The Optoelectronic Swept-Frequency Laser

The Optoelectronic Swept-Frequency Laser and Its Applications in Ranging, Three-Dimensional Imaging, and Coherent Beam Combining of Chirped-Seed Amplifiers - CaltechTHESIS
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The Optoelectronic Swept-Frequency Laser and Its Applications in Ranging, Three-Dimensional Imaging, and Coherent Beam Combining of Chirped-Seed Amplifiers
Citation
Vasilyev, Arseny
(2013)
The Optoelectronic Swept-Frequency Laser and Its Applications in Ranging, Three-Dimensional Imaging, and Coherent Beam Combining of Chirped-Seed Amplifiers.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/YD38-BT07.
Abstract
This thesis explores the design, construction, and applications of the optoelectronic swept-frequency laser (SFL). The optoelectronic SFL is a feedback loop designed around a swept-frequency (chirped) semiconductor laser (SCL) to control its instantaneous optical frequency, such that the chirp characteristics are determined solely by a reference electronic oscillator. The resultant system generates precisely controlled optical frequency sweeps. In particular, we focus on linear chirps because of their numerous applications. We demonstrate optoelectronic SFLs based on vertical-cavity surface-emitting lasers (VCSELs) and distributed-feedback lasers (DFBs) at wavelengths of 1550 nm and 1060 nm. We develop an iterative bias current predistortion procedure that enables SFL operation at very high chirp rates, up to 10^16 Hz/sec. We describe commercialization efforts and implementation of the predistortion algorithm in a stand-alone embedded environment, undertaken as part of our collaboration with Telaris, Inc. We demonstrate frequency-modulated continuous-wave (FMCW) ranging and three-dimensional (3-D) imaging using a 1550 nm optoelectronic SFL.
We develop the technique of multiple source FMCW (MS-FMCW) reflectometry, in which the frequency sweeps of multiple SFLs are "stitched" together in order to increase the optical bandwidth, and hence improve the axial resolution, of an FMCW ranging measurement. We demonstrate computer-aided stitching of DFB and VCSEL sweeps at 1550 nm. We also develop and demonstrate hardware stitching, which enables MS-FMCW ranging without additional signal processing. The culmination of this work is the hardware stitching of four VCSELs at 1550 nm for a total optical bandwidth of 2 THz, and a free-space axial resolution of 75 microns.
We describe our work on the tomographic imaging camera (TomICam), a 3-D imaging system based on FMCW ranging that features non-mechanical acquisition of transverse pixels. Our approach uses a combination of electronically tuned optical sources and low-cost full-field detector arrays, completely eliminating the need for moving parts traditionally employed in 3-D imaging. We describe the basic TomICam principle, and demonstrate single-pixel TomICam ranging in a proof-of-concept experiment. We also discuss the application of compressive sensing (CS) to the TomICam platform, and perform a series of numerical simulations. These simulations show that tenfold compression is feasible in CS TomICam, which effectively improves the volume acquisition speed by a factor ten.
We develop chirped-wave phase-locking techniques, and apply them to coherent beam combining (CBC) of chirped-seed amplifiers (CSAs) in a master oscillator power amplifier configuration. The precise chirp linearity of the optoelectronic SFL enables non-mechanical compensation of optical delays using acousto-optic frequency shifters, and its high chirp rate simultaneously increases the stimulated Brillouin scattering (SBS) threshold of the active fiber. We characterize a 1550 nm chirped-seed amplifier coherent-combining system. We use a chirp rate of 5*10^14 Hz/sec to increase the amplifier SBS threshold threefold, when compared to a single-frequency seed. We demonstrate efficient phase-locking and electronic beam steering of two 3 W erbium-doped fiber amplifier channels, achieving temporal phase noise levels corresponding to interferometric fringe visibilities exceeding 98%.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
Swept-frequency laser; frequency-modulated continuous-wave reflectometry; three-dimensional imaging; optical phase-locked loops, coherent beam combining.
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
Thesis Committee:
Yariv, Amnon (chair)
Crosignani, Bruno
Vahala, Kerry J.
Schwab, Keith C.
Yang, Changhuei
Defense Date:
20 May 2013
Non-Caltech Author Email:
arsenyvasilyev (AT) gmail.com
Funders:
Funding Agency
Grant Number
Defense Advanced Research Projects Agency
UNSPECIFIED
U.S. Army Research Office
UNSPECIFIED
High Energy Laser Joint Technology Office
UNSPECIFIED
Record Number:
CaltechTHESIS:06032013-060508409
Persistent URL:
DOI:
10.7907/YD38-BT07
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
7820
Collection:
CaltechTHESIS
Deposited By:
Arseny Vasilyev
Deposited On:
06 Jun 2013 22:24
Last Modified:
04 Oct 2019 00:01
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The Optoelectronic Swept-Frequency Laser and Its
Applications in Ranging, Three-Dimensional
Imaging, and Coherent Beam Combining of
Chirped-Seed Amplifiers

Thesis by

Arseny Vasilyev
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy

California Institute of Technology
Pasadena, California

2013
(Defended May 20, 2013)

c 2013
Arseny Vasilyev

iii

Acknowledgments
I am deeply thankful to my advisor, Prof. Amnon Yariv, for taking me into his
research group and providing an environment in which I had the freedom to pursue
original ideas. Prof. Yariv’s advice has been key in picking the direction of our work
and his expertise in optical physics has been a continuous source of inspiration.
I thank Profs. Bruno Crosignani, Keith Schwab, Kerry Vahala, and Changhuei
Yang for serving on my thesis committee.
I was trained to conduct experiments in optoelectronics by Dr. Naresh Satyan,
and I am deeply thankful for his instruction and his patience. I also learned a great
deal from Dr. George Rakuljic, and am thankful for his support and for the many
technical discussions that we had over the years.
I would like to acknowledge our collaborators at the United States Army Research
Laboratories. Dr. Jeffrey White made it possible for us to participate in an exciting
research program, and on a few occasions hosted us at the US ARL. I am thankful
to have had the opportunity to work with and learn from Dr. White’s team and his
colleagues: Dr. Eliot Petersen, Dr. Olukayode Okusaga, Dr. Carl Mungan, James
Cahill, and Zhi Yang.
I am also thankful to our collaborators at the Jet Propulsion Laboratory: Dr.
Baris Erkmen, Dr. John Choi, and Dr. William Farr.
My fellow Yariv group members have supported and encouraged me throughout
the years, and I am deeply grateful to have been surrounded with such kind and
talented individuals. I have enjoyed my time with Prof. Bruno Crosignani, Prof. Avi
Zadok, Dr. Naresh Satyan, Dr. George Rakuljic, Dr. Jacob Sendowski, Dr. Christos
Santis, Dr. Hsi-Chun Liu, Dr. Xiankai Sun, Scott Steger, Yasha Vilenchik, Mark

iv
Harfouche, Marilena Dimotsantou, Sinan Zhao, and Dongwan Kim. I am particularly
thankful to Dr. Reg Lee for the excellent technical advice that he has given to us
over the years. I am grateful to Connie Rodriguez for taking care of all of us and for
making sure that I started writing my thesis on time. I would also like to acknowledge
Alireza Ghaari, Kevin Cooper and Mabel Chik for their support.
I want to thank my parents and family for their love, support, and patience with
me on this journey. Lastly, I want to thank my new best friend, Debi, for filling my
life with joy and happiness over these last several months.

Abstract
This thesis explores the design, construction, and applications of the optoelectronic
swept-frequency laser (SFL). The optoelectronic SFL is a feedback loop designed
around a swept-frequency (chirped) semiconductor laser (SCL) to control its instantaneous optical frequency, such that the chirp characteristics are determined solely by
a reference electronic oscillator. The resultant system generates precisely controlled
optical frequency sweeps. In particular, we focus on linear chirps because of their
numerous applications. We demonstrate optoelectronic SFLs based on vertical-cavity
surface-emitting lasers (VCSELs) and distributed-feedback lasers (DFBs) at wavelengths of 1550 nm and 1060 nm. We develop an iterative bias current predistortion
procedure that enables SFL operation at very high chirp rates, up to 1016 Hz/sec. We
describe commercialization efforts and implementation of the predistortion algorithm
in a stand-alone embedded environment, undertaken as part of our collaboration
with Telaris, Inc. We demonstrate frequency-modulated continuous-wave (FMCW)
ranging and three-dimensional (3-D) imaging using a 1550 nm optoelectronic SFL.
We develop the technique of multiple source FMCW (MS-FMCW) reflectometry,
in which the frequency sweeps of multiple SFLs are “stitched” together in order to
increase the optical bandwidth, and hence improve the axial resolution, of an FMCW
ranging measurement. We demonstrate computer-aided stitching of DFB and VCSEL
sweeps at 1550 nm. We also develop and demonstrate hardware stitching, which
enables MS-FMCW ranging without additional signal processing. The culmination
of this work is the hardware stitching of four VCSELs at 1550 nm for a total optical
bandwidth of 2 THz, and a free-space axial resolution of 75 µm.
We describe our work on the tomographic imaging camera (TomICam), a 3-D

vi
imaging system based on FMCW ranging that features non-mechanical acquisition
of transverse pixels. Our approach uses a combination of electronically tuned optical sources and low-cost full-field detector arrays, completely eliminating the need for
moving parts traditionally employed in 3-D imaging. We describe the basic TomICam
principle, and demonstrate single-pixel TomICam ranging in a proof-of-concept experiment. We also discuss the application of compressive sensing (CS) to the TomICam
platform, and perform a series of numerical simulations. These simulations show
that tenfold compression is feasible in CS TomICam, which effectively improves the
volume acquisition speed by a factor ten.
We develop chirped-wave phase-locking techniques, and apply them to coherent
beam combining (CBC) of chirped-seed amplifiers (CSAs) in a master oscillator power
amplifier configuration. The precise chirp linearity of the optoelectronic SFL enables
non-mechanical compensation of optical delays using acousto-optic frequency shifters,
and its high chirp rate simultaneously increases the stimulated Brillouin scattering
(SBS) threshold of the active fiber. We characterize a 1550 nm chirped-seed amplifier
coherent-combining system. We use a chirp rate of 5 × 1014 Hz/sec to increase the
amplifier SBS threshold threefold, when compared to a single-frequency seed. We
demonstrate efficient phase-locking and electronic beam steering of two 3 W erbiumdoped fiber amplifier channels, achieving temporal phase noise levels corresponding
to interferometric fringe visibilities exceeding 98%.

vii

Contents
Acknowledgments

iii

Abstract

List of Figures

List of Tables

xviii

Glossary of Acronyms

1 Overview and Thesis Organization

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

Ranging and 3-D Imaging Applications . . . . . . . . . . . . . . . . .

1.2.1

Optical FMCW Reflectometry . . . . . . . . . . . . . . . . . .

1.2.2

Multiple Source FMCW Reflectometry . . . . . . . . . . . . .

1.2.3

The Tomographic Imaging Camera . . . . . . . . . . . . . . .

1.3

Phase-Locking and Coherent Combining of Chirped Optical Waves

2 Optical FMCW Reflectometry
2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1.1

Basic FMCW Analysis and Range Resolution . . . . . . . . .

2.1.2

Balanced Detection and RIN . . . . . . . . . . . . . . . . . . .

2.1.3

Effects of Phase Noise on the FMCW Measurement . . . . . .

2.1.3.1

Statistics and Notation . . . . . . . . . . . . . . . . .

2.1.3.2

Linewidth of Single-Frequency Emission . . . . . . .

viii

2.1.4

2.1.3.3

Fringe Visibility in an FMCW Measurement . . . . .

2.1.3.4

Spectrum of the FMCW Photocurrent and the SNR

2.1.3.5

Phase-Noise-Limited Accuracy . . . . . . . . . . . .

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 The Optoelectronic Swept-Frequency Laser

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

System Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.1

The Optoelectronic SFL as a PLL . . . . . . . . . . . . . . . .

3.2.2

Small-Signal Analysis . . . . . . . . . . . . . . . . . . . . . . .

3.2.3

Bias Current Predistortion . . . . . . . . . . . . . . . . . . . .

Design of the Optoelectronic SFL . . . . . . . . . . . . . . . . . . . .

3.3.1

SCL Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.2

Amplitude Control . . . . . . . . . . . . . . . . . . . . . . . .

3.3.3

Electronics and Commercialization . . . . . . . . . . . . . . .

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4.1

Precisely Controlled Linear Chirps . . . . . . . . . . . . . . .

3.4.2

Arbitrary Chirps . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

3.4

3.5

3.6

Demonstrated Applications

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

3.5.1

FMCW Reflectometry Using the Optoelectronic SFL . . . . .

3.5.2

Profilometry . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Multiple Source FMCW Reflectometry

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.1

Review of FMCW Reflectometry . . . . . . . . . . . . . . . .

4.2.2

Multiple Source Analysis . . . . . . . . . . . . . . . . . . . . .

4.2.3

Stitching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Experimental Demonstrations . . . . . . . . . . . . . . . . . . . . . .

4.3.1

4.3

Stitching of Temperature-Tuned DFB Laser Sweeps . . . . . .

4.4

4.3.2

Stitching of Two VCSELs . . . . . . . . . . . . . . . . . . . .

4.3.3

Hardware Stitching of Four VCSELs . . . . . . . . . . . . . .

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 The Tomographic Imaging Camera
5.1

5.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1.1

Current Approaches to 3-D Imaging and Their Limitations . .

5.1.2

Tomographic Imaging Camera . . . . . . . . . . . . . . . . . .

5.1.2.1

Summary of FMCW Reflectometry . . . . . . . . . .

5.1.2.2

TomICam Principle

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

5.1.2.3

TomICam Proof-of-Principle Experiment . . . . . . .

Compressive Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.1

Compressive Sensing Background . . . . . . . . . . . . . . . .

5.2.2

TomICam Posed as a CS Problem . . . . . . . . . . . . . . . .

5.2.3

Robust Recovery Guarantees . . . . . . . . . . . . . . . . . . .

5.2.3.1

Random Partial Fourier Measurement Matrix . . . .

5.2.3.2

Gaussian or Sub-Gaussian Random Measurement Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2.4
5.3

Numerical CS TomICam Investigation . . . . . . . . . . . . . 101

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 Phase-Locking and Coherent Beam Combining of Broadband
Linearly Chirped Optical Waves

108

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2

Coherent Beam Combining . . . . . . . . . . . . . . . . . . . . . . . . 109

6.3

Phase-Locking of Chirped Optical Waves . . . . . . . . . . . . . . . . 111

6.4

6.3.1

Homodyne Phase-Locking . . . . . . . . . . . . . . . . . . . . 112

6.3.2

Heterodyne Phase-Locking . . . . . . . . . . . . . . . . . . . . 118

6.3.3

Passive-Fiber Heterodyne OPLL . . . . . . . . . . . . . . . . . 120

Coherent Combining of Chirped Optical Waves . . . . . . . . . . . . 124
6.4.1

Passive-Fiber CBC Experiment . . . . . . . . . . . . . . . . . 124

6.5

6.4.2

Combining Phase Error in a Heterodyne Combining Experiment 127

6.4.3

Free-Space Beam Combining of Erbium-Doped Fiber Amplifiers 127

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7 Conclusion
7.1

Summary of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.1.1

Development of the Optoelectronic SFL

7.1.2

Ranging and 3-D Imaging Applications . . . . . . . . . . . . . 134

7.1.3
7.2

133

. . . . . . . . . . . . 133

7.1.2.1

MS-FMCW Reflectometry and Stitching . . . . . . . 134

7.1.2.2

The Tomographic Imaging Camera . . . . . . . . . . 135

Phase-Locking and CBC of Chirped Optical Waves . . . . . . 136

Current and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 137

Appendices
A Time-Domain Phase Analysis Using I/Q Demodulation

140

B Phase-Noise-Limited Tiled-Aperture Fringe Visibility

142

Bibliography

143

List of Figures
2.1

Time evolution of the optical frequencies of the launched and reflected
waves in a single-scatterer FMCW ranging experiment . . . . . . . . .

2.2

Mach-Zehnder interferometer implementation of the FMCW ranging experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

A balanced Michelson interferometer implementation of the FMCW
ranging experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6

A balanced Mach-Zehnder interferometer implementation of the FMCW
ranging experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5

Michelson interferometer implementation of the FMCW ranging experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

Convergence of the Monte Carlo simulation of the baseband electric
field spectrum (blue) to the theoretical expression (red). The angular
linewidth is ∆ω = 2π(1 MHz). N is the number of iterations used in
calculating the PSD estimate. . . . . . . . . . . . . . . . . . . . . . . .

2.7

Normalized frequency noise spectra (top panel) and corresponding baseband electric field spectra (bottom panel) for ∆ω = 2π(900 kHz) (black),
2π(300 kHz) (blue), and 2π(100 kHz) (green). The spectra are averaged
over N=1000 iterations. The red curves are plots of the theoretical lineshape for the three values of ∆ω. . . . . . . . . . . . . . . . . . . . . .

2.8

Baseband FMCW photocurrent spectra for four different values of τ /τc ,
normalized to zero-frequency noise levels. The scan time is T = 1 ms

2.9

and the coherence time is τc = 1 µs. . . . . . . . . . . . . . . . . . . .

FMCW SNR as a function of τ /τc for three different values of T /τc . .

xii
3.1

Schematic diagram of the SCL-based optoelectronic SFL . . . . . . . .

3.2

Elements of the optoelectronic SFL lumped together as an effective VCO 34

3.3

Small-signal frequency-domain model of the optoelectronic SFL . . . .

3.4

Single predistortion results . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

Iterative predistortion results . . . . . . . . . . . . . . . . . . . . . . .

3.6

Measured optical spectra of DFB and VCSEL SFLs at wavelengths of
1550 nm and 1060 nm . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7

Schematic diagram of the amplitude controller feedback system . . . .

3.8

Comparison between the off(blue) and on(red) states of the SOA amplitude controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.9

Comparison between the off(blue) and on(red) states of the VOA amplitude controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.10

Optoelectronic SFL printed circuit board layouts . . . . . . . . . . . .

3.11

The 1550 nm CHDL system. . . . . . . . . . . . . . . . . . . . . . . . .

3.12

MZI photocurrent spectrum during the predistortion process and in the
locked state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.13

Locked MZI spectra of various SFLs for different values of the chirp rate
ξ. The x-axis in all the plots corresponds to the chirp rate. . . . . . . .

3.14

Quadratic chirp spectrogram . . . . . . . . . . . . . . . . . . . . . . .

3.15

Exponential chirp spectrogram . . . . . . . . . . . . . . . . . . . . . .

3.16

FMCW reflectometry of acrylic sheets using the VCSEL-based optoelectronic SFL with a chirp bandwidth of 500 GHz and a wavelength of
∼ 1550 nm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.17

Depth profile of a United States $1 coin measured using the VCSELbased optoelectronic SFL with a chirp bandwidth of 500 GHz and a

4.1

wavelength of ∼ 1550 nm . . . . . . . . . . . . . . . . . . . . . . . . .

Schematic of an FMCW ranging experiment. PD: Photodetector . . .

xiii
4.2

Schematic representation of single-source FMCW reflectometry. Top
panel: the window function a(ω) corresponding to a single chirp. Bottom
panel: The underlying target function ytarget (ω) (blue) and its portion
that is measured during the single sweep (red) . . . . . . . . . . . . . .

4.3

Schematic representation of dual-source FMCW reflectometry.

Top

panel: the window function a(ω) corresponding to two non-overlapping
chirps. Bottom panel: The underlying target function ytarget (ω) (blue)
and its portion that is measured during the two sweeps (red) . . . . . .
4.4

Multiple source model. (a)ω-domain description. The top panel shows
a multiple source window function aN (ω). This function may be decomposed into the sum of a single-source window function (middle panel)
and a function that describes the inter-sweep gaps (bottom panel). (b)ζdomain description. The three figures show the amplitudes of the ζdomain FTs of the corresponding functions from part (a). . . . . . . .

4.5

Schematic of a multiple source FMCW ranging experiment. A reference
target is imaged along with the target of interest, so that the inter-sweep
gaps may be recovered. BS: Beamsplitter. PD: Photodetector . . . . .

4.6

Proposed multiple source FMCW system architecture. BS: Beamsplitter. PD: Photodetector

4.7

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

Single-sweep and stitched two-sweep photocurrent spectra of a dual reflector target with a separation of 1.49 mm. No apodization was used.

4.10

Single-sweep and stitched two-sweep photocurrent spectra of a dual reflector target with a separation of 5.44 mm. No apodization was used.

4.9

Optical spectra of the two DFB sweeps (blue and red) and the optical
spectrum analyzer PSF (black) . . . . . . . . . . . . . . . . . . . . . .

4.8

Single-sweep and stitched two-sweep photocurrent spectra of a dual reflector target with a separation of 1.00 mm (a microscope slide). No
apodization was used. . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiv
4.11

The gray and black curves correspond to single-sweep and stitched threesweep photocurrent spectra, respectively. No apodization was used. (a)
Single reflector spectrum. (b) Glass slide spectrum. The peaks correspond to reflections from the two air-glass interfaces. The slide thickness
is 1 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.12

Dual VCSEL FMCW reflectometry system diagram. The feedback loop
ensures chirp stability. A reference target is used to extract the intersweep gaps. PD: Photodiode, BS: Beamsplitter . . . . . . . . . . . . .

4.13

Optical spectra of the two VCSEL sweeps in the 250 GHz experiment .

4.14

Optical spectra of the two VCSEL sweeps in the ∼ 1 THz experiment .

4.15

Single-sweep and stitched two-sweep photocurrent spectra of dual reflector targets with various separations. The total chirp bandwidth is
250 GHz. No apodization was used. . . . . . . . . . . . . . . . . . . . .

4.16

Single-sweep and stitched two-sweep photocurrent spectra of dual reflector targets with various separations. The total chirp bandwidth is
∼ 1 THz. No apodization was used.

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

4.17

Four channel 2 THz hardware stitching experiment . . . . . . . . . . .

4.18

Optical spectra of the four 1550 nm VCSEL sweeps in the 2 THz hardware stitching experiment . . . . . . . . . . . . . . . . . . . . . . . . .

4.19

Schematic representation of a family of locked states (red) of the optoelectronic SFL. In lock, the SCL (black) follows the locked state that
most closely matches its free-running chirp. In hardware stitching, temperatures and currents are tuned so that all the MS-FMCW channels
operate in the same locked state (blue). . . . . . . . . . . . . . . . . .

4.20

Top panel: time-domain stitched photocurrent in the hardware stitching experiment. Bottom panel: Single-sweep (black) and stitched foursweep (red) photocurrent spectra of a 150 µm glass microscope coverslip
suspended above a metal surface. The spectra are apodized with a Hamming window. The total chirp bandwidth is 2 THz. . . . . . . . . . . .

xv
5.1

Principle of FMCW imaging with a single reflector . . . . . . . . . . .

5.2

(a) Volume acquisition by a raster scan of a single-pixel FMCW mea-

surement across the object space. (b) Volume acquisition in a TomICam
system. 3-D information is recorded one transverse slice at a time. The
measurement depth is chosen electronically by setting the frequency of
the modulation waveform. . . . . . . . . . . . . . . . . . . . . . . . . .
5.3

(a)Spectrum of the FMCW photocurrent. The peaks at frequencies ξτ1 ,
ξτ2 , and ξτ3 , where ξ is the chirp rate, correspond to scatterers at τ1 ,
τ2 , and τ3 . (b) The beam intensity is modulated with a frequency ξτ1 ,
shifting the signal spectrum, such that the peak due to a reflector at τ1
is now at DC. This DC component is measured by a slow integrating
detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4

(a) Single-pixel FMCW system. The interferometric signal is recorded
using a fast photodetector, and reflector information is recovered at
all depths at once. (b) Single-pixel TomICam. The beam intensity is
modulated with a sinusoid, and the interferometric signal is integrated
using a slow detector. This gives one number per scan, which is used to
calculate the reflector information at a particular depth, determined by
the modulation frequency. . . . . . . . . . . . . . . . . . . . . . . . . .

5.5

A possible TomICam configuration utilizing a CCD or CMOS pixel array in a Michelson interferometer. Each transverse point (x, y) at a
fixed depth (z) in the object space is mapped to a pixel on the camera.
The depth (z) is tuned electronically by adjusting the frequency of the
modulation waveform W (t). . . . . . . . . . . . . . . . . . . . . . . . .

5.6

Schematic diagram of the TomICam proof-of-principle experiment. A
slow detector was modeled by a fast detector followed by an integrating
analog-to-digital converter. The detector signal was sampled in parallel
by a fast oscilloscope, to provide a baseline FMCW depth measurement. 93

xvi
5.7

The custom PCB used in the TomICam experiment. Implemented functionality includes triggered arbitrary waveform generation and high-bitdepth acquisition of an analog signal. . . . . . . . . . . . . . . . . . . .

5.8

Comparison between FMCW (red) and TomICam (blue) depth measurements. The two are essentially identical except for a set of ghost
targets at 13 of the frequency present in the TomICam spectrum. These
ghosts are due to the third-order nonlinearity of the intensity modulator
used in this experiment. . . . . . . . . . . . . . . . . . . . . . . . . . .

5.9

Characterization of the FMCW and TomICam dynamic range. The
signal-to-noise ratio was recorded as a function of attenuation in one of
the interferometer arms. At low attenuations, the SNR saturates due to
SFL phase noise and residual nonlinearity. . . . . . . . . . . . . . . . .

5.10

Flow diagram and parameters of the CS TomICam simulation . . . . . 101

5.11

SER curves for a CS simulation with a Gaussian random matrix . . . . 102

5.12

SER curves for a CS simulation with a waveform matrix given by the
absolute value of a Gaussian random matrix . . . . . . . . . . . . . . . 102

5.13

SER curves for a CS simulation with a waveform matrix whose entries
are uniformly distributed between 0 and 1 . . . . . . . . . . . . . . . . 104

5.14

SER curves for a CS simulation with a waveform matrix whose entries
are uniformly distributed between 0.5 and 1 . . . . . . . . . . . . . . . 104

5.15

SER curves for a CS simulation with a waveform matrix whose entries
take on the values of 0.5 or 1 with equal probabilities . . . . . . . . . . 105

5.16

SER curves for an N = 1000 CS simulation with a waveform matrix
whose entries are uniformly distributed between 0.5 and 1 . . . . . . . 105

6.1

Intuitive description of chirped-seed amplifier coherent beam combining.
A path-length mismatch between amplifier arms results in a frequency
difference at the combining point, and can therefore be compensated
using a frequency shifter placed before amplifier 2. . . . . . . . . . . . 111

xvii
6.2

Passive-fiber chirped-wave optical phase-locked loop in the homodyne
configuration. PD: Photodetector . . . . . . . . . . . . . . . . . . . . . 113

6.3

Small-signal frequency-domain model of the homodyne chirped-wave optical phase-locked loop. The model is used to study the effect noise and
fluctuations (green blocks) on the loop output variable δθ12 (ω). . . . . 113

6.4

Passive-fiber chirped-wave optical phase-locked loop in the heterodyne
configuration. PD: Photodetector . . . . . . . . . . . . . . . . . . . . . 117

6.5

Small-signal frequency-domain model of the heterodyne chirped-wave
optical phase-locked loop. The model is used to study the effect noise
and fluctuations (green blocks) on the loop output variable δθrn (ω). . . 117

6.6

Locked-state Fourier spectrum of the measured beat signal between the
reference and amplifier arms, over a 2 ms chirp interval. The nominal
loop delay parameters are τd = 20 m and τr1 ≈ 0 m. The time-domain
signal was apodized with a Hamming window. . . . . . . . . . . . . . . 120

6.7

(a) Phase difference between the reference and amplifier arms calculated
using the I/Q demodulation technique. The three curves (offset for
clarity) correspond to different values of the loop delay τd and the pathlength mismatch τr1 . (b) Transient at the beginning of the chirp. The
locking time is determined by the loop bandwidth, which is limited by
the AOFS to about 60 KHz. . . . . . . . . . . . . . . . . . . . . . . . . 121

6.8

Schematic diagram of the passive-fiber chirped-seed CBC experiment
with two channels. Heterodyne optical phase-locked loops are used to
lock the amplifier (blue, green) and reference (black) arms. The outputs
of the amplifier arms are coupled to a microlens (µ-lens) array to form
a two-element tiled-aperture beam combiner. The far-field intensity distribution of the aperture is imaged on a CCD camera. . . . . . . . . . 123

6.9

Characterization of the two heterodyne OPLLs in the locked state. τd ≈
0 m, τr1 = τr2 ≈ 0 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.10

Characterization of the two heterodyne OPLLs in the locked state. τd ≈
18 m, τr1 = τr2 ≈ 0 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

xviii
6.11

Characterization of the two heterodyne OPLLs in the locked state. τd ≈
18 m, τr1 = τr2 ≈ 32 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.12

Experimental demonstration of electronic phase control and beam steering of chirped optical waves. (a) Far-field intensity profiles for the unlocked and phase-locked cases. The position of the fringes is controlled
by varying the phase of the electronic oscillator in one loop. (b) Horizontal cross sections of the far-field intensity patterns . . . . . . . . . . 126

6.13

Schematic diagram of the dual-channel CSA coherent-combining experiment. PD: Photodetector, PM: Back-scattered power monitor . . . . . 128

6.14

Far-field intensity distributions of the individual channels and the locked
aperture. τr1 = −19 mm, and τr2 = 1 mm . . . . . . . . . . . . . . . . 130

6.15

Steering of the combined beam through emitter phase control. θos,12 is
the relative DDS phase. . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.16

I/Q-demodulated phase differences between the amplifier channels and
the reference. θos,12 is the relative DDS phase. . . . . . . . . . . . . . . 131

7.1

(a) Hybrid Si/III-V DFB laser bar. (b) Scanning electron microscope
(SEM) image of a 1 × 3 multimode interference (MMI) coupler, (c) SEM
image of a 2 × 2 MMI coupler. (d) SEM closeup of the a spiral delay
line for the loop Mach-Zehnder interferometer (MZI) . . . . . . . . . . 137

7.2

Schematic of the hybrid Si/III-V high-coherence semiconductor laser.
(a) Side-view cross section. (b) Top-view of the laser and the modulatedbandgap resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

7.3

Schematic representation of the label-free biomolecular sensing system

139

xix

List of Tables
5.1

Recent three-dimensional (3-D) camera embodiments . . . . . . . . . .

6.1

Measured OPLL phase error standard deviation and phase-locking effi-

ciency for different values of the loop delay τd and the differential delay
τr1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2

OPLL phase errors and phase-noise-limited fringe visibilities in the dualchannel active CBC experiment . . . . . . . . . . . . . . . . . . . . . . 132

Glossary of Acronyms
2-D two-dimensional
3-D three-dimensional
AOFS acousto-optic frequency shifter
CBC coherent beam combining
CHDL chirped diode laser
CS compressive sensing
CSA chirped-seed amplifier
DDS direct digital synthesis
DFB distributed-feedback laser
EDFA erbium-doped fiber amplifier
FDML Fourier-domain mode-locked
FM frequency modulation
FMCW frequency-modulated continuous-wave
FSR free spectral range
FT Fourier transform
FWHM full width at half maximum

xxi
GRIN gradient-index
I/Q in-phase and quadrature
lidar light detection and ranging
MEMS microelectromechanical
MOPA master oscillator power amplifier
MS-FMCW multiple source FMCW
MZI Mach-Zehnder interferometer
OPLL optical phase-locked loop
PCB printed circuit board
PD photodetector
PLL phase-locked loop
PSD power spectral density
PSF point spread function
radar radio detection and ranging
RF radio frequency
RIN relative intensity noise
SBS stimulated Brillouin scattering
SCL semiconductor laser
SER signal-to-error ratio
SFL swept-frequency laser
SNR signal-to-noise ratio

xxii
SOA semiconductor optical amplifier
SS-OCT swept-source optical coherence tomography
TOF time-of-flight
TomICam tomographic imaging camera
VCO voltage-controlled oscillator
VCSEL vertical-cavity surface-emitting laser
VOA variable optical attenuator

Chapter 1
Overview and Thesis Organization
1.1

Introduction

This thesis focuses on the construction and applications of the optoelectronic sweptfrequency laser (SFL)—a feedback system that enables closed-loop control over the
instantaneous optical frequency of a chirped semiconductor laser (SCL) [1–3]. Even
though our feedback architecture is laser-agnostic, we restrict our attention to SCL
diodes because of their small size, high wall-plug efficiency, and superior sub-MHz
linewidths. The wide gain bandwidth of semiconductor quantum wells, the ability to
fabricate SCLs with precisely controlled emission frequencies [4], and the fact that
SCLs can be frequency tuned with current [5] enable broadband and agile coverage
of the optical spectrum. These properties uniquely position the SCL as the device
of choice for a range of high-fidelity applications, such as optical phase-locking and
coherent combining [6–12], ranging and 3-D imaging [1, 13, 14], and spectroscopy and
chemical sensing [6, 15]. The design and construction of the optoelectronic SFL is
discussed in chapter 3.
The optoelectronic SFL can be configured to generate chirps with any arbitrary
optical frequency vs. time profile, subject to the tunability of the SCL in its core.
Precisely linear frequency sweeps are of particular interest because of their applications in optical frequency-modulated continuous-wave (FMCW) reflectometry and
3-D imaging, as described in chapter 2, and chirped-seed phase-locking, as described
in chapter 6. Building on our group’s expertise in the field of phase and frequency

control of SCLs, we develop applications that take advantage of the unique properties
of the SCL-based optoelectronic SFL. These applications can be subdivided into two
categories: ranging and 3-D imaging using FMCW reflectometry, and coherent beam
combining (CBC) of chirped-seed amplifiers (CSAs).

1.2

Ranging and 3-D Imaging Applications

The fundamental challenge of 3-D imaging is ranging—the retrieval of depth information from a scene or a sample. One way to construct a 3-D imaging system is
to launch a laser beam along a particular axis, and collect the reflected light, in an
effort to determine the depths of all the scatterers encountered by the beam as it
propagates. A 3-D image may then be recorded by scanning the beam over the entire
object space.
A conceptually simple way to retrieve depth information is to launch optical pulses,
and record arrival times of the reflections. Scatterer depth can then be calculated
by multiplying the arrival times by the speed of light c. Implementations based on
this idea, collectively known as time-of-flight (TOF) systems, have been successfully
demonstrated [16, 17]. The depth resolution, also called range resolution or axial
resolution, of TOF methods depends on the system detection bandwidth, with 1 GHz
yielding a resolution of ∆z ∝ c × (1 ns) = 30 cm in free space. Improvement of the
resolution to the sub-mm range requires detectors with 100s of GHz of bandwidth,
and is prohibitively expensive with current technology.

1.2.1

Optical FMCW Reflectometry

The technique of frequency-modulated continuous-wave (FMCW) reflectometry, originally developed for radio detection and ranging (radar), can be applied to the optical
domain to circumvent the detector bandwidth limit by using a swept-frequency optical
waveform. Systems utilizing FMCW reflectometry, also known as swept-source optical
coherence tomography (SS-OCT) in the biomedical optics community, are capable of

resolutions of a few µm with low detection bandwidths. As a result, FMCW reflectometry has found numerous applications, e.g. light detection and ranging (lidar) [18,19],
biomedical imaging [20, 21], non-contact profilometry [22, 23] and biometrics [24, 25].
The FMCW technique is analyzed in full detail in chapter 2, and in chapter
3 we apply the optoelectronic SFL to FMCW imaging and demonstrate a simple
profilometry application.

1.2.2

Multiple Source FMCW Reflectometry

In chapter 4 we describe multiple source FMCW (MS-FMCW) reflectometry—a novel
imaging approach aimed at increasing the effective bandwidth of an FMCW ranging system. This is achieved by combining, or stitching, separate swept-frequency
lasers (SFLs), to approximate a swept-source with an enhanced bandwidth [13,14,19].
The result is an improvement in the range resolution proportional to the increase in
the swept-frequency range. This technique is of particular interest in the context of
the SCL-based optoelectronic SFL. MS-FMCW leverages narrow SCL linewidths to
present a pathway towards long-distance ranging systems with sub-100 µm resolutions.

1.2.3

The Tomographic Imaging Camera

FMCW reflectometry enables the retrieval of depth information from a single location in the transverse plane. One way to acquire a full 3-D data set is through
mechanical raster-scanning of the laser beam across the object space. The acquisition time in such systems is ultimately limited by the scan speed, and for very
high resolution datasets (> 1 transverse mega pixel) is prohibitively slow. Rapid 3-D
imaging is of crucial importance in in vivo biomedical diagnostics [21, 26] because
it reduces artifacts introduced by patient motion. In addition, a high-throughput,
non-destructive 3-D imaging technology is necessary to meet the requirements of several new industrial developments, including 3-D printing and manufacturing [27], 3-D
tissue engineering [28–30], and 3-D cell cultures and tissue models [31].

In chapter 5 we discuss the tomographic imaging camera (TomICam), which combines FMCW ranging with non-mechanical transverse imaging, enabling robust, large
field of view, and rapid 3-D imaging. We also discuss the application of compressive
sensing (CS) to the TomICam platform. CS is an acquisition methodology that takes
advantage of signal structure to compress and sample the information in a single
step. It is of particular interest in applications involving large data sets, such as 3-D
imaging, because compression reduces the volume of information that is recorded by
the sensor, effectively speeding up the measurement.

1.3

Phase-Locking and Coherent Combining of
Chirped Optical Waves

In chapter 6, we switch gears and discuss our work on the phase-locking of and
coherent combining of chirped optical waves. The phase-locking of optical waves
with arbitrary frequency chirps is a difficult problem in general. However, precisely
linear chirps, such as the ones generated by the optoelectronic SFL can be phaselocked with very high efficiency using a frequency shifter. The main application of
this result is the simultaneous stimulated Brillouin scattering (SBS) suppression and
coherent combining of high-power fiber amplifiers.
The output power of optical fiber amplifiers is usually limited by SBS. Conventional methods to suppress SBS by increasing its threshold include the broadening
of the seed laser linewidth through high-speed phase modulation. The increase in
the amplifier SBS threshold comes at the expense of the seed coherence length [32],
which places strict path-length matching requirements on the scaling of optical power
through coherent combining of multiple amplifiers. Efficient coherent combining of
such amplifiers has been demonstrated, but requires careful path-length matching to
submillimeter accuracy [33, 34].
In chapter 6 we explore an architecture capable of SBS suppression and coherent
combining without stringent mechanical path-length matching requirements. Our

approach is to use the optoelectronic SFL as the amplifier seed, in order to reduce
the effective length over which SBS occurs [35, 36]. We develop a chirped phaselocking technique and demonstrate its use in coherent beam combining of multiple
chirped-seed amplifiers. Path-length matching requirements are relaxed due to the
long coherence length (10s of meters) of semiconductor laser based SFLs.
The work described in chapter 6 was performed in collaboration with Jeffrey O.
White’s group at the United States Army Research Laboratory.

Chapter 2
Optical FMCW Reflectometry
2.1

Introduction

The centerpiece and workhorse of the research described in this thesis is the optoelectronic swept-frequency laser (SFL)—a feedback system designed around a frequencyagile laser to produce precisely linear optical frequency sweeps (chirps) [1–3]. This system is studied in detail in chapter 3. In the present chapter, by way of introduction, we
focus on an application of swept-frequency waveforms, optical frequency-modulated
continuous-wave (FMCW) reflectometry, and its use in three-dimensional (3-D) imaging. We examine how chirp characteristics affect application metrics and therefore
motivate the choices made in the design of the optoelectronic SFL.
The fundamental challenge of 3-D imaging is ranging—the retrieval of depth information from a scene or a sample. One way to construct a 3-D imaging system
is to launch a laser beam along a particular axis, and collect the reflected light, in
an effort to determine the depths of all the scatterers encountered by the beam as it
propagates. A 3-D image may then be recorded by scanning the beam over the entire
object space.
A conceptually simple way to retrieve depth information is to launch optical pulses,
and record arrival times of the reflections. Scatterer depth can then be calculated
by multiplying the arrival times by the speed of light c. Implementations based on
this idea, collectively known as time-of-flight (TOF) systems, have been successfully
demonstrated [16, 17]. The depth resolution, also called range resolution or axial

resolution, of TOF methods depends on the system’s ability to generate and record
temporally narrow optical pulses. A state-of-the-art TOF system therefore requires a
costly pulse source, e.g., a mode-locked laser, and a high-bandwidth detector [37]. A
detection bandwidth of 1 GHz yields a resolution of ∆z ∝ c × (1 ns) = 30 cm in free
space. Improvement of the resolution to the sub-mm range requires detectors with
100s of GHz of bandwidth, and is prohibitively expensive with current technology.
The technique of frequency-modulated continuous-wave (FMCW) reflectometry,
originally developed for radio detection and ranging (radar), can be applied to the optical domain to circumvent the detector bandwidth limit by using a swept-frequency
optical waveform. Systems utilizing FMCW reflectometry, also known as swept-source
optical coherence tomography (SS-OCT) in the biomedical optics community, are capable of resolutions of a few µm with low detection bandwidths. Moreover, optical
FMCW is an interferometric technique in which the measured signal is proportional
to the reflected electric field, as opposed to the reflected intensity, as in the TOF
case. The signal levels due to a scatterer with reflectivity R < 1 are therefore propor√
tional to R and R in TOF and FMCW systems, respectively. The combination of
higher signal levels due to electric field dependence, and lower noise due to low detection bandwidths results in a significantly higher dynamic range and sensitivity of the
FMCW system versus a TOF implementation [37,38]. As a result, FMCW reflectometry has found numerous applications, e.g., light detection and ranging (lidar) [18,19],
biomedical imaging [20, 21], non-contact profilometry [22, 23] and biometrics [24, 25].

2.1.1

Basic FMCW Analysis and Range Resolution

Let us first examine the problem of recovering single-scatterer depth information using
a SFL. For simplicity, we consider a noiseless laser whose frequency changes linearly
with time. The normalized electric field at the source, for a single chirp period, is
given by
e(t) = rect

t − T /2

ξt2
cos φ0 + ω0 t +

(2.1)

where T is the scan duration, ξ is the slope of the optical chirp, and φ0 and ω0 are
the initial phase and frequency, respectively. The rect function models the finite
time-extent of the chirp and is defined by:
0, |x| > 1/2
rect(x) ≡ 1/2, |x| = 1/2
 1, |x| < 1/2

(2.2)

The instantaneous optical frequency is given by the time derivative of the argument
of the cosine in equation (2.1)
ωSF L (t) =
dt

ξt2
φ 0 + ω0 t +

= ω0 + ξt

(2.3)

The total frequency excursion of the source (in Hz) is then given by B = ξT /2π.
We illuminate a single scatterer with the chirped field, and collect the reflected light.
The time evolution of the frequencies of the launched and reflected beams is shown
in figure 2.1. Because the chirp is precisely linear, a scatterer with a round-trip time
delay τ (and a corresponding displacement cτ /2 from the source) results in constant
frequency difference ξτ between the launched and reflected waves.
The FMCW technique relies on a measurement of this frequency differences to determine the time delay τ . This is accomplished in a straightforward way by recording
the time-dependent interference signal between the launched and reflected waves on
a photodetector. An FMCW measurement setup based on a Mach-Zehnder interferometer (MZI) is shown schematically in figure 2.2. Another common implementation

ωL

Figure 2.1: Time evolution of the optical frequencies of the launched and reflected
waves in a single-scatterer FMCW ranging experiment

is based on a Michelson interferometer, and is shown in figure 2.3. In both implementations, the sum of the electric fields of the launched and reflected waves is incident
on a photodetector. It is common to call the launched wave a local or a reference
wave, and we will use all three terms interchangeably (hence the reference arm and
reference mirror designations in the MZI and Michelson interferometer figures).
The normalized photocurrent is equal to the time-averaged intensity of the incident
beam, and is given by

e(t) + R e(t − τ )

t − T /2
1+R √
ξτ 2
= rect
+ R cos (ξτ )t + ω0 τ −

i(t) =

(2.4)

where R is the target reflectivity, and we have assumed that τ << T . The averaging,
denoted by h·it , is done over an interval that is determined by the photodetector
response time, and is much longer than an optical cycle, yet much shorter than the
period of the cosine in equation (2.4). In the expressions that follow we drop the
DC term (1 + R)/2 for simplicity. It is convenient to work in the optical frequency

Figure 2.2: Mach-Zehnder interferometer implementation of the FMCW ranging experiment

Figure 2.3: Michelson interferometer implementation of the FMCW ranging experiment

11
domain, so we use equation (2.3) to rewrite the photocurrent as a function of ωSF L .
ωSF L − ω0
y(ωSF L ) ≡ i
ωSF L − ω0 − πB
ξτ 2
cos ωSF L τ −
= R rect
2πB

(2.5)

The delay τ is found by taking the Fourier transform (FT) of y(ωSF L ) with respect
to the variable ωSF L , which yields a single sinc function centered at the delay τ .

ξτ 2
exp [−j(ζ − τ )(ω0 + πB)] sinc [πB(ζ − τ )] ,
Y (ζ) ≡ FωSF L {y(ωSF L )} = πB R exp −j
(2.6)
where ζ is the independent variable of the FT of y(ωSF L ), and has units of time, and
sinc(x) = sinx x . Additionally, we only consider positive Fourier frequencies since the
signals of interest are purely real, and the FT therefore possesses symmetry about
ζ = 0.
A collection of scatterers along the direction of beam propagation arising, for
example, from multiple tissue layers in an SS-OCT application, results in a collection
of sinusoidal terms in the photodetector current, so that equation (2.6) becomes:
Xp
ξτn2
Y (ζ) = πB
Rn exp −j
exp [−j(ζ − τn )(ω0 + πB)] sinc [πB(ζ − τn )],
(2.7)
where τn and Rn are the round-trip time delay and the reflectivity of the n-th scatterer. Each scatterer manifests itself as a sinc function positioned at its delay, with a
strength determined by its reflectivity. The ζ-domain description is therefore a map
of scatterers along the axial direction.
The range resolution is traditionally chosen to correspond to the coordinate of the
first null of the sinc function in equation (2.6) [39]. The null occurs at ζ = τ + 1/B,
which corresponds to a free-space axial resolution
∆z =

(2.8)

12
The first constraint on the SFL is therefore the chirp bandwidth B—a large optical
frequency range is necessary in order to construct a high-resolution imaging system.
SS-OCT applications require resolutions below 10 µm in order to resolve tissue structure, and therefore make use of sources with bandwidths exceeding 10 THz.
An additional constraint on the imaging system is the need for precise knowledge of
the instantaneous optical frequency as a function of time—it was used in transforming
the photocurrent to the ω-domain. In the preceding analysis we have assumed a
linear frequency sweep. While chirp linearity is preferred since it simplifies signal
processing, it is not strictly necessary. As long as ωSF L (t) is known precisely, it is
still possible to transform the measured signal to the optical frequency domain, and
extract the scatterer depth information. Because most SFLs have nonlinear chirps,
it is common practice to measure the instantaneous chirp rate in parallel with the
measurement using a reference interferometer. A related technique relies on what is
called a k-clock—an interferometer that is used to trigger photocurrent sampling at
time intervals that correspond to equal steps in optical frequency [20]. The k-clock is
therefore a hardware realization of the ω-domain transformation.
While nonlinear chirps can be dealt with, they require faster electronics in order
to acquire the higher frequency photocurrents associated with a nonuniform chirp
rate. The optoelectronic SFL described in chapter 3 uses active feedback to enable
precise control of the instantaneous optical frequency. As a result, the chirp can be
programmed to be exactly linear in advance, allowing the use of a lower detection
bandwidth, and hence decreasing electronic noise in an FMCW measurement.

2.1.2

Balanced Detection and RIN

In the preceding FMCW analysis we have simplified the expressions by intentionally
leaving out DC contributions to the photocurrent. This simplification, while valid in
an ideal noiseless laser, needs further justification in a practical measurement. The
output intensity of laser systems varies due to external causes such as temperature and
acoustic fluctuations, and also due to spontaneous emission into the lasing mode [40].

13
These intensity fluctuations scale with the nominal output intensity and are termed
relative intensity noise (RIN). In a laser with RIN, the terms which give rise to the
DC components of equation (2.4), also give rise to a noise component that we call
n(t). Equation (2.4) is therefore modified to
i(t) = rect

t − T /2

1+R
ξτ 2
+ n(t) + R cos (ξτ )t + ω0 τ −

(2.9)

The term n(t) is a random variable whose statistics depend on the environmental
conditions, the type of laser used in the measurement, and on the frequency response
of the detection circuit. While the DC terms are readily filtered out, n(t) is broadband and can corrupt the signal. This corruption is particularly important when the
scatterers are weak and the signal level is low.
Balanced detection is a standard way to null the contribution of the DC terms and
RIN. It relies on the use of a 2x2 coupler and a pair of photodetectors to measure the
intensities of both the sum and the difference of the reference and reflected electric
fields. Mach-Zehnder interferometer (MZI) and Michelson interferometer balanced
FMCW implementations are shown in figure 2.4 and figure 2.5. These measurements
produce pairs of photocurrents
i± (t) = rect

t − T /2

1+R
ξτ 2
+ n(t) ± R cos (ξτ )t + ω0 τ −
. (2.10)

Balanced processing consists of averaging the two photocurrents, yielding
i+ (t) − i− (t)
= rect
idiff (t) ≡

t − T /2

ξτ 2
R cos (ξτ )t + ω0 τ −

(2.11)

The DC and RIN terms are nulled in the subtraction, justifying the simplification
made earlier. However, small gain differences in the photodetector circuitry, as well
as slight asymmetries in the splitting ratio of the 2x2 coupler, result in a small amount
of residual DC and RIN being present in the balanced photocurrent. This places a
further constraint on the SFL—it is desirable that the laser possess a minimal amount
of RIN so as to limit the amount of noise left over after balancing, and therefore

Figure 2.4: A balanced Mach-Zehnder interferometer implementation of the FMCW
ranging experiment

Figure 2.5: A balanced Michelson interferometer implementation of the FMCW ranging experiment

15
enhance the measurement dynamic range.

2.1.3

Effects of Phase Noise on the FMCW Measurement

So far we have assumed an SFL with a perfectly sinusoidal electric field. Practical
lasers, however, exhibit phase and frequency noise. These fluctuations arise due
to both external causes, such as thermal fluctuations, as well as due to spontaneous
emission into the lasing mode [40]. These phenomena are responsible for a broadening
of the spectrum of the electric field of a laser. In this section we analyze the effects
of phase noise on the FMCW measurement. We begin by deriving the linewidth ∆ω
of single-frequency emission with phase noise. We then modify the FMCW equations
to account for phase noise, derive its effects on fringe visibility, and define the notion
of coherence time. To further quantify the effects of phase noise, we calculate the
FMCW photocurrent spectrum. It will turn out that phase noise degrades the signalto-noise ratio (SNR) with increasing target delay, putting a limit on the maximum
range that can be reliably measured. We conclude by deriving statistical properties
of the measurement accuracy, which help quantify system performance in a singlescatterer application (for example, profilometry).

2.1.3.1

Statistics and Notation

We first review some useful statistical results and introduce notation. For a wide-sense
stationary random process x(t), we denote its autocorrelation function by Rx :
Rx (u) = E [x(t)x(t − u)] ,

(2.12)

where E[·] is the statistical expectation value. For an ergodic random process, the
expectation can be replaced by an average over all time, giving:
Rx (u) = hx(t)x(t − u)it ,

(2.13)

16
By the Wiener–Khinchin theorem, the power spectral density (PSD) Sx (ω) and autocorrelation Rx (u) are FT pairs.
Sx (ω) = Fu [Rx (u)] =

Z ∞
−∞

Rx (u)e−iωu du,

(2.14)

where Fu [·] is the Fourier transform with respect to the variable u. We denote the
variance of x(t) by σx2 . For an ergodic process, the variance may be calculated in the
time domain:
σx2 = x(t)2 t − hx(t)i2t .

(2.15)

Alternatively, it may be calculated by integrating the PSD:
σx2 =

2.1.3.2

2π

Z ∞
−∞

Sx (ω)dω.

(2.16)

Linewidth of Single-Frequency Emission

We first derive a standard model for the spontaneous emission linewidth of a singlefrequency laser [41]. The electric field is given by
e(t) = cos [ω0 t + φn (t)] ,

(2.17)

where φn (t) is a zero-mean stationary phase noise term. Plugging this expression into
equation (2.13), we find the autocorrelation.
Re (u) = hcos [ω0 t + φn (t)] cos [ω0 (t − u) + φn (t − u)]it
hhhh

hhhh
= hcos [ω0 u + ∆φn (t, u)]it + hcos [2ω0 t − ω0 uh+hφh
hh+hφh
n (t)
n (t − u)]it ,
hhhh
(2.18)

where the sum term is crossed out because it averages out to zero. ∆φn (t, u) is the
accumulated phase error during time u, defined by
∆φn (t, u) ≡ φn (t) − φn (t − u),

(2.19)

17
and is the result of a large number of independent spontaneous emission events. By
the central limit theorem, ∆φn (t, u) must be a zero-mean Gaussian random variable.
The following identities therefore apply:
(u)
σ∆φ
hcos [∆φn (t, u)]it = exp −
, and hsin [∆φn (t, u)]it = 0.

(2.20)

So, equation (2.18) simplifies to
2
σ∆φn (u)
Re (u) = cos(ω0 u) exp −

(2.21)

Taking the FT of equation (2.21), we find the spectrum of the electric field,
Se (ω) =

1 ◦
[S (ω − ω0 ) + Se◦ (ω + ω0 )] ,
4 e

(2.22)

where Se◦ (ω) is the baseband spectrum given by
Se◦ (ω) = Fu

2
σ∆φn (u)
exp −

(2.23)

To determine the emission lineshape we first consider the variance of the accumulated
phase error. We start by expressing the autocorrelation of ∆φn (t, u) in terms of the
autocorrelation of φn (t). Using equation (2.13) and equation (2.19),
R∆φn (s, u) = h∆φn (t, u)∆φn (t − s, u)it = 2Rφn (s) − Rφn (s + u) − Rφn (s − u). (2.24)
The PSD is given by
S∆φn (ω, u) = Fs [R∆φn (s, u)] = Sφn (ω) 2 + ejωu + e−jωu
= 4Sφn (ω) sin2 (ωu) = 4u2 Sφ.n (ω)sinc2 (ωu),

(2.25)

where Sφ.n (ω) = ω 2 Sφn (ω) is the spectrum of the frequency noise φn . Spontaneous
emission into the lasing mode gives rise to a flat frequency noise spectrum [40, 42],

18
and we therefore assign a constant value to Sφ.n (ω),
Sφ.n (ω) ≡ ∆ω.

(2.26)

We plug equation (2.25) and equation (2.26) into equation (2.16) to calculate the
variance of the accumulated phase error.
Z ∞
S∆φn (ω, u)dω
2π −∞
Z ∞
4u2 ∆ω sinc2 (ωu)dω
2π −∞

σ∆φ
(u) =

(2.27)

= |u|∆ω.
Plugging this result into equation (2.23), we obtain the baseband spectrum of the
electric field.

2
σ∆φn (u)
∆ω
= Fu exp −|u|
exp −
∆ω
(∆ω/2)2 + ω 2

Se◦ (ω) = Fu

(2.28)

The presence of phase noise broadens the baseband spectrum from a delta function
to a Lorentzian function with a full width at half maximum (FWHM), or linewidth,
of ∆ω.
To summarize, a flat frequency noise spectrum with a value of ∆ω corresponds to
a linewidth of ∆ω.
Sφ.n (ω) = ∆ω ⇐⇒ linewidth ∆ω (rad/s)

(2.29)

So far we have been using angular frequency units (rad/s) for both frequency noise and
linewidth. Ordinary frequency units (Hz) are often used, so we convert the relation
in equation (2.29) to
S φ.n (ν) =
2π

∆ν
∆ω
Sφ.n (2πν) =
⇐⇒ linewidth ∆ν =
(Hz),
(2π)
2π
2π

(2.30)

19
where ν = ω/(2π) is the Fourier frequency in Hz. In practice, there are other noise
sources that give rise to a 1/f behavior of the frequency noise spectrum. It has been
shown that such noise sources generate a Gaussian lineshape [43].
As an exercise, we numerically verify equation (2.30) using a Monte Carlo simulation. We model a flat angular frequency noise spectrum by drawing samples from
a zero-mean Gaussian distribution. These frequency noise samples are integrated in
time, and the cosine of the resultant phase noise is calculated. The PSD of this signal
therefore corresponds to half the baseband spectrum of equation (2.28). Each iteration of this procedure is performed over a finite time T , and therefore yields only an
estimate of the true PSD. If the angular frequency resolution 2π/T is much smaller
than the angular linewidth ∆ω, the mean of this estimate, over many iterations, will
converge to equation (2.28) [44].
Estimates of baseband electric field spectra corresponding to Sφ.n (ω) = 2π(1 MHz)
are shown in blue in figure 2.6. As the number of iterations N used in the calculation
is increased, the simulated PSD converges to the true PSD of equation (2.28), shown
in red. Simulated frequency noise spectra and corresponding baseband lineshapes
for three different values of ∆ω are plotted in figure 2.7, illustrating the relation of
equation (2.29).

2.1.3.3

Fringe Visibility in an FMCW Measurement

We continue or analysis by modifying the chirped electric field in equation (2.1) to
include phase noise,
e(t) = rect

t − T /2

ξt2
cos φ0 + ω0 t +
+ φn (t) ,

(2.31)

and assume a perfect reflector (R = 1). The photocurrent is therefore given by
i(t) = rect

t − T /2

ξτ 2
1 + cos (ξτ )t + ω0 τ −
+ ∆φn (t, τ ) ,

(2.32)

Electric field baseband PSD (a.u.)

−2

−1

N=1

N = 10

N = 100

N = 1000

−2 −1
Frequency (MHz)

Figure 2.6: Convergence of the Monte Carlo simulation of the baseband electric field
spectrum (blue) to the theoretical expression (red). The angular linewidth is ∆ω =
2π(1 MHz). N is the number of iterations used in calculating the PSD estimate.

300

Electric field baseband PSD (a.u.)

2π Sφ̇

(kHz, log scale)

900

100

−1

−0.5

Frequency (MHz)

0.5

100

Figure 2.7: Normalized frequency noise spectra (top panel) and corresponding baseband electric field spectra (bottom panel) for ∆ω = 2π(900 kHz) (black), 2π(300 kHz)
(blue), and 2π(100 kHz) (green). The spectra are averaged over N=1000 iterations.
The red curves are plots of the theoretical lineshape for the three values of ∆ω.

22
where ∆φn (t, τ ) is the familiar accumulated phase error during time τ . In the noiseless
case, the oscillations (fringes) in the photocurrent extend from 0 to 2. The presence
of phase noise will add jitter to the locations of the peaks and troughs. The amplitude of the fringes, averaged over many scans, is therefore expected to decrease with
increasing phase noise. To quantify this effect, we define the fringe visibility
V ≡

imax − imin
imax + imin

(2.33)

where imax and imin are the photocurrent values at the peaks and troughs, averaged
over many scans. The visibility takes on a value of 1 in the noiseless case, and goes
to zero as the amount of noise increases. Using the identities in equation (2.20), we
write down expressions for the maximum and minimum currents,
(τ )
σ∆φ
imax = 1 + exp −
, and
2
σ∆φn (τ )
imin = 1 − exp −

(2.34)

Plugging in equation (2.27) and equation (2.34) into equation (2.33), we arrive at an
expression for the phase-noise-limited visibility [45],

∆ω
V = exp −|τ |

|τ |
= exp −
τc

(2.35)

where
τc ≡

∆ω
π∆ν

(2.36)

is the coherence time of the SFL. For delays much shorter than the coherence time,
the visibility decreases linearly with τ . Once τ is comparable to τc , the visibility
drops exponentially. The coherence time is therefore a measure of the longest range
that can be acquired by an FMCW system.

23
2.1.3.4

Spectrum of the FMCW Photocurrent and the SNR

The signal-to-noise ratio (SNR) is more useful in quantifying the effect of phase noise
than the visibility. To determine the SNR we must first calculate the photocurrent
spectrum. We assume a balanced detector and disregard, for now, the rect function
that models the finite chirp bandwidth of the SFL. The photocurrent expression
becomes
i(t) =

ξτ 2
R cos (ξτ )t + ω0 τ −
+ ∆φn (t, τ ) .

(2.37)

Plugging this expression into equation (2.13), we find the autocorrelation,
hcos [(ξτ )u + ∆φn (t, τ ) − ∆φn (t − u, τ )]it
= hcos [(ξτ )u + θ(t, τ, u)]it
2
σθ (τ, u)
= cos [(ξτ )u] exp −

(2.38)

θ(t, τ, u) ≡ ∆φn (t, τ ) − ∆φn (t − u, τ ),

(2.39)

Ri (u) =

where

and we have assumed that θ(t, τ, u) possesses Gaussian statistics. Taking the FT of
equation (2.38), we find the spectrum of the photocurrent.
Si (ω) =

1 ◦
[S (ω − ξτ ) + Si◦ (ω + ξτ )] ,
4 i

(2.40)

where Si◦ (ω) is the baseband spectrum given by
Si◦ (ω) = Fu

exp −

σθ(τ,u)

(2.41)

To find the baseband spectrum and the SNR we need to calculate the variance of
θ(t, τ, u). First we derive a useful identity. Let us write down the variance of ∆φn (t, u),

24
as it is defined in equation (2.15),
(u) = [φn (t) − φn (t − u)]2 t
σ∆φ

(2.42)

= 2σφ2 n − 2 hφn (t)φn (t − u)it .
This gives us an expression for the autocorrelation of φn (t),
Rφ (u) = hφn (t)φn (t − u)it = σφ2 n −

(u)
σ∆φ

(2.43)

We plug this result into equation (2.24),
R∆φn (s, u) = 2Rφn (s) − Rφn (s + u) − Rφn (s − u)

σ∆φ
(s + u) σ∆φ
(s − u)
(s)
− σ∆φ

(2.44)

We are now in a position to calculate the variance of θ(t, τ, u). Beginning with the
definition in equation (2.15),
σθ2 (τ, u) = [∆φn (t, τ ) − ∆φn (t − u, τ )]2 t
= ∆φn (t, τ )2 + ∆φn (t − u, τ )2 − 2∆φn (t, τ )∆φn (t − u, τ ) t

(2.45)

= 2σ∆φ
(τ ) − 2R∆φn (u, τ ).

Plugging in equation (2.44), we arrive at
σθ2 (τ, u) = 2σ∆φ
(τ ) + 2σ∆φ
(u) − σ∆φ
(u + τ ) − σ∆φ
(u − τ ).

(2.46)

Using the result of equation (2.27), we write down a final expression for the variance of θ(t, τ, u),
σθ2 (τ, u) = ∆ω (2τ + 2|u| − |u − τ | − |u + τ |)
4|u|
|u| ≤ τ,
τc
4τ
|u| > τ.
τc

(2.47)

25
The baseband photocurrent spectrum is found by plugging equation (2.47) into equation (2.41), yielding [46, 47]
Si◦ (ω) = Fu

exp −

= 2πδ(ω)e

− 2τ
τc

σθ(τ,u)

τc

ωτc 2

1−e

− 2τ
τc

sin(ωτ ) .
cos(ωτ ) +
ωτc

(2.48)

This expression has two terms—the delta function that represents the beat signal
due to an interference of the reference and reflected beams, and the noise pedestal
that arises as a result of the finite coherence time of the chirped beam. Each FMCW
measurement is performed over a finite time T , and its PSD is therefore only an
estimate of equation (2.48). The expected spectrum is given by the convolution of
equation (2.48) and the PSD of the rect function that accompanies the electric field
of equation (2.31) [44],

1 ◦
Tω
S (ω) ? T sinc
2π i

τc
Tω − 2τ
− 2τ
e c +
sin(ωτ ) .
= T sinc
2 1 − e c cos(ωτ ) +
ωτc
1 + ωτc

Si◦ (ω, T ) =

(2.49)
In performing this convolution we have assumed that the scan time is the slowest
time scale in the model, i.e., T
τ and T
τc , so that the sinc-squared PSD of the
rect function effectively acts as a delta function when convolved with the spectrum of
the noise pedestal. Plots of equation (2.49) for four different values of τ /τc are shown
in figure 2.8. The scan time is T = 1 ms and the coherence time is τc = 1 µs. The
spectra are normalized to the level of the noise at ω = 0. In the coherent regime, i.e.,
τc , the PSD comprises a sinc-squared signal peak and a broad noise pedestal
with oscillations. The period of these oscillations is given by 2π/τ . As the delay is
increased, the signal peak shrinks, and the noise pedestal grows, until we obtain a
Lorentzian profile with a FWHM of 2∆ω. This is what we expect for a beat spectrum
of two uncorrelated beams with a linewidth of ∆ω each.

Photocurrent baseband PSD relative to noise (dBrn/Hz)

40
30

τ/τc = 0.3

τ/τc = 1

τ/τc = 3

τ/τc = 10

20
10
−10
−20
−30
40
30
20
10
−10
−20
−30
−10

−10

Frequency (MHz)
Figure 2.8: Baseband FMCW photocurrent spectra for four different values of τ /τc ,
normalized to zero-frequency noise levels. The scan time is T = 1 ms and the coherence time is τc = 1 µs.

80
60

Signal to noise ratio (dB)

40
20
−20
−40
−60
−80 −2
10

T/τc = 102
T/τc = 103
T/τc = 104
−1

τ/τc

Figure 2.9: FMCW SNR as a function of τ /τc for three different values of T /τc

28
The SNR is readily calculated from equation (2.49), and is given in decibel units
by

SNRdB = 10 log10 

 .
τi e2τ /τc − 1 + 2τ

(2.50)

τc

A plot of the SNR versus τ /τc is shown in figure 2.9 for three different values of
T /τc . In the coherent regime, the SNR decreases at 20 dB/decade with τ /τc , and
drops sharply for τ > τc . This is consistent with the rapid decrease in visibility
for delays longer than the coherence time, as predicted by equation (2.35). As the
current analysis shows, the visibility is not the full story—even low fringe visibilities
can result in a decent SNR, provided that the scan time T is long enough.

2.1.3.5

Phase-Noise-Limited Accuracy

The axial resolution of an FMCW system, ∆z = c/2B, quantifies its ability to tell
apart closely-spaced scatterers. If we assume that the beam only encounters a single
scatterer, as it would in a profilometry application, then the relevant system metric
is the accuracy—the deviation of the measured target delay τ m from the true target
delay τ . We briefly consider statistical properties of the accuracy using the phase
noise model developed above.
The instantaneous photocurrent frequency in a single-scatterer FMCW experiment is given by a derivative of the cosine phase in equation (2.37),
ωP D (t) = ξτ +

∆φn (t, τ ).
dt

(2.51)

The target delay is calculated from an average of the photocurrent frequency over the
scan time T ,
ξτ

Z T
ωP D (t) = ξτ +

∆φn (T, τ ) − ∆φn (0, τ )

(2.52)

29
The accuracy is therefore given by
δτ ≡ τ m − τ =

∆φn (T, τ ) − ∆φn (0, τ )

(2.53)

The accuracy of a single measurement is a zero-mean random process with standard
deviation
σδτ =

4τ
τc

(2.54)

where we have used equation (2.39) and equation (2.47). Likewise, the depth accuracy
δz is characterized by the standard deviation
σδz =
2B

4τ
= ∆z
τc

4τ
τc

(2.55)

Equation (2.55) shows that by operating in the sub-coherent regime, τ
τc , it is
possible to measure spatial features on a scale that is much finer than the axial
resolution. We come back to this idea in section 3.5.2, where we are able to record
surface variations on a scale of a few tens of microns using an FMCW system with
an axial resolution of 300 µm.

2.1.4

Summary

We have introduced the technique of optical frequency-modulated continuous-wave
reflectometry and outlined its advantages over TOF ranging in 3-D imaging applications. We have derived the dependence of axial resolution on the chirp bandwidth
and introduced balanced detection as a way to mitigate intensity noise. We have
shown that SFL linewidth puts an upper limit on the target range, introduced system performance metrics, and derived the dependence of these metrics on the SFL
coherence length, target delay, and scan time.
An ideal SFL will possess a narrow linewidth, linear frequency tuning, high chirp
bandwidth, and a low RIN. The semiconductor laser (SCL)-based optoelectronic SFL
attains these qualities without moving parts, and is studied in detail in chapter 3.

Chapter 3
The Optoelectronic
Swept-Frequency Laser
3.1

Introduction

In this chapter we study the optoelectronic swept-frequency laser (SFL)—a feedback
system that enables closed-loop control over the instantaneous optical frequency of a
chirped semiconductor laser (SCL). Precisely linear frequency sweeps are of particular
interest because of their applications in optical frequency-modulated continuous-wave
(FMCW) reflectometry and 3-D imaging, as described in chapter 2. The SFL is a key
component of an FMCW system since its characteristics directly affects important
performance metrics. Specifically, the axial resolution and the maximum range are
inversely proportional to the laser frequency tuning range and linewidth, respectively.
Mechanically tunable extended cavity lasers with large frequency excursions of
about 10 THz have been used in medical tomographic applications to achieve range
resolutions of about 10 µm [26, 48]. However, linewidths of tens of GHz, which are
typical for such devices, limit ranging depths to just a few mm [49, 50]. Moreover,
the mechanical nature of the frequency tuning limits the scan repetition rate and
adds overall system complexity. Commercially available semiconductor laser (SCL)
diodes, on the other hand, offer superior sub-MHz linewidths, corresponding to ranging depths of a few hundred meters, and can be frequency tuned with current [5],
enabling precise chirp control with closed-loop feedback [1]. The small size and high

31
wall-plug efficiency of these devices makes them attractive for hand-held applications.
The wide gain bandwidth of semiconductor quantum wells and the ability to fabricate
SCLs with precisely controlled emission frequencies [4] make possible sophisticated
imaging modalities such as multiple source FMCW [13, 14], described in chapter 4.
In this chapter we begin by analyzing the SCL-based optoelectronic SFL. We
derive equations governing the SFL closed-loop operation, and describe a bias current predistortion algorithm that improves the SFL linearity. We discuss the SCLs
that were used in our experiments and describe an amplitude control sub-system
that suppresses the intensity modulation of a current-tuned SCL. We demonstrate
closed-loop linear chirps at range of chirp rates and wavelengths, and show that the
optoelectronic SFL is capable of generating arbitrary chirp profiles. We describe our
collaborative efforts with Telaris Inc. to implement the feedback and predistortion
functionality on an embedded electronic platform and commercialize the SFL. We
conclude by demonstrating the use of the optoelectronic SFL in reflectometry and
profilometry applications.

3.2

System Analysis

A schematic diagram of the optoelectronic SFL is shown in figure 3.1. The system
comprises an SCL coupled to a Mach-Zehnder interferometer (MZI), a photodetector
at the MZI output, a mixer that compares the phases of the photocurrent and the
reference oscillator, and an integrator that processes the mixed-down signal and feeds
it back into the SCL. The MZI measures the instantaneous chirp slope, and the feedback loop locks it to a constant value that is determined by the reference frequency,
ensuring a perfectly linear chirp. An amplitude controller is used to keep the SCL
intensity constant as its frequency is tuned with input current. We begin our analysis by noting an analogy between the SFL feedback and a phase-locked loop (PLL).
We then derive its steady-state operating point and analyze small-signal deviations in
the frequency domain. We introduce an iterative predistortion procedure that relaxes
constraints on the optoelectronic feedback and enables locking at high chirp rates.

32
We conclude by discussing different SCL platforms and how they motivate the choice
of an amplitude control element.

System output
MZI

Amplitude
controller

Tap coupler

Photodetector

Semiconductor
laser
Predistorted bias
current

Reference
oscillator

Figure 3.1: Schematic diagram of the SCL-based optoelectronic SFL

3.2.1

The Optoelectronic SFL as a PLL

We first demonstrate that the optoelectronic SFL acts like a phase-locked loop in the
small-signal approximation. We begin by assuming that the SCL bias current predistortion is perfect, so that the output chirp is precisely linear. We will later remove this
assumption by treating post-predistortion residual nonlinearity as additional phase
noise. The electric field of a linear chirp is given by equation (2.1), replicated below
without the rect function that models the chirp’s finite duration:
ξt2
e(t) = cos [φSF L (t)] , and φSF L (t) ≡ φ0 + ω0 t +

(3.1)

where φSF L (t) is an overall electric field phase that is quadratic in time, ξ is the slope
of the optical chirp, and φ0 and ω0 are the initial phase and frequency, respectively.

33
The instantaneous optical frequency is therefore the derivative of φSF L (t):
ωSF L (t) =

dφSF L (t)
dt

(3.2)

A tap coupler is used to launch a small amount of the chirped light into a MZI with
delay τ . The beat signal between e(t) and e(t − τ ) is measured by a photodetector,
so that its output current is given by:
iPD ∝ cos [φSF L (t) − φSF L (t − τ )]
dφSF L
≈ cos τ
= cos [φP D (t)] , and φP D (t) ≡ ωSF L (t)τ,
dt

(3.3)

and we have ignored DC terms for simplicity. Equation (3.3) shows that if the MZI
delay is chosen small enough, the photocurrent phase φP D is proportional to the
instantaneous SCL frequency. Consider a small-signal δs(t) at the input to the integrator in figure 3.1. Assuming that the integrated signal is small enough so that the
SCL tuning remains linear, the associated change in the photocurrent phase δφP D (t)
is given by:
δφP D (t) = δωSF L (t)τ ∝

Z t
δs(u) du,

(3.4)

where δωSF L (t) is the SCL frequency shift due to the additional bias current. The
photocurrent phase shift is proportional an integral of δs(t) in the small-signal approximation, which is the defining characteristic of an ideal voltage-controlled oscillator (VCO). The integrator, the SCL, the MZI, and the photodetector may therefore
be lumped together and treated as a VCO. These elements are highlighted in figure
3.2.
The action of the optoelectronic SFL is therefore to lock the phase of the effective
VCO to a reference electronic oscillator.
φLP D (t) = φREF (t) + 2πn =⇒ ωSF
L (t)τ = ωREF t + φREF (0) + 2πn, n ∈ Z,

(3.5)

where φREF (t) is the overall phase of the reference oscillator and ωREF is its fre-

System output

Effective VCO
MZI

Amplitude
controller

Tap coupler

Photodetector
φPD(t)=ωSFL(t)τ
δφPD(t) tδs(u)du

Semiconductor
laser

δs(t)

φREF(t)=ωREF + φREF(0)

Predistorted bias
current

Reference
oscillator

Figure 3.2: Elements of the optoelectronic SFL lumped together as an effective VCO

quency. We use the superscript L to denote quantities associated with the locked
state. The feedback maintains a precisely linear chirp with a chirp rate and initial
optical frequency given by
ξL =

ωREF
φREF
2π
and ω0L =
+n .

(3.6)

is just the free spectral range (FSR) of the MZI. Equation
We recognize that 2π
(3.5) describes a family of closed-loop linear chirp solutions indexed by the integer
n. The solutions are separated in optical frequency by the MZI FSR, and the choice
of a particular one depends on the free-running chirp parameters. Specifically, the
system will lock to the solution whose initial optical frequency most closely matches
the free-running optical frequency.
At the end of the scan the system is taken out of lock, and the SCL current is
brought back to its original value. The chirp is consequently re-started and lock
re-established. As a result, if the fluctuations in the free-running initial optical frequency are great enough, for example, due to imperfect SCL temperature control,

35
the SFL will lock to a different system solution during subsequent scans. To obtain
repeatable chirps, it is therefore necessary to choose the MZI FSR large enough, so
that fluctuations in the free-running chirp are localized around a single closed-loop
solution.

Locked phase
ΔφSFL
(ω)

SCL phase
noise φn,SCL(ω)

Mixer

SCL
HFM (ω) / jω

MZI
jωτ
Reference phase
noise φn,REF (ω)
MZI fluctuations
φn,MZI(ω)

Integrator
1 / jω

Loop delay
e-jωτd

Loop gain

Figure 3.3: Small-signal frequency-domain model of the optoelectronic SFL

3.2.2

Small-Signal Analysis

The preceding discussion establishes an analogy between the optoelectronic SFL and
a phase-locked loop. We now apply small-signal analysis [1, 51] to study fluctuations
about the locked state in the Fourier domain, with the Fourier frequency denoted
by ω. The small-signal model of the feedback loop is shown in figure 3.3. The loop
variable is the deviation of the optical phase from its steady-state value,
φREF
2π
ωREF t2
φSF L (t) = φ0 +
t+n t+

2τ

(3.7)

M (ω)
The transfer function of the SCL is HFjω
, where HF M (ω) is the frequency modula-

tion (FM) response of the SCL, normalized to unity at DC, and jω
results from the

36
integral relationship between the SCL bias current and the optical phase. The FM
response of single-section SCLs is characterized by a competition between thermal
and electronic tuning mechanisms [52–55]. At low modulation frequencies, the optical frequency decreases with rising bias current due to increased junction heating. At
higher modulation frequencies, carrier tuning dominates, and the optical frequency is
increased with rising bias current. As a result, the FM response of the SCL undergoes
a phase reversal in the Fourier frequency range of 0.1 − 10 MHz. This phenomenon
is the dominant bandwidth limitation in the optoelectronic SFL [7, 9].

For frequencies much smaller than its free spectral range, the MZI can be approximated as an ideal optical phase differentiator with gain τ (this is the same
approximation as in equation (3.3)) [56]. The total DC loop gain K is given by the
product of the gains of all the loop elements—laser, photodetector, mixer, integrator,
and other electronic circuits that are not explicitly shown. The loop propagation
delay τd is on the order of tens of ns. While it does add to the loop phase response at
higher frequencies, around the feedback bandwidth its contribution is small compared
to the phase acquired due to the SCL FM response.

The optical phase noise of the SCL and the optical phase excursion due to residual
nonlinearity are lumped together and denoted by φn,SCL (ω). The phase noise of the
reference oscillator and the phase noise introduced by environmental fluctuations
in the MZI are denoted by φn,REF (ω) and φn,M ZI (ω), respectively. Going around
the loop, we write a frequency-domain expression for the locked phase deviation
∆φLSF L (ω) from the steady state.
∆φLSF L (ω) = φn,SCL (ω) − K

HF M (ω)e−jωτd
[φn,REF (ω) + φn,M ZI (ω)]
ω2

HF M (ω)e−jωτd
+ Kτ
∆φLSF L (ω)
jω

(3.8)

37
Solving for ∆φLSF L (ω) yields
jω
φn,SCL (ω)
jω + Kτ HF M (ω)e−jωτd
Kτ HF M (ω)e−jωτd
[φn,REF (ω) + φn,M ZI (ω)] .
jωτ
jω + Kτ HF M (ω)e−jωτd

∆φLSF L (ω) =

(3.9)

We observe that for frequencies within the loop bandwidth, the residual phase deviation tracks the reference oscillator and MZI noise, suppressed by the term jωτ ,
∆φLSF L (ω
Kτ ) ≈

φn,REF (ω) + φn,M ZI (ω)
jωτ

(3.10)

For frequencies outside the loop bandwidth, the residual phase deviation is given by
the free-running phase noise term,
∆φLSF L (ω
Kτ ) ≈ φn,SCL (ω).

(3.11)

From equation (3.10) it is clear that there are three considerations involved in the
generation of precisely linear chirps: (1) using an electronic oscillator with low phase
noise, (2) stabilizing the MZI against acoustic and thermal fluctuations, and (3)
picking a large τ .
High quality electronic oscillator integrated circuits are widely available. In our
systems we use direct digital synthesis (DDS) oscillators because they offer excellent
phase and frequency stability, precise control of the reference frequency, and broad
frequency tuning. The latter is useful in generating a wide range of chirp rates.
Active and passive interferometer stabilization techniques are well known, and
include locking the delay to a reference laser using a fiber stretcher [57], athermal
design of the MZI waveguides [58], and the use of vibration-damping polymers in
interferometer packaging [59], to name a few. In our systems we use fiber-based MZIs
packaged with sheets of Sorbothane R .
The choice of MZI delay is constrained by the free-running frequency fluctuations
of the SCL, as discussed in section 3.2.1. In our systems we choose the largest

38
τ that yields repeatable chirps from scan to scan. For distributed-feedback laser
(DFB) systems we use a delay from 5ns to as much as 30 ns, depending on the laser
quality. Systems based on vertical-cavity surface-emitting lasers (VCSELs) possess
more frequency jitter, and we therefore use MZIs with delays of about 1 ns.

3.2.3

Bias Current Predistortion

So far we have assumed that the SCL bias current is predistorted so that the chirp
is sufficiently linear for lock acquisition. In this section we describe a predistortion
procedure based on a simple nonlinear tuning model [1]. Even though the model
is inaccurate, and a single use of this procedure does not yield a linear chirp, it is
possible to achieve the desired linearity through iteration.
We model the nonlinear current-frequency relation of an SCL by introducing a
tuning constant K that is a function of the SCL modulation current.
ωSF L (t) = ω0 + i(t)K[i(t)],

(3.12)

where ω0 is the initial SCL frequency due to some bias current, and i(t) is the deviation
from that bias. To characterize the chirp we calculate the spectrogram of the MZI
photocurrent. The spectrogram allows us to extract the instantaneous photocurrent
frequency, found by differentiating the photocurrent phase in equation (3.3)
ωP D (t) =

dφP D (t)
dωSF L (t)
=τ
dt
dt

(3.13)

Plugging in equation (3.12), we arrive at
ωPD (t) = τ

dωSF L (t)
di(t)
di(t)
= τ [K(i) + iK 0 (i)]
= S(i)
dt
dt
dt

(3.14)

where S(i) ≡ K(i) + iK 0 (i). The function S(i) describes the nonlinear tuning of the
SCL and can be measured by launching a linear current ramp into the SCL and using

(a) Current ramp chirp spectrogram

(b) Predistorted chirp spectrogram

Figure 3.4: Single predistortion results

a spectrogram to calculate ωPD (t). Then,
S(i) =

ωPD [t(i)]

(3.15)

where α is the current ramp slope. We use equation (3.14) to write a differential
equation for the SCL modulation current
di(t)
τ dωSF L,d
dt
S(i) dt

(3.16)

where ωSF L,d (t) is the desired optical chirp. We solve this equation numerically to
find the current that will generate the desired tuning behavior ωSF L,d (t).
The outlined procedure was used to predistort the current for a 1.55 µm VCSEL
chirping 475 GHz in 100 µs. According to equation (3.13), a perfectly linear chirp
is described by a flat photocurrent spectrogram. Figure 3.4a shows the spectrogram
corresponding to a current ramp that was used to characterize S(i). The y-axis
has been rescaled by τ to show the instantaneous chirp rate. Figure 3.4b shows the
spectrogram corresponding to the predistorted waveform, confirming the improvement
in linearity.
Figure 3.4b is not perfectly flat, meaning that the corresponding optical chirp is
not perfectly linear. The reason is that the tuning model is incomplete—it fails to cap-

40
ture dynamic tuning behavior, e.g., the competition between thermal and electronic
tuning mechanisms described in section 3.2.2. Even though the model is incomplete,
its application yields an improvement in chirp linearity. It stands to reason that
iterative application of the model may yield additional improvements in chirp linearity. Iterative application means that we use the previous current predistortion
to re-characterize the function S(i), and calculate the next approximation by again
solving equation (3.16). This process is repeated until the ensuing optical chirp is
perfectly linear. We can combine equation (3.16) and equation (3.14) to write down
the differential equation that can be used to calculate the nth -order predistortion in (t)
from the preceding order predistortion, in−1 (t), and the corresponding photocurrent
frequency measurement ωP D,n−1 (t).
din (t)
ξd τ
din−1
dt
ωP D,n−1 [tn−1 (in )]
dt

(3.17)

where ξd is the desired chirp rate, and tn−1 (i) is the inverse of the in−1 -th predistortion.
We have found that using a fourth-order predistortion is sufficient. The results are
shown in figure 3.5. Each successive predistortion results in a chirp that is closer to
the desired tuning characteristic, so that by the fourth order we arrive at a sufficiently
linear chirp.
It is possible to further simplify the predistortion procedure. In equation (3.17)
we evaluate the photocurrent frequency at time tn−1 (in ) = i−1
n−1 (in ). It stands to
reason that once the difference between successive predistortions is small enough, the
photocurrent frequency may be evaluated just at time t,
din (t)
ξd τ
din−1
dt
ωP D,n−1 (t)
dt

(3.18)

This procedure makes sense intuitively—the slope of the previous predistortion
is scaled by the ratio of the desired photocurrent frequency to the instantaneous
photocurrent frequency. If the chirp is too fast, then the ratio will be less than one,
slowing down the sweep at that particular time. Likewise, if the chirp is too slow, the

(a) Zeroth-order predistortion (current
ramp) spectrogram

(b) First-order predistortion spectrogram

(d) Third-order predistortion spectrogram

(e) Fourth-order predistortion spectrogram

Figure 3.5: Iterative predistortion results

42
ratio will be greater than unity, speeding it up.
This procedure is simpler computationally—it does not involve inverses, and therefore avoids interpolation. The differential equation (3.18) itself is simpler as well, since
the right hand side no longer depends on in . We motivated these simplifications with
the assumption that the successive predistortions are already close to each other.
It is the case, however, that the simplified procedure works well in practice, and
demonstrates the same rate of convergence as the original scheme, even if iterated
starting at the linear current ramp. As part of our collaboration with Telaris Inc.
to commercialize the optoelectronic SFL, we have implemented the simplified predistortion procedure on a microcontroller. The full predistortion procedure would have
been significantly more difficult to realize in the computationally-limited embedded
environment.

3.3

Design of the Optoelectronic SFL

3.3.1

SCL Choice

The choice of the semiconductor laser to use in the optoelectronic SFL is dependent
upon the desired chirp bandwidth and linewidth. Distributed-feedback lasers (DFB)
are inherently single-mode, possess a stable polarization, and a narrow linewidth of
hundreds of kHz to a ∼ 1 MHz. The chirp ranges of commercially-available DFB
lasers depend on the emission wavelength. In our experience, 1550 nm DFB lasers
are limited in chirp range to about 100 GHz. The frequency chirp range increases
with decreasing wavelength, and we have found that DFB lasers in the 1060 nm range
can be current-tuned over a spectral range of about 200 GHz. The output power of
DFB lasers is usually in the tens of mW.
When compared to DFBs, vertical-cavity surface-emitting lasers (VCSELs) are
much cheaper, and generally tune over greater regions of the optical spectrum. We
have tested VCSELs at wavelengths of 1550 nm, 1310 nm, 1060 nm, and 850 nm.
We measured chirp bandwidths of ∼ 500 GHz at 1550 nm, ∼ 1 THz at 1310 nm,

1550 nm DFB SFL

1550 nm VCSEL SFL

−10
−20

Optical Spectrum (dBm)

−30
−40
−50
1536
10

1538

1540

1060 nm DFB SFL

1554

1556

1558

1060 nm VCSEL SFL

−10
−20
−30
−40
−50

1064

1066

1068
1064
Wavelength (nm)

1066

1068

Figure 3.6: Measured optical spectra of DFB and VCSEL SFLs at wavelengths of
1550 nm and 1060 nm

44
∼ 400 GHz at 1060 nm, and ∼ 1.5 THz at 850 nm. The 1060 nm VCSEL breaks the
pattern, possibly because this is the least-developed VCSEL wavelength. Because of
their short cavity lengths, VCSELs have broader linewidths of a few tens of MHz,
and suffer from a reduced frequency stability. The reduced stability necessitates the
use of shorter MZIs in the SFL feedback, increasing the closed-loop residual phase
error, as described in section 3.2.2. In addition, VCSELs possess significantly lower
powers of hundreds of µW to a few mW. As a result, VCSEL-based SFLs require
amplitude control elements capable of providing optical gain, as described in section
3.3.2. Due to the circular symmetry of the VCSEL cavity, these devices sometimes
have polarization instability and polarization mode hops [60], limiting their use in
applications requiring polarization control. Nevertheless, VCSELs remain extremely
attractive as swept sources in imaging and ranging applications due to their broad
chirp bandwidths.
The optical spectra of the locked optoelectronic SFLs based on some of the SCLs
discussed above are shown in figure 3.6. Swept sources based on 1310 nm and 850 nm
VCSELs are currently being developed.

Optical
input

SOA/VOA

Optical output
Tap coupler
Photodetector

Gain

Loop
Filter

Amplitude
setpoint

Figure 3.7: Schematic diagram of the amplitude controller feedback system

Amplitude control on

MZI photo−
current (a.u.)

AC photo−
current (a.u.)

Amplitude control off

80 100 120
20
Time (µs)

80 100 120

Figure 3.8: Comparison between the off(blue) and on(red) states of the SOA amplitude controller

Amplitude control off

Amplitude control on

MZI photo−
current (a.u.)

AC photo−
current (a.u.)

Transient

80 100 120
20
Time (µs)

80 100 120

Figure 3.9: Comparison between the off(blue) and on(red) states of the VOA amplitude controller

3.3.2

Amplitude Control

As the SCL current is swept to produce a frequency chirp, the light undergoes undesired amplitude modulation. To overcome this effect, we place an amplitude controller
after the SCL. The amplitude controller is a feedback system, shown in figure 3.7, that
uses an intensity modulator and a tap photodetector to measure the instantaneous
optical intensity, and lock it to a constant value. In our systems we have used two different intensity modulation elements—semiconductor optical amplifiers (SOAs) and
variable optical attenuators (VOAs) based on electro-optical ceramics [61]. The SOAs
provide optical gain, have GHz-range modulation bandwidths [62], but require temperature control and heat sinking. Furthermore, additional optical isolation is necessary to prevent lasing. VOAs solutions are cheaper and more compact because they
do not generate excess heat, but they are also much slower, with sub-MHz modulation
bandwidths. Because VOAs are passive devices, they are only practical for use with
SCLs that emit sufficiently high optical powers. We use SOAs with VCSEL-based
systems, which serves the dual purpose of amplitude control and optical amplification
of the weak VCSEL output, and reserve the use VOAs for DFB-based SFLs.
The effect of the SOA-based amplitude controller on a chirped VCSEL input
is shown in figure 3.8. The amplitude controller feedback signal is shown in the
top panels and the MZI photocurrent is plotted in the bottom panels. When the
amplitude controller is turned on, the intensity of the input into the MZI becomes
fixed, suppressing the fluctuations in the MZI signal envelope. Corresponding plots for
the DFB-VOA combination are shown in figure 3.9. Because the VOA is considerably
slower than the SOA, transient effects appear in the beginning of the scan.

3.3.3

Electronics and Commercialization

As part of our collaboration with Telaris Inc., the company has commercialized the
optoelectronic SFL. The chirped diode laser (CHDL) system offered by Telaris Inc.
is a stand-alone SFL that is controlled by a computer through a USB port. The
feedback electronics are implemented on a pair of printed circuit boards (PCBs),

(a) SFL feedback PCB

(b) Amplitude controller PCB

Figure 3.10: Optoelectronic SFL printed circuit board layouts

Figure 3.11: The 1550 nm CHDL system.

48
shown in figure 3.10. The boards include low-noise current sources and temperature controllers for the SCL and the SOA-based amplitude controller, a direct digital
synthesis (DDS) chip to provide a frequency-agile reference oscillator, a 1 µs sampling rate digital-to-analog converter to generate predistortion waveforms, an offsettrimmed multiplier, and digital potentiometers to provide control over the various
feedback gain and filter parameters. Calculating spectrograms for the predistortion
procedure is a computationally-intensive task. Instead, the MZI signal is digitized
using a comparator, and its instantaneous frequency is calculated by counting the
number of zero-crossings that occur in a specified time window. This hardwareassisted predistortion measurement, along with the simplified algorithm described in
section 3.2.3, enables rapid predistortion of the SCL bias current in an embedded
environment. The entire system is controlled by an 8-core microcontroller. Parallel
cores provide deterministic timing that is necessary for the simultaneous processing
of the MZI signal and generation of the predistortion waveform. The system uses an
acoustically-isolated fiber MZI to generate the feedback signal. The VCSEL-based
1550 nm CHDL system is shown in figure 3.11, and is capable of generating precisely
linear chirps exceeding 500 GHz in bandwidth, at a maximum rate of 10 kHz.

3.4

Experimental Results

3.4.1

Precisely Controlled Linear Chirps

The optoelectronic SFL is turned on by first iterating the predistortion procedure,
as described in section 3.2.3, with the MZI feedback gain set to zero. The MZI
photocurrent spectra at different steps of this process are shown in figure 3.12. The
x-axis has been scaled by the MZI FSR to correspond to the chirp rate ξ. Successive
predistortion steps lead to a narrowing of the signal peak at the desired chirp rate. By
the 3rd predistortion, the signal peak width has achieved the transform limit τ1T , and
additional predistortion steps reduce the noise pedestal. Once sufficient linearity is
achieved with the predistortion, the feedback gain is turned on, and the SFL acquires

Current ramp

1st-order predistortion

2 -order predistortion

3 -order predistortion

4th-order predistortion

Locked

50
40
30
20
10

Photocurrent Spectrum (dB)

−10
50

40
30
20
10
−10
50
40
30
20
10
−10

0.7

1.3
0.7
15
Chirp rate (x10 Hz/sec)

1.3

Figure 3.12: MZI photocurrent spectrum during the predistortion process and in the
locked state

50
lock, yielding a constant chirp slope and a fixed starting frequency. The locked
spectrum is characterized by a transform-limited peak with a low noise pedestal.
The chirp rate of the optoelectronic SFL is controlled by tuning the frequency
of the electronic reference oscillator. The systems that we have built are capable of
generating linear chirps with rates that are tunable over a decade. The locked spectra
at different chirp rates of an optoelectronic SFL based on a 1550 nm DFB laser are
shown in figure 3.13a. Corresponding spectra for a 1550 nm VCSEL system are shown
in figure 3.13b, for a 1060 nm DFB system in figure 3.13c, and for a 1060 nm VCSEL
system in figure 3.13d. The x-axis in all the plots corresponds to the chirp rate.

3.4.2

Arbitrary Chirps

So far we have focused on precisely linear chirps. The optoelectronic feedback technique can be extended in a straightforward way to generate arbitrary frequency
chirps [1]. The predistortion procedure is modified to include time-dependence in
the desired chirp rate ξd in equation (3.17) and equation (3.18). The integral of ξd (t)
gives the desired optical frequency vs. time function. Similarly, the locking frequency
ωREF becomes a function of time. The locked optical frequency evolution of the SFL
will therefore be given by
ωSCL (t) =

Z t
ωREF (t)dt +

φREF
2π
+n ,

(3.19)

where φREF is again the DC phase of the reference oscillator, and the integer n indexes
the family of possible locked behaviors.
We have demonstrated this principle experimentally by generating quadratic and
exponential optical frequency chirps using a DFB-based SFL at 1550 nm. For the
quadratic chirp, we varied the reference frequency between 1.43 and 4.29 MHz, corresponding to a linear variation of the chirp rate from 50 to 150 GHz/ms. The measured
photocurrent spectrogram in figure 3.14 matches the desired chirp characteristic exactly. In the exponential chirp case, we varied the reference frequency according

51
14

~ 1x10 Hz/sec mode

−20

−40

−40
0.1

0.12 0.14

~ 7.5x1014Hz/sec mode

0.15

0.2

0.25

~ 8.5x1014Hz/sec mode

−20
−40
−60

0.4

0.6

0.8

0.6

0.8

−60

0.3

1.2

Photocurrent Spectrum (dB)

0.06 0.08

0.06 0.08 0.1 0.12 0.14

−60

0.3

0.4

−20

−40

−40
−60

−20

−40

~ 6x10 Hz/sec mode

−20
−40
−60

0.2

0.3

0.4

−60

0.25

0.5

1.5

0.4

0.6

1.2

1.4

1.6

~ 5x10 Hz/sec mode

2.5
Chirp rate (x1015 Hz/sec)

0.8

0.06 0.08

0.1

~ 2x10 Hz/sec mode

0.12 0.14

0.15

0.2

0.25

0.3

~ 8x10 Hz/sec mode

−20
−40
−60

0.3

0.4
15

0.5

~ 1.5x10 Hz/sec mode

−20

−40

~ 1x10 Hz/sec mode

0.6 0.8
1.2 1.4
Chirp rate (x1015 Hz/sec)

0.8

~ 4x10 Hz/sec mode

−60

0.6

Photocurrent Spectrum (dB)

0.2
14

~ 3.5x10 Hz/sec mode

0.7

~ 1x10 Hz/sec mode

0.6

~ 2x10 Hz/sec mode

0.15

0.3

(b) 1550 nm VCSEL SFL

0.06 0.08 0.1 0.12 0.14 0.1

0.5

~ 2x10 Hz/sec mode

(a) 1550 nm DFB SFL

−60

0.25

−40

0.6
0.8
1.2
Chirp rate (x1015 Hz/sec)

0.2

~ 1x1015Hz/sec mode

−20

~ 1x10 Hz/sec mode

0.15

~ 9x10 Hz/sec mode

~ 2x10 Hz/sec mode

~ 5x1014Hz/sec mode

−60

~ 1x10 Hz/sec mode

−60

Photocurrent Spectrum (dB)

~ 2x10 Hz/sec mode

−60

1.5

0.6

0.8

1.2

~ 2.6x10 Hz/sec mode

1.5
2.5
Chirp rate (x1015 Hz/sec)

3.5

(d) 1060 nm VCSEL SFL

Figure 3.13: Locked MZI spectra of various SFLs for different values of the chirp rate
ξ. The x-axis in all the plots corresponds to the chirp rate.

Figure 3.14: Quadratic chirp spectrogram

Figure 3.15: Exponential chirp spectrogram

53
to
ωREF (t) = 2π × (4.29 MHz) ×

1.43 MHz
4.29 MHz

t/(1 ms)

(3.20)

This corresponds to an exponential decrease of the optical chip rate from 150 to
50 GHz/ms. The measured photocurrent spectrogram is shown in figure 3.15. A
combination of bias current predistortion and optoelectronic feedback can therefore
be used for arbitrary chirp generation.

3.5

Demonstrated Applications

3.5.1

FMCW Reflectometry Using the Optoelectronic SFL

The development of the optoelectronic SFL was motivated by FMCW reflectometry
and its applications in ranging and 3-D imaging (see chapter 2). The free-space
depth resolution of an FMCW system is given by equation (2.8), and a bandwidth of
500 GHz corresponds to a free-space resolution of 0.3 mm. For a medium with index
of refraction n, the depth resolution is given by
∆z =

2nB

(3.21)

where B is the chirp bandwidth of the SFL. We demonstrated the use of the VCSELbased optoelectronic SFL in FMCW reflectometry by imaging acrylic sheets of varying
thickness and a refractive index of 1.5 using the experimental configuration of figure
2.2. Reflections from the front and back acrylic surfaces show up as peaks in the
FMCW photocurrent spectrum, shown in figure 3.16 for four sheets with nominal
thicknesses of (a) 4.29 mm, (b) 2.82 mm, (c) 1.49 mm, and (d) 1.00 mm. The xaxis has been scaled to distance. The measured peak separations agree well with
the nominal values. The bandwidth of the SFL was 500 GHz, corresponding to a
resolution of 0.2 mm in acrylic. As a result, all of the reflection pairs shown in figure
3.16 are very well resolved.

Figure 3.16: FMCW reflectometry of acrylic sheets using the VCSEL-based optoelectronic SFL with a chirp bandwidth of 500 GHz and a wavelength of ∼ 1550 nm

3.5.2

Profilometry

The range resolution of an FMCW system describes its ability to tell apart reflections
from closely spaced scatterers. In some imaging applications, such as profilometry, it
is a priori known that there is only a single scatterer. The relevant metric then is not
resolution, but accuracy. The accuracy of an FMCW system can be much finer than
its resolution, as described in section 2.1.3.5. We demonstrate this by measuring the
profile of a United States $1 coin using the VCSEL-based optoelectronic SFL with
a chirp bandwidth of 500 GHz at 1550 nm. The coin was mounted on a motorized
two-dimensional translation stage. The light was collimated using a gradient-index
(GRIN) lens with a beam diameter of 0.5 mm. The depth at a particular transverse
location was determined by measuring the strongest photocurrent frequency in a
Michelson interferometer with a balanced detector (figure 2.5). The measured profile
is shown in figure 3.17. As expected, we were able to record features with depth
variations that are much finer than the 0.3 mm axial resolution of a 500 GHz chirp.

3.6

Summary

In this chapter we described the design of the SCL-based optoelectronic SFL. We
derived equations that govern its steady-state operation, and introduced an iterative
predistortion procedure that relaxes constraints on the optoelectronic feedback and
enables locking at high chirp rates. We discussed different SCL platforms and how
they motivate the choice of an amplitude control element. We demonstrated closedloop linear and arbitrary chirps and established the use of the optoelectronic SFL in
reflectometry and profilometry applications.
In the next chapter we examine multiple source FMCW (MS-FMCW) reflectometry, a high-resolution optical ranging technique that is enabled by the starting frequency stability and chirp control of the optoelectronic SFL.

Figure 3.17: Depth profile of a United States $1 coin measured using the VCSELbased optoelectronic SFL with a chirp bandwidth of 500 GHz and a wavelength of
∼ 1550 nm

Chapter 4
Multiple Source FMCW
Reflectometry
4.1

Introduction

In this chapter we describe a novel approach aimed at increasing the effective bandwidth of a frequency-modulated continuous-wave (FMCW) ranging system. This is
achieved by combining, or stitching, separate swept-frequency lasers (SFLs), to approximate a swept-source with an enhanced bandwidth [13, 14, 19]. The result is
an improvement in the range resolution proportional to the increase in the sweptfrequency range. This technique bears resemblance to synthetic aperture radar, in
which radio frequency (RF) signals collected at multiple physical locations are used
to approximate a large antenna aperture, and hence a high transverse resolution.
In multiple source FMCW reflectometry, the synthesized aperture is not physical,
but instead represents the accessible optical frequency range. This technique is of
particular interest in the context of the SCL-based optoelectronic SFL. MS-FMCW
leverages narrow SCL linewidths to present a pathway towards long-distance ranging
systems with sub-100 µm resolutions.
We start our discussion of MS-FMCW by generalizing the results of chapter 2 to
the case of multiple sources. We consider software and hardware implementations
of stitching—the action of synthesizing a high-resolution range measurement from
multiple source data sets—and present a series of experiments that demonstrate the

58
MS-FMCW principle. The culmination of this effort is a four-VCSEL system capable
of ranging with an effective optical bandwidth of 2 THz, and a corresponding freespace axial resolution of 75 µm.
000000 00 00000000 00 00000000 00Target
00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0 00 0 00 0
0 00 0 0 0 00 0 0 0 00 0 00 0000 00 00 00 0000 00 00 00 0000 00 00 00 0000 00 00 00 0000 00 00 00 0000 00 00 00 00 00 00 00 000

1x2
Coupler

2x1
Coupler

Circulator

Laser

Delay Line

Data
Processing

Figure 4.1: Schematic of an FMCW ranging experiment. PD: Photodetector

4.2

Theoretical Analysis

4.2.1

Review of FMCW Reflectometry

We begin our discussion with a brief review of FMCW reflectometry (see chapter 2
for a full discussion). Consider the FMCW ranging experiment shown in figure 4.1.
The normalized electric field of the linearly chirped SFL is given by
e(t) = rect

t − T /2

ξt2
cos φ0 + ω0 t +

(4.1)

where T is the scan duration, ξ is the slope of the optical chirp, and φ0 and ω0 are the
initial phase and frequency, respectively. The total frequency excursion is therefore
given by B = ξT /2π. For a single scatterer with round-trip delay τ and reflectivity
R, the ω-domain photocurrent is given by
y(ω) =

R rect

ω − ω0 − πB
2πB

ξτ 2
cos ωτ −

(4.2)

59
The signal processing consists of calculating the Fourier transform (FT)1 of y(ω) with
respect to the variable ω, which yields a sinc peak centered at the delay τ .
ξτ 2
Y (ζ) = πB R exp −j
exp [−j(ζ − τ )(ω0 + πB)] sinc [πB(ζ − τ )] ,

(4.3)

where ζ is the independent variable of the FT of y(ω), and has units of time. Y (ζ −τ )
is therefore the axial point spread function (PSF) of the FMCW system.
The range resolution is given by the location of the first null of the sinc function
in equation (4.3) [37, 39]. This happens at ζ = τ + 1/B, which corresponds to a
free-space range resolution
∆d =

(4.4)

where c is the speed of light. An equivalent metric of the resolution of the FMCW
system is the full width at half maximum (FWHM) of the sinc function, given by
FWHM ≈

3.79
7.58
= ∆d
πB
πc

(4.5)

Let us now consider the following view of an FMCW imaging system. The target
is characterized by some underlying function of the optical frequency, ytarget (ω), given
by
ytarget (ω) =

ξτn2
Rn cos ωτn −

(4.6)

where τn and Rn are the delays and reflectivities of the multiple reflectors that make
up the target. In deriving equation (4.6) we have assumed highly transparent reflectors (Rn
1), and ignored interference between reflected beams. The FMCW
photocurrent is then given by
y(ω) = a(ω)ytarget (ω),

(4.7)

where a(ω) is the rectangular window function, as in equation (4.2). The function

In the following analysis we use capital letters to denote the FTs of the corresponding lower-case
functions.

Figure 4.2: Schematic representation of single-source FMCW reflectometry. Top
panel: the window function a(ω) corresponding to a single chirp. Bottom panel: The
underlying target function ytarget (ω) (blue) and its portion that is measured during
the single sweep (red)

Figure 4.3: Schematic representation of dual-source FMCW reflectometry. Top
panel: the window function a(ω) corresponding to two non-overlapping chirps. Bottom panel: The underlying target function ytarget (ω) (blue) and its portion that is
measured during the two sweeps (red)

61
ytarget (ω) contains all the information about the target, and perfect resolution is obtained if ytarget (ω) is known for all values of the optical frequency ω. The measurement
in equation (4.7) gives us partial information about ytarget (ω), collected over the frequency excursion defined by a(ω), resulting in a nonzero ∆d. This single-source
measurement is illustrated in figure 4.2.
We next develop the theory of MS-FMCW reflectometry, in which multiple sources
sweep over distinct regions of the optical spectrum. The motivation for this approach
is that the use of multiple sources allows us to further characterize ytarget (ω). Figure
4.3 shows a schematic representation of a dual-source FMCW measurement. The
target information is collected over a larger portion of the optical spectrum, resulting
in an increase in the effective B, and a corresponding decrease in ∆d.

4.2.2

Multiple Source Analysis

Taking the FT of equation (4.7), and equation (4.6), we arrive at the expression
1 Xp
ξτn2
Y (ζ) =
Rn exp −j
A(ζ − τn ),
2 n

(4.8)

which has peaks at ζ = τn . The axial PSF (i.e., the shape of the peaks) is given by
the FT A(ζ) of the window function a(ω). We model the use of multiple sources with
a window function aN (ω) that comprises N non-overlapping rectangular sections, as
shown in the top panel of figure 4.4a. The approach is easily modified to include
overlapping sections. The k-th sweep originates at ω0, k , and is characterized by an
angular frequency excursion 2πBk , where k = 1, . . . , N . As illustrated in the middle
and bottom panels of figure 4.4a, aN (ω) can be decomposed into a rectangular window
hP
PN −1 i
with width 2π B̃ ≡ 2π
k=1 Bk +
k=1 δk , and a set of thin rectangular sections
(gaps). Each gap represents the frequency range 2πδk between the end of the kth sweep and the beginning of the (k+1)-th sweep, across which no photocurrent
is measured. Amplitudes of the ζ-domain FTs of the functions in figure 4.4a are
shown in figure 4.4b. We observe that if the gaps are chosen sufficiently small, their
effect in the ζ-domain can be treated as a small perturbation of the single sweep of

aN (ω)

ωk

|AN (ζ)|

2π B̃

2πBk

2π B̃

2πB̃

2π B̃

2πδk

−1

(a)

(b)

Figure 4.4: Multiple source model. (a)ω-domain description. The top panel shows a
multiple source window function aN (ω). This function may be decomposed into the
sum of a single-source window function (middle panel) and a function that describes
the inter-sweep gaps (bottom panel). (b)ζ-domain description. The three figures
show the amplitudes of the ζ-domain FTs of the corresponding functions from part
(a).

63
bandwidth 2π B̃.
Therefore, an N-source sweep is described by

aN (ω) = rect

ω − ω0,1 − π B̃
2π B̃

−1

rect

k=1

ω − ω0, k+1 + πδk
2πδk

(4.9)

in the ω-domain, and by

AN (ζ) = 2π B̃ exp −jζ(ω0,1 + π B̃) sinc ζ B̃
− 2π

−1
k=1

(4.10)

δk exp [−jζ (ω0, k+1 − πδk )] sinc(ζδk )

in the ζ-domain. To find the range resolution we find the first null of equation (4.10).
P −1
PN
Expanding near ζ = 1/B̃ and using the approximation N

k=1
k=1 Bk yields
−1
ζnull

−1
i N
B̃ exp −jζnull (ω0,1 + π B̃) +
δk exp [−jζnull (ω0,k+1 − πδk )] .

(4.11)

k=1

Equation (4.11) can be solved numerically to find ζnull . We note that an upper bound
on ζnull , and consequently on the range resolution, may be obtained by applying the
triangle inequality to equation (4.11), to yield
∆dMS−FMCW ≤

k=1 Bk

(4.12)

The conclusion is that by sweeping over distinct regions of the optical spectrum, we
collect enough information about the target to arrive at an range resolution equivalent
to the total traversed optical bandwidth, provided that the said bandwidth is much
greater than the inter-sweep gaps.

4.2.3

Stitching

We next consider the problem of stitching, that is, synthesizing a measurement with
enhanced resolution using photocurrents collected from multiple sweeps. In the pre-

64
00 00 00 00 00 00 00 00 00 00 00 00 00Target
000000000000000000000000000000000
0 0 0 0 0 0 0 0 0 0 0 0 0 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

Reference

Optical
Input

1x2
Coupler

2x1
Coupler

Circulator

Delay Line

Figure 4.5: Schematic of a multiple source FMCW ranging experiment. A reference
target is imaged along with the target of interest, so that the inter-sweep gaps may
be recovered. BS: Beamsplitter. PD: Photodetector

ceding sections we have mapped photocurrents from the time domain to the optical
frequency domain. Since the optical frequency is linear in time, this mapping involves
first scaling the time axis by the chirp slope, and then translating the data to the
correct initial frequency. Whereas the rate of each chirp is precisely controlled [1],
the starting sweep frequencies are not known with sufficient accuracy. To reflect this
uncertainty, we omit the translation step, so that the data collected during the k-th
scan is given by
yk (ω) = rect

ω − πBk
2πBk

ytarget (ω + ω0,k ).

(4.13)

The stitched measurement, given by ystitched = aN (ω)ytarget (ω), can be written in
terms of functions yk (ω) using equation (4.9):

ystitched (ω) =

k=1

yk (ω − ω0,k ).

(4.14)

The magnitude of the FT of equation (4.14) may be used for target recognition, and
is given by
|Ystitched (ζ)| =

k=1

exp −j2πζ

k−1
l=1

(Bl + δl ) Yk (ζ) .

(4.15)

65
The uncertainty in the starting frequencies manifests itself as an uncertainty in
the inter-sweep gaps. To recover the gaps, we use a known reference target along with
the target of interest, as shown in figure 4.5. By analyzing the data collected from
the reference target, we are able to extract the parameters δk , and stitch together the
target of interest measurement, according to equation (4.15).
To develop a gap recovery algorithm, we examine a two-sweep system with a
single gap δ. The case of more than two sources may then be treated by applying
this algorithm to adjacent sweeps in a pairwise manner. For simplicity we consider
sweeps of equal slopes ξ, durations T , and therefore, bandwidths B. Suppose the
known reference target consists of a single reflector with reflectivity Ra , located at
the delay τa . The experiment of figure 4.5 generates two photocurrents, one for each
sweep, of the form of equation (4.2). The initial photocurrent phase depends on the
starting frequency of the corresponding sweep, and will change as the inter-sweep gap
varies. Therefore, by considering the phase difference between the two photocurrents,
we can calculate the value of the gap. Formally, let us evaluate the FT of the k-th
photocurrent, at the maximum of the reference target peak. Using equation (4.6) and
equation (4.13),
Yk (τa ) = πBk Ra exp −jξ τ2a + jω0,k τa , k = 1, 2.

(4.16)

The ratio of the two expressions in equation (4.16) yields the phase difference between
the photocurrents:
ψa ≡

Y1 (τa )
= exp [−j2πτa (B + δ)] .
Y2 (τa )

(4.17)

Given the reference reflector delay τa and the frequency excursion B, the gap may be
found using
arg [exp(j2πτa B)ψa ] = −2πτa δ.

(4.18)

The phase of a complex number can only be extracted modulo 2π, so that equation
(4.18) can only be used to recover δ with an ambiguity of 1/τa . Therefore, the gap
needs to be known to within 1/τa before equation (4.18) may be applied. Using a

66
grating-based optical spectrum analyzer would yield the gap value with an accuracy
of a few GHz, and we therefore need 1/τa & 10 GHz. The nonzero linewidth of the
source generates errors in the phase measurement ψa in equation (4.17) (see section
2.1.3.4). According to equation (4.18), the corresponding error in the gap calculation
is inversely proportional to τa , and a large τa is therefore necessary to calculate δ
accurately.
To overcome this issue, we use two reflectors τa and τb , and express the gap size
as a function of the reflector separation. We define two phase factors
ψn ≡

Y1 (τn )
Y2 (τn )

n = a, b

(4.19)

and calculate the two reflector-analog of equation (4.18):
ψa
arg exp [j2π(τa − τb )B]
= −2π(τa − τb )δ.
ψb

(4.20)

From equation (4.20), 1/|τa − τb | can be chosen to be > 10 GHz to determine the
value of δ. The error in this calculation is proportional to 1/|τa − τb |. The accuracy
of the gap calculation can now be improved by using equation (4.18), which yields a
new value of δ with a lower error proportional to 1/τa . Depending on system noise
levels, more stages of evaluation of δ using more than two reference reflectors may be
utilized to achieve better accuracy in the calculations.
A potential MS-FMCW system architecture employing the stitching technique
is shown in figure 4.6. The optical sources are multiplexed and used to image a
target and a reference, as discussed above. The optical output is demultiplexed and
measured using a set of photodetectors to generate the photocurrents of equation
(4.13). The reference data is processed and used to stitch a target measurement
with improved resolution. The multiplexing may be performed in time or optical
frequency, or a combination of the two. The real power of the MS-FMCW technique
then lies in its scalability. One envisions a system that combines cheap off-the-shelf
SCLs to generate a swept-frequency ranging measurement that features an excellent

67
Target

0000 00 00 0000 00 00 0000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

Laser 1

Laser N

Nx1
Mux

1x2
Coupler

Circulator

2x1
Coupler

PD 1
1xN
Demux

Delay Line

PD 2

PD N

i1
i2

Data Processing

Laser 2

Reference

Figure 4.6: Proposed multiple source FMCW system architecture. BS: Beamsplitter.
PD: Photodetector
combination of range resolution, scan speed, and imaging depth.

4.3

Experimental Demonstrations

4.3.1

Stitching of Temperature-Tuned DFB Laser Sweeps

Our first demonstration of the MS-FMCW technique was based on a 1550 nm DFB
optoelectronic SFL (see chapter 3). The source generated a highly linear chirp with a
bandwidth of 100 GHz and a scan time of 1 ms. We used the configuration of figure 4.5
with a two reflector reference characterized by 1/|τa −τb | ∼ 10 GHz (∼ 3 cm free-space
separation). This reference was chosen to accommodate the accuracy with which the
gaps are initially known (∼ 1 GHz). We tuned the SCL temperature through two set
points to generate two 100 GHz sweeps with different starting frequencies. Optical
spectra of the two sweeps (blue and red) are shown in figure 4.7. Even though it
looks like the sweeps have significant overlap in optical frequency, the end of one is
actually aligned to the beginning of the other. The perceived overlap is due to the
nonzero width of the analyzer PSF, shown in black.
These sweeps were launched sequentially into the experiment, and the corresponding photocurrents were recorded. Applying the two-step procedure described in sec-

Figure 4.7: Optical spectra of the two DFB sweeps (blue and red) and the optical
spectrum analyzer PSF (black)

Figure 4.8: Single-sweep and stitched two-sweep photocurrent spectra of a dual reflector target with a separation of 5.44 mm. No apodization was used.

Figure 4.9: Single-sweep and stitched two-sweep photocurrent spectra of a dual reflector target with a separation of 1.49 mm. No apodization was used.

Figure 4.10: Single-sweep and stitched two-sweep photocurrent spectra of a dual
reflector target with a separation of 1.00 mm (a microscope slide). No apodization
was used.

70
tion 4.2.3, we recovered the gaps, and stitched the photocurrent spectra using equation (4.15). To characterize the range resolution of the system, we imaged slabs of
transparent material (acrylic and glass). Reflections from the two slab interfaces were
recorded for three slab thicknesses: 5.44 mm, 1.49 mm, and 1.00 mm (glass microscope slide). Figure 4.8, figure 4.9, and figure 4.10 show the ζ-domain photocurrent
spectra for the three cases. The x-axis has been rescaled to correspond to distance in
a material with refractive index 1.5, i.e., acrylic and glass. Each of the three figures
shows the single-sweep spectra (blue and green), as well as the stitched spectrum
(red). The FWHM of the peaks in the stitched plots is half of the FWHM of the
peaks in the single-sweep plots, as predicted by equation (4.5). Figure 4.10 is of
particular interest because the two peaks in the single-scan spectrum, corresponding
to reflections from the two microscope slide facets, are barely resolved. This is consistent with the theoretical range resolution in glass of 1 mm for a 100 GHz sweep.
The stitched curve shows two prominent peaks, demonstrating our improved ability
to resolve two closely spaced targets.

By using more aggressive temperature and current tuning, we were able to extend the number of sweeps to three, and observe a threefold improvement in range
resolution. The single-scan and stitched photocurrent spectra of a single reflector
are shown in figure 4.11a. The single reflector spectra allows us to reliably measure
the improvement in the FWHM of the axial PSF. The FWHMs are 12.17 ps and
4.05 ps for the single and multiple source cases, respectively. Using equation (4.5) we
calculate the free-space range resolutions to be 1.51 mm and 500 µm. The threefold
range resolution enhancement is consistent with equation (4.12). The improvement
in resolution again allows us to resolve the two reflections from the 1 mm glass microscope slide in figure 4.11b. The measured peak separation of 10 ps is the round-trip
delay between the two slide facets, and indeed corresponds to a glass thickness of 1
mm.

Normalized FT Amplitude (dB)

−5
−10
−15
−20
−25
−30

1.34 1.36 1.38 1.4 1.42 1.44
Time Delay (ns)
(a)

−5
−10
−15
−20
−25
−30

3.22 3.24 3.26 3.28 3.3 3.32
Time Delay (ns)
(b)

Figure 4.11: The gray and black curves correspond to single-sweep and stitched threesweep photocurrent spectra, respectively. No apodization was used. (a) Single reflector spectrum. (b) Glass slide spectrum. The peaks correspond to reflections from the
two air-glass interfaces. The slide thickness is 1 mm.

Optical
Switch

VCSEL 1

Amplitude
controller

VCSEL 2
Predistorted
bias current

90/10 Output
Coupler

Reference
oscillator
Integrator

MZI

PD
Mixer

Target

Circulator

Reference

Data
Processing

2x1
Coupler

Delay Line

Figure 4.12: Dual VCSEL FMCW reflectometry system diagram. The feedback loop
ensures chirp stability. A reference target is used to extract the inter-sweep gaps.
PD: Photodiode, BS: Beamsplitter

Figure 4.13: Optical spectra of the two VCSEL sweeps in the 250 GHz experiment

Figure 4.14: Optical spectra of the two VCSEL sweeps in the ∼ 1 THz experiment

Figure 4.15: Single-sweep and stitched two-sweep photocurrent spectra of dual reflector targets with various separations. The total chirp bandwidth is 250 GHz. No
apodization was used.

Figure 4.16: Single-sweep and stitched two-sweep photocurrent spectra of dual reflector targets with various separations. The total chirp bandwidth is ∼ 1 THz. No
apodization was used.

4.3.2

Stitching of Two VCSELs

In the current section we describe the next phase of our MS-FMCW experiments—the
stitching of two commercial VCSELs at 1550 nm. When compared to DFB lasers,
VCSELs offer increased tunability, a faster chirp rate, as well as a significant cost
reduction (see section 3.3.1).
In the proof-of-principle DFB experiment we used a single laser and tuned its
temperature through multiple setpoints to generate up to three sweeps. The SCL
temperatures needed to equilibrate before data collection, and as a result, the system
scan time was about ten minutes. In this experiment, we used two VCSELs and an
optomechanical switch in a feedback loop to form an optoelectronic SFL, as shown
in figure 4.12. The switch selects a particular VCSEL, and the feedback imposes a
perfectly linear chirp. Each VCSEL completed its chirp in 100 µs, but the total scan
time was limited by the switch to about 20 ms.
We started our experiment with conservative tuning of 125 GHz per channel. The
temperatures of the VCSELs were tuned to align their optical spectra to each other,
as shown in figure 4.13. As before, we included a reference target to aid in the gap
recovery procedure. Figure 4.15 compares single-source and stitched axial scans of
acrylic and glass slabs of varying thicknesses. As expected, the stitched scans have a
higher axial resolution, as evidenced by both the reduced FWHM of the axial PSF,
as well as our ability to resolve the surfaces of the 1 mm glass slide in figure 4.15d.
We continued our experiment with more aggressive temperature and current tuning, which yielded 475 GHz of optical bandwidth per channel, with a total chirp
bandwidth of just under 1 THz, as shown in figure 4.14. This bandwidth corresponds
to a free-space axial resolution of ∼ 150 µm, and a glass resolution of ∼ 100 µm.
Two-target axial scans are shown in figure 4.16. As before, stitching the photocurrents narrowed the axial PSF. The thinnest target we used was a glass microscope
coverslip with a nominal thickness of 150 µm, which showed up as a single broad peak
in the single sweep, but was resolved into two reflections in the stitched scan.

4.3.3

Hardware Stitching of Four VCSELs

Previous stitching experiments relied on a simultaneous measurement of a multireflector reference target to determine the inter-sweep gaps. Our next stitching experiment relied on the optoelectronic SFL to control the starting sweep frequencies
of each channel. We used four 1550 nm VCSEL-based optoelectronic SFLs in the
configuration of figure 4.17. The electric fields of each channel were added using a
4 × 1 fiber coupler. Each VCSEL was chirped 500 GHz in 100 µs, and then turned off.
We allocated 25 µs between adjacent channel time slots to allow the previous laser
to turn off, and the next laser to turn on. The total scan time was therefore 500 µs.
As described in section 3.2.2, locked states of the optoelectronic SFL form a family
of linear chirps, separated by the loop MZI FSR in optical frequency. These locked
states are shown schematically in red in figure 4.19. A locked SFL (shown in black)
aligns itself to the state that most closely matches its free-running chirp. Tuning the
SCL temperature and initial sweep current can therefore be used to shift the SFL to
a particular locked state. We used an MZI with a relatively large FSR of 9.6 GHz
to lock the VCSELs. As a result, the SFLs locked to the same state from scan to
scan, generating precisely repeatable linear chirps. Moreover, because the SFLs used
the same MZI, it was possible to tune all four channels to the same locked state, as
shown in the blue curve in figure 4.19.
The combination of repeatable chirps and the ability to lock the SFLs to the same
chirped state obviates the gap recovery procedure that was necessary in previous iterations of our MS-FMCW systems. The stitching is therefore essentially performed by
the SFL hardware, which enables real-time processing of the MS-FMCW photocurrents. We used a coarse 80.2 GHz FSR MZI to tune the SCL temperatures and ensure
that the SFLs are locked to the same chirped state. This MZI was used purely for
calibration at start-up, and the acquisition of the coarse MZI signal was not necessary
to process the MS-FMCW measurement.
Each channel in the hardware stitching system generated a 500 GHz sweep, for
a total chirp bandwidth of 2 THz, as shown in figure 4.18. This corresponds to a

Figure 4.17: Four channel 2 THz hardware stitching experiment

Figure 4.18: Optical spectra of the four 1550 nm VCSEL sweeps in the 2 THz hardware stitching experiment

Figure 4.19: Schematic representation of a family of locked states (red) of the optoelectronic SFL. In lock, the SCL (black) follows the locked state that most closely
matches its free-running chirp. In hardware stitching, temperatures and currents are
tuned so that all the MS-FMCW channels operate in the same locked state (blue).
range resolution of 75 µm in free space, and 50 µm in glass. We imaged a 150 µm
glass microscope coverslip that was suspended above a metal surface. The timedomain stitched photocurrent is shown in the top panel of figure 4.20. The curve
was generated by measuring the photocurrent during each channel’s time slot, and
appending the four data sets to each other. The SFL hardware enables perfect realtime stitching, and we therefore observe a continuous time-domain curve. Singlesweep (black) and stitched (red) photocurrent spectra are shown in the bottom panel
of figure 4.20. The x-axis is scaled to correspond to distance in glass. The timedomain photocurrents were apodized with a Hamming window before calculating
the FT. The windowing suppressed the sinc sidebands seen in previously calculated
photocurrent spectra, at the cost of broadening the PSF by a factor of 1.37. The
Hamming-broadened glass resolutions are therefore 274 µm for the single-sweep, and
68.5 µm for the stitched measurement.

80
In the single-sweep photocurrent spectrum we observe two broad peaks, one due
to reflections from the coverslip, and the other one due to a reflection from the metal
surface underneath. In the stitched spectrum, the front and back coverslip surfaces
are perfectly resolved, and the peak due to the metal surface is narrowed by a factor
of four.

4.4

Summary

We have analyzed and demonstrated a novel variant of the FMCW optical imaging
technique. This method combines multiple lasers that sweep over distinct but adjacent regions of the optical spectrum, in order to record a measurement with increased
effective optical bandwidth and a corresponding improvement in the range resolution. The MS-FMCW technique is highly scalable and is a promising approach to
realize a wide-bandwidth swept-frequency imaging system that inherits the speed and
coherence of the SCL.
We have described the various phases of our experimental work on MS-FMCW.
We started with a single DFB proof-of-concept experiment that relied on temperature tuning to generate three sweeps of 100 GHz each, for a total chirped bandwidth
of 300 GHz. Because the laser temperature had to equilibrate between sweeps, the
system scan time was about 10 minutes. We then moved on to a two-source VCSELbased system with a bandwidth of 500 GHz per channel, and a total chirp bandwidth
of 1 THz. We used an optomechanical switch to select the particular VCSEL channel,
which limited the minimum scan time to about 20 ms. The last MS-FMCW iteration
took advantage of the starting frequency control of the optoelectronic SFL to essentially perform real-time stitching in hardware. We used four VCSEL channels, and
turned them on one at a time. Each VCSEL chirped 500 GHz in 100 µs, with a total
chirp bandwidth of 2 THz and a scan time of 500 µs.
These results demonstrate the possibility of high-resolution depth imaging, e.g.,
optical coherence tomography, in a SCL-based platform with no moving parts.

Figure 4.20: Top panel: time-domain stitched photocurrent in the hardware stitching
experiment. Bottom panel: Single-sweep (black) and stitched four-sweep (red) photocurrent spectra of a 150 µm glass microscope coverslip suspended above a metal
surface. The spectra are apodized with a Hamming window. The total chirp bandwidth is 2 THz.

Chapter 5
The Tomographic Imaging Camera
5.1

Introduction

So far in our discussion of 3-D imaging we have focused on the retrieval of depth
information from a single location in the transverse plane. One way to acquire a full
3-D data set is through mechanical raster-scanning of the laser beam across the object
space. The acquisition time in such systems is ultimately limited by the scan speed,
and for very high resolution datasets (> 1 transverse mega pixel) is prohibitively slow.
Rapid 3-D imaging is of crucial importance in in vivo biomedical diagnostics [21,
26] because it reduces artifacts introduced by patient motion. In addition, a highthroughput, non-destructive 3-D imaging technology is necessary to meet the requirements of several new industrial developments. The emerging fields of 3-D printing and
manufacturing [27] will require high-precision and cost-effective 3-D imaging capabilities. Advances in 3-D tissue engineering, such as synthetic blood vessels [28], synthetic
tendons [29], and synthetic bone tissue [30], require high-resolution 3-D imaging for
tissue monitoring and quality control. To ensure higher physiological relevance of
drug tests, the pharmaceutical industry is moving from two-dimensional (2-D) to 3-D
cell cultures and tissue models, and high-throughput 3-D imaging will be used as
a basic tool in the drug development process [31]. To date, no imaging technology
exists that meets these industrial demands.
In this chapter we describe our development of a conceptually new, 3-D tomographic imaging camera (TomICam) that is capable of robust, large field of view,

83
and rapid 3-D imaging. We develop the TomICam theory and demonstrate its basic
principle in a proof-of-concept experiment. We also discuss the application of compressive sensing (CS) to the TomICam platform. CS is an acquisition methodology
that takes advantage of signal structure to compress and sample the information in a
single step. It is of particular interest in applications involving large data sets, such as
3-D imaging, because compression reduces the volume of information that is recorded
by the sensor, effectively speeding up the measurement. We use a series of numerical
simulations to demonstrate a reduction in the number of measurements necessary to
acquire sparse scatterer information with CS TomICam.

5.1.1

Current Approaches to 3-D Imaging and Their Limitations

A generic FMCW 3-D imaging system has two important components: an SFL for
ranging and a technique to translate the one-pixel measurement laterally in two dimensions to capture the full 3-D scene. The basic principle of FMCW ranging is
illustrated in figure 5.1. The optical frequency of a single-mode laser is varied linearly with time, with a slope ξ. The output of the laser is incident on a target sample
and the reflected signal is mixed with a part of the laser output in a photodetector (PD). If the relative delay between the two light paths is τ , the PD output is a
sinusoidal current with frequency ξτ . The distance to the sample is determined by
taking a Fourier transform of the detected photocurrent. Reflections from multiple
scatterers at different depths result in separate frequencies in the photocurrent.

ωL

Launched

ω0 + ξt

Reflected

ω0 + ξ(t − τ )

i ∝ cos [ξτ t + ω0 τ ]

Resolution: δz = 2B

Figure 5.1: Principle of FMCW imaging with a single reflector

84
The important metrics of an SFL are first, the sweep linearity—a highly linear
source reduces the data-processing overhead—and, second, the total frequency excursion, B, which determines the axial (z) resolution (see figure 5.1 and equation (5.3)).
State-of-the-art SFL sources for biomedical and other imaging applications are typically mechanically-tuned external-cavity lasers where a rotating grating tunes the
lasing frequency [26, 48, 63]. Excursions in excess of 10 THz, corresponding to axial
resolutions of about 10 µm [26, 48] have been demonstrated for biomedical imaging
applications. Fourier-domain mode locking (FDML) [64] and quasi-phase continuous
tuning [65] have been developed to further improve the tuning speed and lasing properties of these sources. However, all these approaches suffer from complex mechanical
embodiments that lead to a high system cost and limit the speed, linearity, coherence,
size, reliability, and ease of use of the SFL.
Detectors for 3-D imaging typically rely on mechanical scanning of a single-pixel
measurement across the scene [66], as shown schematically in figure 5.2a. The combination of high lateral resolution (< 10 µm) and large field of view (> 1 cm), requires
scanning over millions of pixels, resulting in slow acquisition. The mechanical nature
of the beam scanning is unattractive for high-throughput, industrial applications, due
to a limited speed and reliability. It is therefore desirable to eliminate the requirement
for beam scanning, and obtain the information from the entire field of view in one
shot. This is possible using a 2-D array of photodetectors and floodlight illumination.
However, in a high-axial-resolution system, each detector in the array measures a beat
signal ξτ in the MHz regime. A large array of high speed detectors therefore needs
to operate at impractical data rates (∼THz) and is prohibitively expensive. For this
reason, full-field FMCW imaging systems have been limited to demonstrations with
extremely slow scanning rates [25, 66] or expensive small arrays [67].
A further limitation of FMCW imaging is the need to process the photodetector information. This processing typically consists of taking a Fourier transform of
the photocurrent at each lateral (x, y) position. In applications requiring real-time
imaging, e.g., autonomous navigation [68], it is desirable to minimize the amount of
processing overhead.

85
An ideal FMCW 3-D imaging system will therefore consist of a rapidly tuned
SFL with a large frequency sweep and a detection technique that is capable of measuring the lateral extent of the object in one shot. The system will be inexpensive,
robust, and contain no moving parts. The TomICam platform achieves these goals
through its use of low-cost low-speed detector arrays. It takes advantage of the linearity and starting frequency stability of the optoelectronic SFL (see chapter 3), as
well as our development of SFLs at wavelengths compatible with off-the-shelf silicon
cameras (1060 nm and 850 nm). Moreover, TomICam is inherently compatible with
novel compressive acquisition schemes [69], which leads to further increases in the
acquisition speed.
Various other approaches to 3-D imaging have been described in literature, and
recent work is summarized in table 5.1. Broadly speaking, the depth information
is obtained using time-of-flight (TOF) or FMCW techniques. Transverse imaging
is obtained either by mechanical scanning or using a full-field detector array. In
some embodiments, compressive sensing ideas are used to reduce the number of measurements necessary to obtain the full 3-D image. TOF ranging systems illuminate
the sample with a pulsed light source, and measure the arrival time of the reflected
pulse(s) to obtain depth information. As a result, the axial resolution of TOF systems
is limited by the pulse-width of the optical source, as well as the bandwidth of the detector. Ongoing TOF experiments rely on expensive femto/pico-second mode-locked
lasers and/or acquisition systems with large bandwidths (≃ 10s of GHz), in order to
achieve sub-cm axial resolution [17]. Transverse imaging is typically achieved using
mechanically scanned optics [16]. Full-field imaging systems using specially designed
demodulating pixels have also been demonstrated; however, these systems have significantly lower axial resolution (≃ 10s of cm) and a limited unambiguous depth of
range [70].
FMCW ranging has many advantages over the TOF approach, since it eliminates the need for narrow optical pulses or accurate high-speed optical detectors
and electronics (see chapter 2). Very high resolution systems (< 10 µm) have been
demonstrated, and have found many applications, e.g., swept-source optical coherence

Technology

TOF

Transverse
imaging

Hardware
requirement

Limitations

Compressive
sensing

Not used in
cited work

≃ 2 cm

Mechanical
scanning

Mode-locked
laser, fast
electronics

Slow scanning,
moving parts,
expensive
components,
limited
resolution

≃ 1 cm

Spatial
light
modulator,
single pixel
detector

Mode-locked
laser, fast
electronics,
SLM

Expensive
components,
limited
resolution

Used to
convert the
single-pixel
data into a
3D model

10s of cm

Lock-in
pixel CCD

Poor
resolution,
limited lock-in
CCD size

Not used

SSOCT/CSOCT [71]

1–10 µm

Mechanical
scanning

Slow scanning,
moving parts,
bulky and
fragile

Used to
reduce scan
time

TomICam

10–
100 µm

CCD/
CMOS
array

Specially
engineered
lock-in pixel
CCD
External cavity
chirped laser
with moving
parts, slow
detector
Optoelectronic
SFL (no
moving parts),
standard
CCD/CMOS
sensor

Floodlight
illumination
(higher power)

Reduced
acquisition
time and
power

TOFLIDAR [16]

Single-pixel
TOFLIDAR [17]

Lock-in
TOF [70]

FMCW

Axial
resolution

Table 5.1: Recent 3-D camera embodiments

(a)

(b)

Figure 5.2: (a) Volume acquisition by a raster scan of a single-pixel FMCW measurement across the object space. (b) Volume acquisition in a TomICam system. 3-D
information is recorded one transverse slice at a time. The measurement depth is
chosen electronically by setting the frequency of the modulation waveform.
tomography [71].
The TomICam approach is unique, in that it combines the high resolution of
FMCW ranging, along with full-field imaging using a detector array, thereby eliminating any mechanical beam scanning optics. Moreover, it does not require specially
engineered detectors pixels, unlike the lock-in TOF lidar [70], making it more versatile and scalable. Specifically, state-of-the art lock-in CCDs are limited to tens of
thousands of pixels, while standard low-speed CMOS/CCD sensors with tens of mega
pixels are commercially available. The TomICam technique therefore has significant
advantages over other state-of-the-art high-resolution 3-D imaging modalities.

5.1.2

Tomographic Imaging Camera

In its basic implementation, the TomICam acquires an entire 2-D (x, y) tomographic
slice at a fixed depth z, as shown in figure 5.2b. A full 3-D image is obtained by a set
of measurements where the axial (z) location of the 2-D slice is tuned electronically.
An intuitive description of the TomICam principle is shown in figure 5.3. The
conventional FMCW measurement in figure 5.3a produces peaks in the photocurrent

88
FMCW Target
reflections

TomICam
measurement

Sinusoidal
intensity
modulation

n = xt1

xt1
xt2
xt3
Fourier variable (x × time)

xt2-n xt3-n
Fourier variable (x × time)

(a)

(b)

Figure 5.3: (a)Spectrum of the FMCW photocurrent. The peaks at frequencies ξτ1 ,
ξτ2 , and ξτ3 , where ξ is the chirp rate, correspond to scatterers at τ1 , τ2 , and τ3 . (b)
The beam intensity is modulated with a frequency ξτ1 , shifting the signal spectrum,
such that the peak due to a reflector at τ1 is now at DC. This DC component is
measured by a slow integrating detector.
spectrum, each peak corresponding to a scatterer at a particular depth (z) within the
sample. If a sinusoidal modulation is imposed on the optical intensity, and hence on
the photocurrent, the spectrum is shifted towards DC. In figure 5.3b, the DC component of the shifted spectrum is measured by a slow detector (e.g., a pixel in a CCD
or CMOS array). The entire spectrum is recovered by changing the modulation frequency over several scans. This scheme supplants the need for computing the Fourier
transform and thus effects a reduction in system complexity. Inherent compatibility
with compressive sensing further reduces the number of measurements necessary to
reconstruct the full 3-D scene.
In the following sections we develop the formalism necessary to describe the TomICam principle and its extension with compressive sensing.
5.1.2.1

Summary of FMCW Reflectometry

A detailed description of the FMCW ranging system is presented in chapter 2. Here,
we briefly summarize the FMCW analysis to set the scene for TomICam. Consider
the FMCW experiment shown in figure 5.4a. We analyze the response of this system
under excitation by an SFL with a linear frequency sweep, ω(t) = ω0 + ξt. We
assume that the sample comprises a set of scatterers with reflectivities Rn and round-

Sample

1×2
coupler
SFL

Circulator

2×1
coupler

Reference arm

Integrating
Fast
detector
Fourier
transform

(a)

Sample

SFL
Intensity
modulator

W(t)

1×2
coupler

Circulator

2×1
coupler

Integrating
detector

Reference arm

(b)

Figure 5.4: (a) Single-pixel FMCW system. The interferometric signal is recorded
using a fast photodetector, and reflector information is recovered at all depths at
once. (b) Single-pixel TomICam. The beam intensity is modulated with a sinusoid,
and the interferometric signal is integrated using a slow detector. This gives one
number per scan, which is used to calculate the reflector information at a particular
depth, determined by the modulation frequency.

90
trip delays τn ; and that these delays are smaller than the laser coherence time, so
that any phase noise contribution can be neglected. The normalized photocurrent is
equal to the time-averaged intensity of the incident beam (see chapter 2),
2+

iFMCW (t) =

Xp
Rn e(t − τn )
e(t) +

= rect

t − T /2

ξτn 2
Rn cos (ξτn )t + ω0 τn −

(5.1)

where T is the scan duration, ξ is the slope of the optical chirp, φ0 and ω0 are the
initial phase and frequency, respectively, and only the cross terms were retained for
simplicity. The total frequency excursion of the source (in Hz) is therefore given by
B = ξT /2π. A Fourier transform of this photocurrent results in a map of scatterers
along the direction of beam propagation (e.g., figure 5.3a). The strength of a scatterer
at some delay τ is given by the intensity of the Fourier transform of equation (5.1),
evaluated at a frequency ν = ξτ :

|Y (ν = ξτ )| =

Z T

exp [j(ξτ )t] iFMCW (t)dt .

(5.2)

By the Fourier uncertainty relation, the resolution of this measurement is inversely
proportional to the integration time T . The spatial resolution is, therefore, given by
∆z =

c 2π 1
2 ξ T
2B

(5.3)

where c is the speed of light.1
5.1.2.2

TomICam Principle

The key idea behind TomICam is that the Fourier transform required for FMCW
data processing may be performed in hardware using an integrating photodetector,
e.g. a pixel in a CCD or CMOS imaging array. To this end, we modify the basic
FMCW experiment to include an intensity modulator, as shown in figure 5.4b. The

The scatterer range is given by z = cτ /2.

91
integrating detector is reset at the beginning of every sweep, and sampled at the end.
For a given modulation signal W (t), the beat signal at the detector is given by
yW (t) ∝ W (t) iFMCW (t).

(5.4)

The value sampled at the output of the integrating detector is therefore given by
Z T
W (t) iFMCW (t)dt,

YW =

(5.5)

where YW is the TomICam measurement corresponding to an intensity modulation
waveform W (t), and we assumed an overall system gain of 1 for simplicity. The
TomICam measurement therefore amounts to projecting the FMCW photocurrent of
equation (5.1) onto a basis function described by the modulation W (t).
We consider two modulations: WC = cos [(ξτ )t], and WS = sin [(ξτ )t], which
correspond to the cosine and sine transforms.
Z T
YWC (τ ) =

cos [(ξτ )t] iFMCW (t)dt

(5.6)

sin [(ξτ )t] iFMCW (t)dt

(5.7)

Z T
YWS (τ ) =

Equation (5.2) may therefore be written as:
|Y (ν = ξτ )|2 = |YWC (τ ) + j ∗ YWS (τ )|2 = |YWC (τ )|2 + |YWS (τ )|2 .

(5.8)

The scatterer strength at a delay τ is calculated using two consecutive scans. The
strength of the TomICam platform lies in its ability to generate depth scans using
low-bandwidth integrating detectors, making possible the use of a detector array, such
as a CMOS or CCD camera. A possible extension to a 2-D integrating detector array
is shown in figure 5.5. Each element in the array performs a TomICam measurement
at a particular lateral (x, y) location, as described above. The TomICam platform

92
Reference
wavefront

Reference
mirror
Aperture

Sample

Camera

W(t)
Swept-frequency
laser

Intensity
modulator

Illuminating
wavefront

Figure 5.5: A possible TomICam configuration utilizing a CCD or CMOS pixel array
in a Michelson interferometer. Each transverse point (x, y) at a fixed depth (z) in the
object space is mapped to a pixel on the camera. The depth (z) is tuned electronically
by adjusting the frequency of the modulation waveform W (t).
therefore has the following important features:
• A full tomographic slice is obtained in a time that is only limited by the chirp
duration. This is orders of magnitude faster than a raster-scanning solution,
and enables real-time imaging of moving targets.
• The depth of the tomographic slice is controlled by the electronic waveform
W (t), so that the entire 3-D sample space can be captured without moving
parts.
• It leverages the integrating characteristic of widely available inexpensive CCD
and CMOS imaging arrays to substantially reduce signal processing overhead.
• It is scalable to a large number of transverse pixels with no increase in acquisition
or processing time.
• The TomICam platform is not limited to sinusoidal modulations W (t), making
it inherently suitable for compressive sensing, as described in section 5.2.

5.1.2.3

TomICam Proof-of-Principle Experiment

In order to verify the equivalence of FMCW and TomICam measurements, we have
performed a proof-of-principle experiment, shown schematically in figure 5.6. We

Figure 5.6: Schematic diagram of the TomICam proof-of-principle experiment. A
slow detector was modeled by a fast detector followed by an integrating analog-todigital converter. The detector signal was sampled in parallel by a fast oscilloscope,
to provide a baseline FMCW depth measurement.

Figure 5.7: The custom PCB used in the TomICam experiment. Implemented functionality includes triggered arbitrary waveform generation and high-bit-depth acquisition of an analog signal.

94
used the 1550 nm VCSEL-based optoelectronic SFL, described in section 3.4.1, which
produced a precisely linear chirp with a swept optical bandwidth of 400 GHz, and a
scan time of 2 ms. The beam was modulated using a commercially available lithium
niobate intensity modulator.
The necessary electronic functionality, including an arbitrary waveform generator,
an integrating high-bit-depth analog-to-digital converter, and a microcontroller, was
implemented on a PCB, shown in figure 5.7. The waveform generator was used to
provide sine and cosine waveforms of different frequencies to the intensity modulator.
The amplitude of these waveforms was apodized by a Hamming window, which suppressed the sinc sidebands associated with a rectangular apodization. The integrating
analog-to-digital converter recorded a single number per scan. The microcontroller
was used to coordinate the waveform generation and signal acquisition. The photodetector output was also sampled on a high-speed oscilloscope in order to provide
a baseline FMCW measurement.
We used a sample comprising two acrylic slabs. Reflections from the air-acrylic
and acrylic-air interfaces were recorded and the results are shown in figure 5.8. The
red curve is the intensity of the Fourier transform of the FMCW photocurrent. The
blue curve is constructed by varying the frequencies of the modulation waveforms
WC (t) and WS (t), and applying equation (5.8). As expected, the two curves are
practically identical.
We note that a copy of the signal, scaled in frequency by a factor of 13 , shows up
in the TomICam spectrum in figure 5.8. This ghost replica is due to a third-order
nonlinearity exhibited by our intensity modulator, and can be resolved through the
use of a linear intensity modulator. An example of such a modulator is the amplitude
controller based on an semiconductor optical amplifier in a feedback loop, described
in section 3.3.2.
We characterize the dynamic range of our system by performing FMCW and
TomICam measurements on a fiber Mach-Zehnder interferometer (MZI). We introduce optical attenuation in one of the MZI arms, and measure the signal SNR. The
results for unbalanced and balanced acquisition in FMCW and TomICam configura-

Figure 5.8: Comparison between FMCW (red) and TomICam (blue) depth measurements. The two are essentially identical except for a set of ghost targets at 13 of the
frequency present in the TomICam spectrum. These ghosts are due to the third-order
nonlinearity of the intensity modulator used in this experiment.

Figure 5.9: Characterization of the FMCW and TomICam dynamic range. The signalto-noise ratio was recorded as a function of attenuation in one of the interferometer
arms. At low attenuations, the SNR saturates due to SFL phase noise and residual
nonlinearity.

96
tions are shown in figure 5.9. The dynamic range of our system, defined as the ratio
of the strongest to weakest measurable target reflectivity, is about ∼ 80 dB. For low
attenuation, i.e., large reflectivities, the SNR is limited by the laser coherence and
residual chirp nonlinearity, saturating at a (path-length mismatch dependent) value
of ∼ 50 dB. The fiber mismatch used in this experiment was about 40 mm.

5.2

Compressive Sensing

The total number of tomographic slices, N , used in a 3-D image reconstruction is
given by the axial extent, Lz , of the target divided by the axial resolution, ∆z. We
note that most real life targets are sparse in the sense that they have a limited number
of scatterers, k, in the axial direction. The acquisition of N
k slices to form the 3-D
image is therefore inefficient. In this section, we investigate the use of compressive
sensing (CS) in conjunction with the TomICam platform in order to obtain the 3-D
image with many fewer than N measurements. This has the potential to reduce the
image acquisition time and the optical energy requirement of the TomICam by orders
of magnitude.

5.2.1

Compressive Sensing Background

We briefly state the salient features of CS [69]. Consider a linear measurement system
of the form:
y = Ax

A ∈ Cm×N , x ∈ CN , y ∈ Cm ,

(5.9)

where the vector x is the signal of interest, and the vector y represents the collected
measurements. The two are related by the measurement matrix A. The case of
interest is the highly under-determined case, m
N . The system therefore possesses
infinitely many solutions. Nevertheless, CS provides a framework to uniquely recover
x, given that x is sufficiently sparse, and the measurement matrix A satisfies certain
properties such as restricted isometry and incoherence [69]. The intuition behind CS

97
is to perform the measurements in a carefully chosen basis where the representation
of the signal x is not sparse. The signal is then recovered by finding the sparsest x
that is consistent with the measurement in equation (5.9). Specifically, the recovery
is accomplished by solving a convex minimization problem:
minimize

kxk1

(5.10)

subject to Ax = y,
where kxk1 denotes the l1 norm of x. The use of the l1 norm promotes sparse
solutions, while maintaining convexity of the minimization problem, resulting in a
tractable computation time. Success of recovery depends on the number of measurements m, the sparsity level of x, and the properties of the measurement matrix A.
This approach is of particular interest due to continuous advances in computational
algorithms that improve the reconstruction speed [72].

5.2.2

TomICam Posed as a CS Problem

Fundamentally, the FMCW imaging technique converts the reflection from a given
depth in the z direction to a sinusoidal variation of the detected photocurrent at a
frequency that is proportional to the depth. Scatterers from different depths thus
result in a photocurrent with multiple frequency components. In its basic implementation (section 5.1.2.2), the TomICam uses a single-frequency modulation of the beam
intensity to determine one of these possible frequency components. Full image acquisition requires N measurements (N = Lz /∆z), determined by the axial resolution
of the swept-frequency source. When the number of axial scatterers—and hence the
number of frequency components in the photocurrent—is sparse, the CS framework
enables image acquisition with a smaller number of measurements.
We first show that the TomICam is inherently suited to compressive imaging
and that different types of measurements may be easily performed with almost no
modification to the system. We recast equation (5.5) in a form more suitable for
the discussion of CS. We assume that there are N possible target locations with

98
corresponding delays τn , n = 0, 1, . . . , (N − 1) and target reflectivities Rn . These
target locations are separated by the axial resolution: τn = n/B. We assume that
the target is k-sparse so that only k of the N possible reflectivities are nonzero. The
time axis is discretized to N points given by th = hT
, h = 0, 1, . . . (N − 1). Equation
(5.5) can now be written as

−1 N
−1

W (th )

h=0 n=0

Rn
cos (ξτn th + ω0 τn ).

(5.11)

Each TomICam measurement therefore yields a single value y for a particular W (th )
(per pixel in the lateral plane), as given by equation (5.11). Note that a sinusoidal
variation of W (th ) yields the reflectivity at a particular axial depth, and a tomographic
slice is obtained using a detector array, as described in section 5.1.2.2.
In this section, we will explore other intensity modulation waveforms W (th ) that
are compatible with the CS framework to reduce the number of scans in the axial
dimension. We extend the discussion to include m measurements indexed by s, i.e.,
we will use m different intensity modulation waveforms Ws (th ) to obtain m distinct
measurements ys . Equation (5.11) can be simplified to give

ys =

−1 N
−1
h=0 n=0
−1 N
−1
h=0 n=0

r
ω
2πhn
Rn
· √ exp −j
exp −j n

(5.12)

Wsh · Fhn · xn ,

where Wsh ≡ Ws hT

, Fhn ≡

√1 exp

−j 2πhn

, xn ≡

Rn
exp

−j ωB0 n , and it is

understood that the measurements correspond to the real part of the right hand side.
Rewriting equation (5.12) in matrix notation, we obtain:
y = WFx,

(5.13)

where x is the k-sparse target vector of length N , y is the vector containing the m
TomICam measurements, F is the N ×N unitary Fourier matrix, and W is the m×N

99
matrix that comprises the m intensity modulation waveforms Ws (th ).
Since W is electrically controlled, a variety of measurement matrices can therefore be programmed in a straightforward manner. Each TomICam measurement ys is
obtained by multiplying the optical beat signal with a unique modulation waveform
Ws (th ) and integrating over the measurement interval. If the modulation waveforms
are chosen appropriately, the measurement matrix can be made to satisfy the crucial requirements for CS, i.e., the restricted isometry property and incoherence [69].
This ensures that the information about the target—which is sparse in the axial
dimension—is “spread out” in the domain in which the measurement is performed,
and a much smaller number of measurements is therefore sufficient to successfully
recover the complete image.

5.2.3

Robust Recovery Guarantees

We now consider two possibilities for W that yield a measurement matrix capable
of robust signal recovery. These represent straightforward implementations of CS
TomICam imaging.
5.2.3.1

Random Partial Fourier Measurement Matrix

A random partial Fourier matrix of size m × N is generated by selecting m rows at
random from the N ×N Fourier matrix F. This operation is accomplished by a binary
matrix W that has a single nonzero entry in each row. The location of the nonzero
entry is chosen randomly without replacement. For this class of matrices, robust
signal recovery is guaranteed whenever the number of measurements satisfies [73]
m ≥ Ck log (N/),

(5.14)

where k is the signal sparsity, 1 − is the probability of recovery, and C is a constant
of order unity.
In the TomICam implementation, a random partial Fourier measurement corresponds to pulsing the intensity modulator during the linear chirp, so that only a single

100
optical frequency is delivered to the target per scan, leaving a lot of dead time. As a
result, the optoelectronic SFL is not the most ideal laser candidate, and other sources
that can provide rapid random frequency access, such as sampled grating SCLs, are
more suitable [74]. In these devices, the cavity mirrors are formed using a pair of
sampled gratings, each of which has multiple spectral reflection bands. Current tuning of the mirror sections is used to make these reflection bands overlap, forming a
single band whose position may be varied over a broad spectral range. Further, a
phase section current is applied to align a Fabry-Pérot cavity mode to the middle
of the band in order to optimize lasing properties. Simultaneously tuning all three
sections enables broadband frequency access, approaching 5 THz at 1550 nm [75].

5.2.3.2

Gaussian or Sub-Gaussian Random Measurement Matrix

This class of matrix has the property that any entry Aij in the matrix A is randomly
chosen from independent and identical Gaussian or sub-Gaussian distributions. In
this case, robust signal recovery is guaranteed for
m ≥ Ck log (N/k),

(5.15)

where k is the signal sparsity, and C is a constant of order unity. Moreover, the same
result also applies to a measurement matrix that is a product of a Gaussian or subGaussian random matrix and a unitary matrix. Since F is unitary, a Gaussian random
matrix W results in robust signal recovery when equation (5.15) is satisfied [76]. The
measurements obtained using a Gaussian matrix W may be interpreted as a collection
of conventional TomICam measurements where each measurement queries all possible
depths with different weights.
We want the failure rate to be much less than unity, while the sparsity level k is
at least unity. Therefore, the Gaussian random matrix requires fewer measurements
than the random partial Fourier matrix for correct recovery.

101
M easu rem ent
Generate random
target of given
sparsity xo

Generate random
code matrix W

Repeat 100 times

M inimize Ll norm
of x, subject to
yo= WFx

Inject noise, and
make
"measurement"
yo= WF(xo+ noise)

• Space dimension :
- N=100
• Number of measurements:
- m = 0 to 100
• Sparsity:
- k = [ 1, 3, 5, 7, 9]
• SNR: [40dB , 80dB, 120dB]

Figure 5.10: Flow diagram and parameters of the CS TomICam simulation

5.2.4

Numerical CS TomICam Investigation

Because the partial Fourier matrix is not well-suited for the TomICam platform, we
continue our investigation with the Gaussian random matrix in mind. We evaluate
the performance of a compressively-sampled TomICam through a series of numerical
simulations. The simulation steps and parameters are summarized in figure 5.10.
We consider a signal space with dimension N = 100, and generate a random target
signal x0 of a given sparsity. We generate a Gaussian random matrix W of size m×N ,
where m is the number of measurements. We then make a noisy measurement
y0 = WF(x0 + xn ),

(5.16)

where xn is a randomly generated noise vector. We define the SNR as the ratio of
the signal and noise energies,
SNR ≡

kx0 k2
kxn k2

(5.17)

We then proceed to solve the convex minimization problem in equation (5.10), which
yields the recovered signal x. We define the signal-to-error ratio (SER) as the ratio of

102

140

.. .. ..

W standard gaussian distrubuted, N=1 00

120 .
.-..

cc 100

"'0
._..

.....,
ro
'--

60 .

.....,
ro

C>
(/)

'--

'-'-Q)

0 ·
-20

.. ". ". " . " . " ....... " . " .. " .. " . " . " . ··:.

80
40
60
Number of measurements

100

Figure 5.11: SER curves for a CS simulation with a Gaussian random matrix

140

W abs(standard gaussian) distrubuted, N=1 00

120 .
.-..

cc 100

"'0
._..

.....,
ro
'--

60 .

.....,
ro

C>
(/)

'--

'-'-Q)

.......

-20

.. ". ". " . " . " ....... " . " .. " .. " . " . " . ··:.

80
40
60
Number of measurements

100

Figure 5.12: SER curves for a CS simulation with a waveform matrix given by the
absolute value of a Gaussian random matrix

103
the energy of the recovered signal to the energy of the difference between the recovered
and the original signals.
SER ≡

kxk2
kx − x0 k2

(5.18)

We repeat this procedure 100 times and record the average SER. We consider 0 <
m < 100, and simulate 100 reconstructions for each value of m, resulting in a curve
of SER vs. m. We generate 15 such a curves by considering five sparsity levels
k = [1, 3, 5, 7, 9], and three noise levels SNR = [40dB, 80dB, 120dB].
These curves are plotted in figure 5.11, with the 120 dB SNR shown in red, 80 dB
in blue, and 40 dB in black. We expect that for a small number of measurements,
the reconstructions will fail, yielding a zero SER. Once the number of measurements
satisfies equation (5.15), the reconstruction will essentially always succeed, yielding an
SER that is approximately equal to the SNR. This is the pattern that we see in figure
5.11. The curves corresponding to the different sparsity levels are in order, with the
sparsest case achieving the transition in SER at the lowest number of measurements.
We observe that ∼ 50 measurements are necessary to recover a 9-sparse target, which
corresponds to a factor of two compression, when compared to conventional sampling.

We note that a Gaussian random matrix has negative entries, and is therefore
not physical (we can only modulate the beam intensity with a positive waveform).
To fix this, we investigate numerically random matrices that contain only positive
entries. SER curves for W given by the absolute value of a Gaussian random matrix
are shown in figure 5.12. The qualitative behavior of the curves is unchanged from
the random Gaussian case.
A passive intensity modulator can only provide a modulation between 0 and 1,
and we therefore examine a waveform matrix W with entries that are uniformly
distributed between 0 and 1. The SER curves for this case are shown in figure 5.13,
and follow the trend of the previous simulations.
Realistic intensity modulators have a finite extinction ratio, meaning they cannot
be used to turn the beam completely off. Moreover, it may be desirable to operate the

104

14 0

W uniformly distributed on [0 1], N=1 00
............. "

120 .
.-..

cc 100

"'0
._..

ro 80
'--

e 6o .
'--

'-Q)

C>
(/)

40
20

.. ". ". " . " . " ....... " . " .. " .. " . " . " . ··:.

-20~----~----~------~----~----~

60
80
40
Number of measurements

100

Figure 5.13: SER curves for a CS simulation with a waveform matrix whose entries
are uniformly distributed between 0 and 1

140

.. .. ..

W uniformly distributed on [0.5 1], N=1 00

120
.-..

cc 100

"'0
._..

.....,
ro
'--

60 .

.....,
ro

C>
(/)

'--

'-'-Q)

-20

.. ". ". " . " . " ....... " . " .. " .. " . " . " . ··:.

80
40
60
Number of measurements

100

Figure 5.14: SER curves for a CS simulation with a waveform matrix whose entries
are uniformly distributed between 0.5 and 1

105

140 . . . . . . . . . . . . . . . . . . ..

W Digital [0.5 1], N=100

120 .. .. .. .. .. .. .

.... .... .... .... .... .... .... .... . .................................................... ...................................................
·~··

-20~----~----~------~----~----~

Number of measurements

100

Figure 5.15: SER curves for a CS simulation with a waveform matrix whose entries
take on the values of 0.5 or 1 with equal probabilities

140

.. .. ..

W uniformly distributed on [0 1], N=1 000

120
.-..

cc 100

"'0
._..

.....,
ro
'--

.....,
ro

C>
(/)

'--

'-'-Q)

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

.. .... '.. .... '.. .... '.. .... '.. .. .. i . ..

-20

.. ". ". " . " . " ....... " . " .. " .. " . " . " . ··:.

40
60
80
Number of measurements

100

Figure 5.16: SER curves for an N = 1000 CS simulation with a waveform matrix
whose entries are uniformly distributed between 0.5 and 1

106
intensity modulator away from the zero point to keep its response as linear as possible.
To account for this possibility we ran the simulation using a waveform matrix W with
entries that are uniformly distributed between 0.5 and 1. Again, the transition trends
for the SER curves, shown in figure 5.14 remain essentially unchanged.
The waveform generator has a finite bit depth, and we consider, as an extreme
case, only two modulation levels—0.5 and 1—which corresponds to a waveform matrix
W whose entries can equal either of the modulation levels with equal probabilities.
The SER curves for this simulation are shown in figure 5.15, and again demonstrate
the same behavior.
For our final simulation we increased the dimension of the space to 1000, and used
a waveform matrix W with entries that are uniformly distributed between 0 and 1.
The SER curves for this simulation are shown in figure 5.16. We observe that ∼ 80
measurements are necessary to recover a 9-sparse target, which corresponds to greater
than 10× compression, when compared to conventional sampling.

5.3

Summary

In this chapter we described the basic tomographic imaging camera principle, and
demonstrated single-pixel TomICam ranging in a proof-of-concept experiment. The
TomICam uses a combination of electronically tuned optical sources and low-cost
full-field detector arrays, completely eliminating the need for moving parts traditionally employed in 3-D imaging. This new imaging modality could be useful in a
variety of established and emerging disciplines, including lidar [18], profilometry [22],
biometrics [25], biomedical diagnostics [21, 26], 3-D manufacturing [27], and tissue
engineering [28–31].
We also discussed the application of compressive sensing to the TomICam platform, and performed a series of numerical simulations. These simulations show that
a factor of 10 reduction in the number of measurements is possible with CS if the
number of depth bins is about 1000. Future implementations of TomICam will benefit
from the development of high frame rate, high pixel count silicon CCD and CMOS

107
cameras, rapidly-tunable semiconductor lasers [77], efficient compressive sensing algorithms, and continuous advances in computing performance. As a result, TomICam
has the potential to push 3-D imaging functionality well beyond the state of the art.

108

Chapter 6
Phase-Locking and Coherent Beam
Combining of Broadband
Linearly Chirped Optical Waves
6.1

Introduction

Optical phase-locking has found various applications in the fields of optical communication links [52, 78–81], clock generation and transmission [82, 83], synchronization
and recovery [84, 85], coherence cloning [7], coherent beam combining (CBC) and optical phased arrays [8,86–91], and optical frequency standards [92,93], to name a few.
In these applications, electronic feedback is used to precisely synchronize the phases
of two optical waves. With a few notable exceptions [34, 94], prior demonstrations of
phase-locking and synchronization have been performed using nominally monochromatic optical waves. In this chapter we describe our work on the phase-locking of
optical waves whose frequencies are swept rapidly with time and over large chirp extents. The phase-locking of optical waves with arbitrary frequency chirps is a difficult
problem in general. However, precisely linear chirps, such as the ones generated by
the optoelectronic SFL (see chapter 3) can be phase-locked with very high efficiency
using a frequency shifter. The main application of this result is the simultaneous
stimulated Brillouin scattering (SBS) suppression and coherent combining of highpower fiber amplifiers. Other potential applications include electronic beam steering
for lidar and 3-D imaging systems.

109
We begin our discussion by reviewing CBC approaches to the generation of highpower continuous-wave optical beams. We proceed to describe the basic principle
behind phase-locking of linearly chirped optical waves, and present theoretical analyses of chirped-wave phase-locking in homodyne and heterodyne configurations. We
demonstrate heterodyne phase-locking of chirped optical waves and implement a
passive-fiber CBC system [10, 11]. We conclude with a description of our recent
CBC experiment with two erbium-doped fiber amplifier (EDFA) channels [12]. The
work described in this chapter was performed in collaboration with Jeffrey O. White’s
group at the United States Army Research Laboratory.

6.2

Coherent Beam Combining

The output power of optical fiber amplifiers is usually limited by SBS. Advances in
the design of the geometry and doping profiles of active fiber have enabled increases
in the SBS threshold power [95–97]. Further increases in the SBS threshold of a
single amplifier can be obtained by broadening the linewidth of the seed laser through
phase or frequency modulation [32,33]. A separate approach to achieving high optical
powers is the coherent beam combining of the outputs of multiple laser or amplifier
elements [8, 86–90].
The efficiency of a CBC scheme depends on the matching of the relative amplitudes, phases, polarizations and pointing directions of the multiple emitters in
the array [86, 98, 99]. Phase synchronization of the array elements is a particularly
difficult challenge, which in the past has been addressed with various approaches,
including evanescent wave and leaky wave coupling of emitters [100, 101], common
resonator arrays [102, 103], and phase-locking through optoelectronic feedback [8, 90].
In optoelectronic feedback systems, the phase error between the combined beams is
measured and fed back to a phase actuator, such as an electro-optic phase modulator [87], acousto-optic frequency shifter [90], or a fiber stretcher [34, 89].
Kilowatt-level systems have been demonstrated [33] and rely on the simultaneous
suppression of SBS in high-power fiber amplifiers and the CBC of multiple amplifier

110
channels. The path-length mismatch between array elements in an amplifier CBC
system has to be much smaller than the seed coherence length, in order to prevent
de-phasing due to incoherence. The traditional approach to SBS suppression relies on
a broadening of the seed linewidth, and therefore a reduction of its coherence length.
As a result, SBS suppression in high-power fiber amplifier CBC systems requires
precise channel path-length matching. Recently, Goodno et al. [33] have demonstrated the phase-locking of a 1.4 kW fiber amplifier. This power level was achieved
by increasing the SBS threshold using a modulated seed source with a linewidth of
∼ 21 GHz. Efficient power combining was only possible with precise path-length
matching of active fibers to sub-mm accuracy. Further increases in the power output of a single amplifier will require even broader seed linewidths, and path-length
matching to within ∼ 10s of µm will be necessary. Weiss et al. [34] have recently
demonstrated that coherent combining can still be achieved using a novel feedback
loop that senses the path-length mismatch and corrects it using a fiber stretcher.
In this chapter we explore an architecture capable of SBS suppression and coherent beam combining without stringent mechanical path-length matching requirements [10–12]. Our approach is to use a rapidly chirped (> 1014 Hz/s) sweptfrequency laser (SFL) seed to reduce the effective length over which SBS occurs [35,
36]. The advantage of this approach is that path-length matching requirements are
relaxed due to the long coherence length (several meters) of semiconductor laser
based SFLs. In the following section we describe the basic principle of phase-locking
of linearly chirped optical waves using acousto-optic frequency shifters (AOFSs) to
compensate for static and dynamic optical path-length differences. We proceed with
an analysis of homodyne and heterodyne optical phase-locked loop (OPLL) configurations, and present results of proof-of-concept experiments that demonstrate phaselocking, coherent combining, and electronic phase control in chirped-wave passivefiber systems.

111

Sweptfrequency
laser

Amplifier 1

Combined
output

ξτ

Mismatch l12
Frequency
shifter

Amplifier 2

Time

Figure 6.1: Intuitive description of chirped-seed amplifier coherent beam combining.
A path-length mismatch between amplifier arms results in a frequency difference at
the combining point, and can therefore be compensated using a frequency shifter
placed before amplifier 2.

6.3

Phase-Locking of Chirped Optical Waves

The basic concept of phase-locking multiple chirped-seed amplifiers (CSAs) in a
master oscillator power amplifier (MOPA) configuration is depicted in figure 6.1 [10,
11]. A SFL is used to generate a linear chirp, with an instantaneous optical frequency
given by
ωL (t) = ωL,0 + ξt, 0 ≤ t ≤ T,

(6.1)

where ωL,0 is the initial optical frequency, ξ is the sweep rate, and T is the sweep time.
The SFL is split into multiple amplifier seeds which then undergo amplification and
recombination to form a high-power beam. A difference in the lengths of the fiber
amplifiers 1 and 2 result in a frequency difference ξl12 /c at the locking point, where l12
is the path-length mismatch and c is the speed of light. An acousto-optic frequency
shifter (AOFS) is placed in one of the arms to correct this frequency difference. For
a linear chirp of 2π
= 1015 Hz/s and a path-length mismatch of 10 cm in fiber, the

required frequency shift is 500 kHz, which is well within the dynamic range of AOFSs.
An optical phase-locked loop is formed by recording an interference signal between
the two arms on a photodetector and feeding it back to the AOFS, as shown in figure
6.2 and figure 6.4. In lock, the AOFS synchronizes the optical phases and corrects

112
the fixed path-length mismatches as well as the dynamic length fluctuations arising
from vibrations and temperature drift. The loop bandwidth determines the fastest
fluctuation frequency that is suppressed, and previous work using AOFSs and singlefrequency seeds has shown that sufficient bandwidths can be achieved for efficient
combining of fiber amplifier outputs [90].
SBS suppression in high-power amplifiers scales with the chirp rate [35,36]. Therefore, we limit our attention to SFLs with perfectly linear chirps, in order to ensure
that uniform SBS suppression is obtained throughout the duration of the frequency
sweep. Moreover, a linear chirp enables path-length mismatches to be corrected by
a constant frequency shift, as described above. Deviations from chirp linearity are
corrected using a feedback loop, as long as these deviations are small and at frequencies within the loop bandwidth. It is therefore desirable that the chirp be close to
perfectly linear, particularly at high chirp rates ξ, in order to relax the requirements
on the frequency tuning range of the AOFS and the bandwidth of the feedback loop.
We note that it should be possible to further extend the phase-locking approach to
other sweep profiles, by using a time-varying frequency shift to compensate for the
time-varying slope of the optical frequency chirp, e.g., using the iterative algorithm
of section 3.2.3 to pre-distort the AOFS bias signal.

6.3.1

Homodyne Phase-Locking

We first consider the homodyne phase-locking configuration shown in figure 6.2. The
output of an optoelectronic SFL is split into two arms using a fiber splitter. The
goal of the experiment is to phase-lock the outputs of the two arms by feeding back
the error signal generated using a 2 × 2 fiber coupler and a balanced detector. The
bias frequencies and phase shifts of the two AOFSs are denoted by ω1 , ω2 and φ1 ,
φ2 . The differential delay between the first and second arms is denoted by τ12 . We
also introduce a common delay τd to model the long fiber length inside an optical
amplifier. The feedback loop is very similar to a typical phase-locked loop [51], and
can be analyzed accordingly. We define the DC loop gain KDC as the product of the

113

Sweptfrequency
laser

AOFS 1

AOFS 2

PD 1

Delay τd
Delay τd-τ12

2x2
Coupler

PD 2

Feedback
Gain

Figure 6.2: Passive-fiber chirped-wave optical phase-locked loop in the homodyne
configuration. PD: Photodetector

SCL phase
noise θL(ω)

Arm 1 phase
fluctuations θ1(ω)
Delay e-jωτd
Arm 2 phase
fluctuations θ2(ω)

Balanced
Detector

δθ12(ω)

Delay e-jω(τd-τ12)
Loop transfer function
[KDCKel (ω)/jω] sin δθ12
Figure 6.3: Small-signal frequency-domain model of the homodyne chirped-wave optical phase-locked loop. The model is used to study the effect noise and fluctuations
(green blocks) on the loop output variable δθ12 (ω).

114
optical power in each arm (units: W), and the gains of the balanced detector (V/W),
loop amplifier (V/V), and frequency shifter (rad/s/V). Let the SFL optical frequency
be given by equation (6.1), and let us denote the optical phases of the two arms at
the coupler by θ1 (t) and θ2 (t). The optical phase difference between the two arms is
given by
θ12 (t) ≡ θ1 (t) − θ2 (t)

= (ωL,0 + ω1 )(t − τd ) + (t − τd )2 + φ1 − (ωL,0 + ω2 )(t + τ12 − τd )
2Z
t+τ12 −τd
KDC cos θ12 (u)du =
− (t + τ12 − τd )2 − φ2 −
Z t+τ12 −τd
ξτ12
KDC cos θ12 (u)du,
= ∆ωf r (t − τd ) − (ω2 + ωL,0 )τ12 −
+ φ12 −
(6.2)

where ∆ωf r ≡ ω1 − ω2 − ξτ12 is the free-running frequency difference between the
two arms, and φ12 ≡ φ1 − φ2 . The final term in equation (6.2) represents the phase
shift due to the feedback to the AOFS, which is the integral of the frequency shift.
, obtained by setting the time derivative of θ12 (t) to 0,
The steady-state solution θ12

is given by
θ12
= cos−1

∆ωf r
KDC

(6.3)

We use this result to rewrite equation (6.2),
θ12 (t) = ∆ωf r (t + τ12 − τd ) + θ12

Z t+τ12 −τd
KDC cos θ12 (u)du.

(6.4)

In lock, the optical phases of the two arms differ by θ12
, and there is no frequency

difference.
Next, we linearize the loop about its steady-state solution in order to study dynamic behavior and the effect of fluctuations. We denote the phase noise and residual
nonlinearity of the SFL by θLn (t), and the phase noise introduced in the two arms by
θ1n (t) and θ2n (t), which include noise contributions from the AOFSs and fluctuations
in the optical path lengths. We introduce δθ12 (t), the small-signal fluctuation of θ12 (t)

115
about the steady state, so that
θ12 (t) = θ12
+ δθ12 (t).

(6.5)

We plug equation (6.5) into equation (6.4), and expand about the steady-state point
(equation (6.3)). Solving for δθ12 (t), we arrive at
δθ12 (t) = θ12
(t) + θLn (t − τd ) − θLn (t + τ12 − τd ) + KDC sin θ12

Z t+τ12 −τd
δθ12 (u)du, (6.6)

(t) ≡ θ1n (t) − θ2n (t). Taking the Fourier transform of both sides of equation
where θ12

(6.6), we arrive at a frequency-domain description of the small-signal fluctuations,
δθ12 (ω) = θ12
(ω) + θLn (ω) e−jωτd − e−jω(τd −τ12 )

KDC Kel (ω) sin θ12
e−jω(τd −τ12 ) δθ12 (ω),
jω

(6.7)

where Kel (ω) is the frequency-dependent gain of the loop electronics. This frequencydomain model is shown schematically in figure 6.3. The solution of equation (6.7) is
given by
θL (ω) e−jωτd − e−jω(τd −τ12 )
θ12
(ω)
δθ12 (ω) =
1 + K(ω)
1 + K(ω)

(6.8)

where
KDC Kel (ω) sin θ12
e−jω(τd −τ12 )
jω
2
∆ωf r
KDC Kel (ω) 1 − KDC
e−jω(τd −τ12 )
jω

K(ω) ≡ −

(6.9)

is the total frequency-dependent feedback gain, and we picked the negative root in
calculating sin θ12
in order to achieve negative feedback.

In our experiments, loop bandwidths have been limited to the sub-MHz range by
the AOFS frequency modulation response, and we therefore restrict our attention to
the Fourier frequency range below ∼ 10 MHz. Typical values of τ12 are in the ns

116
range, so ωτ12 . 10−2 , and we can expand equation (6.8) and equation (6.9) near
ωτ12 = 0, yielding
δθ12 (ω) =

θ12
(ω)
θL (ω)e−jωτd
− jωτ12
, and
1 + K(ω)
1 + K(ω)

KDC Kel (ω)
K(ω) =

1−
jω

∆ωf r
KDC

(6.10)

2
e−jωτd .

(6.11)

According to equation (6.10), phase fluctuations in the fiber are reduced by a
factor 1 + K(ω) in the locked state. For frequencies within the loop bandwidth,
K(ω)
1, and significant noise suppression is obtained. The second term describes
the effect of the SFL phase noise and residual chirp nonlinearity. The system behaves
like a frequency discriminator with gain τ12 , and the feedback again suppresses the
measured frequency noise by the factor 1 + K(ω). It is clear that a small differential
delay τ12 and an SFL with a highly linear chirp and low phase noise minimize the
phase error in the loop.
The homodyne phase-locking approach described above has a few shortcomings.
can only be adjusted (within the range
1. The value of the steady-state phase θ12

[0, π]) by varying the bias frequency shifts ω1 and ω2 ; this is not optimal since
it adversely impacts loop gain and therefore performance.
2. The desired operating point for in-phase beam combining is θ12
≈ 0; however,
, and the
according to equation (6.9), the loop gain contains the factor sin θ12

loop therefore loses lock as this operating point is approached. It is desirable
that the loop be locked at quadrature θ12
= π/2, maximizing the gain.

3. Finally, it is not straightforward to scale this approach to multiple phase-locked
arms.
These problems are all addressed by adopting a heterodyne phase-locking architecture, as described in the next section.

117

Sweptfrequency
laser

Reference arm
Delay τd
Amplifier arm n
AOFS

Delay τd – τrn

2x2
Coupler

Mixer

Gain

Offset
Oscillator

Figure 6.4: Passive-fiber chirped-wave optical phase-locked loop in the heterodyne
configuration. PD: Photodetector

SCL phase
noise θL(ω)

Reference phase
fluctuations θr(ω)

Offset oscillator
phase noise θos(ω)

Delay e-jωτd
Amplifier n phase
fluctuations θn(ω)

δθrn(ω)

Delay e-jω(τd-τrn)
Loop transfer function
[KDCKel (ω)/jω] sin δθrn
Figure 6.5: Small-signal frequency-domain model of the heterodyne chirped-wave
optical phase-locked loop. The model is used to study the effect noise and fluctuations
(green blocks) on the loop output variable δθrn (ω).

118

6.3.2

Heterodyne Phase-Locking

In a heterodyne chirped-seed CBC experiment, the SFL output is split into a reference
and multiple amplifier arms. The goal of the experiment is to lock the phases of all
the amplifier arms to the reference, at an offset frequency ωos . The heterodyne OPLL
formed between the reference and the n-th amplifier is shown in figure 6.4. The bias
frequency and phase shift of the AOFSs is denoted by ωn and φn . The differential
delay between the reference and amplifier arms is denoted by τrn , and we again
introduce a common delay τd . The optical phase difference between the two arms is
given by
ξτ 2
θrn (t) = (−ωn − ξτrn )(t − τd ) − (ωn + ωL,0 )τrn − rn − φn
Z t+τrn −τd
KDC cos [θrn (u) + ωos u + θos,n ] du,

(6.12)

where θos,n is the phase of the offset oscillator in the n-th OPLL. The steady-state
(t), obtained by setting the time derivative of the mixer phase θrn (t) +
solution θrn

ωos t + θos to 0, is given by
θrn
(t) = −ωos t − θos,n + cos−1

∆ωf r
KDC

(6.13)

where ∆ωf r = ωos − ωn − ξτrn . We use this result to rewrite equation (6.12),
θrn (t) = ∆ωf r (t + τrn − τd ) + θrn
(t) −

Z t+τrn −τd
KDC cos [θrn (u) + ωos u + θos,n ] du.

(6.14)
If we acquire lock at a zero free-running frequency difference, the steady-state optical
phase difference between the n = 1 and n = 2 amplifier arms is given by
θ12
= θr2
(t) − θr1
(t) = θos,1 − θos,2 ≡ θos,12 .

(6.15)

The steady-state phase difference between the two amplifier arms can now be controlled electronically by setting the relative offset oscillator phase θos,12 . Loop oper-

119
ation off quadrature is therefore no longer required. The electronic phase control is
also important for beam-steering and phase-controlled optical apertures.
Next, we linearize the loop about its steady-state solution. We denote the phase
noise introduced in the reference and amplifier arms by θrn (t) and θnn (t), and offset
(t). We introduce δθrn (t), the small-signal fluctuation
oscillator phase noise by θos,n

of θrn (t) about the steady state, so that
(t) + δθrn (t).
θrn (t) = θrn

(6.16)

We plug equation (6.16) into equation (6.14), and expand about the steady-state
point (equation (6.13)). Solving for δθrn (t), we arrive at
δθrn (t) = θrn
(t) + θLn (t − τd ) − θLn (t + τrn − τd )
2 Z t+τrn −τd
∆ωf r
− KDC 1 −
δθrn (u) + θos,n
(t) du.
KDC

(6.17)

When locked at quadrature, the frequency-domain description of the small-signal
fluctuations is given by
δθrn (ω) = θrn
(ω) − jωτrn e−jωτd θLn (ω) − K(ω) δθrn (ω) + θos,n
(ω) ,
where

K(ω) ≡

KDC Kel (ω) 1 −
jω

∆ωf r
KDC

(6.18)

2
e−jω(τd −τrn ) ,

(6.19)

and we have introduced the frequency-dependent electronic gain Kel (ω). This smallsignal model is shown schematically in figure 6.5. The solution of equation (6.18) is
given by
δθrn (ω) =

K(ω)θos,n
(ω)
θL (ω)e−jωτd
θrn
(ω)
− jωτrn
1 + K(ω)
1 + K(ω)
1 + K(ω)

(6.20)

As in the homodyne case, for frequencies within the loop bandwidth, the OPLL
reduces the phase error due to fiber fluctuations and SFL phase noise by a factor
1 + K(ω). The factor multiplying the offset phase noise term θos,n
(ω) goes to 1 for

120

Figure 6.6: Locked-state Fourier spectrum of the measured beat signal between the
reference and amplifier arms, over a 2 ms chirp interval. The nominal loop delay
parameters are τd = 20 m and τr1 ≈ 0 m. The time-domain signal was apodized with
a Hamming window.
large K(ω). The offset oscillator phase noise is transferred to the optical wave, and
should be kept as small as possible.

6.3.3

Passive-Fiber Heterodyne OPLL

The heterodyne phase-locking experiment of figure 6.4 was performed at 1550 nm
using a VCSEL-based optoelectronic SFL with a chirp rate of 2 × 1014 Hz/s (see
chapter 3 for a summary of its operation). We used polarization-maintaining fiberoptic components, and an AOFS (Brimrose Corporation) with a nominal frequency
shift of 100 MHz and a frequency modulation bandwidth of ∼ 75 kHz. We used a DDS
integrated circuit to provide the 100 MHz offset signal. The circuit can rapidly switch
the output amplitude, phase and frequency when driven by an external trigger, which
allowed us to use different locking parameters for the up and down chirps. Similarly,
we designed a triggered arbitrary waveform generator in order to vary the AOFS bias
during the up and down chirps. The experiment was performed for different values

121

(a)

(b)

Figure 6.7: (a) Phase difference between the reference and amplifier arms calculated
using the I/Q demodulation technique. The three curves (offset for clarity) correspond
to different values of the loop delay τd and the path-length mismatch τr1 . (b) Transient
at the beginning of the chirp. The locking time is determined by the loop bandwidth,
which is limited by the AOFS to about 60 KHz.
of the loop propagation delay τd and path-length mismatch τr1 .1
We measured the beat signal between the reference and amplifier arms in order
to characterize the performance of the heterodyne OPLL. The locked-state beat
signal phase fluctuations, δθr1 (t), are described in the frequency domain by equation
(t)it , is the critical metric
(6.20). The variance of these phase fluctuations, hδθr1

of loop performance since it determines the fraction of the amplifier power that is
coherent with the reference path [8, 104]. The spectrum of the beat signal over one
2 ms chirp duration is calculated using a Fourier transform with a Hamming window,
and is shown in figure 6.6. The delay parameters were τd = 20 m and τr1 ≈ 0. The
spectrum comprises a transform-limited peak at 100 MHz and a small noise pedestal.
The loop bandwidth is about 60 kHz, limited by the AOFS. The residual noise may be
calculated by integrating the noise in the spectral measurement [8, 104]. From figure
6.6, the standard deviation of the phase fluctuations is calculated to be 0.08 rad,
which corresponds to 99.4% of the amplifier optical power being coherent with the
reference wave.
An alternative means of analysis is to use the in-phase and quadrature (I/Q)

The optical delay is reported here in units of length, and is to be understood as the time taken
for light to propagate along that length of polarization-maintaining Panda fiber.

122
demodulation technique, as described in appendix A. It allows us to extract the
(t)it .
time-domain phase fluctuations δθr1 (t), and directly calculate the variance hδθr1

The locked-state phase fluctuations during one 2 ms chirp are plotted in figure 6.7a
for three different values of the loop delay τd and the differential delay τr1 (the curves
are offset from each other for clarity). The locking transient is shown in figure 6.7b.
The locking time is determined by the loop bandwidth, which is limited by the AOFS
to about 60 KHz.
We calculated the phase error standard deviations and locking efficiencies for
different delays, and the results are tabulated in table 6.1. For a given differential
delay, the addition of a large loop delay τd = 20 m slightly reduces the bandwidth of
the loop, resulting in a marginally lower phase-locking efficiency. On the other hand,
for a given loop delay, the addition of τr1 = 32 cm of differential delay results in
an increased amount of SFL phase noise affecting the loop, as predicted by equation
(6.20). This reduces the locking efficiency from ∼ 99% to ∼ 90%. Differential delays
much smaller than 32 cm are trivially achieved in practice, and correspond to phaselocking efficiencies larger than 90%.

Loop delay Differential delay

Phase error std. dev.
1/2

σr1 = hδθr1
(t)it

(mrad)

Locking efficiency
η = 1+σ

τd (m)

τr1 (cm)

99.8%

279

92.8%

99.4%

315

91.0%

Table 6.1: Measured OPLL phase error standard deviation and phase-locking efficiency for different values of the loop delay τd and the differential delay τr1

123

Reference arm 1
Delay τd
AOFS 1

Amplifier
arm 1

PD 1

Delay τd-τr1

Mixer
Gain

Offset
Oscillator

Reference arm 2
Delay τd
Amplifier
arm 2
AOFS 2

Sweptfrequency
laser

μ-lens
array
PD 2

Delay τd-τr2
Mixer
Gain

Offset
Oscillator

Camera

Figure 6.8: Schematic diagram of the passive-fiber chirped-seed CBC experiment with
two channels. Heterodyne optical phase-locked loops are used to lock the amplifier
(blue, green) and reference (black) arms. The outputs of the amplifier arms are
coupled to a microlens (µ-lens) array to form a two-element tiled-aperture beam
combiner. The far-field intensity distribution of the aperture is imaged on a CCD
camera.

124

6.4

Coherent Combining of Chirped Optical
Waves

6.4.1

Passive-Fiber CBC Experiment

To demonstrate beam combining and electronic beam steering, we constructed two
separate heterodyne OPLLs, as shown in figure 6.8. The SFL output was split into
a reference arm and two amplifier channels. The reference arm was further split into
two, and delivered to the two OPLLs. The two loops were locked using electronic offset
signals that were provided by a pair of synchronized DDS oscillators, with individually
controllable amplitudes and phases. We measured the OPLL photocurrents in each
loop for three values of the loop delay τd and differential delays τr1 and τr2 . The
calculated spectra and demodulated phases are shown in figure 6.9 for τd ≈ 0 m, τr1 =
τr2 ≈ 0 cm, figure 6.10 for τd ≈ 18 m, τr1 = τr2 ≈ 0 cm, and figure 6.11 for τd ≈
18 m, τr1 = τr2 ≈ 32 cm. The performance of the two loops is essentially identical.
The same trend that is described above is evident in these figures—a large loop delay
τd only slightly affects the loop bandwidth and marginally increases the measured
noise levels, while the addition of a differential delay τr1 or τr2 increases the effect
of SFL phase noise, causing a noticeable increase in the spectra pedestals and phase
deviations.
The outputs of the two amplifier paths (after the AOFSs and additional fiber
delays) were used to form a coherent aperture using a fiber V-groove array placed
at the focal plane of a microlens array. The emitter spacing was 250 µm. A CCD
camera was used to image the far-field intensity distribution of the aperture over
many chirp periods. The delays in the fiber paths that deliver the amplifier channels
to the microlens array are not compensated for by the OPLLs, which yields an optical
frequency difference between the two channels at the aperture. We solved this issue
by simply phase-locking the two loops at slightly different offset frequencies, so as to
get a stable fringe pattern on the camera. Moreover, we isolated these fibers using a
vibration-damping polymer sheet, in order to minimize the fluctuations in their path

125

(a) OPLL spectra.

(b) I/Q-demodulated OPLL phases.

Figure 6.9: Characterization of the two heterodyne OPLLs in the locked state. τd ≈
0 m, τr1 = τr2 ≈ 0 cm.

(a) OPLL spectra.

(b) I/Q-demodulated OPLL phases.

Figure 6.10: Characterization of the two heterodyne OPLLs in the locked state.
τd ≈ 18 m, τr1 = τr2 ≈ 0 cm.

(a) OPLL spectra.

(b) I/Q-demodulated OPLL phases.

Figure 6.11: Characterization of the two heterodyne OPLLs in the locked state.
τd ≈ 18 m, τr1 = τr2 ≈ 32 cm.

126
Unlocked

Locked, in phase

Locked, out of phase
250
150
100
50

Power (a.u.)

200

Unlocked
Locked, in phase
Locked, out of phase 250

150
100

Power (a.u.)

200

Position (a.u.)

Figure 6.12: Experimental demonstration of electronic phase control and beam steering of chirped optical waves. (a) Far-field intensity profiles for the unlocked and
phase-locked cases. The position of the fringes is controlled by varying the phase
of the electronic oscillator in one loop. (b) Horizontal cross sections of the far-field
intensity patterns

lengths. It is important to note that these efforts are not necessary in the free-space
experiment of section 6.4.3
The far-field intensity distributions of the aperture in the locked and unlocked
states are shown in figure 6.12. We observe a narrowing of the central lobe in the
locked case vs. the unlocked case, and a corresponding increase in its intensity by
a factor of 1.6. We also demonstrate electronic steering of the far-field intensity
pattern by varying the phase of one of the offset oscillators, as shown in figure 6.12.
The demonstrated coherent-combining approach also scales well to larger systems,

127
since the combination of coherent signal gain and incoherent phase errors leads to an
increasing interferometric visibility with increasing number of array elements [105].

6.4.2

Combining Phase Error in a Heterodyne Combining
Experiment

We briefly revisit the small-signal residual phase error analysis. So far we have focused
on measuring phase errors between the reference and amplifier arms, which is useful
in characterizing the OPLL performance. However, in a dual-channel combining
experiment, the relevant phase error is the combining error δθ12 (ω), given by
δθ12 (ω) ≡ δθr2 (ω) − δθr1 (ω)

(6.21)

Plugging in equation (6.20), we arrive at
δθ12 (ω) =

(ω)
θ12
θL (ω)e−jωτd
− jωτ12
1 + K(ω)
1 + K(ω)

(6.22)

is the relative path-length fluctuation of the two amplifier arms. In deriving
where θ12

equation (6.22), we have assumed equal gains in the two OPLLs, and neglected the
contribution of the offset oscillator noise.
In the experiment of section 6.4.1, we learned that the amount of differential pathlength mismatch essentially determines the locked-state noise levels. From equation
(6.22), it is clear that the combining noise level is actually determined by τ12 =
τr2 − τr1 , the path-length mismatch between the two amplifier arms, and not by τr1
or τr2 alone.

6.4.3

Free-Space Beam Combining of Erbium-Doped Fiber
Amplifiers

A schematic of the dual-channel chirped-seed amplifier (CSA) CBC experiment is
shown in figure 6.13. An optoelectronic SFL based on a 1550 nm VCSEL is linearly

128

Reference

50/50

Channel 1 50/50 Channel 2

AOFS 1

PM 1

Swept-frequency laser

Fiber
amp.

From
OPLL 1

PM 2

AOFS 2

Fiber
amp.

From
OPLL 2

To
OPLL 1

Reflector
PD 1

Reflector
PD 2

Camera

Beam dump

To
OPLL 2

Figure 6.13: Schematic diagram of the dual-channel CSA coherent-combining experiment. PD: Photodetector, PM: Back-scattered power monitor

129
chirped over a bandwidth of 500 GHz in 1 ms, resulting in a sweep rate ξ/(2π) =
5 × 1014 Hz/sec. At the end of the 1 ms sweep time, the laser is chirped in reverse at
the same rate, bringing it back to its original starting frequency. Channels 1 and 2
are boosted to powers of ∼ 3 W each with commercially available erbium-doped fiber
amplifiers.
The back-scattered power from the 5 m final amplifier stage and the 45 m delivery
fiber is recorded for each channel. We define the stimulated Brillouin scattering
threshold as the power level at which the ratio of the back-scattered power to the
forward power is 10−4 . We report a threefold increase in the SBS threshold for the
5 × 1014 Hz/sec chirp rate, when compared to a single-frequency seed.
Synchronized DDS circuits are used as offset oscillators in the two heterodyne
OPLLs. An offset frequency of 100 MHz is chosen to match the nominal acoustooptic frequency shift. A tiled-aperture is formed using a 90◦ prism with reflecting legs,
and its far-field distribution is imaged onto a phosphor-coated CCD camera with a
lens.
Intensity distributions of the individual channels, as well as that of the locked
aperture are shown in figure 6.14. The path lengths are nominally matched, with
l12 = 20 mm. This level of path-length matching is easily achieved. We observe, in
the locked state, a twofold narrowing of the central lobe and an associated increase
in the peak lobe intensity. The phases of the individual emitters track the phases of
the DDS oscillators, and we are therefore able to electronically steer the combined
beam. Intensity distributions corresponding to relative DDS phases of θos,12 = 0, π/2,
π, and 3π/2 radians are shown in figure 6.15.
We extract the time-dependent phase differences between the reference and amplifier channels from the two photodetector signals. The phase differences corresponding
to the four values of θos,12 are shown figure 6.16. As expected, the OPLL phases, and
hence the phases of the individual chirped waves track the DDS setpoint.
To characterize performance, we consider three path-length matching cases, summarized in Table 6.2. The I/Q technique yields the residual phase errors, δθr1 (t) and
δθr2 (t). The time-domain combining phase error is then calculated using equation

Channel 2

Locked

Channel 1
Channel 2
Locked

250
200
150
100
50

250
200
150
100

Power (a.u.)

Channel 1

Power (a.u.)

130

Position (a.u.)

θos,12 = π/2

θos,12 = π

θos,12 = 3π/2

250
200
150
100
50
250
200
150
100
50

Power (a.u.)

θos,12 = 0

Power (a.u.)

Figure 6.14: Far-field intensity distributions of the individual channels and the locked
aperture. τr1 = −19 mm, and τr2 = 1 mm

Position (a.u.)

Figure 6.15: Steering of the combined beam through emitter phase control. θos,12 is
the relative DDS phase.

131
θos,12 = 0

θos,12 = π/2

θos,12 = π

θos,12 = 3π/2
3π/2

Channel 1
Channel 2

π/2

1 0

1 0
Time (ms)

1 0

Figure 6.16: I/Q-demodulated phase differences between the amplifier channels and
the reference. θos,12 is the relative DDS phase.
(6.21). The standard deviations σxy =

2 (t)i of all three phase errors, along
hδθxy

with the phase-noise-limited fringe visibilities are listed in table 6.2. The visibilities
are calculated from the standard deviations σ12 using a Gaussian phase noise model,
as described in appendix B.
The first case (nominally path-length-matched) has the lowest combining error,
which is consistent with equation (6.22). The second and third cases have nearly
identical amplifier path-length mismatches and exhibit nearly identical combining
phase errors. This is consistent with the prediction that the residual combining error
is determined solely by the mismatch between the amplifier channels.
The phase-noise-limited fringe visibility for the path-length-matched case is almost 99%, yet the fringe visibility in figure 6.14 is only about 80%. We believe the
discrepancy is due to the wavefront distortions introduced by the collimators and the
prism reflectors.

6.5

Summary

We have analyzed and experimentally demonstrated the phase-locking of chirped
optical waves in a master oscillator power amplifier configuration. The precise chirp
linearity of the optoelectronic SFL enables non-mechanical compensation of optical
delays using acousto-optic frequency shifters, and is at the heart of our chirped phaselocking and coherent-combining systems.

132
We have demonstrated heterodyne phase-locking of optical waves with a chirp
rate of 5 × 1014 Hz/sec at 1550 nm, achieving a loop bandwidth of 60 kHz and a

phase error variance less than 0.01 rad2 . We used the heterodyne OPLL architecture
to construct a dual-channel passive-fiber coherent beam combining experiment, and
have demonstrated coherent combining and electronic beam steering of chirped optical
waves.
We have also implemented and characterized a 1550 nm chirped-seed amplifier
coherent-combining system. We used a chirp rate of 5 × 1014 Hz/sec, which resulted
in a threefold increase of the amplifier SBS threshold, when compared to a singlefrequency seed. We demonstrated efficient phase-locking and electronic beam steering
of two 3 W erbium-doped fiber amplifier channels. We achieved temporal phase noise
levels corresponding to fringe visibilities exceeding 90% at path-length mismatches of
≈ 300 mm, and exceeding 98% at a path-length mismatch of 20 mm.
The optoelectronic SFL has the potential to significantly increase the achievable
output power from a single fiber amplifier by increasing its SBS threshold. Coherent
beam combining techniques developed in this chapter can be used to efficiently combine multiple chirped amplifier outputs, without imposing strict path-length matching
requirements, presenting a viable path towards high-power continuous-wave sources.

Case Differential delay (mm)a Phase error (mrad) Fringe visibility

τr1

τr2

τ12

σr1

σr2

σ12

V = e−σ12 /2

−19

118 79.3

160

98.7%

110

450

340

184 531

428

91.3%

−118 220

338

150 273

410

92.0%

These are fiber lengths corresponding to the time delays between the different
paths. Actual mismatches have both free-space and fiber components.

Table 6.2: OPLL phase errors and phase-noise-limited fringe visibilities in the dualchannel active CBC experiment

133

Chapter 7
Conclusion
7.1

Summary of the Thesis

7.1.1

Development of the Optoelectronic SFL

We have demonstrated the use of optoelectronic feedback for precise control over the
optical chirp of a semiconductor laser diode. This system, the optoelectronic SFL,
formed the backbone of all the work described in this thesis. The development of
the optoelectronic SFL was guided by optical FMCW reflectometry and 3-D imaging
applications. Specifically, we aimed to build a swept-source with narrow linewidth (for
long-range imaging), linear frequency tuning (to reduce the processing overhead), and
high chirp bandwidth (for high axial resolution), all on a compact platform without
moving parts.
The optoelectronic SFL works like a PLL. A portion of the SCL light is launched
into a Mach-Zehnder interferometer, and the loop locks the sinusoidal intensity fluctuation at the interferometer output to a reference electronic oscillator. The optoelectronic SFL, just as a regular PLL, only achieves lock if the feedback bandwidth
is larger than the unlocked beat signal linewidth, which is determined by the freerunning SCL chirp nonlinearity. As the SCL is chirped faster, the nonlinearity is
increased, which lead to poor locking—our initial experiments were limited to a chirp
rate of 1014 Hz/s for DFB lasers and 5×1014 for VCSELs. To improve the free-running
sweep nonlinearity, we developed a bias current predistortion algorithm. Even though

134
the algorithm was based on a very naive nonlinear tuning model, it yielded impressive results when iterated. Using iterative predistortion we were able to significantly
increase the chirp rates of our systems, up to 1015 Hz/s for DFB lasers and 1016 for
VCSELs. We developed SFLs based on VCSELs and DFB lasers at wavelengths of
1550 nm and 1060 nm, and demonstrated their use in reflectometry and profilometry applications. Electronic development of the SFL undertaken as part of our work
eventually lead to its commercialization by Telaris, Inc.
A key feature of the optoelectronic SFL, albeit not one that we recognized until
after the first system was built and tested, is that successive chirps are exactly repeatable. The PLL locks not just the beat signal frequency, i.e., the instantaneous chirp
rate, but also the beat signal phase, i.e., the starting chirp frequency. This means
that each frequency sweep starts at the exact same point. As it turned out, stability
of the starting sweep frequency was crucial for our work on MS-FMCW reflectometry
and TomICam.

7.1.2

Ranging and 3-D Imaging Applications

7.1.2.1

MS-FMCW Reflectometry and Stitching

In an effort to increase the axial resolution of an SCL-based ranging system, we
developed a novel variant of the FMCW optical imaging technique. This method,
MS-FMCW reflectometry, uses multiple lasers that sweep over distinct but adjacent
regions of the optical spectrum, in order to “stitch” a measurement with increased
optical bandwidth and a corresponding improvement in the axial resolution. This
technique bears resemblance to synthetic aperture radar, in which RF signals collected
at multiple physical locations are used to approximate a large antenna aperture, and
hence a high transverse resolution. In MS-FMCW reflectometry, the synthesized
aperture is not physical, but instead represents the accessible optical frequency range.
The culmination of this work was an MS-FMCW system with four VCSEL channels, yielding a total chirp bandwidth of 2 THz and a scan time of 500 µs. This
particular demonstration relied on hardware stitching to remove the need for addi-

135
tional signal processing that was present in our early MS-FMCW work. In a hardware
stitching system, the SCL sweeps are locked to the same MZI with an electronic reference oscillator whose phase is not reset during channel switching. Because the
starting frequencies of the sweeps are controlled exclusively by the reference oscillator phase, this configuration allowed perfect stitching to be performed in hardware.
Each channel’s chirp started precisely where the previous one ended!
7.1.2.2

The Tomographic Imaging Camera

One of the goals of our work is to enable rapid, high-resolution, and low-cost 3-D
imaging without moving parts. The tomographic imaging camera was our solution to
the problem of non-mechanical acquisition of transverse pixel information. TomICam
uses low-cost full-field detector arrays to acquire depth information one transverse
slice at a time. This is achieved by modulating the intensity of the transmitted beam
with sinusoidal function, which shifts the signal spectrum to DC, allowing the use
of low-speed integrating detector arrays, i.e., CCD and CMOS cameras. The depth
of the slice is determined by the modulation frequency, and can therefore be tuned
electronically. As a result, TomICam completely eliminates the need for moving parts
traditionally employed in 3-D imaging.
We demonstrated basic TomICam functionality in a single-pixel proof-of-concept
experiment at 1550 nm, and showed that the depth scan retrieved with TomICam
is identical to the traditional FMCW measurement. It turns out that multiple measurements (two to four, depending on whether or not the imaging interferometer is
balanced) at the same modulation frequency but different modulation phases are necessary to extract the depth information. This means that TomICam imaging would
not be possible if there was appreciable starting frequency jitter between subsequent
SFL sweeps. For TomICam, as for MS-FMCW, precise repeatability of the frequency
sweeps generated by the optoelectronic SFL turned out to be a necessary requirement. We also discussed the application of compressive sensing to the TomICam
platform, and showed, through computer simulations, that a tenfold improvement in
the volume acquisition speed is possible for sufficiently sparse depth signals.

136

7.1.3

Phase-Locking and CBC of Chirped Optical Waves

Out group’s current focus on the phase and frequency control of SCLs started a
few years ago with phase-locking and coherent beam combining experiments that
used commercially available, single-frequency semiconductor laser diodes. We have
generalized these experiments to the case of chirped optical waves. The precise chirp
linearity of the optoelectronic SFL enables non-mechanical compensation of optical
delays using acousto-optic frequency shifters, and is at the heart of our chirped phaselocking and coherent-combining systems.
We have demonstrated heterodyne phase-locking of optical waves with a chirp rate
of 5 × 1014 Hz/sec at 1550 nm, and constructed a dual-channel passive-fiber coherent
beam combining experiment. We achieved efficient combining and demonstrated electronic beam steering of chirped optical waves by tuning the electronic offset oscillator
phase in one of the heterodyne OPLLs.
The key physical result driving this work is that swept-frequency optical waveforms
suppress stimulated Brillouin scattering (SBS) in fiber by reducing the effective length
over which SBS occurs. This has the potential to increase the maximum output of
high-power fiber amplifiers; and the chirped phase-locking techniques developed in
this thesis can be used to form coherent amplifier arrays, further scaling the optical
power. Conventional SBS suppression techniques result in a decrease of the seed
laser coherence length, and coherent combining therefore requires very strict pathlength matching. In practice, sub-mm matching is necessary at the kW power level.
The chirped-seed combining approach developed in this thesis does not have strict
matching requirements, due to the comparatively long coherence lengths of SCLbased SFLs, and therefore presents a viable path towards high-power continuous-wave
sources.
We have also performed, for the first time, an active CBC experiment using a
chirp rate of 5 × 1014 Hz/sec and two 3 W erbium-doped fiber amplifier channels.
We recorded a threefold increase of the amplifier SBS threshold, when compared
to a single-frequency seed. We demonstrated efficient phase-locking and electronic

137
beam steering of amplified chirped beams, and achieved temporal phase noise levels
corresponding to fringe visibilities exceeding 90% at path-length mismatches of ≈
300 mm, and exceeding 98% at a path-length mismatch of 20 mm.

7.2

Current and Future Work

The ground for continuing SFL development is fertile. One of the projects undertaken
in our group, led by Yasha Vilenchik, is the integration of the optical components
of the optoelectronic SFL on a hybrid Si/III-V integrated platform. Images of the
subcomponents fabricated to date are shown in figure 7.1. The hybrid platform
has the potential to bring photonic and electronic components together on a single
bonded chip, and continuing development will one day yield a chip-scale chirped
LIDAR transmitter.
Another interesting development in our group is the recent demonstration of a
hybrid Si/III-V high-coherence semiconductor laser based on a modulated-bandgap
design, shown in figure 7.2 [106]. The laser’s high-Q resonator, designed and fabricated by Christos Santis, is contained entirely in silicon, and is therefore subject to
much lower optical loss than traditionally used III-V resonators. This laser’s chirp
bandwidth is comparable to that of commercially available DFBs, while its linewidth
is inherently superior. The use of this laser in an optoelectronic SFL will enable
3-D imaging systems that simultaneously possess long imaging range and high axial
resolutions.

(a)

(b)

(c)

(d)

Figure 7.1: (a) Hybrid Si/III-V DFB laser bar. (b) Scanning electron microscope
(SEM) image of a 1 × 3 multimode interference (MMI) coupler, (c) SEM image of a
2 × 2 MMI coupler. (d) SEM closeup of the a spiral delay line for the loop MZI

138

(a)

(b)

Figure 7.2: Schematic of the hybrid Si/III-V high-coherence semiconductor laser.
(a) Side-view cross section. (b) Top-view of the laser and the modulated-bandgap
resonator

Development of narrow-linewidth swept-frequency lasers will also contribute to
the group’s label-free biomolecular sensing project, led by Jacob Sendowski. The
sensor comprises an ultra-high-Q SiN microdisk resonator and a microfluidic analyte
delivery system [15], as shown in figure 7.3. Biomolecular binding events shift the
microdisk resonance frequency, which is detected using the optoelectronic SFL. Longterm repeatability of the starting frequency of SFL sweeps was a deciding factor in
using it to interrogate the biomolecular sensor. The use of narrow-linewidth SFLs has
the potential to improve measurement sensitivity by enhancing the sensor’s ability to
resolve small resonant frequency shifts. Moreover, integration of the SFL will enable
a complete chip-scale high-sensitivity biomolecular sensor.
Recent developments in the field of microelectromechanical (MEMS) VCSELs hold
promise for SFLs with extremely high chirp rates [107]. These devices are based on
an electrically-tunable MEMS mirror, and are capable of sweeping a bandwidth of
100 nm at a wavelength of 1060 nm, with repetition rates exceeding 100 kHz. This
corresponds to a chirp rate > 1018 Hz/sec, which is two orders of magnitude higher
than the fastest SFLs constructed with conventional SCLs.
Our chirped-waveform CBC experiments are currently being repeated at 1060 nm
using the VCSEL-based SFL. This is the wavelength of choice for high-power laser
sources because of the extremely efficient Yb-doped fiber amplifier technology. The
development of an SFL based on the 1060 nm MEMS VCSEL will yield unprecedented
chirped-seed SBS suppression results, due to the extremely high chirp rates attainable
with these devices.

139

Figure 7.3: Schematic representation of the label-free biomolecular sensing system
TomICam experiments aimed at demonstrating full 3-D imaging capability using
a low-cost silicon CCD camera are currently being performed in our group. These
experiments rely on our 1060 nm DFB and VCSEL SFLs for illumination. A preferred
wavelength for silicon sensors is 850 nm, and we are currently developing an 850 nm
VCSEL-based SFL to address this demand. Recently-demonstrated 850 nm MEMS
VCSELs [108] can be used to build SFLs that will enable µm-scale axial resolutions
in our TomICam systems. An alternative path towards increasing TomICam axial
resolution is through the use of MS-FMCW. Hardware stitching can be adopted
to the TomICam platform in a very straightforward way, and an array of 850 nm
VCSELs can therefore be used for broadband swept-frequency illumination.
In summary, electronic control over the frequency of semiconductor lasers enables
a range of swept-frequency applications, from spectroscopy and biomolecular sensing,
to ranging and 3-D imaging, to stimulated Brillouin scattering suppression in, and
coherent combining of high-power fiber amplifiers. Continuing development and integration of the SFL technology holds promise for chip-scale coherent sensing and 3-D
imaging systems.

140

Appendix A
Time-Domain Phase Analysis
Using I/Q Demodulation
In this appendix we describe the in-phase and quadrature (I/Q) demodulation technique which is used for time-domain analysis of the locked-state OPLL phase error
in chapter 6.
The goal of the technique is to separate the amplitude modulation A(t) from the
phase modulation θ(t) of a sinusoidal signal y(t) with a known frequency ω0 ,
y(t) = A(t) sin [ω0 t + θ(t)] .

(A.1)

We form the in-phase signal yi (t) and the quadrature signal yq (t) by multiplying
y(t) with sine and cosine waveforms at a frequency of ω0 , and low-pass filtering the
results.
yi (t) = h(t) ? [y(t) sin ω0 t]
A(t)
A(t)
= h(t) ?
cos θ(t) −
cos [2ω0 t + θ(t)] , and

(A.2)

yq (t) = h(t) ? [y(t) cos ω0 t]
A(t)
A(t)
= h(t) ?
sin θ(t) +
sin [2ω0 t + θ(t)] ,

(A.3)

where h(t) is the impulse response of the low-pass filter, and ‘?’ denotes the convolution operation. The filter is designed to average out the sum frequency terms at

141
frequency 2ω0 , while retaining the difference frequency terms at DC, yielding
yi (t) =

A(t)
cos θ(t), and

(A.4)

A(t)
sin θ(t).

(A.5)

yq (t) =

The amplitude and phase modulations are recovered using
A(t) = 2 yi2 (t) + yq2 (t), and

(A.6)

θ(t) = atan2 [yq (t), yi (t)] ,

(A.7)

where atan2(yq , yi ) is the four-quadrant inverse tangent function defined below.

−1 yq
tan
yi
−1 yq
tan
+π
yi
 tan−1 yq − π
yi
atan2(yq , yi ) ≡
pi
+2
− pi2
undefined

yi > 0
yq ≥ 0, yi < 0
yq < 0, yi < 0
yq > 0, yi = 0
yq < 0, yi = 0
yq = 0, yi = 0

(A.8)

142

Appendix B
Phase-Noise-Limited
Tiled-Aperture Fringe Visibility
We consider the case of tiled-aperture CBC with two emitters. We assume that
the emitters have equal intensities and are phase-locked with a residual phase error
δθ12 (t). The far-field intensity at location r is then given by:

I ∝ h|1 + exp [jθ12 (r ) + jδθ12 (t)]|2 it = 2 + 2e−σ12 /2 cos θ12 (r ),

(B.1)

where θ12 (r ) is the mean phase difference between the beams at the point r and hit
denotes an average over time. We assumed that δθ12 (t) is a zero-mean Gaussian

, so that hejδθ12 (t) it = e−σ12 /2 . Intensity extrema
random variable with variance σ12

are found at points of constructive and destructive interference, with cos θ12 (r ) = ±1.
The fringe visibility is therefore given by:

V ≡ (Imax − Imin )/(Imax + Imin ) = e−σ12 /2

(B.2)

Strictly speaking, this derivation applies only to single-frequency beams, since in
the chirped case the propagation phase θ12 is a function of both r and t. However,
equation (B.2) still applies to the chirped-seed CBC experiments of chapter 6, because
the frequency ranges considered there are ∼ 0.25% of the nominal lasing frequency.
Chirp ranges that constitute a significant fraction of the lasing frequency require a
more sophisticated analysis based, for example, on chirped Gaussian modes [109].

143

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The Optoelectronic Swept-Frequency Laser and Its
Applications in Ranging, Three-Dimensional
Imaging, and Coherent Beam Combining of
Chirped-Seed Amplifiers

Thesis by

Arseny Vasilyev
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy

California Institute of Technology
Pasadena, California

2013
(Defended May 20, 2013)

c 2013
Arseny Vasilyev

iii

vii

Contents
Acknowledgments

iii

Abstract

List of Figures

List of Tables

xviii

Glossary of Acronyms

1 Overview and Thesis Organization

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2

Ranging and 3-D Imaging Applications . . . . . . . . . . . . . . . . .

1.2.1

Optical FMCW Reflectometry . . . . . . . . . . . . . . . . . .

1.2.2

Multiple Source FMCW Reflectometry . . . . . . . . . . . . .

1.2.3

The Tomographic Imaging Camera . . . . . . . . . . . . . . .

1.3

Phase-Locking and Coherent Combining of Chirped Optical Waves

2 Optical FMCW Reflectometry
2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1.1

Basic FMCW Analysis and Range Resolution . . . . . . . . .

2.1.2

Balanced Detection and RIN . . . . . . . . . . . . . . . . . . .

2.1.3

Effects of Phase Noise on the FMCW Measurement . . . . . .

2.1.3.1

Statistics and Notation . . . . . . . . . . . . . . . . .

2.1.3.2

Linewidth of Single-Frequency Emission . . . . . . .

viii

2.1.4

2.1.3.3

Fringe Visibility in an FMCW Measurement . . . . .

2.1.3.4

Spectrum of the FMCW Photocurrent and the SNR

2.1.3.5

Phase-Noise-Limited Accuracy . . . . . . . . . . . .

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 The Optoelectronic Swept-Frequency Laser

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

System Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2.1

The Optoelectronic SFL as a PLL . . . . . . . . . . . . . . . .

3.2.2

Small-Signal Analysis . . . . . . . . . . . . . . . . . . . . . . .

3.2.3

Bias Current Predistortion . . . . . . . . . . . . . . . . . . . .

Design of the Optoelectronic SFL . . . . . . . . . . . . . . . . . . . .

3.3.1

SCL Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3.2

Amplitude Control . . . . . . . . . . . . . . . . . . . . . . . .

3.3.3

Electronics and Commercialization . . . . . . . . . . . . . . .

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4.1

Precisely Controlled Linear Chirps . . . . . . . . . . . . . . .

3.4.2

Arbitrary Chirps . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

3.4

3.5

3.6

Demonstrated Applications

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

3.5.1

FMCW Reflectometry Using the Optoelectronic SFL . . . . .

3.5.2

Profilometry . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Multiple Source FMCW Reflectometry

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2.1

Review of FMCW Reflectometry . . . . . . . . . . . . . . . .

4.2.2

Multiple Source Analysis . . . . . . . . . . . . . . . . . . . . .

4.2.3

Stitching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Experimental Demonstrations . . . . . . . . . . . . . . . . . . . . . .

4.3.1

4.3

Stitching of Temperature-Tuned DFB Laser Sweeps . . . . . .

4.4

4.3.2

Stitching of Two VCSELs . . . . . . . . . . . . . . . . . . . .

4.3.3

Hardware Stitching of Four VCSELs . . . . . . . . . . . . . .

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 The Tomographic Imaging Camera
5.1

5.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1.1

Current Approaches to 3-D Imaging and Their Limitations . .

5.1.2

Tomographic Imaging Camera . . . . . . . . . . . . . . . . . .

5.1.2.1

Summary of FMCW Reflectometry . . . . . . . . . .

5.1.2.2

TomICam Principle

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

5.1.2.3

TomICam Proof-of-Principle Experiment . . . . . . .

Compressive Sensing . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.1

Compressive Sensing Background . . . . . . . . . . . . . . . .

5.2.2

TomICam Posed as a CS Problem . . . . . . . . . . . . . . . .

5.2.3

Robust Recovery Guarantees . . . . . . . . . . . . . . . . . . .

5.2.3.1

Random Partial Fourier Measurement Matrix . . . .

5.2.3.2

Gaussian or Sub-Gaussian Random Measurement Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2.4
5.3

Numerical CS TomICam Investigation . . . . . . . . . . . . . 101

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6 Phase-Locking and Coherent Beam Combining of Broadband
Linearly Chirped Optical Waves

108

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2

Coherent Beam Combining . . . . . . . . . . . . . . . . . . . . . . . . 109

6.3

Phase-Locking of Chirped Optical Waves . . . . . . . . . . . . . . . . 111

6.4

6.3.1

Homodyne Phase-Locking . . . . . . . . . . . . . . . . . . . . 112

6.3.2

Heterodyne Phase-Locking . . . . . . . . . . . . . . . . . . . . 118

6.3.3

Passive-Fiber Heterodyne OPLL . . . . . . . . . . . . . . . . . 120

Coherent Combining of Chirped Optical Waves . . . . . . . . . . . . 124
6.4.1

Passive-Fiber CBC Experiment . . . . . . . . . . . . . . . . . 124

6.5

6.4.2

Combining Phase Error in a Heterodyne Combining Experiment 127

6.4.3

Free-Space Beam Combining of Erbium-Doped Fiber Amplifiers 127

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7 Conclusion
7.1

Summary of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.1.1

Development of the Optoelectronic SFL

7.1.2

Ranging and 3-D Imaging Applications . . . . . . . . . . . . . 134

7.1.3
7.2

133

. . . . . . . . . . . . 133

7.1.2.1

MS-FMCW Reflectometry and Stitching . . . . . . . 134

7.1.2.2

The Tomographic Imaging Camera . . . . . . . . . . 135

Phase-Locking and CBC of Chirped Optical Waves . . . . . . 136

Current and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 137

Appendices
A Time-Domain Phase Analysis Using I/Q Demodulation

140

B Phase-Noise-Limited Tiled-Aperture Fringe Visibility

142

Bibliography

143

List of Figures
2.1

Time evolution of the optical frequencies of the launched and reflected
waves in a single-scatterer FMCW ranging experiment . . . . . . . . .

2.2

Mach-Zehnder interferometer implementation of the FMCW ranging experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3

A balanced Michelson interferometer implementation of the FMCW
ranging experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6

A balanced Mach-Zehnder interferometer implementation of the FMCW
ranging experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5

Michelson interferometer implementation of the FMCW ranging experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

2.7

2.8

Baseband FMCW photocurrent spectra for four different values of τ /τc ,
normalized to zero-frequency noise levels. The scan time is T = 1 ms

2.9

and the coherence time is τc = 1 µs. . . . . . . . . . . . . . . . . . . .

FMCW SNR as a function of τ /τc for three different values of T /τc . .

xii
3.1

Schematic diagram of the SCL-based optoelectronic SFL . . . . . . . .

3.2

Elements of the optoelectronic SFL lumped together as an effective VCO 34

3.3

Small-signal frequency-domain model of the optoelectronic SFL . . . .

3.4

Single predistortion results . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

Iterative predistortion results . . . . . . . . . . . . . . . . . . . . . . .

3.6

Measured optical spectra of DFB and VCSEL SFLs at wavelengths of
1550 nm and 1060 nm . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7

Schematic diagram of the amplitude controller feedback system . . . .

3.8

Comparison between the off(blue) and on(red) states of the SOA amplitude controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.9

Comparison between the off(blue) and on(red) states of the VOA amplitude controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.10

Optoelectronic SFL printed circuit board layouts . . . . . . . . . . . .

3.11

The 1550 nm CHDL system. . . . . . . . . . . . . . . . . . . . . . . . .

3.12

MZI photocurrent spectrum during the predistortion process and in the
locked state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.13

Locked MZI spectra of various SFLs for different values of the chirp rate
ξ. The x-axis in all the plots corresponds to the chirp rate. . . . . . . .

3.14

Quadratic chirp spectrogram . . . . . . . . . . . . . . . . . . . . . . .

3.15

Exponential chirp spectrogram . . . . . . . . . . . . . . . . . . . . . .

3.16

3.17

Depth profile of a United States $1 coin measured using the VCSELbased optoelectronic SFL with a chirp bandwidth of 500 GHz and a

4.1

wavelength of ∼ 1550 nm . . . . . . . . . . . . . . . . . . . . . . . . .

Schematic of an FMCW ranging experiment. PD: Photodetector . . .

xiii
4.2

4.3

Schematic representation of dual-source FMCW reflectometry.

Top

4.5

4.6

Proposed multiple source FMCW system architecture. BS: Beamsplitter. PD: Photodetector

4.7

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

Single-sweep and stitched two-sweep photocurrent spectra of a dual reflector target with a separation of 1.49 mm. No apodization was used.

4.10

Single-sweep and stitched two-sweep photocurrent spectra of a dual reflector target with a separation of 5.44 mm. No apodization was used.

4.9

Optical spectra of the two DFB sweeps (blue and red) and the optical
spectrum analyzer PSF (black) . . . . . . . . . . . . . . . . . . . . . .

4.8

xiv
4.11

4.12

4.13

Optical spectra of the two VCSEL sweeps in the 250 GHz experiment .

4.14

Optical spectra of the two VCSEL sweeps in the ∼ 1 THz experiment .

4.15

4.16

Single-sweep and stitched two-sweep photocurrent spectra of dual reflector targets with various separations. The total chirp bandwidth is
∼ 1 THz. No apodization was used.

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

4.17

Four channel 2 THz hardware stitching experiment . . . . . . . . . . .

4.18

Optical spectra of the four 1550 nm VCSEL sweeps in the 2 THz hardware stitching experiment . . . . . . . . . . . . . . . . . . . . . . . . .

4.19

4.20

xv
5.1

Principle of FMCW imaging with a single reflector . . . . . . . . . . .

5.2

(a) Volume acquisition by a raster scan of a single-pixel FMCW mea-

5.4

5.5

5.6

xvi
5.7

5.8

Comparison between FMCW (red) and TomICam (blue) depth measurements. The two are essentially identical except for a set of ghost
targets at 31 of the frequency present in the TomICam spectrum. These
ghosts are due to the third-order nonlinearity of the intensity modulator
used in this experiment. . . . . . . . . . . . . . . . . . . . . . . . . . .

5.9

5.10

Flow diagram and parameters of the CS TomICam simulation . . . . . 101

5.11

SER curves for a CS simulation with a Gaussian random matrix . . . . 102

5.12

SER curves for a CS simulation with a waveform matrix given by the
absolute value of a Gaussian random matrix . . . . . . . . . . . . . . . 102

5.13

SER curves for a CS simulation with a waveform matrix whose entries
are uniformly distributed between 0 and 1 . . . . . . . . . . . . . . . . 104

5.14

SER curves for a CS simulation with a waveform matrix whose entries
are uniformly distributed between 0.5 and 1 . . . . . . . . . . . . . . . 104

5.15

SER curves for a CS simulation with a waveform matrix whose entries
take on the values of 0.5 or 1 with equal probabilities . . . . . . . . . . 105

5.16

SER curves for an N = 1000 CS simulation with a waveform matrix
whose entries are uniformly distributed between 0.5 and 1 . . . . . . . 105

6.1

xvii
6.2

Passive-fiber chirped-wave optical phase-locked loop in the homodyne
configuration. PD: Photodetector . . . . . . . . . . . . . . . . . . . . . 113

6.3

6.4

Passive-fiber chirped-wave optical phase-locked loop in the heterodyne
configuration. PD: Photodetector . . . . . . . . . . . . . . . . . . . . . 117

6.5

6.6

6.7

6.8

6.9

Characterization of the two heterodyne OPLLs in the locked state. τd ≈
0 m, τr1 = τr2 ≈ 0 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.10

Characterization of the two heterodyne OPLLs in the locked state. τd ≈
18 m, τr1 = τr2 ≈ 0 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

xviii
6.11

Characterization of the two heterodyne OPLLs in the locked state. τd ≈
18 m, τr1 = τr2 ≈ 32 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.12

6.13

Schematic diagram of the dual-channel CSA coherent-combining experiment. PD: Photodetector, PM: Back-scattered power monitor . . . . . 128

6.14

Far-field intensity distributions of the individual channels and the locked
aperture. τr1 = −19 mm, and τr2 = 1 mm . . . . . . . . . . . . . . . . 130

6.15

Steering of the combined beam through emitter phase control. θos,12 is
the relative DDS phase. . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.16

I/Q-demodulated phase differences between the amplifier channels and
the reference. θos,12 is the relative DDS phase. . . . . . . . . . . . . . . 131

7.1

7.2

7.3

Schematic representation of the label-free biomolecular sensing system

139

xix

List of Tables
5.1

Recent three-dimensional (3-D) camera embodiments . . . . . . . . . .

6.1

Measured OPLL phase error standard deviation and phase-locking effi-

ciency for different values of the loop delay τd and the differential delay
τr1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2

OPLL phase errors and phase-noise-limited fringe visibilities in the dualchannel active CBC experiment . . . . . . . . . . . . . . . . . . . . . . 132

Chapter 1
Overview and Thesis Organization
1.1

Introduction

1.2

Ranging and 3-D Imaging Applications

1.2.1

Optical FMCW Reflectometry

1.2.2

Multiple Source FMCW Reflectometry

1.2.3

The Tomographic Imaging Camera

1.3

Phase-Locking and Coherent Combining of
Chirped Optical Waves

Chapter 2
Optical FMCW Reflectometry
2.1

Introduction

2.1.1

Basic FMCW Analysis and Range Resolution

t − T /2

ξt2
cos φ0 + ω0 t +

(2.1)

(2.2)

The instantaneous optical frequency is given by the time derivative of the argument
of the cosine in equation (2.1)
ωSF L (t) =
dt

ξt2
φ 0 + ω0 t +

= ω0 + ξt

(2.3)

ωL

Figure 2.1: Time evolution of the optical frequencies of the launched and reflected
waves in a single-scatterer FMCW ranging experiment

e(t) + R e(t − τ )

t − T /2
1+R √
ξτ 2
= rect
+ R cos (ξτ )t + ω0 τ −

i(t) =

(2.4)

Figure 2.2: Mach-Zehnder interferometer implementation of the FMCW ranging experiment

Figure 2.3: Michelson interferometer implementation of the FMCW ranging experiment

11
domain, so we use equation (2.3) to rewrite the photocurrent as a function of ωSF L .
ωSF L − ω0
y(ωSF L ) ≡ i
ωSF L − ω0 − πB
ξτ 2
cos ωSF L τ −
= R rect
2πB

(2.5)

The delay τ is found by taking the Fourier transform (FT) of y(ωSF L ) with respect
to the variable ωSF L , which yields a single sinc function centered at the delay τ .

(2.8)

2.1.2

Balanced Detection and RIN

t − T /2

1+R
ξτ 2
+ n(t) + R cos (ξτ )t + ω0 τ −

(2.9)

t − T /2

1+R
ξτ 2
+ n(t) ± R cos (ξτ )t + ω0 τ −
. (2.10)

Balanced processing consists of averaging the two photocurrents, yielding
i+ (t) − i− (t)
= rect
idiff (t) ≡

t − T /2

ξτ 2
R cos (ξτ )t + ω0 τ −

(2.11)

Figure 2.4: A balanced Mach-Zehnder interferometer implementation of the FMCW
ranging experiment

Figure 2.5: A balanced Michelson interferometer implementation of the FMCW ranging experiment

15
enhance the measurement dynamic range.

2.1.3

Effects of Phase Noise on the FMCW Measurement

2.1.3.1

Statistics and Notation

We first review some useful statistical results and introduce notation. For a wide-sense
stationary random process x(t), we denote its autocorrelation function by Rx :
Rx (u) = E [x(t)x(t − u)] ,

(2.12)

where E[·] is the statistical expectation value. For an ergodic random process, the
expectation can be replaced by an average over all time, giving:
Rx (u) = hx(t)x(t − u)it ,

(2.13)

16
By the Wiener–Khinchin theorem, the power spectral density (PSD) Sx (ω) and autocorrelation Rx (u) are FT pairs.
Sx (ω) = Fu [Rx (u)] =

Z ∞
−∞

Rx (u)e−iωu du,

(2.14)

(2.15)

Alternatively, it may be calculated by integrating the PSD:
σx2 =

2.1.3.2

2π

Z ∞
−∞

Sx (ω)dω.

(2.16)

Linewidth of Single-Frequency Emission

We first derive a standard model for the spontaneous emission linewidth of a singlefrequency laser [41]. The electric field is given by
e(t) = cos [ω0 t + φn (t)] ,

(2.17)

hhhh
= hcos [ω0 u + ∆φn (t, u)]it + hcos [2ω0 t − ω0 uh+hφh
hh+hφh
n (t)
n (t − u)]it ,
hhhh
(2.18)

where the sum term is crossed out because it averages out to zero. ∆φn (t, u) is the
accumulated phase error during time u, defined by
∆φn (t, u) ≡ φn (t) − φn (t − u),

(2.19)

(2.20)

So, equation (2.18) simplifies to
2
σ∆φn (u)
Re (u) = cos(ω0 u) exp −

(2.21)

Taking the FT of equation (2.21), we find the spectrum of the electric field,
Se (ω) =

1 ◦
[S (ω − ω0 ) + Se◦ (ω + ω0 )] ,
4 e

(2.22)

where Se◦ (ω) is the baseband spectrum given by
Se◦ (ω) = Fu

2
σ∆φn (u)
exp −

(2.23)

(2.25)

where Sφ.n (ω) = ω 2 Sφn (ω) is the spectrum of the frequency noise φn . Spontaneous
emission into the lasing mode gives rise to a flat frequency noise spectrum [40, 42],

18
and we therefore assign a constant value to Sφ.n (ω),
Sφ.n (ω) ≡ ∆ω.

(2.26)

We plug equation (2.25) and equation (2.26) into equation (2.16) to calculate the
variance of the accumulated phase error.
Z ∞
S∆φn (ω, u)dω
2π −∞
Z ∞
4u2 ∆ω sinc2 (ωu)dω
2π −∞

σ∆φ
(u) =

(2.27)

= |u|∆ω.
Plugging this result into equation (2.23), we obtain the baseband spectrum of the
electric field.

2
σ∆φn (u)
∆ω
= Fu exp −|u|
exp −
∆ω
(∆ω/2)2 + ω 2

Se◦ (ω) = Fu

(2.28)

(2.29)

∆ν
∆ω
Sφ.n (2πν) =
⇐⇒ linewidth ∆ν =
(Hz),
(2π)
2π
2π

(2.30)

2.1.3.3

Fringe Visibility in an FMCW Measurement

We continue or analysis by modifying the chirped electric field in equation (2.1) to
include phase noise,
e(t) = rect

t − T /2

ξt2
cos φ0 + ω0 t +
+ φn (t) ,

(2.31)

and assume a perfect reflector (R = 1). The photocurrent is therefore given by
i(t) = rect

t − T /2

ξτ 2
1 + cos (ξτ )t + ω0 τ −
+ ∆φn (t, τ ) ,

(2.32)

Electric field baseband PSD (a.u.)

−2

−1

N=1

N = 10

N = 100

N = 1000

−2 −1
Frequency (MHz)

300

Electric field baseband PSD (a.u.)

2π Sφ̇

(kHz, log scale)

900

100

−1

−0.5

Frequency (MHz)

0.5

100

imax − imin
imax + imin

(2.33)

(2.34)

Plugging in equation (2.27) and equation (2.34) into equation (2.33), we arrive at an
expression for the phase-noise-limited visibility [45],

∆ω
V = exp −|τ |

|τ |
= exp −
τc

(2.35)

where
τc ≡

∆ω
π∆ν

(2.36)

23
2.1.3.4

Spectrum of the FMCW Photocurrent and the SNR

ξτ 2
R cos (ξτ )t + ω0 τ −
+ ∆φn (t, τ ) .

(2.37)

(2.38)

θ(t, τ, u) ≡ ∆φn (t, τ ) − ∆φn (t − u, τ ),

(2.39)

Ri (u) =

where

and we have assumed that θ(t, τ, u) possesses Gaussian statistics. Taking the FT of
equation (2.38), we find the spectrum of the photocurrent.
Si (ω) =

1 ◦
[S (ω − ξτ ) + Si◦ (ω + ξτ )] ,
4 i

(2.40)

where Si◦ (ω) is the baseband spectrum given by
Si◦ (ω) = Fu

exp −

σθ(τ,u)

(2.41)

To find the baseband spectrum and the SNR we need to calculate the variance of
θ(t, τ, u). First we derive a useful identity. Let us write down the variance of ∆φn (t, u),

24
as it is defined in equation (2.15),
(u) = [φn (t) − φn (t − u)]2 t
σ∆φ

(2.42)

= 2σφ2 n − 2 hφn (t)φn (t − u)it .
This gives us an expression for the autocorrelation of φn (t),
Rφ (u) = hφn (t)φn (t − u)it = σφ2 n −

(u)
σ∆φ

(2.43)

We plug this result into equation (2.24),
R∆φn (s, u) = 2Rφn (s) − Rφn (s + u) − Rφn (s − u)

σ∆φ
(s + u) σ∆φ
(s − u)
(s)
− σ∆φ

(2.44)

(2.45)

= 2σ∆φ
(τ ) − 2R∆φn (u, τ ).

Plugging in equation (2.44), we arrive at
σθ2 (τ, u) = 2σ∆φ
(τ ) + 2σ∆φ
(u) − σ∆φ
(u + τ ) − σ∆φ
(u − τ ).

(2.46)

(2.47)

25
The baseband photocurrent spectrum is found by plugging equation (2.47) into equation (2.41), yielding [46, 47]
Si◦ (ω) = Fu

exp −

= 2πδ(ω)e

− 2τ
τc

σθ(τ,u)

τc

ωτc 2

1−e

− 2τ
τc

sin(ωτ ) .
cos(ωτ ) +
ωτc

(2.48)

Si◦ (ω, T ) =

Photocurrent baseband PSD relative to noise (dBrn/Hz)

40
30

τ/τc = 0.3

τ/τc = 1

τ/τc = 3

τ/τc = 10

20
10
−10
−20
−30
40
30
20
10
−10
−20
−30
−10

−10

80
60

Signal to noise ratio (dB)

40
20
−20
−40
−60
−80 −2
10

T/τc = 102
T/τc = 103
T/τc = 104
−1

τ/τc

Figure 2.9: FMCW SNR as a function of τ /τc for three different values of T /τc

28
The SNR is readily calculated from equation (2.49), and is given in decibel units
by

SNRdB = 10 log10 

 .
τi e2τ /τc − 1 + 2τ

(2.50)

τc

2.1.3.5

Phase-Noise-Limited Accuracy

∆φn (t, τ ).
dt

(2.51)

The target delay is calculated from an average of the photocurrent frequency over the
scan time T ,
ξτ

Z T
ωP D (t) = ξτ +

∆φn (T, τ ) − ∆φn (0, τ )

(2.52)

29
The accuracy is therefore given by
δτ ≡ τ m − τ =

∆φn (T, τ ) − ∆φn (0, τ )

(2.53)

The accuracy of a single measurement is a zero-mean random process with standard
deviation
σδτ =

4τ
τc

(2.54)

where we have used equation (2.39) and equation (2.47). Likewise, the depth accuracy
δz is characterized by the standard deviation
σδz =
2B

4τ
= ∆z
τc

4τ
τc

(2.55)

2.1.4

Summary

Chapter 3
The Optoelectronic
Swept-Frequency Laser
3.1

Introduction

3.2

System Analysis

32
We conclude by discussing different SCL platforms and how they motivate the choice
of an amplitude control element.

System output
MZI

Amplitude
controller

Tap coupler

Photodetector

Semiconductor
laser
Predistorted bias
current

Reference
oscillator

Figure 3.1: Schematic diagram of the SCL-based optoelectronic SFL

3.2.1

The Optoelectronic SFL as a PLL

(3.1)

where φSF L (t) is an overall electric field phase that is quadratic in time, ξ is the slope
of the optical chirp, and φ0 and ω0 are the initial phase and frequency, respectively.

33
The instantaneous optical frequency is therefore the derivative of φSF L (t):
ωSF L (t) =

dφSF L (t)
dt

(3.2)

(3.3)

Z t
δs(u) du,

(3.4)

(3.5)

where φREF (t) is the overall phase of the reference oscillator and ωREF is its fre-

System output

Effective VCO
MZI

Amplitude
controller

Tap coupler

Photodetector
φPD(t)=ωSFL(t)τ
δφPD(t) tδs(u)du

Semiconductor
laser

δs(t)

φREF(t)=ωREF + φREF(0)

Predistorted bias
current

Reference
oscillator

Figure 3.2: Elements of the optoelectronic SFL lumped together as an effective VCO

quency. We use the superscript L to denote quantities associated with the locked
state. The feedback maintains a precisely linear chirp with a chirp rate and initial
optical frequency given by
ξL =

ωREF
φREF
2π
and ω0L =
+n .

(3.6)

Locked phase
ΔφSFL
(ω)

SCL phase
noise φn,SCL(ω)

Mixer

SCL
HFM (ω) / jω

MZI
jωτ
Reference phase
noise φn,REF (ω)
MZI fluctuations
φn,MZI(ω)

Integrator
1 / jω

Loop delay
e-jωτd

Loop gain

Figure 3.3: Small-signal frequency-domain model of the optoelectronic SFL

3.2.2

Small-Signal Analysis

2τ

(3.7)

M (ω)
The transfer function of the SCL is HFjω
, where HF M (ω) is the frequency modula-

tion (FM) response of the SCL, normalized to unity at DC, and jω
results from the

HF M (ω)e−jωτd
[φn,REF (ω) + φn,M ZI (ω)]
ω2

HF M (ω)e−jωτd
+ Kτ
∆φLSF L (ω)
jω

(3.8)

37
Solving for ∆φLSF L (ω) yields
jω
φn,SCL (ω)
jω + Kτ HF M (ω)e−jωτd
Kτ HF M (ω)e−jωτd
[φn,REF (ω) + φn,M ZI (ω)] .
jωτ
jω + Kτ HF M (ω)e−jωτd

∆φLSF L (ω) =

(3.9)

We observe that for frequencies within the loop bandwidth, the residual phase deviation tracks the reference oscillator and MZI noise, suppressed by the term jωτ ,
∆φLSF L (ω
Kτ ) ≈

φn,REF (ω) + φn,M ZI (ω)
jωτ

(3.10)

For frequencies outside the loop bandwidth, the residual phase deviation is given by
the free-running phase noise term,
∆φLSF L (ω
Kτ ) ≈ φn,SCL (ω).

(3.11)

3.2.3

Bias Current Predistortion

(3.12)

dφP D (t)
dωSF L (t)
=τ
dt
dt

(3.13)

Plugging in equation (3.12), we arrive at
ωPD (t) = τ

dωSF L (t)
di(t)
di(t)
= τ [K(i) + iK 0 (i)]
= S(i)
dt
dt
dt

(3.14)

where S(i) ≡ K(i) + iK 0 (i). The function S(i) describes the nonlinear tuning of the
SCL and can be measured by launching a linear current ramp into the SCL and using

(a) Current ramp chirp spectrogram

(b) Predistorted chirp spectrogram

Figure 3.4: Single predistortion results

a spectrogram to calculate ωPD (t). Then,
S(i) =

ωPD [t(i)]

(3.15)

where α is the current ramp slope. We use equation (3.14) to write a differential
equation for the SCL modulation current
di(t)
τ dωSF L,d
dt
S(i) dt

(3.16)

(3.17)

(3.18)

(a) Zeroth-order predistortion (current
ramp) spectrogram

(b) First-order predistortion spectrogram

(d) Third-order predistortion spectrogram

(e) Fourth-order predistortion spectrogram

Figure 3.5: Iterative predistortion results

3.3

Design of the Optoelectronic SFL

3.3.1

SCL Choice

1550 nm DFB SFL

1550 nm VCSEL SFL

−10
−20

Optical Spectrum (dBm)

−30
−40
−50
1536
10

1538

1540

1060 nm DFB SFL

1554

1556

1558

1060 nm VCSEL SFL

−10
−20
−30
−40
−50

1064

1066

1068
1064
Wavelength (nm)

1066

1068

Figure 3.6: Measured optical spectra of DFB and VCSEL SFLs at wavelengths of
1550 nm and 1060 nm

Optical
input

SOA/VOA

Optical output
Tap coupler
Photodetector

Gain

Loop
Filter

Amplitude
setpoint

Figure 3.7: Schematic diagram of the amplitude controller feedback system

Amplitude control on

MZI photo−
current (a.u.)

AC photo−
current (a.u.)

Amplitude control off

80 100 120
20
Time (µs)

80 100 120

Figure 3.8: Comparison between the off(blue) and on(red) states of the SOA amplitude controller

Amplitude control off

Amplitude control on

MZI photo−
current (a.u.)

AC photo−
current (a.u.)

Transient

80 100 120
20
Time (µs)

80 100 120

Figure 3.9: Comparison between the off(blue) and on(red) states of the VOA amplitude controller

3.3.2

Amplitude Control

3.3.3

Electronics and Commercialization

(a) SFL feedback PCB

(b) Amplitude controller PCB

Figure 3.10: Optoelectronic SFL printed circuit board layouts

Figure 3.11: The 1550 nm CHDL system.

3.4

Experimental Results

3.4.1

Precisely Controlled Linear Chirps

Current ramp

1st-order predistortion

2 -order predistortion

3 -order predistortion

4th-order predistortion

Locked

50
40
30
20
10

Photocurrent Spectrum (dB)

−10
50

40
30
20
10
−10
50
40
30
20
10
−10

0.7

1.3
0.7
15
Chirp rate (x10 Hz/sec)

1.3

Figure 3.12: MZI photocurrent spectrum during the predistortion process and in the
locked state

3.4.2

Arbitrary Chirps

Z t
ωREF (t)dt +

φREF
2π
+n ,

(3.19)

51
14

~ 1x10 Hz/sec mode

−20

−40

−40
0.1

0.12 0.14

~ 7.5x1014Hz/sec mode

0.15

0.2

0.25

~ 8.5x1014Hz/sec mode

−20
−40
−60

0.4

0.6

0.8

0.6

0.8

−60

0.3

1.2

Photocurrent Spectrum (dB)

0.06 0.08

0.06 0.08 0.1 0.12 0.14

−60

0.3

0.4

−20

−40

−40
−60

−20

−40

~ 6x10 Hz/sec mode

−20
−40
−60

0.2

0.3

0.4

−60

0.25

0.5

1.5

0.4

0.6

1.2

1.4

1.6

~ 5x10 Hz/sec mode

2.5
Chirp rate (x1015 Hz/sec)

0.8

0.06 0.08

0.1

~ 2x10 Hz/sec mode

0.12 0.14

0.15

0.2

0.25

0.3

~ 8x10 Hz/sec mode

−20
−40
−60

0.3

0.4
15

0.5

~ 1.5x10 Hz/sec mode

−20

−40

~ 1x10 Hz/sec mode

0.6 0.8
1.2 1.4
Chirp rate (x1015 Hz/sec)

0.8

~ 4x10 Hz/sec mode

−60

0.6

Photocurrent Spectrum (dB)

0.2
14

~ 3.5x10 Hz/sec mode

0.7

~ 1x10 Hz/sec mode

0.6

~ 2x10 Hz/sec mode

0.15

0.3

(b) 1550 nm VCSEL SFL

0.06 0.08 0.1 0.12 0.14 0.1

0.5

~ 2x10 Hz/sec mode

(a) 1550 nm DFB SFL

−60

0.25

−40

0.6
0.8
1.2
Chirp rate (x1015 Hz/sec)

0.2

~ 1x1015Hz/sec mode

−20

~ 1x10 Hz/sec mode

0.15

~ 9x10 Hz/sec mode

~ 2x10 Hz/sec mode

~ 5x1014Hz/sec mode

−60

~ 1x10 Hz/sec mode

−60

Photocurrent Spectrum (dB)

~ 2x10 Hz/sec mode

−60

1.5

0.6

0.8

1.2

~ 2.6x10 Hz/sec mode

1.5
2.5
Chirp rate (x1015 Hz/sec)

3.5

(d) 1060 nm VCSEL SFL

Figure 3.13: Locked MZI spectra of various SFLs for different values of the chirp rate
ξ. The x-axis in all the plots corresponds to the chirp rate.

Figure 3.14: Quadratic chirp spectrogram

Figure 3.15: Exponential chirp spectrogram

53
to
ωREF (t) = 2π × (4.29 MHz) ×

1.43 MHz
4.29 MHz

t/(1 ms)

(3.20)

3.5

Demonstrated Applications

3.5.1

FMCW Reflectometry Using the Optoelectronic SFL

2nB

(3.21)

Figure 3.16: FMCW reflectometry of acrylic sheets using the VCSEL-based optoelectronic SFL with a chirp bandwidth of 500 GHz and a wavelength of ∼ 1550 nm

3.5.2

Profilometry

3.6

Summary

Figure 3.17: Depth profile of a United States $1 coin measured using the VCSELbased optoelectronic SFL with a chirp bandwidth of 500 GHz and a wavelength of
∼ 1550 nm

Chapter 4
Multiple Source FMCW
Reflectometry
4.1

Introduction

1x2
Coupler

2x1
Coupler

Circulator

Laser

Delay Line

Data
Processing

Figure 4.1: Schematic of an FMCW ranging experiment. PD: Photodetector

4.2

Theoretical Analysis

4.2.1

Review of FMCW Reflectometry

t − T /2

ξt2
cos φ0 + ω0 t +

(4.1)

R rect

ω − ω0 − πB
2πB

ξτ 2
cos ωτ −

(4.2)

(4.3)

(4.4)

where c is the speed of light. An equivalent metric of the resolution of the FMCW
system is the full width at half maximum (FWHM) of the sinc function, given by
FWHM ≈

3.79
7.58
= ∆d
πB
πc

(4.5)

Let us now consider the following view of an FMCW imaging system. The target
is characterized by some underlying function of the optical frequency, ytarget (ω), given
by
ytarget (ω) =

ξτn2
Rn cos ωτn −

(4.6)

(4.7)

where a(ω) is the rectangular window function, as in equation (4.2). The function

In the following analysis we use capital letters to denote the FTs of the corresponding lower-case
functions.

4.2.2

Multiple Source Analysis

Taking the FT of equation (4.7), and equation (4.6), we arrive at the expression
1 Xp
ξτn2
Y (ζ) =
Rn exp −j
A(ζ − τn ),
2 n

(4.8)

aN (ω)

ωk

|AN (ζ)|

2π B̃

2πBk

2π B̃

2πB̃

2π B̃

2πδk

−1

(a)

(b)

63
bandwidth 2π B̃.
Therefore, an N-source sweep is described by

aN (ω) = rect

ω − ω0,1 − π B̃
2π B̃

−1

rect

k=1

ω − ω0, k+1 + πδk
2πδk

(4.9)

in the ω-domain, and by

AN (ζ) = 2π B̃ exp −jζ(ω0,1 + π B̃) sinc ζ B̃
− 2π

−1
k=1

(4.10)

δk exp [−jζ (ω0, k+1 − πδk )] sinc(ζδk )

in the ζ-domain. To find the range resolution we find the first null of equation (4.10).
P −1
PN
Expanding near ζ = 1/B̃ and using the approximation N

k=1
k=1 Bk yields
−1
ζnull

−1
i N
B̃ exp −jζnull (ω0,1 + π B̃) +
δk exp [−jζnull (ω0,k+1 − πδk )] .

(4.11)

k=1

k=1 Bk

(4.12)

4.2.3

Stitching

We next consider the problem of stitching, that is, synthesizing a measurement with
enhanced resolution using photocurrents collected from multiple sweeps. In the pre-

Reference

Optical
Input

1x2
Coupler

2x1
Coupler

Circulator

Delay Line

ω − πBk
2πBk

ytarget (ω + ω0,k ).

(4.13)

The stitched measurement, given by ystitched = aN (ω)ytarget (ω), can be written in
terms of functions yk (ω) using equation (4.9):

ystitched (ω) =

k=1

yk (ω − ω0,k ).

(4.14)

The magnitude of the FT of equation (4.14) may be used for target recognition, and
is given by
|Ystitched (ζ)| =

k=1

exp −j2πζ

k−1
l=1

(Bl + δl ) Yk (ζ) .

(4.15)

(4.16)

The ratio of the two expressions in equation (4.16) yields the phase difference between
the photocurrents:
ψa ≡

Y1 (τa )
= exp [−j2πτa (B + δ)] .
Y2 (τa )

(4.17)

Given the reference reflector delay τa and the frequency excursion B, the gap may be
found using
arg [exp(j2πτa B)ψa ] = −2πτa δ.

(4.18)

Y1 (τn )
Y2 (τn )

n = a, b

(4.19)

and calculate the two reflector-analog of equation (4.18):
ψa
arg exp [j2π(τa − τb )B]
= −2π(τa − τb )δ.
ψb

(4.20)

67
Target

0000 00 00 0000 00 00 0000 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00

Laser 1

Laser N

Nx1
Mux

1x2
Coupler

Circulator

2x1
Coupler

PD 1
1xN
Demux

Delay Line

PD 2

PD N

i1
i2

Data Processing

Laser 2

Reference

Figure 4.6: Proposed multiple source FMCW system architecture. BS: Beamsplitter.
PD: Photodetector
combination of range resolution, scan speed, and imaging depth.

4.3

Experimental Demonstrations

4.3.1

Stitching of Temperature-Tuned DFB Laser Sweeps

Figure 4.7: Optical spectra of the two DFB sweeps (blue and red) and the optical
spectrum analyzer PSF (black)

Figure 4.8: Single-sweep and stitched two-sweep photocurrent spectra of a dual reflector target with a separation of 5.44 mm. No apodization was used.

Figure 4.9: Single-sweep and stitched two-sweep photocurrent spectra of a dual reflector target with a separation of 1.49 mm. No apodization was used.

Figure 4.10: Single-sweep and stitched two-sweep photocurrent spectra of a dual
reflector target with a separation of 1.00 mm (a microscope slide). No apodization
was used.

Normalized FT Amplitude (dB)

−5
−10
−15
−20
−25
−30

1.34 1.36 1.38 1.4 1.42 1.44
Time Delay (ns)
(a)

−5
−10
−15
−20
−25
−30

3.22 3.24 3.26 3.28 3.3 3.32
Time Delay (ns)
(b)

Optical
Switch

VCSEL 1

Amplitude
controller

VCSEL 2
Predistorted
bias current

90/10 Output
Coupler

Reference
oscillator
Integrator

MZI

PD
Mixer

Target

Circulator

Reference

Data
Processing

2x1
Coupler

Delay Line

Figure 4.12: Dual VCSEL FMCW reflectometry system diagram. The feedback loop
ensures chirp stability. A reference target is used to extract the inter-sweep gaps.
PD: Photodiode, BS: Beamsplitter

Figure 4.13: Optical spectra of the two VCSEL sweeps in the 250 GHz experiment

Figure 4.14: Optical spectra of the two VCSEL sweeps in the ∼ 1 THz experiment

Figure 4.15: Single-sweep and stitched two-sweep photocurrent spectra of dual reflector targets with various separations. The total chirp bandwidth is 250 GHz. No
apodization was used.

Figure 4.16: Single-sweep and stitched two-sweep photocurrent spectra of dual reflector targets with various separations. The total chirp bandwidth is ∼ 1 THz. No
apodization was used.

4.3.2

Stitching of Two VCSELs

4.3.3

Hardware Stitching of Four VCSELs

Figure 4.17: Four channel 2 THz hardware stitching experiment

Figure 4.18: Optical spectra of the four 1550 nm VCSEL sweeps in the 2 THz hardware stitching experiment

4.4

Summary

Chapter 5
The Tomographic Imaging Camera
5.1

Introduction

5.1.1

Current Approaches to 3-D Imaging and Their Limitations

ωL

Launched

ω0 + ξt

Reflected

ω0 + ξ(t − τ )

i ∝ cos [ξτ t + ω0 τ ]

Resolution: δz = 2B

Figure 5.1: Principle of FMCW imaging with a single reflector

Technology

TOF

Transverse
imaging

Hardware
requirement

Limitations

Compressive
sensing

Not used in
cited work

≃ 2 cm

Mechanical
scanning

Mode-locked
laser, fast
electronics

Slow scanning,
moving parts,
expensive
components,
limited
resolution

≃ 1 cm

Spatial
light
modulator,
single pixel
detector

Mode-locked
laser, fast
electronics,
SLM

Expensive
components,
limited
resolution

Used to
convert the
single-pixel
data into a
3D model

10s of cm

Lock-in
pixel CCD

Poor
resolution,
limited lock-in
CCD size

Not used

SSOCT/CSOCT [71]

1–10 µm

Mechanical
scanning

Slow scanning,
moving parts,
bulky and
fragile

Used to
reduce scan
time

TomICam

10–
100 µm

CCD/
CMOS
array

Specially
engineered
lock-in pixel
CCD
External cavity
chirped laser
with moving
parts, slow
detector
Optoelectronic
SFL (no
moving parts),
standard
CCD/CMOS
sensor

Floodlight
illumination
(higher power)

Reduced
acquisition
time and
power

TOFLIDAR [16]

Single-pixel
TOFLIDAR [17]

Lock-in
TOF [70]

FMCW

Axial
resolution

Table 5.1: Recent 3-D camera embodiments

(a)

(b)

5.1.2

Tomographic Imaging Camera

88
FMCW Target
reflections

TomICam
measurement

Sinusoidal
intensity
modulation

n = xt1

xt1
xt2
xt3
Fourier variable (x × time)

xt2-n xt3-n
Fourier variable (x × time)

(a)

(b)

Summary of FMCW Reflectometry

Sample

1×2
coupler
SFL

Circulator

2×1
coupler

Reference arm

Integrating
Fast
detector
Fourier
transform

(a)

Sample

SFL
Intensity
modulator

W(t)

1×2
coupler

Circulator

2×1
coupler

Integrating
detector

Reference arm

(b)

iFMCW (t) =

Xp
Rn e(t − τn )
e(t) +

= rect

t − T /2

ξτn 2
Rn cos (ξτn )t + ω0 τn −

(5.1)

|Y (ν = ξτ )| =

Z T

exp [j(ξτ )t] iFMCW (t)dt .

(5.2)

By the Fourier uncertainty relation, the resolution of this measurement is inversely
proportional to the integration time T . The spatial resolution is, therefore, given by
∆z =

c 2π 1
2 ξ T
2B

(5.3)

where c is the speed of light.1
5.1.2.2

TomICam Principle

The scatterer range is given by z = cτ /2.

91
integrating detector is reset at the beginning of every sweep, and sampled at the end.
For a given modulation signal W (t), the beat signal at the detector is given by
yW (t) ∝ W (t) iFMCW (t).

(5.4)

The value sampled at the output of the integrating detector is therefore given by
Z T
W (t) iFMCW (t)dt,

YW =

(5.5)

cos [(ξτ )t] iFMCW (t)dt

(5.6)

sin [(ξτ )t] iFMCW (t)dt

(5.7)

Z T
YWS (τ ) =

Equation (5.2) may therefore be written as:
|Y (ν = ξτ )|2 = |YWC (τ ) + j ∗ YWS (τ )|2 = |YWC (τ )|2 + |YWS (τ )|2 .

(5.8)

92
Reference
wavefront

Reference
mirror
Aperture

Sample

Camera

W(t)
Swept-frequency
laser

Intensity
modulator

Illuminating
wavefront

5.1.2.3

TomICam Proof-of-Principle Experiment

In order to verify the equivalence of FMCW and TomICam measurements, we have
performed a proof-of-principle experiment, shown schematically in figure 5.6. We

Figure 5.7: The custom PCB used in the TomICam experiment. Implemented functionality includes triggered arbitrary waveform generation and high-bit-depth acquisition of an analog signal.

5.2

Compressive Sensing

5.2.1

Compressive Sensing Background

We briefly state the salient features of CS [69]. Consider a linear measurement system
of the form:
y = Ax

A ∈ Cm×N , x ∈ CN , y ∈ Cm ,

(5.9)

kxk1

(5.10)

5.2.2

TomICam Posed as a CS Problem

−1 N
−1

W (th )

h=0 n=0

Rn
cos (ξτn th + ω0 τn ).

(5.11)

ys =

−1 N
−1
h=0 n=0
−1 N
−1
h=0 n=0

r
ω
2πhn
Rn
· √ exp −j
exp −j n

(5.12)

Wsh · Fhn · xn ,

where Wsh ≡ Ws hT

, Fhn ≡

√1 exp

−j 2πhn

, xn ≡

Rn
exp

−j ωB0 n , and it is

understood that the measurements correspond to the real part of the right hand side.
Rewriting equation (5.12) in matrix notation, we obtain:
y = WFx,

(5.13)

where x is the k-sparse target vector of length N , y is the vector containing the m
TomICam measurements, F is the N ×N unitary Fourier matrix, and W is the m×N

5.2.3

Robust Recovery Guarantees

We now consider two possibilities for W that yield a measurement matrix capable
of robust signal recovery. These represent straightforward implementations of CS
TomICam imaging.
5.2.3.1

Random Partial Fourier Measurement Matrix

(5.14)

5.2.3.2

Gaussian or Sub-Gaussian Random Measurement Matrix

(5.15)

101
M easu rem ent
Generate random
target of given
sparsity xo

Generate random
code matrix W

Repeat 100 times

M inimize Ll norm
of x, subject to
yo= WFx

Inject noise, and
make
"measurement"
yo= WF(xo+ noise)

• Space dimension :
- N=100
• Number of measurements:
- m = 0 to 100
• Sparsity:
- k = [ 1, 3, 5, 7, 9]
• SNR: [40dB , 80dB, 120dB]

Figure 5.10: Flow diagram and parameters of the CS TomICam simulation

5.2.4

Numerical CS TomICam Investigation

(5.16)

where xn is a randomly generated noise vector. We define the SNR as the ratio of
the signal and noise energies,
SNR ≡

kx0 k2
kxn k2

(5.17)

We then proceed to solve the convex minimization problem in equation (5.10), which
yields the recovered signal x. We define the signal-to-error ratio (SER) as the ratio of

102

140

.. .. ..

W standard gaussian distrubuted, N=1 00

120 .
.-..

cc 100

"'0
._..

.....,
ro
'--

60 .

.....,
ro

C>
(/)

'--

'-'-Q)

0 ·
-20

.. ". ". " . " . " ....... " . " .. " .. " . " . " . ··:.

80
40
60
Number of measurements

100

Figure 5.11: SER curves for a CS simulation with a Gaussian random matrix

140

W abs(standard gaussian) distrubuted, N=1 00

120 .
.-..

cc 100

"'0
._..

.....,
ro
'--

60 .

.....,
ro

C>
(/)

'--

'-'-Q)

.......

-20

.. ". ". " . " . " ....... " . " .. " .. " . " . " . ··:.

80
40
60
Number of measurements

100

Figure 5.12: SER curves for a CS simulation with a waveform matrix given by the
absolute value of a Gaussian random matrix

103
the energy of the recovered signal to the energy of the difference between the recovered
and the original signals.
SER ≡

kxk2
kx − x0 k2

(5.18)

104

14 0

W uniformly distributed on [0 1], N=1 00
............. "

120 .
.-..

cc 100

"'0
._..

ro 80
'--

e 6o .
'--

'-Q)

C>
(/)

40
20

.. ". ". " . " . " ....... " . " .. " .. " . " . " . ··:.

-20~----~----~------~----~----~

60
80
40
Number of measurements

100

Figure 5.13: SER curves for a CS simulation with a waveform matrix whose entries
are uniformly distributed between 0 and 1

140

.. .. ..

W uniformly distributed on [0.5 1], N=1 00

120
.-..

cc 100

"'0
._..

.....,
ro
'--

60 .

.....,
ro

C>
(/)

'--

'-'-Q)

-20

.. ". ". " . " . " ....... " . " .. " .. " . " . " . ··:.

80
40
60
Number of measurements

100

Figure 5.14: SER curves for a CS simulation with a waveform matrix whose entries
are uniformly distributed between 0.5 and 1

105

140 . . . . . . . . . . . . . . . . . . ..

W Digital [0.5 1], N=100

120 .. .. .. .. .. .. .

.... .... .... .... .... .... .... .... . .................................................... ...................................................
·~··

-20~----~----~------~----~----~

Number of measurements

100

Figure 5.15: SER curves for a CS simulation with a waveform matrix whose entries
take on the values of 0.5 or 1 with equal probabilities

140

.. .. ..

W uniformly distributed on [0 1], N=1 000

120
.-..

cc 100

"'0
._..

.....,
ro
'--

.....,
ro

C>
(/)

'--

'-'-Q)

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

.. .... '.. .... '.. .... '.. .... '.. .. .. i . ..

-20

.. ". ". " . " . " ....... " . " .. " .. " . " . " . ··:.

40
60
80
Number of measurements

100

Figure 5.16: SER curves for an N = 1000 CS simulation with a waveform matrix
whose entries are uniformly distributed between 0.5 and 1

5.3

Summary

108

Chapter 6
Phase-Locking and Coherent Beam
Combining of Broadband
Linearly Chirped Optical Waves
6.1

Introduction

6.2

Coherent Beam Combining

111

Sweptfrequency
laser

Amplifier 1

Combined
output

ξτ

Mismatch l12
Frequency
shifter

Amplifier 2

Time

6.3

Phase-Locking of Chirped Optical Waves

(6.1)

6.3.1

Homodyne Phase-Locking

113

Sweptfrequency
laser

AOFS 1

AOFS 2

PD 1

Delay τd
Delay τd-τ12

2x2
Coupler

PD 2

Feedback
Gain

Figure 6.2: Passive-fiber chirped-wave optical phase-locked loop in the homodyne
configuration. PD: Photodetector

SCL phase
noise θL(ω)

Arm 1 phase
fluctuations θ1(ω)
Delay e-jωτd
Arm 2 phase
fluctuations θ2(ω)

Balanced
Detector

δθ12(ω)

is given by
θ12
= cos−1

∆ωf r
KDC

(6.3)

We use this result to rewrite equation (6.2),
θ12 (t) = ∆ωf r (t + τ12 − τd ) + θ12

Z t+τ12 −τd
KDC cos θ12 (u)du.

(6.4)

In lock, the optical phases of the two arms differ by θ12
, and there is no frequency

115
about the steady state, so that
θ12 (t) = θ12
+ δθ12 (t).

(6.5)

Z t+τ12 −τd
δθ12 (u)du, (6.6)

(t) ≡ θ1n (t) − θ2n (t). Taking the Fourier transform of both sides of equation
where θ12

(6.6), we arrive at a frequency-domain description of the small-signal fluctuations,
δθ12 (ω) = θ12
(ω) + θLn (ω) e−jωτd − e−jω(τd −τ12 )

KDC Kel (ω) sin θ12
e−jω(τd −τ12 ) δθ12 (ω),
jω

(6.7)

(6.8)

where
KDC Kel (ω) sin θ12
e−jω(τd −τ12 )
jω
2
∆ωf r
KDC Kel (ω) 1 − KDC
e−jω(τd −τ12 )
jω

K(ω) ≡ −

(6.9)

is the total frequency-dependent feedback gain, and we picked the negative root in
calculating sin θ12
in order to achieve negative feedback.

116
range, so ωτ12 . 10−2 , and we can expand equation (6.8) and equation (6.9) near
ωτ12 = 0, yielding
δθ12 (ω) =

θ12
(ω)
θL (ω)e−jωτd
− jωτ12
, and
1 + K(ω)
1 + K(ω)

KDC Kel (ω)
K(ω) =

1−
jω

∆ωf r
KDC

(6.10)

2
e−jωτd .

(6.11)

loop therefore loses lock as this operating point is approached. It is desirable
that the loop be locked at quadrature θ12
= π/2, maximizing the gain.

117

Sweptfrequency
laser

Reference arm
Delay τd
Amplifier arm n
AOFS

Delay τd – τrn

2x2
Coupler

Mixer

Gain

Offset
Oscillator

Figure 6.4: Passive-fiber chirped-wave optical phase-locked loop in the heterodyne
configuration. PD: Photodetector

SCL phase
noise θL(ω)

Reference phase
fluctuations θr(ω)

Offset oscillator
phase noise θos(ω)

Delay e-jωτd
Amplifier n phase
fluctuations θn(ω)

δθrn(ω)

118

6.3.2

Heterodyne Phase-Locking

(6.12)

where θos,n is the phase of the offset oscillator in the n-th OPLL. The steady-state
(t), obtained by setting the time derivative of the mixer phase θrn (t) +
solution θrn

ωos t + θos to 0, is given by
θrn
(t) = −ωos t − θos,n + cos−1

∆ωf r
KDC

(6.13)

where ∆ωf r = ωos − ωn − ξτrn . We use this result to rewrite equation (6.12),
θrn (t) = ∆ωf r (t + τrn − τd ) + θrn
(t) −

Z t+τrn −τd
KDC cos [θrn (u) + ωos u + θos,n ] du.

(6.15)

The steady-state phase difference between the two amplifier arms can now be controlled electronically by setting the relative offset oscillator phase θos,12 . Loop oper-

of θrn (t) about the steady state, so that
(t) + δθrn (t).
θrn (t) = θrn

(6.16)

(6.17)

K(ω) ≡

KDC Kel (ω) 1 −
jω

∆ωf r
KDC

(6.18)

2
e−jω(τd −τrn ) ,

(6.19)

and we have introduced the frequency-dependent electronic gain Kel (ω). This smallsignal model is shown schematically in figure 6.5. The solution of equation (6.18) is
given by
δθrn (ω) =

K(ω)θos,n
(ω)
θL (ω)e−jωτd
θrn
(ω)
− jωτrn
1 + K(ω)
1 + K(ω)
1 + K(ω)

(6.20)

120

6.3.3

Passive-Fiber Heterodyne OPLL

121

(a)

(b)

The optical delay is reported here in units of length, and is to be understood as the time taken
for light to propagate along that length of polarization-maintaining Panda fiber.

122
demodulation technique, as described in appendix A. It allows us to extract the
(t)it .
time-domain phase fluctuations δθr1 (t), and directly calculate the variance hδθr1

Loop delay Differential delay

Phase error std. dev.
1/2

σr1 = hδθr1
(t)it

(mrad)

Locking efficiency
η = 1+σ

τd (m)

τr1 (cm)

99.8%

279

92.8%

99.4%

315

91.0%

Table 6.1: Measured OPLL phase error standard deviation and phase-locking efficiency for different values of the loop delay τd and the differential delay τr1

123

Reference arm 1
Delay τd
AOFS 1

Amplifier
arm 1

PD 1

Delay τd-τr1

Mixer
Gain

Offset
Oscillator

Reference arm 2
Delay τd
Amplifier
arm 2
AOFS 2

Sweptfrequency
laser

μ-lens
array
PD 2

Delay τd-τr2
Mixer
Gain

Offset
Oscillator

Camera

124

6.4

Coherent Combining of Chirped Optical
Waves

6.4.1

Passive-Fiber CBC Experiment

125

(a) OPLL spectra.

(b) I/Q-demodulated OPLL phases.

Figure 6.9: Characterization of the two heterodyne OPLLs in the locked state. τd ≈
0 m, τr1 = τr2 ≈ 0 cm.

(a) OPLL spectra.

(b) I/Q-demodulated OPLL phases.

Figure 6.10: Characterization of the two heterodyne OPLLs in the locked state.
τd ≈ 18 m, τr1 = τr2 ≈ 0 cm.

(a) OPLL spectra.

(b) I/Q-demodulated OPLL phases.

Figure 6.11: Characterization of the two heterodyne OPLLs in the locked state.
τd ≈ 18 m, τr1 = τr2 ≈ 32 cm.

126
Unlocked

Locked, in phase

Locked, out of phase
250
150
100
50

Power (a.u.)

200

Unlocked
Locked, in phase
Locked, out of phase 250

150
100

Power (a.u.)

200

Position (a.u.)

127
since the combination of coherent signal gain and incoherent phase errors leads to an
increasing interferometric visibility with increasing number of array elements [105].

6.4.2

Combining Phase Error in a Heterodyne Combining
Experiment

(6.21)

Plugging in equation (6.20), we arrive at
δθ12 (ω) =

(ω)
θ12
θL (ω)e−jωτd
− jωτ12
1 + K(ω)
1 + K(ω)

(6.22)

is the relative path-length fluctuation of the two amplifier arms. In deriving
where θ12

6.4.3

Free-Space Beam Combining of Erbium-Doped Fiber
Amplifiers

A schematic of the dual-channel chirped-seed amplifier (CSA) CBC experiment is
shown in figure 6.13. An optoelectronic SFL based on a 1550 nm VCSEL is linearly

128

Reference

50/50

Channel 1 50/50 Channel 2

AOFS 1

PM 1

Swept-frequency laser

Fiber
amp.

From
OPLL 1

PM 2

AOFS 2

Fiber
amp.

From
OPLL 2

To
OPLL 1

Reflector
PD 1

Reflector
PD 2

Camera

Beam dump

To
OPLL 2

Figure 6.13: Schematic diagram of the dual-channel CSA coherent-combining experiment. PD: Photodetector, PM: Back-scattered power monitor

Channel 2

Locked

Channel 1
Channel 2
Locked

250
200
150
100
50

250
200
150
100

Power (a.u.)

Channel 1

Power (a.u.)

130

Position (a.u.)

θos,12 = π/2

θos,12 = π

θos,12 = 3π/2

250
200
150
100
50
250
200
150
100
50

Power (a.u.)

θos,12 = 0

Power (a.u.)

Figure 6.14: Far-field intensity distributions of the individual channels and the locked
aperture. τr1 = −19 mm, and τr2 = 1 mm

Position (a.u.)

Figure 6.15: Steering of the combined beam through emitter phase control. θos,12 is
the relative DDS phase.

131
θos,12 = 0

θos,12 = π/2

θos,12 = π

θos,12 = 3π/2
3π/2

Channel 1
Channel 2

π/2

1 0

1 0
Time (ms)

1 0

Figure 6.16: I/Q-demodulated phase differences between the amplifier channels and
the reference. θos,12 is the relative DDS phase.
(6.21). The standard deviations σxy =

2 (t)i of all three phase errors, along
hδθxy

6.5

Summary

132
We have demonstrated heterodyne phase-locking of optical waves with a chirp
rate of 5 × 1014 Hz/sec at 1550 nm, achieving a loop bandwidth of 60 kHz and a

Case Differential delay (mm)a Phase error (mrad) Fringe visibility

τr1

τr2

τ12

σr1

σr2

σ12

V = e−σ12 /2

−19

118 79.3

160

98.7%

110

450

340

184 531

428

91.3%

−118 220

338

150 273

410

92.0%

These are fiber lengths corresponding to the time delays between the different
paths. Actual mismatches have both free-space and fiber components.

Table 6.2: OPLL phase errors and phase-noise-limited fringe visibilities in the dualchannel active CBC experiment

133

Chapter 7
Conclusion
7.1

Summary of the Thesis

7.1.1

Development of the Optoelectronic SFL

7.1.2

Ranging and 3-D Imaging Applications

7.1.2.1

MS-FMCW Reflectometry and Stitching

The Tomographic Imaging Camera

136

7.1.3

Phase-Locking and CBC of Chirped Optical Waves

7.2

Current and Future Work

(a)

(b)

(c)

(d)

138

(a)

(b)

Figure 7.2: Schematic of the hybrid Si/III-V high-coherence semiconductor laser.
(a) Side-view cross section. (b) Top-view of the laser and the modulated-bandgap
resonator

139

140

(A.1)

(A.2)

yq (t) = h(t) ? [y(t) cos ω0 t]
A(t)
A(t)
= h(t) ?
sin θ(t) +
sin [2ω0 t + θ(t)] ,

(A.3)

where h(t) is the impulse response of the low-pass filter, and ‘?’ denotes the convolution operation. The filter is designed to average out the sum frequency terms at

141
frequency 2ω0 , while retaining the difference frequency terms at DC, yielding
yi (t) =

A(t)
cos θ(t), and

(A.4)

A(t)
sin θ(t).

(A.5)

yq (t) =

The amplitude and phase modulations are recovered using
A(t) = 2 yi2 (t) + yq2 (t), and

(A.6)

θ(t) = atan2 [yq (t), yi (t)] ,

(A.7)

where atan2(yq , yi ) is the four-quadrant inverse tangent function defined below.

−1 yq
tan
yi
−1 yq
+π
tan
yi
 tan−1 yq − π
yi
atan2(yq , yi ) ≡
pi
+2
− pi2
undefined

yi > 0
yq ≥ 0, yi < 0
yq < 0, yi < 0
yq > 0, yi = 0
yq < 0, yi = 0
yq = 0, yi = 0

(A.8)

142

I ∝ h|1 + exp [jθ12 (r ) + jδθ12 (t)]|2 it = 2 + 2e−σ12 /2 cos θ12 (r ),

(B.1)

where θ12 (r ) is the mean phase difference between the beams at the point r and hit
denotes an average over time. We assumed that δθ12 (t) is a zero-mean Gaussian

random variable with variance σ12
, so that hejδθ12 (t) it = e−σ12 /2 . Intensity extrema

are found at points of constructive and destructive interference, with cos θ12 (r ) = ±1.
The fringe visibility is therefore given by:

V ≡ (Imax − Imin )/(Imax + Imin ) = e−σ12 /2

(B.2)

143