm Si

8 nm SiO2

BOX

69

[42], the modal gain would be 3.2 dB/cm. This is attractive as it could provide sufficient

gain for lasing while potentially allowing for electrical excitation of the nm scale slot

medium. Another critical thing to notice in the plot are the spikes of the TM weighting

function at the interfaces between different layers which make it more susceptible to

surface and interface roughness scattering losses. This makes achieving ultra smooth

waveguide surfaces all that more critical.

4.2 Experimental Evaluation

We evaluated the waveguide propagation losses of the horizontal slot waveguides

using the weakly coupled ring resonator geometry as discussed Sec. 2.5. Compared to the

cut-back or Fabry---Pérot fringe methods, this provides a more reliable loss measurement

due to unambiguous and precise resonance characteristic measurement of a single device

made using a scanning single-frequency laser. With negligible contribution from

coupling loss, the loss would be given by

( )g

FWHMelogcmdB

ννπα Δ⋅⋅⋅=⎟

⎠⎞

⎜⎝⎛ 210 10 (4.8)

where ΔνFWHM is the measured drop port full-width at half maximum line width and νg

is

the waveguide group velocity. The experiment is conducted using an external cavity laser

diode with a continuous single longitudinal mode fine wavelength scanning feature. Fig.

4.4 shows the smallest observed linewidth of Δν=625MHz, yielding a resonator of

Q~3X105.

The measured group index of ng=4.1, obtained from the free-spectral range of the

70

Figure 4.4. Scan of ring resonance illustrating ΔνFWHM =625 MHz with a corresponding Q~3 X 105

resonator is quite close to the numerically modeled value of 4.0, which is significantly

larger than the calculated phase index of nphase=2.1 for the TM mode of this slot structure.

Using the experimental group index, the corresponding loss is 2.33 dB/cm if all the loss

is ascribed to propagation. However, the calculated coupling coefficient for the

intentionally weak directional couplers in this resonator is 2.15% per coupler, and the two

couplers combined thus contribute a loss of 0.50 dB/cm for our 600 µm radius ring,

leading to a waveguide transmission loss of 1.83 dB/cm [39, 40].

In addition to loss variation with wavelength that is expected due to the ‘‘magic

width’’ effect, we also see small variations in successive resonances. We believe these

may be due to instrumentation and facet reflectivity effects and are under investigation.

But even if averaged, these yield a loss increase of only ~25% above the minimum value.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1566.15 1566.2 1566.25 1566.3 1566.35 1566.4 1566.45 1566.5

QTM=3.06 x 105

ΔνFWHM,TM=625 MHzαTM=2.33 dB/cm

ΔνFSR,TM= 19.40 GHzD

rop

port

resp

onse

(arb

.)

Wavelength (nm)

600 µm Ring, 1.51 µm “Magic width” Slot Waveguide

71

An estimate of loss of our amorphous silicon film was also made. For this we refer

back to our result for the TM mode loss of 0.94 dB/cm in shallow ridge waveguides[19]

(Sec. 2.6). If we now attribute all the excess loss of 0.89 dB/cm in going from ridge

waveguide to the slot waveguide to a-Si and consider the 32.5 % weighting function

confinement in the 100 nm a-Si film in the slot waveguide, we get an upper limit of 2.8

dB/cm for the a-Si loss. This loss value for a-Si puts it in the class of some of the best

results reported[38] so far for low loss or good optical quality amorphous silicon films.

TM slot mode losses from conventional bending radiation are vanishingly small at

our large ring radius. Calculations reveal that a 0.1 dB/cm excess loss would be incurred

for a reduced radius of ~320 µm. However, with a modest increases in ridge height to

only 15 or 30 nm, which is not expected to increase loss significantly, the calculated 0.1

dB/cm conventional bending loss radius reduces to 82 and 24 µm respectively due to the

excellent lateral bend tolerance of TM modes. TE mode bending losses at these shallow

ridge heights are exorbitantly high and hence TE mode drop ports couldn’t be measured.

In conclusion, we have demonstrated a-Si---SiO2---c-Si 8.3-nm slot waveguides with

quasi-TM-mode loss of 1.83 dB/cm using an ultra smooth, low-loss shallow ridge

fabrication technique. This result illustrates that low-loss slot waveguides can be

achieved with nm-scale slots using deposited amorphous silicon for the upper layer of the

slot structure, providing flexibility in designing high-index-contrast, high-confinement

passive and active horizontal slot waveguide devices.

72

Chapter 5. Erbium Doped 8 nm Horizontal Slot Waveguides

Erbium (Er) has played a key role in advancing the long haul telecommunication

optical networks by providing means for repeaterless amplification of optical signals at

the telecommunication wavelength of 1.55 μm. Trivalent Erbium Er3+ (the preferred

bonding state of Er) has an incomplete 4f electronic shell that is shielded from the outside

world by the closed 5s and 5p shells. As a result, rather sharp optical intra-4f transitions

can be achieved from erbium doped materials. The transition from the first excited state

to the ground state in Er3+ occurs at an energy of 0.8 eV, corresponding to a wavelength

of 1.54 μm, a standard telecommunication wavelength at which the standard silica based

optical fibers have maximum transparency. This precise attribute of Erbium is utilized in

Erbium doped fiber amplifiers (EDFAs)[43] to amplify optical signals at 1.54 μm. A

continuous pump laser is used to create population inversion between the ground state

and the first excited state in Er3+, whereupon stimulate emission serves to amplify a high

frequency telecommunication signal at 1.54 μm. The availability of miniature

semiconductor diodes to pump the EDFAs, coupled with its exquisite gain characteristics,

polarization insensitivity, temperature stability, quantum limited noise figure and

immunity to interchannel crosstalk meant EDFAs could be deployed widely in a cost

effective manner while at the same time setting a high standard for network performance.

73

It is perhaps safe to argue that it is mostly because of the availability of the EDFAs, that

the optical telecommunication networks are as well developed as they are today.

With the long haul conquered the next logical frontier for Erbium is chip based

applications where it is being investigated as a source of optical gain. However, the

challenges now are even more formidable than before as has been indicated in Sec. 1.1.

In spite of these, considerable progress has been made in devising erbium doped

waveguide amplifiers (EDWAs) [44, 45] and more recently chip based erbium doped

lasers [46-48]. However these structures are designed to operate under optical pumping

conditions and provide no clear direction for extension to electrical drivability in a non-

invasive manner. As a result they have not been widely employed in chip scale

applications.

We believe horizontal slot waveguides with ultra narrow slots in principle provide

means of electrical excitation of the slot embedded gain media like Erbium through non-

invasive mechanisms like tunneling, polarization transfer and transport. However, as a

first step towards evaluating the suitability of such a device to provide optical gain under

electrical excitation it is instructive to characterize the behavior of the device under

optical pumping scheme. Having achieved encouraging loss values for the horizontal slot

waveguide structures for TM mode operation, erbium implantation of the slot was

undertaken. The samples were characterized for photoluminescence and erbium lifetime.

In the process, an unexpected physical phenomenon - Purcell enhancement[49] of

luminescence decay associated with erbium luminescence centers in ultra thin slots came

to forefront. The same was thoroughly analyzed theoretically and experimentally and

74

could have significant positive implications on the realization of novel light sources in

silicon[50, 51].

5.1 Erbium Doped Slot Waveguide Fabrication

For Erbium doped slot waveguides the challenge was incorporation of erbium into the

ultra thin slot with minimal exposure of the underlying crystalline silicon layer to

Erbium. This entailed utilization of extremely low implantation energy of just 2 keV and

a 45° implantation angle. A relatively higher dosage of erbium of 6X1020 cm-3 was used

to compensate for the low implantation energies. The Erbium doped slot samples

followed the same fabrication sequence as was utilized for the formation of the passive

slot structures described in Sec. 4.1 (Fig. 4.1) except for the implantation of Erbium in

the ultra thin slot followed by Erbium activation anneal at 700 °C in nitrogen ambient for

15 minutes prior to amorphous silicon film deposition. Fig. 5.1 shows the implantation

conditions and the final Erbium doped slot structure obtained after all the processing

steps, with structural attributes same as the final passive slot structure shown in Fig. 4.1.

Figure 5.1. (a) Implant and anneal conditions and (b) cross sectional view of Erbium doped slot

waveguide

(a) (b)

100 nm a-Si

Si Substrate

BOX 143 nm c-Si

8 nm Er/SiO2

7.5 nm Ridge

W=1.51 µm (magic width)

Amorphous Silicon deposition

Erbium Implantation2 KeV 6E20 cm-3

Activation anneal700°C, N2 ambient, 15 min

75

5.2 Photoluminescence Measurements

Photoluminescence measurements were done in full waveguide configuration by end

pumping straight slot waveguides using a high power 1480 nm pump laser. Fig. 5.2

shows a schematic of the experimental setup used to measure the photoluminescence.

Light was coupled in and out of the waveguides using lens tipped fibers precision aligned

to the waveguide using six axis piezo stages. The pump was TM polarization to utilize

the higher confinement characteristic of the TM mode in the slot. Cascaded band pass

filters with centre wavelength of 1480 nm and 1550 nm with 30 nm bandwidth were used

on the input and output respectively to provide more than 60 dB of out of band isolation.

The luminescence spectrum was measured on an ANDO AQ6317 optical spectrum

analyzer with a resolution bandwidth (RBW) of 0.5 nm. Fig. 5.3 shows the measured

luminescence spectra at different waveguide coupled pump powers which shows clear

Erbium characteristics. This we believe is the first observation of Erbium luminescence

in ultrathin slot waveguides in full 3D waveguide configuration primarily made possible

due to the low losses of these waveguides. The waveguide transmission loss is ~3 dB/cm

for the TM mode as determined from the ring resonator measurements.

Figure 5.2. Setup for measurement of Erbium doped slot waveguide luminescence

Pump1480 nm

Polarization controller

CascadedFilters OSA

1550 nm Band pass

Six Axis Piezo stages

Er doped Slot waveguides

CascadedFilters

1480 nm Band pass

76

Figure 5.3. Erbium doped slot waveguide photoluminescence at different waveguide coupled pump

powers, expressed as power spectral density (PSD)

5.3 Time Resolved Photoluminescence Measurements

Considerable effort was put into accurately measuring the Erbium spontaneous

emission lifetime to quantify the radiative and non-radiative processes associated with

Erbium in the ultra narrow 8 nm slot. The measurement is made complex by the fact that

only 2.6 nW of integrated luminescence power could be collected from these waveguides

even under saturation conditions. In the absence of a sensitive detector like photo

multiplier tube (PMT), a setup had to be designed to increase the signal to noise ratio of

the luminescence.

-85

-83

-81

-79

-77

-75

1525 1535 1545 1555 1565

~ 8mW

~ 40mW

Wavelength (nm)

PSD

(dB

m)

RBW=0.5 nm

Waveguide Coupled Pump Power

77

The setup for lifetime measurement was similar to that showed in Fig 5.2, except that

the luminescence was now detected using a Thorlabs InGaAs PIN fiber coupled detector

instead of OSA. The detector has a responsivity of 0.83 at 1550 nm, a capacitance of 0.7

pf and a noise equivalent power of 10-15 W/(Hz)0.5. Even with a load resistance of 10 kΩ

which corresponds to a bandwidth of 22.7 MHz, we can expect to see only 21.25 μV at a

signal to noise ratio (SNR) of 0.12 for 2.6 nW of optical power coupled to the detector,

which would be impossible to see on a digitizing oscilloscope. To improve the signal to

noise ratio, a scheme using a lock-in amplifier was initially implemented to quantify the

Erbium lifetimes using the frequency domain technique. The method yielded very fast

lifetimes in the range of ~150 μsec which seemed unusually fast compared to the

expected millisecond range lifetimes for Erbium in silica matrix. The results were also

convoluted by the presence of multiple life times for Erbium making it challenging to

interpret the frequency domain data. To corroborate the fast lifetimes, a more direct time

domain technique was implemented using a gated integrator and BOXCAR averager

system with preamplifiers. The system pre-amplifiers supplied with the BOXCAR have a

gain of 5 and a noise of 6.4 nV/(Hz)0.5. The system allows cascading of five such

amplifiers together. The best expected signal output with 2.6 nW of coupled optical

power is 0.013 V with a SNR of 12.7 by using 15 μsec gate width (GW) and ten thousand

sample averaging. Fig. 5.4 shows the pump pulse and the luminescence signal obtained

under these conditions. Although the luminescence can be seen, the decay is not smooth.

This meant a further improvement in SNR was required to get a quantitative

measurement of the Erbium lifetime. This was only possible if we improved on our pre-

amplifiers as we were already operating at the limit of the BOXCAR at ten thousand

78

Figure 5.4. (a) Pump pulse and (b) Luminescence signal using gated integrator and BOXCAR

averager with system preamplifiers. Observed peak luminescence matches calculated value

samples averaging. The system pre-amplifiers were replaced by a ultra low noise

(3pA/(Hz)0.5) trans- impedance amplifier (TIA) with a bandwidth from DC to 2 MHz and

a gain of 10 MV/A. With this TIA, the expected signal was 4.25 V with a SNR of 543.

Fig. 5.5 (a) shows the measured luminescence signal under these new conditions and the

signal is finally clean enough to measure the Erbium lifetime. Fig. 5.5 (b) shows the

luminescence decay on a semi-log scale under saturation conditions. The measured

Erbium lifetime is 152 μsec, similar to what we had obtained using the frequency domain

method using a lock-in amplifier.

A fast lifetime of ~152 μsec suggests a strong non-radiative decay channel for

erbium. However it is also possible that other factors might me contributing to the fast

lifetime including the waveguide structural impact on the radiative spontaneous emission

rates. This will be the topic of the next section.

(a) (b)

-5.0E-03

-1.0E-17

5.0E-03

1.0E-02

1.5E-02

0 1 2 3 4 5 6 7-0.20.00.20.40.60.81.01.2

0 1 2 3 4 5 6 7Time (x100 μsec)

Box

car o

/p (V

olts

) Er Luminescence

Nor

mal

ized

Box

car o

/p (V

olts

) Pump pulse @1480 nm

Time (x100 μsec)

79

Figure 5.5. (a) Photoluminescence measured with the system pre-amplifiers replaced by a low noise

high gain TIA in conjunction with gated integrator and BOXCAR averager (b) Luminescence decay

signal on a semi-log plot with a lifetime of ~ 152 μsec

0.00.51.01.52.02.53.03.54.04.55.0

0 2 4 6 8 10Time (x100 μsec)

Box

car o

/p (V

olts

)

1 µsec GW,1K Sample Avg.

15 µsec GW,10K Sample Avg.

Er Luminescence

(b)

(a)

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

3.445 5.445 7.445 9.445Time (x100 μsec)

152 μsec

Nor

mal

ized

Lum

ines

cenc

e D

ecay

Sig

nal

80

5.4 Spontaneous Emission in Waveguide Media

The atomic spontaneous emission rate can be expressed in terms of the zero-point

fluctuations of the electromagnetic field at the position of the atom. The local zero-point

field fluctuations depend on the photon density of states and on the electromagnetic field

strengths of the modes. As the electromagnetic fields of the modes depend strongly on

the configuration and on the electromagnetic properties of the materials, the spontaneous

emission rate can be either increased or decreased, depending on the electromagnetic

properties of the atom’s environment. The possibility of modifying the spontaneous

emission rate by changing the electromagnetic environment was first pointed out by E.M.

Purcell in 1946 [49], where it was implicitly assumed that a sizable modification in

spontaneous emission rates could only be brought about by putting the atom in a high Q

resonator. However very recently pronounced broadband enhancements of spontaneous

emission rates has been modeled and observed in Er-doped Si/SiO2 slot waveguides [52,

53] i.e. even in a non-resonator configuration. Due to the high index contrast at the

Si/SiO2 interfaces, the electromagnetic field strongly concentrates in the narrow slot

region (for the TM mode), leading to an increase in the zero-point fluctuations. The zero-

point field fluctuations can be calculated by applying quantum electrodynamic formalism

which requires determination of the complete set of electromagnetic modes which can

exist in the configuration so as to quantize the electromagnetic field [54, 55]. The

modification of the spontaneous emission rate can also be obtained from a semi classical

treatment by calculating the emitted power from a classical dipole source which is also

equivalent to the work done on a dipole by its own reflected field [56, 57].

81

However, here we show that Purcell enhancement in waveguides is already correctly

captured in conventional spontaneous emission expressions[58]. A quantum

electrodynamic derivation of enhanced spontaneous emission rate in waveguides is also

shown to prove the equivalence of the two approaches[58]. We also show how this result

for a waveguide naturally maps into the traditional expression for Purcell factor which is

usually associated with a resonator[58]. The results are then compared with those

obtained using a full numerical QED calculation which includes the contribution from the

radiation modes. Finally the experimental results presented above are analyzed in light of

this new information[50, 51].

5.4.1 Spontaneous Emission into a Particular Waveguide Mode

This sub-section closely follows materials from unpublished notes used in ECE 450,

Optoelectronics Physics and Lightwave Technology II, taught by Prof. Thomas L. Koch

[58].

Elementary treatments of spontaneous emission into waveguides often invoke the

fraction of spontaneous emission emitted into a waveguide mode by dividing the solid

angle ΔΩ mode of the mode by the total 4π emission. From simple diffraction theory

modemode An2

2λ=ΔΩ (5.1)

where Amode is the mode effective area, which is assigned the value

Γ= active

modeA

A (5.2)

82

by considering the confinement factor Г of the mode within the active emitting area

Aactive. This leads to a fraction of spontaneous emission emitted into both directions of

one waveguide polarization mode to be given by

activeactivesp AnAn

F Γ=Γ⋅⋅= 2

2

2

2

441

212

πλλ

π (5.3)

where the confinement factor is given by the classical expression based upon mode field

amplitudes E and H ,

( )∫ ∫

∫∞

∞−

∞

∞−

∗

∗

⋅×+×

⋅≡Γ

dxdy

dxdyncactiveA

activeo

zHEHE

EE2

ˆ*

ε (5.4)

The spontaneous emission rate (SER) into one waveguide polarization mode is then

30

30

221

2

2

3411

cn

AnF

activemediumhomospsp

WGntoisp επωμ

πλ

ττ⋅Γ=⋅= (5.5)

which includes the standard “free space” SER for a medium of index n for a transition

with a dipole moment µ21. It is remarkable, however, that when applied to slot

waveguides we find that the “fraction of spontaneous emission” Fsp into the waveguide

mode given above can easily exceed unity. While this is seemingly a nonsensical result,

it is actually rigorously correct and the factor Fsp above is exactly the Purcell

enhancement factor for emission into a waveguide mode. This follows rigorously from a

quantum electrodynamic treatment of inhomogeneous dielectrics as derived for example

by Dalton, et al. [59]. The quantized electric field operator in an inhomogeneous

dielectric medium )x(rε is

( )∑ +− −⋅=n

nti†

nnti

nn eaieait )()(),( * xAxA

2xE ωωω (5.6)

83

where the vector basis An(x) set satisfies the self-adjoint operator eigen value wave

equation

( ) nn

nnr c

AAAx

12

22 ω

ε=⋅∇∇−∇

)( (5.7)

along with the modified radiation gauge criterion 0Ax =⋅∇ nr )(ε .These vectors,

proportional to the vector potential, are thus complete under the orthogonalization

condition

mnV

nmrnm xd ,*)( δεε =⋅≡ ∫ 3

0 AAxAA (5.8)

Assuming a transition dominated by dipole emission, the spontaneous transition rate from

level b to level a into mode An (x) is given by Fermi’s Golden rule as

( ) )()(ˆ

)(,),(,

* ωδμπω

ωδπ

−−⋅⋅+=

−−⋅+=−

abnbann

abnnspontba

EEN

EEbNetaNW

22

2

xAρ1

xxE12

(5.9)

where Nn is the photon occupancy of the mode and ρ is the unit vector in the dipole

direction. For a channel waveguide of length L and propagation constant βn we have

Lyxe nzi

nn

0

φxAε

β ),()( = (5.10)

where φn (x, y) is scaled so that An (x) satisfies the normalization condition above. The

density of states for forward and backward emission is given by

( ) cnLE geff −⋅⋅=

πρ

22)( (5.11)

So the spontaneous emission rate averaged over dipole orientations becomes

84

⎟⎟⎠

⎞⎜⎜⎝

⎛×=

= −−

),(

),(),(

yxAn

yxc

nyxW

modemediumhomosp

nbabageff

spontba

1411

φ3

2

2

2

0

2

λπτ

εμω

(5.12)

where the effective area of the mode for an emitter at (x, y) is

2φ1

),(),(

yxnnyxA

ngeffmode

−

≡ (5.13)

The first term in Eq. 5.12 is the standard homogeneous medium spontaneous emission

rate and the second term (enclosed in brackets) which accounts for the impact on the

emission rate due to the waveguide configuration is referred to as the Purcell factor.

Using the equivalence between the Poynting vector and the group velocity times the

energy density; it can be shown that the confinement factor defined earlier is also equal to

∫ −=ΓactiveA

ngeff dxdyyxnn 2φ ),( (5.14)

If we then average the SER across the active emitting region we find that indeed

Γ= active

modeA

yxA ),( (5.15)

and hence we obtain the same expression for the Purcell factor as the one we wrote for

Fsp in Eq. 5.3

activesp An

FFactorPurcell Γ== 2

2

4πλ (5.16)

We thus see that the conventional expressions for emission into waveguide modes still

apply, and indeed predict a SER into the waveguide mode alone that exceeds the total

homogenous medium result whenever we satisfy

85

( )

22

zHEHE

EE21⎟⎠⎞

⎜⎝⎛⋅>

⋅×+×

⋅=

∫ ∫∞

∞−

∞

∞−

∗

∗

λπε n

dxdy

yxncyxA

o

mode ˆ

),(),( *

(5.17)

for an emitter at the position (x, y)[58]. The equivalent 1-D slab waveguide analysis can

be carried out similarly and yields a spontaneous enhancement factor

)(1

221

1 xtnnn

Fmode

effDsp ⋅⎟

⎠⎞

⎜⎝⎛=−

λ (5.18)

where [tmode (x)]-1 i.e. the inverse modal thickness is the function above in Eq. 5.17 with

the denominator integrated over just the vertical dimension x.

So for the ultra-thin 8.3 nm slot waveguide shown in the inset of Fig. 5.6, Fig. 5.6

illustrates a dramatic enhancement of emission expected into the TM slot mode for

Figure 5.6. (Inset) Ultra-thin 8.3 nm slot slab waveguide structure. Plot of effective inverse modal

thickness in the vertical direction as a function of position for both the TE and TM modes

-0.5

-0.4

-0.3

-0.2

-0.10

0.1

0.2

0.3

0.4

0.5

0 10 20 30Inverse Modal Thickness (µm-1)

Wav

egui

de P

ositi

on (

µm)

TE TM

150 nm Si

100 nm Si

8.3 nm SiO2

BOX

86

emitters placed in the slot. The average effective modal thickness across the active slot

layer in the vertical direction is calculated to be nmtt activemode 35/ =Γ= which is 1/30th

of a wavelength in the medium, yielding a ~11X emission rate enhancement into just the

slot TM mode compared to homogeneous silica.

5.4.2 Total Spontaneous Emission Rate Including All Modes

To obtain the total SER enhancement, the calculation must also include emission into

radiation modes that may also get modified by the structure. To do this rigorously, a

quantum electrodynamic approach was used to calculate the zero-point fluctuations of the

electromagnetic field and the spontaneous emission rate from a non-absorbing dielectric

film. Our formulation closely follows the one given in [54] where the authors analyze the

spontaneous emission rate from a lossless dielectric film bounded by two non-absorbing

dielectric half spaces of arbitrary refractive indices. We extended this methodology to the

five layer slot structure i.e. Si-SiO2-Si slabs bounded by air and SiO2 dielectric half

spaces.

The process requires the quantization of the electromagnetic field which in turn

requires that the complete set of spatial electromagnetic modes that exist in the

configuration be determined. These modes must be orthonormalized with respect to the

scalar product corresponding to the electromagnetic energy density. The set consists of

radiation modes pertaining to plane waves that are incident from either dielectric half

space for both orthogonal polarizations. In addition, guided modes of either polarization

may exist. These modes are evanescent in both dielectric half spaces. Spontaneous

87

emission rate and zero-point field fluctuations are then calculated by incorporating the

quantized field in the Fermi’s Golden Rule. Here we summarize the key results of the

formulation. For a complete derivation, the reader is referred to Appendix A –

Spontaneous Emission Rate in a Slot Waveguide.

Suppose that an atom makes a spontaneous dipole transition from a state 2 to a

state 1 thereby emitting a photon of energy 0ω . The spontaneous emission can occur

in any mode of the electromagnetic field of frequency 0ω , or equivalently of wave

number ck /00 ω= in vacuum. The transition rate into a particular electromagnetic mode

with parameters μλκ ,, say, and with wave number k in vacuum, is given by [60]

)(ˆ kkifc ED −Η 0

2

2

2 δπ (5.19)

where i and f are the initial and final states of the combined atom-radiation system

and EDΗ is the electric-dipole interaction part of the complete Hamiltonian of the atom -

radiation system. In the initial state of the electromagnetic field, no photons are present.

In the final state there is one photon in the mode with parameters ( )μλκ ,, .

According to Fermi’s Golden Rule, the total spontaneous emission rate is obtained by

integrating Eq. 5.19 over all the final states. Suppose that the dipole moment of the atom

is parallel to one of the axes of the Cartesian coordinate system and let j be x, y, or z. Let

jE be the jth component of the transverse electric field operator. Then the total

spontaneous emission rate of an atom at position r whose dipole moment is parallel to

the jth axis is [61]

88

)(zFDc

e j2

122

221 πτ

= (5.20)

where e is the electron charge, 12D is the dipole matrix element of the atomic transition,

and

( )

∑ ∑ ∫∫

∑ ∑ ∫∫∫

=

∞

= >

= =<+

−+

−

=

PSyx

PSnkkk

yxj

j

min

yx

dkdkkkrE

dkdkdkkkrE

zF

,

,

,

))(()(

)()(

)(

λ ν ββ

λννλκ

λ μμλκ

λν

μ

βδεω

δεω

10

2

0

5

40

2

0

2

221

22

(5.21)

is the contribution (per wave number) of all modes of wave number 0k to the jth

component of the zero-point field fluctuations.

When polar coordinates are introduced in the ( )yx kk , plane:

)(insk),(cosk yx ϕβϕβ == (5.22)

Eq. 5.21 becomes

∑ ∑ ∫ ∫

∑ ∑ ∫ ∫ ∫

=

∞

=

∞

=

= =

∞

=

−+

−

=

PS sincos

PS

nk

ksincos

j

j

min

ddkkrE

ddkdkkrE

zF

, ))(),((

, )),(),((

,

))(()(

)()(

)(

λ ν

π

β

λν

ϕβϕβκνλκ

λ μ

π

ϕβϕβκμλκ

λν

μ

ϕβββδεω

ϕββδεω

1

2

00

2

0

5

4

2

0 0 00

2

0

2

2

(5.23)

∑ ∑ ∫

∑ ∑ ∫ ∫

=<

≥ =

= = =

+

=

PSkk

sinkcosk

PS

nk

ksincos

j

j

min

dkdk

dkrE

ddrE

zF

, ))()(),()((

, )),(),((

,

)()()(

)(

)(

λ ν

π λνλ

νϕβϕβκ

νλκ

λ μ

π

ϕβϕβκμλκ

λν

λν

λν

μ

ϕββεω

ϕββεω

000

0

1

2

000

2

0

0

5

4

2

0 0

2

0

0

2

2

(5.24)

89

The integral over ϕ for the three components j=x, y, z in Eq. 5.24 can be computed

explicitly. For the mean of the three components,

[ ])()()()( zFzFzFzF zyx ++=31 (5.25)

the result of the integral over ϕ yields a rather concise formula. The quantity )(zF is the

zero-point field fluctuation of a randomly oriented atom at depth z and is therefore of

special interest. After further simplifications it is given by

∑ ∑

∑ ∑ ∫

=<

≥ =

= = =

+

=

PSkk

k

PS

nk

k

j

min

kdk

dkrE

drE

zF

, )),((

, ),,(

,

)()()(

)(

)(

λ ν

λνλ

νβκ

νλκ

λ μ βκμλκ

λν

λν

μ

ββεωπ

ββεωπ

00

0

100

0

2

0

0

5

4 0 0

2

0

0

232

232

(5.26)

The remaining integrals with respect to β have to be computed numerically.

Consider the zero-point field fluctuations in a homogeneous dielectric with a

refractive index n1. The components of the zero point fluctuations in this homogeneous

case are given by

freezyx FnFFF 1=== (5.27)

where freeF denotes the vacuum field fluctuations (in free space):

20

2

30

6 cF free

επω= (5.28)

For an atom at depth z inside the slab of index n1 in a multi-dielectric slab structure,

having a dipole moment parallel to the jth coordinate axis, the spontaneous emission rate

relative to the value in a bulk n1 index medium will be

free

j

FnzF

1

)( (5.29)

while for a randomly oriented atom the relative transition rate is

90

freeFnzF

1

)( (5.30)

with )(zF being defined by Eq. 5.25.

Finally, the mean spontaneous emission rate for a film of thickness d relative to bulk

film medium is obtained by computing the mean of Eq. 5.30 over the thickness of the

film.

Figure 5.7 shows a plot of the ( )freeFznzF )()( for the 8.3 nm slot structure shown in

the inset. It is clear that an emitter placed in the silica slot will see large enhancement in

the SER relative to that in a bulk silica medium. Furthermore, Fig 5.8 shows the mean

relative SER enhancement for an emitter placed in a T nm thick silica slot. The thickness

Figure 5.7. Plot of F(z)/(n(z)Ffree) as defined in Eq. 5.30 as function of position in the 8.3 nm slot

structure shown in the inset.

-250

-200

-150

-100

-50

0

50

100

150

200

250

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

8.3 nm SiO2

Slab

Pos

ition

(nm

)

( )freeFznzF )()(

BOX

150 nm c-Si

100 nm a-Si

Air

91

of the a-Si and c-Si have been fixed at 100 nm and 150 nm respectively. In addition to the

total enhancement, the contributions from TM guided, TE guided and combined radiation

modes have been plotted separately to elucidate the contributions of each to the total SER

enhancement. Based upon these results, for T=8.3 nm we can expect a 13.1X

enhancement in the SER for an emitter placed in the silica slot relative to that in a bulk

silica medium. This means our analytical result of above of ~11.06X is quite accurate

even though we ignored the contributions of the radiation and TE modes to it. This is

Figure 5.8. Plot of total spontaneous emission rate enhancement, averaged over polarizations, as a

function of slot thickness. Also shown are all the contributions, dominated by the TM slot guided

mode, along with the efficiency of emission into just the TM slot mode. Values indicated for the 8.3

nm slot case

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0

2

4

6

8

10

12

14

16

18

20

0 10 20 30 40 50 60 70 80 90 100

TM G

uided Efficiency

Slot Thickness, T(nm)

SER

Enh

ance

men

t in

Slot

Total

TE

TM

Radiation

13.07

0.86

92

because in the slot structure the dominant SER is into the TM polarized guided mode.

The efficiency into the TM guided mode is plotted on the right vertical axis in Fig. 5.8

and is as high as ~ 90% in this structure for the thinnest of slots.

5.4.3 Relation to Purcell Factor

The traditional expression for the Purcell factor is different from the one derived

above, which raises the question about the relationship between the two. This sub-section

also closely parallels the notes from Optoelectronics Physics and Lightwave Technology

II [58].

The Purcell enhancement factor F is usually associated with resonators and is written

in terms of resonator Q and mode volume V as

VQ

nF

3

2 26

⎟⎠⎞

⎜⎝⎛= λ

π (5.31)

In this resonator picture the spontaneous emission rate enhancement is usually cast

in-terms of very high density of states at the resonance as captured by the resonator Q

value. However, the actual basis for this effect is due to the anomalously large amplitude

of the modal field in such a resonator, and correspondingly, the anomalously large

amplitude of the vacuum field fluctuations that can be viewed as being responsible for the

spontaneous emission. The spontaneous emission changes we have seen in the

waveguide are due to this exact same phenomenon. To see this, we can very easily

extend the waveguide treatment to include a resonator, and get the Purcell ratio

above[58].

93

To do this, rather than viewing the Purcell effect as arising from an increase in the

density of states, one can embed the resonator in a volume large enough such that the

phase variance upon transmitting through the resonator does not significantly affect the

density of states of the large volume, even at the resonant frequency. We will show here

that this same requirement on the size of the volume being large is also sufficient to

guarantee that the energy inside the resonator, while possibly large at resonance, is still

vanishingly small compared to the energy outside the resonator in the large volume. This

means that the large volume modes can still be used without changing their normalization

to the large volume, and the density of states is also unchanged for these modes.

As an application of this, we can consider the two-dimensional channel waveguide.

We already derived the spontaneous emission rate into such a waveguide which is given

by

2

0

2

ρ ),(ˆ * yxc

nW n

babageffspontba ϕ

εμω

⋅= −− (5.32)

where we have not averaged over all angles of the atomic dipole moment. If we place two

reflectors of power reflectivity R in this waveguide separated by a length Lc as shown in

Fig. 5.9, this makes a small Fabry-Perot resonator of length Lc with the transmission

characteristic given by

c

c

Li

Li

FP eReTt β

β

ω 21 ⋅−=)( (5.33)

and if we let R = 1−δ where δ is understood to be small, we obtain the power

transmission of the mirror as T = δ and the Q of the resonator is given by

δω

cnL

Q geffc −= (5.34)

94

Figure 5.9. A Fabry-perot resonator of length Lc formed by inserting mirrors of reflectivity R in

an arbitrarily long waveguide of length L

We now want to stipulate that the total external length of the waveguide L is so large

that the number of external modes that “sample” the spectral width of the resonance for a

mode in question is large. If we consider propagation along the z-axis as illustrated in the

Fig. 5.9, the usual periodic boundary condition, or phase condition, for the modes of the

huge cavity in the z direction would be

MLL resonatorc πϕβ 2=+− )( (5.35)

and the number of modes in an interval dβ is then

πβ

βϕ

2d

dd

LLdM resonatorcz ⎟⎟

⎠

⎞⎜⎜⎝

⎛ +−= )( (5.36)

We know that the resonator phase will vary rapidly in the region of the resonance, so

to significantly oversample the resonance, we must require that, even in the vicinity of

the resonance,

)( cresonator LLd

d −<<β

ϕ (5.37)

At resonance, the phase derivative above is given by

ωβϕ

geff

resonator

ncQ

dd

−

= 2 (5.38)

Eout EoutEin

Lc

R R

L

z

95

So to have a high density of external modes to smoothly and effectively sample through

the resonance we need

cLLn

cnL

Q cgeffgeffc

2)( −

<<= −− ωδ

ω (5.39)

which says that we just need to ensure that δ

cLL

2>> , which is easy to do

mathematically, recalling that L is just the large, otherwise arbitrary dimension of our

waveguide.

We also want to stipulate that the normalization of the long waveguide modes is not

impacted by the presence of the resonator. Note that this normalization is given by

1A22 =⋅∫∫ dxdyndz

An

L

(5.40)

where A covers the area of the lateral mode and L is the very long external length of the

waveguide. Because22

AE nn ∝ , clearly if the overall normalization is not impacted we

would require that the integrated value of2

En inside the resonator (i.e., the energy) is

negligibly small compared to the total integrated value of 2

E n along the entire length of

the waveguide. To compute this, we need to evaluate the field enhancement of an

external mode that is coupled into the resonator. The ratio of the field outside to the field

inside the resonator is simply given by

δ== TEE

in

out (5.41)

96

Thus the energy density outside compared to the energy density inside will be

proportional tocQnL geffc ω

δ −= . Then if the energy density inside is U, the energy density

outside is cQnL

U geffc ω− , and we require for negligible energy in the resonator

LcQnL

ULU geffcc

ω−<<⋅ (5.42)

which thus requires that

ωgeffncQL

−

>> or δ

cLL >> (5.43)

and which we notice is identical to the condition we needed to make sure the phase

distortion of the resonator transmission was not strong enough to impact the sampling of

the resonance feature with many external long waveguide modes.

From this point, the enhancement inside the resonator is obvious because the long

waveguide modes normalization is unaffected, and the density of states is identical to

what we have already used to calculate spontaneous emission into the waveguide modes.

The only difference is the fact that the amplitude of the electric field locally inside the

mode is now larger by a factor as noted above,

δ1=

out

in

EE

(5.44)

and since the spontaneous rate scaled as 2

E ),( yxn , the spontaneous rate is just increased

by exactly a factor of

ωδ geffcguidechannelsp

resonatorsp

nLcQ

WW

−−−

− == 1 (5.45)

97

Referring to our previous calculation of spontaneous emission rate into the channel

waveguide mode, we then have our result for the enhancement of spontaneous emission

inside the resonator, as compared to free space, as being

( )tEnhancemenResonatorWaveguideW

W

guidechannelspontba

resspontba

×= −−−

−−

or

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅=

−

−−− ω

ϕε

μω

geffcn

babageffresspontba nL

cQyxc

nW

2

0

2

ρ ),(ˆ *

(5.46)

and the ratio to the free space spontaneous emission rate is then

1

30

322

0

2

3ρ

−−

−

−−−

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛⋅⋅=

cnyx

cn

nLcQ

WW

baban

babageff

geffc

spacefreespontba

resspontba

επωμϕ

εμω

ω),(ˆ *

or

23

2 ρ12

3 ),(ˆ~ * yxnnL

Qnn

nF

WW

ngeffcgeff

spacefreespontba

resspontba

ϕλπ

⋅⎟⎠⎞

⎜⎝⎛== −

−

−−−

−−

(5.47)

which we recognize as essentially the Purcell factor F quoted earlier when we equate

[ ] 12 −

−≡ ),( yxnnA ngeffeff ϕ (5.48)

for an atom located inside the resonator at a position (x, y) to get

effgeff VnnQ

nF 1

26~ 3

2−

⋅⎟⎠⎞

⎜⎝⎛= λ

π (5.49)

where 2

ceffeff

LAV = . The additional factor of ½ arises if we consider the position of the

atom to be at the peak of a longitudinal standing wave inside the cavity rather than its

average value along the cavity, as would be clear if the cavity was only half a wavelength

98

long and we used the same method to define the cavity length as we have for the cavity

area, and the factor of geffn

n

−

can be seen as a correction for dispersive effects.

5.5 Analysis of Experimental Results and Discussion

The measured lifetime of 152 μsec for Erbium in the slot waveguide structures

suggests a strong non-radiative decay. However in spite of this we could still a see an

integrated collected luminescence power of 2.6 nW from these waveguides under

saturation conditions at room temperature. We now ask the question: How much power

do we expect to see coming out of these waveguides under the Purcell enhancement

condition?

Using the rigorous calculation of the emission rate into the forward direction of the

waveguide mode, including the Purcell factor, we have

active

eff

bulkradmode-waveguide-spont An

WΓ

⎟⎠⎞

⎜⎝⎛⋅⋅=

−

2

41

211 λ

πτ (5.50)

The total emitted power into the waveguide mode is

active

effact

bulkrademitted An

ALhfN

PΓ

⎟⎠⎞

⎜⎝⎛⋅⋅⋅⋅⋅

⋅=

−

22

41

21 λ

πν

τ (5.51)

where f is the fraction of erbium that is optically active.

To determine N2, the excited state erbium population, we make use of the fact that we

are pumping erbium to saturation. Using the experimentally determined value of nsp~1.6

at 1.48 μm which is known from erbium amplifiers we have

99

2

22

2

12

2

2

161

NNNNN

NNN

Nntottot

sp

−=

−−=

−≈=

)(.

totNN 72702 .≈⇒

(5.52)

The collected power would then be

collection

active

effactive

bulkrad

tot

expected

AnALhfN

P

ηλπ

ντ

⋅⋅Γ

⎟⎠⎞

⎜⎝⎛⋅⋅⋅⋅⋅⋅=

−

64041

217270 2

.. (5.53)

where we have included a factor of 0.64 to account for the integrated waveguide

transmission loss of 3 dB/cm and a ηcollection term which accounts for 10-13 dB of

waveguide to output lens tipped fiber coupling losses.

So the power which we can expect to collect under the Purcell enhancement condition

would be

[ ] fnW2613Pexpected ⋅−~ (5.54)

This figure seems quite reasonable and in comparing with the experimentally

measured value of ~2.6 nW suggests that only ~10-19% of the high density implanted Er

is optically active. This is highly likely as the erbium concentration in the slot

waveguides is quite high at 6X1020 cm-3. So our experimental results are consistent with

the calculated Purcell enhancement in the slot channel waveguides.

These results are quite interesting because if we remind ourselves of the reasons for

the desirability of lasers at the first place like their high modulation bandwidths, high

optical efficiency into the guided mode and for some applications their exteremely

narrow spectrum, we have seen that these conditions are met in slot waveguides even

without a resonator or threshold behavior. Perhaps the only quality missing is the

coherence. This implies that erbium doped slot waveguides can find application in very

100

efficient LED sources where all the emission is into a single preferred waveguide mode,

with improved efficiency resulting from the increased dominance of radiative decay over

any nonradiative decay channels, and in some cases increased modulation bandwidth

from the faster decay times.

101

Chapter 6. Future Work and Outlook

The purpose here is to set a definitive direction for future work which can be carried

out based on the theoretical and experimental findings of this thesis. A new

photolithography mask set was designed to further improve on the already verified

experimental results and to also put the new theoretical conclusions on sound

experimental footing. The new mask was designed with an aim to corroborate the “magic

radius” phenomenon in shallow ridge waveguides, seek optical gain in erbium doped

silica slot waveguides and to comprehensively verify Purcell enhancement in slot

waveguides so as to pave way for an electrically driven light source in Silicon. Here I

discuss in the design of the new mask.

We first start with a brief overview of the mask and then go into the details of its

various parts. The designs are aided by comprehensive simulation results which are

presented in detail. Finally the key findings of this thesis are summarized and outlook of

their role in silicon photonics is provided.

102

6.1 Mask Overview

The mask primarily consists of designs for (a) Passive ridge waveguides (b) Passive

horizontal Si---SiO2---Si slot waveguides (c) Critical coupling designs in ring resonators and

(d) Active horizontal Si---Er doped SiO2---Si slot waveguide devices

The mask was laid using Tanner EDA L-edit software using its programming feature

to automate the mask layout. Fig 6.1 shows the image of the mask with dimensions of 9

cm by 10 cm which is used to expose a 6’’ SOI wafer using a 1X projection photolitho-

Figure 6.1. Image showing the mask set consisting of 54 blocks arranged in a 9 by 6 grid. An

exemplary block is marked

9 cm

10 cm

103

Figure 6.2. Image of a typical individual mask block as marked in Fig 6.1

graphy system. The mask consists of 54 blocks which are arranged in a 9 by 6 grid. An

exemplary block is marked in Fig 6.1. Each block which in turn translates into a 1 cm by

1.6 cm SOI chip consists typically of 60 to 80 unique devices. Figure 6.2 shows the

image of a typical block.

6.2 Passive Ridge Waveguides

Passive ridge waveguides in the form of ring resonators were designed to verify the

“magic width” and “magic radius” phenomenon in shallow ridge waveguides, the

theoretical aspects of which were detailed in chapter 2 and 3. The base SOI to work with

1.6 cm

1.0

cm

104

now has a crystalline silicon thickness of 220 nm and a BOX thickness of 3 μm. Fig 6.3

shows the transverse TM mode count as a function of waveguide width W and ridge

height for the ridge waveguide shown in the inset with a fixed silicon core thickness of

220 nm. At a waveguide width around 1.45 μm where we expect the second order “magic

width” to be, we see that we require a ridge height of < 7.5 nm for single TM-like mode

operation. A single mode operation is preferable to minimize noise in the drop port

responses, resulting in a cleaner signal to infer the waveguide losses from. To ascertain

the sensitivity of the “magic width” to nm level ridge height variations, Fig. 6.4 shows

the loss of the fundamental TM-like mode for a straight waveguide as a function of

Figure 6.3. Transverse TM mode count for a 220 nm fixed Si core ridge waveguide shown in the

inset as a function of waveguide width and ridge height. TM mode count is indicated

10

20

30

40

50

60

70

80

0.5 1.0 1.5 2.0 2.5 3.0

Rid

ge h

eigh

t (nm

)

Waveguide Width W (µm)

Si SubstrateBOX

220 nmc-SiRidge W

1 2 3

< 7.5 nm

105

Figure 6.4. Fundamental TM mode loss for the shallow 220 nm fixed Si core ridge waveguide shown

in the inset of Fig 6.3 as a function of waveguide width for ridge heights at and in the vicinity of 7 nm

waveguide width for ridge heights at and in the vicinity of 7 nm. We see that the second

order magic width for this straight 220 nm Si core structure is 1.458 μm and shows little

variation with ridge height fluctuations.

Figure 6.5 shows the TM mode loss as a function of ring radius around 400 μm.

Traces are shown for two waveguide widths in the ring. One is for a ring waveguide

width of 1.458 μm, which is also the magic width for a straight waveguide and second

one is for 1.35 μm, which is an anti-magic width for the straight waveguide. We also

show these traces for three ridge heights of (a) 6 nm (b) 7 nm and (c) 8 nm. We see that

the TM loss is a strong function of both the ring radius and ring waveguide width. Low

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.3 1.35 1.4 1.45 1.5 1.55 1.6

6 nm7 nm9 nm

TM m

ode

Loss

(db/

cm)

Waveguide Width W (µm)

2nd order MW1.458 µm

Ridge Height

106

Figure 6.5. (Color) Fundamental TM mode loss for the shallow 220 nm fixed Si core ridge waveguide

shown in the inset of Fig 6.3 in ring resonator configuration as a function of ring radius. The width of

the waveguide in the ring (WR) is 1.458 μm and 1.35 μm. The ridge height is (a) 6 nm (b) 7 nm and

(c) 8 nm

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

399 399.143 399.286 399.429 399.572 399.715 399.858 400.001 400.144 400.287 400.43 400.573 400.716 400.859

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

399 399.143 399.286 399.429 399.572 399.715 399.858 400.001 400.144 400.287 400.43 400.573 400.716 400.859

1.E-02

1.E-01

1.E+00

1.E+01

399 399.143 399.286 399.429 399.572 399.715 399.858 400.001 400.144 400.287 400.43 400.573 400.716 400.859

7 nm

8 nm

Ridge Height 6 nm

Ring Radius (µm)

Loss

(dB

/cm

)Lo

ss (d

B/c

m)

WR 1.35 µm

WR 1.458 µm

Loss

(dB

/cm

)

(a)

(b)

(c)

107

losses can only be achieved at particular ring radii –“magic radii”, the positions of which

also depend on the width of the waveguide in the ring. These results also indicate that the

“magic radii” are sensitive to ridge height variations. Considering the case of 1.35 μm

width, at 6 nm the “magic radius” for the structure is 399.858 μm with loss <0.01 dB/cm,

however for a ridge height of 7 nm the TM mode will incur a loss of ~10 dB/cm for the

same radius.

With the goal of tracing out the curves shown in Fig. 6.5 experimentally thereby

verifying the “magic radius” phenomenon, weakly coupled ring resonators were laid out

as shown in Fig 6.2. The ring radius was varied from 399 μm to 401 μm in 0.143 μm

steps. The width of the through and drop port bus waveguides was always fixed at 1.458

μm while separate blocks were laid out for 1.458 μm and 1.35 μm ring waveguide width

case respectively. Each block also consists of different coupling gaps to vary the coupling

strengths from weak (0.06%) to strong (5.29%). This would allow us to remove the

contribution of the coupling loss to the overall loss as ascertained by measuring the drop

port responses. In addition to this, separate blocks were also laid out to account for over

and under exposure during the photolithography process as over or under exposure results

in a change in the waveguide widths from their nominal values.

6.3 Passive Horizontal Si-SiO2-Si Slot Waveguides

Ring resonator structures were laid out to determine the fundamental TM mode loss

in passive horizontal Si – SiO2 – Si slot waveguides. The intrinsic resonator quality factor

determined from these passive structures is a key input to the active slot structure design

108

as discussed later in this chapter. The passive structure was also designed with the

intention of using it as a test vehicle for the active devices. Again, starting with a base

SOI with 220 nm of crystalline silicon thickness and 3 μm BOX, the two design

parameters are thickness of various layers i.e. the c-Si layer, the SiO2 slot layer and the

amorphous silicon (a-Si) layer and the ridge height. The thicknesses of different layers

should be selected to achieve preferably single TM mode operation. The field and

weighting function confinement should be maximized in the ultra thin (5 to 15 nm) slot

layer to maximize gain; on the other hand the same should be reduced in the amorphous

silicon layer to minimize the absorption losses. From the electrical design point of view,

it is preferable to have thicker c-Si and a-Si layers. So the a-Si thickness has to be

selected to balance the electrical conductivity and absorption losses. It was also agreed

upon to avoid the long blanket thermal oxidation step utilized for thinning out the c-Si

layer thickness before the slot oxidation as it was noticed to be causing pin holes in the c-

Si layer. The designed structure should also minimize substrate leakage losses. A shallow

ridge height should be selected to minimize the side wall scattering losses and also to aid

conductive lateral electrical access to the waveguide core. The ridge height will also be a

criterion in setting the minimum ring radius that can be utilized along with the waveguide

width. It must also be pointed out that the above characteristics of the structure are

desired at both the signal wavelength (~1530 nm) and the pump wavelength (~1480 nm).

In circumstances where it is difficult or cumbersome to satisfy the conditions for both the

wavelengths, the signal wavelength design takes precedence over the pump as the signal

emission from Er is anticipated to be much weaker compared to the excitation pump.

109

Figure 6.6 (a) and (b) show the weighting function confinement in the a-Si and 10 nm

SiO2 slot layer for a 5 layer (Air – a-Si – SiO2 – c-Si – BOX) slab structure as a function

of c-Si thickness and a-Si thickness at a wavelength of 1550 nm. The darker blue region

in the top right corner of the plots corresponds to the design space where there are >1 TM

guided modes. Based on various criterion mentioned above, it is preferable to work in the

top right corner of this c-Si and a-Si layer thickness design space. However due to

practical limitations on the quality control of the a-Si at thicknesses >100 nm, an

amorphous silicon thickness of 100 nm was selected for the design. A 100 nm of a-Si and

216 nm c-Si (thickness of c-Si after 10 nm slot oxidation and avoiding any blanket

oxidation) provides 17% weighting function confinement in the 10 nm slot and 15%

confinement in the a-Si layer. The corresponding values for the 100 nm a-Si –10 nm SiO2

– 150 nm c-Si structure which closely mimics the old design are 17% and 19 %

respectively. Therefore the new design will minimize the confinement in the amorphous

silicon layer with no penalty on the confinement in the SiO2 slot. Calculations were also

performed for a 15 nm SiO2 slot structure which yielded weighting function confinement

values of 22% in SiO2 slot and 13% in a-Si for a 100 nm a-Si – 15 nm SiO2 – 213 nm c-Si

structure. Similar calculations were also performed at a wavelength of 1480 nm. The

confinement values at 1480 nm are very close to that at 1550 nm wavelength.

With the layer thicknesses fixed, a ridge height of 10 nm was chosen. Figure 6.7 (a)

and (b) show the analytical bending loss for the 3D slot waveguide structure for a ring

radius of 400 μm and 200 μm respectively. The bending loss is shown as a function of

thickness of c-Si in the core and the amorphous silicon which is uniform in both the core

as well as the cladding. Therefore, with the 10 nm ridge height, the thickness of the c-Si

110

Figure 6.6. Weighting function confinement % at 1550 nm in (a) a-Si and (b) 10 nm SiO2 slot as a

function of c-Si and a-Si thickness

WF Confinement in a Si

cSi 216nmaSi 100nm

15%

cSi 150nmaSi 100nm

19%

WF Confinement in slot

cSi 216nmaSi 100nm

17%

cSi 150nmaSi 100nm

17%

(a)

(b)

111

Figure 6.7. Analytical Bending loss at 1480 nm for a 10 nm ridge slot waveguide structure with a

radius of (a) 400 μm and (b) 200 μm as a function of the c-Si thickness in the core and amorphous

silicon thickness with the waveguide width equaling the “magic width” at 1550 nm

Log10 of Bending Loss in dB/cm

cSi 216nmaSi 100nm

1.04

cSi 150nmaSi 100nm

-1.51

Log10 of Bending Loss in dB/cm

cSi 216nmaSi 100nm

-2.33

cSi 150nmaSi 100nm

-7.53

Radius 400 µm

Radius 200 µm

(a)

(b)

112

layer in the cladding is 10 nm less than that in the core. The figures show the bending

loss at 1480 nm wavelength with the waveguide width at the corresponding 1550 nm

wavelength “magic width” of the structure as obtained using the magic width equation.

For the selected layer thicknesses of 100 nm a-Si – 10 nm SiO2 – 216 nm c-Si and a 10

nm ridge, the bending loss at 400 μm is 4.67X10-3 dB/cm and for 200 μm is 10.96 dB/cm.

The corresponding values at 1550 nm are slightly less severe at 3.09X10-3 dB/cm and

8.71 dB/cm respectively and the values for the 15 nm slot layer structure (100 nm a-Si –

15 nm SiO2 – 213 nm c-Si and a 10 nm ridge) are slightly better than the 10 nm slot case.

We also notice that the old slot structure has much lower bending loss values compared

to the new design as indicated in the figure. For the 10 and 15 nm slot structures with 10

nm ridge height, for the bending loss to be <1X10-3 dB/cm requires that the ring radius be

>410 μm. With this in mind, radii of 400 μm and 450 μm were selected for the slot

waveguide structures.

Figure 6.8 (b) shows the fundamental TM mode loss of the 15 nm slot, 10 nm ridge

waveguide (Fig 6.8 (a)) as determined using COMSOL FEM. The actual slanted profile

of the waveguide side walls as determined using AFM was utilized in the simulation. The

loss is plotted as a function of the width of the c-Si ridge waveguide, as this can be

directly related to the width on the mask. The loss is shown at both the signal (1530 nm)

and the pump (1480 nm) wavelength. Material dispersion was taken into account in doing

these calculations. As we can see the TM mode loss minima for the two wavelengths are

at slightly different waveguide widths. Choosing to design at the “magic width” of 1.605

μm at the signal wavelength, the loss at 1480 nm at this waveguide width is ~0.1 dB/cm.

113

Figure 6.8. (a) Schamatic of a 10 nm ridge, 100 nm a-Si – 15 nm SiO2 – 213.25 nm c-Si slot waveguide

structure (b) Fundamental TM mode loss of this straight waveguide at 1530 nm and 1480 nm as a

function of the width W of the c-Si ridge

Figure 6.9 (a) and (b) show the TM mode loss for the 15 nm slot, 10 nm ridge

structure shown in Fig 6.8 (a) as a function of ring radius around 400 μm and 450 μm

respectively at a wavelength of 1530 nm. The width of the c-Si ridge waveguide in the

ring is 1.605 μm (“magic width” for the straight waveguide at 1530 nm). Traces are

100 nm a-Si

Si Substrate

BOX 203.25 nm c-Si

15 nm SiO2

10 nm Ridge

W

Waveguide Width W, C-Si Ridge (µm)

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

1.E+02

1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8

1.48 mic 1.53 mic

Loss

(dB

/cm

)

1.605 µm @1530 nm1.585 µm @1480 nm

15 nm Slot10 nm Ridge

(a)

(b)

114

shown for 9 nm, 10 nm and 11 nm ridge heights. Similar to the passive ridge waveguides,

we see that the positions of the “magic radii” are sensitive to the ridge height variations.

To verify the positions of the “magic radii”, weakly coupled ring resonators were laid out

using the same layout principle as detailed in Sec. 6.2 for passive ridge waveguides.

Structures were also designed for the 10 nm slot case.

Figure 6.9. (Color) Fundamental TM mode loss for the 15 nm slot, waveguide shown in Fig. 6.8 (a) in

ring resonator configuration as a function of ring radius around (a) 400 μm and (b) 450 μm. The

width of the c-Si ridge waveguide in the ring is 1.605 μm. Traces shown for ridge heights of 9 nm, 10

nm and 11 nm

(a)

(b) 1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

451 451.273 451.546 451.819 452.092 452.365 452.638 452.911 453.184 453.457 453.73

9 nm 10 nm 11 nm

Ring Radius (µm)

Loss

(dB

/cm

)

Ridge Height

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

400 400.231 400.462 400.693 400.924 401.155 401.386 401.617 401.848 402.079 402.31 402.541 402.772

9 nm 10 nm 11 nm

Loss

(dB

/cm

)

Ring Radius (µm)

Ridge Height

115

6.4 Ring Resonators for Critical Coupling in Horizontal Si-SiO2-Si Slot

Waveguides

A ring resonator is a travelling-wave-resonator (TWR). Therefore by appropriate

design of the waveguide-resonator coupling, a complete 100% transmission of energy

from a bus waveguide to the resonator is possible[29]. From the active device

perspective, critical coupling at the pump wavelength can be beneficial for the judicious

utilization of the pump power for pumping erbium in the slot in a ring resonator

configuration. Engineering of the waveguide-resonator coupling structure to achieve

critical coupling can be quite challenging. Here we discuss our attempts to achieve

critical coupling in the shallow ridge slot waveguides.

Fig 6.10 (a) shows the TWR ring resonator coupled to a bus waveguide. In such

TWR-waveguide structures, in the weak coupling regime, the power transmission and the

total loaded quality factor (QL) at resonance are expressed as

2

0 11

CO

CO

in

out

QQQQ

PP

T//

)(+−

==ω (6.1)

111 −−− += COL QQQ (6.2)

where QO and QC are the intrinsic and coupling quality factors respectively. The coupling

quality factor is used as a measure of the rate of energy decay from the cavity to the

waveguide. Hence, stronger the waveguide-resonator coupling, smaller is the QC. As can

be inferred from Eq. 6.1, in order to completely transfer energy from the waveguide to

the resonator at resonance, requires QO to be equal to QC and as a result the loaded

quality factor QL becomes half of QO. This condition is referred to as the critical

116

coupling. Therefore from Eq. 6.1 and 6.2, by changing QC, both QL and the power

transmission through the bus waveguide can be controlled as shown in Fig. 6.10 (b).

Figure 6.11 shows the coupling quality factor for the 15 nm slot, 10 nm ridge

waveguide structure as a function of the minimum gap between the straight bus

waveguide and the ring as obtained using RSOFT 2D simulations. The bus waveguide is

Figure 6.10. (a) Schematic of TWR ring resonator coupled to a bus waveguide (b) Power

transmission through the bus waveguide as a function of the normalized loaded quality factor (QL)

Pin Pout

ωOQO

-50

-40

-30

-20

-10

0

0.1 0.3 0.5 0.7 0.9

QC <QO QC >QO

Over Coupled

UnderCoupled

Pow

er T

rans

mis

sion

(dB

)

QL/ QO

(a)

(b)

117

Figure 6.11. Coupling Quality factor for the 15 nm slot, 10 nm ridge structure as a function of the

minimum gap between the tangential bus waveguide and the ring as shown in Fig 6.12 (a)

Figure 6.12. Section of a bus waveguide-resonator system showing the coupler configuration (a)

Regular coupler - Bus waveguide running tangentially to the ring with gap of G μm at minimum

separation between the two (b) Curved coupler - Bus waveguide wrapped around the ring to increase

the interaction length and hence to reduce the QC at a gap of G μm. The curved interaction length is

parameterized by the arc angle θ

Regular Coupler

Ring

BusCurved Coupler

Ring

Bus

θ

(a) (b)

1.E+05

1.E+06

1.E+07

1.E+08

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

RR400,Lam1.53RR400,Lam1.48RR450,Lam1.53RR450,Lam1.48

Cou

plin

g Q

, QC

Gap, G (µm)

4.E+05

118

running tangentially to the ring as shown in Fig 6.12 (a). The QC is shown for ring radii

of 400 μm and 450 μm at both the pump (1480 nm) and signal (1530 nm) wavelengths.

The width of the waveguide in both the ring and the straight sections is at the “magic

width” at 1530 nm. Assuming an intrinsic quality factor of 4X105, we see that the

coupling gap needs to be ~0.8 μm to achieve the critical coupling condition i.e. QO= QC.

But, our photolithography system limits the minimum achievable gap for this 15 nm slot

structure to 1.152 μm at which QC > QO and hence the ring will be under coupled at this

gap. So in order to satisfy the critical coupling condition at the larger gap of 1.152 μm

requires increased coupling strength at this gap. This can be achieved by increasing the

interaction length between the bus waveguide and the ring using the configuration shown

in Fig 6.12(b). Figure 6.13 now shows the QC for the same structure evaluated in Fig 6.11

Figure 6.13. (Color) Coupling Quality factor for the 15 nm slot, 10 nm ridge waveguide-resonator

system with a curved coupler as a shown in Fig 6.12 (b). Arc angle θ is defined in Fig. 6.12(b). Gap is

fixed at 1.152 μm

1.00E+05

1.00E+06

0 1 2 3 4 5 6 7

RR400,Lam1.53RR400,Lam1.48RR450,Lam1.53RR450,Lam1.48

Arc Angle (º)

4.E+05

Cou

plin

g Q

, QC

119

as a function of the arc angle θ as defined in Fig 6.12 (b). The gap is now fixed at 1.152

μm. We notice that for arc angles > 4.2° and > 5.8°, a QC of < 4X105 can be achieved at

1530 nm and 1480 nm respectively for both 400 and 450 μm radius rings for this 15 nm

slot structure.

Similar calculations were also performed for the 10 nm slot, 10 nm ridge structure.

Structures to verify the critical coupling were laid out consisting of ring resonators with

curved coupler configurations with the arc angle θ increased incrementally from 1° to 7°.

6.5 Active Horizontal Si-Er doped SiO2-Si Slot Waveguides

Designs for erbium doped silica slot waveguides were incorporated with the aim of

optically pumping these structures at 1480 nm to achieve optical gain in a ring resonator

configuration. The key design parameter here is the intrinsic quality factor Q of the

resonator which is obtained by doing loss measurements on a passive horizontal slot ring

resonator. Assuming we can achieve an intrinsic resonator Q of 4X105, we summarize

here results predicting the behavior of our erbium doped silica slot waveguide ring laser.

The analysis is based on coupled mode formalism for a resonator coupled to an

external waveguide[47]. As an exemplary case, we consider 100 nm a-Si – 15 nm Er

doped SiO2 – 213 nm c-Si, 10 nm ridge horizontal slot structure configured as a 450 μm

ring resonator coupled to a straight bus waveguide as shown in Fig 6.14. A pump at 1480

nm is coupled to the ring through the bus waveguide to excite the erbium in the slot so as

to collect erbium signal emission at the output of the bus waveguide. Figure 6.15 shows

120

Figure 6.14. (a) Exemplary erbium doped slot waveguide ring resonator coupled to a bus waveguide

to aid pumping of erbium in the slot (b) Cross section of the slot waveguide constituting the ring

Figure 6.15. (Color) Minimum erbium concentration required in the slot for the structure shown in

Fig. 6.14 to reach threshold as a function of the coupling gap G for different intrinsic resonator Qs

the minimum erbium concentration required in the slot to reach threshold at a particular

coupling gap G for different intrinsic resonator Qs. The minimum erbium concentration

required to reach threshold increases as the gap decreases for a given intrinsic Q.

(a) (b)

100 nm a-Si

Si Substrate

BOX 203.25 nm c-Si

15 nm Er/SiO2

10 nm Ridge

1.605 µm

PPin PPout

Q

450 µm

SPout

λ1480 nm

Gap, G (µm)

Erbi

um C

once

ntra

tion

(m-3

)

7.6763e26 5.6642e26

121

Assuming an intrinsic Q of 4X105, which is in the range of best results we have achieved

so far for the TM-like mode in horizontal slot waveguides, we see that we require the

minimum erbium concentration in the slot to be 7.7X1020 cm-3 to reach threshold for all

gaps >1.1 μm. As the achievable resonator Q improves, threshold can be reached for

lower erbium concentrations in the slot. Figure 6.16 shows the pump threshold power as a

function of the coupling gap with the erbium concentration fixed at 7.7X1020 cm-3 and

resonator Q of 4X105. Threshold powers < 10 mw can be achieved for gaps in the range

of 1.2 μm to 1.9 μm. It must however be pointed out that in these calculations we have

neglected co-operative upconversion, excited state absorption and pair induced quenching

effects which can be prevalent at such high erbium concentrations. Fig. 6.17 shows the

expected lasing power at 1530 nm from this 450 μm ring resonator with a Q of 4X105 and

7.7X1020 cm-3 slot erbium concentration as a function of coupling gap and pump power.

Figure 6.16. Threshold pump power as a function of coupling gap for the ring resonator

configuration shown in Fig. 6.14 with Q = 4X105 and slot erbium concentration of 7.7X1020 cm-3

Gap, G (µm)

Thre

shol

d Po

wer

(mW

) >

122

Figure 6.17. Output lasing power at 1530 nm as a function of pump power and coupling gap for the

configuration shown in Fig. 6.14 with Q = 4X105 and slot erbium concentration of 7.7X1020 cm-3

The white portion of the plot corresponds to the region where the structure doesn’t reach

threshold for the given parameters. We see that we can expect lasing power of ~ 90 μW

at a gap of 1.3 μm when pumped at 40 mW.

Ring resonator structures at “magic radii” and “magic width” with single coupler and

coupling gaps in the range of 1.1 to 1.7 μm were laid out in a similar scheme as shown in

Fig 6.2. Calculations were also done for the 10 nm slot case where minimum erbium

concentration to reach threshold is higher due smaller pump/signal confinement in the

thinner slot compared to the 15 nm case.

Pum

p Po

wer

(m

W)

Gap, G (µm)

µW

123

6.6 Summary and Future Work

A horizontal slot waveguide operating in TM mode is arguably the most practical

architecture for achieving an electrically pumped silicon based light source. The design

allows for fabrication using standard CMOS planar processing and provides the

necessary means for electrical excitation of the slot embedded gain media. This compared

to other architectures like micro-toroids or disks on pedestals which require elaborate

fabrication and testing mechanisms and provide absolutely no means for electrical

excitation of the gain media. Usually such complex fabrication techniques are resorted to

for essentially one simple reason – to reduce the loss i.e. to achieve ultra high Q

resonators. However doing so makes them unsuitable for wide deployment.

Minimizing the losses in waveguide architectures suitable for light sources in silicon

photonics i.e. a horizontal slot waveguide, has been the overarching theme of this thesis.

This has required overcoming some stringent conceptual barriers, devising novel

fabrication techniques and improvements in material quality.

Before our work, it was usually argued that the TM like mode in shallow ridge

waveguides is always leaky and hence high loss. However, TM mode operation is

fundamentally critical to our application due to its high confinement in the slot. A

comprehensive analysis of this problem was presented and we showed that this TM mode

leakage loss can be mitigated by simple width control of the waveguide to “magic

widths”.

Achieving smooth waveguide surfaces is absolutely critical for achieving low loss in

a high index contrast system like SOI. However the challenge here is to do so using

techniques which still fit into the frame work of standard CMOS planar processing.

124

Novel fabrication techniques based on thermal oxidation were devised which yielded

ultra-smooth waveguide surfaces. Shallow ridge waveguides fabricated using this method

were demonstrated with a Q of 1.6X106 for the TE-like mode and when combined with

the “magic width” designs yielded a Q of 6.8X105 for the TM-like mode, the highest yet

reported for such shallow ridge structures with tight vertical confinement.

Comprehensive full vector numerical models were also developed for rigorous

analysis and quick design of these structures. These techniques, namely mode matching

(MM) and finite element method (FEM) were then utilized to analyze the lateral leakage

phenomenon in structures which usually don’t lend themselves to simple

phenomenological methods such as ring resonators and directional couplers, which are

key components in integrated photonics. In analyzing these, new anomalous losses, over

and above any reported values came forth. To counteract these losses, new design

principles like the “magic radii” were conceived and elucidated to achieve highest

performance in such devices.

These novel design concepts and fabrication techniques were then applied to the

fabrication of horizontal slot waveguides. As the design required utilization of amorphous

silicon as the top cladding layer which can be high loss, considerable improvement in the

optical quality of the PECVD deposited amorphous silicon had to be achieved. With ~3

dB/cm loss for our amorphous silicon, puts it in the league of the best results reported by

other groups. This combined with a carefully engineered design at “magic widths”

resulted in a Q of 3X105 for the TM-like mode in a ultra thin 8 nm horizontal slot

waveguide.

125

Erbium was then implanted into a silica slot just 8 nm thick. A relatively higher

dosage of erbium at 6X1020 cm-3 was utilized as a safety net against the very low

implantation energies of just 2 keV. Nevertheless, erbium photoluminescence was

observed in full waveguide configuration primarily made possible due to the low TM-like

mode losses of these waveguides. A noise analysis based test setup was designed to

extract the temporal characteristics from the very weak erbium luminescence signal

which indicated some unusually fast erbium lifetime which didn’t seem to fit the

conventional wisdom. Other groups working on similar structures also reported seeing

similar results around the same time and attributed these fast lifetimes in slot waveguides

to enhanced spontaneous emission rate or Purcell enhancement, a phenomenon usually

associated with high Q resonators and explained it using a quantum electrodynamic

formalism. We however showed that the conventional simple expressions for

spontaneous emission into a waveguide mode correctly capture the Purcell enhancement

in slot waveguides and showed our experimental results in slot waveguides to be

consistent with this phenomenon. A full numerical quantum electrodynamic analysis of

these slot structures was also carried out to further buttress the simple “fraction of

spontaneous emission into waveguide mode” model. This enhanced spontaneous

emission rate characteristic of the slot waveguides make them even more interesting as it

means that laser like characteristics – preferential emission into single waveguide mode,

high modulation bandwidth can now be achieved in slot waveguides even with out a

resonator and threshold behavior. This means slot waveguides can be the basis for very

efficient LEDs, without the very stringent constraint of achieving gain.

126

The novel waveguide architectures detailed in this thesis can play a crucial role for

photonic integration in silicon. The ultra low loss shallow ridge waveguides are ideal

candidates for high performance sensors, tunable filters and ultra small foot print

polarizers. Comprehensive simulation techniques which predict their behavior with great

accuracy have been put in place to allow their comprehensive analysis and seamless

integration.

In line with the findings of this thesis, the future work will be geared towards putting

not yet experimentally proven theoretical aspects like the “magic radius” phenomenon on

sound experimental footing. It is also anticipated that by using a much lower erbium

concentration in the slot compared to the present case of 6X1020 cm-3 would allow a more

robust verification of Purcell enhancement in slot waveguides and evaluating its

consequences on light sources in silicon. To study these and to further improve on the

already established results, an improved mask was designed and laid out.

127

Appendix A

Spontaneous Emission Rate in a Slot Waveguide

The formulation presented here closely follows the one given in [54] where the

authors apply quantum electrodynamics to calculate the zero-point fluctuations of the

electromagnetic field and the spontaneous emission rate from a non-absorbing dielectric

film bounded by two non-absorbing dielectric half spaces of arbitrary refractive indices.

Here we extend their methodology to the five layer slot structure i.e. Si-SiO2-Si slabs

bounded by air and SiO2 dielectric half spaces. The entire derivation is given here for

completeness.

The quantization of the electromagnetic field requires that the complete set of spatial

electromagnetic modes that exist in the configuration be determined. These modes must

be orthonormalized with respect to the scalar product corresponding to the

electromagnetic energy density. For the dielectric film the set of modes is well known. It

consists of radiation modes pertaining to plane waves that are incident from either

dielectric half space for both orthogonal polarizations. In addition, guided modes of either

polarization may exist. These modes are evanescent in both dielectric half spaces.

128

We will list the radiation and guided modes in Sec. A.2 and A.3 respectively. The

electromagnetic field will be quantized in Sec. A.4. In Sec. A.5 formulas for the

spontaneous emission rate and the zero-point field fluctuations will be derived. It should

be mentioned that the local-field effects on a molecular scale may be very important in

determining the spontaneous emission rate [61]. However as we are only interested in the

emission rate of the film relative to the bulk value, the spatial electromagnetic modes do

not have to be corrected for local field effects.

A.1 Notations

Considering an electromagnetic plane wave in a medium that is translation invariant

in the x and y directions of a cartesian co-ordinate system (x, y, z). Let the medium have a

refractive index n. The time dependence of time harmonic fields will be given by the

factor exp(-iωt), where ω>0 is the frequency. Let k =(kx , ky , kz) be the wave vector of the

plane wave so that

22222 nkkkk zyx =++ (A.1)

where k= ω(ε0 μ0)1/2 is the wave number in vacuum. The components kx and ky are

assumed real but kz = (k2 n2 - kx2

– ky2)1/2 is purely imaginary when kx

2 + ky

2 > k2 n2. We

choose the branch of the complex square root such that the cut is along the negative real

axis so that when a is positive, a½ is positive and (-a) ½ = + i a½. Hence kz is positive

imaginary when kx2

+ ky2 > k2 n2 and the wave is then evanescent in the z direction. The

plane wave can be written in a unique way as a linear combination of two orthogonally

polarized plane waves, namely TE or S polarized wave whose electric field vector is

129

parallel to the (x, y) plane and a TM or P polarized plane wave for which the magnetic

field vector is parallel to the (x, y) plane. We introduce the following two unit vectors

corresponding to the electric field vector of the two polarizations:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛−

+=

0

1

21

22x

y

yx

kk

kkSk

)(),(ι (A.2)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

+−+++=

)()()(),(ˆ

2221

2221

222

1

yx

zy

zx

yxzyx kkkkkk

kkkkkPkι (A.3)

Then the S- polarized plane wave is given by

),(ˆ)( Sk)rkexp(iArE ι⋅= (A.4)

),(ˆ)()(

)( Pkrkiexpk

kkkArH yxz ι

με

⋅++

⎟⎟⎠

⎞⎜⎜⎝

⎛=

21

22221

0

0 (A.5)

where A is the amplitude of the electric field strength. Similarly for the TM or P-

polarized waves we have

),(ˆ)( Pk)rkexp(iArE ι⋅= (A.6)

),(ˆ)()(

)( Skrkiexpkkk

nkArH

yxz

ιμε

⋅++

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

21

222

221

0

0 (A.7)

Figure A.1 shows the configuration we will study here. The Cartesian coordinate system

(x, y, z) is chosen with its origin in the middle of the layer with index n1 which in our case

will be the slot layer. The z axis is perpendicular to the interfaces. For given real kx and ky

we will use the notation:

( ) 54321212222 ,,,,

/=−−≡ jforkknkk yxjjz (A.8)

130

Figure A.1. A five layer dielectric slab structure representative of the slot waveguide

where the branch of the square root was described above. In addition +jk and −

jk are the

wave vectors given by

),,(,),,( jzyxjjzyxj kkkkkkkk −== −+ (A.9)

As the time dependence of the fields is assumed to be exp(-iωt), it implies that when kjz is

real, +jk and −

jk are the wave vectors of plane waves that propagate in the positive and

the negative z directions respectively. When kjz is imaginary, these plane waves decay

exponentially in the positive and the negative z directions respectively.

n5

n3

n1

n2

n4

z=d/2xy

z

z=-d/2

z=-(d/2)-b

z=(d/2)+aAir

Si

SiO2

Si

SiO2

131

Finally, we recall that for a plane wave impinging on the interface between two

dielectrics with refractive indices n1 and n2 incident from dielectric 1 as shown in the Fig.

A.2, the reflection and transmission coefficients are given by

onpolarizatiSforkk

krt

kkkkr

zz

z

zz

zz

21

11212

21

2112

21+

=+=

+−=

(A.10)

and

.onpolarizatiPfor

nk

nk

nk

rt

nk

nk

nk

nk

r

zz

z

zz

zz

22

22

1

1

21

1

1212

22

22

1

1

22

22

1

1

12

21

+=+=

+

−=

(A.11)

Figure A.2. Illustration of the definition of the reflection and transmission coefficients r12 and t12

n1

n2

z

t12

r12

132

A.2 Radiation Modes

We first start by describing the radiation modes that can exist in the configuration of

Fig. A.1. We distinguish between modes that are incident from medium 4 and modes that

are incident from medium 5 and between S and P polarizations.

Mode incident from medium 4

For every frequency ω and every kx and ky, satisfying

24

222 nkkk yx <+ (A.12)

where k= ω(ε0 μ0)1/2, there are S and P polarized electromagnetic fields in the structure

consisting of plane waves and standing waves of which kx and ky are the components of

the wave vectors in the x and y directions. The electric field of the S polarized mode is

given by

[ ]

[ ]

[ ][ ]

[ ]⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛ +−≤⋅+⋅

⎟⎠⎞

⎜⎝⎛−≤≤⎟

⎠⎞

⎜⎝⎛ +−⋅+⋅

⎟⎠⎞

⎜⎝⎛≤≤⎟

⎠⎞

⎜⎝⎛−⋅+⋅

⎟⎠⎞

⎜⎝⎛ +≤≤⎟

⎠⎞

⎜⎝⎛⋅+⋅

⎟⎠⎞

⎜⎝⎛ +≥⋅

=

−−++

−−++

−−++

−−++

++

bdzforSkrkiexpBSkrkiexpA

dzbdforSkrkiexpDSkrkiexpCA

dzdforSkrkiexpFSkrkiexpEA

adzdforSkrkiexpHSkrkiexpGA

adzforSkrkiexpIA

rE

SS

SSS

SSS

SSS

SS

2

22

22

22

2

444444

2242244

1141144

3343344

5544

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(

)(

ιι

ιι

ιι

ιι

ι

(A.13)

where

133

[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] [ ]

122

2222

222222

222222

22

4212

42113

424

41241312

432135

432351312

431352412

43352413

4 Functionkbdbkiexpr

kbdbkdkiexprkbdiexpr

kbddkiexprrrkbdakbkdkiexpr

kbdakbkiexprrrkbdakdkiexprrr

kbdakiexprrr

B zz

zzz

z

zz

zzzz

zzz

zzz

zz

S )()(

)()(

)()()(

)(

+−−+−++

+−−+−+

+−++++−+−+−++

+−−

=

(A.14)

122

222

22

22

22

2

4242

421131242

4321351242

432351342

4 Function

kbdkbdiexpt

kbdkbddkiexprrt

kbdakkbddkiexprrt

kbdakkbdiexprrt

Czz

zzz

zzzz

zzz

S⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ ++

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ ++−

⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛ ++−

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛ +

=

(A.15)

122

222

22

22

22

2

421242

4211342

43213542

43235131242

4 Function

kbdkbdiexprt

kbdkbddkiexprt

kbdakkbddkiexprt

kbdakkbdiexprrrt

Dzz

zzz

zzzz

zzz

S⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ −−−

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ −−+

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛ −−+

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛ −−−

=

(A.16)

12

22

22

432135134221

4214221

4 Function

kbdakbkkdiexprrtt

kbdbkkdiexptt

Ezzzz

zzz

S⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−++⎟

⎠⎞

⎜⎝⎛+

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛

=

(A.17)

134

12

22

3

223

4321354221

421134221

4 Function

kbdakbkkdiexprtt

kbdbkkdiexprtt

Fzzzz

zzz

S⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−++⎟

⎠⎞

⎜⎝⎛+

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛

=

(A.18)

122 4321422113

4 Function

kbdkdbkdkiexptttG

zzzzS

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛−+

= (A.19)

12

22 432135132142

4 Function

kbdkadbkdkiexprtttH

zzzzS

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ +++

=

(A.20)

122 432135132142

4 Function

kadkbdakbkdkiexpttttI

zzzzS

⎢⎣

⎡

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ +−++

=

(A.21)

and

[ ] [ ] [ ] [ ] [ ] [ ] [ ]z

zzz

zzzz

zzzz

bkiexprrakdkiexprrakiexprr

akbkdkiexprrdkiexprrbkdkiexprrakbkiexprrrr

Function

22412

31351233513

321352411312

2124133235241312

22222

2211

++−+

++−−+−++

=

(A.22)

with the reflection and transmission coefficients defined by Eq. A.10. When the factor

SA4 is chosen such that

( )21

4234 2

1⎟⎟⎠

⎞⎜⎜⎝

⎛=

z

PorS

kkA /π

(A.23)

it follows that for every k~ , xk~ and yk~ satisfying Eq. A.12, with E~

being the electric field

of the corresponding S polarized mode, we have

135

∫∫∫ −−−=⋅ )~()~()~(~

* kkkkkkrdEE yyxx δδδεε

0

(A.24)

where ε denotes the piecewise constant function, which is equal to εj in medium j and

where δ is the Dirac’s delta function. All radiation modes are parameterized by the triplet

kx , ky , k.

The electric field of the P-polarized mode corresponding to kx, ky, k and satisfying Eq.

A.12 is given by

[ ]

[ ]

[ ]

[ ]

[ ]⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛ +−≤⋅+⋅

⎟⎠⎞

⎜⎝⎛−≤≤⎟

⎠⎞

⎜⎝⎛ +−⋅+⋅

⎟⎠⎞

⎜⎝⎛≤≤⎟

⎠⎞

⎜⎝⎛−⋅+⋅

⎟⎠⎞

⎜⎝⎛ +≤≤⎟

⎠⎞

⎜⎝⎛⋅+⋅

⎟⎠⎞

⎜⎝⎛ +≥⋅

=

−−++

−−++

−−++

−−++

++

bdzforPkrkiexpBPkrkiexpA

dzbdforPkrkiexpDPkrkiexpCnnA

dzdforPkrkiexpFPkrkiexpEnnA

adzdforPkrkiexpHPkrkiexpGnnA

adzforPkrkiexpInnA

rE

PP

PPP

PPP

PPP

PP

2

22

22

22

2

444444

2242242

44

1141141

44

3343343

44

5545

44

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(

)(

ιι

ιι

ιι

ιι

ι

(A.25)

where PPPPPPPP IandHGFEDCB 44444444 ,,,,,, are given by Eqs.A.14 through A.22 but now

the reflection and transmission coefficients are given by Eq. A.11. The choice of PA4 as

given by Eq. A.23 again leads to the satisfaction of the orthonormality relation given by

Eq. A.24 for the electric fields of the two P-polarized modes.

Mode incident from medium 5

For all values of k, kx and ky satisfying

25

222 nkkk yx <+ (A.26)

there are S and P polarized electromagnetic fields that are incident from medium 5 and

consist of plane waves and standing waves of which kx and ky are the components of the

136

wave vectors in the x and y directions. The electric field of the S polarized mode is given

by

[ ]

[ ]

[ ][ ]

[ ]⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛ +−≤⋅

⎟⎠⎞

⎜⎝⎛−≤≤⎟

⎠⎞

⎜⎝⎛ +−⋅+⋅

⎟⎠⎞

⎜⎝⎛≤≤⎟

⎠⎞

⎜⎝⎛−⋅+⋅

⎟⎠⎞

⎜⎝⎛ +≤≤⎟

⎠⎞

⎜⎝⎛⋅+⋅

⎟⎠⎞

⎜⎝⎛ +≥⋅+⋅

=

−−

++−−

++−−

++−−

++−−

bdzforSkrkiexpIA

dzbdforSkrkiexpHSkrkiexpGA

dzdforSkrkiexpFSkrkiexpEA

adzdforSkrkiexpDSkrkiexpCA

adzforSkrkiexpBSkrkiexpA

rE

SS

SSS

SSS

SSS

SS

2

22

22

22

2

4455

2252255

1151155

3353355

555555

),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

)(

ι

ιι

ιι

ιι

ιι

(A.27)

where

[ ] [ ] [ ]

[ ] [ ]

[ ] [ ] [ ]

1222

222222

2222

2222

222

53112

5313

532124

532241312

535

51351312

52352412

521352413

5 Functionkadakdkiexpr

kadakiexprkadakbkdkiexpr

kadakbkiexprrrkadiexpr

kaddkiexprrrkadbkiexprrr

kadbkdkiexprrr

B zzz

zz

zzzz

zzz

z

zz

zz

zzz

S )()(

)()(

)()()(

)(

+−+++−−

+−++++−+−

+−−+−++−−

+−+

=

(A.28)

137

122

222

222

2222

5353

531131253

532241253

5321241353

5 Function

kadkadiexpt

kadkaddkiexprrt

kadkadbkiexprrt

kadkadbkdkiexprrt

Czz

zzz

zzz

zzzz

S⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ ++

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ ++−

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ +++

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ +++−

=

(A.29)

122

2

22

2222

222

5311253

531353

53212453

53224131253

5 Function

kadkaddkiexprt

kadkadiexprt

kadkadbkdkiexprt

kadkadbkiexprrrt

Dzzz

zz

zzzz

zzz

S⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ −−+

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ −−−

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ −−++

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ −−−

=

(A.30)

12

22

22

532124125331

5315331

5 Function

kadakbkkdiexprrtt

kadakkdiexptt

Ezzzz

zzz

S⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−++⎟

⎠⎞

⎜⎝⎛+

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛

=

(A.31)

122

3

22

23

531125331

5321245331

5 Function

kadakkdiexprtt

kadakbkkdiexprtt

Fzzz

zzzz

S⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛+

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−++⎟

⎠⎞

⎜⎝⎛

=

(A.32)

122 5321533112

5 Function

kadakkddkiexptttG

zzzzS

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛−

= (A.33)

138

12

22 532124533112

5 Function

kadakkbddkiexprtttH

zzzzS

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛ ++

= (A.34)

122 5432124123153

5 Function

kadkbdakbkdkiexpttttI

zzzzzS

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛ +−++

=

(A.35)

and Function1 is same as defined in Eq. A.22. When the factor SA5 is chosen such that

( )21

5235 2

1⎟⎟⎠

⎞⎜⎜⎝

⎛=

z

PorS

kkA /π

(A.36)

the electric field is orthonormal in the sense of Eq. A.24. Similarly, for the P-polarized

modes that are incident from medium 5, the electric field is

[ ]

[ ]

[ ]

[ ]

[ ]⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛ +−≤⋅

⎟⎠⎞

⎜⎝⎛−≤≤⎟

⎠⎞

⎜⎝⎛ +−⋅+⋅

⎟⎠⎞

⎜⎝⎛≤≤⎟

⎠⎞

⎜⎝⎛−⋅+⋅

⎟⎠⎞

⎜⎝⎛ +≤≤⎟

⎠⎞

⎜⎝⎛⋅+⋅

⎟⎠⎞

⎜⎝⎛ +≥⋅+⋅

=

−−

++−−

++−−

++−−

++−−

bdzforPkrkiexpInnA

dzbdforPkrkiexpHPkrkiexpGnnA

dzdforPkrkiexpFPkrkiexpEnnA

adzdforPkrkiexpDPkrkiexpCnnA

adzforPkrkiexpBPkrkiexpA

rE

PP

PPP

PPP

PPP

PP

2

22

22

22

2

4454

55

2252252

55

1151151

55

3353353

55

555555

),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

)(

ι

ιι

ιι

ιι

ιι

(A.37)

where PPPPPPPP IandHGFEDCB 55555555 ,,,,,, are given by Eqs.A.28 through A.35 but now

the reflection and transmission coefficients are given by Eq. A.11. The choice of PA5 as

given by Eq. A.36 again leads to the satisfaction of the orthonormality relation given by

Eq. A.24 for the electric fields of the two P-polarized modes.

139

A.3 Guided Modes

As we are specifically interested in slot waveguides – we have a specific layered

configuration. We have n4 (Silica -BOX) =1.444, n2 (Crystalline silicon) =3.4797, n1 (Slot

oxide) =1.444, n3 (Amorphous silicon) =3.4797 and n5 (Air) =1 at 1550 nm wavelength.

The guided modes may exist when

( ) ( ) ( )

( ) ( ) ( )4797.3444.1

..2/122

23

2/12241

kkkk

einornkkknornk

yx

yx

<+<

<+<

(A.38)

The guided modes are evanescent in n5(Air) and n4(BOX-SiO2). Hence k5z and k4z must

be purely imaginary. Also in our case of the slot structure, k1z will also be purely

imaginary. The length of the projection of the wave vector on the (x, y) plane,

( ) 2/122yx kk +≡β , is called the propagation constant of the guided mode. For a given

value of the wave number k in vacuum, or equivalently for a given frequency ω, there are

at most a finite number of propagation constants in the interval defined by Eq. A.38. The

set of βs are different for the two polarizations. We will consider S and P polarization

separately.

S Polarized Guided Modes

The relation to be satisfied by the propagation constant β of S-polarized guided mode

is

[ ] [ ] [ ] [ ] [ ] 12

2222

22

22412

31351233513

321352411312

2124133235241312

=−++−

+++++++−

z

zzz

zzzz

zzzz

bkiexprrakdkiexprrakiexprr

akbkdkiexprrdkiexprrbkdkiexprrakbkiexprrrr ][][

(A.39)

where the reflection coefficients are given by Eq. A.10.

140

The electric field of an S-polarized guided mode is given by

[ ]

[ ]

[ ][ ]

[ ]⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛ +−≤⋅

⎟⎠⎞

⎜⎝⎛−≤≤⎟

⎠⎞

⎜⎝⎛ +−⋅+⋅

⎟⎠⎞

⎜⎝⎛≤≤⎟

⎠⎞

⎜⎝⎛−⋅+⋅

⎟⎠⎞

⎜⎝⎛ +≤≤⎟

⎠⎞

⎜⎝⎛⋅+⋅

⎟⎠⎞

⎜⎝⎛ +≥⋅

=

−−

−−++

−−++

−−++

++

bdzforSkrkiexpJE

dzbdforSkrkiexpHSkrkiexpGE

dzdforSkrkiexpFSkrkiexpE

adzdforSkrkiexpDSkrkiexpCE

adzforSkrkiexpAE

rE

Sg

Sg

Sg

Sg

Sg

Sg

Sg

Sg

Sg

Sg

Sg

Sg

2

22

22

22

2

44

2222

1111

3333

55

),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(

)(

ι

ιι

ιι

ιι

ι

(A.40)

where

[ ]z

zzzSg akiexprr

kadakkdiexpttA

33513

5313513

2122

+

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛

= (A.41)

[ ]z

zzSg akiexprr

kdkdiexptC

33513

3113

2122

+

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛

= (A.42)

[ ]z

zzSg akiexprr

kadkdiexprtD

33513

313513

21

222

+

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛

= (A.43)

[ ] [ ] [ ]z

zzzSg akiexprr

dkiexprakdkiexprF33513

1133135

212

+++= (A.44)

[ ]z

zzSg

bkiexprr

kdkdiexptG

224

12

2112

21

22

−+

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛+⎟

⎠⎞

⎜⎝⎛−

= (A.45)

141

[ ]z

zzSg

bkiexprr

kbdkdiexprt

H2

24

12

2124

12

21

222

−+

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−⎟

⎠⎞

⎜⎝⎛−

= (A.46)

[ ]z

zzzSg

bkiexprr

kbdbkkdiexprtt

J2

24

12

42124

2412

21

22

−+

⎥⎦

⎤⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ +−−⎟

⎠⎞

⎜⎝⎛−

= (A.47)

When

( )( )

( ) ( ) ( )[ ]( ) ( )[ ]

( ) ( ) ( )

( ) ( ) ( )[ ]( ) ( )[ ]

( )( )

21

4

424

22

22

2

22

11

21

33

33

3

23

5

525

22

21

21

21

212

1

12

12

21

22

21

−

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

+−+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+

−−−−+++

⎥⎥⎦

⎤

⎢⎢⎣

⎡++++

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−−+

−−+++

+−

=

z

zSg

Sg

zzSg

Sg

zzSg

Sg

z

Sg

Sg

Sg

Sg

Sg

Sgz

Sg

Sg

z

zzSg

Sg

zzSg

Sg

z

Sg

Sg

Sg

Sg

z

zSg

Sg

Sg

kbdkexp

JJn

bkiexpdkiexpHGi

bkiexpdkiexpHGi

kbHHGGn

dFFdkSinhFFk

n

akiexpdkiexpDCi

akiexpdkiexpDCi

kaDDCCn

kadkexp

AAn

E

*

*

***

**

*

***

*

π

(A.48)

is chosen, the electric fields become normalized in the sense that for two S-polarized

guided modes corresponding to xk , yk ,ν and xk~ , yk~ ,ν~ with electric fields E and

E~

respectively, we have

∫∫∫ −−=⋅ ννδδδεε

~* )~()~(

~yyxx kkkkdrEE

0

(A.49)

where 1=ννδ ~ if νν ~= and = 0 otherwise. ν is the mode label.

142

It is also important to note that in deriving SgE above we have assumed a particular

index structure. In particular we have assumed an oxide slot. As a result k1z is taken to be

imaginary.

P Polarized Guided Modes

The relation to be satisfied by the propagation constant β of P-polarized guided

modes is same as that for S-polarized mode i.e. Eq. A.39, but with the reflection and

transmission coefficients given by Eq. A.11.

The electric field of a P-polarized guided mode is given by

[ ]

[ ]

[ ]

[ ]

[ ]⎪⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪⎪

⎨

⎧

⎟⎠⎞

⎜⎝⎛ +−≤⋅

⎟⎠⎞

⎜⎝⎛−≤≤⎟

⎠⎞

⎜⎝⎛ +−⋅+⋅

⎟⎠⎞

⎜⎝⎛≤≤⎟

⎠⎞

⎜⎝⎛−⋅+⋅

⎟⎠⎞

⎜⎝⎛ +≤≤⎟

⎠⎞

⎜⎝⎛⋅+⋅

⎟⎠⎞

⎜⎝⎛ +≥⋅

=

−−

−−++

−−++

−−++

++

bdzforPkrkiexpJnnE

dzbdforPkrkiexpHPkrkiexpGnnE

dzdforPkrkiexpFPkrkiexpE

adzdforPkrkiexpDPkrkiexpCnnE

adzforPkrkiexpAnnE

rE

Pg

Pg

Pg

Pg

Pg

Pg

Pg

Pg

Pg

Pg

Pg

Pg

2

22

22

22

2

444

1

22222

1

1111

33333

1

555

1

),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(),(ˆ)(

),(ˆ)(

)(

ι

ιι

ιι

ιι

ι

(A.50)

where Pg

Pg

Pg

Pg

Pg

Pg JandGFDCA ,,,, are given by Eqs.A.41 through A.47 but now the

reflection and transmission coefficients are given by Eq. A.11.

When

143

( )( )

( ) ( ) ( )[ ]( ) ( )[ ]

( ) ( ) ( )

( ) ( ) ( )[ ]( ) ( )[ ]

( )( )

21

4

4

22

22

2

11

33

33

3

5

5

1

22

21

21

21

212

1

12

12

21

22

21

−

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎝

⎛

+−+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−+

−−−−+++

++++

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−−+

−−+++

+−

=

z

zPg

Pg

zzPg

Pg

zzPg

Pg

z

Pg

Pg

Pg

Pg

Pg

Pgz

Pg

Pg

z

zzPg

Pg

zzPg

Pg

z

Pg

Pg

Pg

Pg

z

zPg

Pg

kbdkexp

JJ

bkiexpdkiexpHGi

bkiexpdkiexpHGi

kbHHGG

dFFdkSinhFFk

akiexpdkiexpDCi

akiexpdkiexpDCi

kaDDCC

kadkexp

AA

nE

*

*

***

**

*

***

*

π

(A.51)

is chosen, orthonormality condition given by Eq. A.49 is satisfied. Again, in deriving PgE

above we have assumed the slot slab structure, where k1z is purely imaginary.

A.4 Field Quantization

The radiation modes are labeled by the triplet ),,( kkk yx=κ where ck /ω= is the

wave number in vacuum, by the index λ, which specifies the polarization (λ = S or P) and

by the index μ which specifies the medium from which the radiation mode is incident (i.e.

μ = 4 or 5). The electric fields of the radiation modes are thus denoted by μλκE .

The guided modes are labeled by ),( yx kk=κ , λ and ν, where ν is the index of

propagation constant of the guided mode. The electric field of the guided mode is

denoted by νλκE .

144

The orthogonality relations, Eqs. A.24 and A.49 for the modes can be summarized as

follows:

∫∫∫ −=⋅2121222111 21

0μμλλμλκ δδκκδ

εε

μλκ)(* drEE (A.52a)

∫∫∫ −=⋅2121222111 21

0ννλλνλκ δδκκδ

εε

νλκ)(* drEE (A.52b)

∫∫∫ =⋅ 0222111

0

drEE *μλκνλκε

ε (A.52c)

The set of electromagnetic fields of the radiation and guided modes is a complete

orthonormal set of eigen functions of Maxwell’s equations in the space of all pairs

),( HE of transverse square integrable vector fields. By definition transverse fields satisfy

00 =⋅∇=⋅∇ HE ,)(ε (A.53)

Let μλκa and *ˆ μλκa be the destruction and creation operators corresponding to the

radiation modes and let νλκa and *ˆ νλκa be the corresponding operators for the guided

modes. These operators satisfy the commutation relations

2121222111 21 μμλλμλκ δδκκδμλκ

)(]ˆ,ˆ[ * −=aa (A.54a)

2121222111 21 ννλλνλκ δδκκδνλκ

)(]ˆ,ˆ[ * −=aa (A.54b)

0222111222111

== ]ˆ,ˆ[]ˆ,ˆ[ **μλκμλκ μλκμλκ aaaa (A.54c)

0222111222111

== ]ˆ,ˆ[]ˆ,ˆ[ **νλκνλκνλκνλκ aaaa (A.54d)

0222111222111222111

=== ]ˆ,ˆ[]ˆ,ˆ[]ˆ,ˆ[ ***μλκνλκμλκνλκμλκνλκ aaaaaa (A.54e)

The transverse electric field operator is

),(ˆ),(ˆ),(ˆ trEtrEtrE +− += (A.55)

where

145

( )

∑ ∑ ∫∫

∑ ∑ ∫∫∫

=

∞

= >=

= =<+ =

+

−⎟⎟⎠

⎞⎜⎜⎝

⎛+

−⎟⎟⎠

⎞⎜⎜⎝

⎛

=

PSyxkc

PSnkkk

yx

kc

min

yx

dkdkatiexprEi

dkdkdkatiexprEi

trE

, )(

,

,

ˆ)()(

ˆ)()(

),(ˆ

λ ν ββνλκβωνλκ

λ μμλκ

ω

μλκ

λν

λν

μ

ωεω

ωεω

1

21

0

5

4

21

0

2

221

22

(A.56)

and ),(ˆ trE − is the adjoint of ),(ˆ trE + .

When Eqs. A.52 and A.54 are used it follows that the contribution of the transverse

electric field operator to the Hamiltonian for the total electromagnetic field is given by

( )

∑ ∑ ∫∫

∑ ∑ ∫∫∫

∫∫∫

=

∞

= >=

= =<+

=

⎥⎦⎤

⎢⎣⎡ ++

⎥⎦⎤

⎢⎣⎡ +

=⋅

PSyxkc

PSnkkk

yxkc

min

yx

dkdkaa

dkdkdkaa

rdtrEtrE

,

*)(

,

*

*

,

ˆˆ

ˆˆ

),(ˆ),(ˆ

λ ν ββνλκνλκβω

λ μμλκμλκω

λν

λν

μ

ω

ω

ε

1

5

4

21

21

21

2121

21

22

(A.57)

The contribution to the Hamiltonian of the magnetic field operator is identical to that of

the electric field operator. Hence the total Hamiltonian of the electromagnetic field is

given by the well known superposition of the number operators for all electromagnetic

modes.

A.5 Spontaneous Emission and Zero Point Field Fluctuations

Suppose that an atom makes a spontaneous dipole transition from a state 2 to a

state 1 thereby emitting a photon of energy 0ω . The spontaneous emission can occur

in any mode of the electromagnetic field of frequency 0ω , or equivalently of wave

146

number ck /00 ω= in vacuum. The transition rate into a particular electromagnetic mode

with parameters μλκ ,, say, and with wave number k in vacuum, is given by[60]

)(ˆ kkifc ED −Η 0

2

2

2 δπ (A.58)

where i and f are the initial and final states of the combined atom-radiation system

and EDΗ is the electric-dipole interaction part of the complete Hamiltonian of the atom -

radiation system. In the initial state of the electromagnetic field, no photons are present.

In the final state there is one photon in the mode with parameters μλκ ,, . Because the

transition rate is much smaller than the transition frequency, the Lorentzian line is

extremely peaked. Hence, in a good approximation only modes of energy identical to the

transition energy contribute to the transition.

According to Fermi’s “golden rule”, the total spontaneous emission rate is obtained

by integrating Eq. A.58 over all the final states. Suppose that the dipole moment of the

atom is parallel to one of the axes of the Cartesian coordinate system and let j be x, y, or

z. Let jE be the jth component of the transverse electric field operator. Then the total

spontaneous emission rate of an atom at position r whose dipole moment is parallel to

the j axis is [61]

)(zFDc

e j2

122

221 πτ

= (A.59)

where e is the electron charge, 12D is the dipole matrix element of the atomic transition,

and

147

( )

∑ ∑ ∫∫

∑ ∑ ∫∫∫

=

∞

= >

= =<+

−+

−

=

PSyx

PSnkkk

yxj

j

min

yx

dkdkkkrE

dkdkdkkkrE

zF

,

,

,

))(()(

)()(

)(

λ ν ββ

λννλκ

λ μμλκ

λν

μ

βδεω

δεω

10

2

0

5

40

2

0

2

221

22

(A.60)

where in the integrand of the first term to the right, kc=ω whereas in the integrand of

the second term )(βω λνkc= . The function jF clearly depends only on the coordinate z

and is independent of x and y. Note that without the delta function in the integrands, Eq.

A.60 is the jth component of the zero-point electromagnetic field

fluctuations 00 2)(ˆ rE j . Hence )(zF j is the contribution (per wave number) of all

modes of wave number 0k to the jth component of the zero-point field fluctuations. We

will refer to )(zF j as the jth component of the zero-point field fluctuations in the

understanding that only the contributions of the modes of wave number 0k are meant.

When polar coordinates are introduced in the ( )yx kk , plane:

)(insk),(cosk yx ϕβϕβ == (A.61)

Eq. A.60 becomes

∑ ∑ ∫ ∫

∑ ∑ ∫ ∫ ∫

=

∞

=

∞

=

= =

∞

=

−+

−

=

PS sincos

PS

nk

ksincos

j

j

min

ddkkrE

ddkdkkrE

zF

, ))(),((

, )),(),((

,

))(()(

)()(

)(

λ ν

π

β

λν

ϕβϕβκνλκ

λ μ

π

ϕβϕβκμλκ

λν

μ

ϕβββδεω

ϕββδεω

1

2

00

2

0

5

4

2

0 0 00

2

0

2

2

(A.62)

148

∑ ∑ ∫

∑ ∑ ∫ ∫

=<

≥ =

= = =

+

=

PSkk

sinkcosk

PS

nk

ksincos

j

j

min

dkdk

dkrE

ddrE

zF

, ))()(),()((

, )),(),((

,

)()()(

)(

)(

λ ν

π λνλ

νϕβϕβκ

νλκ

λ μ

π

ϕβϕβκμλκ

λν

λν

λν

μ

ϕββεω

ϕββεω

000

0

1

2

000

2

0

0

5

4

2

0 0

2

0

0

2

2

(A.63)

The integral over ϕ for the three components j=x, y, z in Eq. A.63 can be computed

explicitly. The results for the x and y components are identical as they should be.

For the mean of the three components,

[ ])()()()( zFzFzFzF zyx ++=31 (A.64)

the result of the integral over ϕ yields a rather concise formula. The quantity )(zF is the

zero-point field fluctuation of a randomly oriented atom at depth z and is therefore of

special interest. It is easy to infer from the formulas for the electric fields of the modes in

Sec. A.2 and Sec. A.3 that for )),(),(( 0ksincos ϕβϕβκ = and

))()(),()(( ϕβϕβκ λν

λν sinkcosk 00= the squared amplitudes

22∑=

j

j rErE )()(μλκμλκ (A.65)

and

22∑=

j

j rErE )()(νλκνλκ (A.66)

are independent of ϕ . Hence we may take 0=ϕ in these sums. It then follows from Eq.

A.63 that

149

∑ ∑

∑ ∑ ∫

=<

≥ =

= = =

+

=

PSkk

k

PS

nk

k

j

min

kdk

dkrE

drE

zF

, )),((

, ),,(

,

)()()(

)(

)(

λ ν

λνλ

νβκ

νλκ

λ μ βκμλκ

λν

λν

μ

ββεωπ

ββεωπ

00

0

100

0

2

0

0

5

4 0 0

2

0

0

232

232

(A.67)

The remaining integrals with respect to β have to be computed numerically.

Consider now the zero-point field fluctuations in a homogeneous dielectric with a

refractive index n1. By substituting n5=n3=n2=n4=n1 into the expressions for the radiation

modes listed in Sec. A.2, we find

⎪⎩

⎪⎨⎧

===⋅

===⋅=

−−

++

),,(,,,),(ˆ)(

),,(,,,),(ˆ)()(

kkkkPSforkirkiexpA

kkkkPSforkirkiexpArE

yx

yx

λμλ

λμλμλκ

5

4

11

11

(A.68)

where 12122 nkkk yx <+ /)( and

( )21

1232

1⎟⎟⎠

⎞⎜⎜⎝

⎛=

zkkA /π

(A.69)

to ensure the modes are orthogonal. Hence the radiation modes reduce to plane waves. In

a homogeneous space the guided modes obviously don’t exist. By substituting Eq. A.68

and A.69 into Eq. A.63, we find for the components of the zero-point fluctuations in a

homogeneous dielectric of refractive index n1:

freezyx FnFFF 1=== (A.70)

where freeF denotes the vacuum field fluctuations (in free space):

20

2

30

6 cF free

επω= (A.71)

150

For an atom at depth z inside the slab of index n1 in a multi-dielectric slab structure,

having a dipole moment parallel to the jth coordinate axis, the spontaneous emission rate

relative to the value in bulk n1 index medium will be

free

j

FnzF

1

)( (A.72)

while for a randomly oriented atom the relative transition rate will be

freeFnzF

1

)( (A.73)

with )(zF being defined by Eq. A.64.

Finally, the mean spontaneous emission rate for the film of thickness d relative to

bulk film medium is obtained by computing the mean of Eq. A.73 over the thickness of

the film.

Results for the slot waveguide are shown in Fig. 5.7, which shows the mean relative

spontaneous emission rate for an emitter embedded in a silica slot relative to that in a

bulk silica medium.

The structure was also analyzed using a formulation presented in [55] where a transfer

matrix approach is utilized to determine the fields in the dielectric layers. Here the

authors also utilized a modified set of radiative modes characterized by a single outgoing

component as it is claimed that using the conventional set of radiative modes (used

above) gives rise to quantum interference between different outgoing modes. For the slot

structure under analysis, both the methods however, yielded exactly the same results.

151

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155

RAVI S. TUMMIDI, Ph.D.

Room 206 D, Sinclair laboratory, Center for Optical Technologies, Lehigh University, Bethlehem, PA 18015Ph: (484) 767 5920 [email protected] http://www.lehigh.edu/~rst205/ravi.html

EDUCATION

Lehigh University, Bethlehem, PA Candidate for Doctor of Philosophy in Electrical Engineering June 2007 to August 2011 Thesis Advisor: Prof. Thomas L. Koch • Cumulative GPA: 3.92/4.00 • Air Force office of Scientific Research (AFOSR), MURI Silicon Laser Project Graduate Research

Assistantship at Lehigh University from January 2007 till present • 6 Journal and 10 Conference publications, 1 Patent Filed, >25 Citations

Lehigh University, Bethlehem, PA Master of Science in Electrical Engineering August 2005 to May 2007 Advisor: Prof. Nelson Tansu • Cumulative GPA: 3.97/4.00 • Phi Beta Delta honors (Inducted in 2006), Lehigh University • Graduate Teaching Assistantship at Lehigh University from January 2006 till December 2006 • 3 Conference publications

North Maharashtra University, India Bachelor of Engineering in Electronics and Telecommunication August 1999 to June 2003 • First Class with Distinction (Top 2% of 180 students) • Implemented an automated telephone call routing circuitry. Routes calls based on a person’s

location identification in a premises using a transceiver chip

RESEARCH AND PROFESSIONAL EXPERIENCE

Lightwire, Inc., Allentown, PA June 2011 till Present Photonic Device Engineer • Developing components for high speed, low power CMOS photonic interconnect technology

LehighUniversity, Center for Optical Technologies Jan 2007 to August 2011 Ph.D. Candidate and Research Assistant • Principal design responsibilities in an AFOSR funded Silicon Laser MURI led by MIT, where

eight universities are targeting a CMOS compatible electrically driven silicon laser technology for nano-photonic interconnect applications

• Proposed, analyzed and demonstrated ultra low loss (very high Q) shallow ridge horizontal slot waveguides at “Magic Widths” operating in TM mode which is a critical ingredient in our overall design for an electrically driven photon source in Silicon

• Developed new fabrication techniques to achieve ultra smooth waveguide surfaces to substantially reduce waveguide scattering losses in high index contrast systems like SOI

• Combined the new fabrication techniques and novel resonator designs to achieve resonators with Qs of 1.6×106 for the TE-like mode in non-slot configurations and 3×105 for the TM-like mode in full slot configuration, the highest yet reported for this type of structure and close to our design requirements for a laser

• Incorporated Erbium into a slot just 8.3 nm thick and observed photoluminescence in full waveguide configuration. Performed Quantum electro dynamic (QED) numerical analysis which predicted >10 times Purcell enhancement in these non-resonator structures. Verified the same experimentally

• Collaborated with a research group at RMIT, Australia on low loss resonator designs which resulted in >8 joint publications

• Presented bimonthly project progress reports and annual reviews totaling >20 since January 2007 to MURI team comprising of research groups at MIT, Caltech, Stanford, Cornell, Boston U., U. Delaware & U. Rochester. Only student working on the project at Lehigh University

156

Lehigh University, Center for Optical Technologies Aug 2005 to Jan 2007 Master’s Student and Research Assistant • Performed extensive numerical modeling of VCSELs and quantum well structures • Executed design and MOCVD epitaxy of III-Nitride semiconductor optoelectronic devices for

mid-IR applications • Strong knowledge in material characterization using photoluminescence, atomic force microscopy

(AFM), scanning electron microcopy (SEM) and Hall measurement • Liaised with external vendors for the installation and commissioning of the MOCVD system at

Lehigh University and gained experience in system and support equipment maintenance

Videsh Sanchar Nigam Limited (VSNL), India Nov 2003 to July 2005 Transmission Engineer • Maintained international telecommunication links, fiber optic communication equipment –

MUX/DEMUX, digital cross connects, SDH and PDH equipment • Gained familiarity with usage of various transmission test equipment – Advanced network tester,

PFA 30/35, spectrum analyzer, HP communication analyzer • Selection for the post was on all India basis. Was one among fifty who got selected from the initial

34,000 candidates who applied for the positions

ACTIVITIES

• Reviewer for IEEE Photonics, OSA Optics Express, OSA Optical Materials Express journals • Founding member of the IEEE Photonics Society student chapter at Lehigh University in 2008

and Vice President for the chapter • Student member of IEEE Photonics Society, Optical Society of America (OSA), International

Society for Optics and Photonics (SPIE) • Captain of the University Badminton and Volleyball teams during undergraduate studies

REFEREED JOURNAL PUBLICATIONS

1. Naser Dalvand, Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch, and Arnan Mitchell, “Thin-ridge Silicon-on-Insulator waveguides with directional control of lateral leakage radiation”, Optics Express, 19, (6), pp. 5635-5643, March 2011

2. Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch and Arnan Mitchell, “Lateral leakage in TM-like mode Thin-ridge Silicon-on-Insulator Bent waveguides and Ring Resonators”, Optics Express 18, (7), pp. 7243-7252, March 2010

3. Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch and Arnan Mitchell, “Lateral leakage in TM-like whispering gallery mode of thin-ridge silicon-on-insulator disk resonators”, Optics Letters, 2009, 34, (7), pp. 980-982

4. Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch and Arnan Mitchell, “Rigorous Modeling of Lateral Leakage Loss in SOI Thin-Ridge Waveguides and Couplers”, Photonics Technology Letters, IEEE, 2009, 21, (7), pp. 486-488

5. Robert M. Pafchek, Jinjin Li, Ravi S. Tummidi and Thomas L. Koch, “Low Loss Si – SiO2 – Si 8 nm Slot Waveguides”, Photonics Technology Letters, IEEE, 2009, 21, (6), pp. 353-355

6. Robert M. Pafchek, Ravi S. Tummidi, Jinjin Li, Mark A. Webster, E. Chen and Thomas L. Koch, “Low-loss SOI Shallow-Ridge TE and TM Waveguides Formed Using Thermal Oxidation,” Applied Optics, 2009,48,(5), pp. 958-963

CONFERENCE PUBLICATIONS

1. Ravi S. Tummidi, Thach G. Nguyen, Arnan Mitchell and Thomas L. Koch, “An Ultra-compact waveguide polarizer based on “Anti-Magic-Widths””, IEEE Group Four Photonics (GFP), London, UK, September, 2011 (Accepted)

2. Naser Dalvand, Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch, and Arnan Mitchell, “Directional control of lateral leakage loss in Thin-ridge Silicon-on-Insulator waveguides”, IEEE/OSA Conference on Lasers and Electro-Optics (CLEO) 2011, Baltimore MD, May, 2011

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3. Ravi S. Tummidi, Robert M. Pafchek, Kangbaek Kim and Thomas L. Koch, “Purcell-enhanced Erbium Emission in Low-loss Horizontal nm-Scale Slot Silicon Photonic Waveguides”, Invited Talk V1.2, 2011 Materials Research Society Spring Meeting and Exhibit, San Francisco, April 25 - 29, 2011

4. Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch and Arnan Mitchell, “”Magic Radius” Phenomenon in Thin-ridge SOI Ring Resonators: Theory and Preliminary Observations”, IEEE/OSA Conference on Lasers and Electro-Optics (CLEO) 2010, San Jose CA, May, 2010

5. Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch, and Arnan Mitchell, “Mode-Quenching in Thin Ridge Silicon on Insulator Disk Resonators with Lateral Leakage”, Australian Conference on Optical Fiber Technology (ACOFT), The University of Adelaide, Australia, Nov-Dec 2009

6. Ravi S. Tummidi*, Robert M. Pafchek, Kangbaek Kim and Thomas L. Koch, “Modification of spontaneous emission rates in shallow ridge 8.3 nm Erbium doped silica slot waveguides”, IEEE Group Four Photonics (GFP), San Francisco, CA, September, 2009

7. Thach G. Nguyen, Ravi S. Tummidi, Thomas L. Koch and Arnan Mitchell, “Thin-Ridge SOI Disk and Ring Resonators with "Magic Radius" and "Magic Width" Phenomena ”, IEEE/OSA Conference on Lasers and Electro-Optics (CLEO) 2009, Baltimore MD, June, 2009

8. Ravi S. Tummidi*, Thach G. Nguyen, Arnan Mitchell and Thomas L. Koch, “Anomalous losses in curved waveguides and directional couplers at “magic widths” ”, in Proc. of the IEEE Lasers and electro-optics society (LEOS), Newport Beach, CA, November, 2008

9. Thach G. Nguyen, Ravi S. Tummidi, Mark A. Webster, Thomas L. Koch, and Arnan Mitchell, “Analysis of Lateral Leakage Loss in Silicon-On-Insulator Thin-Rib Waveguides”, Australian Conference on Optical Fiber Technology (ACOFT), Sydney, Australia, July 2008

10. Robert M. Pafchek, Jinjin Li, Ravi S. Tummidi* and Thomas L. Koch, “ Low Loss Si – SiO2 – Si 8 nm Slot Waveguides”, in Proc. of the IEEE/OSA Conference on Lasers and Electro-Optics (CLEO) 2008, San Jose CA, May 2008

11. R. A. Arif, Ravi S. Tummidi, Y. K. Ee, and N. Tansu, “Design Analysis of Lattice-Matched AlInGaN-GaN Quantum Wells for Optimized Intersubband Absorption in the Mid-IR Regime,” in Proc. of the SPIE Photonics West 2007, Physics and Simulation of Optoelectronics Devices XV, San Jose, CA, Jan 2007

12. H. Li, J. T. Perkins, H. M. Chan, R. P. Vinci, Y. K. Ee, R. A. Arif, Ravi S. Tummidi, J. Li, and N.Tansu, “Nanopatterning of Sapphire for GaN Heteroepitaxy by Metalorganic Chemical Vapor Deposition,” in Proc. of the MRS Fall Meeting 2006: Symposium I: Advances in III-V Nitride Semiconductor Materials and Devices, Boston, MA, USA, November-December 2006

13. Z. Jin, Ravi S. Tummidi*, Y. P. Gupta, D. M. Schindler, and N. Tansu, “Quasi-Guided-Optical-Waveguide VCSELs for Single-Mode High-Power Applications,” in Proc. of the IEEE/OSA Conference on Lasers and Electro-Optics (CLEO) 2006, Long Beach, CA, May 2006

* Oral Presenter

INVITED SEMINARS

1. Ravi S. Tummidi, “Novel waveguide architectures for light sources in Silicon Photonics”, Lightwire Inc., Allentown, PA, February 2011

2. Ravi S. Tummidi, “Novel waveguide architectures for light sources in Silicon Photonics”, GE Global Research, Niskayuna, NY, March 2011

PATENTS

1. “Chip-Based Slot Waveguide Spontaneous Emission Light Sources”, US Patent Application 12/799,171 filed April 20, 2010 claiming priority of provisional patent application 61/214,313 filed April 22, 2009. Inventors: Harry A. Atwater, Jr., Ryan M. Briggs, Mark L. Brongersma, Young Chul Jun, Thomas L. Koch and Ravi S. Tummidi

novel waveguide architectures for light sources in silicon photonics

Documents