Narrow-Linewidth Si/III-V Lasers: a Stud

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Narrow-Linewidth Si/III-V Lasers: a Study of Laser Dynamics and Nonlinear Effects
Citation
Vilenchik, Yaakov
(2015)
Narrow-Linewidth Si/III-V Lasers: a Study of Laser Dynamics and Nonlinear Effects.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/Z9513W57.
Abstract
Narrow-linewidth lasers play an important role in a wide variety of applications, from sensing and spectroscopy to optical communication and on-chip clocks. Current narrow-linewidth systems are usually implemented in doped fibers and are big, expensive, and power-hungry. Semiconductor lasers compete favorably in size, cost, and power consumption, but their linewidth is historically limited to the sub-MHz regime. However, it has been recently demonstrated that a new design paradigm, in which the optical energy is stored away from the active region in a composite high-Q resonator, has the potential to dramatically improve the coherence of the laser.
This work explores this design paradigm, as applied on the hybrid Si/III-V platform. It demonstrates a record sub-KHz white-noise-floor linewidth. It further shows, both theoretically and experimentally, that this strategy practically eliminates Henry’s linewidth enhancement by positioning a damped relaxation resonance at frequencies as low as 70 MHz, yielding truly quantum limited devices at frequencies of interest.
In addition to this empirical contribution, this work explores the limits of performance of this platform. Here, the effect of two-photon-absorption and free-carrier-absorption are analyzed, using modified rate equations and Langevin force approach. The analysis predicts that as the intra-cavity field intensity builds up in the high-Q resonator, non-linear effects cause a new domain of performance-limiting factors. Steady-state behavior, laser dynamics, and frequency noise performance are examined in the context of this unique platform, pointing at the importance of nonlinear effects.
This work offers a theoretical model predicting laser performance in light of nonlinear effects, obtaining a good agreement with experimental results from fabricated high-Q Si/III-V lasers. In addition to demonstrating unprecedented semiconductor laser performance, this work establishes a first attempt to predict and demonstrate the key impact of nonlinear effects on silicon-based lasers.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
nonlinear effect; Si/III-V; Narrow-linewidth lasers; TPA; Narrow-linewidth; Two-photon-absorption Free-carrier-absorption; FCA;
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Yariv, Amnon
Group:
Kavli Nanoscience Institute
Thesis Committee:
Vahala, Kerry J. (chair)
Yariv, Amnon
Painter, Oskar J.
Faraon, Andrei
Defense Date:
4 June 2015
Record Number:
CaltechTHESIS:06042015-232226135
Persistent URL:
DOI:
10.7907/Z9513W57
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
8991
Collection:
CaltechTHESIS
Deposited By:
Yaakov Vilenchik
Deposited On:
14 Nov 2016 23:04
Last Modified:
04 Oct 2019 00:08
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Narrow-linewidth Si/III-V lasers: A study of laser
dynamics and nonlinear eects

Thesis by

Yaakov (Yasha) Vilenchik

In Partial Ful
llment of the Requirements
for the Degree of
Doctor of Philosophy

California Institute of Technology
Pasadena, California

2015
(Defended June 4, 2015)

c 2015
Yaakov (Yasha) Vilenchik

iii

Acknowledgments
These last few years at Caltech have been a time of immense personal and intellectual
growth for me. I would like to take the opportunity to thank the many people I've
learned from and with.
First and foremost, I would like to thank and acknowledge my advisor, Professor
Amnon Yariv.

Professor Yariv provided me with the opportunity to work at the

forefront of science, and has succeeded in convincing me that nothing is impossible.
His guidance has shaped me into a researcher who strives to combine a top-level
scienti
c bird's-eye-view, with a down-to-earth understanding of technical challenges.
I deeply appreciate the time and eort of my thesis committee members, Professors
Kerry Vahala, Oskar Painter and Andrei Faraon. This committee is truly the dream
team for my thesis, and I am very privileged to have them in my corner.
I would like to thank Professor Bruno Crosignani for his support and encouragement over the years.

His unique approach to problem-solving in all
eld from

physics to politics and philosophy has deeply enriched me and made me a better
researcher and critical thinker.
I cannot thank enough the members of the Yariv group. I couldn't have done any
of the work presented in this thesis without them. Scott Steger has guided me through
the III-V fabrication process, and played a huge role in the fabrication of the lasers
used in this thesis. Many of the important insights in this work are a result of our long
discussions. Christos Santis has always been a model for hard-work and perfectionism
in the clean-room. His contribution was crucial both in the conception of some of the
initial ideas, and the implementation of the high-Q silicon resonators that were used in
this work. I would also like to thank Naresh Satyan, whom I always sought for advice,

iv
and his input was crucial time after time. Our collaboration has yielded an order of
magnitude improvement in both the characterization setup, and my experimental
skills.

I must thank Dongwan Kim for his help in the fabrication eort, and for

our many conversations which helped me to constantly improve my understanding
of every aspect of this project.

I would also like to thank Mark Harfouche for his

frequent advice, that always provided me with new insights. This project bene
ted a
lot from his wide range of expertise, that never ceased to amaze me. Other members
of the Yariv group Jacob Sendowski, Arseny Vasilyev, Sinan Zhao, Xiankai Sun,
Hsi-Chun Liu, Avi Zadok, Paula Popescu and Marilena Dimotsantou have all
contributed to this work, each in their unique way, and for that I am thankful. A
special thanks goes to group-members in the broader sense, Reg Lee and George
Rakuljic, for their useful input and advice.
The sta members at KNI were a source of both knowledge and help. The support and advice I got from Guy DeRose, Melissa Melendes and Steven Martinez was
indispensable. I would like to thank Christy Jenstad and Mabel Chik for helping me
to navigate life and research at Caltech. I would especially like to thank Connie Rodriguez, who enabled me to do science, without needing to worry about the immense
amount of paperwork. She made sure that my environment never got too cold, too
warm or too loud both literally and metaphorically.
The Israeli community at Caltech has provided an important support net, and
helped me transition smoothly to a new country and culture.

My time at Caltech

would not have been as vibrant without them. I also thank Chabad of Pasadena and
the Pasadena Jewish Community for enriching my spiritual life here.
Last, but absolutely not least, I would like to thank my family.

My parents,

who have supported me over the years, and encouraged me to seek knowledge and
happiness. My children, Daniel and Mattan, who were

the source of joy and pride in

my life, and helped me keep my priorities straight. Most of all I would like to thank
my partner in life, Neta, who has been an amazing pillar of support and strength
in my life.

She even helped proofread this entire work a true test of spousal

dedication. Thank you!

Abstract
Narrow-linewidth lasers play an important role in a wide variety of applications,
from sensing and spectroscopy to optical communication and on-chip clocks.

Cur-

rent narrow-linewidth systems are usually implemented in doped
bers and are big,
expensive, and power-hungry. Semiconductor lasers compete favorably in size, cost,
and power consumption, but their linewidth is historically limited to the sub-MHz
regime. However, it has been recently demonstrated that a new design paradigm, in
which the optical energy is stored away from the active region in a composite high-Q
resonator, has the potential to dramatically improve the coherence of the laser.
This work explores this design paradigm, as applied on the hybrid Si/III-V platform.

It demonstrates a record sub-KHz white-noise-
oor linewidth.

It further

shows, both theoretically and experimentally, that this strategy practically eliminates Henry's linewidth enhancement by positioning a damped relaxation resonance
at frequencies as low as 70 MHz, yielding truly quantum limited devices at frequencies
of interest.
In addition to this empirical contribution, this work explores the limits of performance of this platform. Here, the eect of two-photon-absorption and free-carrierabsorption are analyzed, using modi
ed rate equations and Langevin force approach.
The analysis predicts that as the intra-cavity
eld intensity builds up in the highQ resonator, non-linear eects cause a new domain of performance-limiting factors.
Steady-state behavior, laser dynamics, and frequency noise performance are examined in the context of this unique platform, pointing at the importance of nonlinear
eects.
This work oers a theoretical model predicting laser performance in light of non-

vi
linear eects, obtaining a good agreement with experimental results from fabricated
high-Q Si/III-V lasers. In addition to demonstrating unprecedented semiconductor
laser performance, this work establishes a
rst attempt to predict and demonstrate
the key impact of nonlinear eects on silicon-based lasers.

vii

Contents
Acknowledgments

iii

Abstract

1 Introduction

1.1

Narrow-linewidth semiconductor lasers

. . . . . . . . . . . . . . . . .

1.2

Laser sources for coherent communication

. . . . . . . . . . . . . . .

1.3

Linear and non-linear performance limiting factors . . . . . . . . . . .

2 Hybrid Si/III-V as a platform for narrow linewidth

2.1

Noise in conventional semiconductor lasers . . . . . . . . . . . . . . .

2.2

Hybrid Si/III-V platform . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Noise reduction in hybrid Si/III-V . . . . . . . . . . . . . . . . . . . .

2.3.1

High-Q silicon resonator

. . . . . . . . . . . . . . . . . . . . .

2.3.2

Modal gain and loss

. . . . . . . . . . . . . . . . . . . . . . .

2.3.2.1

General description . . . . . . . . . . . . . . . . . . .

2.3.2.2

The spacer lasers . . . . . . . . . . . . . . . . . . . .

2.3.3

Schawlow-Townes linewidth

. . . . . . . . . . . . . . . . . . .

3 Non-linear eects in hybrid Si/III-V
3.1

Two-photon-absorption in silicon

3.2

Free-carrier-absorption

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.1

General methodology . . . . . . . . . . . . . . . . . . . . . . .

3.2.2

Bulk recombination . . . . . . . . . . . . . . . . . . . . . . . .

viii
3.2.3

Surface recombination

. . . . . . . . . . . . . . . . . . . . . .

3.2.4

Carrier diusion . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.5

Eective carrier lifetime

. . . . . . . . . . . . . . . . . . . . .

4 Modi
ed rate equations

4.1

Flat-mode approximation . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Working with densities or total numbers? . . . . . . . . . . . . . . . .

4.3

Pump

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

Linear loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5

Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.1

The quantum-well: a two or three dimensional creature?

. . .

4.5.2

Active con
nement factor

. . . . . . . . . . . . . . . . . . . .

4.5.3

Material gain

. . . . . . . . . . . . . . . . . . . . . . . . . . .

Spontaneous emission . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6.1

Model for the population inversion factor . . . . . . . . . . . .

4.7

Two-photon-absorption . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

Spontaneous recombination in the QW

. . . . . . . . . . . . . . . . .

4.9

Rate equation for free-carriers in silicon . . . . . . . . . . . . . . . . .

4.6

4.10 Free-carrier-absorption

. . . . . . . . . . . . . . . . . . . . . . . . . .

4.11 Total loss rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.12 The modi
ed rate equations

. . . . . . . . . . . . . . . . . . . . . . .

5 Steady-state operation - Theoretical analysis

5.1

Steady-state carrier density in silicon

. . . . . . . . . . . . . . . . . .

5.2

Gain saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3

Threshold current . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4

Output power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4.1

Wall-plug e
ciency . . . . . . . . . . . . . . . . . . . . . . . .

5.4.2

L-I curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4.3

Slope e
ciency

5.5

. . . . . . . . . . . . . . . . . . . . . . . . . .

Schawlow-Townes linewidth

. . . . . . . . . . . . . . . . . . . . . . .

6 Steady-state operation - Experimental results

6.1

Threshold current . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2

L-I curves

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.3

Schawlow-Townes noise
oor . . . . . . . . . . . . . . . . . . . . . . .

7 Dynamic operation - Theoretical analysis

7.1

Small-signal analysis

. . . . . . . . . . . . . . . . . . . . . . . . . . .

7.2

Intensity modulation response . . . . . . . . . . . . . . . . . . . . . .

7.2.1

. . . . . . . . . . . . . . . . . . . . .

7.2.1.1

Low nonlinear loss regime: . . . . . . . . . . . . . . .

7.2.1.2

High nonlinear loss regime:

. . . . . . . . . . . . . .

Numerical investigation . . . . . . . . . . . . . . . . . . . . . .

7.2.2
7.3

Analytical investigation

Frequency modulation response
7.3.1

7.3.2

7.3.3

. . . . . . . . . . . . . . . . . . . . .

Eect of Quantum Well carriers . . . . . . . . . . . . . . . . .

7.3.1.1

Gain compression . . . . . . . . . . . . . . . . . . . .

7.3.1.2

Henry's alpha parameter . . . . . . . . . . . . . . . .

7.3.1.3

Frequency modulation response curve . . . . . . . . .

The eects of free-carriers in silicon . . . . . . . . . . . . . . .

7.3.2.1

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

Plasma eects in silicon

The total frequency chirp

8 Dynamic operation - Experimental results
8.1

Intensity modulation response . . . . . . . . . . . . . . . . . . . . . .

8.2

Frequency modulation response

. . . . . . . . . . . . . . . . . . . . .

9 Noise performance - Theoretical analysis

99
99
105

110

9.1

Methodology - Langevin noise sources . . . . . . . . . . . . . . . . . .

111

9.2

Source of noise -
uctuations . . . . . . . . . . . . . . . . . . . . . . .

112

9.2.1

Photon density

116

9.2.2

Carriers in the quantum wells

9.2.3

Free-carriers in silicon

. . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .

119

. . . . . . . . . . . . . . . . . . . . . .

120

9.2.4
9.3

Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . .

122

Frequency noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

124

9.3.1

Spontaneous emission . . . . . . . . . . . . . . . . . . . . . . .

125

9.3.2

Henry's linewidth enhancement

126

9.3.3

Noise due to the plasma eect in silicon

9.3.4

Noise due to the thermo-optic eect in silicon

. . . . . . . . .

131

9.3.5

Total noise spectrum . . . . . . . . . . . . . . . . . . . . . . .

133

. . . . . . . . . . . . . . . . .
. . . . . . . . . . . .

127

10 Noise performance - Experimental results

138

11 Conclusion

144

11.1 Summary of key results . . . . . . . . . . . . . . . . . . . . . . . . . .

144

11.2 Future directions

146

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Bibliography

149

A Fabrication process

165

A.1

A.2

A.3

Silicon processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165

A.1.1

Chrome deposition

. . . . . . . . . . . . . . . . . . . . . . . .

165

A.1.2

Lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . .

168

A.1.3

Etch

168

A.1.4

Oxidation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169

A.2.1

Surface treatment . . . . . . . . . . . . . . . . . . . . . . . . .

170

A.2.2

Bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

170

A.2.3

Substrate removal . . . . . . . . . . . . . . . . . . . . . . . . .

170

III-V processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

172

A.3.1

Ion implantation

172

A.3.2

P-metal deposition

. . . . . . . . . . . . . . . . . . . . . . . .

172

A.3.3

Mesa formation . . . . . . . . . . . . . . . . . . . . . . . . . .

173

A.3.4

N-Metal deposition . . . . . . . . . . . . . . . . . . . . . . . .

175

A.3.5

Cleaving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

176

Wafer bonding

. . . . . . . . . . . . . . . . . . . . . . . . .

B Characterization setups
B.1

B.2

B.3

B.4

B.5

177

Mounting and probing the lasers . . . . . . . . . . . . . . . . . . . . .

178

B.1.1

Mounting of laser bars

. . . . . . . . . . . . . . . . . . . . . .

178

B.1.2

Thermal management

. . . . . . . . . . . . . . . . . . . . . .

178

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179

B.2.1

CW excitation . . . . . . . . . . . . . . . . . . . . . . . . . . .

179

B.2.2

Pulsed excitation

. . . . . . . . . . . . . . . . . . . . . . . . .

179

Intensity modulation response . . . . . . . . . . . . . . . . . . . . . .

180

B.3.1

Setup and equipment . . . . . . . . . . . . . . . . . . . . . . .

180

B.3.2

Calibration and measurement procedures

. . . . . . . . . . .

181

L-I curves

B.3.2.1

Photodetector response

. . . . . . . . . . . . . . . .

181

B.3.2.2

Driving circuitry response . . . . . . . . . . . . . . .

181

B.3.2.3

Calculating the small-signal current . . . . . . . . . .

182

B.3.2.4

Delay compensation

. . . . . . . . . . . . . . . . . .

183

. . . . . . . . . . . . . . . . . . . . .

183

B.4.1

Setup and equipment . . . . . . . . . . . . . . . . . . . . . . .

183

B.4.2

Calibration and measurement procedure

. . . . . . . . . . . .

185

Frequency modulation response

B.4.2.1

Balancing photodetectors

. . . . . . . . . . . . . . .

185

B.4.2.2

Photodetector response

. . . . . . . . . . . . . . . .

186

B.4.2.3

Voltage swing

. . . . . . . . . . . . . . . . . . . . .

186

B.4.2.4

Delay compensation

B.4.2.5

Measurement procedure

B.4.2.6

Calculating the frequency response from the measure-

. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .

188
188

ment . . . . . . . . . . . . . . . . . . . . . . . . . . .

189

Frequency noise spectrum measurement . . . . . . . . . . . . . . . . .

191

B.5.1

Setup and equipment . . . . . . . . . . . . . . . . . . . . . . .

192

B.5.2

Calibration and measurement procedures . . . . . . . . . . . .

192

B.5.2.1

Balanced PD and ampli
er

. . . . . . . . . . . . . .

192

B.5.2.2

Measurement procedure

. . . . . . . . . . . . . . . .

194

B.5.2.3

Calculating the noise spectrum from the measurement 196

xii

List of Figures
1.1

Constellation diagrams. (a) Binary Amplitude Phase Shift Keying (b)
16 Quadrature Amplitude Modulation.

1.2

. . . . . . . . . . . . . . . . .

Phasor diagram demonstrating the eect of a spontaneous emission
event

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

Hybrid Si/III-V laser schematics

2.2

Band structure of the mode-gap resonator. The parabolic potential-well

. . . . . . . . . . . . . . . . . . . . .

supports one optical mode . . . . . . . . . . . . . . . . . . . . . . . . .
2.3

(a) Quality factor of hybrid Si/III-V composite resonator (b) Mode pro
le of a traditional III-V laser (c) Pro
le of a high-Q hybrid Si/III-V
laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

Con
nement in III-V and in QWs vs. spacer thickness

. . . . . . . . .

3.1

Schematic description of two-photon-absorption . . . . . . . . . . . . .

3.2

Model for FCA in silicon. Model #1 by [58, 101]. Model #2 by [70] . .

3.3

Waveguide structure used for analysis of FCA . . . . . . . . . . . . . .

3.4

Model for the ambipolar diusion coe
cient of silicon . . . . . . . . . .

3.5

Carrier density pro
le for several dierent surface recombination pro
les

3.6

Eective lifetime of carrier in silicon for S=1

. . . . . . . . . . . . .

3.7

Eective lifetime of carriers in Si (low photon excitation regime) . . . .

4.1

Flat mode approximation for the optical mode and the electron density

4.2

Symmetrical quasi-Fermi level model . . . . . . . . . . . . . . . . . . .

cm
sec

xiii
4.3

Comparison between dierent loss mechanisms vs. photon density. Qsi =

106 , typical absorption in III-V is assumed (i.e., 10cm−1 ), con
nement
factor in III-V of 1% and eective lifetime of carriers in Si of τef f = 30ns.

5.1

Threshold current vs. con
nement factor in III-V for dierent quality
factors of the Si resonator

5.2

. . . . . . . . . . . . . . . . . . . . . . . . .

int
ciency (left axis) and total Q (right axis) vs. mirror Q. Qsi = 10 ;

ηi = 1 ; IItr = 10 (a) 30nm spacer (b) 100nm spacer (c) 150nm spacer

. . . . . . . . . . . . .

5.3

ciency of dierent spacer design vs. total-Q

5.4

L-I curves for dierent values of con
nement factors with and without

int
nonlinear eects for Qsi = 10 (a) spacer 150nm (b) spacer 100nm (c)
spacer 30nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5

Normalized slope e
ciency at I = 4 · Ith

5.6

Schawlow-Townes linewidth vs. con
nement in III-V for dierent silicon

. . . . . . . . . . . . . . . . .

resonators, with and without nonlinear eects at I = 4 · Ith . . . . . . .
5.7

Impact of nonlinear eects on linewidth for changing quality factors.
Calculated at I = 5 · Ith

5.8

. . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

Threshold current for dierent spacer designs.

Normalized L-I curves for three spacer designs.

(a) 1560nm lasers (b)

1575nm lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2

Impact of nonlinear eects on linewidth for changing pump current.

Calculated at I = 5 · Ith for QSi = 10

6.1

The non-normalized

output powers at I=150mA are: 0.89mW, 0.62mW, and 10.2mW for the
200nm, 100nm, and 30nm spacers, respectively
6.3

L-I curves of the 150nm spacer. (a) For varying stage temperatures (b)
In pulsed operation (duty cycle = 1%)

6.4

. . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

Schawlow-Townes linewidth vs current oset from threshold for the three
spacer lasers. The dotted lines represent expected 1/P dependence

. .

xiv
7.1

Intensity modulation response curves with and without nonlinear eects.

I = 4Ith , QSi = 106 (a) 30nm spacer (b) 100nm spacer (c) 150nm spacer
7.2

Intensity modulation response curves for dierent spacer thicknesses.

I = 4Ith , QSi = 106 (a) amplitude (b) phase . . . . . . . . . . . . . . .
7.3

. . . . . . . . . . . . . .

. . . .

(a) 30nm spacer (b) 100nm spacer (c) 150nm spacer

. . . . . . . . . .

Frequency modulation response for several dierent spacer thicknesses

for αH = 7, I = 2 · Ith , QSi = 10
7.8

Frequency modulation response for several dierent spacer thicknesses,
with and without nonlinear eects for αH = 7, I = 2 · Ith , QSi = 10

7.7

Frequency modulation response due to free carriers in silicon for dierent

values of spacer thickness. αH = 7, I = 2 · Ith , QSi = 10
. . . . . . . .
7.6

Frequency modulation response due to quantum well electrons for dif-

ferent values of spacer thickness. αH = 7, I = 2 · Ith , QSi = 10
7.5

Intensity modulation response curves for dierent spacer thicknesses.

I = 10Ith , QSi = 106 (a) amplitude (b) phase
7.4

. . . . . . . . . . . . . . . . . . . . .

Eect of pump current on frequency modulation response for two dier-

ent spacer thicknesses αH = 7, I = 2 · Ith , QSi = 10 . (a) 30nm spacer
(b) 150nm spacer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.1

Intensity modulation response of 30nm spacer laser (Chip 1, bar 5, Slot
1, device 7) for dierent bias currents. 3.5mA current modulation. Measured with New-Focus 1544B photodetector and HP 8722C RF network
analyzer (a) Normalized magnitude (b) Phase . . . . . . . . . . . . . .

8.2

101

Intensity modulation response of 100nm spacer laser (chip 1, bar 1, slo
t2, device 19) for dierent bias currents. 6mA current modulation. Low
frequency response (<50MHz) was measured using New-Focus 1544B
photodetector and Agilent 4395A network analyzer. High frequency response (>50 MHz) was measured using HP 8722C RF network analyzer
(a) Normalized magnitude (b) Phase

. . . . . . . . . . . . . . . . . . .

102

xv
8.3

Intensity modulation response of 150nm spacer laser (chip 1, bar 1,
slot 2, device 19) for dierent bias currents. 3.3mA current modulation.
Measured using New-Focus 1544B photodetector and Agilent 4395A net-

8.4

work analyzer (a) Normalized magnitude (b) Phase

. . . . . . . . . .

Intensity modulation response (magnitude in a.u.)

of dierent spacer

lasers for pump current oset of 40mA.
8.5

. . . . . . . . . . . . . . . . .

103

104

Frequency modulation response of 30nm spacer laser (Chip 1, bar 5, Slot
1, device 7) for dierent bias currents. 0.1mA current modulation. Measured with Optilab BPR-20-M balanced photodetector and HP 8722C
RF network analyzer, using MZI with FSR = 7.06GHz . . . . . . . . .

8.6

105

Frequency modulation response of 100nm spacer laser (Chip 1, bar 1,
Slot 2, device 19) for dierent bias currents. 35nA current modulation.
Measured with Optilab BPR-20-M balanced photodetector MZI with
FSR = 1.56GHz.

Low frequencies (<500 MHz) were measured using

Agilent 4395A network analyzer and high frequencies (>500MHz) using
HP 8722C analyzer. Curves from the two analyzers are plotted together
without any additional post-processing (stitching)
8.7

. . . . . . . . . . .

106

Frequency modulation response of 150nm spacer laser (Chip 1, bar 1,
Slot 2, device 19) for dierent bias currents. 0.1mA current modulation.
Measured with Optilab BPR-20-M balanced photodetector and Agilent
4395A network analyzer, using MZI with FSR = 1.56GHz

8.8

. . . . . . .

106

Frequency modulation response of dierent spacer lasers for pump current oset of 50mA.

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.1

Particle reservoir picture of the system

. . . . . . . . . . . . . . . . .

9.2

PSD of the photon density (150nm spacer, QSi = 10 , I = 4Ith ) (a) Com-

107

113

parison of conventional terms (terms which don't involve TPA-generated
free-carriers in Si) (b) Comparison of these conventional terms to freecarrier related terms

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

118

xvi
9.3

Comparison of the dierent terms of the PSD of QW-carrier density

(150nm spacer, QSi = 10 , I = 4Ith )
9.4

. . . . . . . . . . . . . . . . . . .

Noise spectrum of silicon carrier density for shot-noise model (150nm

spacer, QSi = 10 , I = 4Ith ) . . . . . . . . . . . . . . . . . . . . . . . .
9.5

120

Frequency noise due to spontaneous emission and QW carrier density

uctuations using Langevin shot-noise model.

αH = 3
9.6

119

QSi = 106 , I = 4Ith ,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127

The eect of Si-carrier
uctuation on the frequency noise spectrum:
Eective recombination model vs recombination-diusion model. 150nm

Spacer, QSi = 10 , I = 2Ith
9.7

. . . . . . . . . . . . . . . . . . . . . . . .

Frequency noise spectrum due to Si carrier
uctuations for dierent

pump currents. 150nm Spacer, QSi = 10
9.8

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . .

136

Predicted frequency noise spectrum for the 150nm spacer designs at

dierent pump powers (QSi = 10 )

10.1

135

Predicted frequency noise spectrum of dierent spacer designs (QSi =

106 , I = 2 · Ith ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.13

134

The dierent components of the total frequency noise spectrum (QSi =

106 , I = 2 · Ith ). (a) 150nm spacer (b) 100nm spacer (c) 30nm spacer .
9.12

133

Frequency noise related to temperature
uctuations for dierent spacer

designs (QSi = 10 , I = 2 · Ith ) . . . . . . . . . . . . . . . . . . . . . .
9.11

131

The dierent components of the frequency noise spectrum due to tem-

perature
uctuations (150nm spacer, QSi = 10 , I = 2 · Ith )
9.10

130

Frequency noise spectrum due to Si carrier
uctuations for dierent

spacers. QSi = 10 , I = 2 · Ith
9.9

129

. . . . . . . . . . . . . . . . . . . .

136

Eect of BOA ampli
cation on frequency noise spectrum. All data was
taken with the same 150nm spacer at constant laser pump current. Only
BOA pump current changed from curve to curve

10.2

. . . . . . . . . . . .

Frequency noise of 150nm spacer, ampli
ed using BOA and EDFA

. .

139
140

xvii
10.3

Frequency noise spectrum vs. pump current for dierent spacer designs
(a) 150nm spacer (threshold @ 66mA) (b) 100nm spacer (threshold @
28mA) (c) 30nm spacer (threshold @ 55mA) . . . . . . . . . . . . . . .

141

10.4

Frequency noise spectrum for dierent spacer designs at I − Ith = 33mA 142

A.1

Spacer laser device Schematics

. . . . . . . . . . . . . . . . . . . . . .

166

A.2

Flow diagram of the silicon processing steps . . . . . . . . . . . . . . .

167

A.3

Flow diagram of the III-V processing steps . . . . . . . . . . . . . . . .

172

A.4

SEM images of mesa formed using the two mask process (a) Entire mesa
and Si waveguide (b) Sidewalls of mesa and InGaAs layers (c) Etch and
under-cut of the QW layer . . . . . . . . . . . . . . . . . . . . . . . . .

175

B.1

Schematics of L-I curve characterization setup . . . . . . . . . . . . . .

179

B.2

Schematics of the experimental setup used for intensity modulation response measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B.3

Measured response of NF1544B photodetector.

The PD response was

subtracted from measured laser response . . . . . . . . . . . . . . . . .
B.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193

Intensity noise measured for dierent input powers by the balanced photodetector (Optilab BPM-20)

B.9

187

Schematics of the experimental setup used for frequency noise spectrum
measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B.8

185

Response curves of balanced PDs (a) New Focus 1817 (b) Optilab BPR
20-M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

B.7

184

Schematics of the electronic feedback PCB used to lock the MZI to
quadrature

B.6

182

Schematics of the experimental setup used for frequency modulation
response measurements . . . . . . . . . . . . . . . . . . . . . . . . . . .

B.5

180

. . . . . . . . . . . . . . . . . . . . . . .

194

Intensity noise vs calculated dark+shot noise for dierent input power
to the balanced PD (a) 0.25mW per detector (b) 0.5mW (c) 1 mW

. .

195

xviii

List of Tables
4.1

Parameters used for rate equations

. . . . . . . . . . . . . . . . . . . .

A.1

Device dimensions. Notations is based on Figure A.1

. . . . . . . . . .

166

A.2

III-V wafer structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167

A.3

Conditions of Cr evaporation

. . . . . . . . . . . . . . . . . . . . . . .

168

A.4

Steps for E-Beam lithography . . . . . . . . . . . . . . . . . . . . . . .

168

A.5

ICP etch steps and conditions . . . . . . . . . . . . . . . . . . . . . . .

169

A.6

Oxidation/Anneal steps and conditions . . . . . . . . . . . . . . . . . .

170

A.7

Steps and conditions for wafer bonding . . . . . . . . . . . . . . . . . .

171

A.8

Substrate removal steps and conditions . . . . . . . . . . . . . . . . . .

171

A.9

Parameters used for ion implantation . . . . . . . . . . . . . . . . . . .

173

A.10

Steps and conditions used to deposit P metal stack

. . . . . . . . . . .

174

A.11

Steps and conditions for lift-o photo-lithography . . . . . . . . . . . .

174

A.12

steps and conditions used to form the mesa

175

A.13

Steps and conditions used for N-metal deposition

. . . . . . . . . . . . . . .
. . . . . . . . . . . .

176

Chapter 1
Introduction
For decades, semiconductor lasers have been considered reliable, inexpensive everyday
light sources.

However, they lack high quality coherence properties.

On the other

hand, narrow-linewidth lasers, i.e.,
ber or solid-state devices, have been considered
exotic, bulky, and expensive.

A new technology the hybrid Si/III-V platform

[75]; and a new laser design paradigm [106], were recently combined to demonstrate
a record narrow-linewidth semiconductor laser [88].

This demonstration might be

the
rst milestone towards the penetration of semiconductor lasers to the narrowlinewidth laser industry, with a potentially dramatic impact on the narrow-linewidth
industry.
Work pioneered by John Bowers et al.

(i.e., [76, 75, 113, 34]) has enabled the

integration of direct-bandgap semiconductor devices with a low-loss material, mainly
silicon.

The work by Amnon Yariv et al.

(i.e., [132, 88]) used this platform to

demonstrate a hybrid Si/III-V laser with 18 KHz white-noise
oor linewidth.

The

mechanism of noise reduction was further developed by Yariv et al., who portrayed
a path towards further reduction [106]. However, experimental results in that work
were inconclusive, and did not overcome the ∼18 KHz barrier.
The work presented in this thesis is a direct continuation of the work by Steger
and Christos Santis [89]. It relies on the design and fabrication of high-Q Si resonators
[88], and on the linewidth reduction approach by modal engineering and the control
of spontaneous emission into the mode [106]. The novelty and contribution of this
work is three-fold:

1. It provides a conclusive experimental validation for the new design approach
- experiments demonstrating a record sub-KHz linewidth are presented in this
work for the
rst time. A conclusive linewidth reduction trend is also presented,
portraying the path for further improvement.

2. A theoretical analysis and experimental results of laser dynamics of this unique
platform are also presented in this work for the
rst time. Intensity and frequency modulation response measurements were performed, showing unique
characteristics of the hybrid platform, i.e., relaxation resonance frequencies as
low as 100 MHz.

3. It provides a theoretical model and experimental validation of the impact of
nonlinear eects, i.e., two-photon-absorption and the attendant free-carrierabsorption on laser performance. It is shown that as the intra-cavity intensity
builds up these nonlinear processes limit the achievable linewidth and alter the
laser dynamics.

Others have shown that free-carrier-absorption in Si aects

device performance [85, 57, 37, 112, 133]. However, this is the
rst time these
eects are considered and experimentally demonstrated in the context of highQ Si-based lasers. The analysis in this work provides a limit to the achievable
device performance, and also oers directions to overcome these limits in silicon.

1.1 Narrow-linewidth semiconductor lasers
A laser is a light source that emits coherent radiation. The temporal coherence of a
laser refers to the spectral purity of the electromagnetic
eld emitted by it. A related
concept the spatial coherence refers to the correlation between the laser
eld at
dierent points in space. The spatial coherence properties of a laser allow for tight
beam focusing, as well as low-divergence beams. A convenient metric to quantify the
coherence of a laser device is by the width of the spectral line of the electromagnetic

eld, also known as the linewidth. In this work, the use of this metric will imply a
single mode laser, unless stated otherwise.

Semiconductor lasers (SCLs), in the context of this work, are high-coherence light
sources constructed using direct band-gap semiconductor materials. These lasers can
be pumped either optically or electrically.

However, the main advantage of SCLs

is their ability to be pumped electrically using standard current sources.

For this

reason, in the remainder of this work I will assume electrical pumping whenever SCLs
are described. SCLs have many other advantages: they are lightweight, e
cient in
converting electrical energy into optical energy, and inexpensive, as well as have a
small footprint. They can also be easily integrated with other electronic components
to allow for control and manipulation of the laser
eld.

However, the coherence

properties of SCLs are usually inferior in comparison to other laser platforms, such
as
ber lasers, solid-state lasers or gas lasers. The linewidth of SCLs is historically
limited to the sub-MHz regime. The reasons for that will be described in detail in
Chapter 2.
Narrow-linewidth lasers have been used in many
elds.

High-coherence light

sources are so useful since there are many systems in which the narrow-linewidth
sources translate directly into high-resolution.

For example, in an interferometric

ranging experiment high coherence translates into superior depth resolution. In spectroscopy, narrow-linewidth operation results in higher spectral resolution. In atomic
clocks that are based on coherent population trapping (CPT), low-noise operation is
essential for good temporal resolution of the clock.
To understand the basis behind this linewidth-resolution relationship it is useful
to refer to a simple experiment: the Mach-Zehnder interferometer (MZI). Consider
a laser beam (assume perfect spatial coherence for simplicity) that is split into two
arms.

One arm is delayed by a time delay of

τ compared to the other, and the

two beams are then combined and detected on a photodetector (PD). Assume the
detected signal is measured for a time period much longer than the coherence time,
such that the signal Fourier-limited width is much narrower than the linewidth of the
laser. Neglecting amplitude noise, the complex laser
eld can be expressed as:

E(t) = Aej(ωt+δφ(t))

(1.1)

The power at the PD can be written as (assume equal power and polarization in both
arms):

A2
P (t) = |E(t) + E(t − τ )|2 =
(1 + cos (ωτ + δφ(t) − δφ(t − τ )))

(1.2)

If we assume for simplicity that the constant phase term is biased such that ωτ

2πm + π2 , where m is an integer, we get
A2
P (t) ≈
(1 + ∆φ(t, τ ))

(1.3)

∆φ(t, τ ) = δφ(t) − δφ(t − τ )

(1.4)

where we have de
ned

and assumed ∆φ(t, τ )
π . The power spectral density (PSD) of the stationary and
stochastic phase change term ∆φ is related to the PSD of the laser frequency noise

Wφ̇ by [78]:
W∆φ = Wφ̇

sin (ωτ /2)

(ω/2)2

(1.5)

Assuming a white frequency noise, the phase change can be found analytically by
integration [78]:

< ∆φ(t, τ )2 >= Wφ̇ τ

(1.6)

It can be seen from 1.6 and 1.5 that the noise at the PD output scales with the laser
frequency noise.

If, for example one wishes to detect small changes of the optical

delay τ , the laser's noise term will eventually limit the achievable resolution.

The

same line of arguments can be made for many other systems, relating laser linewidth
to resolution in the broader sense.

Implementation of such high resolution devices

using semiconductors can potentially bring these devices to the hand-held consumer
electronics industry, such as cell-phones.

(a)

(b)

Figure 1.1: Constellation diagrams. (a) Binary Amplitude Phase Shift Keying (b) 16
Quadrature Amplitude Modulation.

1.2 Laser sources for coherent communication
Semiconductor lasers have been driving the optical communication industry for several
decades.

Direct (intensity) modulation is the most common modulation scheme.

However, as the demand for data increases rapidly, phase modulation schemes gain
popularity. Among other advantages, phase modulation scheme can pack more bits
per symbol, hence increasing the data content e
ciency of the channel. The concept
of high data content per symbol is demonstrated in Figure 1.1. The bit error rate will
be a function of the phase noise accumulated within one modulation period [134] and
the distance between two consecutive states on the phasor diagram. In the realm of
communication networks, this puts a linewidth requirement for the laser source, for
a given modulation scheme.
Another way to increase data rates is to increase the modulation speed. Going this
route would relax the linewidth requirement discussed above. This is due to the short
time period between consecutive symbols. As modulation speed increases, the phase
noise accumulated during one period decreases. This can be seen from Equation 1.6:
as τ decreases the RMS phase deviation ∆φ(t, τ ) decreases by the same factor. For
a given modulation scheme, i.e., 16-QAM, the system can tolerate higher laser phase

noise, wφ̇ , if τ is smaller. These two competing trends merit some discussion when
trying to evaluate the requirements from future optical communication light sources.
The increase in modulation speed is mostly driven by advances in the miniaturization of light modulators. Low capacitance is often required and sets a demand for
smaller footprints devices. On the other hand, the increase of data rate by packing
more bits per symbol is driven by advancements in digital signal processing (DSP).
Faster parallel electronics is needed to encode multi-bit signals. This in turn is also
driven by progress in high-speed electronics. These two market trends, faster modulators and faster DSPs, often go hand in hand.

In the context of light sources

for future optical communication networks, this means that the rate of increase in
demand for narrower linewidth lasers for telecom should stagnate somewhat due to
these competing trends.
The argument above implies that in the near future, it might not be necessary to
portray a fast path towards constant reduction of laser linewidth. However, today's
semiconductor lasers are incompatible with the transition to coherent communication,
requiring alternative solutions. For example [94], to transmit 64 bits per symbol at

−4
frequency of 20 GHz, while requiring Bit Error Rate (BER) of 10
and sensitivity
penalty of 2dB using feed-forward phase estimation techniques would require a laser
with linewidth of 1.2 KHz. Today's state of the art DFB lasers that are commonly
employed in communication networks are limited to a linewidth of about 100 KHz.
Therefore, an improvement of about 2 orders of magnitude in linewidth is required
to support near-future coherent communication.

1.3 Linear and non-linear performance limiting factors
The performance metrics for a laser is strongly dependent on the speci
c application.
For narrow linewidth lasers, the important metric is often the amount of phase noise at
the relevant frequency range. However, other metrics such as power output, side mode

suppression, and wall plug e
ciency are often as important, for practical reasons.
Laser linewidth, or noise, are fundamentally related to losses. Here I refer to loss
in the broader sense by de
ning it as a mechanism that couples useful energy
laser mode electromagnetic energy or pump electrical energy to a thermal bath
or a continuum of modes. For example, optical loss through scattering couples the
mode to the free-space continuum.

Dissipation through resistive heating couples

pump current to the thermal bath.

A very fundamental theorem, the
uctuation-

dissipation theorem, relates the dissipation through these channels to
uctuations in
the cavity. This theorem implies that the same mechanism that couples energy to the
outside world also couples
uctuation from the outside world to the cavity.
Despite the universal nature of the
uctuation-dissipation theorem, it is not very
useful for describing laser noise in a quantitative way. Text-book laser theory usually
attacks noise by engaging the speci
c physics of a laser system. Losses in the cavity
force us to pump the laser harder, such that gain overcomes loss. This in turn will
yield larger density of excited electrons. These excited electrons will relax randomly
to the ground state, adding random phase to the laser
eld through the process of
spontaneous emission. In this case, the relaxation of excited electrons to the ground
state is due to the interaction with the vacuum state corresponding to the laser mode
and not the free-space continuum. This approach can be shown to be directly related
to the the
uctuation-dissipation theorem [52].
In addition to
uctuations, losses limit the amount of energy we can store in the
cavity. The addition of random phase through spontaneous emission will add noise
to the laser
eld, and will have bigger impact if there are fewer stored photons. This
is demonstrated in the phasor diagram of Figure 1.2. No matter which approach or
argument is used to calculate the exact noise characteristics, it is obvious that loss is
a fundamentally root cause of noise or
uctuations. We will divide loss mechanisms
into two categories:

1. Linear loss: this is the most common and basic loss mechanism.

In general,

Figure 1.2: Phasor diagram demonstrating the eect of a spontaneous emission event

linear loss can be described through the equation:

dI
= −αI
dt

(1.7)

where I is a generic quantity, e.g., photon density or EM energy. The decay in
this case is exponential with a rate that depends only on the constant α.

2. Non-linear loss:

this loss refers to a mechanism in which the decay rate

depends on the quantity I through some non-constant function:

dI
= −α(I) · I
dt

(1.8)

In the context of this work, non-linear loss refers to optical losses that depend
on the photon density itself.

Despite the more complex nature of this loss

mechanism, the same arguments implied in the
uctuation-dissipation theorem
are valid, and non-linear loss will have an impact on laser coherence.

In Chapter 2 I will analyze linear loss and will describe a strategy to decrease it for
narrow-linewidth operation. We will see in chapter 3 that low-loss high-Q operation
will yield an increase of non-linear loss. In Chapters 3-9 I will analyze the eect of
these non-linear processes on noise performance and other important metrics.

Chapter 2
Hybrid Si/III-V as a platform for
narrow linewidth
Semiconductor lasers are notorious for their low coherence properties.

Despite the

tremendous progress in micro fabrication and wafer growth this limit has hardly been
broken. Some of the more successful attempts include a 64 KHz laser [32], a 28 KHz
laser [38], and a 3.6 KHz laser [71]. Common to these results is the implementation of
a large-mode-area, small con
nement factors, and low-loss cavities. In [71] linewidth
improvement was achieved also by suppression of spectral hole burning.
As we shall see in this chapter a key requirement for linewidth suppression is
the reduction of modal loss. However, semiconductor lasers based on common III-V
epistructures require high density of dopants and carriers to maintain a low resistance
current path for carrier injection into the quantum wells (QW). The high carrier density interacts with the mode and induces extra loss through radiation and dissipation.
The fact that the III-V-only platform requires current path through the same volume
in which the optical mode is stored renders this platform inconsistent with low-loss.
Even though some improvement can be made for III-V lasers by decreasing QW
con
nement and reducing losses elsewhere, this platform is inherently limited.
In science and technology, a big milestone is often achieved only after improvements are made on the material-science front. New materials can overcome barriers
that earlier were considered a fundamental limit.

As we shall see in this chapter,

the developement of a new platform at the University of California, Santa Barbara

10
(UCSB), the hybrid Si/III-V platform [75] opened up a path to narrow linewidth semiconductor lasers. This path was followed by researchers from Caltech [88], demonstrating 18 KHz lasers.

The work by Yariv et al.

further linewidth reduction on this platform.
were inconclusive .

has portrayed a path towards

However, experimental results [106]

In this work I will provide theory describing the limits of per-

formance of this platform and conclusive experimental results supporting the theory
and demonstrating record-breaking performance.

2.1 Noise in conventional semiconductor lasers
There is a vast amount of research on the physics of noise in semiconductor lasers.
[131] attacks the problem from fundamental quantum mechanics, [13] has a nice discussion on some phenomenological aspects of noise analysis, [78] have a detailed discussion on characterization and measurement of noise in lasers. Detailed derivations
of semiconductor noise is outside the scope of this work. However, it is important to
understand a few key concepts and results.
The electromagnetic laser
eld has magnitude and phase. In general, noise can
appear in both.

In common laser systems amplitude noise is vastly suppressed by

gain saturation, providing a restoring force for the laser amplitude. In fact, in many
practical laser systems, amplitude noise is at or very close to the shot-noise limit. On
the other hand, the phase of the electromagnetic
eld doesn't have the same restoring
force. Laser phase is
uctuating in a random walk process. For this reason, for most
applications laser noise is dominated by phase noise. Thus, in this work I will refer
to phase noise as laser noise, unless otherwise stated.
Laser frequency noise spectrum has several components. Equation 2.1 can be used
to describe it in a general way:

Wφ̇ =

ai
i=1

(2.1)

The
rst sum in Equation 2.1 is often called one over f noise. The source of these

11
noise terms can be traced to several mechanisms. Some of them tend to be technical
(e.g., mechanical vibration of the stage), and some of them are more fundamental
(e.g., thermal diusion). A nice review of generic 1/f noise can be found in [66]. Due
to the illusive nature of these 1/f noise terms they are often treated as technical
noise. In this work I will use the same notation even though in some cases the source
of this noise could be fundamental. The constant C of the second term in Equation
2.1 is referred to as Schawlow-Townes white noise
oor. In most semiconductor lasers,
this term becomes dominant over 1/f terms at frequencies as low as few KHz. For this
reason, laser noise analysis often emphasizes this white noise
oor level. The ideas
and strategies laid out in this chapter will focus on the reduction of this dominant
term.
The analysis of white noise in semiconductor lasers can be laid out in several steps:

1. The main source of noise in SCLs is spontaneous emission. This refers to the
stochastic radiative decay of excited electrons to the ground state. Every spontaneous emission event adds on the average one photon to the laser mode [31].
The
eld of this photon has a random phase compared to the laser coherent

eld and is thus source of phase
uctuations.

The lack of restoring force for

the laser's phase means that over time the phase will go through a random
walk process. The mean square of the phase deviation will be proportional to
observation time τ . If the frequency noise is white (setting ai = 0 in Equation
2.1 ) the mean square of the phase deviation will follow Equation 1.6.

2. Above laser threshold the modal gain of the laser is clamped at the modal loss
level.

A gain term that is exceeding the losses would imply the unphysical

situation of diverging exponential photon density. In fact, due to spontaneous
emission, the gain is only approaching losses from below; however, for any
practical purposes we can set them as equal:

ΓGm (n, ν) = α

(2.2)

12
Here Gm is material gain in units of sec

−1

and is a function of both the las-

ing frequency ν and the density of excited electrons n .

Γ is the con
nement

coe
cient that describes the con
nement of optical energy in the active (gain)
region, such that ΓGm is the modal gain. α describes modal loss in inverse time
units.

This includes intrinsic loss due to absorption and scattering, but also

external mirror loss.

3. There is a fundamental relationship between stimulated and spontaneous emission. The gain, or stimulated emission rate, depends on the level of population
inversion, i.e., the dierence between the density of excited electrons and ground
state electrons. The spontaneous emission process depends only on that of the
number of excited electrons. The inversion level and the density of excited carriers are tied together using the quasi Fermi levels of the pumped semiconductor
[13]:

Rsp = ΓGm nsp

(2.3)

Where Rsp is the spontaneous emission rate into the mode (photon number per
unit time) and nsp is the population inversion factor de
ned as:

nsp =

1 − e(E21 −∆Ef )/KT

(2.4)

where E21 is the bandgap energy and ∆Ef is the dierence between the quasi
Fermi levels.

The value of this population inversion factor is on the order of

1, but can quickly diverge in cases where the laser is operated very close to
transparency (E21 ≈ ∆Ef ).

4. Modal loss α can be expressed using the cavity/mode quality factor Q:

(2.5)

5. Laser white noise level due to spontaneous emission can be expressed using the
spontaneous emission rate Rsp and the total number of stored photons in the

13
cavity Np [31, 131, 13]:

Wφ̇ =

Rsp
2Np

(2.6)

6. The photon storage capability of the resonator is a function of its quality factor.
At threshold, the stimulated emission rate will reach the loss rate to maintain
steady state. From Equation 2.5 it is clear that this rate is de
ned by the quality
factor. Above threshold, each added photon due to injection will contribute to
the growth of photon number in the cavity in a rate set by the
xed gain:

Np = ηi

(I − Ith ) Q

(2.7)

where q is the electron charge and ηi is the injection internal e
ciency and Ith
is the threshold current.

7. The linewidth is enhanced beyond the quantum limit due to relationship between the imaginary and real parts of the refractive index [31, 116]. This relationship exists in semiconductors due to the non-symmetrical gain spectrum.
As discussed previously, photon number
uctuation induces gain
uctuations.
These in turn cause
uctuations in the refractive index that induce phase noise.
The eect of that process is a broadening of the Schawlow-Townes linewidth by

a factor of (1 + α ), where α is a material-dependent parameter usually in the
range of 2-5 [107].

8. Putting together Equations 2.2, 2.3, 2.5, 2.6, and 2.7 we get the famous modi
ed
Schawlow-Townes formula that relates laser white noise PSD to the inverse of
the square of the quality factor and the inverse of the output power:

Wφ̇ =

qω 2 nsp
(1 + α2 )
2Q (I − Ith )ηi

(2.8)

Equation 2.8 shows that increasing the quality factor of a resonator would yield a
power-law reduction of noise. As we shall see in the next sections, this can be done
in the hybrid Si/III-V platform. Moreover, this platform will lend itself towards the

14
eective elimination of the linewidth enhancement factor.

2.2 Hybrid Si/III-V platform
In the past decade the amount of academic research on silicon photonics devices and
systems has grown rapidly.

Silicon-based devices were demonstrated commercially,

and much industrial eort has been devoted to this platform. This rapid growth was
mainly motivated by several elements:

• CMOS compatibility - Realizing complex photonic integrated circuits using mature CMOS processes has the potential to considerably decrease cost and to
increase performance while maintaining scalability.

• Integration with silicon-based electronics - Integration of optoelectronic devices
with high-speed electronics can yield unprecedented control and manipulation
capabilities of light in devices and systems.

• The high index contrast of Si can yield small footprint devices and support
further miniaturization of standard devices.

• Silicon is transparent at telecom wavelengths - The low absorption loss together
with reduced scattering thanks to advanced processing capabilities can yield
very low-loss waveguides and devices.

The typical loss
gure is around 1-3

dB/cm; however, devices with losses as low as 0.3 dB/cm and lower have been
demonstrated [9].

• CMOS compatible integration of Germanium with Si - The integration of Ge
(strained [2] or SiGe [72]) for light detection is crucial for many applications.

The explosion of this
eld in recent years was followed by an attempt to integrate
laser light sources on to this platform.

Despite some success in monolithic growth

of active material on Si [64, 124, 10] , the most common technology for laser-Si
integration is the hybrid Si/III-V bonded platform. In this technology Si is patterned

Figure 2.1: Hybrid Si/III-V laser schematics

using standard nanofabrication techniques and then a bare III-V chip is bonded to
it. The bonded Si/III-V acts as a composite material, where the Si devices are an
integral part of the active device. Bonding was demonstrated using direct bonding
[75], adhesive bonding [42], or fusion bonding [110]. After bonding, the III-V stack
is patterned and processed to create a current injection mechanism to the device's
speci
c location. An elaborate explanation of the fabrication technique is found in
appendix A. A cross-section schematic of a hybrid Si/III-V laser is shown in Figure
2.1
Hybrid Si/III-V lasers have been demonstrated by several groups [76, 105, 109].
Early work [132] on hybrid lasers attempted to maximize optical gain by pulling the
mode to the III-V's quantum wells for ampli
cation, and pushing it back to Si for
useful output in Si waveguides. Even today, many groups still utilize this strategy.
However, it was recently demonstrated [88] that the opposite approach minimizing
overlap with III-V and utilizing the low-loss Si platform yields lasers with superior
performance.

2.3 Noise reduction in hybrid Si/III-V
In terms of coherence, the hybrid platform has many advantages over conventional
III-V lasers. Silicon is a low-loss material for light of frequency above its bandgap.
Linear absorption of light at telecom wavelength is practically negligible. Advanced
silicon fabrication techniques together with a well-behaved silicon oxide allow for the
fabrication of waveguides with minimal sidewall roughness, yielding reduced scattering losses. The hybrid structure is constructed such, that no current
ows through
silicon so that carriers and dopants don't have to be introduced into silicon to support
the current
ow. We shall see in the next sections that all these properties can be
exploited for narrow-linewidth through careful design of the laser.

2.3.1 High-Q silicon resonator
As can be inferred from Equation 2.8, the
rst building block of a high-coherence
laser is a high-Q resonator. Passive micro Si resonators with quality factors close to
or bigger than a million were reported in the literature [25, 88] .

These integrated

Si-based implementations can yield small cavity sizes, compared to those used in
commercial narrow-linewidth lasers.
Several resonator design topologies are available for high-Q. Common to all of
them is the minimization of scattering through optimization of sidewall roughness
and disorder. This is often done by designing waveguides that are weakly guiding,
such that the mode is minimally interacting with the patterned core.

One of the

most common high-Q resonator designs is the ring [113] or disc resonator.
the disadvantages of such a design is its large footprint.

One of

While the quality factor

scales with the diameter of the ring, its area scales with the diameter squared. Since
expensive III-V gain media should have a similar footprint, this strategy can quickly
become prohibitively expensive.

Two-dimensional photonic crystals were also uti-

lized to demonstrate extremely high-Q cavities [95]. However, the commercialization
viability of such technology is questionable.
One-dimensional photonic crystal oers good compromise in regards to footprint,

Figure 2.2: Band structure of the mode-gap resonator. The parabolic potential-well
supports one optical mode

fabrication and design complexity and high-Q. Quality factors as high as 1.1 million
were demonstrated [88, 46]. These inline resonators can easily be converted into dense
arrays of lasers, suitable for applications, such as phase array and wavelength-division
multiplexing (WDM).
In this work, we implement a mode gap resonator design, in which a 1-D grating
is patterned.

The grating has constant pitch, however, the dimensions of grating

elements vary along the resonator length. The coupling coe
cient κ is chirped such
that the resulting photonic bandgap has a
nite parabolic potential well. This well
is designed such that only one mode is supported [100] . The photonic band diagram
with the supported mode can be seen in Figure 2.2. Details on the exact structure
and design methodology can be found in [89] and in [106].

2.3.2 Modal gain and loss
The availability of high-Q resonators on silicon does not immediately guarantee a
narrow-linewidth laser. When the III-V is bonded to the high-Q resonator, it becomes
an integral part of the resonator's structure. Losses induced by the bonded III-V can
considerably lower the composite total Q and eliminate any bene
t from the the low-

18
loss silicon. For that reason, it is important to engineer the structure such that the

composite resonator is still high-Q, while supporting laser operation. In this section,
I will describe such a design methodology. A detailed analysis of such an approach
can be found in [106].

2.3.2.1 General description
Common doped III-V stacks have intrinsic losses with equivalent quality factor of

QIII−V = 104 [13].

The silicon resonators in this work may have unloaded Q as

high as QSi = 10 . The composite bonded structure will have quality factors in the
range between these two extremes, depending on the exact modal composition. Total
modal loss, which is related to the imaginary part of the eective refractive index,
can be well estimated by using a weighted sum of material absorbance weighted by
the con
nement factor:

αmode =

αi Γi

(2.9)

For TE modes, as in our case, the appropriate approximation can be written as [121]:

Γi =

ni (r)|E(r)|2 d3 r
nef f all |E(r)|2 d3 r

(2.10)

where nef f is the mode's eective index (real part), and the index i represents the
desired material or region. Ignoring mirror loss for output coupling, we can express
the total Q using:

ΓIII−V
1 − ΓIII−V
QIII−V
QSi
A graphic representation of Equation 2.11 is shown in Figure 2.3.

(2.11)

It is evident from

Figure 2.3 (a) that one can reduce losses by reducing the overlap with the lossy III-V
material, a process shown schematically in 2.3(b-c). In the high III-V con
nement
regime, the reduction in loss is a quasi-linear function of the con
nement factor. At
around the point where

ΓIII−V
QIII−V

= 1−ΓQIII−V
the total quality factor becomes sub-linear
Si

and eventually saturates at QSi .
The modal gain can also be expressed using the same overlap integral as in Equa-

(a)

(b)

(c)

Figure 2.3: (a) Quality factor of hybrid Si/III-V composite resonator (b) Mode pro
le
of a traditional III-V laser (c) Pro
le of a high-Q hybrid Si/III-V laser

20
tion 2.10. Since gain is only present at the quantum wells, we can express the gain
(sec

−1

) G using:

G = ΓQW Gm

(2.12)

It is worth noting that even as the mode composition changes, i.e., is pushed down
to Si, the ratio between ΓIII−V and ΓQW stays constant for any practical purpose, as
shown in Figure 2.4.
To reach lasing the gain only needs to compensate for the loss, as is evident from
Equation 2.2. Since the modal gain is a linear function of ΓIII−V , in the regime where
the loss is quasi-linear (Q <

Q ), the gain is reduced by the exact same amount
2 Si

that the loss is reduced and the threshold current remains nearly a constant as ΓQW
is reduced, yet the attendant reduction in loss translates to high coherence as implied
from Equation 2.8.

2.3.2.2 The spacer lasers
Reducing the overlap of the mode with the III-V requires a physical mechanism to
push the mode further down into the silicon.
SiO2 spacer layer [106].

In this work, this is achieved using

The oxide is thermally grown on the Si device layer and

separates between the high index silicon and the high index III-V. The thicker this
oxide separation layer, the further the mode is pushed into Si. This is a very e
cient
method for modal engineering.

Small changes in oxide thickness in the order of

10's of nanometers can change the con
nement factor by orders of magnitude. For
comparison, the size of an external cavity laser would have to scale by the required
reduction in active con
nement, yielding devices larger by orders of magnitude. The
spacer transverse modal control allows us to achieve the same eect with very little
compromise in footprint (the device might need to be slightly longer if the grating
is weaker due to the spacer). The eect of spacer thickness on III-V con
nement is
shown in Figure 2.4.

Figure 2.4: Con
nement in III-V and in QWs vs. spacer thickness

2.3.3 Schawlow-Townes linewidth
The

dependance of the modi
ed Schawlow-Townes linewidth formula of Equation
Q2

2.8 suggests that high quality factor will yield narrow linewidth. As we decrease losses
further and further, the excited carrier density at threshold approaches the density
required to reach transparency. Operating the laser close to the transparency point
will be manifested in Equation 2.8 as very large population inversion factor nsp . At
this regime, the population inversion factor is very sensitive to changes in threshold
gain, and we can no longer treat nsp as constant. Since operating close to transparency
is realistic and even desired in our case, I will introduce an approximation for nsp that
is more suited for this regime.
The spontaneous emission rate into the mode can be expressed using the phenomenological expression:

ΓQW βsp N2
RSP =
τsp

(2.13)

where N2 is number of excited electrons, τsp is the spontaneous radiative decay time of

excited electrons into all modes, and βsp is the spontaneous emission factor. ΓQW βsp
describes the fraction of photon that is coupled to the lasing mode out of all modes

(notice that some textbooks de
ne that as βsp ).

The quantity

βsp
is material and
τsp

22
structure dependent.

Very small changes in spacer thickness, which hardly change

the cavity volume, are not expected to aect this quantity. The number of excited
electrons can be broken into carriers needed to get to transparency and carriers needed
to overcome loss:

N2 = Ntr + (N2 − Ntr )

(2.14)

We can then use a linearized expression for the gain (which is more accurate the closer
we get to transparency):

G = ΓQW Gm (N2 − Ntr )

(2.15)

where Gm is the material dierential gain. We can now use Equations 2.6, 2.7, 2.13,
2.14, and 2.15 to express the linewidth using:

βsp
qω
Wφ̇ =
τsp (I − Ith )ηi

Ntr
ΓQW
+ 0 2
Gm Q

(2.16)

Equation 2.16 implies some important conclusions:

• Far from transparency, where the second term in the bracket is dominant, increasing the quality factor Q yields a square power law improvement in linewidth.
This is because the reduction in needed threshold gain lowers spontaneous emission rates, and because the photon density increases with increasing Q.

• As we get closer to transparency by increasing Q the square law improvement
saturates into a linear improvement, as the
rst term in the bracket becomes
the dominant one.

This is because the spontaneous emission rate is now ap-

proximately a constant, set by the transparency carrier density, and only the
photon number, NP , increases with increasing Q.

• A reduction in ΓQW will lower the spontaneous emission rate into the mode.
The total spontaneous emission rate to all modes is constant, but less will be
coupled to the lasing mode due to the weak interaction of the quantum well
with this mode.

• In the hybrid platform, ΓQW is our knob to reduce Q (see Equation 2.11).

23
Equation 2.16 teaches us that this strategy guarantees a

dependance, both
Γ2

close and far from transparency, as long as we have not saturated the total Q.

In this chapter, I discussed noise in conventional lasers, and introduced hybrid
Si/III-V as a platform for low-noise lasers. The spacer laser design was introduced
as a simple mechanism to push noise performance to the limit of the platform. In
the next chapter, I will describe the nonlinear eects that may be responsible for this
limit.

Chapter 3
Non-linear eects in hybrid Si/III-V
As described in section 2.3 moving the energy of the optical mode (pushing it) into
low-loss silicon resonator yields lasers with a very high-Q lasing mode. This mode
exists mostly in silicon and interacts very weakly with the active (and lossy) III-V.
The high intra-cavity
eld intensities that are a consequence of the high-Q and the
small cavity volume enhance the probability of non-linear and multi-photon processes
in silicon.
Si has a large non-linear

χ(3) coe
cient compared to commonly used low-loss

materials such as SiO2 or Si3 N4 .

Both Raman scattering and optical Kerr eects

have been reported in silicon. Brillouin scattering, which is often dominant in optical

bers, is much weaker than the Raman process in silicon, and is rarely observed [97].
Raman scattering is a non-linear process involving interaction with optical phonon.
The Raman gain in silicon has a narrow linewidth response (∼100GHz) centered at
a frequency of 15.6 THz (at room temperature) [29]. Silicon Raman lasers have been
demonstrated by several authors, both pulsed [7] and CW [84]. Cascaded Si Raman
lasers have also been demonstrated [86]. However, the single mode operation of our
hybrid Si/III-V lasers and the narrow gain bandwidth of the quantum wells relative
to the Raman frequency eliminates (
rst order) Raman scattering from interfering
with the laser operation.
Kerr non-linearity involves light intensity-dependent alterations to the refractive index, both real and imaginary parts. Writing the third-order polarization for

Figure 3.1: Schematic description of two-photon-absorption

monochromatic light E(t) = cos(ωt) as [6]:

P (3) (t) = χ(3) E(t)3 = χ(3) 3 cos(3ωt) + χ(3) 3 cos(ωt)

(3.1)

The second term of Equation 3.1 represents a contribution to polarization at the same
frequency of light and hence an intensity-dependent refractive index. The real part
of the refractive index can be expressed as:

n = n0 + n2 I

(3.2)

where n0 is the linear (low-intensity) refractive index and I is the light intensity, such
that [6]

n2 =
The imaginary part of χ

(3)

12π 2 (3)
Re χ
n20 c

(3.3)

can phenomenologically describe two-photon-absorption

(TPA) , a process that is schematically described in Figure 3.1. The total attenuation
in the presence of the nonlinearities now has the form [30]:

dI(z)
= −αI(z) − βT I 2 (z)
dz

(3.4)

where α is regular linear attenuation and βT is the so-called two-photon-absorption
coe
cient.

26
Si has a bandgap that corresponds to a wavelength of 1.1µm.

Therefore TPA,

which is wavelength dependent, can occur at wavelengths of 1.12.2µm, and is around
its peak at telecom wavelengths. In the hybrid Si/III-V laser, the
eld intensity builds
up in the presence of gain and the high-Q resonator. As a consequence, non-linear
processes such as TPA add extra loss [4] that would otherwise be negligible in the lowintensity passive version of the same resonator. TPA is a non-resonant fast process. Its
time-constant is considerably smaller than any time-constant of interest in our system.
We therefore treat it as instantaneous. As depicted in Figure 3.1 the absorption of
two cavity photons conserves energy by exciting an electron to the conduction band.
Since silicon is not a direct bandgap semiconductor, the electron is likely to recombine
non-radiatively. As we shall see in the next sections, the conduction-band electrons
can live quite a long time in the excited states before they recombine or diuse away
from the mode, and act as free carriers that can interact with the mode and add extra
loss through Free-Carrier-Absorption (FCA) processes.
In this chapter I analyze phenomena related to non-linear loss mechanisms, such
as TPA and FCA. In the following chapter I will analyze the eect of these processes
on the high-Q hybrid Si/III-V lasers. A thorough review of non-linear eects in Si
can also be found in [58].

3.1 Two-photon-absorption in silicon
Kerr nonlinearities impact the functionality of devices in many systems.

In some

cases these nonlinear eects have been used to create new functionality. Among other
applications, the Kerr eect has been used to demonstrate all-optical modulation [69],
continuum generation [73], pulse compression, and mode-locking [112]. In the context
of laser noise TPA has been used to suppress relative intensity noise (RIN) [3], and
to generate photon-number squeezed-light [36].
In other cases these nonlinearities have degraded or impacted performance.
silicon TPA often induce free carriers.

In many examples of performance limiting

nonlinearities it is those free carriers that are responsible for performance degradation.

27
The next section is devoted to FCA and some examples will be given there. In this
section we will discuss only the physics of two-photon-absorption.
As has been discussed in previous sections TPA is manifested as the imaginary
part of the index of refraction, and is non-zero whenever the photon's energy is larger
than half the band-gap of silicon, as in the case of telecom wavelengths. The coe
cient

βT relates the change of the imaginary part of the refractive index ∆ni to the
eld's
intensity I using [58]:

∆ni =

βT I
2ω

(3.5)

The coe
cient βT is wavelength-dependent, and its value at 1.55µm has been measured by several authors. Femto-second pulses and balanced Z-scan technique have
been used to measure TPA coe
cient experimentally.

An empiric value of βT

cm
0.8 GW
has been obtained [17] at 1.54µm using that technique.

Similar value was

measured by [8]. Experimental data from nonlinear power transmission was also used
to extract a value of βT = 0.45

cm
at the same wavelength [114]. Evidence of crystal
GW

anisotropy was also found for the third order susceptibility tensor, where a factor of
2.36 between tensor components was obtained [136]. In this work I will use the value:

βT = 0.8

cm
GW

(3.6)

Optical loss induced by two-photon absorption in Si can be described by a nonlinear rate equation term for the average photon density np :

dnp
= −βT hνvg2 Γ2Si n2p
dt

(3.7)

where hν is the photon energy, vg the group velocity, and ΓSi is the con
nement factor
in Si, de
ned as in Equation 2.10 (with layer i being the Si layer).

The nonlinear

nature of TPA is manifested in this equation by the np square power low in the right
half side.

3.2 Free-carrier-absorption
For each TPA absorption event in Si, an electron-hole pair is generated. Long carrier recombination life time of conduction-band electrons in intrinsic silicon allows
these carriers to accumulate and interact with the electromagnetic
eld. This interaction induces both extra loss through FCA and plasma-eect-related refractive index
changes.
TPA-induced free-carrier-absorption was shown to alter and degrade the performance of many devices. In high-Q resonators, free-carriers have caused self-induced
modulation of the transmitted light [37]. In Raman lasers and ampli
ers, free-carriers
alter the Raman gain and induce loss [85]. This loss has limited the available pump
power in Raman lasers [57].
Free-carriers recombine and diuse through a time-dependent process. Therefore,
Unlike TPA, which is considered instantaneous, the interaction with free-carriers often
adds some frequency response that would alter the dynamics of devices.

FCA has

been found to limit switching speeds in all-optical switching schemes [73]. It has been
found to broaden intense pulses [112] and to alter spectrum of self-phase-modulation
experiments in Si [133].
The electron and hole momentum relaxation times are longer than that of an
optical cycle. Therefore, carriers can easily oscillate at optical frequencies. Optical
energy will be attenuated through both radiation and heat.

The strength of these

damping eects can be well modeled using the Drude model [101] where the loss
coe
cient per unit distance αF CA (as in Equation 3.4) and refractive index change

nF CA can be directly related to the electron and hole concentrations nSi and pSi ,
respectively.
There have been empirical measurements of absorption in doped p-type [92] and ntype [104, 92] Si. Comparison of the empirical results to the Drude model predictions
was done by [101] and reviewed by [58]. The Drude model predictions were found to
underestimate absorption by about a factor of two. [58] derived an empirical formula
for free-carrier-absorption. For intrinsic silicon at 1.55µm the cross-section parameter

Figure 3.2: Model for FCA in silicon. Model #1 by [58, 101]. Model #2 by [70]

[58] σa is used to relate FCA loss (per unit length) to electron concentration nSi :

(z)

αF CA = σa nSi

(3.8)

−3
Where nSi is the electron and hole concentration in units of cm
and the value of σa
is given by:

σa = 1.45 · 10−17 cm2

(3.9)

A more recent attempt to characterize FCA in Si was done by [70]. For wavelength
of 1.55µm the following formula was obtained for electron and hole concentration ne
and nh , respectively:

(z)

+ 5.84 · 10−20 · n1.109
αF CA = 8.88 · 10−21 · n1.167

(3.10)

A graphic comparison between these two models is shown in Figure 3.2.
As can be seen from Figure 3.2, the two models are not drastically dierent,
especially at the high carrier concentrations. In this work, I will use the model given
by Equation 3.8.

Figure 3.3: Waveguide structure used for analysis of FCA

3.2.1 General methodology
TPA-induced free carriers are generated instantaneously. Once generated, they can
recombine at the bulk of the silicon or at its surface. The generated carrier density
will vary along the waveguide since TPA depends on light intensity. More electrons
will be generated at the peak of the mode than at its tail. Therefore, free carriers will
also undergo a diusion process. To properly model the optical loss due to FCA we
will need to estimate how many carriers interact with the mode in the steady state,
and what is their temporal response.
To estimate the number of free electrons in the steady state operation of the laser
we need to solve the diusion-recombination-generation equation. Since recombination at the bulk is a rather slow process in intrinsic silicon, surface recombination
is expected to be dominant.

The magnitude of the eect of surface recombination

will depend on the speci
c waveguide geometry. For example, if the mode is located
far away from any surface or interface then surface recombination will play a smaller
role. For that reason, we need to take into account the exact waveguide geometry to
account for surface recombination properly. The model I will use for our analysis is
as depicted in Figure 3.3. Where the Si slab thickness is 500nm and the width of the
waveguide is 2µm.

3.2.2 Bulk recombination
Several physical processes are responsible for the carrier recombination at the bulk of
the silicon (away from the surfaces and interfaces). The lifetime of an electron-hole

31
pair can be expressed using a contribution from all these processes as [33]:

τB
τSRH τAug τrad

(3.11)

where τSRH is lifetime associated to ShockleyRead Hall (SRH) recombination, τAug
is associated to Auger-type recombination, and τrad is radiative recombination lifetime.
Radiative recombination, where the annihilation of an electron-hole pair is accompanied by generation of a photon, can be modeled using the rate (per unit volume):

Rrad = Bn2

(3.12)

−14 cm3
where B ≈ 1 · 10
[22], and n is the conduction band electron density. Since
sec

silicon is a non-direct bandgap material, radiative recombination is usually negligible
compared to other processes.
The Auger recombination is a three-particle collision process, where the energy
of the electron-hole recombination is transferred to a third electron or hole.

The

third electron or hole is then promoted to a higher energy in the same band, and
relaxes back to the band edge by thermalization.

The parametrization of Auger

recombination was done by [40]. An approximate expression can be used for Auger
recombination time constant [33] :

τAug

= γn2

(3.13)

where the parameter γ is carrier-concentration dependent:

γ = γmin +

γmax − γmin
1 + nref

(3.14)

−31 cm6
−30 cm6
17
−3
where γmin = 4 · 10
, γmax = 1.35 · 10
, and nref = 7 · 10 cm . As evident
sec

sec

from Equation 3.13, the Auger recombination rate is higher at high carrier-densities.
ShockleyReadHall recombination refers to recombination through a trap in the

32
electronic bandgap [98]. For low carrier concentration this is the dominant recombination process. The rate of recombination can be described using the expression[98]:

RSRH =

n · p − n2i
τp (n + n1 ) + τn (p + p1 )

(3.15)

where n1 , p1 are auxiliary variables that depend on the trap energy level, τp,n is a time
constant related to the density of traps and temperature [21], and ni is the equilibrium

p · n product. As we shall see in this chapter, the bulk recombination lifetime is of
secondary importance in our devices. The eective lifetime in our rib structure will
be dominated by surface recombination, and will be in the range of 1-50ns.

The

bulk recombination lifetime is typically in the range of tens of µsec [16] to as high
as 30msec [41]. For that reason the exact form and values in our approximations are
of little to no consequence. I will therefore use a simpler approximation for the SRH
recombination lifetime:

τSRH

n2 − n2i
2τt (n2 + n · ni )

(3.16)

where I estimate a fairly large eective time τt = 100µsec based on the assumption
of a high-resistance, low-defect-density silicon.

3.2.3 Surface recombination
Dangling bonds and defects at the surfaces and interfaces of the silicon slab act as
traps that promote recombination at the surface. The exact expression for the surface
recombination rate is very similar to the SRH recombination, except that the density
of traps is now expressed in two dimensions [93]. Those surfaces and interfaces act
as carrier sink, which deplete the carrier density at the surface.

Carriers from the

bulk will therefore diuse toward the low-concentration regime at the surface. The
rate at which carriers diuse toward the surface is characterized using the surface
recombination velocity S in units of

cm
sec

, such that the two-dimensional rate is given

2D
by RS = S · n [93].
The optical mode in our lasers is con
ned to a 500nm thick Si slab. Carriers are

33
generated through TPA in the thin Si slab and diuse quickly towards the surface
where they recombine via surface traps. The fact that the slab thickness is narrower
than the typical diusion length of Si allows us to approximate the carrier density
within the slab thickness as constant [16]. The recombination can be expressed using

a three-dimensional rate (

cm−3
sec

) by distributing the two-dimensional recombination

process at the surface to the entire slab thickness using[16]:

RS =

(3.17)

where H is the Si slab thickness as in Figure 3.3 and n is the excited carrier density
in Si. This approximation neglects the spatial variation of the mode in the thin Si
slab.
The value of the surface recombination velocity can vary a great deal depending
on the quality of the Si surfaces and interfaces.

For non-passivated surfaces, the

velocity can reach several thousands [74]. Passivated Si wafers can have much lower
surface recombination velocities, but the exact conditions of the passivization play an
important role. Velocities of the order of 500-1800

cm
were reported for non-optimized
sec

oxidation processes [47]. For optimized oxidation, such as that commonly employed
in the fabrication of Si solar cells, velocities as low as 1-50

cm
are reported [41].
sec

Our lasers go through high temperature anneal and oxidation during the fabrication process. A very thick (up to 150nm) thermal oxide layer terminates the Si slab.
The oxide is further sealed with the III-V chip.

No direct measurement of surface

recombination velocity was performed prior to bonding and it is therefore di
cult to
predict the resulting surface recombination velocity. However, given historical data
from oxidized surfaces, a very low velocity will not be surprising. In later chapters,
I will use an indirect measurement to estimate the recombination velocity and will
show that its value is on the lower end of previously reported values.

3.2.4 Carrier diusion
Carriers in silicon interact with the optical mode and induce loss and phase changes.
Among the recombination processes described above, carriers that were generated
locally by TPA will also diuse to low carrier density areas. Therefore, the interaction
time of the mode with these carriers will be limited to the length of time they spend
in the vicinity of the mode. It is therefore important to add diusion to our model.
The diusion equation governing this process is given by:

dn
= D∇2 n
dt

(3.18)

where D is the diusion coe
cient for silicon. For high carrier-densities we will have
to use the ambipolar diusion coe
cient given by [87]:

2KT
Da =

µe µp

−1
(3.19)

where µe,p is the mobility of electrons/holes. When the carrier density is high, carriercarrier scattering will increase. This will reduce the mobility of the carriers further
at high densities. I will model the eect of carrier density on mobility using [39]:

−0.667
∆ne,p · ln(1 + 4.54 · 1011 ∆ne,p
= (0) +
µe,p
1.428 · 1020
µe,p

(3.20)

−3
where ∆ne,p is electron/hole density in [cm ] and the low-density mobility is taken
to be:

cm
µe = 1430
V·sec
cm
µ0p = 495
V·sec

(3.21)

(3.22)

The eect of carrier density on the diusion coe
cient of silicon is shown in Figure
3.4

Figure 3.4: Model for the ambipolar diusion coe
cient of silicon

3.2.5 Eective carrier lifetime
The optical mode is con
ned within the three-dimensional cavity. In the longitudinal
direction, the mode is designed to be a Gaussian with half-width in the order of
hundreds of microns. In the transverse direction, the mode is con
ned to only a few
microns. The orders of magnitude change in con
nement between longitudinal and
transverse direction motivates us to approximate the longitudinal direction as having
a uniform photon distribution.

The gradient in carrier density in the longitudinal

direction is much weaker than that of the transverse.

Therefore, diusion in the

longitudinal direction could be neglected. For the thin Si slab I can assume a uniform
carrier density since the diusion length is much larger than that thickness (y direction
in Figure 3.3). For that reason, I can approximate the whole problem using a 1-D
diusion equation in the x direction only.
The generation-recombination-diusion equation describing the dynamic of the
carriers in Si can be expressed as [16] :

βT hνΓ2Si Vg2 2 nSi
dnSi
d2 nSi
np −
− 2 nSi + Da
dt
τb
dx2

(3.23)

where the
rst term on RHS represents generation due to TPA; the second term rep-

Figure 3.5: Carrier density pro
le for several dierent surface recombination pro
les

resents bulk recombination; the third term represents surface recombination, and the
last term represents diusion. To gain more insight into the typical carrier distribution I will assume that the generation of carriers is due to TPA induced by an optical
mode with a Gaussian spatial pro
le:

x2
np,0
np = √ e− 2σ2
2π

(3.24)

where the width of the Gaussian is chosen to be equal to the waveguide ridge width

σ = 2µm (see Figure 3.3). np,0 would be the average photon density, if the mode were
uniform within the waist area (np,0 =
np (x)dx).
A numerical solution to Equation 3.23 is shown in Figure 3.5. A few conclusions
can be drawn from this
gure:

• Even for very fast surface recombination, the carrier pro
le is much broader than
the mode's pro
le. This will motivate us to approximate the carrier density as
uniform in the vicinity of the mode.

• Surface recombination is the main mechanism of recombination. Mode pro
le
depends almost solely on the surface recombination velocity.

• The steady-state carrier-density peak-height strongly depends on the surface

Figure 3.6: Eective lifetime of carrier in silicon for S=1

cm
sec

recombination velocity. However, as the surface recombination velocity becomes
slower and slower the carrier-density peak eventually saturates.

Once I have numerically solved this equation for a given photon density pro
le, I can
de
ne an eective lifetime for carriers in Si for the given optical mode:

τef f =
βT hνvg2 n2p

nSi (x)np (x)dx
np (x)dx

(3.25)

The parameter τef f represents the average time in which generated electrons interact
with the optical mode before they recombine at the surface or the bulk, or diuse
away from the mode's area.

A similar approach was taken by [16].

The eect of

surface recombination velocity on this time constant is shown in Figure 3.6. We can
identify three regimes in this
gure:

1. Low carrier densities - The eective lifetime is constant for a wide range of rather
low densities.

This corresponds to the
at area in the diusion coe
cient of

Figure 3.4.

2. Intermediate carrier densities - The eective lifetime increases. This is due to
carrier-carrier scattering that reduces the carrier mobility and hence slows down

Figure 3.7: Eective lifetime of carriers in Si (low photon excitation regime)

diusion.

3. High carrier densities - Eective lifetime reduces with density. This is due to
Auger recombination at the bulk, which becomes dominant at high densities.

This behavior also explains the saturation of peak density shown in Figure 3.5
Auger recombination processes limit the accumulation of carriers.
Finally, the eective lifetime of carriers in Si is shown in Figure 3.7 vs the surface
recombination velocity, for low photon density excitation. According to this
gure,
the expected lifetimes are in the range 130ns for reported values of recombination
velocities. Though the exact dynamics of carriers in Si is hard to monitor, the eective
lifetime is often much easier to measure, or can be estimated indirectly. For example,
eective lifetime of 25ns was estimated in Raman ampli
ers [85]. In that work FCA
associated with this relatively high time constant was shown to yield non-linear loss
that eected Raman gain at high pump levels. In a dierent structure, lifetimes in
the order of 10ns were observed in an all-optical modulation experiment [69], where
free-carriers have aected modulation speed and quality at high powers.
Photo-generated free-carriers have been shown to aect the performance of many
devices. The free carriers' eective lifetime, de
ned in this chapter, is a convenient

39
way to lump many physical processes into one time constant.

It was shown that

micron-scale devices with highly passivated surfaces, as in the case of our Si high-Q
laser-resonator, can have lifetimes in the order of tens of nanoseconds.
In this chapter I have laid the foundation for the consideration of nonlinear eects
in the context of high-Q Si/III-V lasers. In the next chapter, I will incorporate TPA
and FCA in a model describing our lasers. Later chapters will show that these eects
have a considerable impact on laser performance.

Chapter 4
Modi
ed rate equations
As described in the last chapter, TPA and FCA can impact laser performance and
dynamics. To investigate these eects, I will start with the classical rate equations.
These equations can be found in almost any laser physics textbook (i.e., [13, 131, 99]).
For the purpose of noise estimation, I shall use a spontaneous emission source term
in the rate equation. The origin of spontaneous emission is quantum mechanical in
nature; however, once an expression was derived using quantum theory, it is plugged
into the rate equation as a classical term.

The same is done for noise calculation:

quantum mechanical results are implemented in the framework of these classical rate
equations. Modi
cations to the classical rate equations will be done to account for
non-linear eects. Terms for TPA and FCA will be added to the rate equations, and
another equation for the density of free-carriers in silicon will be introduced.
This chapter will describe the general methodology and will construct, step by
step, the modi
ed rate equations. This will be the foundation for the laser performance analysis in later chapters.

4.1 Flat-mode approximation
Formal rate equations include spatial variation of the electromagnetic
eld, the electron density, and the semiconductor mesoscopic polarization.

The phase of both

the optical
eld and polarization should also be taken into account in a rigorous
study.

Therefore, at least
ve variables are required.

A system of such rate equa-

41
tions based on a density matrix approach was constructed and discussed by several
authors [111, 1, 129]. One of the
rst discussions, by C.L. Tang, has shown that in
the limit where the coherence time of the atomic system is much shorter than both
the photon lifetime in the cavity and the relaxation time of the atomic transition,
the classical rate equation is obtained and two equations are su
cient [111]. In semiconductor lasers it was shown that for very short pulses, such that the polarization
can't respond fast enough, the classical rate equations don't hold [129] (in fact, gain
and refractive index can't even be de
ned in such cases). Moreover, in semiconductor lasers, intra-band relaxation has to be assumed faster than other lifetimes in the
system in order to use the classical rate equations [5]. It is also the case in our lasers
that the optical mode and electron density pro
le vary along the resonator's volume.
However, the resonator is designed such that these spatial variations are slow to avoid
spatial hole burning, and thus local spikes, which might change the entire dynamics,
are not an issue in our lasers.
Though spatially-dependent density matrix formulation is possible in our system,
it will make the numerical analysis di
cult and might shadow some of the important
nonlinear physics.

Therefore, in order to numerically investigate non-linear laser-

dynamics, it is advantageous to simplify the model. To that end, I will neglect all
spatial variations and approximate our mode as constant within a rectangular box.
The size of the box will be equivalent to the full-width-half-max (FWHM) volume
of the mode, and the amplitude will be taken as the average amplitude within that
rectangle.
The electron density in the QWs will also be taken constant within a box. Since
I am only interested in the QW electrons that have signi
cant contribution to the
modal gain, I will choose the size of that box to correspond to the dimension of
the optical mode, in the x and z dimensions (referring to Figure 4.1).

The third

dimension (y) will represent the thickness of the QW layer, since the optical mode is
approximately
xed over nanometer scale.
The interaction strength between photons and electrons will be calculated using
the overlap integral for TE mode de
ned in Equation 2.10. This methodology, ex-

Figure 4.1: Flat mode approximation for the optical mode and the electron density

pressed graphically in Figure 4.1, will allow me to neglect any spatial pro
le of both
photons and electrons, while accounting for the interaction between the two.

It is

worth noting that in many textbooks the quantum wells are localized at the peak
of the mode. In these cases it is common to use the volume of the quantum well to
calculate the strength of the interaction. This approach is not valid in our case, since
the QWs are located at the far tail of the optical mode.

4.2 Working with densities or total numbers?
After neglecting any spatial dependence, one has to decide whether to express the
rate equations using the total numbers of photon/electrons or their density. There is
a certain elegance in expressing the rate equations in total number, as the equation
for photon and electron becomes symmetrical. For example, the rate of generation
(absorption) of the total number of photons Np in the box will be calculated using
the material gain (loss) Gm :

dNp
= ΓQW Gm Np
dt

(4.1)

Conservation of energy during radiative recombination of QW electrons and the quantum nature of photons will force the rate equation for the generation of total number

43
of electrons to be exactly the same (with an opposite sign):

dNe
= −ΓQW Gm Np
dt

(4.2)

Furthermore, one could associate
uctuations due to spontaneous emission as addition
of a single photon to the cavity [31], which will be easily evident in the total number
representation.
However, my main goal here is to consider the eect of non-linear eects. TPA,

(3)
and other χ
processes depend on the

intensity of light. For that reason, the size of

the box does matter. Furthermore, the magnitude of the gain term Gm will depend
on

density of QW electrons. For all these reasons, writing the rate equations in terms

of total numbers will be futile. On the other hand, some features that distinguish our
hybrid spacer lasers from conventional lasers will become more evident in the density
representation. I will therefore use the

density of photons/electrons to construct the

rate equations.

4.3 Pump
Our laser is electrically pumped through the contact pads shown in Figure 2.1. The
current
ow through the path is de
ned by the ion implantation mask. Fabrication
tolerances and non-optimized ion-implant processes will be manifested as leakage: a
fraction of the pump electrons will leak to areas of little interest (outside the mode
area).

Furthermore, the QWs themselves have some quantum e
ciency associated

with them. In this work, I will lump all factors that reduce the pump e
ciency to
one parameter: ηi . The rate of change of the density of QW electrons ne due to pump
current I will be given by:

dne
ηi I
dt
qVQW

(4.3)

where VQW is de
ned as the eective volume of the QW electron box (see Figure 4.1).

4.4 Linear loss

The linear loss in our laser is de
ned as the combination of intrinsic loss, dominated
by scattering, and mirror loss. It is convenient to describe the total loss α (in units
of sec

−1

) using the quality factor Q:

ΓIII−V
1 − ΓIII−V
= int
+ mirror
int
QIII−V
QSi
2πν
= photon decay rate [sec−1 ]
α=

(4.4)

(4.5)

int
int
Where QSi and QIII−V are the intrinsic losses in Si and in III-V, respectively and

Qmirror is the external mirror loss. Qint
Si is dominated by scattering loss and will
depend on the mode pro
le. I can therefore interpret the term
to sidewall scattering in Si experienced by the

1−ΓIII−V
as loss due
Qint
Si

composite hybrid Si/III-V mode.

For designs that mostly utilize Si, the intrinsic loss of the composite mode will be
very close to loss in the passive Si-only resonator. I therefore use the passive loss to

int
approximate QSi .
The output power will depend on how much light we couple as useful output
through the mirrors. If the mirrors are too strong, the intra-cavity intensity will be
higher, but the laser will be ine
cient since most of the loss is not due to useful
output. If the mirror loss is too high, the total loss will be dominated by it, and the
Q will be low. A good balance between output power and linewidth can be achieved
in the optimal coupling point, de
ned as the point in which [99]:

= mirror
Qc

(4.6)

At this point, the intrinsic loss is equal to the external loss. Beyond optimal coupling,
we can only increase the total Q by at most a factor of two, at the expense of a major
reduction in output power (approaching zero at the limit).

4.5 Gain

The gain for the laser is provided by the monolithically grown quantum wells in the
III-V. The amount of gain depends on the strength of the interaction between the carriers in the quantum wells and the lasing mode. The theoretical treatment describing
QW gain in the literature often has somewhat of a dissonance; the density of states
is calculated using quantum mechanics and is considered to be a two-dimensional
quantity, while the gain is often calculated using the con
nement factor, which assigns
nite volume to the quantum well. In the next sub-section, I will address this
ambiguity and describe my approach to expressing the modal gain.

4.5.1 The quantum-well: a two or three dimensional creature?
A `textbook' quantum well is usually described using a potential well of small dimension in one axis (i.e., z direction) with in
nite energy barriers. The k-vectors of the
resulting quantized electron wave-function will be sparsely spaced in the kz direction.
The energy levels in the well will also be discrete where each level corresponds to a
well-de
ned wave-function in the z-direction. Therefore in a QW laser, where most
transitions are from the ground state of the QW, the wave-function of the electrons in
the z-direction is well-de
ned, and the electrons behave like a two-dimensional electron gas. For this reason, a quantum well is inherently a two-dimensional creature.

unit
area ρa(2D) and the interaction with the optical mode will be restricted to a single
slice z0 . The modal gain will be proportional to the product of the density of state
If we treat it as two-dimensional, we should express the density of states per

and the intensity [77]:

G ∼ ρ(2D)
E 2 (z0 )

(4.7)

(2D)
where ρa
doesn't depend directly on the thickness of the QW WQW .
In reality, we are using a stack of several QWs (
ve in our case) and they don't have
in
nite potential barriers. The separation between them is large enough so that the
wave-function of the electron is approximately a linear combination of the separate

46
wells. Each wave-function will be spread out due to the
nite barrier size. In this case
the QW stack is no longer a well-de
ned two-dimensional creature, though we can still
treat it this way. Treating it as a three-dimensional object and assigning a volume to
it will not change any of the results. The density of states per

unit volume ρ(2D)

will be inversely proportional to WQW , but now the gain will be calculated using
an overlap integral between the mode and the electron wave-function, and will be
summed over all QWs in the volume.
It is a matter of choice how to represent the gain, either by using the
eld at a

point E (z0 ), or by using the overlap integral and the con
nement factor ΓQW . Doing
it with the latter provides more general results, because they could be easily converted
to a bulk gain medium, or quantum dots. This representation is also more common
in the literature. I will therefore adhere to the representation using the con
nement
factor.

4.5.2 Active con
nement factor
The rate of the total electron and photon number of Equations 4.1 and 4.2 can be
converted to densities by dividing the equation by the volume of the optical cavity
and the QW, respectively. Referring to Figure 4.1, it is useful to de
ne a geometrical
con
nement factor:

Γgeom =

WQW
Wm

(4.8)

The rate equation for the densities representing gain could be written as:

dnp
= ΓQW Gm (ne )np
dt
dne
ΓQW
=−
Gm (ne )np
dt
Γgeom

(4.9)
(4.10)

As expected, the representation using densities is no longer symmetrical.
lasers, the QWs are located exactly at the peak of the mode.

In most

In this case, the

47
con
nement factor can be well-approximated using the geometrical con
nement:

ΓQW
≈1
Γgeom

(4.11)

which is what most, if not all, textbooks show. The fact that this approximation is
no longer valid in our special case will have major consequences on the small signal
dynamics and on gain saturation.

4.5.3 Material gain
It is common to use the empirical formula for the material gain [13]:

Gm (ne ) = G0 ln

ne
ntr

(4.12)

Gm is the material gain in units of sec−1 , where the dierntial gain is given by dG
dne

Go
, and ntr is the transparency carrier-density.
ne

The low losses due to the high-Q

resonator enable the laser to operate close to transparency. This allows us to de
ne

Gm = nGtr0 and write the material gain using the approximate linear expression:

Gm = Gm (ne − ntr )

(4.13)

This will make some of the calculations and numerics easier.

4.6 Spontaneous emission
The rate of spontaneous emission into the mode is given by Equation 2.3. The rate
for the photon density can be expressed by dividing the equation by modal volume:

Rsp =

ΓQW Gm (ne )nsp
Vp

(4.14)

(sp)

dnp
dt

= Rsp

(4.15)

48
It is evident here that the reduced QW con
nement factor ΓQW , i.e., reduced rate
of spontaneous emission, which is a characteristic of our high-Q platform, minimizes
the spontaneous emission rate into the mode. It is also important to note that our
rate equations only express photon density, and completely ignore the phase.

The

spontaneous emission term in the rate equations is important because it provides
an initial photon source for lasing, and a straightforward way to estimate the ST linewidth.

In Chapter 9 I replace this term with properly normalized Langevin

forces.

4.6.1 Model for the population inversion factor
It is common in the literature to set the population inversion coe
cient nsp to a
constant.

Values of 1-2 are common [13].

gross errors.

However, in our case this might lead to

Since our low-loss lasers are working very close to transparency this

term can reach very high values. To calculate nsp exactly we need to know the exact
form of the quasi-Fermi levels.

In general, the two quasi-Fermi levels are dierent

and are not symmetrical with respect to the center of the electronic bandgap. This
could be a consequence of doping or the dierent eective masses for electrons and
holes [77]. However, to simplify our calculation while still getting the right trends and
physics, I shall assume that the two quasi-Fermi levels are exactly symmetrical with
respect to the center of the bandgap, and that the Fermi-level is exactly at the center
of the bandgap. This situation is depicted in Figure 4.2. Exploiting the symmetry
in this model, I can express the distance of the electrons quasi-Fermi level from the
conduction band using the bandgap and quasi-Fermi level separation:

2 (E2 − Ef,c ) = E21 − ∆Ef
At transparency (in any SCL) ∆Ef

(4.16)

= E21 , and due to the symmetry I can further

write Ef,c = E2 . I will use this relationship to express the conduction band electrons
above transparency using:

ne =

2 · ntr
(E
1 + e 2 −Ef,c )/KT

(4.17)

Figure 4.2: Symmetrical quasi-Fermi level model

Finally, I can use Equation 2.4, 4.16 and 4.17 to express the population inversion
factor using useful macroscopic quantities:

nsp =

1−

2ntr
−1

(4.18)

Though this is a very approximated expression that should be used with caution, it
gives a simple way to calculate the population inversion factor that gives the correct
behavior. It is unity in complete inversion (in this model full inversion is at n = 2ntr ),
and blows up very close to transparency.

4.7 Two-photon-absorption
The
at-mode approximation makes our calculations much simpler since it neglects
spatial variations and assigns an average photon density to a
xed box. In terms of
TPA, this approximation is a bit problematic. TPA depends on the local intensity of
the mode, and loss rate scales as the intensity square. By taking an average value lower
than the peak's intensity, we might underestimate TPA. The exact (local) equation

50
in Si:

dnp (r)
= −βT hνvg2 n2p (r)
dt

(4.19)

can be averaged by integration:

dnp (r) 3
d r = −βT hνvg2
dt
Vp

n2p (r)d3 r

(4.20)

Notice that the integral in the RHS is with respect to the density

squared. It is

therefore dierent from the de
nition of the con
nement factor, which is linear in
the photon density. To express the averaged equation I de
ne the factor MT P A and
express the integral on the RHS by:

MT P A
n2p (r)d3 r =
Vp2
Si

np (r)d r

(4.21)

such that now the integral in the RHS can be expressed using the con
nement factor
in Si ΓSi and the average photon density np :

n2p (r)d3 r = MT P A Γ2Si n2p

(4.22)

Where MT P A is de
ned from Equation 4.21 as:

|E(r)|4 d3 r
2
|E(r)|2 d3 r
Si

MT P A = Vp R Si

(4.23)

The total loss rate for TPA will be given by:

dnp
= −βT hνvg2 MT P A Γ2Si n2p
dt

(4.24)

4.8 Spontaneous recombination in the QW
The excited electrons in the QWs have a
nite lifetime, as was the case in Si (see
section 3.2.2).

However, unlike the case in silicon, InGaAsP is a direct bandgap

51
semiconductor. Therefore, radiative spontaneous recombination (spontaneous emission) will play a bigger role. Other recombination mechanisms are also important:
non-radiative recombination via SRH or Auger processes. I will lump all the recombination processes, radiative and non-radiative, into one time constant τr that will
enter the rate equation for electrons in the QWs:

dne ne
dt
τr

(4.25)

It is possible to capture the dependence of the time constant on the density of electrons
to better account for radiative and Auger recombination. However, this will clutter
our expressions and slow down the numerical analysis. Furthermore, the goal of this
work is to evaluate the eect of non-linear TPA and FCA. Adding nonlinearities in
the form of a density-dependent lifetime might shadow some important physics.

therefore treat this lifetime as constant.

4.9 Rate equation for free-carriers in silicon
Following the discussion in section 3.2 I will use the eective lifetime τef f of carriers
in silicon to account for recombination in the bulk, the surface, and diusion of the
carriers away from the mode. The total number of carriers generated by TPA can
be given by half the total number of photons absorbed by TPA (since two photons
generate one carrier).

When writing a rate equation for the

density of carriers

in silicon, I consider only the volume that is occupied by silicon.

Therefore, I can

construct a rate equation for the density of silicon carriers nSi :

dnSi
nSi
= βT hνvg2 MT P A ΓSi n2p −
dt
τef f

(4.26)

This will add an extra rate equation for the system of equations, which is another
contribution of this work compared to conventional lasers analysis.

4.10 Free-carrier-absorption
As described in section 3.2, I will use the empirically measured cross-section σa to
account for optical absorption by free-carriers in silicon. Only portions of the mode
con
ned in Si will be aected by silicon FCA so that the loss scales with the con
nement factor ΓSi . The rate equation for the loss in photon density due to FCA is
therefore given by:

dnp
= −vg σa nSi ΓSi np
dt

(4.27)

It is worth noting that on
rst glance, this equation looks like an extra linear loss
term.

However, the loss rate term is also proportional to the nSi . It is clear from

Equation 4.26 that in the steady state (

= 0) the number of carriers in silicon is

proportional to np :

nSi,0 = τef f βT hνvg2 MT P A ΓSi n2p,0

(4.28)

Therefore the FCA absorption loss rate in the steady state will take the form:

(FCA loss)

dnp

|steady state = − vg σa Γ2Si τef f βT hνvg2 MT P A n3p,0

(4.29)

It is now evident that FCA can in fact be considered, at least mathematically, as a

three-photon-absorption process since it is proportional in the steady-state to np . It
is therefore expected that when the photon density is high, FCA might be dominant.

4.11 Total loss rate
So far I have identi
ed the main three loss mechanisms in our lasers: linear loss, TPA,
and FCA. From sections 4.4, 4.7, and 4.10 I can compare these three mechanisms to
gain some insight on the magnitude of each. For example, a comparison of dierent
loss mechanisms is shown in Figure 4.3. As can be shown in Figure 4.3, loss due to
FCA can exceed that due to the linear loss at su
ciently high photon densities. To
gain some insight on the order of magnitude of photon densities it is worth noting

that for typical waveguide dimensions (i.e., cross-section of (1µm) ), photon density

Figure 4.3: Comparison between dierent loss mechanisms vs. photon density. Qsi =
106 , typical absorption in III-V is assumed (i.e., 10cm−1 ), con
nement factor in III-V
of 1% and eective lifetime of carriers in Si of τef f = 30ns.

17
−3
of 10 [cm ] represents intra-cavity power of about 1 Watt.

4.12 The modi
ed rate equations
I can now express the complete system of rate equations:

ΓQW
ηi I
ne
dne
Gm (ne )np +
=− −
dt
τr
ΓGeom
qVQW
dnp
= (ΓQW Gm (ne ) − α) np − βT hνvg2 MT P A Γ2Si n2p
dt

(4.30)

(4.31)

− vg σa nSi ΓSi np + Rsp
dnSi
nSi
= βT hνvg2 MT P A ΓSi n2p −
dt
τef f

(4.32)

The parameters I will use for the rate equation analysis are summarized in table 4.1.
This chapter formally incorporated TPA and FCA in the rate equations and pinpointed some unique characteristics of the high-Q hybrid platform. This analysis will

1 Measured experimentally. Measurement technique and details appear in later chapters.

54
Parameter

Description

Value

Units

τr

Recombination life time

50 · 10−9

sec

(including non-radiative)

Gm
Gm
ntr

Material gain

Equation 4.13
−19

sec

−1

QWs transparency density

1 · 10
2 · 1024

−3

Quantum e
ciency

0.3

Dierential material gain

(including current leakage)

VQW
Vp
ΓQW
Γgeom
vg
βT

Pump current

(sweep)

Amper

Electron charge

Eective mode volume

1.6 · 10−19
4.2 · 10−17
6 · 10−16

Linear loss rate in the cavity

Equation 4.5

Coulombs
−1
sec

QW con
nement factor

Equation 2.10

Geometrical con
nement factor

0.07

c/3.53
8 · 10−12

Quantum well eective volume

Mode's group velocity
TPA coe
cient

[17, 8]

MT P A
σa

TPA magni
cation factor
FCA cross section

Equation 4.23
−21

1.45 · 10

[58]

ΓSi
Rsp

Si con
nement factor

eq. 2.10

Spontaneous emission rate into

Equation 4.14

−1

sec

mode

τef f

Si carriers eective lifetime

30ns1

sec

Table 4.1: Parameters used for rate equations

lay the foundation for a detailed exploration of laser performance in the following
chapters.

Chapter 5
Steady-state operation - Theoretical
analysis
In the previous chapter, I developed a set of rate equations that describes the laser
operation and included nonlinear eects such as TPA and FCA. The
rst step towards
understanding the eect of the nonlinear terms on the laser performance is to solve
the set of equations for the steady-state point. I will therefore set all

= 0 in the
dt

LHS of Equations 4.30-4.32.

5.1 Steady-state carrier density in silicon
The third equation for the Si carriers is the simplest to solve:

nSi,0 = βT hνvg2 MT P A ΓSi n2p τef f

(5.1)

The average density of the Si carriers is proportional to the square of the density of
photons and to τef f . As expected, increasing the intra-cavity intensity would quickly
increase the Si carrier density and will yield increased losses through FCA. From
Chapter 3.2 we know that the eective lifetime of carriers in Si can vary by two
orders of magnitude, depending on the surface quality and device dimensions. It is
now clear that the carrier density in Si (and therefore FCA) scale with this time
constant.

5.2 Gain saturation

I will now deal with the
rst rate Equation 4.30 for the electron density in the QW. To
make the expressions analytical I will use the linear material gain model of Equation
4.13. setting

dne
= 0, I get the expression for the modal gain:
dt

ΓQW Gm (npump − ntr )

ΓQW Gm (n − ntr ) =

QW
1 + G0m Γgeom
τr np,0

(5.2)

where I have de
ned

npump =

ηi I
τr
qVQW

(5.3)

Notice that the density de
ned by npump is the steady-state density if we set np = 0.
Therefore we can interpret npump as the steady-state carrier density we pump into the
QWs. Equation 5.2 is a familiar saturated gain term, with gain saturation coe
cient:

s = Gm

ΓQW
τr
Γgeom

such that:

(5.4)

Gm
Gm (np ) =
1 + s np

(5.5)

There is one noticeable dierence in the gain saturation coe
cient compared to conventional lasers: the dependence on the ratio
is usually unity.

ΓQW
. In a conventional laser, this term
Γgeom

Since the QWs in our lasers are located at the tail of the optical

mode, the gain saturation coe
cient is much lower than the value in a conventional
laser.

5.3 Threshold current
Just below threshold, the intra-cavity intensity is very low and nonlinear eects are
negligible. Therefore, the threshold current is not eected by TPA and FCA. Above
transparency, the current has to be increased to the point in which the gain overcomes
loss in order to start lasing oscillation. Dierent spacer designs have dierent modal

Figure 5.1: Threshold current vs.

con
nement factor in III-V for dierent quality

factors of the Si resonator

gain values, since the the interaction with the QWs is reduced with thicker spacers.
However, the modal loss is also reduced as a consequence of the increased overlap
with the low-loss material and the reduced overlap with the lossy materials. This is
one of the key features and strengths of this platform and design approach.
As long as the total-Q remains lower than the silicon-Q (intrinsic and external),
a small increase in spacer thickness will yield a linear reduction in the total loss.
This is demonstrated in the quasi-linear portion of the plot in Figure 2.3. When the
thickness is increased to a point in which the total-Q is approaching the silicon-Q,
the reduction in loss starts to saturate. Since the reduction in gain is always linear
with the con
nement factor, the lasing condition will now be met at a higher level,
and threshold is increased.
This behavior is demonstrated in Figure 5.1, which was obtained using the numerical solution to the rate equations. Figure 5.1 demonstrates a very important feature
of the hybrid Si-III-V platform: starting with a high-Q silicon resonator allows us
to increase the total-Q by increasing spacer thickness without trading o threshold
current.

However, if we overdo it by working too close to the silicon-Q, we pay a

penalty in threshold current.

5.4 Output power

Output power is one of the most important metrics for almost every application.
Narrow-linewidth lasers are often used for sensing or ranging, where output power is
crucial to ensure enough photons fall on the photodetector so the signal is well above
the instrumental noise. In coherent communication, a combination of output power
and linewidth is used to calculate bit-error-rate. External ampli
cation of the output
power might result in the corruption of the initial high coherence.

5.4.1 Wall-plug e
ciency
Every laser has an unavoidable intrinsic loss, as well as a mirror loss.

The mirror

loss couples the stored energy in the cavity to the outside world as useful output.
The energy lost due to the intrinsic part is connected to heat or to the modes of the
free-space continuum, and is thus wasted. The total quality factor of a generic laser
can be written as:

= int + mirror

(5.6)

The rate of output photons PN (photons/sec) given using the product of the total
number of photons stored Np and the mirror loss rate is:

PN =

ωNp
Qmirror

(5.7)

The total number of photons in the cavity can be calculated using Equation 2.7. We
can de
ne the external e
ciency ηe as the ratio between the rate of output photons
and input electrons:

ηe = ηi (1 −

Ith
) mirror
I Q

(5.8)

We can further express the threshold current Ith using the transparency current and
total loss from the lasing condition (gain = loss) and assuming a linear gain model
as in 4.13:

Ith =

qVQW
(ntr +
ηi τr
QΓQW G0m

(5.9)

The external e
ciency will now take the form:

ηe = ηi (1 −

Qint
Itr
Itr
) · int
· mirror
− ηi ξ
mirror
Q +Q
I Q

(5.10)

where Itr is the transparency current and we have de
ned the parameter ξ as:

ξ=

ΓQW G0m ntr

(5.11)

Our hybrid Si/III-V lasers have a unique form for the intrinsic and mirror quality

int
factors given by Equation 4.4. For a given Qsi and a given ΓIII−V , I can calculate
the external e
ciency using Equation 5.10. An example is shown in Figure 5.2 for

Qint
si = 10 and for several values of con
nement factors.
As can be seen from Figure 5.2 there is some trade o between e
ciency and
total Q. For the e
ciency, there is an optimal mirror-Q. Increasing it to increase the
total-Q for narrower linewidth will take its toll on the output power. The de
nition
of optimal coupling de
ned in Equation 4.6 can now be better understood: at the
optimal coupling point, where the total Q is exactly half the intrinsic Q, the e
ciency
is below optimal levels, but only slightly. Increasing the mirror-Q above that point
can only yield a factor of two improvements in total-Q, but e
ciency will quickly go
to extremely low values.
The total quality factor is a measure of the resulting linewidth of the laser. Dierent spacer designs yield dierent linewidths and e
ciencies. A comparison between
dierent designs and the exact trade-os between quality factor and e
ciency of each
design are shown in Figure 5.3.

5.4.2 L-I curve
I can now solve the system of rate equations numerically in the steady state for
dierent silicon quality factors and dierent con
nement factors in III-V. The solution

(a)

(b)

(c)

int
Figure 5.2: E
ciency (left axis) and total Q (right axis) vs. mirror Q. Qsi = 10 ;
ηi = 1 ; Itr = 10 (a) 30nm spacer (b) 100nm spacer (c) 150nm spacer

Figure 5.3: E
ciency of dierent spacer design vs. total-Q

will yield the steady states photon density in the cavity, which we can convert to

mirror
output power by assuming optimal coupling (Q
= Qint ) and using:

Pout =

2hν 2 np Vp
Qint

int
Numerically obtained L-I curves for Qsi

(5.12)

= 106 for dierent values of con
nement

factors are shown in Figure 5.4, with and without the inclusion of nonlinear eects.
As the intra-cavity intensity builds up, TPA and FCA processes become more and
more dominant, and introduce excess loss.

When the current pump increases, the

intensity-dependent loss increases, and the total-Q of the cavity is reduced.

As a

result, the cavity cannot increase its photon storage at the same rate as the current
pump and the L-I curve becomes nonlinear.

As can be seen in Figure 5.4, though

TPA itself aects the linearity of the L-I curve, FCA has a much bigger impact.

5.4.3 Slope e
ciency
It is evident from the L-I curves of Figure 5.4 that the slope of the curve reduces
with pump current due to nonlinear eects such as TPA and FCA. For high pump
currents, the output power and the slope of the L-I curves are both highly aected

(a)

(b)

(c)
Figure 5.4: L-I curves for dierent values of con
nement factors with and without
int
nonlinear eects for Qsi = 10 (a) spacer 150nm (b) spacer 100nm (c) spacer 30nm

Figure 5.5: Normalized slope e
ciency at I = 4 · Ith

by nonlinear eects. I can de
ne a local slope e
ciency at a pump value I as the
increase in number of photons at the output for every electron at the input.

ηs (I) =

dNpout
dNein

(5.13)

Using this de
nition, I can examine the impact of nonlinear eects on the power
extraction e
ciency for dierent spacer designs.

For ease of interpretation of this

gure of merit I will assume that the mirror-Q is exactly half the total cold-cavity-Q.
In such a setup, if there were no nonlinear eects, exactly half the input energy above
threshold would go to useful output. The eect of nonlinear loss is shown in Figure
5.5 for dierent silicon-Qs. For more aggressive designs (i.e., higher silicon-Q or lower
con
nement in III-V) I expect to have higher-Q, and therefore nonlinear loss would
be higher for these designs. The nonlinear loss will reduce the slope e
ciency and the
external e
ciency, as shown in Figure 5.5. Therefore, the e
ciency plots of Figures
5.2 and 5.3 will change with pump current in the presence of strong TPA and FCA.

5.5 Schawlow-Townes linewidth
If a laser has only white frequency noise with a constant power spectral density Wφ̇
the spectrum of the
eld E(t) is a Lorentzian with spectral width of [78]:

∆ν =

Wφ̇
2π

(5.14)

This linewidth is also known as the Schawlow-Townes linewidth.
A realistic laser will have other noise components.

However, these components

usually decay with frequency and have the form 1/ν α for some positive α.

At the

high frequencies, the laser noise will usually be dominated by the S-T linewidth. If
we are only interested in the high frequencies components of the signal, as in optical
communication, the S-T linewidth will set an eective linewidth for the purpose of
that measurement. If the frequency range of interest is below the relaxation oscillation

frequency then the S-T
oor is multiplied by a factor of (1 + αH ) due to coupling
between amplitude and phase
uctuations [31] (the Henry linewidth enhancement).
To estimate the S-T noise
oor I will use Equation 2.6 that relates the spontaneous
emission rate into the mode to the frequency noise PSD
oor.

We estimate the

spontaneous emission rate using Equation 4.14 that relates spontaneous emission to
the laser's gain. The population inversion factor is calculated using the symmetrical
quasi-Fermi level approximation of Equation 2.4. For a given pump power, the gain
and the photon density are calculated numerically from the steady-state rate equation,
and the S-T noise
oor is derived.
The S-T Linewidth [91] is known to be inversely proportional to
versely proportional to the output power P .

Q2 and in-

For laser resonators with nominally

high quality factors, the nonlinear loss eectively limits the Q by introducing excess
intensity-dependent loss, thus preventing the intra-cavity intensity from rising.

demonstrated in Figure 5.6 , this eect increases the linewidth compared to the ST linewidth of an equivalent resonator without TPA and FCA. For a given silicon
resonator quality factor, reducing con
nement in III-V increases the total-Q and re-

Figure 5.6: Schawlow-Townes linewidth vs. con
nement in III-V for dierent silicon
resonators, with and without nonlinear eects at I = 4 · Ith

duces spontaneous emission to the lasing mode. In the absence of nonlinear loss, this
process will result in quick reduction of linewidth. However, nonlinearities, such as
TPA and FCA, limit the total-Q and the resulting linewidth is somewhat clamped.
For example, Figure 5.6 shows that for 200nm spacer increasing the silicon-Q from

106 to 5 · 106 should result in an improvement of an order of magnitude in linewidth,
and should yield S-T linewidth as low as 4Hz. However, due to TPA and FCA the
improvement is limited to a factor of two, down to a value of 60 Hz. At silicon-Qs
of about one million, the linewidth is limited by nonlinearities to around 100 Hz,
and performs very similar to a device with quality factor of half a million. It is thus
demonstrated that nonlinear loss limits the achievable linewidth, to a point where it

is no longer so attractive to fabricate ultra-high-Q (Q > 10 ) resonators in silicon
for hybrid lasers.

This argument is further demonstrated in Figure 5.7.

The 1/Q2

dependency of the linewidth is retrieved in the numerical calculation in the absence
of nonlinear eects. However, in the presence of TPA and subsequent FCA this form
no longer holds for high-Q. As the quality factor increases, the linewidth tends to
saturate at a few tens of Hz.
The saturation of stored energy due to TPA and FCA also occurs when the pump
current increases. As a consequence, the linewidth would decrease at a smaller rate

Figure 5.7:

Impact of nonlinear eects on linewidth for changing quality factors.

Calculated at I = 5 · Ith

Figure 5.8: Impact of nonlinear eects on linewidth for changing pump current. Cal6
culated at I = 5 · Ith for QSi = 10

−1
than the expected (I − Ith ) . This can be seen in Figure 5.8. Moreover, not only do
nonlinear eects saturate the linewidth improvement with increased power, but even
cause re-broadening of the linewidth. This is due to the increased total loss by TPA
and FCA; the gain has to compensate for the excess non-linear loss by increasing the
QW carrier density. This, in turn, increases the spontaneous emission rate into the
mode. This eect is accompanied by the saturation of the stored energy in the cavity
and leads to a broadening of the linewidth at higher currents.
In this chapter, I analyzed the steady-state performance of the lasers in light of
nonlinear eects such as TPA and FCA. It was predicted that L-I curves of narrowlinewidth lasers might show nonlinear behavior. As we saw, the frequency noise-
oor
of these lasers will deviate from the familiar S-T formula, and may saturate or even
broaden at high-power. It was also shown that although some tradeo between wallplug e
ciency and linewidth exists, it is possible to design ultra-narrow linewidth
lasers with reasonable e
ciency.

Chapter 6
Steady-state operation Experimental results
We have fabricated and characterized hybrid Si/III-V spacer lasers with several laser
designs. Details on the laser design and fabrication process can be found in appendix
A, while details on the measurement techniques and procedures can be found in
appendix B. Steady-state performance of similar hybrid spacer lasers are also reported
in [106]. Here, I will focus on results that are relevant to the observation of nonlinear
eects, or new results that are absent in [106].
In the design and fabrication of the spacer lasers we have swept several parameters:

1. Wavelength - Three dierent grating periods 240nm, 242.5nm, and 245nm
corresponding to the three dierent lasing wavelengths were fabricated.

2. Spacer thicknesses - Four dierent spacers were fabricated 30nm, 100nm,
150nm and 200nm. The most aggressive design the 200nm spacer thickness
didn't lase at all. It is possible that the modal gain for this spacer thickness
was too low, indicating that the total-Q was past the point of saturation at the
silicon-Q (as suggested by Figure 5.1).

In this chapter, I will present results

from the remaining operational three spacer designs.

3. Mirror section length - Devices were cleaved to yield
ve dierent bar lengths,
corresponding to varying mirror-Qs. The exact mirror quality factor value depends strongly on etch depth and pro
le, which
uctuate considerably between

69
dierent fabrication runs. Therefore, a quantitative estimation of the mirror-Q
is unreliable. Furthermore, we ended up with lasers with relatively low power,
and for experimental reliability we had to work with lasers that have reasonable
output. Therefore, this chapter will only include results from the shortest bars.
Despite this fact, there is evidence to indicate that mirror-Q is still high enough
to be considered under-coupled (in the regime to the right of the optimum peak
in Figure 5.2). For example, overall low e
ciency and decreasing output power
with increasing mirror-Q, both indicate under-coupling.

4. Potential-well depth - Two designs were fabricated, with 100GHz and 120GHz
well-depth (see Figure 2.2). However, designs with deeper potential wells are
more con
ned to the defect region and therefore have an eectively stronger
mirror grating. Since all devices turned under-coupled, we chose to work with
the shallow-well designs to obtain more output power.

6.1 Threshold current
Threshold current was extracted from the L-I curves by locating the intersection of
the slopes before and after threshold. All threshold data presented here are for stage

temperature of 20 C . The dierent designs are of comparable physical dimensions.
The same current channel was de
ned by the ion implantation steps, and the same
III-V wafer was used in all cases.

I therefore expect that the biggest impact on

threshold current will be due to variation in modal loss, compared to modal gain
between designs.

The scatter plot of Figure 6.1 compares threshold currents for

dierent spacer designs for two dierent lasing frequencies. Several conclusions can
be drawn from Figure 6.1:

1. The spread of threshold current values is much bigger for thicker spacers. This
is expected, since as the spacer thickness increases the total-Q is dominated by
the silicon-Q, which varies considerably due to small variations in dimensions
due to fabrication conditions. The laser bar of the 150nm spacer had only three

(a)

(b)
Figure 6.1:

Threshold current for dierent spacer designs.

1575nm lasers

(a) 1560nm lasers (b)

71
lasing devices. We would have most likely seen a spread with higher thresholds
as well; however, these didn't lase due to thermal eects decreasing the material
gain.

2. The low threshold devices of the 100nm spacer have the same threshold as the
30nm spacer. This indicates that for these devices, the silicon-Q is high enough
to be in the regime where threshold is no longer aected by reduced active
con
nement, as suggested by Figure 5.1.

3. The spread of threshold current values is very small for the thin 30nm spacer
design.

This indicates that losses are dominated by the III-V intrinsic loss,

which has very little variation.

6.2 L-I curves
Dierent spacer designs were fabricated in separate fabrication runs. Small and unavoidable variations in etch depth can have a big impact on the grating strength, and
therefore on the amount of output coupling in the mirror sections of the lasers. The
fabricated lasers are all under-coupled. The overall e
ciency is low, and laser bars
with longer mirror sections have low output power. Furthermore, there is variation
in the quality of the facets, as they were cleaved but not polished. For these reasons,
it is very hard to compare the e
ciency between dierent spacer lasers: the arbitrary
output coupling acts as an unknown scaling factor. It is therefore nearly impossible
to draw conclusions that are based on the absolute output power across lasers. It is
worth noting that though this is true for the absolute power, it is not the case for the
linewidth; all the lasers are under-coupled, and therefore the loss is dominated by the
intrinsic-Q and not the mirror-Q. This fact causes the linewidth of dierent lasers to
be almost independent of the mirror transmission.
Though the absolute output power of dierent lasers is somewhat arbitrary, the
shape of the L-I curve within a single laser contains much information. Figure 6.2
shows a comparison between typical L-I curves of the three dierent spacer designs.

Figure 6.2:

Normalized L-I curves for three spacer designs.

output powers at I=150mA are:

The non-normalized

0.89mW, 0.62mW, and 10.2mW for the 200nm,

100nm, and 30nm spacers, respectively

The absolute magnitude of dierent curves was altered and normalized to ensure that
all curves are on the same scale for better comparison. It is evident from Figure 6.2
that while the 30nm and 100nm spacers have a nearly linear L-I curve, the 150nm
spacer is very nonlinear.
One has to be careful, though, in attributing nonlinear L-I curves to TPA and
FCA. Power roll-os are common in many laser systems.

In fact, every laser will

roll o (or even burn) at su
ciently high pump currents. Standard power roll-o
is usually attributed to thermal eects [79]. Increase in pump current elevates the
temperature through Joule heating, Thomson Heating, and non-radiative recombination heating. The elevated temperature lowers the gain due to the widening of the
Fermi spreading of carriers. To maintain lasing at this lower gain, the carrier density
increases. This in turn elevates electron leakage, Auger recombination, spontaneous
recombination, and SRH recombination. It was suggested [80] that electron leakage
is the main source of power roll-o.
The chain of events causing power roll-o begins with Joule heating. It was found
[80] that Thomson heating, which is a consequence of the capture of electrons in
the QW, is of lesser impact and is compensated by Thomson cooling (due to escape

73
of electrons from the QW). Figure 6.3(a) shows the eect of stage temperature on
the L-I curve. Furthermore, to isolate the eect of thermal power roll-o, a pulsed
current source was used. Figure 6.3(b) shows L-I curves from a pulsed power source,
with pulse width of 1µsec and 10µsec.

The short pulses and the small duty cycle

guarantee that thermal eects are minimized. It is shown in Figure 6.3 that even in
pulsed operation, the L-I curve of the thick spacer laser is nonlinear, in agreement
with the theoretical analysis of Chapter 5 that attributed the nonlinearity to freecarrier-absorption.

6.3 Schawlow-Townes noise
oor
The Schawlow-Townes linewidth represents the spectral width of the electromagnetic

eld when the frequency noise spectrum consists of white noise only. In a practical
laser, the noise spectrum is not a
at white noise curve. Though the noise
oor is
limited by the S-T linewidth, there are other noise sources that dominate in some
frequency ranges. The low frequency is dominated by technical noise and thermal
uctuations noise.

This is common to almost all laser systems.

Since our lasers

had relatively low output power (∼1mW), we needed an ampli
er to get a reliable
measurement of the frequency noise. For that reason, the high-frequency noise was
dominated by phase-noise added by the ampli
er. The eect of the ampli
er on the
noise measurements is discussed in detail in Chapter 10.
The S-T noise
oor was extracted from an experimental noise spectrum from the
intermediate regime of 100MHz 1GHz. The single sided PSD of the frequency noise
was multiplied by π according to Equation 5.14 (notice that in the equation Wφ̇ is
the PSD of the angular frequency), to obtain the equivalent white noise linewidth.
Figure 6.4 shows the S-T linewidth for dierent pump currents and for the three
dierent spacer designs. It is evident from Figure 6.4 that the 1/(I−Ith ) dependence
that is expected from the S-T linewidth formula is maintained at the lowest threshold,
at least when the pump current is not too high.
the linewidth saturates.

As the pump current increases,

The more aggressive designs, 100nm and 150nm spacers,

(a)

(b)
Figure 6.3: L-I curves of the 150nm spacer. (a) For varying stage temperatures (b)
In pulsed operation (duty cycle = 1%)

Figure 6.4: Schawlow-Townes linewidth vs current oset from threshold for the three
spacer lasers. The dotted lines represent expected 1/P dependence

deviate considerably from the 1/(I−Ith ) dependence, in agreement with the theoretical
predictions made in regards to TPA and FCA. The rate of linewidth reduction with
pump current slows down below 1/(I−Ith ) and at high pump currents the linewidth
even broadens. It is worth noting that measured linewidth in the 150nm spacer is
limited by the measurement setup, due to the presence of the ampli
er. Therefore,
the white noise
oor could not be observed and the speci
ed values for these devices
(150nm spacer) only represent upper bound and should be taken with a grain of salt.
In this chapter I have presented experimental results regarding the steady-state
operation of narrow-linewidth hybrid Si/III-V lasers. One of the most striking experimental results in the steady-state operation of these lasers is their noise performance.
The 150nm spacer, in which less than 1 percent of the light resides in the III-V material, yielded lasers with Schawlow-Townes linewidth of ∼1 KHz. In fact, this
gure is
limited by the measurement, and the actual laser linewidth is sub-KHz. This is, to the
best of our knowledge, the lowest noise ever reported for a stand-alone semiconductor
laser, without an externally-coupled cavity. The trend from Figure 6.4 is also very
clear: pushing the mode into silicon yields lasers with superior noise performance.
In the context of nonlinear phenomena in silicon there are several indications for

76
the signi
cance of these eects.

First, the aggressive spacer designs showed very

nonlinear L-I curves as expected from the theoretical analysis.

Moreover, the im-

provement in linewidth with pump current saturates quickly for the thick spacer
lasers.

These observations are in agreement with theoretical analysis that predicts

the same eect due to nonlinear loss in the Si. These two facts are initial indications
for the impact of TPA and FCA on laser performance. In the next sections, I will
analyze the dynamical behavior of the lasers in the presence of nonlinear loss, both
theoretically and experimentally.

Chapter 7
Dynamic operation - Theoretical
analysis
The semiconductor laser is one of the pillars of the optical communication
eld and
of other high-speed applications. Their fast output power vs. injection current response is being exploited for direct modulation of the laser using its pump current.
Modulation speeds of tens of GHz are commonly achieved [62, 125, 35, 137] in direct
(current) modulation communication schemes. The modulation response of the laser
is also important for its noise characteristics. Random
uctuations in the input current will cause intensity and phase
uctuations, which follow the same modulation
response transfer functions.
On the other hand, in coherent communication, where information is encoded in
the phase of the
eld, external modulators are usually used. In this case, a slow
response, which doesn't extend to modulation frequencies, is advantageous since it
will suppress phase and amplitude
uctuations due to inevitable current-source noise
at the high frequencies of interest. In this chapter, I use the rate equations developed
in Chapter 4 to examine how the laser responds to a small perturbation of the pump
current around the steady-state, which was analyzed in Chapter 5.

7.1 Small-signal analysis
In Chapter 5 I set all time-derivatives to zero to
nd the steady-state. To account for

uctuations around the steady-state we will need to keep those time derivatives in our

78
current analysis. Unfortunately, the rate equations are nonlinear, even without TPA,
and cannot be solved analytically. To get a closed-form solution, I will assume small

uctuations compared to the average values and linearize all the equations, neglecting
higher-order terms.
I will start by expressing all the dynamic variables using the steady-state value
plus small perturbation. Since the resulting system of equations is linearized, I will
then analyze the equations in the complex Fourier domain, using the de
nitions of
section 4.12:

ne = ne,0 + ∆ne · eiωt

(7.1)

np = np,0 + ∆np · eiωt

(7.2)

nSi = nSi,0 + ∆nSi · eiωt

(7.3)

I = I0 + ∆I · eiωt

(7.4)

The resulting linearization of 4.30-4.32 yields the following system of linear equations:



∆I
 qVQW i 

A12 A13
∆ne
 11


 
A21 A22 A23   ∆np  = 

 
A31 A32 A33
∆nSi

(7.5)

79
Where

A11 = iω +
A12 =

ΓQW 0
G np,0
τr ΓGeom m

ΓQW 0
G (ne,0 − ntr )
Γgeom m

A13 = A31 = 0

(7.6)

(7.7)

(7.8)

A21 = −ΓQW Gm np,0
A22 = iω − ΓQW Gm (ne,0 − ntr ) − α + 2Bnp,0 + CnSi,0

(7.10)

A23 = Cnp,0

(7.11)

A32 = −2Dnp,0

(7.12)

A33 = iω +

(7.9)

(7.13)

τef f

And I have de
ned the parameters:

B = βT hνvg2 MT P A Γ2Si

(7.14)

C = vg σa ΓSi

(7.15)

D = βT hνvg2 MT P A ΓSi

(7.16)

The steady-state gain term ΓQW Gm (ne,0 − ntr ) will be calculated using the clamped
gain (gain = loss) to eliminate ne,0 from the expression:

ΓQW Gm (ne,0 − ntr ) = α + Bnp,0 + CnSi,0

(7.17)

Notice the spontaneous emission term was omitted, since well above threshold it is
negligible compared to the stimulated emission.

7.2 Intensity modulation response
During modulation of the input current, the intensity changes based on the intensitymodulation transfer-function. In most semiconductor lasers, the transfer-function is

80
a second-order low-pass
lter, of the generic form H(s) =

ωn
2 . The low fres2 +2ξωn s+ωn

quencies (ω
ωn ) propagate without distortion, while frequencies above the natural
frequency (ω > ωn ) are suppressed. Notice, that in most laser systems the response
is under-damped, such that the natural frequency ωn provides a good estimate for
the relaxation-oscilation frequency. It is worth noting that that are two factors that
aect the modulation response [65]:

1. Internal laser dynamics - The interplay between QW carriers and cavity photons
that yields the relaxation oscillation.

2. Capacitance-related response - The capacitance due to package parasitics will
alter the way input-current propagates through the device.

Though these mechanisms are fundamentally dierent, in practice it is hard to distinguish between the two. In this section, I will analyze only the basic internal laser
dynamics, and will ignore parasitics, which are package-dependent.

Experimental

results will validate that in the frequency range of interest, this is justi
ed in our
case.
Most semiconductor laser designs attempt to push the relaxation resonance to as
high a frequency as possible, so that modulation frequencies fall within the constant
portion of the response. This is usually done by choosing materials that have large
dierential gain (e.g., AlGaInAs), working at low temperatures, decreasing the active
region volume, and working at high powers [128]. Pushing the resonance frequency
further by driving the laser harder has an important yet limited eect. Thermal issues
and increased damping usually limit the bandwidth to ∼30 GHz. Optical injection
locking was found useful in taking care of damping eects, and resonance at 70 GHz
or more was demonstrated [127].
On the other hand, narrow-linewidth lasers, e.g.,
ber lasers, have a very low relaxation oscillation resonance frequency, often sub-MHz, or tens of KHz. The resonance
in these lasers is also typically highly peaked, at 20dB or more. These characteristics
often limit their usefulness, and
ber-laser designs attempt to suppress the resonance
peak. Several techniques are used to that end: active feedback [26], intra-cavity loss

81
elements with feedback [138], locking to a master laser [59, 27], and incorporation
of non-linear loss elements [11, 108, 3]. Out of these techniques, the latter is most
relevant to our study. TPA, which is a nonlinear loss mechanism, is capable of suppressing intensity noise.

When the intensity noise is very high, as in the case of a

relaxation peak, the nonlinear loss is also high, and the noisy peak is suppressed.

7.2.1 Analytical investigation
The matrix in Equation 7.5 can be solved analytically, and simpli
ed using Equation
7.17. The transfer function H(s) can be de
ned using the Laplace transform :

∆np (s) = H(s)∆I(s)

(7.18)

which relates modulation of intra-cavity photon density to input current modulation .
To relate it to the output power, one can use Equation 5.12. The resulting expression
has the form:

ΓQW Gm np,0 τef1 f + s qVηQW
i
H(s) =
(s + 2ξωn ) (s + Bnp,0 ) τef1 f + s + 2CDn2p,0 + τef1 f + s ωn2

(7.19)

Where the natural frequency ωn and the damping factor ξ are de
ned as:

ωn2 =
ξ=

ΓQW 0
G np,0 αT
Γgeom m
ΓQW
+ Γgeom
Gm np,0
τr

2ωn

(7.20)

(7.21)

and the total loss rate, linear and nonlinear, is de
ned as:

αT = α + Bnp,0 + CnSi,0

(7.22)

The transfer function of Equation 7.19 has in general one zero and three poles.
Unfortunately, the roots of the third-order polynomial in the denominator cannot be

82
expressed analytically. I will study this transfer function by looking at two dierent
regimes: low and high nonlinear losses.

7.2.1.1 Low nonlinear loss regime:
Written explicitly, the second term in the denominator is:

2CDn2p,0 =

τef f

vg σa nSi,0 ΓSi =

τef f

αF CA

(7.23)

Where we have used the steady-state silicon carrier-density from Equation 5.1, and
de
ned the loss rate due to FCA αF CA in units of [sec
terms in the square bracket of Equation 7.19.

−1

]. I can now compare a few

If the FCA loss rate is slower than

some frequency ω of interest:

2αF CA
then the term 2CDnp,0 can be neglected. In this case, the

(7.24)

+s
τef f

term cancels

out everywhere, and the transfer function is reduced to the familiar form:

ΓQW Gm np,0
H(s) = 2
s + 2ξωn s + ωn2

(7.25)

which is the typical second-order low-pass
lter.
The exact location of the poles ωp of the transfer-function is given by:

ωp = ωn −ξ ± ξ 2 − 1

(7.26)

When the damping factor is smaller than unity, the knee frequency can be wellapproximated by the value of ωn and the resonance is under-damped.

When the

damping factor is larger than unity, there will be two real poles and the system will
be highly damped.
It is interesting to study how the relaxation resonance behavior changes with
dierent spacer designs. Most textbooks have a similar expression as Equation 7.20,
except that for a traditional laser

ΓQW
Γgeom

≈ 1, so the dependance on ΓQW is often

83
obscured. I will use the relationship between the photon density and the total loss
rate:

np,0 = η

(I − Ith )
eVp αT

(7.27)

such that:

ωn2 =

ΓQW 0 (I − Ith )
G η
Γgeom m
eVp

(7.28)

It is now clear that in our spacer design, where unlike the case of a conventional laser,
the quantum wells reside deeply in the exponential tail of the mode, such that

ΓQW

Γgeom

1, and we expect to see very small relaxation oscillation resonance frequencies with
the trend:

ωn ∼

ΓQW

(7.29)

The damping factor in our system can also be evaluated. Assuming we are operating in a regime where

τr

QW
< Γgeom
Gm np,0 , which is a reasonable assumption for

practical photon densities and con
nements, we can express the damping factor as:

ξ≈

ΓQW np,0
Γgeom αT

(7.30)

Since the total loss is also a function of ΓIII−V in the regime where the total-Q is not
saturated by the silicon-Q, we can write: np,0 ∼

ξ∼p

ΓQW

αT

∼ ΓQW
and get the form:

(7.31)

and we expect an increasingly damped response with reduced III-V con
nement.

7.2.1.2 High nonlinear loss regime:
When the FCA loss is high and the second term in the square bracket of Equation
7.19 cannot be neglected, we are back to the regime in which an analytical expression
cannot be obtained.

However, there are some conclusions we can draw from the

general form of the transfer function:

84
1. The existence of a zero of the transfer function at ωz =

. In the low nonlinear
τef f

regime, this zero was eectively masked by a pole at the same frequency. In the
high nonlinear regime this is no longer the case, and we expect to see a zero of
the transfer function.

2. The existence of three poles of the transfer function. In general, these can be
three real poles, or a pair of complex-conjugate poles and a real pole. Several
types of behavior will be possible, depending on strength of FCA, and on the
location of the zero/poles. For example, if the system is highly damped we can
expect to see a pole of the transfer function at low frequency, then a zero at
an intermediate frequency, and the pole pair at high frequencies. If the system
is less damped, we expect to see the zero at low frequency and three poles at
higher frequencies. Due to the complex nature of the transfer function, and the
fact that np,0 , ΓQW , QSi , and τef f all depend on each other, a numerical analysis
is needed.

7.2.2 Numerical investigation
In this section, I will study the small-signal response numerically. First, a steady-state
solution is obtained as in Chapter 5. Then, the linearized small-signal rate equations
are solved numerically using Cramer's rule, and the previously obtained steady-state
values. This process is repeated for several values of III-V con
nement.
Figures 7.1(a)-(c) demonstrate the impact of nonlinear eects on the response
curve. To isolate the eect of both TPA and FCA on the transfer function, I have
repeated the study three times:
rst with all nonlinearities, then without FCA by
setting σa = 0, and
nally without TPA by setting βT P A = 0. In 7.1(a) the 30nm
spacer results in a relatively low-Q. Nonlinear eects are small and the three curves,
including TPA and FCA and without them, are roughly equivalent. In Figure 7.1(c)
the 150nm spacer yields a fairly high-Q, and the impact of nonlinear eects is evident.
Without nonlinearities, the typical shape of the relaxation resonance curve is restored.
The resonance peaks a few dB above the DC response. When TPA is turned on, but

(a)

(b)

(c)
Figure 7.1: Intensity modulation response curves with and without nonlinear eects.
I = 4Ith , QSi = 106 (a) 30nm spacer (b) 100nm spacer (c) 150nm spacer

86
still without FCA, the shape of the curve is maintained, but the resonance peak is
suppressed.

This is due to the stronger TPA at higher intensities.

turned on, the entire curve changes:

When FCA is

the zero of the transfer function appears at

f = 2πτ1ef f = 5.3M Hz and changes the resonance peak into a broad hill peak.
Furthermore, the entire response curve drops. This can be understood by looking at
the L-I curves of Figure 5.4: The DC value of the response curve represents the local
slope at the working point, and with the nonlinear L-I curve this slope is reduced.
Another interesting feature of Figure 7.1 is the trend in the relaxation oscillation
frequency. The bigger the spacer, the lower the resonance frequency. This is better
demonstrated in Figure 7.2, where we compare both amplitude and phase of the three
spacers on the same plot with nonlinear eects. In fairly good agreement with the
anlytical analysis of section 7.2.1, the relaxation resonance frequency scales with the
square root of the con
nement. Resonance frequencies as low as a few hundred MHz
are expected from thick spacer lasers. Figure 7.3 shows the response curve for a higher
bias point. Here, due to the stronger pump, nonlinear eects are evident, even in the
thinner spacer design.

7.3 Frequency modulation response
When the input current to the laser is being modulated, the lasing frequency changes
as well as the intensity. The low-frequency response is usually dominated by thermal
eects; the medium's refractive index is temperature-dependent, and the modulation
of the input current changes the laser's temperature. The thermo-optic coe
cient is
of the same order of magnitude in III-V and silicon [14, 12], about

dn
≈ 2 · 10−4 [K−1 ].
dT

We therefore expect the hybrid silicon laser to perform similarly to a conventional
III-V laser as far as low-frequency thermal response is concerned. For that reason,
I will ignore thermal eect in the following analysis, and will focus on the unique
characteristics of the hybrid platform.
The change in lasing frequency of the laser will depend on changes of the eective
refractive index of the lasing mode. Assuming these changes are small enough, we can

(a)

(b)
Figure 7.2:

Intensity modulation response curves for dierent spacer thicknesses.
I = 4Ith , QSi = 106 (a) amplitude (b) phase

(a)

(b)
Figure 7.3:

Intensity modulation response curves for dierent spacer thicknesses.
I = 10Ith , QSi = 106 (a) amplitude (b) phase

89
safely neglect changes in the modal pro
le and connect local changes of the refractive
index to the eective index using the con
nement factor. For example, if the index

(Si)
(QW )
of silicon changes by ∆nr
, and the index of the QWs changes by ∆nr
, we can
calculate the change to the eective index using:

∆nr = ΓQW ∆n(QW
+ ΓSi ∆n(Si)

(7.32)

The resulting frequency chirp ∆ν is approximated using [13, 126]:

∆ν = −

vg
∆nr
λ0

(7.33)

7.3.1 Eect of Quantum Well carriers
The modulation of input current yields a change of the carrier density in the quantum
wells. This in turn causes refractive index modulation through the plasma eect [44],
which results in frequency chirping. In an ideal laser, the carrier density is clamped
to its threshold value. This fact would mean that a DC modulation should result in
zero frequency chirp. In practical laser systems this is not the case. Even at DC, the
plasma eect causes frequency tuning, typically few hundred MHz per mA of input
modulation [123, 78, 13, 126].

To explain this discrepancy I will have to consider

gain compression in our model. This eect was of no signi
cance in previous
analysis. However, since the laser is extremely sensitive to refractive index changes,
it is important to consider it in the analysis of the frequency modulation response.

7.3.1.1 Gain compression
It is an approximation to view the gain as clamped to its threshold value. In practice,
the high photon density will compress the (unsaturated) gain. This non-linearity of
the gain is often attributed to spectral hole burning and carrier heating (intra-band
re-absorption of photons) [129].

A good model for this eect can be given by the

90
expression:

Gm
Gm (np ) =
1 + ΓQW c np

(7.34)

c is the gain compression coe
cient, which is derived empirically.

When

where

the laser pump current is modulated, the photon density responds, as described in
section 7.2. The resulting compression of the gain would force the QW carriers to
follow in order to compensate for the change in dierential gain, and frequency chirp
will be observed.

In light of this modi
cation to the model, the dierential rate

equations should be altered. Gain compression should be included in the dierential
rate Equations 7.6-7.13 by making the dierential gain dependent on the photon

density, thus making the substitution Gm → Gm (np ). Moreover, the derivative of the
gain with respect to the photon density has to be included. Speci
cally, this results
in changes to two terms in the small-signal matrix 7.5:

ΓQW 0
ΓQW c np,0
A12 =
G (np ) · (ne,0 − ntr ) 1 −
Γgeom m
1 + ΓQW c np,0

(7.35)

ΓQW c np,0
A22 = iω + α + 2Bnp,0 + CnSi,0 − ΓQW Gm (np ) · (ne,0 − ntr ) 1 −
1 + ΓQW c np,0

(7.36)

7.3.1.2 Henry's alpha parameter
In laser analysis, it is useful to express changes to the refractive index nr using changes
to the imaginary part of the refractive index ni , which is related to the material gain

g . Henry's alpha parameter αH can be de
ned as:
dn

dni
αH = dnre / dn

(7.37)

and is used to connect the two using the expression [115]:

dnr
λ0 Gm
= −αH
dne
4π 2vg

(7.38)

Figure 7.4: Frequency modulation response due to quantum well electrons for dierent
values of spacer thickness. αH = 7, I = 2 · Ith , QSi = 10

where we have used the explicit linear form of the gain (Equation 2.15) to calculate

dg
. The minus was added to force αH to be positive for the expected blue shift with
dne
carrier-density.

7.3.1.3 Frequency modulation response curve
I can now calculate the eect of quantum well carriers on the frequency modulation
response. By combining Equations 7.33 and 7.38 I can express the frequency response
as:

∆νQW
vg dnr
αH
0 ∆ne
=−
∆ne =
ΓQW Gm
∆I
λ0 dne
8π
∆I
where

(7.39)

∆ne
is calculated directly from the small signal matrix 7.5. The exact expression
∆I

can be presented analytically using the same assumption we have used in section 7.1:

∆νQW
(s) =
∆I

h
c np,0
s + Bnp,0 + αT 1+ΓQW
+ 2CDn2p,0
QW c np,0
i
(s + 2ξωn ) (s + Bnp,0 ) τef1 f + s + 2CDn2p,0 + τef1 f + s ωn2

αH
Γ G ηi
8π QW m eVQW

s + τef1 f

(7.40)
The resulting response due to the plasma eect in the quantum wells can be seen
in Figure 7.4. Several prominent new features are evident:

92
1. For the thin 30nm spacer, the response resembles that of a conventional semiconductor laser [43]. A
at response up to a few hundred MHz, of magnitude
of roughly few hundred MHz/mA are very common for III-V lasers.

2. As the spacer thickness increases, say in the case of 100nm, the entire curve
maintains its general shape, but decreases in magnitude.

This is due to the

decreased overlap between the the mode and the QW. Changes in the QW's
refractive index have a smaller eect on the mode due to the low con
nement
factor.

3. The resonance frequency decreases with increasing spacer thickness, for the
same reasons that were discussed in section 7.2.

4. For very thick spacers, i.e., 150nm, the response curve changes: a shallow dip
due to FCA is revealed at ω =

τef f

5. In the case of the 150nm spacer, the magnitude of the response at low frequencies is comparable to the 100nm spacer, despite the reduced overlap with the
quantum wells. This is due to nonlinear loss. TPA and FCA act as eective gain
compression mechanisms; increased photon density increases nonlinear loss, and
QW carrier density has to grow to increase the gain, such that gain=loss.

7.3.2 The eects of free-carriers in silicon
As we have seen in the previous section, the plasma eect due to quantum well carriers
has a small impact on the frequency modulation response as we push the mode further
and further into the silicon. Figure 7.4 suggests an order of magnitude reduction in
DC frequency modulation compared to the typical 300 MHz/mA response [123, 78,
13, 126]. In the hybrid Si/III-V platform, the bulk of the mode is in silicon, especially
in our narrow-linewidth design approach. It is therefore important to consider the
impact of free-carriers in silicon on the frequency modulation.
As the pump current is modulated, the intra-cavity photon density follows the
intensity modulation response. TPA, which is considered instantaneous, tracks the

93
changes in the photon density and the density of the free-carriers in silicon are modulated. This in turn causes refractive index changes, due to the plasma eect, which
yields frequency chirping.

7.3.2.1 Plasma eects in silicon
Silicon modulators are studied intensively in the literature. A very common empirical
model that relates the carrier density in silicon to the refractive index at 1550nm has
the form [101, 81]:

∆nr = ξnSi = −8.8 · 10−22 nSi,e − 8.5 · 10−18 (nSi,p )0.8

(7.41)

−3
where nSi,e/p are the electrons and hole densities respectively in units of cm . In our
intrinsic silicon, I shall set the hole and electron densities as equal nSi,e = nSi,p = nSi .
The dierential refractive index change depends on the carrier density due to the
dierent eective masses of electrons and holes and is given by:

dnr
= ξ(nSi ) = −8.8 · 10−22 − 8.5 · 10−18 (nSi )−0.2
dnSi

(7.42)

in units of cm .

Frequency modulation response curve
The frequency chirp due to the refractive index modulation in silicon is given by using
Equations 7.33 and 7.41:

∆νSi
vg
∆nSi
= − ΓSi ξ
∆I
λ0
∆I

(7.43)

It can be calculated analytically from the system of dierential rate equations:

−2 λvg0 ΓSi ξ(nSi ) eVηQW
ΓQW Gm Dn2p,0
∆νQW
i
(s) =
∆I
(s + 2ξωn ) (s + Bnp,0 ) τef f + s + 2CDnp,0 + τef f + s ωn2

(7.44)

The eect of silicon free-carriers on the frequency modulation response is demonstrated in Figure 7.5. The response follows the three-pole transfer function discussed

Figure 7.5: Frequency modulation response due to free carriers in silicon for dierent
values of spacer thickness. αH = 7, I = 2 · Ith , QSi = 10

in Chapter 7.2. The
rst pole is at ω ≈

, while the other two poles are at the resoτef f

nance frequency, which decreases with increasing spacer thickness. The magnitude of
the response increases with spacer thickness. Since TPA is a nonlinear process and is

proportional to np , the dierential response is dependent on the photon density. The
thicker spacers have higher-Q and therefore higher stored photon density. Therefore,
the plasma eect in silicon is expected to be stronger for higher-Q designs, such as
the thick spacer.

7.3.3 The total frequency chirp
The combined frequency chirp due to QW and silicon plasma eects can be calculated
using the partial derivatives:

vg
∆ν = −
λ0

dnr
dnr
ΓQW
∆ne + ΓSi
∆nSi
dne
dnSi

(7.45)

The exact phases of ∆nQW and ∆nSi have to be taken into account in the addition.
Figures 7.6(a)-(c) show the (total) frequency modulation response vs. the same response in the absence of nonlinear eects.

TPA and free-carriers in silicon have a

(a)

(b)

(c)
Figure 7.6: Frequency modulation response for several dierent spacer thicknesses,
with and without nonlinear eects for αH = 7, I = 2 · Ith , QSi = 10 . (a) 30nm
spacer (b) 100nm spacer (c) 150nm spacer

Figure 7.7: Frequency modulation response for several dierent spacer thicknesses for
αH = 7, I = 2 · Ith , QSi = 106

major impact on the frequency response in the thicker spacer designs. A very clear
dip in the response curve, followed by a resonance, is a unique consequence of freecarrier-dispersion in silicon, and is very dierent from a conventional III-V, where
TPA is negligible.

The frequency modulation curves of the dierent spacer thick-

nesses are compared in Figure 7.7 The eect of pump current on the response curve
is shown in Figure 7.8(a)-(b) for thin and thick spacers.

In the thin 30nm spacer,

as the pump current increases, the eect of free-carriers becomes more prominent, as
indicated by the appearance of the extra pole before the resonance frequency. In the
thick 150nm spacer, as the pump current increases, the resonance is pushed to higher
frequencies, and the frequency response curve is altered accordingly.
In this chapter, I analyzed the the laser dynamics in both the intensity and the frequency response. The analysis predicts much lower relaxation resonance frequencies
than a conventional semiconductor laser. This is due to grossly reduced con
nement
in the QW, which reduces the induced transition rate. This is the base of this platform. The eect of free-carriers in silicon was also analyzed. It was predicted that
it will add a zero to the intensity modulation transfer function, and a unique dip to
the frequency modulation response. In the next chapter I will present experimental

(a)

(b)
Figure 7.8: Eect of pump current on frequency modulation response for two dierent
spacer thicknesses αH = 7, I = 2 · Ith , QSi = 10 . (a) 30nm spacer (b) 150nm spacer

98
results from fabricated devices and compare them to these predictions.

Chapter 8
Dynamic operation - Experimental
results
In this chapter, I will discuss the modulation response experiments that were performed with a number of dierent spacer designs. The experimental setup and procedures are described in detail in appendix B. The experimental results presented in
this chapter are the
rst published results for the dynamics of low active con
nement
hybrid Si/III-V lasers. They will be used to point out some of the special characteristics of the low-noise spacer design, and to probe and quantify some of the nonlinear
eects described in earlier chapters.

8.1 Intensity modulation response
Intensity modulation response experiments were conducted using the setup described
in appendix B.3. The intensity modulation transfer function HIM (ν) was calculated
by computing the ratio between the relative intensity modulation and the laser's input
current at a given modulation frequency:

HIM (ν) =

∆Pout 1
∆I
P0

(8.1)

The response of the driving circuitry was divided out from this calculation to isolate
the response of the laser only.
appendix B.3.2.

Details on this calibration process can be found in

100
Experimental results from intensity modulation experiments of dierent spacer
lasers and at dierent bias currents are shown in Figures 8.1-8.3. Each
gure shows the
measured magnitude and phase of the amplitude modulation (AM) transfer function.
Several interesting features are present in the AM transfer function of dierent spacer
lasers:

1. Resonance frequency position vs spacer thickness - As expected from the theoretical analysis, the resonance frequency shifts towards lower frequencies as the
overlap with the QW is reduced. The 150nm spacers show relaxation frequency
as low as ∼100MHz. This is one to two orders of magnitude lower than conventional III-V lasers, and to the best of our knowledge, the lowest-ever reported
for a semiconductor laser.

Figure 8.4 compares the AM response of dierent

spacer lasers for the same oset current from threshold. It is worth noting that
though the trend is in perfect agreement with the theory, the theory predicts
that the ratio between the resonance frequency at the 30nm spacer and the
150nm spacer should be:

ωn (30nm spacer)
ωn (150nm spacer)

ΓQW (30nm spacer)
≈4
ΓQW (150nm spacer)

(8.2)

where the con
nement factors used are based on Comsol simulation. Based on
the experimental AM curve, the ratio is

ωn (30nm spacer)
≈ 8, . Since theory preωn (150nm spacer)

dicts a square root relationship between con
nement and resonance frequency,
it might indicate that the 150nm spacer has lower con
nement ΓIII−V (by about
a factor of 4) than estimated by the simulation. If this is the case, and the simulation over-estimates the con
nement in the QW, it can explain the low yield
of the 150nm spacer and the zero yield of the 200nm spacer (the gain is reduced
and is no longer compensated by an increase of Q).

2. The presence of a zero of the transfer function for the 150nm spacer (Figure 8.3)
- What might look like very broad resonance in Figure 8.3 is in fact a zero of
the transfer function. This observation is consistent with both the shape of the

101

(a)

(b)
Figure 8.1: Intensity modulation response of 30nm spacer laser (Chip 1, bar 5, Slot
1, device 7) for dierent bias currents. 3.5mA current modulation. Measured with
New-Focus 1544B photodetector and HP 8722C RF network analyzer (a) Normalized
magnitude (b) Phase

102

(a)

(b)
Figure 8.2: Intensity modulation response of 100nm spacer laser (chip 1, bar 1, slo
t2, device 19) for dierent bias currents. 6mA current modulation. Low frequency
response (<50MHz) was measured using New-Focus 1544B photodetector and Agilent
4395A network analyzer. High frequency response (>50 MHz) was measured using
HP 8722C RF network analyzer (a) Normalized magnitude (b) Phase

103

(a)

(b)
Figure 8.3: Intensity modulation response of 150nm spacer laser (chip 1, bar 1, slot
2, device 19) for dierent bias currents. 3.3mA current modulation. Measured using
New-Focus 1544B photodetector and Agilent 4395A network analyzer (a) Normalized
magnitude (b) Phase

104

Figure 8.4:

Intensity modulation response (magnitude in a.u.)

of dierent spacer

lasers for pump current oset of 40mA.

magnitude response curve, and the phase-lead in the phase response. This is in
very good agreement with the theoretical analysis (see Figure 7.2), attributing
the zero to free-carrier-absorption. The very clear zero of the transfer function
at ν ≈ 5M Hz, can be used to experimentally estimate the eective lifetime of
carriers in Si, by comparing it to the theoretical transfer function of Equation
7.19.

For the AM response of the 150nm spacer, the resulting experimental

carrier lifetime is estimated to be:

τef f ≈ 30ns

(8.3)

This is, to the best of our knowledge, a novel experimental technique to measure
the eective lifetime of carriers in Si, and the
rst time it is measured in Si/IIIV lasers. As discussed in section 3.2.5, this value is comparable to previously
reported values in the literature, but is at the high end of the range. It indicates
that surface recombination velocity is diminished due to the high-quality of
surfaces and interfaces induced by the high-temperature anneal and oxidation.

3. Resonance frequency position vs pump current - The resonance frequency is

105

Figure 8.5:

Frequency modulation response of 30nm spacer laser (Chip 1, bar 5,

Slot 1, device 7) for dierent bias currents.

0.1mA current modulation.

Measured

with Optilab BPR-20-M balanced photodetector and HP 8722C RF network analyzer,
using MZI with FSR = 7.06GHz

pushed to higher frequencies with pump current, as expected from classic laser
dynamics theory.

4. Resonance frequency damping - All spacer lasers have damped relaxation resonance. This is expected due to the high-Q and TPA that suppresses intensity
peaks.

8.2 Frequency modulation response
The setup used to perform frequency modulation response measurements is described
in appendix B.4. The network analyzer's output was converted to units of GHz/mA
using the calibration procedure described in appendix B.4.2. Figures 8.5-8.7 show experimental frequency response curves from the three spacer designs, and for dierent
bias currents.

The frequency response curves show some very unique characteristics

that are very dierent from conventional III-V lasers. Some of the key observations
are:
1. The thin 30nm spacer has a classic response curve for semiconductor lasers.

106

Figure 8.6: Frequency modulation response of 100nm spacer laser (Chip 1, bar 1,
Slot 2, device 19) for dierent bias currents.

35nA current modulation.

Measured

with Optilab BPR-20-M balanced photodetector MZI with FSR = 1.56GHz.

Low

frequencies (<500 MHz) were measured using Agilent 4395A network analyzer and
high frequencies (>500MHz) using HP 8722C analyzer. Curves from the two analyzers
are plotted together without any additional post-processing (stitching)

Figure 8.7: Frequency modulation response of 150nm spacer laser (Chip 1, bar 1,
Slot 2, device 19) for dierent bias currents. 0.1mA current modulation. Measured
with Optilab BPR-20-M balanced photodetector and Agilent 4395A network analyzer,
using MZI with FSR = 1.56GHz

107

Figure 8.8: Frequency modulation response of dierent spacer lasers for pump current
oset of 50mA.

It shows a response of few hundred MHz/mA of plasma dispersion eect, and
a resonance peak at few GHz.

The relaxation follows a zero of the transfer

function as predicted from the (classic) rate equation analysis. This behavior is
very typical of III-V lasers, as expected from thin spacer designs.

2. In all lasers, the very low frequency (<1MHz) is dominated by 1/f-like response
that is attributed to thermal frequency drift.

3. The thicker spacer designs have a qualitatively dierent response: they feature
dips of the frequency response curve, prior to the relaxation frequency. This is
in very good qualitative agreement with the nonlinear rate equation analysis,
which attributes this feature to Si free-carrier-dispersion.

4. The thick 150nm spacer in Figure 8.7 demonstrates a wide range of bias currents
with a close to threshold measurement curve. This
gure shows a much lower
frequency response than at the higher bias currents. This is consistent with the
nonlinear rate equation analysis that attributed the rise in the response to the
accumulation of free-carriers in Si.

Figure 8.8 compares the three spacer design for a given oset from threshold.

108
was also evident from the intensity modulation response, the resonance frequency
shifts to extremely low frequencies when the mode is pushed further into silicon. It
also demonstrates the dierent dynamics of the 30nm spacer, which is closer to the
conventional design, compared to the thicker 100nm and 150nm spacer designs.
Despite the very good qualitative agreement with the theoretical analysis, there
are still some characteristics that are not explained by the nonlinear model:

1. The position of the frequency response dip - Theory predicts that this unique
dip will move to slightly higher frequencies with spacer thickness. In the experimental results, the dip is in fact pushed to lower frequencies.

2. The relative frequency response magnitude between the 100nm and 150nm spacers - The nonlinear model predicts that the thicker spacer will have a stronger
frequency response at the low frequencies than the thinner spacer. In the experimental results of Figure 8.8 the 100nm spacer
attens at value of ∼700MHz/mA
while the 150nm spacer does so at 350MHz/mA

It is worth noting that at the root of these discrepancies lies the assumption that
dierent spacer laser are identical in all parameters but spacer thicknesses. This is
very unlikely. The two lasers were fabricated separately; they have dierent threshold
currents, and hence operate at dierent temperatures.

Heat management is less

cient for the thick spacer design, due to the thermally isolating oxide layer. They
may also have dierent internal e
ciencies, due to fabrication variations in the ionimplant. For all of these reasons, it is di
cult to compare absolute values in dierent
spacer lasers. However, the unique characteristics that were outlined in the theoretical
analysis in Chapter 7 are observed in these lasers, and the trends within a speci
spacer design are in agreement with the theory.
In this chapter I presented experimental results demonstrating the modulation
response behavior of our lasers. The relaxation resonance was found to be at frequencies as low as ∼100MHz, validating the theoretical predictions. Predictions regarding
the role of free-carriers in Si were also validated: a zero of the intensity modulation

109
transfer function and a unique dip of the frequency modulation response were both
empirically observed. The location of the transfer function's zero was used to estimate that the eective lifetime of carriers in Si is ∼30ns. This is the
rst time this
lifetime has been measured in the context of hybrid Si/III-V lasers.

110

Chapter 9
Noise performance - Theoretical
analysis
Chapter 5 considered the noise due to the spontaneous emission process. The analysis there was based on the Schawlow-Townes formula, which was derived by using
Fermi's golden rule to calculate the rate of spontaneous decay and the coupling of this
spontaneous radiation to the lasing mode. The resulting frequency noise spectra from
the S-T formula is white. Further enhancement of the noise was considered by taking
into account the coupling between the imaginary and real parts of the refractive index through Henry's alpha parameter. The modi
ed Schawlow-Townes formula was
derived, but was still considered white noise.
As was seen in Chapter 7, the dynamic response of the laser cannot be simpli
ed by
making it a constant, especially at high frequencies, where resonance eects appear.
Furthermore, though the modi
ed Schawlow-Townes formula withstood the test of
time and was proven useful, our high-Q hybrid lasers have a new component which
is absent from conventional lasers: the free-carriers in silicon.

Fluctuations in the

number of free-carriers in Si are bound to add excess noise to the system, and must
be considered in the context of narrow-linewidth Si/III-V lasers. This chapter will
use a dierent method to carry out the analysis, the Langevin noise source approach,
to reconsider laser frequency noise in the presence of
uctuations of carriers, photons,
and temperature. Experimental results are compared to the theoretical predictions
in Chapter 10.

111

9.1 Methodology - Langevin noise sources
The use of Langevin noise terms in the context of lasers has many analogies to the
original use of Paul Langevin (1872-1946) in the context of Brownian motion of particles. The same results that Albert Einstein derived by using advanced mathematical
tools were derived by Langevin by using a simple, yet very dierent, technique. In his
own words, Langevin describes his work in
nitely more simple than Einstein's [54].
Despite the simpli
cation, his method is considered even more general than Einstein's
[54].

Langevin introduced a stochastic force that pushes the Brownian particle (in

the velocity space), a concept that since then was generalized into a very useful class
of methods in the study of continuous random processes.
Laser noise formulas were derived using the advanced mathematical tools of quantum mechanics.

The inclusion of quantum-mechanical operators in that analysis

makes this approach very cumbersome, and many simpli
cations must be made to
obtain analytical results.

The treatment of McCumber [63] in a paper from 1965

suggested the interpretation of quantum
uctuations of a system that is described
by rate equations in terms of
ctitious Langevin noise sources. This approach was
later justi
ed both in the context of classical self-sustained oscillators [53] and for
laser oscillators [51]. The same approach was further justi
ed and used to analyze
quantum noise of semiconductor lasers [67, 28, 130, 117].

Carrier
uctuations and

temperature diusion were also included to successfully predict the noise spectrum of
semiconductor lasers [50]. This approach provides a strong, yet simple, mathematical
tool to estimate laser noise spectrum and was found to agree well with experimental
results [28].
The strength of the Langevin noise approach is its simplicity:

one can simply

add a stochastic driving force term to the rate equations. These somewhat
ctitious
forces are engineered to recreate noise predicted from master equations or other
fundamental approaches. In the context of recombination and generation of photons
and carriers, the Langevin noise force that recreates quantum mechanical results is
simply shot-noise [63]. Quantum
uctuations can be described using delta-function

112
impulses with integrated intensity of one (in the total number description).

This

implies that one can infer the magnitude of the noise-term just by looking at the form
of the dynamic equations. A Langevin noise source suitable for diusion processes
can also be derived.

In the next section, Langevin noise terms are added to the

dierential rate equations, and their statistical properties are discussed.

9.2 Source of noise -
uctuations
Fluctuations of the carriers and photon density can be considered by inserting Langevin
noise driving terms in the system of rate Equations 7.5. I will set the current modulation to zero (∆I

= 0) and write the noise-driven rate equations in the matrix

form:

A11 A12 A13



∆ne

Fne


 

 
A21 A22 A23   ∆np  =  Fnp 

 
A31 A32 A33
∆nSi
FnSi

(9.1)

where the RHS represents the stochastic noise terms, and the matrix elements are the
same as in Equations 7.6-7.13. The statistical properties of the dierent noise terms
will be discussed in the following subsections. Since shot-noise describes the statistics
of many of these noise terms, it is convenient to look at a particle reservoir picture
of the dierent variables, as shown in Figure 9.1. In Figure 9.1 α is linear loss, αT P A
is loss due to TPA, αF CA is loss due to FCA, ηi is the internal e
ciency, and ηo is
output mirror coupling. R21 is the rate of stimulated emission and R12 is the rate of
stimulated absorption. The two are related to the modal gain using:

(R21 − R12 ) Vp = ΓQW Gm (ne )np Vp

(9.2)

113

Figure 9.1: Particle reservoir picture of the system

The following relationships can be inferred from the steady-state solution:

Np αT P A = βT hνvg2 MT P A Γ2Si n2p Vp

(9.3)

Np αF CA = vg σa nSi ΓSi np Vp

(9.4)

RSP VP = ΓQW Gm (ne )nsp

(9.5)

R21 VP = RSP VP · np VP

(9.6)

NP (αT P A + αF CA + α) + R12 VP = R21 VP + RSP VP

(9.7)

By normalizing the dierential rate equations to total numbers, as done in the
particle reservoir picture, we can infer the magnitude of the noise simply through the
shot-noise. The PSD of a shot-noise process can be deduced from the delta function
correlation assumed for shot-noise:

< Fi (t)Fj (t − τ ) >= Sij δ(τ )

(9.8)

where Sij de
nes the correlation strength between the two noise sources. The spectral
density of such a process is white (constant) with magnitude equal to the correlation

114
strength:

WFi Fj = Sij

(9.9)

Our small-signal analysis was conducted in the Fourier domain, such that all quantities are frequency dependent. A convenient way to calculate spectral densities using
quantities that are expressed in the frequency domain can be obtained by
rst considering the ensemble average:

< Fi (ω)Fj∗ (ω ) >=<

−jωτ1

Fi (τ1 )e

Z Z

Fj∗ (ω )ejω τ2 dτ2 >

dτ1

(9.10)

< Fi (τ1 )Fj∗ (τ2 ) > e−j(ωτ1 −ω τ2 ) dτ1 dτ2

Assuming stationary and ergodic processes the cross-correlation only depends on the
time dierence τ = τ1 − τ2 and we can write:

< Fi (ω)Fj∗ (ω ) >=

< Fi (t + τ )Fj∗ (t) > e−jωτ dτ

e−j(ω−ω )t dt
(9.11)

< Fi (t + τ )Fj∗ (t) > e−jωτ dτ · 2πδ(ω − ω )

Using the WienerKhinchin theorem and the the expression in Equation 9.11 we can
express the PSD of the dynamic variables ∆ni using [13]:

W∆ni =
2π

< ∆ni (ω)∆ni (ω )∗ > dω

(9.12)

for which the correlations of the Langevin noise source terms are given by:

< Fi (ω)Fj (ω )∗ >= 2πSij δ(ω − ω )

(9.13)

If we write the solution of the matrix 9.1 using a sum:

∆ni =

Aik (ω)Fk

(9.14)

then when we calculate the PSD using Equation 9.12, we end up with a summation

115
of the form:

W∆ni =

Sjj |Aij |2 + 2

j=1

Re {Aik Aij } Skj

(9.15)

k6=j

The correlations Sii can be calculated simply by inspecting the rates of particle

in
out
owing in (Ri ) and out (Ri ) of the reservoir:

< Sii >=
and the cross correlations

Riin +

Riout

(9.16)

Sij by the rates in which the two reservoirs, i and j ,

exchange particles:

< Sij >= −

Ri→j +

Rj→i

(9.17)

All in units of total number of particles, and the minus sign is due to the negative
correlation (reservoir i receives a particle, while reservoir j loses one).

For exam-

ple, inspection of the particle reservoir picture in Figure 9.1 suggests the following
correlation strength for the photon density
uctuations:

Vp2 < F nP FnP >= (R12 + R21 ) VP + RSP VP + np Vp (α + αT P A + αF CA )

(9.18)

Simplifying the expression using the steady-state conditions and following this procedure for the rest of the correlations yields:

< FnP FnP > = 2RSP nP +
VP
2R np 2ΓQW Gm np
2ηi I
+ 2SP −
< Fne Fne > =
eVQW
Γgeom
Γgeom VQW

βT hνvg2 MT P A n2p
< FnSi FnSi > =
VP

(9.19)

(9.20)

(9.21)

Notice that the auto-correlation of the Si carriers assumes that the eective lifetime

τef f describes recombination-like processes only. This is an approximation that assigns a recombination lifetime to a diusion process. Re
ned approximation will be
introduced in relevant sections.

116
The cross-correlations can be shown to be:

ΓQW Gm nP
2RSP nP
1+
< Fne FnP > = −
Γgeom
2nP VP
Γgeom VP
βT hνvg ΓSi MT P A np
< FnP FnSi > = −
2VP
< Fne FnSi > = 0

(9.22)

(9.23)

(9.24)

The square root in the second equation is due to the fact that for each two-photon
pair, only one Si electron is generated. The zero correlation between QW and silicon
electrons (third equation) is due to the fact that the two only interact through the
mediation of photons in a
nite bandwidth, and the shot-noise impulses are assumed
instantaneous.
Notice that we have assumed shot-noise statistics for all processes that involve
generation or annihilation of particles.

Though this is considered generally correct

in the linear regime (or at least provides correct results), it wasn't proven in this
work that this holds for nonlinear loss processes as well. In fact, it is not accurate
for multi-particle processes such as TPA. For example, two-photon-absorption has
been shown to produce sub-Poissonian light [23, 36]. An improved statistical model
can be derived for TPA, at the cost of increased complexity. However, it is shown in
this work that TPA is almost negligible compared to FCA, which is a linear process.
Moreover, strong squeezing is only evident in resonant nonlinearities, or ultra-high
light intensities [18]. The analysis implies that any squeezing-like eects in this platform would be weak, and in any case masked by free-carrier eects. I therefore treat

uctuations due to TPA as any other loss with shot-noise statistics.

9.2.1 Photon density
Next, I will solve the small-signal, Langevin-force-driven, linear set of equations (matrix 9.1) to get the photon density ∆nP , and calculate its PSD using Equation 9.12
and the correlation above. Recalling that A31 = A13 = 0, and using Cramer's rule,

117
we get the following expression:

Wnp =

{|A21 A33 |2 < Fne Fne > +|A11 A33 |2 < Fnp Fnp >
|H0 (ω)|2

+ |A23 A11 |2 < FnSi FnSi > −2Re {A21 A33 A∗11 A∗33 } < Fne Fnp >

(9.25)

− 2Re {A11 A33 A∗11 A∗23 } < Fnp FnSi >}
where the function H0 (ω) is de
ned:

H0 (ω) = (jω + 2ξωn ) (jω + Bnp,0 )

τef f

+ jω

+ 2CDn2p,0

τef f

+ jω ωn2
(9.26)

for which parameters are de
ned in Equations 7.20-7.22.
The expression for the PSD of the photon density in Equation 9.25 has two terms
which are unique to the hybrid platform, and link
uctuations of Si carrier density
to
uctuations of photon density.

The other terms, which don't involve the TPA-

generated carriers, are compared in Figure 9.2(a). This
gure demonstrates that the
negative cross-correlation between QW-carrier and photons decreases
uctuations of
photon density below its intrinsic level. This eect, which is related to gain saturation,
is the reason that lasers have suppressed intensity noise.

Figure 9.2(b) adds the

contribution of free-carriers in Si. It shows that
uctuations of density of carriers in
Si has a minor eect, and hardly changes the intensity noise from what we would
expect in a conventional III-V laser.

118

(a)

(b)
Figure 9.2:

PSD of the photon density (150nm spacer,

QSi = 106 , I = 4Ith ) (a)

Comparison of conventional terms (terms which don't involve TPA-generated freecarriers in Si) (b) Comparison of these conventional terms to free-carrier related terms

119

Figure 9.3:

Comparison of the dierent terms of the PSD of QW-carrier density
(150nm spacer, QSi = 10 , I = 4Ith )

9.2.2 Carriers in the quantum wells
The same method that was used for the photon density will now be used for the QW
carrier density. The following expression for the PSD is derived after some algebra:

Wne =

{|A22 A33 − A23 A32 |2 < Fne Fne > +|A12 A33 |2 < Fnp Fnp >
|H0 (ω)|

+ |A12 A23 |2 < FnSi FnSi > −2Re {(A22 A33 − A23 A32 ) A∗12 A∗33 } < Fne Fnp >
− 2Re {A12 A33 A∗12 A∗23 } < Fnp FnSi >}
(9.27)

Figure 9.3 shows the magnitude of the dierent terms aecting the PSD of the density
of QW carriers. Interestingly, the dominant term in that spectrum is the one due to

uctuation of photon-density < FnP FnP

> . Inherent
uctuations of QW electrons

(the < Fne Fne > term) are small due to gain clamping. We will shortly see that this
behavior gives the Henry linewidth-enhancement a multiplicative nature, rather than
an additive one.

120

Figure 9.4:

Noise spectrum of silicon carrier density for shot-noise model (150nm
spacer, QSi = 10 , I = 4Ith )

9.2.3 Free-carriers in silicon
In principle, the same treatment that was used above for the photons and the QW
carriers could be used for the free-carriers in Si. However, this approach would incorrectly treat the eective lifetime in Si, τef f , as a time constant for recombination. As
we recall from Chapter 3, this time constant also captures the average time in which
carriers diuse out of the mode's area. Though this was suitable for the steady-state
analysis, it might not be adequate for noise estimation.

Despite that fact, we will

ignore diusion for the time being, and examine the spectral content of the noise, assuming shot-noise statistics. The resulting PSD is shown in Figure 9.4. Note that in
the case of the Si carriers, the dominant contribution is from the intrinsic
uctuation
of Si carriers (the < FnSi FnSi > term ). The other correlation terms contribute very
little and can be neglected.
In the context of
uctuations, diusion of particles have completely dierent statistics than recombination [118].

Recombination-generation has shot-noise statistics,

with zero correlation between dierent points of space and time. Diusion, however,
a process that is stochastic in nature, has a unique and dierent correlation. If the

121
carrier density obeys the diusion equation:

dnSi
− D∇2 nSi (t, r) =fD (x, t)
dt

(9.28)

Where the term on the RHS is the Langevin force, then the appropriate correlation
for the diusion-driven noise source is [122, 119]:

< fD (t, r)fD∗ (t , r ) >= −2D∇r · ∇r0 nSi (r)δ 3 (r − r ) δ(t − t )

(9.29)

From the previous discussion of Figure 9.4, we have concluded that only inherent

uctuations of the Si carrier should be considered, and all other contributions can be
neglected. Therefore, I adopt the following strategy to re
ne the spectra of Si carrier
density and consider diusion, rather than just recombination:

1. Return to the original generation-recombination-diusion equation (see Equation 3.23):

βT hνΓSi Vg2 2
dnSi
np −
dt

+2
τb

nSi + Da ∇2 nSi

(9.30)

2. Take the Fourier transform in both time and space of a linearized small-signal
equation, and introduce two Langevin forces:

∆nSi =

Where now

fnSi (ω, k) + fD (ω, k)
+ Da |k|2
jω + τr,Si

(9.31)

is time constant for recombination only (sum of bulk and surface
τr,Si

recombination as in the bracket in Equation 9.30).

fnSi is the Langevin force

due to generation-recombination, while fD is due to diusion. Notice that for
purposes of simplicity, we have decoupled this equation from the other rate
equations, by neglecting the dependence on the photon density. The generation
term is considered a constant in this treatment. It will contribute to shot noise,
but its dynamics are ignored. This is motivated by the previous analysis, that

122
showed that the spectra is dominated by the low-pass
lter function, and only
minor adjustments are required around the relaxation resonance.

I will use the result of Equation 9.31 to calculate the frequency noise in the following
sections.

9.2.4 Temperature
Silicon has a large thermo-optic coe
cient of about ηT

dn
= dT
= 1.8 · 10−4 [K−1 ] at

room temperature [45], comparable to that of InP and other III-V materials. Thermal

uctuations are therefore easily coupled to the frequency noise through the thermooptic eect. This section will oer a model, based on the Langevin force approach, to
take temperature
uctuations into account. I will consider both inherent temperature

uctuations, and those induced by thermal dissipation.
The temperature pro
le in the laser cavity obeys the non-steady-state heat equation [96]:

ch ρ

∂T
= q + κ∇2 T
∂t

where ch is the speci
c heat in units[

(9.32)

Joul
kg·K

] κ is the thermal conductivity in units

[ secJoul
], and ρ is the density in units [ mkg3 ]. The variable q represents the heat source.
·m·K
In this work, I will only consider heat that is generated due to dissipation in Si, such
that q can be represented using the absorbance αz (units of [

q = αz I

] ) and the intensity I :

(9.33)

I will consider the following heat-generating mechanisms:

1. Every non-radiative recombination event generates heat by the amount of energy
quanta absorbed.

2. Energy dissipated through free-carrier absorption fully converted to heat.

I will also assume that the temperature gradients are small enough within the mode's
volume, such that it will be approximated as constant. No knowledge of the exact

123
temperature pro
le is required.
Expressing the two heat mechanisms explicitly, we get the following heat term:

q = hνvg σa nsi ΓSi np +

2hνnsi
τr,Si

(9.34)

where in the second term we used the two-photon energy instead of the silicon's
bandgap energy, to take into account the thermalization process of carriers before
they relax to the band-edge.
Linearizing the heat equation and introducing a Langevin noise force, we can
write:

∂∆T
= q1 ∆np + q2 ∆nSi + DT ∇2 ∆T − ∆T + fT
∂t
τT
where

DT =

is the temperature diusion coe
cient,
ch ρ

(9.35)

fT is the temperature

Langevin force. The parameters q1 and q2 are de
ned as:

q1 =
q2 =
The parameter

hνvg σa nsi,0 ΓSi
ch ρ
hνvg σa ΓSi np,0 + τ2hν
r,Si

(9.36)

(9.37)

ch ρ

is the temperature decay rate [55]. It was arti
cially inserted in the
τT

equation, to account for heat
ow out of the cavity and into the heat sink. This was
necessary since we don't treat the boundary conditions in the following derivation.
The temperature
uctuations are eected by both photon density
uctuations and
Si-carrier-density
uctuations, as described by the above equation. The correlations
of the Langevin force for the temperature can be expressed as in other diusion
processes:

< fT (t, r)fT∗ (t , r ) >= A · ∇r · ∇r0

T (r)δ (r − r ) δ(t − t )

(9.38)

To
nd the unknown A, we require that for constant average temperature the root
mean square (RMS) calculated using the above expression would agree with familiar

124
statistical physics results for a body in contact with a thermal bath [49]:

< ∆T 2 >=

KB T 2
ρch V

(9.39)

By integrating expression 9.38 over volume V, for constant temperature T, and equating the result to the above RMS, we
nd the constant A:

2DT KB T
ρch

(9.40)

9.3 Frequency noise
The lasing frequency is set by the average refractive index, as sensed by the optical
mode.

Fluctuations in the local refractive index will be averaged over the mode's

area, and the
uctuations of the eective refractive index can be well-approximated
using:

ZZZ
∆nef f (t) =

|e(x, y, z)|2 ∆n(x, y, z, t)dxdydz

(9.41)

Where e(x, y, z) is the electric
eld pro
le of the mode, normalized such that:

ZZZ

|e(x, y, z)|2 dxdydz = 1

(9.42)

Fluctuations in the frequency can be approximated using the eective refractive index,
as was done for the frequency response calculations:

∆ν = −

vg
∆nef f
λ0

(9.43)

These
uctuations can be broken apart to contributions from the dispersion plasma
eect (of both QW carriers and Si carriers) and the thermo-optic eect. This can be
approximated using the respective con
nement factors:

vg
∆ν = −
λ0

∂∆nef f
∂∆nef f
∂∆nef f
ΓQW
∆ne + ΓSi
∆nSi + ΓSi
∆T
∂ne
∂nSi
∂T

(9.44)

125
Expressing the dispersion plasma eect in the QW using Henry's alpha parameter (see
Equation 7.38), we get the following PSD of the frequency noise due to
uctuations
in ne , nSi and T :

(
2
2
0 2
Gm
vg
vg
W∆ν =
ΓQW αH
W∆ne +
ξΓSi W∆nSi +
ΓSi ηT W∆T
4π
λ0
λ0

(9.45)

We have seen in previous sections that correlations between ∆ne and ∆nSi are very
weak and hardly contribute to the total noise spectrum, and so I neglected them in the
above expression. In the following sub-sections, I will use this expression to examine
the contribution of the dierent
uctuation mechanisms, and attempt to predict the
resulting PSD of the frequency noise. I will use the results from previous sections for
the noise spectrum of carriers.

9.3.1 Spontaneous emission
The expression for the frequency noise that I have derived in the previous section
considers the contribution of carriers and temperature
uctuations. However, it lacks
a very important component: spontaneous emission. Spontaneous emission aects the
frequency spectrum by injecting photons with random phase, which is uncorrelated
to the mode's phase. In the spirit of this chapter, I will introduce a phase Langevin
force, Fφ , to account for this random-walk of the phase:

d∆φ
= 2π∆ν = Fφ
dt

(9.46)

The RMS of the phase
uctuations is related to the photon density
uctuations using
[13, 31]:

< Fφ Fφ >=

< FnP FnP >
4n2p

(9.47)

126
Recall that we have previously calculated the correlation on the RHS of this equation.
We therefore obtain a Lorentzian linewidth:

∆ν =

2RSP nP + V1P
2π · 4n2p

RSP
4πnp

(9.48)

recreating the familiar S-T linewidth (see Equations 5.14,2.6).

9.3.2 Henry's linewidth enhancement
I have calculated the
uctuations of QW carrier density in section 9.2.2 and we have
seen that the dominant term is the one proportional to < FnP FnP

>. The fact that

the spontaneous emission noise is also proportional to this term gives rise to the
linewidth enhancement factor. We can write the contribution of both spontaneous
emission and dispersion plasma eect in the QW using:

0 2
|A12 |2 |iω + τef1 f |2
1 + 4n2p ΓQW αH m
|H0 (ω)|2

W∆ = W∆νS−T

(9.49)

where W∆νS−T is the S-T white noise level, and the term in bracket is the Henry
enhancement. Figure 9.5 shows the contribution of spontaneous emission (which we
derived using shot-noise model for the photon density) plus the contribution of QW
carrier density
uctuations. This
gure shows that, as expected, QW carrier
uctuations give rise to the Henry linewidth enhancement, which drops o to the SchawlowTownes linewidth for frequencies above the relation resonance. It also demonstrates
the advantage of having the resonance at very low frequencies, as in the thick spacer
designs; the noise becomes truly quantum-mechanically limited (the S-T limit) at
frequencies as low as ∼1GHz. Interestingly, the zero of the transfer function, which
was predicted for the intensity modulation, doesn't appear in the noise spectrum.
This is also evident in Figure 9.3, where the cross-correlation between
uctuations
in photon and electron density (the < Fne FnP
transfer function in the dominant term.

> term ) cancels out the zero of the

127

Figure 9.5: Frequency noise due to spontaneous emission and QW carrier density

uctuations using Langevin shot-noise model. QSi = 10 , I = 4Ith , αH = 3

9.3.3 Noise due to the plasma eect in silicon
In this section I will consider only noise due to
uctuations of free-carriers in Si that
aect the frequency noise spectrum through the plasma eect (free-carrier-dispersion):

vg
∆ν = −
λ0

∂∆nef f
ΓSi
∆nSi
∂nSi

(9.50)

I will compare the resulting spectrum from two dierent models:

1. Shot-noise only model - In this model I treat the eective lifetime of carriers in
Si as a recombination lifetime. I will neglect carrier diusion altogether, and
assign recombination properties to the carriers that diuse away from the mode.

2. Shot-noise and diusion noise - I consider diusion of carriers together with recombination (bulk and surface). However, I neglect noise due to cross-correlation
of photons and Si carriers, and only treat intrinsic carrier
uctuations.

In the second model, we need to take into account spatial variation. The correlation
of Langevin noise force (shot-noise) now becomes [50]:

< fnSi (t, r)fn∗Si (t , r ) >=

nSi (r) βT hνΓSi Vg2 2
np (r) δ 3 (r − r )δ(t − t )
τr,Si

(9.51)

128
while the correlation for the Langevin diusion source is:

< fD (t, r)fD∗ (t , r ) >= −2Da ∇r · ∇r0

nSi (r)δ (r − r ) δ(t − t )

(9.52)

To estimate the frequency
uctuation, I need to know the optical mode pro
le i =

|e(r)|2 . For purposes of simplicity, I will assume a Gaussian pro
le with the same
FWHM as the simulated mode.

I will then compute the spatial Fourier transform

of the energy distribution I(kx , ky , kz ). Notice that the Langevin force correlations in
the Fourier domain have the form:

< fnSi (ω, k)fn∗Si (ω , k ) >=

nSi (k + k ) βT hνΓSi Vg2
G(k + k ) 2πδ(ω − ω )
τr,Si
(9.53)

< fD (ω, k)fD∗ (ω , k ) > = −2Da k · k nSi (k + k )2πδ(ω − ω )

(9.54)

where nSi (k), G(k) are the Fourier transforms of nSi (r), nP (r), respectively.

The

frequency
uctuations can be calculated by taking the Fourier transform of Equation
9.50, and by using the Fourier-domain Si carrier
uctuations from Equation 9.31.
Finally, I will calculate the PSD of the frequency
uctuations using Equation 9.12,
and plug in the correlations of Equations 9.51 and 9.52. Notice that this can be done
numerically using no other assumptions. However, this brute-force approach requires
calculating six nested integrals numerically. Performing this with su
cient resolution
to obtain a reliable result is almost prohibitive, especially if one wishes to sweep
other parameters, e.g., the pump power. I will therefore simplify this six-dimensional
integral by using the following assumptions:

1. Instead of assuming a Gaussian pro
le I assume a three-dimensional sinc function for the optical mode with the same width. The Fourier transform of a sinc
function is a window function. This will change the limits of integration to a

nite window (an ellipsoid), and make the integrand managable.

2. I assume that the Si carrier density pro
le nSi (r) is constant over the mode

129

Figure 9.6:

The eect of Si-carrier
uctuation on the frequency noise spectrum:

Eective recombination model vs recombination-diusion model.
QSi = 106 , I = 2Ith

150nm Spacer,

pro
le. We have previously seen in Chapter 3 that this is a good approximation
due to the large diusion length of carriers in Si. The Fourier transform will
be a delta function (compared to the Fourier transform of the optical mode),
which will reduce the dimensionality of the resulting integral.

3. I will work in spherical coordinates since the term |k| appears everywhere in
the integration. The only non-spherical symmetric part of the integration is its
boundary; the integration is over an ellipsoid and not a sphere.

Figure 9.6 compares frequency noise from the recombination-only model and the
recombination-diusion model. Some interesting conclusions can be drawn from this

gure:

1. At the very low frequencies they both have same order of magnitude
at response. In the steady-state, the rate of carrier generation due to TPA equals
the rate of carrier (eective) recombination.

In both models, the generation

term is identical so the shot-noise generated by this process is likewise identical.

2. The recombination-diusion model has a pole at a much lower frequency. This

130

Figure 9.7: Frequency noise spectrum due to Si carrier
uctuations for dierent pump
currents. 150nm Spacer, QSi = 10

is due to the much lower recombination lifetime of carriers in Si, compared to
the eective lifetime that includes diusion.

3. The high-frequency noise due to diusion drops o much more slowly than that
of the recombination-only model.

This is a characteristic feature of diusion

noise. It demonstrates why it is so important to take the diusion noise into
consideration; at the high frequencies (∼1 GHz) it might be dominant.

In the example of Figure 9.6 the diusion noise at 1GHz for the 150nm spacer corresponds to a linewidth of 200Hz. In the high-Q platform, this is comparable to the S-T
linewidth we would expect from this laser. This suggests that diusion noise might
be a dominant noise-source in our platform.
Figure 9.7 shows the eect of pump current on frequency noise due to Si carrier

uctuations.

As we increase the pump current, the intra-cavity optical intensity

increases together with the two-photon-absorption rate. This increases the free-carrier
density in silicon, and their
uctuations increasingly degrade the frequency spectrum.
The increase in noise level with pump current highlights an important feature of
the high-Q hybrid Si/III-V platform: Noise that stems from
uctuations of carriers
generated at the silicon slab due to TPA, does not behave as the noise of a conventional

131

Figure 9.8: Frequency noise spectrum due to Si carrier
uctuations for dierent spac6
ers. QSi = 10 , I = 2 · Ith

laser. In a conventional laser, the linewidth scales inversely with power. Fluctuation
of the QW carriers yield the Henry linewidth enhancement, but the overall noise
spectrum goes down at high frequencies with increasing power. However, in the highQ hybrid platform the noise due to silicon carriers increases with power. This work
therefore predicts that as we increase the laser's power, the linewidth will eventually
start to broaden.

This model predicts a sub-KHz equivalent linewidth as a rough

limit. A better estimate will follow in the next sections. Figure 9.8 shows that this
conclusion also holds for spacer thickness. For the thicker spacer, as we attempt to
reduce loss and store more photons in the laser cavity,
uctuations of Si carriers limit
the achievable linewidth.

9.3.4 Noise due to the thermo-optic eect in silicon
In previous sections I have considered the role of
uctuations in QW and Si carrier
densities that contribute to the frequency noise through the plasma eect. However,
the index of refraction of the laser's medium is also temperature-dependent, and it is
important to investigate the role of temperature
uctuations on the frequency noise
spectrum. Following the discussion on components of the frequency noise spectrum

132
in section 9.3, the contribution of temperature
uctuations can be written as:

W∆ν =

vg
ΓSi ηT
λ0

2
W∆T

where ηT is the thermo-optic coe
cient of silicon, and W∆T is the temperature
uctuations PSD. The temperature diusion formula in Equation 9.35 can be expressed
in the Fourier plane as:

∆T (ω, k) =

q1
q2
∆n
(ω,
k)
∆nSi (ω, k)
jω + τT + DT |k|2
jω + τT + DT |k|2
fT (ω, k)
jω + τ1T + DT |k|2

(9.55)

The PSD of the temperature
uctuations is calculated as before, using Equation 9.12,
with the Fourier domain correlation for the temperature derived from Equation 9.38:

< fT (ω, k)fT∗ (ω , k ) >=

2DT KB T 2
k · k (2π)3 δ 3 (k + k ) 2πδ(ω − ω )
ρch

(9.56)

where I have assumed a constant temperature pro
le.
Equation 9.55 explicitly shows the contribution of three components to the temperature
uctuations:

1. Temperature
uctuations due to photon density
uctuations - Since heat is generated in the process of FCA,
uctuations in the photon density also contribute
to temperature
uctuations.

2. Temperature
uctuations due to Si carrier density
uctuations - Heat is generated during the FCA process and non-radiative recombination in Si. Fluctuations in carrier density and the discrete nature of heat-generating recombination
induce temperature
uctuations.

3. Inherent temperature
uctuations - The temperature of the laser cavity is set
by the coupling to the thermal bath through a stochastic process.

A cavity

of
nite volume will have a temperature probability distribution with a
nite

133

Figure 9.9: The dierent components of the frequency noise spectrum due to temper6
ature
uctuations (150nm spacer, QSi = 10 , I = 2 · Ith )

width.

I can now numerically calculate the resulting frequency noise spectrum.

I use the

same approximations as in section 9.3.3. Figure 9.9 shows the dierent components
of the frequency noise that are temperature related. It demonstrates that the noise
at low frequencies is dominated by
uctuation of Si carriers that induce temperature

uctuations.

The low frequency behavior quickly decreases with frequency, and at

intermediate frequencies the noise is dominated by the inherent temperature
uctuations of the cavity. Figure 9.10 shows the temperature-related noise spectrum for
dierent spacer designs. This
gure shows that due to Si carriers, the low-frequency
component of the frequency noise spectrum is higher at thicker spacers.

9.3.5 Total noise spectrum
I can now combine all the sources of
uctuations and calculate the total frequency
noise spectrum.

In previous sections I have broken the spectrum into three parts:

spontaneous emission and QW carriers noise, free-carrier dispersion noise in Si, and
temperature related noise.

The latter included both temperature
uctuations that

are inherent to the cavity and those which stem from Si carriers that have generated

134

Figure 9.10: Frequency noise related to temperature
uctuations for dierent spacer
designs (QSi = 10 , I = 2 · Ith )

heat. In this section, I will re-order those components and group together all the noise
components that originate from free-carriers in Si, both through the plasma eect
and through the thermo-optic eect. Figure 9.11 shows the predicted frequency noise
spectrum and the contribution of dierent elements to that noise for dierent spacer
designs.

This
gure demonstrates that for the thick spacer designs, temperature-

related noise masks the S-T noise, which is revealed only at high frequencies (∼1
GHz). Figure 9.12 compares predictions from dierent spacer designs, and reveals that
thick spacer designs have very similar noise performance except at high frequencies,
where the S-T is reached. Finally, Figure 9.13 compares predictions for the 150nm
spacer at dierent pump powers. It is evident from this
gure that for the aggressive
150nm design, this model predicts that increasing the pump power will broaden the
linewidth instead of narrowing it. The S-T
oor is not reached even at frequencies
as high as 10GHz. It suggests that an equivalent linewidth of a few hundred Hertz
might be the limit of this platform, due to free-carriers in silicon.
In this chapter I used the Langevin force approach to predict the noise performance of our lasers. It was found that both inherent temperature
uctuations and

uctuations of free-carriers in Si aect noise performance. It was predicted that free-

135

(a)

(b)

(c)
Figure 9.11: The dierent components of the total frequency noise spectrum (QSi =
106 , I = 2 · Ith ). (a) 150nm spacer (b) 100nm spacer (c) 30nm spacer

136

Figure 9.12: Predicted frequency noise spectrum of dierent spacer designs (QSi =
106 , I = 2 · Ith )

Figure 9.13:

Predicted frequency noise spectrum for the 150nm spacer designs at
dierent pump powers (QSi = 10 )

137
carriers in Si will limit the achievable noise
oor, especially at high pump powers. In
the next chapter, I will present experimental frequency noise spectra and compare
them to these predictions.

138

Chapter 10
Noise performance - Experimental
results
Frequency noise spectra were measured for dierent spacer designs at dierent pump
powers. The exact experimental setup and calibration process are described in detail
in appendix B.5, while the laser design fabrication process is described in A.
The spectrum I measured using the RF spectrum analyzer is a single sided spectrum, whereas the analysis in the previous section is given as a double sided spectrum.
In this section, I will convert the y-axis of the PSD of the frequency noise to an equivalent Lorentzian white-noise linewidth. The physical meaning of this metric can be
interpreted using the following analogy: suppose that our laser has a certain level of
noise at a given frequency. A (theoretical) laser, that has only white frequency noise
of that level, would have had a Lorentzian line-shape, with width equivalent to the
new y-axis. Such a conversion requires only multiplication by π for the single-sided
spectrum, or by 2π for the double-sided spectrum (the reader can convert the predictions from the previous chapter to these units by multiplying the predicted spectrum
by 2π ).
The output power of the lasers varied from design to design. However, for many
of the measured narrow-linewidth lasers, the output power (in the
ber) was less
than 1mW. Since these narrow-linewidth lasers have very little noise at the high
frequencies we are interested in, external ampli
cation was needed to overcome the
balanced-photodetector dark noise (for further discussion of that point, see appendix

139

Figure 10.1: Eect of BOA ampli
cation on frequency noise spectrum. All data was
taken with the same 150nm spacer at constant laser pump current. Only BOA pump
current changed from curve to curve

B).

We have tested our devices with both booster-optical-ampli
er (BOA) and

Erbium-doped-
ber-ampli
er (EDFA). As can be seen in Figure 10.1, the BOA adds
phase noise to the signal. For the same input signal, higher ampli
cation yields higher
frequency noise. This noise is manifested as an increasing noise spectrum curve , that
is only evident at the high frequencies. Unfortunately, this noise artifact due to ampli
cation is at frequencies and noise levels that are of interest to us.

Figure 10.1

also demonstrates the measurement calibration process. The red curves in the
gure
contain the MZI sinc

transfer function. The blue and pink curves are the calibrated

spectra after deconvolving the sinc

from the measurement. This calibration process

is further described in appendix B.5. In the spectra I present in the remainder of this
chapter I will omit the raw oscillating data, and will only present the calibrated data.
Figure 10.2 shows that the contaminating ampli
er phase noise is also present with
the EDFA ampli
cation. However, the noise level is slightly lower with the EDFA.
This is expected, as the EDFA has a lower speci
ed noise-
gure than the BOA. For
that reason, all the results presented from now on are taken with EDFA ampli
cation. However, the measurement is still limited by EDFA-induced noise for the 150nm
spacer, as was evident from Figure 10.2, and the laser noise-
oor cannot be observed.

140

Figure 10.2: Frequency noise of 150nm spacer, ampli
ed using BOA and EDFA

Therefore, the results presented here for the 150nm spacer should be interpreted as
an upper limit of the noise.
Figure 10.3 shows frequency noise spectra at dierent pump powers for the three
spacer designs. Several interesting observations can be made from this
gure:
1. The 150nm spacer has an equivalent S-T linewidth of ∼1KHz. In fact, this is
only limited by the ampli
er noise. Since the ampli
er noise becomes dominant
for frequencies higher than the knee frequency, we can estimate that the laser's
noise is sub-KHz (few hundred Hz). Such low noise for a semiconductor laser is
unprecedented.

2. For both the 100nm and the 150nm spacers, increasing the pump current doesn't
lower the noise
oor at all. In fact, it increases the lower frequency noise level,
in agreement with the theory that attributed that behavior to
uctuations of
free-carriers in Si.

3. The 30nm spacer laser shows behavior that is closer to that predicted for a
conventional laser. The (modi
ed) S-T noise
oor is observed, and decreases
with increasing pump power, as expected from theory.

4. The 30nm spacer shows the relaxation resonance when operated close to thresh-

141

(a)

(b)

(c)
Figure 10.3: Frequency noise spectrum vs. pump current for dierent spacer designs
(a) 150nm spacer (threshold @ 66mA) (b) 100nm spacer (threshold @ 28mA) (c)
30nm spacer (threshold @ 55mA)

142

Figure 10.4: Frequency noise spectrum for dierent spacer designs at I − Ith = 33mA

old. We can't see the quantum S-T noise
oor due to ampli
er noise, but we
do see that the enhanced S-T level drops by at least a factor of 16.

We can

therefore estimate a lower bound for the Henry linewidth enhancement factor
for this laser:

αH > 4

(10.1)

Figure 10.4 compares the three spacer designs for a constant oset from threshold.
This
gure demonstrates a few important trends:
1. As the spacer thickness increases, the noise
oor level decreases. Spacers 30nm,
100nm and 150nm have equivalent linewidths at the noise
oor of 30KHz, 3KHz,
and 1 KHz, respectively, where the noise
oor for the 150nm spacer is limited
only by the measurement setup.

2. The low frequency noise (< 100MHz) decreases slowly with frequency.

The

thick 150nm spacer has a higher noise at the low frequencies than the 100nm
spacer, in agreement with the nonlinear loss model presented in this work. Only
at the high frequencies is it evident that the thicker spacer has lower noise.
The Schawlow-Townes noise
oor is observed clearly in the 30nm spacer.

In the

100nm spacer it seems that the noise has started to
atten at the knee area, though

143
this is not conclusive. The 150nm spacer clearly doesn't reach the noise
oor before
noise from the measurement setup becomes dominant.
In this chapter I have presented experimental frequency noise spectra from narrowlinewidth hybrid Si/III-V lasers. A record sub-KHz noise
oor was demontrated with
a conclusive trend showing the eect of spacer thickness on the noise performance.
It was also shown that though the noise
oor is lower in the thicker spacer designs,
the low-frequency components are noisier, in agreement with theoretical predictions
attributing that trend to free-carriers in Si. The same trend was also evident when
the pump power increased, as was predicted by the theory.

144

Chapter 11
Conclusion
11.1 Summary of key results
This work presented a theoretical model that takes into account extremely low III-V
con
nement, and nonlinear eects, such as TPA and FCA, that become increasingly
dominant in the presence of a high-Q Si cavity.

Predictions from this model were

compared to experimental results from fabricated devices. Some of the key results in
this work include:

1. Demonstration of a record-breaking sub-KHz semiconductor laser.

2. Demonstration of ultra-low frequency (∼100 MHz) relaxation resonance. The
theory developed in this work shows that this is due to the very low active
con
nement, and not due to the long cavity lifetime, as often suggested.

was argued and experimentally demonstrated that for frequencies above the
resonance, the noise drops to the quantum noise
oor.

The low resonance

frequency therefore lends itself to the realization of truly quantum-limited noise
sources, at frequencies that are useful for optical communication.

3. The intensity modulation response of the laser was investigated experimentally.
It was demonstrated that a zero of the transfer function is present at the thick
spacer designs. This is in agreement with the theoretical model that attributes
this phenomenon to the response of free-carriers in Si.

145
4. The frequency modulation response of the laser was investigated experimentally. It was shown that the response curve has a unique dip in addition to the
conventional resonance peak. This was also explained by the theoretical model,
which attributed these eects to free-carrier-dispersion in Si.

5. The frequency noise spectrum of these lasers was measured up to frequencies
of a few GHz.

It was shown that a very low S-T noise
oor is reached, but

only at high frequencies (>100 MHz). Furthermore, it was demonstrated that
the low frequency portion of the spectrum becomes noisier with pump-power
increase or with higher-Q. This is in agreement with the theoretical model, that
attributes this behavior to
uctuation of free-carriers in Si.

6. The L-I curve for the high-Q designs was shown experimentally to be nonlinear, in agreement with the model that attributes this
nding to nonlinear loss
mechanisms, mainly TPA and FCA.
Qualitatively, all the predictions from the nonlinear loss model were observed experimentally, including the nonlinear L-I curve, the unique characteristics of laser
dynamics, and the noise performance. The most convincing evidence for the role of
free-carriers in Si may lie in the frequency response curve. Other predictions are less
striking, and one could argue that although the experiments agree with the model,
results may stem from other ignored physical processes. However, it is hard to argue that for the frequency response curve. The frequency response is due to changes
of the eective refractive index of the lasing mode. In the high-Q hybrid platform,
the optical energy of the lasing mode is about 99% con
ned in silicon. Since we are
seeing a response that is comparable in magnitude to a conventional laser, it must
come from the Si.

Otherwise, we would have seen an orders of magnitude smaller

response. And since the response has high frequency components, which cannot be
explained by slow thermal processes, the result must be attributed to carriers in Si.
This logical argument, together with the good overall agreement between theory and
experiments, strengthen the case for the validity of the theoretical model, and for the
role of free-carriers in Si.

11.2 Future directions

146

The record-low noise
oor presented in this work is only a an upper limit. The real
laser noise could not be observed experimentally due to noise from the measurement
setup, induced by necessary optical ampli
ers due to the low output power of the
lasers. A better estimation of the laser noise
oor could be made by removing the
optical ampli
er altogether from the measurement setup. This would require higher
power lasers.

In Chapter 5 it was shown that the low output power is not a fun-

damental limit of the platform. Even though some trade-o between linewidth and
wall-plug e
ciency exists, it is predicted that achieving high-coherence lasers with
reasonable e
ciency is feasible.
The theoretical model in this work predicts that nonlinear eects such as TPA and
photo-generated FCA will eventually limit the achievable linewidth of this platform.
This prediction is a result of several physical processes:

1. Nonlinear loss that limits the achievable quality factor - When the intra-cavity
photon density increases, the nonlinear loss processes become increasingly more
dominant.

The nonlinear loss limits the number of photons that are stored

in the cavity, eectively lowering the quality factor.

This will aect the S-T

linewidth, which scales as the inverse of the quality factor squared.

2. Fluctuations of free-carries in Si It was shown theoretically that
uctuations
of free-carriers in Si will couple to the frequency noise through both the freecarrier-plasma eect and the thermo-optic eect. Increase of free-carrier density
due to increase in TPA will yield higher noise.

This noise mainly aects the

low and intermediate frequency range, but can still lower the achievable noise

oor at frequencies of interest (∼1 GHz).

The two performance-limiting mechanisms can be addressed using several approaches:

1. Lowering the temperature - Working at cryogenic temperature will dramatically
lower the noise due to the thermo-optic eect. Inherent temperature
uctuations scale as T

[49], at temperatures above 100K the speci
c heat and the

147
thermo-optic coe
cient scale roughly as T [20, 45], and the thermal conductivity scales roughly as T

−2

[24]. This suggests that a factor of two reduction

in temperature (i.e., from 300K to 150K) can result in noise reduced by an
order of magnitude or more. Although operating the laser at cryogenics temperature might not be a commercially viable solution, it would provide a better
understanding of the noise mechanisms and might reveal the S-T noise
oor at
lower frequencies. In this case, the noise
oor might not be limited by ampli
er
noise. Furthermore, the measurement setup in this case might be constructed
using a longer MZI that will provide higher gain, and might render the ampli
er
unnecessary.

2. The above approach only treats the temperature-related noise. It doesn't tackle
noise due to the free-carrier plasma eect, which was estimated theoretically
to have an observable impact on the noise performance.

One can tackle all

these noise limiting processes (except for inherent temperature
uctuations)
by reducing the free-carrier density in silicon. The most straightforward path
towards such a reduction is by reducing the eective lifetime of carriers in Si.
Fortunately, since the carrier pro
le is much broader than the optical mode's
pro
le, one can strongly aect the carrier population without introducing excess
loss into the mode, by manipulating the carriers only outside the mode's area.
This could be accomplished through two means:

(a) Deliberate deterioration of surface quality - It was argued in Chapter 3
that surface recombination is a dominant factor in the determination of
the eective lifetime. A reduction of carrier lifetime can be achieved by
deliberately introducing surface defects. For example, dry etching arrays
of deep holes or trenches could dramatically impact the eective lifetime.
Another possibility is by ion implanting dopants to introduce excess SRH
defects. As discussed above, this can be done outside the mode's area, so
that the low-loss properties of this platform are maintained.
(b) Construction of PN junction in Si - Another possible way to reduce the

148
lifetime of carriers in Si is by a fast sweep-out of carriers via a reversedbiased PN junction [16, 15].

An order of magnitude or more reduction

in lifetime can be accomplished using this method [15]. Furthermore, the
reverse bias voltage can provide a new knob for investigating the role of
free-carriers in silicon.

In this work, Hybrid Si/III-V lasers were shown to have extremely unique characteristics. Not only do they support ground-breaking low-noise operation, which might
render them the main candidate to replace the DFB laser, but they also provide a
fruitful platform for research of new and exciting scienti
c phenomena.

This work

highlights the intriguing properties of this platform through both theory and experimentation. It also provides an attempt to estimate the limits of this platform, and
oers several directions to overcome them.

The hybrid Si/III-V platform promises

to play an increasingly important and exciting role both in industry and in scienti
research.

149

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165

Appendix A
Fabrication process
This work followed previous endeavors to fabricate narrow-linewidth lasers on the
hybrid Si/III-V platform. A detailed description of the design of the Si resonator can
be found in [89], while details on the spacer platform and the III-V processing can
be found in [106]. This appendix will summarize the fabrication process and design
used in this current work, but the reader is referred to [106] for more details.
The device's schematics is shown in Figure A.1 for which the dimensions are
detailed on table A.1 .

The III-V wafer was obtained from Archcom Technology inc.

Details on the wafer structure are described in table A.2.

A.1 Silicon processing
ow diagram of the Si processing steps are shown in Figure A.2. Detailed description of the resonator's design methodology can be found in [89], and details on the
design and dimensions can be found in [106].

A.1.1 Chrome deposition
Chrome was deposited using CHA Industries (custom made tool based on the mark-40
model) e-beam evaporator. Conditions are depicted in table A.3.

166

Figure A.1: Spacer laser device Schematics

Description

Notation

Dimension

BOX thickness

1µm

Silicon slab thickness

0.5µm

Silicon etch depth

100nm for spacer
30nm; 50nm for other
spacers

Spacer thickness

30nm, 100nm, 150nm,
200nm

Waveguide width

2.5µm

Trench width

15µm

Mesa-metal separation

7.5µm

N-metal stripe width

75µm

Ion implant window

5µm

P-metal stripe width

40µm

P-contact layer width

45µm

Mesa width

60µm

Table A.1: Device dimensions. Notations is based on Figure A.1

167

Layer

Thickness Comment

n-InP buer

500nm

p-In0.53 Ga0.47 As
19
−3
(p>10 cm ) contact layer

200nm

p-InP cladding (p graded
1018 − 5 · 1017 cm−3 )

1.5µm

SCL InGaAsP (1.15Q)

40nm

SCL InGaAsP (1.25Q)

40nm

QW 1% compressive strain

7nm

stripped in substrate
removal step

InGaAsP
QW barrier 3% tensile

Con
nement layer
5 QWs, and 4 barriers
in between each well

10nm

strain InGaAsP
SCL InGaAsP (1.25Q)

40nm

SCL InGaAsP (1.15Q)

40nm

n-InP contact layer
18
−3
(n=10 cm )

110nm

Super Lattice,

7.5nm

n-In0.85 Ga0.15 As0.327 P0.673
Super Lattice, n-InP

7.5nm

n-InP bonding layer

10nm

Con
nement layer

18
−3
x2; n=10 cm

18
−3
n=10 cm

Table A.2: III-V wafer structure

Figure A.2: Flow diagram of the silicon processing steps

168

Parameter

Value

Cr thickness

20nm

Temperature

Uncontrolled

Rate

0.5 A/s

Base pressure

7E-6 Torr

Table A.3: Conditions of Cr evaporation

Step

Conditions

Spin Coat ZEP 520A

5000 RPM for 90 seconds

Soft bake

180 C for 10 minutes

E-beam lithography (Leica

Grating and 1µm of waveguide

EBPG 5000+)

trench close to the sidewall
(sleeve) at 340 µC/cm with 300
pA current and 2.5nm beam step
size and resolution. The rest
using 10 nA beam with 270

µC/cm2 , and 10 nm beam step
size and resolution
Resist developement

1 minute dip of ZED-50N
developer followed by 30 sec
IPA-MIBK solution

Table A.4: Steps for E-Beam lithography

A.1.2 Lithography
E-Beam lithography was performed to pattern the hard Cr mask. All features (gratings, waveguides, markers, etc...) were patterned in a single e-beam lithography run.
Steps and conditions depicted in table A.4.

A.1.3 Etch
The e-beam resist pattern was transferred to the Cr hard mask using ICP etch.
The resist was stripped and the Oxide and Si layers were further etched. Steps and
conditions are described in table A.5.

169

Step

Conditions

Cr etch

using Oxford ICP380 III-V
etcher, at 15 C; chamber
pressure 60mTorr; 60 sccm of
Cl2, 3 sccm of O2;1000 W ICP
power, 100W RF power

Resist strip

using Oxford ICP380 III-V
etcher, at 15 C; chamber
pressure 10mTorr; 100 sccm of
O2; 3000 W ICP power, 20W RF
power

Silicon and SiO2 etch

using Oxford ICP380 III-V
etcher, at 15 C; chamber
pressure 7mTorr; 35 sccm of
C4F8, 5 sccm of O2; 2100 W
ICP power, 200W RF power

Cr strip (wet etch)

Submerge in CR-7S chrome etch
until fully removed

Table A.5: ICP etch steps and conditions

A.1.4 Oxidation
Following etch and clean steps the device was annealed and oxidized to reduce sidewall roughness and minimize optical losses [103, 48]. Preparation steps and furnace
conditions are described in table A.6.

A.2 Wafer bonding
Direct low-temperature wafer bonding [19, 76] is used to bond the InP to Si/SiO2 .
The Si and SiO2 were patterned with an array of 10µm x 10µm out-gassing channel
(50 µm pitch) to allow hydrogen to outgas and avoid formation of interface bubbles
[56]. Further details on the exact bonding recipe and considerations can be found in
[106].

170

Steps

Conditions

Surface wet chemical clean

80 C Sulfuric acid / Hydrogen
peroxide mix, 3:1, for 10
minutes;

Surface plasma clean

using Oxford ICP380 III-V
etcher, at 15 C; chamber
pressure 10mTorr; 100 sccm of
O2; 3000 W ICP power, 20W RF
power

Furnace oxidation

Ramp from 700C to 1000C in 60
minutes with 100 sccm O2
ow,
and 3000 sccm N2
ow. 15
minutes at 1000C with 3000
sccm O2, followed by 30 minutes
of 3000 sccm N2 anneal

Table A.6: Oxidation/Anneal steps and conditions

A.2.1 Surface treatment
Both Si and InP surfaces were Solvent cleaned and activated/cleaned using oxygen
plasma treatment. The same surface plasma clean recipe that is described in table
A.6 was used.

A.2.2 Bonding
Chip were aligned manually by holding the InP back-side with vacuum tweezers and
pressing it against the Si chip. The chips were partially bonded at this stage using
Van der Waals forces, and were then covalently bonded at elevated temperature and
pressure using Suss SB6 wafer bonder. Condition are described in table A.7.

A.2.3 Substrate removal
The InP handle was removed using chemical etch. The III-V sidewalls were protected
using a thick wax layer to prevent under cutting. Two layers of protective resist were
spun to prevent acid from
owing in the waveguide trenches and undercutting the
bonding layer. Exact steps and conditions are depicted in table A.8.

171

Step

Conditions

Set pressure

205 mBar tool pressure (force)
for one hour. Chamber air
−3
pressure lower than 3 · 10 mBar
−4
(typically ∼ 10 )

Low temperature

ramp to 150 C and hold for one

High temperature

Ramp to 285 C and hold for 5

hour
hours
Table A.7: Steps and conditions for wafer bonding

Step

Conditions

Spin resist protection coating

spin PMMA A4 at 2000 RPM
followed by 5 minutes hotplate
soft bake at 180C. Spin PMMA
A11 at 2000 RPM followed by 10
minutes hotplate soft bake at
180 C

Wax protection layer

Melt wax on the entire chip at

Wax removal from III-V

Scrape wax and resist from

∼150C
bonded III-V using razor blade
Chemical etch

Etch using HCL/DI water mix
(3:1) at room temperature until
bubbles stop (typically 45
minutes)

Table A.8: Substrate removal steps and conditions

172

Figure A.3: Flow diagram of the III-V processing steps

A.3 III-V processing
After the bonding process, ion implantation, mesa formation, and metalization are
performed.

The mesa and contact dimensions are not critical, and therefore wet

chemical etch is used.

ow diagram describing the III-V processing is shown in

Figure A.3.

A.3.1 Ion implantation
The mesa and metal contacts are tens of microns wide, while the optical mode is
only a few microns wide. To make a more e
cient electrical pump process, where the
QWs are pumped only at the vicinity of the optical mode, ion implantation is used to
de
ne narrow current path. We have patterned ion-implant photo-mask and sent the
chips to be ion implanted at Kroko Inc. The steps and conditions used are described
in table A.9.

A.3.2 P-metal deposition
For ohmic contacts of low resistance a highly-doped layer of InGaAs was used for
the p-metal. In0.53 Ga0.47 As has the smallest bandgap among the InP lattice-matched
materials [135] . This yields a small energy barrier for carriers and allows for e
cient

eld-thermionic emission of carriers from the metal to the semiconductor [83].

173

Step

Conditions

Spin photo-resist

Spin AZ5214 positive photoresist
at 2000 RPM. Bake 90 seconds
at 95 C

Exposure

expose 9 seconds using 25
@g-line; hard contact

mW/cm

mode
Develop

30 seconds in CD-26 developer
puddle

Hard bake
Ion implant

Strip resist

115 C for 90 seconds on hotplate
Implant of Protons (H ) with a
14
−2
dose of 5·10 cm
and 170KeV
energy at 7 degrees tilt with
20µA
ion beam
97.5cm2
Solvent clean with hot (80 C)
Remover PG, and O2 plasma
(same conditions as plasma clean
in table A.5)

Table A.9: Parameters used for ion implantation

Ti/Pt/Au metal stack is used with low anneal temperature that is compatible with
the bonded platform. A temperature higher than 300 C that is often encountered in
metal contact recipes would jeopardize the integrity of the bonding interface.

The

recipe depicted in table A.10 have been tested using a circular test pattern setup [61]

−5
and yielded contact resistance 5 · 10 [Ωcm ] with anneals temperature of 200 C. This
is comparable with values reported in literature for similar contacts [60].

A.3.3 Mesa formation
The mesa formation is done using a three-steps etch that utilizes two dierent photomasks. After the
rst etch process, in which the p-contact layer is etched, a larger
photo-mask is spun and patterned for the mesa. The second mask is of bigger area
than the
rst one to make sure the sidewalls of the InGaAs layer are protected, such
that it is not attacked by the QW etch solution. Photo-lithography is used despite
the aggressive acids used in the etch that erode the mask.

The etch time is short

enough that mask erosion is tolerable. Table A.12 describes the mesa formation wet

174

Step

Conditions

Pattern lift-o mask

See table A.11

Descum

Using Branson barrel asher, 1
minute at 150 W in 0.6 Torr O2
plasma

De-oxidize

15 seconds in HCl:DI water
(1:10)

Deposit metals

Using CHA e-beam evaporator
deposit 20nm of Ti at 1 A/sec ,
50nm of Pt at 1 A/S and 150nm
of Au at 2 A/S. Pressure lower
than 8E-7.

Lift-o
Anneal

Overnight in Acetone puddle
using Jiplec RTA, anneal
30seconds at 200 C (N and P
metals annealed together after
N-metal deposition)

Table A.10: Steps and conditions used to deposit P metal stack

Step

Conditions

Spin Coat

spin AZ5214 @ 3000 RPM for 45
seconds

Soft bake
expose

hotplate 50 sec at 110 C
expose 2 seconds using 25
@g-line; hard contact

mW/cm

mode
Image reversal bake
Flood expose

hot plate 2 minutes at 107 C
16 seconds using 25 mW/cm
@g-line

Develop

30 seconds dip in CD-26
developer puddle

Table A.11: Steps and conditions for lift-o photo-lithography

175

Step

Conditions

Pattern mask for InGaAs layer

Same PL process as in table A.9

etch
InGaAs etch

7 seconds etch in
room-temperature Sulfuric
acid/Hydrogen peroxide/DI
water (1:1:10)

Strip photo-mask

Acetone and hot (80 C) remover
PG clean

Pattern mask for InP mesa

Same PL process as in table A.9

InP mesa etch

22 seconds in HCl(bottle
strength = 37%)

QW etch

45 seconds etch in
room-temperature Sulfuric
acid/Hydrogen peroxide/DI
water (1:1:10)

Strip resist

Acetone and hot (80 C) remover
PG clean

Table A.12: steps and conditions used to form the mesa

etch process. The resulting mesa pro
le is shown in the SEM images of Figure A.4.
The InGaAs contact layer has rough sidewall due to resist erosion in the Piranha etch
process. However, only its sidewalls, which are of no signi
cance, are aected.

A.3.4 N-Metal deposition
For the N-metal we are using alloyed Ge/Ni/Au contacts [68]. The metal structure
and deposition condition are described in table A.13. The metal contacts fabricated

(a)

(b)

(c)

Figure A.4: SEM images of mesa formed using the two mask process (a) Entire mesa
and Si waveguide (b) Sidewalls of mesa and InGaAs layers (c) Etch and under-cut of
the QW layer

176

Step

Conditions

Pattern lift-o mask

See table A.11

Descum

Using Branson barrel asher, 1
minute at 150 W in 0.6 Torr O2
plasma

Deoxidize

5 seconds in BHF water (1:10)

Deposit metals

Using CHA e-beam evaporator
deposit 30nm of Ge at 0.5 A/sec
, 50nm of Au at 0.5 A/S, 12nm
of Ni at 0.5 A/S and 225nm of
Au at 1 A/S. Pressure lower
than 8E-7.

Lift-o

Overnight in Acetone puddle

Anneal

using Jiplec RTA, anneal
30seconds at 200 C

Table A.13: Steps and conditions used for N-metal deposition

using this low-anneal temperature recipe were tested using the setup described in [82].

−5
Contact resistance of 5 · 10 [Ωcm ] was measured, comparable to industry standard
for similar technique and materials.

A.3.5 Cleaving
The devices were cleaved using a diamond scribe, and were broken into bars manually.
The diamond tip only scribed ∼1-2mm of the bar, and the cleave propagated along
the crystal axis. This yielded devices with optically smooth facets that need not be
polished.

177

Appendix B
Characterization setups
One of the major challenges in this work is the characterization of the lasers. The
lasers were fabricated at Caltech, and were not packaged in a any way. The lasers
in each bar were probed and measured one by one manually without any automated
aligning process. Furthermore, the task at hand was fundamentally challenging for
several reasons:

1. Measurement of phase - The phase and frequency of the lasers had to be measured to characterize both the noise and the frequency response . Measuring
phase required the usage of an MZI that had to be stabilized and locked to the
laser. Locking the MZI to a free running, unpackaged laser was challenging.

2. Noise measurements - The lasers were designed and demonstrated to be extremely clean. Sub-KHz linewidth was demonstrated. Measuring noise in devices with very little noise signal is a major challenge. Special attention to the
minimization of excess instrumentation noise was required. On top of that the
laser's output power was low, making it even harder to measure, and forced us
to fanatically worry about minimizing optical losses in the measurement setup.

3. Modulation response - The lasers were designed for high coherence and not for
high speed modulation.

The devices are big, about 1 mm of length, and no

special attention was given to its capacitance, or to impedance matching in the
laser design. Despite those facts high speed (few GHz) modulation experiments
were performed.

178
In this chapter we will divide the lasers measurement setups into dierent tasks. In
each task a full description of the methodology, experimental setup, equipment and
calibration processes are discussed.

B.1 Mounting and probing the lasers
Each laser bar was roughly 1mm wide and 5mm long. All the measurements in this
work used the same probe station and thermal management techniques described in
this chapter.

B.1.1 Mounting of laser bars
Two techniques were used to mount laser bars to the probe station: mounting on a
C-mount using indium soldier, or on a copper block using thermal paste. Characterization of spacer 30nm and 150nm was done using C-mounts, and of spacer 100nm
using thermal paste. The lasers were probed using Cascade Microtech's ACP40-GSG100 RF probe tip and MPHM micro-positioner. All measurements in this work were
taken with this probe setup. The whole setup was enclosed in a
ber-glass box and
sat on a Sorbothane anti-vibration pad to reduce
uctuations due to the environment.

B.1.2 Thermal management
In all mounting options the laser or the C-mounts were in thermal contact with a large
copper block using thermal paste. Only the bottom side of the laser bar (the silicon
wafer) was in thermal contact with the temperature-controlled stage.

The copper

block was in thermal contact with a Peltier cooler that was connected to a Newport
350 thermoelectric cooler (TEC). All measurements in this work were performed at

20o C unless otherwise stated.

179

Figure B.1: Schematics of L-I curve characterization setup

B.2 L-I curves
L-I curves were measured in both CW and pulsed operation. In both cases a freespace integrating sphere ILX Lightwave OMM 6810B power meter was positioned at
the output facet of the laser under test. It was veri
ed that the alignment is robust
to small spatial deviations, such that thermal expansion of the laser under test did
not result in measured power loss. A schematic of the measurement setup is shown
in Figure B.1.

B.2.1 CW excitation
All experiments were performed using CW excitation, unless otherwise stated. Laser
diode current driver Newport 525B was controlled using a personal computer via
USB. The computer also sampled the detected power on the power meter using GPIB.
Five measurements were averaged to obtain the measured power, and outliers due to
communication or transient problems were removed from the averaging. The power
reading from the power-meter was assumed to be calibrated, and no post processing
was done to the measured value.

B.2.2 Pulsed excitation
Spacer 150nm was also measured in pulsed operation to minimize thermal power rollos. We used ILX-Lightwave LDP 3840B pulsed current source, with a 40Ωimpedance
matching resistor. The power meter was set on slow operation mode and averaged
the detected pulses. The measured average power was then divided by the duty-cycle

180

Figure B.2: Schematics of the experimental setup used for intensity modulation response measurements

to obtain an estimate of the average pulse power.

B.3 Intensity modulation response
The lasers were designed for narrow-linewidth operation and not for high speed.
However, they were found to operate reasonably well under modulation of up to
a few GHz.

This section describes the experimental setup used to take intensity

modulation response, as well the calibration processes used to isolate the response of
the laser from that of auxiliary equipment.

B.3.1 Setup and equipment
A Schematics of the experimental setup appears in Figure B.2. A bias Tee (ZFBT6GW, 0.1-6000MHz) was used to separate DC and high-frequencies and to allow
for biasing of the laser above threshold.

The laser was biased using E3611A DC

voltage power supply. The input current to the laser was monitored using the built
in current monitor. The RF+DC output of the bias-tee could be routed to either the
laser probe or back to the network analyzer for calibration (see next section). Two
dierent network analyzers (NA) were used interchangeably:

1. HP 8722C NA - for the high frequency components we used this 50MHz-40GHz
analyzer. Each trace was averaged four times for consistent low-noise measurements.

181
2. Agilent 4395A NA - for the low frequency components we used this 10Hz to
500MHz analyzer.

The output of the laser was aligned to a lensed
ber (tapered SM
ber, AR coated
TSMJ-X-1550-9/125-0.25-7-2.5-14-2-AR by OZ Optics), which was mounted on a
ve
degrees-of-freedom (DOF) stage. The
ber was aligned with the aid of a power meter.
The output SM
ber was connected to a photodetector (New focus 1544B, DC-12GHz,
-600V/W peak conversion factor). The RF output signal of the PD connected to the
return port in the NA. The NA measurement was captured using a computer through
GPIB.

B.3.2 Calibration and measurement procedures
B.3.2.1 Photodetector response
The PD response, both magnitude and phase, was measured and subtracted from the
laser response. The calibration of the PD response was performed by using a stable
commercial laser, an amplitude modulator, and a separate broadband PD. The input
of the PD was modulated using the NA and the response was measured. The same
experiment was repeated with a broadband PD (HP 11982A. DC-15GHz). The HP
11982A PD was assumed (according to spec) to have very
at response, and so the
NF 1544B was calibrated using it. The resulting response curve is shown in Figure
B.3.

B.3.2.2 Driving circuitry response
The response of the driving circuitry (coax cables, bias tee, probe tip) was measured
and subtracted from the measurement of the laser response. The following procedure
was used:

1. The output of the bias tee was connected to return port of the NA (the dotted
line in Figure B.2), and a modulation response curve was taken.
same cables that drive the laser were used.

The exact

182

Figure B.3: Measured response of NF1544B photodetector.

The PD response was

subtracted from measured laser response

2. The response was registered into the internal memory of the network analyzers
and was used as a calibration trace.

3. The output of the bias tee was disconnected from the NA and connected to the
laser's probe tip. The circuit was then closed as in Figure B.2 and a measurement was taken (calibration trace used by the tool).

B.3.2.3 Calculating the small-signal current
To estimate the input current to the laser the following calculation was used:

Ilaser = √

PdBm
10 20
10 (Rlaser [Ω] + 50Ω)

(B.1)

where Rlaser is the laser's small-signal resistance calculated from the slope of the I-V
curve at the working current, and PdBm is the set-point output power of the NA in
Decibel mW, which gives the power falling on a 50Ωload.

183

B.3.2.4 Delay compensation
To accurately measure the phase of the transfer function compensation of the delay
had to be performed. There are several meters of optical
ber and electrical wires
and without delay compensations many 2π cycles of phase will accumulate at the
high frequencies. As a
rst step, a guess of the delay length was used as an input
to the NA delay compensation parameter. Fine tuning was done manually until the
corrected phase response curve looked as expected (i.e., corresponding to a second
order low-pass-
lter).

It is worth noting that this procedure leaves some room for

interpretation by the user, and that dierent users might end up with slightly dierent
response phases. However, experienced users will still get qualitatively similar results.

B.4 Frequency modulation response
The frequency response measurement is based on frequency-domain network analysis
approach[126]. A schematics of the setup used is shown in Figure B.4. An MZI is used
as a frequency discriminator that convert frequency modulation to intensity. One of
the arms of the MZI is mounted on a
ber stretcher piezo. The piezo is used to lock the
dierential phase between the two MZI arms such that the MZI is locked in quadrature
[102]. The two output of the MZI are connected to a fast balanced photodetector.
The balancing is important in order to make sure that intensity modulation is not
measured, and only the frequency response is detected. A network analyzer is used
to modulate the laser and to register the balanced PD output.
resulting response to units of [

Conversion of the

GHz
] is done through post processing.
mA

B.4.1 Setup and equipment
The driving circuitry (network analyzer, voltage source, bias tee, TEC) is similar to
the one described in the intensity modulation setup of section B.3.1.

The optical

output of the laser is collected through a lensed
ber (tapered SM
ber, AR coated
TSMJ-X-1550-9/125-0.25-7-2.5-14-2-AR by OZ Optics), which is mounted on a
ve

184

Figure B.4:

Schematics of the experimental setup used for frequency modulation

response measurements

DOF stage.

The output is then ampli
ed using an EDFA and launched into the

input port of the piezo-driven MZI. The
ber in one of the MZI arms is mounted
on a piezo
ber stretcher (Evanscent Optics 915B) . The two outputs of the MZI (Q
and I) are connected to a pair of couplers that couple 5% of the light to a (slow)
balanced PD (New-Focus 1817 80 MHz photoreciever), and 95% to a fast balanced
PD (23 GHz Optilab BPR 20M) . Care was given to make sure that input
bers to
the balanced PDs are length-matched to within a mm, to allow for maximal intesity
modulation rejection, even at high frequencies. The slow balanced PD is connected
to an electronic feedback PCB. A schematic of the feedback electronics is shown
in Figure B.5. It was constructed such that it provides variable gain, and variable
bandwidth through potentiometer and switchable capacitors. It also biases the MZI
piezo driver at 2.5 Volts (the driver operate at 0-5 Votls).

Another potentiometer

that is connected between ±5V allows for compensation of oset voltage at the input
(ideally the slow balanced PD should have zero mean signal, but this is not always
the case). The output of the feedback electronics PCB is connected to the MZI piezo
driver (Evanscent Optics 914) to close the feedback circuit. The fast balanced PD is
connected to the return port of the NA.

185

Figure B.5:

Schematics of the electronic feedback PCB used to lock the MZI to

quadrature

B.4.2 Calibration and measurement procedure
B.4.2.1 Balancing photodetectors
There are two photodetectors in the setup: a fast PD to measure the frequency noise,
and a slow PD to lock the MZI in quadrature.

The fast PD has to be balanced

to eliminate the intensity noise, especially due to the presenance of the ampli
er,
which introduces excess intensity noise.

The slow PD has to be balanced to get a

zero-mean small-signal feedback signal. It is worth noting that the feedback circuit
can be constructed without balancing by comparing the DC output of an unbalanced
PD to a reference voltage. However, drift of the output power, mainly due to drift
of the optical alignment during the experiemt will cause the MZI to drift away from
quadrature. For that reason the balanced PD feedback setup was used.
Arbitrary loss in the
ber network yiedls unbalanced PD reading. To compensate
for that the following procedure was used for balancing:

1. Modulate the MZI piezo at 100Hz and monitor the output of the slow balanced
PD on the scope.

2. Manually adjust the
ber coupling to the PD until the scope reading has zero

186
mean.

3. Turn o the slow modulation and bring the system to lock. Adjust the oset
potentiometer to make sure the system is locked at zero voltage.

4. Monitor the output power of the two PDs in the fast balanced detector using
the power monitor connection.

5. Manually adjust the
ber coupling to the PDs until balanced.

This procedure should yield a balanced detection setup for both the measurement
and the feedback.

B.4.2.2 Photodetector response
The photodetector response was measured and used to calibrate the frequency response, as well to calcualte the voltage swing of the balanced PD. The response
curves of Figure B.6 were obtained using the process described in section B.3.2.1.

B.4.2.3 Voltage swing
The output of the fast balanced PD oscilates in response to the frequency modulation
induced by the network analyzer. If the dierential phase between the two MZI arms
is ∆φ then the output voltage of the PD is:

VP D = Vq + Vg sin(∆φ)

(B.2)

where Vq is some oset voltage (ideally zero in the balanced setup and in lock), and

Vg is voltage swing.
For accurate calibration of the measurement, knowledge on Vg is required. This
can conceptually be done by perturbing the system slightly and measuring the voltage
swing. However, the balanced PD is equipped with a DC-block
lter at its output,
which alters the low-speed (<100 KHz) response. We therefore go through the following calibration procedure:

187

(a)

(b)
Figure B.6: Response curves of balanced PDs (a) New Focus 1817 (b) Optilab BPR
20-M

188

15 Hz
1. Replace the laser under test with a fast (∼ 5 · 10
) frequency chirped laser.
sec

We used a home-made frequency chirped semiconductor laser [90, 120].

2. Measure the resulting voltage swing on both fast and slow balanced PDs, using
the two channels of the scope (make sure 50Ωload, as in the NA).

3. Register the ratio between the responses of the two PDs voltage swing:

γ=

Vf ast
Vslow

(B.3)

4. Reconnect the laser under test to the setup.

5. Use a function generator, connected to the piezo driver, to modualte the MZI
at 100Hz (make sure full voltage swings are achived, and that PD is not satuarated).

6. Measure voltage swing Vpp of slow balanced PD at 100 Hz modulation using the
scope.

7. Obtain a conversion coe
cient between the voltage swing read using the fast
balanced PD at ∼2MHz and the slow balanced PD at 100 Hz.

Vg = γ

Vpp
·R

(B.4)

where R is the ratio between the slow PD response at ∼2MHz and 100 Hz
known from the response curve of the PD (Figure B.6a).

B.4.2.4 Delay compensation
The same procedure used in section B.3.2.4 is used .

B.4.2.5 Measurement procedure
The following procedure is used to obtain a frequency response curve:
1. Turn on TEC and bias the laser using the voltage source.

189
2. Balanced the PDs as in section B.4.2.1.

3. Repeat calibration steps of section B.4.2.3 to obtain Vg .

4. Calibrate for the response of the driving ciruitry as in B.3.2.2.

5. Calibrate for the delay as in B.3.2.4.

6. Take a calibrated measurement using the netwrok analyzer. Make sure modulation current is small enough to maintain linearity and not to throw the system
o lock.

B.4.2.6 Calculating the frequency response from the measurement
When the laser's pump current changes by a small amount ∆I(t) the frequency of
the laser will change according to:

f ≈ f0 +

∂f
∆I(t) ≡ f0 + Gf ∆I(t)
∂I

(B.5)

The output of the fast balanced PD which is connected to the MZI is given by:

Z t
Gf ∆I(t )dt
VBP D = Vg sin 2πf0 τ + 2π

(B.6)

t−τ
where

Vg is the voltage swing of the PD determined by the input power and its

internal transimpedance gain, τ is the MZI dierntial delay, and we have neglected
noise. When the MZI is locked at quadrature:

f0 τ = m

(B.7)

for some integer m. For small signal sinusodial modulation, as in the one imposed by
the NA:

∆I(t) = ∆I · cos(2πνt + φ0 )

(B.8)

190
the laser's frequency is:

f ≈ f0 + Gf (ν)∆I(t)cos(2πνt + φ(ν) + φ0 )

(B.9)

If the signal is small enough such that:

Z t

≈ 2πVg

Gf ∆I(t )dt

(B.10)

Gf (ν)cos (2πνt − πντ + φ(ν))

(B.11)

Gf ∆I(t )dt

VBP D = Vg sin 2π

Z t

t−τ

t−τ
the resulting PD output is:

VP D = Vg ∆I ·

sin (πντ )

πτ

Equation B.11 shows that the detected signal contains both the desired contributions
from the lasers response (Gf (ν), φ(ν)), but also contributions from the MZI
nite
FSR and its dierential delay. These contributions will have to be deconvolved from
the measurement to obtain calibrated result.
After going through the calibration process of subtracting the response of the
driving circuitry and the PD response, we are left with system response in power dBm.
The NA return port is terminated with a 50ohm resistor, such that the measured

power for the sinusodial modulation is given by

1 VP D
. The input current to the laser
2 50

is related to the NA output power as described in Equation B.1. We therefore use
the following transformation to convert to the desired units:

Pcal (ν)
πν(Rlaser + 50)
G(ν) =
· 10 20
Vg |sin (πντ )|

φ(ν) = φcal (ν) + πντ

(B.12)

(B.13)

Where Pcal is the already calibrated trace in power dBm, and φcal is the calibrated
phase after delay compensation.

191

B.5 Frequency noise spectrum measurement
The frequency noise measurement setup described in this section enabled us to measure sub-KHz equivalent noise at frequencies of > 1 GHz. Many of the features of the
measurement setup are similar to that of the frequency modulation response measurement setup described in section B.4. In both setups the measured signal is the laser's
frequency. This required the use of an MZI as a frequency discriminator. In both setups an electronic feedback was used to lock the MZI in quadrature for the duration
of the measurement.

However, the noise measurement experiments posseses some

unique challanges. The frequency noise level of these lasers is very low (sub-KHz),
especially at the high frequencies where the noise reaches, or get close to the quantum
noise
oor. The transfer function of the MZI used as frequency discriminator has the
form:

HM ZI = τ 2

(πντ )
(πντ )2

sin

(B.14)

we can identify two regimes of the sinc function response:

1. The
at response - for frequencies below the FSR of the MZI the respnse is

at to a very good degree. The MZI at this regime has a gain of τ .

2. The decaying oscillation - frequencies above the FSR will decay and oscilate with
a period that is related to the FSR. The peaks of the oscillation are decaying
at a rate of

ν2

This response sets a very fundamental limit on our ability to measure high frequencies
with high gain due to the proportionality:

gain ∼

(∆ν)2

(B.15)

Increasing the bandwidth ∆ν results in a reduction of available gain for the measurement. Notice that even if one works with high-gain (long) interferometer and wishes
to extrapolate the laser frequency noise from the oscillating sinc function peaks at
frequencies above its FSR, the response drops as

at that regime, bringing the
ν2

192
gain down again. The low MZI gain at high frequencies, together with the relatively
low power of our lasers, and the unavoidable noise from the measurement setup's
instruments make it a challanging task.

B.5.1 Setup and equipment
The setup for frequency noise spectrum measurement is shown in Figure B.7.

ultra-low-noise current source (ILX lighwave LDX-3620) is used to bias the laser. A
common mode
lter (LNF-320) is connected to its output to reduce RF noise pickup.
The optical output of the laser is collected through a lensed
ber (tapered SM
ber,
AR coated TSMJ-X-1550-9/125-0.25-7-2.5-14-2-AR by OZ Optics), which is mounted
on a
ve DOF stage. The output is ampli
ed using an EDFA (to bring the signal
above the detector noise
oor) and launched into the input port of the piezo-driven
MZI. The
ber in one of the MZI arms is mounted on piezo
ber strecher (Evanescent
Optics 915B) . The two outputs of the MZI (Q and I) are connected to a pair of
couplers that couple 5% of the light to a (slow) balanced PD (New-Focus 1817 80
MHz photoreciever), and 95% to a fast balanced PD (23 GHz Optilab BPR 20M) .
The input
bers to the balanced PDs are length-matched to within a mm, to allow
for maximal intensity modulation rejection. The slow balanced PD is connected to an
electronic feedback PCB. A schematic of the feedback electronics is shown in Figure
B.5, and is described in section B.4.1. The output of the feedback electronics PCB
is connected to the MZI piezo driver (Evanescent Optics 914) to close the feedback
circuit. The fast balanced PD is connected to an RF spectrum analyzer (calibrated
HP 8565E, 30Hz-50GHz) and the resulting spectrum is grabbed using a PC and post
processed.

B.5.2 Calibration and measurement procedures
B.5.2.1 Balanced PD and ampli
er
PD balancing procedure is done prior to measurement as described in section B.4.2.1.
The response of the balanced PD used in the experiment is shown in Figure B.6b. The

193

Figure B.7: Schematics of the experimental setup used for frequency noise spectrum
measurements

dark noise of the PD set the ultimate noise signal that can be measured, and is shown
in Figure B.8 (green curve).

Due to the low power of the laser at test the desired

noise signal is well below the dark noise, and therefore cannot be measured without ampli
cation. Ampli
cation is performed using an EDFA, which demonstrated
superior phase noise performance over BOA. However, this limits our measurement
setup to the C-band. Intensity noise of the laser+EDFA system is eliminated by the
use of balanced PD. Figure B.8 shows intensity noise spectrums of the balanced PD
for dierent input power. This was performed using the same laser seed power with
changing EDFA gain. Figure B.9 shows the measured intensity noise vs the calculated
dark+shot noise. The shot noise was calculated using the input power and assuming
the internal trans-impedance (TI) ampli
er of the balanced PD has the same shape
as the dark noise curve. The comparison of the calculated noise
oor vs the measured
intensity noise (Figure B.9) shows that at high frequencies (>100MHz) the measurement is shot-noise limited. This veri
es that the system is balanced, since otherwise
the noise would have been higher due to the ampli
cation (EDFA has noise-
gure of
at least 3dB). This also shows how crucial balancing is in this measurement. At low
frequencies the high input power has a toll, and the TI ampli
er add excess noise.
However, the frequency noise at low frequencies is high enough, and easy to measure
even with the excess TI noise. We therefore choose to work with high input powers of
1 mW per detector, for which the PD is not saturated yet, and is shot noise limited

194

Figure B.8: Intensity noise measured for dierent input powers by the balanced photodetector (Optilab BPM-20)

at the high frequencies.
The full output voltage swing of the PD is required to calculate voltage/frequency
ratio to calibrate the noise measurement. This is done as described in section B.4.2.3.

B.5.2.2 Measurement procedure
The following procedure is used to obtain the frequency noise spectrum:
1. Turn on TEC and bias the laser using the voltage source.

2. Balance the PDs as in section B.4.2.1.

3. Adjust EDFA gain to obtain 1mW of power per detector (measured using the
the monitor of the balanced PD).

4. Repeat calibration steps of section B.4.2.3 to obtain Vg .

5. Make sure the MZI is locked to quadrature and take a spectrum measurement
using RF spectrum analyzer.

6. Post-process the measured spectrum to convert to meaningful units using the
procedure in section B.5.2.3.

195

(a)

(b)

(c)
Figure B.9: Intensity noise vs calculated dark+shot noise for dierent input power to
the balanced PD (a) 0.25mW per detector (b) 0.5mW (c) 1 mW

196

B.5.2.3 Calculating the noise spectrum from the measurement
When the MZI is locked in quadrature and the phase noise signal is small (RMS

π),

as in our case (since the MZI delay is much shorter than the coherence time), the
output of the balanced PD is given by:

VBP D = Vg ∆φ(t, τ )

(B.16)

The resulting spectrum at the RF spectrum (single sided PSD in W/Hz) is related
to the spectrum of ∆φ by calculating the power that falls on its 50 ohm termination
load:

WBP D =

Vg2 (single-sided)
50 ∆φ

(B.17)

and the PSD of the frequency noise is related to the PSD of ∆φ(t, τ ) using [78]:

2 sin (πντ )
W∆φ = Wφ̇ τ

(πντ )

(B.18)

The frequency noise can be obtained directly from the PD spectrum at frequencies
well below the FSR of the MZI. However, there is information about the frequency
noise even at frequency above the FSR, where the response is falling and oscillating.
To obtain that information in a readable form we need to deconvolve the sinc function
from the measurement. The following procedure is used for that end:

1. Omit all points that are within 5dB of the intensity noise curve. These points
are contaminated too much with AM noise and cannot be reliably used.

2. Subtract the intensity noise level from the remaining spectra. This results in a

fBP D that contains only phase noise information.
signal W
3. Calculate the single sided PSD of the frequency noise using:

(single-sided)

Wφ̇
in units of

rad
sec

/Hz.

50 (πν)2 f
= 2 2
WBP D
Vg sin (πντ )

(B.19)

197
4. One can present the resulting PSD as the equivalent white noise linewidth by
dividing by the resulting W ˙ by 4π (see Equation 5.14, and recall that this is a
φ̇
single-sided PSD).