studies of slow light with applications in optical beam steering

Studies of Slow Light With Applications in

Optical Beam Steering

By

Aaron Schweinsberg

Submitted in Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy

Supervised by

Professor Robert W. Boyd

The Institute of Optics

Arts, Sciences and Engineering

Edmund A. Hajim School of Engineering and Applied Sciences

University of Rochester

Rochester, NY

2012

ii

Dedicated to my parents

iii

Biographical Sketch

Aaron Schweinsberg was born in Lewisburg, Pennsylvania. He attended Cornell

University from 1996 to 2000, graduating with a B.S. degree in applied and engineering

physics. Later in 2000, he began the PhD program at the Institute of Optics at the

University of Rochester. He joined the research group of Robert Boyd, and under his

supervision has conducted research on nonlinear optical materials, optical resonators and

biosensors, fiber optics, and slow and fast light.

Publications

J. E. Vornehm, A. Schweinsberg, Z. Shi, D. J. Gauthier, and R. W. Boyd, “Phase locking of multiple optical fiber channels for a slow-light-enabled laser radar system” (in preparation for submission). S. Jarabo, A. Schweinsberg, N. N. Lepshkin, M. S. Bigelow, R. W. Boyd, “Theoretical model for superluminal and slow light in Erbium doped fibers: enhancement of the frequency response by pump modulation” Applied Physics B, 107, 717–732 (2012). A. Schweinsberg, Z. Shi, J. E. Vornehm Jr., R. W. Boyd, “A slow-light laser radar system with two-dimensional scanning,” Optics Letters, 37, 329–331 (2012). A. Schweinsberg, J. Kuper, R. W. Boyd, “Loss of spatial coherence and limiting of focal plane intensity by small-scale laser-beam filamentation,” Physical Review A, 84, 053837 (2011). A. Schweinsberg, Z. Shi, J. E. Vornehm Jr., R. W. Boyd, “Demonstration of a slow-light laser radar,” Optics Express, 19, 15760–15769 (2011). Z. Shi, A. Schweinsberg, J. E. Vornehm Jr., M. A. M. Gamez, R.W. Boyd, “Low distortion, continuously tunable, positive and negative time delays by slow and fast light using stimulated Brillouin scattering,” Physics Letters A, 374, 4071–4074 (2010).

iv

H. Shin, A. Schweinsberg, R. W. Boyd, “Reducing pulse distortion in fast-light pulse propagation through an Erbium-doped fiber amplifier using a mutually incoherent background field,” Optics Communications, 282, 2085–2087 (2009). A. Schweinsberg, S. Hocdé, N. N. Lepeshkin, R. W. Boyd, C Chase, J. E. Fajardo, “An environmental sensor based on an integrated optical whispering gallery mode disk resonator,” Sensors and Actuators B – Chemical, 123, 727–732 (2007). H. Shin, A. Schweinsberg, G. Gehring, K. Schwertz, H. J. Chang, R. W. Boyd, Q. H. Park, D. J. Gauthier, “Reducing pulse distortion in fast-light pulse propagation through an Erbium-doped fiber amplifier,” Optics Letters, 32, 906–908 (2007). R.M. Camacho, M. V. Pack, J. C. Howell, A. Schweinsberg, R. W. Boyd, “Wide-bandwidth, tunable, multiple-pulse-width optical delays using slow light in cesium vapor,” Physical Review Letters, 98, 153601 (2007). D. D. Smith, N. N. Lepeshkin, A. Schweinsberg, G. Gehring, R. W. Boyd, Q. H. Park, H. Chang, D. J. Jackson, “Coupled-resonator-induced transparency in a fiber system,” Optics Communications, 264, 163–168 (2006). G. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, R. W. Boyd, “Observation of backward pulse propagation through a medium with a negative group velocity,” Science, 312, 895–897 (2006). A. Schweinsberg, N. N. Lepeshkin, M. S. Bigelow, R. W. Boyd, “Observation of superluminal and slow light propagation in Erbium-doped optical fiber,” Europhysics Letters, 73, 218–224 (2006). Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. M. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical fiber,” Physical Review Letters, 94, 153902 (2005). N. N. Lepeshkin, A. Schweinsberg, G. Piredda, R. S. Bennink, R. W. Boyd, “Enhanced nonlinear optical response of one-dimensional metal-dielectric photonic crystals,” Physical Review Letters, 93, 123902 (2004). J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, D. J. Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE Journal of Quantum Electronics, 40, 726–730 (2004). J. E. Heebner, N. N. Lepeshkin, A. Schweinsberg, G. W. Wicks, R. W. Boyd, R. Grover, P. T. Ho, “Enhanced linear and nonlinear optical phase response of AlGaAs microring resonators,” Optics Letters, 29, 769–771 (2004).

v

R. W. Boyd, J. E. Heebner, N. N. Lepeshkin, Q. H. Park, A. Schweinsberg, G. W. Wicks, A. S. Baca, J. E. Fajardo, R. R. Hancock, M. A. Lewis, R. M. Boysel, M. Quesada, R. Welty, A. R. Blier, J. Treichler, R. E. Slusher, “Nanofabrication of optical structures and devices for photonics and biophotonics,” Journal of Modern Optics, 50, 2543–2550 (2003).

Presentations A. Schweinsberg, Z. Shi, J. E. Vornehm, and R. Boyd, “Demonstration of a Slow-Light Laser Radar with Two-Dimensional Scanning,” in Slow and Fast Light, (Optical Society of America, 2011), paper SLMB2. A. Schweinsberg, R. M. Camacho, M. V. Pack, R. W. Boyd, and J. C. Howell, “Tunable Slow Light in Cesium Vapor,” in Frontiers in Optics, (Optical Society of America, 2006), paper FWS5. G. Piredda, A. Schweinsberg, and R. W. Boyd, “Room Temperature Slow Light with 17 GHz Bandwidth in Semiconductor Quantum Dots,” in Slow and Fast Light, (Optical Society of America, 2006), paper MC6. A. Schweinsberg, S. Hocde, N. N. Lepeshkin, R. W. Boyd, C. Chase, and J. E. Fajardo, “An integrated optical disk resonator for sensing applications,” SPIE Optics East, Boston, MA (2005) [Invited]. A. Schweinsberg, S. Hocde, N. N. Lepeshkin, R. W. Boyd, C. Chase, and J. E. Fajardo, “Demonstration of an environmental sensor based on an integrated optical whispering gallery mode resonator,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications Systems Technologies, (Optical Society of America, 2005), paper CTuH1. A. Schweinsberg, M. S. Bigelow, N. N. Lepeshkin, R. W. Boyd, and S. Jarabo, “Fast and slow light propagation in Erbium-doped fiber,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2004), paper FTuJ4.

vi

Acknowledgements

This thesis has only been possible with the insight and aid of many. First, I would like

to thank my advisor Dr. Robert Boyd, not only for his personal guidance, patience, and

support, but for the environment he has created with his research group. In his group I

have been able to work on a wide array of fascinating projects (several of which could

not really be collected into a thesis on slow light), and perhaps more importantly, I have

been able work with a wide array of insightful and creative people, including both fellow

group members and collaborators from elsewhere.

Those many co-workers also have my thanks, as all of my work has been done with

the help of my fellow researchers. I would especially like to thank John Heebner for

taking me under his wing when I was a beginning grad student and teaching me the ways

of experimental fiber-optics. I thank Yoshi Okawachi, and Ryan Camacho for their key

role in a couple of productive out-of-group collaborations. Many of my other

collaborators over the years are in need of special regards, including Zhimin Shi, Joe

Vornehm, Heedeuk Shin, George Gehring, Nick Lepeshkin, Sandrine Hocdé, David

Smith, Matt Bigelow, and Vin Wong. I also thank the many other members of the Boyd

group I have known over the years.

Finally, I would like to thank friends in Rochester and my family for their support

throughout my long graduate student career.

vii

Abstract

This thesis presents a variety of work on the topic of slow light. It contains research

that surveys the many different physical systems capable of producing slow light, and

culminates in a project that demonstrates a novel application of slow light related to all-

optical phased-array beam steering. It is first shown that high-bandwidth pulses can be

substantially delayed with minimal absorption and broadening when transmitted through

a cesium vapor cell and tuned between the Cs D2 hyperfine resonances. Next,

possibilities for the production of slow light in fiber resonator structures are explored.

The delay of a pulse on transmission through a low-finesse fiber ring is measured, and

coupled-resonator-induced transparency, a phenomenon analogous to the

electromagnetically-induced transparency that is commonly used to create slow light, is

demonstrated in a fiber-based coupled-resonator structure. It is then shown that slow light

can be created using stimulated Brillouin scattering in an optical fiber, and how specific

control of the pump beam’s intensity and frequency spectrum can enable slowing of high

bandwidth pulses as well as pulse delays and advancements that are tunable over a wide

range. A series of experiments that explore the phenomenon of pulse delays and

advancements using saturable media are then presented, including work with Erbium-

doped fiber and PbS quantum dots. Finally, it is shown how slow light can be applied to

the problem of temporal pulse mismatch in a pulsed, scanning, multi-aperture laser radar.

viii

Contributors and Funding Sources

All of the research in this thesis was performed under the guidance of my thesis

advisor Prof. Robert W. Boyd. In addition, I collaborated with many other students, post-

docs, and professors on the work presented herein.

The research in chapter 2 was published in Physical Review Letters. [R.M. Camacho,

M. V. Pack, J. C. Howell, A. Schweinsberg, R. W. Boyd, “Wide-bandwidth, tunable,

multiple-pulse-width optical delays using slow light in cesium vapor,” Physical Review

Letters, 98, 153601 (2007).] Ryan Camacho was the primary author and I worked with

him on collection of the data involving 275 ps pulses and assisted with the writing of the

manuscript. The theoretical work was done by Michael Pack and Ryan Camacho, and the

work as a whole was conducted under the supervision of Prof. John Howell. It was

supported by the DARPA/DSO Slow Light Program, the National Science Foundation,

and the Research Corporation.

The work discussed in chapter 3, section 1, was published in the IEEE Journal of

Quantum Electronics. [J. E. Heebner, V. Wong, A. Schweinsberg, R. W. Boyd, D. J.

Jackson, “Optical transmission characteristics of fiber ring resonators,” IEEE Journal of

Quantum Electronics, 40, 726–730 (2004).] I assisted the primary author, John Heebner

with collection of the data. The work was sponsored by the National Reconnaissance

Office, DARPA, and the State of New York NYSTAR Program as part of the Alliance

for Nanomedical Technologies.

ix

The work in chapter 3, section 2, was included in a publication in Optics

Communications. [D. D. Smith, N. N. Lepeshkin, A. Schweinsberg, G. Gehring, R. W.

Boyd, Q. H. Park, H. Chang, D. J. Jackson, “Coupled-resonator-induced transparency in

a fiber system,” Optics Communications, 264, 163–168 (2006).] The concept and theory

behind the paper were developed by the first author David Smith and other co-authors. I

collected the data for the experimental section of the paper with assistance from David

Smith, Nick Lepeshkin, and Deborah Jackson. The FDTD modeling was done by George

Gehring. The work was supported by the NASA Marshall Space Flight Center

Institutional Research and Development Grants CDF03-17 and CDDF04-08, and the

United Negro College Fund Office of Special Programs.

The research in chapter 4, section 1, was published in Physical Review Letters. [Y.

Okawachi, M. S. Bigelow, J. E. Sharping, Z. M. Zhu, A. Schweinsberg, D. J. Gauthier, R.

W. Boyd, A. L. Gaeta, “Tunable all-optical delays via Brillouin slow light in an optical

fiber,” Physical Review Letters, 94, 153902 (2005).] I assisted the primary author, Yoshi

Okawachi and other co-authors with collection of the data. The work was supervised by

Prof. Alex Gaeta. Financial support was provided by the DARPA/DSO Slow-Light

Program, and the Center for Nanoscale Systems under Grant No. EEC-0117770.

The work in chapter 4, section 2 was published in Physics Letters A. [Z. Shi, A.

Schweinsberg, J. E. Vornehm Jr., M. A. M. Gamez, R.W. Boyd, “Low distortion,

continuously tunable, positive and negative time delays by slow and fast light using

stimulated Brillouin scattering,” Physics Letters A, 374, 4071–4074 (2010).] I worked

closely with the first author, Zhimin Shi, and other co-authors in collecting the data and

x

in assisting with the production of the manuscript. The work was supported by the

DARPA/DSO Slow Light program and by the NSF.

The research presented in chapter 5, section 1, was published in Europhysics Letters.

[A. Schweinsberg, N. N. Lepeshkin, M. S. Bigelow, R. W. Boyd, “Observation of

superluminal and slow light propagation in Erbium-doped optical fiber,” Europhysics

Letters, 73, 218–224 (2006).] I was the primary author of this work, though I was aided

by my co-authors in data collection and in preparing the manuscript.

The work in chapter 5, section 2 was published in Science. [G. Gehring, A.

Schweinsberg, C. Barsi, N. Kostinski, R. W. Boyd, “Observation of backward pulse

propagation through a medium with a negative group velocity,” Science, 312, 895–897

(2006).] I collected the data for this experiment with the primary author, George Gehring,

and other co-authors and also assisted in the preparation of the manuscript. The work was

supported by NSF grant ECS-0355206 and by DARPA/DSO.

The data presented in chapter 5, section 3, comes from work that was published in

Optics Letters and in Optics Communications. [H. Shin, A. Schweinsberg, G. Gehring, K.

Schwertz, H. J. Chang, R. W. Boyd, Q. H. Park, D. J. Gauthier, “Reducing pulse

distortion in fast-light pulse propagation through an Erbium-doped fiber amplifier,”

Optics Letters, 32, 906–908 (2007).] [H. Shin, A. Schweinsberg, R. W. Boyd, “Reducing

pulse distortion in fast-light pulse propagation through an Erbium-doped fiber amplifier

using a mutually incoherent background field,” Optics Communications, 282, 2085–2087

(2009).] I assisted the primary author, Heedeuk Shin, with the experimental setup and

also with writing the manuscripts. The numerical modeling was done by Heedeuk, along

xi

with the creation of the figures. The work in this section was supported by the Korea

Research Foundation, NSF grant ECS-0355206, and the DARPA/DSO Slow Light

Program.

The work described in chapter 5, section 4, was published only in a conference paper,

which I presented at the OSA’s Slow and Fast Light conference in 2006. [G. Piredda, A.

Schweinsberg, and R. W. Boyd, “Room Temperature Slow Light with 17 GHz Bandwidth

in Semiconductor Quantum Dots,” in Slow and Fast Light, (Optical Society of America,

2006), paper MC6.] Giovanni Piredda was the first author of this paper. He and I worked

together closely on all aspects of the project. It was supported by the DARPA/DSO Slow

Light Program.

Much of the SLIDAR work presented in chapter 6 has been published in two papers

(with a third in preparation). [A. Schweinsberg, Z. Shi, J. E. Vornehm Jr., R. W. Boyd,

“Demonstration of a slow-light laser radar,” Optics Express, 19, 15760–15769 (2011).]

[A. Schweinsberg, Z. Shi, J. E. Vornehm Jr., R. W. Boyd, “A slow-light laser radar

system with two-dimensional scanning,” Optics Letters, 37, 329–331 (2012).] I was the

first author on these papers but all aspects of the work were the result of a close

collaboration between the co-authors. The design and characterization of the phase

locking circuit was the work of Joe Vornehm, who is the primary author of the yet

unpublished third paper. [J. E. Vornehm, A. Schweinsberg, Z. Shi, D. J. Gauthier, and R.

W. Boyd, “Phase locking of multiple optical fiber channels for a slow-light-enabled laser

radar system” (in preparation for submission).] Joe produced figure 6.6, which illustrates

xii

the circuit’s inner workings. The work in this section was supported financially by the

DARPA/DSO Slow Light Program

xiii

Contents

Acknowledgements vi

Abstract vii

Contributors and Funding Sources viii

List of Figures xi

1 Introduction 1

1.1 The group velocity................................................................................... 2

1.2 Figures of merit for slow light.................................................................. 6

1.3 Material slow light................................................................................... 7

1.4 Structural slow light................................................................................. 8

1.5 Scattering slow light ................................................................................ 8

1.6 Pulse delay in saturable media ................................................................. 9

1.7 Dispersive delay .................................................................................... 10

1.8 Applications of slow light ...................................................................... 11

2 Material Slow Light 12

2.1 Wide-bandwidth slow light in cesium vapor........................................... 12

3 Structural Slow Light 22

3.1 Slow light in a ring resonator ................................................................. 23

3.2 Coupled-resonator-induced transparency in a fiber system ..................... 28

xiv

4 Scattering Slow Light 35

4.1 Demonstration of stimulated Brillouin scattering slow light ................... 35

4.2 Continuously tunable slow and fast light using SBS............................... 43

5 Pulse Delay in Saturable Media 52

5.1 Slow and fast light propagation in Erbium-doped fiber........................... 53

5.2 Observation of backwards pulse propagation ......................................... 64

5.3 Reducing distortion in saturable gain pulse advancement ....................... 70

5.4 Saturable absorption pulse delay with 17 GHz bandwidth ...................... 75

6 Slow Light Detection and Ranging (SLIDAR) 80

6.1 Background and introduction ................................................................. 80

6.2 Theory ................................................................................................... 83

6.3 Setup ..................................................................................................... 86

6.4 Phase control ......................................................................................... 88

6.5 Slow light demonstration and system tests ............................................. 95

6.6 A two-dimensional SLIDAR.................................................................103

7 Conclusion 113

Bibliography 116

xv

List of Figures

Figure Title Page

1.1 Lorentzian gain line ................................................................................. 5

2.1 Transmission spectrum of cesium D2 hyperfine resonances.................... 14

2.2 Experimental setup for observing slow light in Cs vapor........................ 16

2.3 Temperature-dependent pulse delays in Cs vapor................................... 18

2.4 Delay and broadening of slow 740 ps pulses in Cs vapor ....................... 19

2.5 Rapidly-tunable pulse delays in Cs......................................................... 20

2.6 Plot of the turn-on and turn-off times of switchable slow light ............... 20

3.1 Geometry of a fiber ring resonator ......................................................... 24

3.2 Theory curves for ring resonator transmission, phase shift, and delay..... 25

3.3 Optical pulse delay in a ring resonator ................................................... 27

3.4 Diagram of two coupled optical ring resonators ..................................... 29

3.5 Experimental setup for observing CRIT ................................................. 31

3.6 Mode-splitting in two coupled resonators............................................... 32

3.7 Close up of CRIT, with FDTD simulations ............................................ 33

4.1 Illustration of SBS gain spectrum in the vicinity of a strong pump ......... 37

4.2 Experimental setup for observing SBS slow light................................... 40

xvi

4.3 Observation of slow light in optical fiber using SBS .............................. 41

4.4 Delay vs. gain for SBS slow light........................................................... 42

4.5 Illustration of how pump modulations affects the shape of SBS gain...... 46

4.6 Schematic of bi-directionally tunable SBS slow light setup.................... 47

4.7 Measured SBS gain for slow and fast light............................................. 48

4.8 Pulse delays and advancement as a function of SBS pump power .......... 50

4.9 SBS delayed and advanced pulse shape compared to a reference............ 50

5.1 Setup for observing pulse advancements and delays in a EDF................ 57

5.2 Frequency and pump power dependence of slow and fast light in EDF .. 58

5.3 Numerical model of slow and fast light in EDF...................................... 59

5.4 Data for modulation gain vs. modulation frequency in an EDFA............ 60

5.5 Tuning an EDFA from slow to fast light ................................................ 61

5.6 Power dependence of slow light in an EDFA ......................................... 62

5.7 Fractional advancement and delay of zero-background pulses in EDF.... 63

5.8 Setup for observing “backwards” pulse propagation in EDF .................. 66

5.9 Input and output waveforms for fast light in EDF................................... 67

5.10 Time evolution of a backwards propagating pulse in EDF...................... 69

5.11 Demonstration of pulse-distortion minimization in EDF ........................ 73

5.12 Incoherent background effects for slow light in EDF.............................. 75

5.13 Absorption spectrum for PbS quantum dots ........................................... 76

5.14 Setup to measure pulse delay through a quantum dot sample.................. 77

5.15 Measured pulse delay through a QD sample........................................... 78

xvii

6.1 Illustration of enhanced resolution with a wide emitter........................... 85

6.2 Conceptual diagram of a slider system ................................................... 86

6.3 Partial schematic of a SLIDAR .............................................................. 87

6.4 Circuit diagram for a simple phase lock ................................................. 88

6.5 Results with a simple phase lock............................................................ 89

6.6 Diagram of the snap-back circuit............................................................ 92

6.7 Demonstration of 2-channel phase lock with snap-back ......................... 93

6.8 3-channel phase locking......................................................................... 94

6.9 Measurements of dispersive delay for 3 channels ................................... 96

6.10 Pulse shapes through dispersive delay.................................................... 96

6.11 SLIDAR setup for system tests .............................................................. 98

6.12 Eye diagrams showing phase locking and delay ..................................... 99

6.13 Distance detection using the SLIDAR...................................................101

6.14 Spatial resolution measurements of SLIDAR interference patterns........102

6.15 Diagram of the 2-D SLIDAR ................................................................105

6.16 Measured SBS gain spectrum................................................................108

6.17 Tests showing SLIDAR simulated 2-D beam steering...........................109

6.18 Diagram of possible large-scale SLIDAR..............................................111

1 INTRODUCTION 1

Chapter 1

Introduction

“Slow light” is the term given to the phenomenon where the group velocity of a pulse

traveling in a given medium is substantially lower than the phase velocity of

electromagnetic waves at the pulse’s component frequencies. Because the effect

produced is that a light pulse exits an optical medium at a time later than would be

predicted by simple consideration of the medium’s length and phase index, the

terminology has sometimes been extended to encompass phenomena that produce tunable

pulse delays, even if they do not do so strictly via the mechanism of group velocity

adjustment. Slow light techniques can be either material, where dispersive spectral

features of homogeneous materials are exploited, or structural, where optical filters such

as integrated optical devices or etalons are used to produce similar dispersive effects.

The reasons for studying slow light are many. At the most fundamental level, slow

light is the essentially the study of how light pulses interact with materials, a subject

inherently of interest to optical physicists. Additionally, there are many possible

applications that have been proposed for slow light effects. Several possibilities within

the field of communications include proposals for optical pulse re-centering,

synchronization of time-division-multiplexers, and optical correlation [1]. Additionally, it

1 INTRODUCTION 2

has been demonstrated that slow light can be used to enhance the sensitivity of spectral

interferometers [2,3].

Chapters 2-5 of this dissertation will be an experimental survey of different

techniques used for producing slow light, while chapter 6 will focus on a new application,

the use of slow light for synchronizing the multiple output channels of a phased-array

beam-steering device.

1.1 The group velocity

In a dispersive medium, as the different frequency components of the field propagate

at different velocities, the envelope profile produced by their interference may vary

drastically, possibly in non-intuitive ways. For a well-defined pulse of light with its

frequency bandwidth tightly centered at a frequency ω0, we can describe the effects of

dispersion with a Taylor series expansion of the wavenumber about that frequency.

k !( ) = k0 +dkd! !0

! "!0( ) + 12d 2kd! 2

!0

! "!0( )2 + ... (1.1)

Here, the phase velocities of the component frequencies will be given by ω/k. If higher

order terms are small, the pulse will travel unchanged with the velocity of its peak given

by the “group velocity” given in the first order term, defined as

1vg

!dkd" "0

=n0 +"0

dnd"

c!ngc

, (1.2)

1 INTRODUCTION 3

where we have also defined a “group index” ng, as c/vg in a manner analogous to the

familiar phase index n. It is significant that the group index has a term that scales linearly

with the dispersion, because this means that in spectral regions of very large normal or

anomalous dispersion, the group velocity can be extremely small, or can even go

negative.

The above expression informs us that to find or create instances of extreme group

velocity, we must find regions of large dispersion, but we are further aided by

considering the mathematical relationship known as the Kramers-Kronig condition. This

condition states that if two functions constitute the real and imaginary parts of a causal

response function, they must be mathematically related by the integral transform pair:

! "( ) = 1+ 1

#P $! $"( )

$" %"%&

&

' d $" ,

$! "( ) = %1#P

! $"( ) %1$" %"%&

&

' d $" (1.3a,1.3b)

where we have used ε and ε’, the real and imaginary parts of the dielectric constant as

example functions. Similarly, the real and imaginary parts of the refractive index,

corresponding to phase advancement and absorption/gain, are related in such a way. An

intuitive argument for how this relation can be understood is presented by Toll [4]

Intuitively, this can be understood by imagining a signal that begins only after some

appointed time t0. This signal will be composed of many Fourier components, each of

extending from –∞ to ∞, with amplitudes and phases such that they sum to zero for all

time before t0. If we absorb exactly one frequency component perfectly without adjusting

any phases, it is akin to a time-domain mathematical subtraction of that frequency

1 INTRODUCTION 4

component from the signal, which, since each component is a sinusoid that exists for all

time, would leave a sinusoidal signal in the region of t<t0. Since this result is non-causal,

we understand that for a response function to be causal, it must influence the phase

components of the neighboring frequencies in such a way as to cancel the aforementioned

effect in the t<t0 region.

Let us consider the case of a Lorentzian gain line

g !( ) = g01+ ! 2 / " 2 , (1.4)

where δ is the detuning from the center frequency ω0, g0 is the gain at ω0, and γ is the

width of the resonance. Since the phase index n and the gain are related to the real and

imaginary parts of the complex refractive index as given by ñ(ω) = n(ω) – i g(ω), we can

use equation 1.3a to find the phase index corresponding to the Lorentzian gain given

above:

n !( ) = n0 +g0c2"0

#$%

&'(

! / )1+ ! 2 / ) 2 (1.5)

And differentiating the phase index with respect to the detuning δ helps us find the

associated group index as defined in equation 1.2:

ng !( ) = n !( ) + g0c2"

#$%

&'(1) ! 2 / " 2

1+ ! 2 / " 2( )2[5]. (1.6)

These expressions are illustrated in figure 1.1.

What we can see from this example is that an isolated Lorentzian line creates a region

of modified group index in its near vicinity. A gain line will produce slow light, whereas

1 INTRODUCTION 5

an absorption line will produce a low group index, or “fast” light. One important point to

notice is that the bandwidth of the slow light feature is limited, effectively smaller than

the linewidth of the original resonance, and this limit is significant in that it also

represents a lower bound on the shortness of pulses that can be slowed (or advanced) in a

given system. Another point is that the variation of the absorption spectrum with

frequency appears in the form of the group index, but a frequency-independent offset

term would not.

Figure 1.1: Illustration of a Lorentzian gain line and the real part of the refractive index and group index profile it implies [5].

As a result, we need not look for slow light exclusively at the center of a gain line; it

would work equally well to work in a dip in an absorption line, or in the space between

two absorption lines. These possibilities also have the advantage that the signal pulses

being delayed are not distorted by excessive gain, a point that will be explained further in

later chapters.

1 INTRODUCTION 6

1.2 Figures of merit for slow light

There are many figures of merit used to describe slow light systems. The most

obvious is the group index ng or the group delay vg, defined above in equation 1.2. The

operating bandwidth is another important consideration, especially in terms of choosing

the right slow light technique for a desired application. It is especially common for

systems to be evaluated by combining these parameters in terms of the fractional delay,

or the delay-bandwidth product. This is often defined by dividing the delay of the pulse

peak (relative to the delay with the slow-light effect “off”) by the full-width-at-half-

maximum (FWHM) of the input pulse

Tfrac =!T"

(1.7)

although sometimes twice the FWHM (perhaps corresponding to the width of a

telecommunications bit slot), or the width of the output pulse are used. The fractional

delay is not only the most technically difficult parameter to expand, but it is usually the

most significant for applications as well.

Because slow light is almost always researched as a tunable effect, another important

parameter is the tuning speed, also sometimes called the reconfiguration rate. Slow light

effects usually distort transmitted pulses to some degree, so often the broadening B is

quoted, usually expressed as a fraction of the input pulse width. As broadening is not

nearly a complete description of pulse distortion, sometimes more complicated metrics

are used, one of which will be discussed in section 5.3.

1 INTRODUCTION 7

Slow light can also be limited by its pulse accommodation. For example, some

systems can provide huge tunable delays to a few pulses, but fundamentally cannot

handle more than this number. “Stopped light” effects in atomic media and tunable

resonators often have this trait.

Additionally, slow light devices can be evaluated according to traditional technology

metrics such as physical size, cost, and power consumption. Along this line, the early

prominence of slow light using electromagnetically-induced transparency in a Bose-

Einstein condensate and later in a cold doped glass led to an emphasis on the operating

temperature of slow light [6,7]. Few reports on slow light will explicitly cover all of these

parameters (including the reports in this thesis!), but they are very useful to keep in mind

when reviewing new research.

1.3 Material slow light

Here, we define the term “material slow light” to be applicable in situations where

the velocity of light pulses in the medium is described completely by a frequency-

dependent refractive index that is spatially uniform over the optical path, and where the

dispersive term in the group index is significant [8]. As noted in the previous section, the

change in a medium’s group index in the vicinity of a resonance scales with the strength

of the absorption coefficient. This fact implies that resonances in atomic vapor, with

extremely strong and narrow linewidths, are a promising system for extreme group

indices. Furthermore, by tuning the signal wavelength between two nearby resonances,

severe absorption can be avoided.

1 INTRODUCTION 8

In addition to materials with natural resonances, slow light has been observed in

materials where optical pumping or coherent effects are used to create the desired

dispersive features. Electromagnetically induced transparency (EIT), and Raman

absorption based slow light are examples that, at least in principle, have the properties of

material slow light [6,9].

1.4 Structural slow light

By structural slow light, we refer to situations where the apparent propagation speed

of light pulses is modified by structures with a spatially varying refractive index. Often,

these devices are analyzed as filters, in terms of their transmission and phase output

characteristics for CW input fields. Care must be taken, since these properties do not

always fully describe the optical behavior of the medium. Devices that have been used

for structural slow light include Fabry-Perot resonators, interference filters, distributed

Bragg gratings, ring and microsphere resonators, and photonic crystal waveguides [10–

15]. While slow light produced in this way does not have all the properties of material

slow light, such as a proportional effect on the energy density and spatial extent of the

pulse, structural slow light does have some advantages, such as the potential ease of

physical integration into developing applications, and the potential to enhance nonlinear

optical interactions by enhancing the electric field strength of the pulse (a trait not shared

by material slow light) [8].

1.5 Scattering slow light

1 INTRODUCTION 9

Some types of stimulated optical scattering can produce the transmission spectrum

necessary to obtain slow light. Stimulated Brillouin scattering (SBS) is a stimulated

scattering process that is often used for this purpose, especially in optical fibers. The

effect can be described as a coupling of two optical fields, a laser field (pump), a Stokes

field, and an acoustic wave in the medium that is created through the effect of

electrostriction. In the field of slow light, it is most commonly used in optical fibers,

where a field at the Stokes frequency experiences resonant gain, taking energy from the

counterpropagating pump field. Slow light created by SBS is covered in chapter 4.

Stimulated Raman scattering (SRS) is also used to produce slow light [16]. In this

effect, energy is scattered into the Stokes field from the pump via an interaction with the

vibrational energy states in the medium. Because of the high bandwidth of the Raman

gain feature, it does not produce a very large group index, but it can be used to delay sub-

ps pulses. Slow light via SRS is not covered further in this thesis.

1.6 Pulse delay in saturable media

A very simple technique that can produce relative advancement or delay of a light

pulse is that of saturable gain or absorption. Here, the effect of saturation creates a time-

dependent response that can advance or delay a pulse essentially via reshaping, but often

while maintaining the original pulse shape to a surprisingly good degree. For example, in

the case of absorption, the front edge of the pulse might contain enough energy to

saturate the absorption response, allowing the back half to travel through the medium

relatively unattenuated, resulting in a pulse that appears retarded in time. The reverse is

1 INTRODUCTION 10

true in the case of saturable gain. In this case of an optical gain medium, we can

dynamically tune the degree of advancement or delay by varying the strength of

pumping. Research on pulse delay and advancement in media with saturable absorption

and saturable gain is presented in chapter 5.

1.7 Dispersive delay

Dispersive delay is the term used to describe techniques that use the natural

frequency-dependence of an optical medium’s refractive index to produce tunable pulse

delay. If the laser source is tunable and fixing the exact frequency of the output light is

not required, all that is needed to produce tunable delay is the laser and a dispersive

medium. In other cases, it may be required to use a more complicated setup where the

wavelength of the optical source is shifted, the light is sent through the dispersive

element, and is then re-converted back to the original wavelength on the far side of the

medium. This technique is known as conversion-dispersion. Possible methods to perform

the re-conversion include four-wave mixing, cross-gain modulation, and self-phase

modulation with filtering.

While dispersive delay does not use the resonance-based group-delay tailoring of

conventional slow-light techniques, it has several advantages when considered for many

of the potential applications traditionally associated with slow light. For pulses of

nanosecond length or longer, the dispersive medium will cause little pulse distortion, and

fractional delays of several thousand have been reported [17].

1 INTRODUCTION 11

Our experimental work on dispersive delay will be presented not in its own chapter,

but in the applications section of chapter 6.

1.8 Applications of slow light

Slow light has always appeared to be a natural fit to applications in

telecommunications, because of the need to manipulate the timing of optical pulses.

While it appears that the high capacities of the buffers needed for optical pulse switching

in modern telecom networks is likely to exceed the capabilities of all-optical slow light

systems [18], there remain possible applications in the realm of retiming and recentering

of communications signals [19].

As the techniques for creating slow light become better known, applications have

branched out into other fields. Slow light devices have been considered for the purposes

of enhancing other optical nonlinearities [11]. It has been proposed that slow light is

useful to quantum information processing, as slowing or stopping light could be one way

to achieve the long storage times needed to perform quantum operations [20,21]. Shi et

al. demonstrated a novel use for slow light by showing that a slow light medium could be

used to enhance the sensitivity of spectral interferometer [3]. Finally, we note that in

laboratory settings slow light has been used to produce true time delay (TTD) to

synchronize the radio-frequency emitters of a phased-array radar system [22,23]. In

chapter 6, we will present a novel application related to this last idea, namely the use of

slow light to synchronize the optical frequency signals in coherently-combined phased

array beam-steering.

2 MATERIAL SLOW LIGHT 12

Chapter 2

Material Slow Light

In this chapter we demonstrate slow light by making use of the natural resonances of

an atomic vapor. Instead of looking for slow light on resonance, we tune the laser

frequency between the two ground-state hyperfine resonances of a hot atomic cesium

vapor cell in order to obtain large fractional time delays of high-bandwidth pulses with

low distortion and high transparency. Specifically, we obtain tunable delays of 275 ps

input pulses up to 6.8 ns and 740 input ps pulses up to 59 ns (group index of

approximately 200) with little pulse distortion. The delay is made tunable with a fast

reconfiguration time (hundreds of ns) by optically pumping out of the atomic ground

states.

2.1 Wide-bandwidth slow light in cesium vapor

There is considerable practical interest in developing all optical delay lines that can

tunably delay short pulses by much longer than the pulse duration. Slow light (i.e., the

passage of light pulses through media with a small group velocity) has long been

considered a possible mechanism for constructing such a delay line. Most commonly, the


steep linear dispersion associated with a single gain or transparency resonance provides

the group delay. In addition to single-resonance systems, double gain resonances have

been used for pulse advancement [24–29] and delay [30]. Widely spaced gain peaks

create a region of anomalous dispersion, resulting in pulse advancement. When the

spacing between the gain peaks is small, a region of normal dispersion is created,

resulting in pulse delay. Pulse advancement is also possible by the proper spacing of two

absorbing resonances [28]. The possibility of pulse delay between two absorbing

resonances has also received some attention [31–35].

Ideally, an optical delay line would delay high bandwidth pulses by many pulse

lengths in a short propagation distance without introducing appreciable pulse distortion

and be able to tune the delay continuously with a fast reconfiguration rate. Minimal pulse

absorption is also desirable, but not necessary because absorption can be compensated

through amplification. Relatively few experiments [6,7,36–39] have directly measured

pulse delays longer than the incident pulse duration, and of these, none has used pulses

shorter than 2 ns or reported reconfiguration rates approaching the inverse pulse delay

time.

Because the dispersive contribution to the group index scales with the strength of a

resonance’s absorption coefficient, resonances in atomic vapor, with extremely strong

and narrow linewidths, are a promising system for creating long optical pulse delays.

Furthermore, by tuning the signal wavelength between two nearby resonances, severe

absorption can be avoided. Use of this particular technique was demonstrated by

Camacho, Pack, and Howell [39]. By transmitting a signal between the resonances of the


rubidium85 D2 hyperfine resonances, they were able to delay 2.4 ns input pulses by 106 ns

when the Rb-containing cell was heated to 140˚C, with broadening of approximately

40%.

As an extension of the above work, we attempted to further the slow light effect by

using a vapor with more widely spaced resonances that could be used to accommodate

shorter pulses, or similar length pulses could be delayed more due to the reduction in

higher-order dispersion. Cesium was chosen, in part because the spacing between the D2

lines of cesium is approximately 9.2 GHz, compared to the roughly 3 GHz spacing of the

same lines in rubidium, enabling the accommodation of shorter pulses and possibly

longer fractional delays.

Figure 2.1: (a) cw signal transmission (asterisks—measured, solid line—fit) overlaid with the spectrum (dashed line) of a 275 ps pulse and (b) index of refraction (solid line) and group velocity (dashed line), all vs. signal detuning for cesium at approximately 114˚C. All theory curves taken from equation 1.1 with A = 4 x 105 rad/s, g1 = 7/16, and g2 = 9/16. High-fidelity optical delay is observed for light pulses passing through the nearly transparent window between the two resonances.


In a medium such as cesium with two Lorentzian absorption resonances, as illustrated

in figure 1, the complex index of refraction can be approximated as

n !( ) = 1" A2

g1! + #+ + i$

+g2

! " #" + i$%&'

()*

(2.1)

where 2γ is the homogeneous linewidth (full width at half maximum, FWHM), g1 and g2

account for the possibility of different strengths for the two resonances, ! =" #"0 # $

is the detuning from peak transmission, !0 = !1 +!2( ) / 2 , ω1 (ω2) is the resonance

frequency for transition 1 (transition 2), !± ="21 ± ! , !21 = !2 "!1( ) / 2 , and

! =g11/3 " g2

1/3

g11/3 " g2

1/3#21 . (2.2)

Pulse propagation can be described in terms of various orders of dispersion, which

can be determined through use of the real part of Eq. (1) as

! j "1cd j# $n #( )d# j

# =#0 +%

(2.3)

where !n "( ) is the real part of the refractive index. Thus the group velocity is given by

vg = 1/β1, and the group-velocity dispersion (GVD) and third-order-dispersion are given

respectively by β2 and β3. In our experiment, we find that the contribution from β2 is

negligible, as the central point between the resonances is also a local minimum for GVD.

The third order term β3 is actually the dominant contribution to pulse broadening, and

scales as 1/τ3, where τ is the initial pulse duration.


Figure 2.2: Experimental schematic. A signal pulse passes through a heated cesium vapor cell. Two pump beams combine on a beam splitter and counterpropagate relative to the signal beam through the vapor to provide tunable delay of the signal pulse.

Our experimental setup is shown in figure 2.1. The signal laser is a cw diode laser

with a wavelength of 852 nm. The signal frequency is tuned to obtain maximum

transmission between the two Cs D2 hyperfine resonances and is pulsed at a pulse

repetition frequency of 100 kHz using a fast electro-optic modulator (EOM). The signal

beam is collimated to a diameter of 3 mm, and two different pulse widths are used, 275

ps or 740 ps FWHM, with a peak intensity of less than 10 mW=cm2. The pulses then

pass through a heated 10-cm-long glass cell containing atomic cesium vapor. The 275 ps

pulses are measured using a 7.5 GHz silicon photodiode, and the 740 ps pulses are

measured with a 1 GHz avalanche photodiode. All electrical signals are recorded with a

30 GHz sampling oscilloscope triggered by the pulse generator. The pump beams are

turned off except for the experiments reported in figures 2.5 and 2.6.


Figure 2.1(a) shows the transmission of a cw optical beam as a function of frequency

near the two cesium hyperfine resonances, overlaid with the spectrum of a 275 ps

Gaussian pulse. The data points are measured values and the solid line fits these points to

the imaginary part of equation 2.1. The entire pulse spectrum lies well within the

relatively flat transmission window between the resonances, resulting in little pulse

distortion from absorption. Figure 2.1(b) shows the real part of the index of and

frequency-dependent group velocity associated with the absorption shown in figure

2.1(a). We note that, in the region of the pulse spectrum, the curvature of the frequency-

dependent group velocity is greater than that of the absorption, suggesting that dispersion

is the dominant form of pulse distortion. This is not the case for single-Lorentzian

systems, where the spectral variation of absorption is the dominant form of distortion

[40]. While most slow light experiments have worked by making highly dispersive

regions transparent, we have worked where a highly transparent region is dispersive.

As shown above, the delay of a pulse is proportional to the optical depth of the vapor.

Figure 2.3 shows that we can control the delay by changing the temperature (and thus

optical depth) of the Cs cell. Using a 10 cm cell, and varying the temperature between

approximately 90˚C and 120˚C, we were able to tune the delay of a 275 ps pulse between

1.8 ns and 6.8 ns. The theory curves in figure 2.3 were obtained using

I z,t( ) = !n 0( )c"0 E z,t( ) 2 / 2 , where the electric field is given by

E z,t( ) = E0! exp "i #0 + $( )t%& '(2)

* d+ exp i#n +( )c

z " +t,-.

/01"+ 2! 2

2%

&2

'

(3

"4

4

5 (2.4)

and where we have used equation 2.1 for the index of refraction.


Figure 2.3: Pulse shapes of 275 ps input pulses transmitted through a cesium vapor cell. Delays a large as 25 pulse widths are observed. The temperature range from 90˚C to 120˚C.

The atomic density N has been chosen separately to fit each measured pulse. We note that

a pulse may be delayed by many pulse widths relative to free-space propagation with

little broadening.

Longer pulses lead to delay with reduced pulse broadening because pulse broadening

is approximately proportional to 1/τ3. To study the larger fractional delays enabled by this

effect, we used longer 740 ps input pulses for which the dispersive broadening is

significantly reduced. Figures 2.4(a) and 2.4(b) show the delay and broadening of a 740

ps pulse after passing through a sequence of three 10 cm cesium vapor cells. The plots

correspond to a temperature range of approximately 110˚C to 160˚C. Even though the

pulse experiences strong absorption at large delays, the fractional broadening of the pulse

FWHM remains relatively low.


Figure 2.4: (a) Output pulse shapes and (b) fractional broadening as functions of fractional delay for a 740 ps input pulse. Fractional delay is defined as the total delay over divided by τin. Fractional broadening is defined as (τout– τin) / τin.

In addition to temperature tuning, the optical depth can be changed much more

rapidly by optically pumping the atoms into the excited state using two pump lasers. As

shown in figure 2.2 each pump laser is resonant with one of the D2 transitions in order to

saturate the atoms without optical pumping from one hyperfine level to the other. The

power of each pump beam is approximately 30 mW, and both pump beams are focused at

the cell center. The signal beam overlaps the pump beams and is also focused to a 100

µm beam diameter. The pump beams are turned on and off using an 80 MHz acousto-

optic modulator with a 100 ns rise or fall time. Being on resonance with the D2

transitions, the pump fields experience significant absorption (αL ≈ 300), and are entirely

absorbed despite having intensities well above the saturation intensity.

With the pump beams on, the decreases in effective ground-state atomic density leads

to smaller delay. Figure 2.5 shows a delayed pulse waveform consisting of two 275 ps

input pulses separated by 1 ns, with the pump on and off. We note that pump fields create


no noticeable change in the waveform shape or amplitude. Also, we measured that the

change in delay is essentially proportional to the pump power.

Figure 2.5: Pulse output waveforms with auxiliary pump beams on (blue dotted line) and off (solid line). Two 275 ps input pulses separated by 1 ns are delayed by approximately 5.3 ns without pumping, but only 4.3 ns with pumping (a change of 1 bit slot) with little change in pulse shape.

In Figure 2.6 the measured signal delay is shown as a function of the difference

between arrival time ts of the signal at the cell and the turn-on time tp of the pump. The

rise and fall times lie in the range 300–600 ns and vary slightly depending on the relative

detunings of the pumps.

Figure 2.6: Pulse delay vs. time following pump turn-on and turnoff, showing the reconfiguration time for optically tuning the pulse delay. The two pump beams are tuned to separate cesium hyperfine resonances and are switched on at the time origin and switched off 24 µs later.


In summary, we have demonstrated the tunable delay of a 1.6-GHz-bandwidth pulse

by up to 25 pulse widths and the tunable delay of a 600-MHz-bandwidth pulse by up to

80 pulse widths by making use of a double absorption resonance in cesium. Furthermore,

we showed that the delay can be tuned with a reconfiguration time of hundreds of

nanoseconds.

3 STRUCTURAL SLOW LIGHT 22

Chapter 3

Structural Slow Light

Most devices used to create structural slow light are in the category of infinite

impulse response filters, defined as filters where the response to an impulse input is non-

zero over an infinite duration. Optical filters that are based on reflections or recirculation

of part of the incident light generally have an impulse response that resembles an infinite,

but decaying, ringing. As described in a paper by Lentz et al. [41], filters are often

analyzed in terms of their complex frequency response H, where

Y !( ) = H !( )X !( ) (3.1)

and Y and X are the output and input power spectra of the signal. If H is written as

H !( ) = H !( ) exp i" !( )#$ %& (3.2)

then the frequency dependent delay is defined as

! "( ) = #d$ "( )d"

. (3.3)

Since the frequency-dependent phase and frequency-dependent index of a non-absorbing

linear medium are related by

! "( ) = "cn "( )L (3.4)


we can use the definition of the group index to associate it with the frequency dependent

delay (and by extension the phase) as

! "( ) = Lcng "( ) . (3.5)

However, it is important for us to note that the comparison to a material slow light

medium is not exact and this relation only holds by analogy. For example, pulse delay

(and pulse advancement) do not always scale with length, as the delay is the result of a

complicated interaction of multiple reflections. One simple but well-known example of

this occurs with the generalized Hartmann effect, where pulse delay is seen to saturate

and becomes independent of length [42]. Indeed, “length” is not even always a well-

defined property for a structural slow light device.

In this chapter, we present the results of two experiments related to the development

of structural slow light. The first is a demonstration of slow light in a fiber ring resonator,

possibly a component of a longer delay line. The second experiment demonstrates in fiber

rings a phenomenon analogous to EIT, which is commonly used to produce slow light in

atomic systems.

3.1 Slow light in a ring resonator

In this subsection, we present our work demonstrating the capability of a fiber ring-

resonators for use in slow-light delay lines. We begin by analyzing the device illustrated

in figure 3.1 as follows [43,44]. We describe the coupling of light into and out of the

resonator in terms of generalized beam splitter relations of the form


E2 = rE1 + itE3 (3.6)

E4 = rE3 + itE1 (3.7)

Figure 3.1: Geometry of a fiber ring resonator.

where r and t are taken to be real quantities that satisfy the relation r2 + t2 = 1, and the

fields are defined with respect to the reference points indicated in figure 3.1. In addition,

we describe the circulation of light within the resonator in terms of the round-trip phase

shift ϕ and the amplitude transmission factor τ such that

E2 = !E4 exp i"( ) (3.8)

The round-trip phase shift ϕ can be interpreted as ϕ = kL, where k = 2πn/λ, n is the

effective refractive index of the fiber mode λ is the vacuum wavelength of the incident

light, and L is the circumference of the fiber ring. Equations 3.6-3.8 can be solved

simultaneously to find that the input and output fields are related by the complex

amplitude transmission

E2E1

= exp i ! + "( )#$ %&' ( r exp (i"( )1( 'r exp i"( ) (3.9)

The intensity transmission factor T is given by the squared modulus of this quantity

T =E2E1

2

=! 2 " 2!r cos #( ) + r21" 2!r cos #( ) + r2! 2 (3.10)


Note that the on-resonance transmission (ϕ = 0) drops to zero for the situation r = τ. In

this case, the internal losses are equal to the coupling losses, and the resonator is said to

be critically coupled. For r < τ, the resonator is said to be undercoupled and for r > τ the

resonator is said to be overcoupled. The phase of the transmitted light is given by the

argument of E2/E1 as follows:

! = " + # + atanr sin #( )

$ % r cos #( )&'(

)*++ atan

$r sin #( )1% $r cos #( )

&'(

)*+

(3.11)

Figure 3.2: Theoretically predicted net transmission, phase shift, and group delay for an ideal ring resonator with single-pass transmissivity τ = 0.95 and varying values of the finesse. The normalized detuning is the angular frequency difference between the light field and the nearest cavity resonance multiplied by the ring circulation time.

Near each resonance, the phase undergoes a rapid variation with respect to the round-

trip phase shift. This round-trip phase shift may be interpreted as a frequency detuning


normalized through multiplication by the round-trip time. The radian-frequency

derivative of the phase shift is the group delay which is proportional to both the finesse

F ! " / 1# r *$( )( ) and the quality factor Q = nFL / !( ) in the overcoupled regime. In

the undercoupled regime, the group delay can take on positive or negative values. Figure

3.2 displays the transmission, phase, and group delay for a ring resonator.

The fractional delay Tfrac is defined as the group delay TD normalized by the FWHM

pulse duration. It is a convenient measure of the number pulsewidths by which a pulse

can be delayed. Because for a single resonator there is a tradeoff between group delay

and bandwidth given by Δν ≈ 2/(πTD), the fractional delay imparted by a resonator is

limited. For a Gaussian pulse with a bandwidth equaling the resonator bandwidth, the

fractional delay is limited to a maximum of roughly 1/ln(2). In reality, however, when the

pulse bandwidth exactly matches the resonator bandwidth, it becomes distorted due to a

nonuniform group delay across the pulse spectrum. Practically, for operation below a

fractional delay of unity, a pulse is delayed and transmitted with low distortion. Larger

fractional delays may be obtained by cascading resonators in a serial manner. It should be

noted that this result is independent of the scale size and nearly independent of the finesse

of the resonator. Thus, in order to maximize the fractional delay for a given pulse, the

resonator bandwidth should be chosen appropriately. This still leaves a choice between a

small, high-finesse resonator or a larger and proportionally lower finesse resonator. If

both suffer the same loss per round trip dictated by splices and coupling-related losses,

the lower finesse device will more closely approximate a phase-only device having a

more uniform transmission across a free-spectral range.


Figure 3.3: (a) A ring-resonator optical delay line. (b) One element of the delay line. (c) Demonstration of controllable optical time delay in the setup of (b). A 51-ns pulse is delayed by 27 ns.

We constructed a ring resonator made of a fiber loop for the purpose of demonstrating

tunable optical true time delays. A dye laser tuned near 589 nm was coupled into one port

of a fixed four-port single-mode (single mode in the visible spectral region) directional

coupler. One output port of the coupler was directly fusion spliced to the other input port

to form a fiber ring resonator with a 2.8-m circumference. A coupling coefficient t2 of 3/4

was chosen to produce a resonator with a low finesse of 5. A low finesse was used to

approximate a phase-only device by maintaining a nearly uniform transmission across a

free-spectral range. The light emerging from the output port was directed onto a silicon

PIN detector and the waveform was collected by a digital oscilloscope. Thermal drifts in

the ambient temperature caused the resonance frequencies to drift on a slow (millisecond)

time scale. These drifts were partially stabilized by immersing the fiber ring into a room-

temperature water bath. An acoustooptic modulator was used to generate a continuous


train of 50-ns pulses from the dye laser output. A beamsplitter picked off a fraction of the

pulse train for use as a trigger reference. As the dye laser frequency was swept through a

free-spectral range (70 MHz) of the fiber ring resonator, the pulses emerging from the

ring resonator showed a variable time delay. Figure 3.3 compares the timing of an off-

resonance pulse to that of an on-resonance pulse. In this experiment, a 51-ns pulse was

delayed by 27 ns for a maximum fractional delay slightly greater than 1/2. By

concatenating a sequence of resonance-locked fiber ring resonators, as shown in (a) of the

figure, it would be possible to create much longer fractional time delays [10,41].

3.2 Coupled-resonator-induced transparency in a fiber system

A historically common method for producing slow light has been to employ the

technique of electromagnetically-induced transparency (EIT) [6,45–47]. EIT is a nonlinear

optical phenomenon wherein a strong pump field coupled to a resonance creates a

transparency window for a weak probe coupled to a resonance that shares one of its levels

with the pump transition. It can be used in several different configurations; in the

configuration called the Λ system, there are two lower energy levels (1,3) and one higher

energy level (2). A probe tuned to the 1–2 transition may see induced transparency in the

presence of a pump providing strong coupling between levels 2 and 3, with transitions

between the two lower states dipole disallowed. The bandwidth of the hole induced in the

absorption line depends on the lifetime of the atomic coherence between the two lower

levels. Since these coherence times can be quite long for atomic media, they create a sharp

spectral feature (linewidths as narrow as 1.3 Hz have been reported in paraffin-coated cells

[48]), whose large induced dispersion makes it a promising feature for observing slow light.


Just as the dipolar response of electrons in an atom can be modeled by a single

mechanical or electrical oscillator [49], quantum coherence effects in atoms can also be

modeled effectively by classical systems of coupled oscillators [50–52]. Not surprisingly

then, structures composed of two coupled optical resonators have been predicted to display

photonic coherence effects such as coupled-resonator-induced absorption (CRIA) and

transparency (CRIT) [53,54] in direct analogy with electromagnetically-induced absorption

[55] and transparency [56,57] in driven three-level atomic systems. These phenomena

were later observed in the whispering-gallery modes of coupled fused-silica microspheres

[58].

In this section we experimentally demonstrate coupled-resonator-induced transparency

(CRIT) in coupled fiber ring resonators. These systems have the advantage that the coupling

between resonators is significantly easier to control than that between micro-resonators, and

that the resonators are easily assembled from commercial off-the-shelf components.

Figure 3.4: Electric field designations for two coupled ring resonators.

The steady-state response of a system two coupled-resonators, one of which is coupled

to an excitation waveguide as shown in figure 3.4, has been investigated extensively by

Smith et al., who, in a separate work, derived the following results (eqns. 3.11-3.16)

highlighting the similarities between its behavior and that of a medium exhibiting EIT [54].


They calculate that for a Λ EIT system as describe above, with zero detuning for the strong

control field and zero decay rate for the 1-3 transition (i.e. Γ31 = Γ13 = 0), the transition rate

for the absorption of an arbitrarily detuned probe beam is given by

W !( ) ="p2 / #$% &'

1+ 4#2 ! (

"c / 2( )2!

$

%))

&

'**

(3.11)

where Ωp and Ωc are the Rabi frequencies of the probe and control fields respectively, Δ is

the frequency detuning of the probe field, and Γ is the decay rate from the excited state to

levels 1 and 3.

For a two-ring coupled resonator system as depicted in figure 3.4, the power absorbance

A for the second ring (i.e. absorbance through the bus waveguide) is found to be given by

!A2 !2 ,!1( ) " 1# !T2 =!A2(env)

1+ !F2 sin2!!1(eff ) + !22

$%&

'()

(3.12)

where

!A2(env) !2 ,!1( ) "

1# r22( ) 1# a22 !$1 2( )1# r2a2 !$1%& '(

2 , (3.13)

is an absorbance envelope function, and

!F2(env) !2 ,!1( ) " r2a2 !#1

1$ r2a2 !#1%& '(2 (3.14)

is a function related to the resonator finesse. r2 is the reflection coefficient of the second

coupler, and ϕ and a are the single-pass phase shift and attenuation factors for the

indicated ring. The factors


!!1 "1( ) # E2

+

E2$ =

r1 $ a1ei"1

1$ r1a1ei"1

(3.15)

and

!!1(eff ) " arg !#1( ) = $ + !1 + arg

a1 % r1e% i!1

1% r1a1ei!1

&'(

)*+

(3.16)

are the complex transmission coefficient and effective phase shift for the first (outer)

ring. Note that quantities denoted with a tilde have a dependence on single-pass phase

shifts. We can see the similarity between CRIT and EIT that we wish to demonstrate

experimentally by comparing equations 3.11 and 3.12. It is true that EIT involves split

Lorentzian resonances, while CRIT involves split Airy resonances, but this distinction

becomes less relevant for high finesse resonators [54].

Figure 3.5: Experimental setup for the observation of CRIT in a fiber system.

Our experimental setup is shown in figure 3.5. A tunable external-cavity diode laser

operating at 1550 nm was used to probe the transmission of a system consisting of two

coupled fiber rings, L1 = 1.22 m and L2 = 0.97 m in circumference. The laser output was

isolated and end-fire coupled into a straight section of a single-mode fiber that was

weakly coupled to one of the rings by means of a 90/10 coupler. A polarization controller


was used to select TE polarization. The two rings were inter-coupled by a tunable coupler

and submerged into a water bath for thermal stabilization.

Figure 3.6 Mode splitting in the transmission spectrum of two coupled fiber ring resonators. The couplings are r1 = 1.0, 0.999, 0.995, 0.99, 0.96 and .01 from top to bottom. The fixed parameters are: a1 = 0.98, a2 = 0.82, and r2 = 0.95.

The output was detected with an InGaAs PIN photodiode, and recorded on a digital

storage oscilloscope as the laser was scanned over a frequency range spanning several

resonances (typically 1 GHz) at a scan rate of several hundred hertz. Whispering-gallery

modes were initially observed in the second ring (the innermost ring, closest to the

excitation guide), with the first ring being de-coupled from the system. Then, changes in

the resonances were observed as the coupling between the rings was increased as shown

in figure 3.6, which shows the transmission of the system as a function of laser detuning


for a variety of couplings. In the trace in figure 3.6 with r1 = 0.99 (4th from the top),

resonances associated with both the innermost (deep, broad dips) and the outermost

(shallow, narrow dips) rings can be observed. Because the circumferences of the rings are

different, the two types of resonances do not overlap on every free spectral range.

However, when they do, a sharp spike in transmission emerges and splits the resonance.

A blow up of this effect is shown in figure 3.7.

Figure 3.7: A close-up view of CRIT, from figure 3.6. The solid line is a theoretical fit where a1, a2, and r1 are fitting parameters. FDTD simulations are shown for the symmetric and anti-symmetric modes after steady-state is achieved.

Note that the transition from the limit of weak coupling to that of strong coupling is

accompanied by a halving in free spectral range (FSR), i.e., the FSR changes from about

200 MHz to 100 MHz. This occurs because, in the limit of strong coupling, the spectrum

simply becomes identical to that of a ring with twice the optical path length of the

individual rings. The couplings r1 were determined by fits to the experimental data using

the fixed parameters: a1 = 0.98, a2 = 0.82, r2 = 0.95.

In summary, we experimentally demonstrated CRIT in two coupled fiber-ring

resonators. Narrow, sub-linewidth spectral features associated with the splitting were


observed. The linewidth of the transmission peaks is determined by the coupling between

the two resonators and thus could be made very small, potentially narrower than the

finesse-limited resonance linewidth of the constituent resonators. We note that while we

did not observe slow light in this system, researchers have more recently been able to

successfully produce slow light using CRIT resonances [59–61].

4 SCATTERING SLOW LIGHT 35

Chapter 4

Scattering Slow Light

In this chapter, we report on two experiments. The first experiment lays out the

essential concept and establishes the effectiveness of using stimulated Brillouin scattering

to produce tunable slow light. We show that we can obtain pulse delays of 25 ns for a 63

ns long (FHWM) optical pulse, and a delay of 19 pulse widths for a 15 ns pulse at

telecommunication wavelengths. The second expands greatly on that initial idea, using an

improved technique for generating the pump and signal fields to create a highly-

reconfigurable gain medium that can accommodate shorter pulses and tune from pulse

delay to pulse advancement over a full pulse-width of range.

4.1 Demonstration of stimulated Brillouin scattering slow

light.

Stimulated Brillouin scattering is a stimulated scattering process that is often used to

obtain slow light, especially in optical fibers. The effect can be described as a coupling of

two optical fields, a laser field (pump), a Stokes field, and an acoustic wave in the

medium. In an optical fiber, the phenomenon of electrostriction will cause glass to


compress in regions of high electric field strength. For counter-propagating beams of the

same frequency, this would result in a standing density pattern that does not couple to any

acoustic mode. However, if the fields are separated in frequency, the pattern created by

their interference will propagate continuously in the direction of the higher frequency

field. If they are spaced by the “Brillouin frequency”, ΩB, which depends on the speed of

sound in the medium and the difference in laser field wavevectors, this traveling wave

will couple to acoustic waves in the medium, as the acoustic wave and lower frequency

field are amplified at the expense of energy in the higher frequency field. If one particular

pump field in the fiber is very strong, a counter-propagating field lower in frequency by

ΩB, is termed the “Stokes” field and experiences gain, while one that is higher in

frequency by ΩB and experiencing loss is called “Anti-Stokes”.

The SBS gain has a Lorentzian spectrum given by

gB !( ) = gp "B / 2( )2!#!B( )2 + "B / 2( )2

, (4.1)

where the detuning between pump and signal is defined by Ω = ωp – ωs, ΓB is the

Brillouin linewidth, and gp is the peak Brillouin gain, given by:

gp =8! 2" e

2

np#p2$0cvA%B

. (4.2)

Here, γe is the electrostrictive constant of the medium, np is the phase index at the pump

frequency, ρ0 is the density of the medium, and va is the speed of sound in the medium

[62,63]. The spectral geometry is illustrated in figure 4.1.


Figure 4.1: SBS gain spectrum in the vicinity of a strong optical pump. In a fiber, light counterpropagating to the pump at the Stokes frequency will see strong SBS gain as well as high refractive index dispersion, leading to slow light.

As we have established, a Lorentzian gain peak leads to slow light on line center, so a

Stokes pulse sent counter to the pump field should experience a slow light effect. Slow

light created through SBS has several advantages over other techniques. Naturally, the

delay can be easily tuned by adjusting the pump power, and the medium can be used to

create slow light at almost any wavelength. This is possible because the explicit

wavelength dependence in the Brillouin gain in equation 4.2 is cancelled by an inverse

square dependence of the Brillouin linewidth ΓB on the pump wavelength. Also, while the

SBS gain in ordinary glass is not large enough to lead to extreme refractive indices, even

a small increase in n can become a contribute fractional pulse delay due to the long

propagation lengths possible with the fiber geometry.

The contribution to the phase refractive index of the Stokes resonance is given by:

!nR =gpI p2" s

1# $1+ $ 2

%&'

()*

(4.3)

where δ is a normalized detuning parameter δ = 2(Ω – ΩB) / ΓB, and Ip is the pump

intensity [62]. We can use the standard equation for group index to obtain


ng = ng0 +gpI p!B

1" # 2

1+ # 2( )2$

%&&

'

())

(4.4)

where ng0 is the standard group index of the fiber disregarding the SBS contribution. If

we wish to consider the case where the pulse bandwidth is small relative to the SBS

linewidth (δ2 << 1), then we can expand the SBS contribution to the group index in a

Taylor series about the point δ = 0:

!ng "gpI p#B

1+ $$ % 2( )

1& % 2

1+ % 2( )2'

())

*

+,,

% =0

% 2 + ...'

())

*

+,,

(4.5)

which simplifies quickly to

!ng "gpI p#B

1$ 3% 2( ) (4.6)

if we ignore the higher order terms.

We can easily see that maximum differential delay at the stokes frequency is

approximately equal to

!Td =gpI pL"B

(4.7)

where L is the fiber length. If we are limited by group velocity dispersion, we can

estimate the maximum possible fractional delay as follows [64]. A pulse of duration τ has

a bandwidth of 1/τ and therefore the fastest and slowest frequency components of the

pulse will have a corresponding differential delay. If we allow such a delay to broaden

the pulse by a factor of 2, we obtain the expression


!Tfrac =!Td"

= 2 =gpI pL#B

3 1"

$%&

'()2$

%&'

(). (4.8)

If we define the total gain: G = gpIpL, we find the maximum allowable gain to be:

G =!B3" 3

3 (4.9)

If we then insert this value into the expression for total delay in equation 4.7, and divide

by the pulse duration to calculate fractional delay, we obtain a fractional delay of

Tfrac =!B2" 2

3 (4.10)

which is likely to be the largest we can obtain without exceeding our self-imposed

fractional broadening limit of two.

In doing experiments with SBS slow light, it is necessary to keep the signal field at

the appropriate Stokes frequency. The simplest way to accomplish this is to split the field

from a single source laser. One path will be amplified and sent into the slow-light fiber to

become the SBS pump. The other will also be amplified, but it will be sent to another

long fiber of the same type as the first. If the field is adequately intense, (over a critical

“Brillouin threshold” intensity where gpIpL ~ 21) the SBS process will be seeded from

noise, and a Stokes field will be generated. Pulses can then be carved from this field, and

it can be sent through the slow light fiber counterpropagating to the pump and switched

out of the fiber with circulators.

Our experimental setup for observing slow light via SBS is diagrammed in figure 4.2.

Light from a 3 mW, 300 kHz linewidth, 1550 nm wavelength laser is sent into a 1 W

Erbium doped fiber amplifier. The amplified continuous wave signal is divided equally


into two 250 mW beams which are sent to high power circulators. The output from the

one of the circulators is sent to an SBS generator consisting of a 1 km long fiber (Corning

SMF-28e) to produce the Stokes shifted light. Reflections from the fiber ends provide

feedback to pro-duce Brillouin lasing near the peak of the Brillouin gain spectrum, where

the linewidth of the emitted beam is comparable to the linewidth of the pump beam [65].

The amplitude of the generated beam is modulated to form the Stokes pulse. Fiber

polarization controllers (FPC) are inserted at various locations to optimize the extinction

ratio of the Stokes pulse. The pulses have a peak power of 1 W.

Figure 4.2: Experimental setup for observing SBS slow light. FPC: fiber polarization controller. EDFA: Erbium doped fiber amplifier.

The Stokes pulse enters a 500 m long laser-pumped optical fiber which serves as the

slow-light medium, where the fiber (Corning SMF-28e) has angled ends to prevent SBS

laser oscillation. From measurements described below, we find that ΓB/2π ≈ 70 MHz,

which is about a factor of 2 larger than that measured in SMF-28 fibers [66]. The output

from the other circulator counterpropagates through the slow-light fiber and serves as the

SBS pump beam, where a FPC is used to optimize the gain. The delayed and amplified

pulses emerging from the slow-light fiber are recorded using an InGaAs photodiode (35


ps rise time) and a digital oscilloscope. The value of G is obtained by measuring the

continuous wave gain of Stokes light propagating through the slow-light fiber.

Figure 4.3 shows our observation of SBS mediated slow light at telecommunications

wavelengths input Stokes pulses of two different pulse durations. Figure 4.3(a) shows the

output shape of a Gaussian pulse with τin = 63 ns in the presence and absence of a pump

beam with G = 11. We measure a total delay of 25 ns, or a fractional delay of Tfrac = 0.4,

and a small amount pulse broadening (roughly 5% of the input width). We find that it is

possible to induce larger relative slow-light pulse delay using an input pulse width that is

moderately short, with the input pulse duration on the order of the inverse Brillouin

linewidth. Figure 4.3(b) shows the system output when using Gaussian-shaped pulses

with τin = 15 ns. Here, we are able to measure a fractional delay of 1.3. This improvement

in relative pulse delay comes at the price of increased pulse broadening, a we measure

approximately 40% pulse broadening in this case.

Figure 4.3: Observation of slow light in optical fiber using SBS. Output pulses are normalized to the level of the input. Dotted lines are in the absence of the pump beam, solid lines are traces taken with the SBS pump turned on (G ~ 11). Figure (a) depicts a 63-ns long input pulse; (b) represents is data for a 15-ns pulse.


Finally, we show how changing the pump power, and therefore the SBS gain can be

used to tune the slow light effect. Figure 4.4 shows the variation in SBS-induced pulse

delay as a function of G, for pulses of 63 and 15 ns. We observe significant pulse

distortion when the input pulse duration is less than 10 ns. Although we vary the pump

power slowly using a manually adjusted attenuator, the reconfiguration time of the delay

can, in principle, be much faster and is limited by the longer of either the transit time of

the pump beam through the fiber (2.5 µs in out experiment) or the SBS lifetime (1/ΓΒ =

2.3 ns).

Figure 4.4: Demonstration of optically controllable slow-light pulse delays. Induced delay as a function of the Brillouin gain parameter G for 63 ns long (square) and 15 ns long (circle) input Stokes pulses.

In conclusion, we have demonstrated that SBS in a single mode fiber can be used to

induce optically-tunable pulse delays with a fractional delay on the order of the input

pulse width. The delays can be obtained for pulses at nearly any wavelength, since we are

not forced to work near any particular material resonance.


4.2 Continuously tunable slow and fast light using SBS

Slow and fast light, which refer to technologies that control the group velocity of

light, have promising applications in telecommunication systems and all-optical signal

processing [40]. So far, tunable temporal delay elements at telecommunication

wavelengths have been demonstrated using various slow-light processes [67–70], one

particular example of which is stimulated Brillouin scattering (SBS), as we saw in the

previous section. Furthermore, many techniques have been proposed to improve aspects

of an SBS-based slow-light delay element, such as its operating signal bandwidth [71–74]

and maximal achievable fractional delay [30, 64, 75–79]. Fast light has also been

investigated using, saturable media [68] and SBS [80–83], but most fast-light modules

reported so far have very limited fractional advancement.

Note that most demonstrated slow-light devices cannot be reconfigured easily to work

in the fast-light regime, and vice-versa. Yet in practice, the temporal position of an

optical signal train can experience random delay or advancement as compared to a

reference clock. This indicates that applications such as jitter correction and data

resynchronization require bidirectional temporal adjustment of the optical signal train. In

such circumstances, a unidirectional time adjustment module with only delay or only

advancement is not enough, and it would be much more useful if a single element could

provide temporal adjustment seamlessly from delay to advancement with a total tuning

range comparable to the temporal extent of a data bit.

It has been shown recently [80] that one can achieve tunable delay and advancement

by adjusting the separation between two Lorentzian gain lines. However, in the process


of tuning the separation, both the gain and group index profiles become highly frequency

dependent over the signal bandwidth, which lead to significant pulse distortion. In this

paper, we construct a low-distortion bidirectionally-tunable optical timing element using

a reconfigurable multiple-gain-line medium. In particular, we use two low-distortion gain

profiles for slow- and fast-light operations, and the tunable delay and advancement are

achieved by controlling the pump power.

The complex refractive index for a medium with a single Lorentzian-shaped gain line

is given by

!n !( ) = nbg +

cg4"!0

#! $ !0 + i#

(4.11)

where ν0 is the center frequency of the gain feature, nbg is the background index of

refraction, c is the speed of light in vacuum, and g and γ are the peak gain coefficient and

the half-width at 1/e linewidth of the gain line, respectively. The real part of ñ has a large

swing in the vicinity of the resonance, resulting in slow and fast light respectively in the

center and in the wings of the resonance. The principle of switching our device between

slow- and fast-light operations is to reshape the gain profile quickly, so that the signal

spectrum lies either within the center region of a single broad gain feature or within the

transparent window between two separated gain features. Tunable delay or advancement

is then achieved by controlling the strength of the corresponding gain feature in each

working regime. Under many circumstances, the primary figure of merit of slow-light

delay devices is the maximum achievable fractional delay ΔTmax, also known as the

delay-bandwidth product [18,64]. In this paper, we define the fractional delay as


ΔTfrac = ΔT/τp, where ΔT is the absolute delay of the peak position of the pulse as

compared to that of a reference, and τp is the full width at half maximum (FWHM) of the

input pulse. In practice, ΔTmax is often limited by the maximum distortion or change in

power level that a signal is allowed to acquire in passing through such a material [84].

Since both the group index and the gain in the vicinity of a Lorentzian gain line are

highly frequency dependent, the maximum achievable fractional delay of a delay element

based on a single-Lorentzian-gain-line medium is very limited.

In this experiment, we use multiple closely-spaced gain lines, and we optimize the

spacing and the relative strength to form a broad, flat-top gain profile for slow-light

operation [75,76,85]. One can also extend this concept to the use of a continuous pump

spectrum to achieve a gain profile as broad as tens of gigahertz [86].

To switch from slow-light to fast-light operation, we split the single broad gain

profile into two separated gain features. Such a separated double gain profile leaves a

transparent window in between, over which the reduced group index, ng_r = v(dn/dv), is

negative. Note that the maximum achievable fractional advancement of such a separated

double gain medium is determined by factors that can be different from those for a slow-

light medium [84]. First, since the signal spectrum sits between two gain features, the

center frequency of the signal sees minimum gain. As a result, the output pulse can

become narrower due to spectrum broadening. However, residual frequency components

due to optical noise in the input signal or spontaneous emission from the gain medium

that fall on the two gain peaks get amplified much more strongly than the signal. Such

amplified noise can form a broad, beating pattern in the time domain, which leads to


pulse distortion and inter-symbol interference [62]. Thus, the maximum achievable

fractional advancement of such an element depends not only on the maximum pump

power the system can provide, but also on the noise level of the input signal as well as the

noise properties of the gain medium. This noise constraint determines the maximum

continuous-wave (CW) gain the system can have at the two gain peaks. Given such a

limit on maximum spectral gain, one way to achieve greater fractional advancement is to

broaden each gain feature to increase the magnitude of the group index within the

transparency window. This concept is illustrated in figure 4.5

Figure 4.5: Illustration of how pump modulation affects the shape of the Stokes and anti-Stokes gain profiles. Since the gain lineshape is effectively a convolution of the SBS lineshape with the pump’s spectral profile, we can use this fact to tailor the gain seen by a pulse near the Stokes or anti-Stokes resonance. Splitting the pump once (a) allows us to observe fast light in between the gain resonances. Further splitting of each resonance allows us to reduce distortion. (Compare with figure 4.1)


To demonstrate our scheme, we use stimulated Brillouin scattering with dual-stage

pump field modulation. Figure 4.6 shows the schematic diagram of the experiment. We

start with a stable laser source (Koshin LS-601A) at a frequency ν0 near 1550 nm, and we

modulate the field using a sinusoidally driven Mach-Zehnder (MZ) intensity modulator

(IM 1 in figure 4.6), which is biased for minimum DC transmission. The modulator

creates two frequency sidebands, ν0 ± ΩB, where ΩB ≈ 10.6 GHz is the SBS Stokes shift

frequency of our single-mode fiber (SMF). The modulated field then propagates through

6 km of SMF with a strong counter-propagating pump field at ν0. The SBS process

amplifies the component at the Stokes frequency ν0 – ΩB and attenuates the anti-Stokes

frequency ν0 + ΩB. The optical field after this SBS purification stage is checked with an

optical spectrum analyzer, and the

power of the Stokes field at ν0 – ΩB is at least 20 dB larger than those at ν0 and ν0 + ΩB. A

second MZ intensity modulator (IM 2) is then used to carve out a train of 6.5 ns FWHM

Gaussian pulses before the signal is sent into the SBS temporal adjustment module.

Figure 4.6: Schematic diagram of a bidirectionally tunable optical timing element using stimulated Brillouin scattering. TL: tunable laser; IM: intensity modulator; AFG: arbitrary function generator; SMF: single-mode fiber; EDFA: Erbium doped fiber amplifier; VOA: variable optical attenuator.


A two-stage pump modulation is used to create a single or a double flattened gain

feature for slow- and fast-light operation, respectively. We use one sinusoidally driven

MZ intensity modulator (IM 3 in Fig. 4.6) to create three closely spaced frequency lines

leading to a single, approximately 80 MHz wide, flattened gain feature [76]. Note that

one can use more complicated intensity or phase modulation and create more lines to

form a broader gain feature [77,78]. A second MZ intensity modulator (IM 4) is used to

configure the final gain profile for slow- or fast-light operation. The modulator is always

biased at minimum DC transmission, and it is sinusoidally modulated at frequency fs.

This setting splits the gain feature produced by IM 3 into two, which are separated by 2fs

from each other. When our device works in the slow-light regime, fs is approximately 34

MHz, and the two gain features are not fully separated but form a broader, flat-top gain

feature [see Fig. 4.7(a)]. For fast-light operation, fs is approximately 148 MHz, so the two

gain features are well separated, leaving a transparent fast-light window in between [see

Fig. 4.7(c)].

Figure 4.7: Measured small signal gain and calculated induced refractive index change as functions of frequency detuning for slow light [(a) and (b)] and fast light [(c) and (d)] configurations. The black dotted lines in (a) and (c) show the power spectrum of Gaussian pulses with FWHM of 6.5 ns.


The modulated pump profile is amplified using an Erbium doped fiber amplifier

(EDFA 3 in Fig. 4.6) and then launched into 4 km of SMF counter-propagating with the

signal field. The amount of delay or advancement is adjusted by controlling the output

power level of EDFA 3. Figures. 4.7(a) and 4.7(c) show the measured small signal gain

for the fast- and slow- light configurations, respectively. The black dotted lines show the

power spectrum of fractional delay Gaussian pulses with a FWHM of 6.5 ns, and one

sees that the flattened gain feature and the transparent window is adequately broad for

this signal spectrum. Figures. 4.7(b) and 4.7(d) show the corresponding refractive index

change calculated according to the Kramers–Kronig relations. One clearly sees the slow-

and fast-light regimes, indicated by positive and negative slopes of n, in the vicinity of

the center frequency for the two respective configurations. Note that the chosen two gain

profiles and their corresponding group index profiles for slow- and fast-light operations

are quite uniform over our signal bandwidth, and therefore one can achieve large

fractional delay and advancement with very low pulse distortion.

Figure 4.8 shows the measured delay and advancement as functions of the pump

power. As shown in the figure, using a maximum pump power of 130 mW, we have

achieved a fractional delay and a fractional advancement of 0.82 and 0.31, respectively,

giving a total continuous tuning range of 1.13 pulse widths, or about 7.35 ns for 6.5-ns

pulses.


Figure 4.8: Measured small signal gain and calculated induced refractive index change as functions of frequency detuning for slow light [(a) and (b)] and fast light [(c) and (d)] configurations. The black dotted lines in (a) and (c) show the power spectrum of Gaussian pulses with FWHM of 6.5 ns.

Note that our setup can switch between slow- and fast-light regime very quickly without

rearranging the components. The FWHM of the delayed and advanced pulses are

approximately 8.7 ns and 6.7 ns, respectively (see Fig. 4.9). The fractional advancement

can be further improved if each of the double gain features is broadened further and the

noise level of the input signal is reduced.

Figure 4.9: Output pulse as a function of time for reference, slow-light, and fast-light configurations.

In summary, we have described a bidirectionally tunable, low-distortion module for

delaying or advancing optical pulses using a reconfigurable gain medium, in which slow-


or fast-light operation is realized using a single flat-top gain profile or two well-separated

gain features, respectively. We have experimentally demonstrated our scheme using

stimulated Brillouin scattering in a single-mode fiber. The reconfigurable optimum gain

profiles have been realized using a two-stage pump modulation method, and we have

continuously tuned the temporal position of 6.5 ns FWHM Gaussian pulses from a

fractional advancement of 0.31 to a fractional delay of 0.82. With better optical signal-to-

noise ratio and broader gain features, more advancement can be expected. Such a device

can be used for bidirectional jitter correction and data resynchronization.

5 PULSE DELAY IN SATURABLE MEDIA 52

Chapter 5

Pulse Delay in Saturable Media

This chapter contains several experiments exploring pulse propagation in saturable

absorbers and saturable gain media. In these materials, the population lifetime T1 is many

orders of magnitude larger than the coherence lifetime T2, enabling their optical

properties to be described with a rate-equation analysis. Ultimately, their effects can be

thought of as a time-dependent gain, with the front and back parts of a given pulse

experiencing different levels of absorption or gain, due to the saturation. With that simple

picture, it is remarkable how closely the behavior of saturable absorbers can follow that

of traditional material slow light, which we will see in the following sections. We will

show how pulse advancement and delay in Erbium-doped fiber is seen alongside an

apparent spectral hole that affects the amplitude and phase of modulation in a manner

analogous to the Kramers-Kronig relations of material slow light. We will also see that

the negative group delay in EDF does create a genuine backwards-traveling pulse shape

inside the medium. Finally, we will explore some techniques for reducing distortion in

these systems, and we will also demonstrate saturable-absorption-based slow light in a

high bandwidth material system, PbS quantum dots.


5.1 Slow and fast light propagation in Erbium-doped fiber

In this section, we show that both pulse delays and pulse advancements can occur in

Erbium-doped optical fiber, occurring through saturation of the absorption and/or gain.

The widespread use of erbium-doped fiber amplifiers at the 1550-nm signal wavelength

used in the telecommunications industry suggests that slow and fast light effects in this

system could lead to important applications. The technique described here works at room

temperature. Additionally, the fiber system has a very simple experimental setup and

allows for the possibility of long interaction lengths and high intensities, both of which

can lead to large time delays. Moreover, while pulses can be delayed without a separate

pump field, the easy integration of a pump at 980nm enables the propagation speed to be

tuned continuously, leading to either significant delay or significant advancement for

appropriate pulse widths. Previous workers had observed phase delays of modulated light

fields in EDFA’s [87–89]. The present research extends this work by showing that both

delays and advancement are possible for either modulated or pulsed light fields. We also

develop a theoretical model that describes our experimental results with high accuracy.

Perhaps the surest insight may be gained by considering the problem in the time

domain, where the pulse delay effect can be seen as the saturation of the medium by the

leading edge of the pulse, allowing the remainder to be transmitted with less attenuation.

The resulting pulse in this case would be delayed, but reduced in overall intensity.

Oppositely, a medium exhibiting saturable gain produces an advanced pulse. The ground

state recovery time of the system is determined by the lifetime of the metastable state,


which places a lower bound on the pulse duration for which anomalous propagation

effects can be observed. This explanation of the effect was first used by Basov et al. in

1965 [91] and was explored in more theoretical detail by Selden in following years

[92,93]. We can also consider the problem in the frequency domain. In this case, if we

consider the modulated part of the input field separately from the background, we can see

a spectral hole burnt into the modulation gain (or modulation absorption), producing slow

light in a manner analogous to the gain or absorption resonances in a conventional

material slow light system [68,94]. However, it should also be noted that this picture

works effectively only when we have a large background that allows us to “linearize” our

treatment of the problem.

We can model the propagation of intensity-modulated 1550 nm light through an

erbium-doped fiber in the presence of a 980 nm pump using a rate equation analysis [88].

The energy levels in erbium can be approximated as a three-level system, and under the

additional approximation of rapid decay from the upper pumping state to the metastable

state, we obtain the rate equation for the ground state population density for the ground-

state population density n:

dndt

=! " n#

+ 1" n!

$%&

'()*s Is "

n!+ s Is "

n!+ pI p (5.1)

where ρ is the Er3+ ion density, τ is the metastable level lifetime (10.5 ms), Ip is the pump

intensity in units of photons/area/time, Is is the signal intensity, βs is the signal emission

coefficient, and αp and αs are the pump and signal absorption coefficients [68]. The


steady-state solution for n to eq. (1) is given by n0 = (ρ/τ +βsIs)/ωc, where we have

defined the “center frequency” as

! c =1"+# pI p$

+# s + %s( ) Is

$. (5.2)

The inverse of this frequency corresponds to the metastable-state lifetime, with additional

terms that allow for power broadening by the pump and signal fields. Because a single

intensity-modulated beam can be expressed in the frequency domain as a field with two

sidebands separated by the modulation frequency, it can provide the input fields for CPO.

If we modulate the signal intensity as Is = I0 + Im cos(∆t), we find that the ground state

population density oscillates as n(t) = n0 + δn(t), where

!n t( ) = " c cos #t( ) + # sin #t( )" c2 + #2

$%&

'()Img (5.3)

and we have additionally defined the gain coefficient as

g = !n0"

# s + $s( ) + $s . (5.4)

We then find that, neglecting second-order terms in the modulated signal, the

propagation equation can be written as

dImdz

= gIm !"1I0 (5.5)

where

!1 Im , I p , I0( ) = ! s + "s

#$%&

'()

* c

* c2 + +2

$%&

'()Img . (5.6)

The phase shift of the modulation is described by


d!dz

=I0Im

"2 Im , I p , I0( ) (5.7)

where

!2 Im , I p , I0( ) = ! s + "s

#$%&

'()

*+ c2 + *2

$%&

'()Img . (5.8)

Here we notice that the modulation absorption coefficient α1 is taken from the

cosinusoidal part of δn, while the phase shift α2 comes from the sinusoidal part, which is

out of phase with the original signal modulation. We also notice that the signs of both α1

and α2 are fixed by the sign of the gain coefficient g, which itself is determined by the

balance between the net gain and absorption experienced by the signal. Furthermore,

from the expression for α2, we can see that the peak of the phase shift will occur when ∆

= ωc. This fact also implies that the pulse of light seeing the largest fractional delay or

advancement will be one where the pulse duration is approximately ωc-1. In this

simplified approach, we have neglected the effect that the time-dependent part of the

signal field will have on the pump field, including only the spatial modulation due to I0.

We also remind the reader that it is implicit in this analysis that the dipole dephasing time

T2 of the material system is much smaller than the ground state recovery time T1 (here

represented as τ) and the inverse of the modulation frequency Δ [95].

The experimental setup used to observe this effect consists essentially of an erbium-

doped fiber amplifier in the reverse-pumped configuration. The signal source is a tunable

diode laser, operating at 1550 nm. The beam is coupled into a fiber and sent through 13

m of EDF that is pumped by a counter-propagating beam from a 980 nm diode laser. Two

percent of the input light is split off before the EDF for use as a reference. For


experiments using a sinusoidally modulated input, the current of the laser is modulated

directly by a function generator. To create Gaussian pulses, the beam was passed through

a rotating wheel with a narrow slit, before being coupled into the single-mode fiber. This

process converts a continuous beam with a Gaussian spatial profile into a series of pulses

with Gaussian time profiles. In each experiment, we subtracted from the raw data a trace

taken with 13 m of undoped silica fiber replacing the EDF, to compensate for any

inherent response difference between the InGaAs photodiodes used to detect the signal

and reference fields. A diagram of the setup is shown in figure 5.1

Figure 5.1: Diagram showing the experimental setup for observing pulse advancement and delay in an optically saturable erbium-doped fiber. It should be noted that the function generator (used to provide a sinusoidal modulation of the laser current), and the chopper (used to carve zero-background pulses out of a DC signal from the laser) were not in use at the same time.

A series of experiments showing the fractional delay or advancement of a modulated

input field is shown in figure 5.2. Traces are plotted against the log of the modulation

frequency and are taken with varying pump powers, with delays being recorded with little

or no pump and superluminal propagation being demonstrated for higher pump powers.


Anomalous propagation speeds are observed over roughly 1.5 decades of bandwidth,

with the peak of the effect being pushed to higher frequencies with increasing pump

power. However, the effect appears to saturate, with a doubling of pump power from 49

mW to 97.5 mW producing only a modest increase in peak frequency, and we do not

observe large modulation advancements above 10 kHz for any value of pump power.

Figure 5.2: Frequency and pump power dependence of the fractional delay observed in propagation through erbium-doped fiber. The input was sinusoidally modulated beam at 1550 nm with an average power of 0.8 mW. The different curves represent different pump powers. In all cases, the ratio Im/I0 = 0.08. The circles represent data points with the curves themselves being splines fit through data. The shaded circles represent the measurements with the fastest and slowest effective group velocities (−c/5600 and c/(1.2 × 104), respectively).

Notably, for a given frequency, increasing the pump power can significantly increase the

fractional advancement. For example, at 31 Hz a signal with a fractional delay of 0.08

unpumped will have a fractional advancement of 0.04 with 20 mW of 980-nm pump, and

can be tuned continuously by varying the pump power in this range. The largest


fractional advancement recorded was 0.125 at a pump power of 97.5 mW, while the

largest fractional delay shown is 0.075 and is obtained with no pump. By increasing the

signal power from 0.8 mW to 1.2 mW, the fractional delay can be increased to 0.089

demonstrating the signal power dependence predicted theoretically.

Theoretical curves are produced through numerical solution of the propagation

equations 5.5-5.8 and show good agreement with the experimental data. They are

displayed in figure 5.3. The effective absorption and emission coefficients and the erbium

ion density are found by a fit to the fractional advancement data. Parameters used for the

calculation are αp = 0.11 m−1, αs = 0.54 m−1, βs = 1.00 m−1, ρ = 1.78 × 1024 m−3, τ =

10.5 ms, mode field diameter = 2.75 μm. By examining the gain or loss on the modulated

signal, we can directly observe the hole in the absorption or gain spectra.

Figure 5.3: Numerically modeled curves for the same pump powers as figure 5.2.


Figure 5.4 shows the relative modulation attenuation plotted against the log of the

modulation frequency for different pump powers. Interestingly, the modulation

experiences relative gain over the DC signal for low pump powers, when the EDF is

acting as a saturable absorber, and relative loss at higher pump powers, when the medium

behaves as an amplifier.

Figure 5.4: Modulation gain, lnIm / I0( )outIm / I0( )in

!

"##

$

%&&

showing the holes in the absorption and

gain spectra for different pump powers. Again, the circles are data points shown representatively for one of the traces, and the curves themselves are splines fit through data.

It is instructive to look directly at the dependence of advancement and delay on the

pump power. In figure 5.5, this is done for two different modulation frequencies, one

relatively fast (1 kHz) relative to the metastable state lifetime, and another that is much

slower (31 Hz). We can see in particular that for the slow signal, the system demonstrates


slow light for low powers, then with increasing pump power shifts relatively quickly to a

pulse-advancement regime, and then the pulse advancement effect slowly fades as the

spectral hole gets broadened out past the ideal bandwidth of the signal. For the faster

signal, we see fractional advancement increasing steadily with pump power, but we

expect that the effect would eventually saturate and diminish at higher pump powers than

we were able to observe in the laboratory.

Figure 5.5: Fractional advancement as a function of pump power for two modulation frequencies. The trace at 31 Hz shows very well how the system behavior at a given frequency can be tuned from slow light to fast light by changing the pump power.

In all of the above figures 5.2–5.5, we can clearly observe the effect that higher 980-

nm pump powers broaden the spectral hole seen by the modulated gain and allow for

higher frequency modulations, or shorter pulses, to be advanced. As we observed in

equation 5.2, the center frequency that determines the bandwidth of the delay or

advancement effect also contains a term proportional to the signal power. We have also


observed this effect experimentally and it is shown in the data illustrated in figure 5.6,

where several traces of fractional delay versus modulation frequency are shown for

different values of the background signal power.

Figure 5.6: Frequency and pump power dependence of the fractional delay observed in propagation through erbium-doped fiber. There is no 980-nm pump field. The signals are sinusoidally modulated with Im/I0 = 0.08

In accordance with what would be expected from the modulation data, we observe

that Gaussian pulses propagate through an EDF with an effective velocity that is either

slow or superluminal depending on the pump power. Figure 5.7 shows plots of fractional

advancement as a function of pulse width. In one case, the EDF was pumped at 12mW,

and in the other, the pump was turned off. Insets show sample time traces of the advanced

and delayed pulses (with normalized intensity) and windows indicating the corresponding

domains of the modulation data for comparison with figure 5.3. For the unpumped case,

the maximum fractional delay recorded was 0.055, corresponding to an effective pulse


velocity of c/(1.2 × 104). The largest fractional advancement observed was 0.092, or a

pulse velocity of c/(−5600). For pulses slightly shorter than the inverse of the hole

bandwidth, we notice significant distortion in the transmitted pulse envelope.

Figure 5.7: Fractional advancement vs. the log of the inverse pulse width in the regimes of slow light (a) and superluminal propagation (b). The x-axis is the log of the inverse of the pulse width (in seconds), so that the width data may be easily compared to modulation frequency data, as shown in the inset. Other insets show sample time traces of the pulses. For the slow light case, the peak input power was 0.8 mW and there was no pump. Power transmission was about 0.1%. For the fast light data, the pump power was 12 mW and the signal power was weak enough to make negligible its contributions to hole broadening and pulse delay. In this case, the signal experienced a total gain factor of 5. Fractional advancement/delay was calculated with respect to twice the FWHM.

In our work we have observed only relatively small fractional advancements and

delays. Even these small timing changes could be useful for certain technological

applications, such as centering a data pulse into a time window, an important step in the

process known as data stream regeneration. For certain other applications, larger


fractional delays or advancements are desirable. The delays and advancements that we

observed are limited primarily by pump depletion effects. Greater delays and

advancements could be obtained by pumping the erbium- doped fiber from both ends or

by cascading several fiber stages.

In summary, we have demonstrated delayed and advanced pulse propagation in an

erbium-doped fiber, the effect being tunable from one regime to the other by varying the

pump power. We were able to achieve significant fractional delays and advancements for

pulses at the technologically important wavelength of 1550 nm.

5.2 Observation of backwards pulse propagation

In the previous section, we showed how 1550-nm pulses could actually experience

temporal advancement when propagating through an optically pumped erbium-doped

fiber. Significantly, the effective group velocity is seen to be not greater than c but is

instead actually negative, a phenomenon that as we recall occurs whenever the dispersive

term of the group velocity, !0dnd!

, is negative and of greater magnitude than n. In such a

situation, it is predicted that the peak of the transmitted pulse will exit the material before

the peak of the incident pulse enters the material, and furthermore that the pulse will

appear to propagate in the backward direction within the medium [96–98]. Although the

first of these effects, equivalent to the occurrence of negative time delays, has been

observed by previous workers [27, 68, 99–103], the second of these effects has

apparently not been previously observed. Indeed, it has not been entirely clear whether


the theoretical prediction of backward propagation is sufficiently robust that it could be

observed under actual laboratory conditions. This question is particularly relevant to the

case of saturable-gain-based fast light. It is understood that the physics of pulse

propagation in the gain medium differ from those of purely material fast light media, and

we have seen previous cases of pulse advancement where there is an apparent effective

group velocity based on the input and output pulse properties that does not correspond to

the position and movement of any real pulse peak inside the medium [42].

We report on our investigations of backward propagation of an optical pulse through

an erbium-doped optical fiber (EDOF) that is pumped in such a manner as to produce a

negative value of the group velocity. By measuring the time evolution of the pulse

intensity at many points within the fiber, our results demonstrate that the peak of the

pulse does indeed propagate in the backward direction within the fiber. However, the

energy flow is always in the forward direction, as the velocity of energy transport is equal

to the group velocity only under special cases, notably the absence of gain or loss in the

medium [104]. We note that even when the group index is negative, it is not necessarily

easy to observe backward propagation, because this effect can be limited or obscured by

competing effects such as pulse broadening and breakup as a result of dispersion of the

group velocity [64] or by severe spectral reshaping of the pulse [101].

Using an EDOF in the amplifier configuration enables us to control the values of the

unsaturated gain g0 and the spectral width of the dip in the gain profile by varying the

power of the 980-nm laser beam used to pump the amplifier. We were thus able to

optimize the value of the time advancement associated with the group velocity [30]. The


large physical length of the material enabled us to readily measure the time evolution of

the optical pulse at many locations within the fiber.

In the experimental setup, seen in figure 5.8, the 1550-nm diode laser produces the

probe pulses for our measurements. The probe laser beam is sent through an isolator,

after which part of the beam is split off and sent to an InGaAs photodiode for use as a

reference.

Figure 5.8: Experimental setup. The 980-nm laser acts as a pump to establish gain in the EDOF amplifier. Pulses or modulated waveforms from the 1550-nm laser probe the propagation characteristics of the fiber. A WDM combines these beams before the fiber, and an optical filter isolates the 1550-nm beam after the fiber. The time evolution of the pulse within the fiber is monitored by successively cutting back the length of the fiber and measuring the output waveform. (b) Alternative experimental setup used to determine the direction of energy transport in a material with a negative group index. Three 3-m sections of EDOF were placed in series, with bidirectional 1% taps placed between each pair of sections. A WDM was connected to each of the tap outputs to separate the signal wavelength from the pump wavelength. The bidirectional taps allowed measurement of energy flow in both directions.


The remaining light is combined with the 980-nm pump beam with the use of a

wavelength-division multiplexer (WDM) and the two beams are sent through the EDOF

coil. The exiting beam is collimated by a microscope objective and filtered to remove the

980-nm pump light before being focused onto a germanium photodetector. This signal

and the reference are recorded by a digital storage oscilloscope.

Signal and reference traces were recorded for two different waveforms: a 1-kHz sine

wave and a 0.5-ms (full width at half maximum) Gaussian pulse. In both cases, the pulse

or sinusoidal waveform was superposed on a large constant background of an intensity 10

times that of peak modulation height. The presence of a large background reduces pulse

distortion effects. The powers of the 980-nm pump and 1550-nm signal fields at the input

to the EDOF were 128 mW and 0.5 mW, respectively. Example traces are shown in

figure 5.9.

Figure 5.9: Input and output waveforms after propagation through a 6-m length of erbium-doped fiber. The output waveform is seen to be advanced in time and to experience slight distortion. Pulse heights are normalized to facilitate comparison of input and output pulse shapes.

In this experiment, we measured an effective group velocity of –75 km/s and a group

index of –4000. Measurements were first taken using a 9-m length of EDOF. The fiber


length was then reduced by about 25 cm by cutting the fiber, and the measurement was

repeated. This procedure was continued until there were only several centimeters of fiber

remaining. In this way, the time evolution of the pulse could be determined at many

points along the length of the fiber.

By arranging the traces according to position and viewing them back in time

sequence, we can observe the pulse evolution within the fiber. Images displaying the

relevant behavior are shown in figure 5.10. In constructing figure 5.10(a), the waveforms

at each spatial location have been normalized, removing the effects of the gain within the

fiber; figure 5.10(b) presents the data without this normalization. In each case, the

background has been removed.

In both figures, the peak of the transmitted pulse is seen to leave the fiber before the

peak of the incident pulse enters the fiber. We also see that as the pulse exits the fiber, a

small peak is created inside the fiber that moves in the backward direction, linking the

input and output pulses. The apparent backward propagation occurs because of reshaping

of the pulse profile within the gain medium as a consequence of time-dependent energy

transfer between the pulse and the gain medium. There is no energy flow in the backward

direction. Also, there is no violation of causality, as the peak that exits the material grows

out of the rising edge of the input pulse, just as the peak of the input pulse becomes part

of the tail of the exiting pulse.

As a further investigation into the nature of negative group velocities, we performed a

second experiment to determine the direction of energy flow within the medium. The

layout is shown in figure 5.8(b). We observed that the signal strength measured at output


ports A and C of the bidirectional 1% taps was barely above the noise floor of our

detection system and was consistent with the small amount of back-reflection expected

from the large number of splices present in this configuration.

Figure 5.10: Time evolution of the pulse as it propagates through the fiber. The peak of the transmitted pulse is seen to exit the fiber before the peak of the incident pulse enters the fiber, and inside the fiber the peak moves from right to left as time increases. The arrows mark the peak of the pulse before entering the fiber (left), within the fiber (center), and after leaving the fiber (right). The time intervals are in milliseconds. In (a), the data have been normalized at each point in the fiber to remove the effects of gain. In (b), the pulse is still seen to propagate in the backward direction, but with a different value of the pulse velocity from that of in figure (a) as a consequence of the influence of gain. The time intervals are in milliseconds.

In contrast, strong signals were measured from ports B and D, thus demonstrating that the

energy flow was only in the forward direction, even though the group velocity was


negative. We also observed that the peak of the pulse arrived at port D before it did at

port B, thus confirming the backward motion of the peak of the pulse within the optical

fiber.

Our experiment shows that within a medium with a negative group velocity, the peak

of a propagating pulse does in fact move in the backward direction, even though energy

flow is always in the forward direction. These results can be understood in terms of the

time dependence of the saturation of the gain of the material, whereby the leading edge of

the incident pulse experiences more gain than does the trailing edge. Thus, the peak of

the pulse within the medium occurs initially at the distant end of the fiber and

progressively moves toward the front end of the fiber. Furthermore, all of these results

are consistent with the principle of causality in that these effects are initiated by the far

leading edge of the pulse.

5.3 Reducing distortion in saturable gain pulse advancement

In the above two sections, we have presented two different experiments in which we

used saturable gain in an erbium-doped fiber to cause a temporal advancement of the

transmitted pulses. Time traces of these pulses compared to their references can be seen

in the inset of figure 5.7(b) and figure 5.9. Observers may note that there is a difference

in the shape of the output pulses in each case. In figure 5.7(b), the pulse has been slightly

broadened by its interaction with the erbium-doped fiber, but in figure 5.9, we see a pulse

that has been narrowed by about 10%. The key experimental difference lies in the fact

that the first pulse was carved with a chopper that left no signal background power,


meaning that the slow light effect had to operate in a largely nonlinear regime. The

second pulse was transmitted on a constant background of roughly ten times the

modulation depth, creating a case where the linearized analysis more closely

representative of a traditional material slow light medium can be applied.

We can describe these two effects, pulse broadening and pulse compression, in terms

of two competing mechanisms. Pulse broadening is caused by a time-dependent

saturation of the amplifier gain. Without any signal background power, the amplifier gain

is quickly depleted by the leading edge of a pulse, but a strong applied pump field can re-

excite the medium. This gain recovery occurs over a characteristic time that depends on

both the lifetime of the metastable state and the applied pump intensity. If the input pulse

duration is comparable to or longer than the gain recovery time, the trailing edge of the

pulse may experience this recovered gain, broadening the pulse [105].

Pulse compression, on the other hand, can be induced by superposing a pulse on a cw

background that is larger than the saturation intensity of the EDFA (Isat ~ 3 kW/cm2 at

1550 nm). This creates a situation analogous to a conventional fast light pulse on a

material absorption resonance. In this case the wings of the pulse spectrum experience a

larger gain than the central frequency components, thereby broadening the pulse

spectrum and compressing the pulse in the time domain. Cao et al. explained this pulse

compression with the negative second derivative of the absorption coefficient and derived

the pulse compression and the advancement factors [103]. This mechanism, which we

refer to as pulse spectrum broadening, competes with gain recovery.


In this section, we will show that by balancing these two mechanisms, we can

regulate the amount of broadening or compression experienced by a pulse during

propagation through an amplifier. Compensating for pulse distortion is frequently the

major limiting factor in the effectiveness of slow and fast light systems, and many

investigations of slow and fast light systems have focused on different techniques for

distortion reduction and compensation [30, 37, 85, 106]. Here, we conduct a series of

experiments to show how pulse distortion in saturable-gain fast light can be minimized

with an appropriate selection of the background power to pulse power ratio.

The experimental setup is similar to those used in the above two sections. Here, the

Gaussian pulses are carved out of the laser field with an electro-optic modulator. A

reference of 2% of the signal power is split off to be used as a reference. The 980-nm

pump field is arranged to co-propagate with the signal field in the EDF and after

transmission through the EDF, is filtered from the signal with a WDM.

As a figure of merit, we will take the distortion parameter D to given by [107]

D =Pout t + !t( ) " Pref t( ) dt

"#

+#

$Pref t( )dt

"#

+#

$

%

&''

(

)**

1/2

"Pref t + +t( ) " Pref t( ) dt

"#

+#

$Pref t( )dt

"#

+#

$

%

&''

(

)**

1/2

(5.9)

where Pout(t) and Pref(t) are the normalized output and reference power envelopes,

respectively, Δt is the measured time advancement of the pulse peak, and δt is the

temporal resolution of the detection system. The first term represents the distortion

caused by pulse reshaping and the second term is a subtraction of the noise contribution.

It is instructive to look at fractional advancement and pulse shape distortion D, as

plotted against the background to pump power ratio. In figure 5.11, we plot these


quantities for Gaussian pulses 10 ms in duration for several different values of the pulse

power. Pump power is fixed at 35 mW; the 1550 background power is varied to change

the ratio along the horizontal axis. The theoretical curves are produced by computer

simulations of pulse propagation in EDF using the five level system rate equations for

erbium ions, including amplified spontaneous emission [108].

Figure 5.11: (a) Experimentally measured (symbols) and theoretically predicted (curves) fractional advancement versus background-to-pulse power ratio for different pulse powers. (b) Experimentally measured (symbols) and theoretically predicted (curves) pulse-shape distortion versus background-to-pulse power ratio. Inset: experimentally measured input (dashed curve) and output (solid curve) pulse waveforms illustrate distortion accompanied by (1) broadening, (2) no pulse-width distortion, and (3) compression.

We can see several interesting things from these plots. First, we observe that for pulse

powers in the vicinity of 50 µW to 100 µW there is very little dependence of the

fractional advancement on the power ratio, except for very small ratios (no background).

This effect can be understood as the pulse’s ability to effectively provide its own


background to a small extent becoming relevant when the real background goes away.

Also, it is apparent from looking at figure 5.11(b) and its insets that there is indeed a

compensating effect between the two causes of pulse distortion. For low power ratios,

there is large distortion due to gain recovery broadening, at high power ratios, we also see

distortion, but because of pulse narrowing from pulse spectrum broadening. In between,

around a power ratio of 0.75:1, we see pulses with minimized distortion as the effects

appear to cancel out.

In many systems, it may be more convenient to use a separate laser for generating the

background field than to use a single laser with an electro-optic modulator (EOM), and in

this case the background field and pulse are generally incoherent with each other. When

the signals, for instance, are return-to-zero pulses, adding the background field from

another laser is more convenient than modifying the signals with an EOM. If we assume

that the dipole dephasing time T2 of the material system is much smaller than the ground

state recovery time T1 and the inverse of the detuning frequency (or modulation

frequency) Δ [95], the pulse advancement is sensitive only to intensity modulation of the

incident laser power. In this case, a background field incoherent to the pulse should have

same properties as coherent background field in the pulse-on-background method.

To test this idea, we performed an experiment with an incoherent background field. A

Gaussian pulse of 10 ms duration and 60 µW pulse power on no background is generated

by using an EOM at a wavelength of 1550 nm, another cw laser is used for a background

power at 1547 nm, they are combined at a y-junction, and then co-propagate. The 980-nm

pump power Ppump was fixed at 17.5 mW. In figure 5.7 we plot the pulse-shape distortion,


as well as the fractional advancement against the background-to-pulse power ratio for

coherent and incoherent background fields.

Figure 5.12: Experimentally measured pulse-shape distortion (left axis) and fractional advancement (right axis) versus background-to-pulse power ratio for coherent (circles) and mutually incoherent (diamonds) background fields to the pulse. The curves are guides for the eye. Inset: Input pulse waveform. Ppulse and Pbg represent the power of the pulse and the background field.

It can be seen from the traces in figure 5.12 that the pulse-on-background method

works identically within experimental error whether or not the background 1550 power

comes from a source that is coherent with the signal pulse.

5.4 Saturable absorption pulse delay with 17 GHz bandwidth

So far, we have used only erbium-doped fiber as our saturable medium. However, in

slow and fast light based on saturable absorption and gain, the pulse bandwidth that can

be accommodated is limited by the inverse of the gain or absorption recovery time of the

medium. For EDF, this metastable state lifetime is about 10.5 ms, and while this is

effectively power broadened by both the pump and signal, as expressed in equation 5.2, it


still is unlikely to allow the delay or advance of pulses shorter in duration than 100 µs.

Pulses used in telecommunications have a duration of a few to tens of picoseconds, and in

order to build delay lines for them, for example for retiming applications, materials with

corresponding saturable absorption recovery times are necessary. Bulk semiconductors

are not suitable because they have two recombination times, one in the tens to hundreds

of fs (due to intraband thermalization) and one in the ns range (recombination time)

[109].

We have directed our attention towards PbS quantum dots (QDs); the recovery time

of their absorption is around 25 ps [110]. Other demonstrations of slow light in QDs have

been presented in [111] and [112]; we apply the same concept to a different pulse

duration. For our study we used PbS QDs produced by Evident Technologies (“Snake

Eyes NIR”) suspended in toluene and with an absorption peak centered at 795 nm. Figure

5.13 shows the absorption spectrum of the 1 cm thick sample, as measured with a

spectrophotometer.

Figure 5.13: Absorption spectrum for PbS quantum dots suspended in toluene.


As we have seen in previous sections, while the existence of a continuous background

signal power can be useful for, the dynamics of saturable pulse delay can be realized with

a single pulse experiment; the delay of a single pulse through the material as a function of

the pulse energy is related to the saturation intensity and tests whether the recovery time

is well matched to the pulse duration [92].

We tested our material with 25 ps pulses at 795 nm produced by an OPA pumped by

the third harmonic of a mode locked Nd:YAG regenerative amplifier system. The

saturation fluence with our pulses is on the order of 4 mJ/cm2, and for higher fluences the

material shows saturation of the absorption. Because do not have detectors and

electronics capable of timing a 25 ps pulse directly, we measured the timing of the pulses

with a correlation technique.

Figure 5.14: Correlation setup used to measure the delay of 25 ps pulses through the quantum dot sample. Two separate optical paths lead to a focus on a GaAsP LED. One path is transmitted in a single pass through the quantum dot sample, the other is reflected off a scanning stage so that pulse delay can be measured. The LED responds to the two photon signal, and therefore indicates clearly when pulses from each path are overlapped in time.


We placed the 1 cm thick QD sample in one arm of an intensity autocorrelator and we

took autocorrelation traces of pulses of different energies, with the expectation that low

energy pulses will not see any saturation or slowing effect and will therefore constitute an

effective reference. Figure 5.14 shows a diagram of the experimental setup.

Figure 5.15 shows a scan at low intensity, in which there are no nonlinear effects, and

a scan at an energy that corresponds to the saturation fluence. The comparison between

the two scans shows that the high energy pulse is delayed by 2.8 ps and broadened with

respect to the low energy pulse. The delay (10% of the pulse FWHM) is in good

agreement with the model of [92] for a saturation recovery time of 30 ps.

Figure 5.15: Correlation traces of high intensity and low intensity pulses transmitted through the QD sample. We observe a delay of 2.8 ps in the higher intensity pulses, approximately 10% of the pulse width.

The concept demonstrated in this research is easily extendable to a wavelength in the

telecommunication range using quantum dots of a different size (and therefore a different


absorbance). Additionally the dots could be embedded in a resin or glass host to form a

solid-state delay medium. Such samples would be more stable than quantum dots

suspended in toluene.

6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 80

Chapter 6

Slow Light Detection and Ranging (SLIDAR)

In this chapter, we propose and demonstrate a new application for slow light. Namely

we develop a proof-of-concept system for a coherently combined multi-aperture slow-

light laser radar. By employing slow-light delay elements in short-pulse-emitting systems

to ensure synchronized pulse arrival at the target, we show that it is possible to

simultaneously achieve high resolution in the transverse and the lateral dimensions with a

wide steering angle. We also show that this concept can be used in a two-dimensional

system, with two independent slow-light mechanisms, namely dispersive delay and

stimulated Brillouin scattering, used to dynamically compensate the group delay

mismatch among different apertures

6.1 Background and introduction

Light Detection and Ranging (lidar) systems have been used for many applications,

including atmospheric sensing, chemical and biological agent detection, and aerial

surveying [113]. Similar to radar, lidar technology offers the key advantage of improved

spatial resolution because of the shorter wavelength of the radiation used. Systems that


require detection over an angular range typically mount their emitting optics on a gimbal.

However, coherent optical phased arrays have drawn interest for their ability to generate

and sweep an optical beam without the use of large, mechanical steering elements

[114,115]. In many current optical sensing systems, the large mass associated with these

elements restricts the ability of the beam to steer quickly, and even where rapid steering

is possible, it is generally power-intensive [114]. Phased-array steering is an alternative

without these disadvantages and may also be more compatible with the aerodynamic

requirements of aircraft-based sensor systems. The development of these arrays relies on

several other technologies including coherent beam combining, optical phase locking,

microwave phased-array beam steering, and in our case, slow light, each with their own

paths of development.

Coherent combining of the output from two 10-W ytterbium-doped fiber amplifiers

was accomplished by Augst et al., in which the channels were phased-locked to a

reference with active feedback loops using acousto-optic modulators [116]. Serati et al.

implemented side-by-side beam combining in two dimensions using a phased array of

phased arrays (PAPA). In this case, phase locking was achieved via a digital loop with

feedback from a camera in the far field and a liquid crystal phase control element [117].

Another method of phase control was developed and demonstrated by Shay et al., who

locked 9 channels with a phase accuracy of λ/20 using a technique they call LOCSET

[118,119]. In LOCSET, no independent reference is required as each emitter is phase-

modulated at a different frequency. The central point of the far field is imaged onto a


detector and, with signal processing, the appropriate error signal is recovered and sent to

a phase modulator in each channel.

Optically-induced true time delay has long been recognized to be a promising

application for slow light in the field of microwave beam steering [120]. Johns et al.

showed that highly-dispersive optical fiber and a tunable diode laser source could be used

to produce variable time delays in microwave signals [121]. A more recent system for

incorporating optical dispersion techniques into microwave phased-array antennas was

presented by Muszkowski and Sędek, who model the effect of differential channel

dispersion on emitted beam direction [122]. Bashansky et al. have recently demonstrated

that stimulated Brillouin scattering based slow light can be incorporated into the design

of a phased-aperture microwave radar [123–125].

In the optical wavelength regime, beam steerers have been made based on different

types of phase-shifting arrays, including liquid crystal phase shifters, and a device based

on an integrated-optical array of AlGaAs waveguides with indium tin oxide/AlGaAs

Shottky junctions [114,126]. Also, Xiao et al. have made use of a tunable source and

dispersive delay to steer a beam created by a coherent array of emitting waveguides on a

chip [127,128].

In this chapter, we propose and demonstrate a proof-of-concept multi-aperture laser

radar system operating in the pulsed mode. Our system uses optical phase locking

techniques to control the phase relation among multiple signal channels (sub-emitters).

We also incorporate slow light to control the relative delay of pulsed signals among the

channels to ensure that the emitted signals always arrive at the target simultaneously.


6.2 Theory

There are many applications where one may wish to probe the environment via active

scanning of a laser beam. In such cases, various factors will determine the longitudinal

and transverse resolutions of the system. When using time-of-flight detection, the

longitudinal resolution of a laser radar will be primarily limited by the duration of the

emitted pulses of light, a shorter pulse giving more precise positional information. On the

other hand, the transverse spatial resolution of a laser radar will be limited roughly by the

focused beam size in the far field. The smallest spot to which an optical beam can be

focused is roughly the diffraction-limited spot size:

Rfar field =1.22L!D (6.1)

where λ is the wavelength, L is the distance from the emitter to the target plane and D is

the diameter of the emitting aperture. We can see that the far-field transverse resolution

will scale inversely with the aperture size, and therefore a large aperture is desirable.

However, use of a single large aperture requires bulky opto-mechanical components for

focusing and steering. One method for obtaining a large effective aperture without

requiring a single large emitter is to use a spaced, phased array of smaller emitters. With

careful design of the sub-aperture spacing, the resolution of a multi-aperture system can

be comparable to that of a single large-aperture system [129,130].

In phased-array beam-steering, the individual phases of an array of wave-emitters

need to be carefully synchronized so that the fields from different apertures will interfere


in a controllable way and produce the desired transverse beam profile in the far field. In

the commonly desired case of a concentrated lobe of power at a given angle, the phases

are adjusted to produce constructive interference at that angle and destructive interference

in other directions. Phased array technology potentially offers improvements in steering

speed and removes the need for a large, moving mechanical component. If we disregard

fine phase control within apertures, the situation can be considered as an N-slit diffraction

pattern, which has a far-field intensity given by:

I !( ) = I0sin "( )"

#$%

&'(

2 sin N)( )sin )( )

#$%

&'(

2

(6.2)

where ! = "a / #( )sin $( ) , and ! = "d / #( )sin $( ) , with a and d representing the slit width

and spacing between neighboring slits, respectively. The first sinc2 factor is a broad

envelope whose primary lobe width scales as λ/a. The fine fringes are defined by the

second factor, and have a central lobe width proportional to the inverse of D ≡ (N–1)d,

the total width of the aperture array. This relationship tells us that if we want to build a

scanning system with high angular resolution, we must use an appropriately wide array of

emitters, bearing in mind that too low a fill fraction a/d will result in an unsuitably low

fraction of the power going to the central lobe.

An illustration of the enhanced resolution effect can be seen in figure 6.1. It can be

seen that increasing the total aperture width increases for a sparse-aperture system while

holding the emitting area constant can indeed reduce the angular spread of the narrowest

emission feature, although this comes at the expense of spreading the power out over

many such lines. The practical difficulties for a LIDAR system caused by this effect can


be reduced by using a slightly randomized aperture spacing. Under such a configuration,

all side-lobes other than the central peak are blurred together, so that the central peak can

be much more easily distinguished from the background power.

Figure 6.1: Illustration of the far-field power profiles for three different layouts of 9 identically-sized apertures (each of the nine apertures is 3 units by 3 units). The plots are made of power vs. angle. Units are arbitrary, but the scale is the same along both axes for all plots. In (a), the aperture is a single 3 unit x 27 unit bar. In (b), the 9 3-unit apertures are spread evenly over a 200 unit width. In (c), the apertures are spread over the same total width as in (b), but with a slightly randomized spacing.

While phased-array systems have been used extensively for radio and microwave

frequencies, the short wavelength of optical radiation presents special challenges. The

requirement for a significant fill fraction and a tolerance to phase error at the target that

may be on the order of λ/10 means that for large-scale devices, we are likely forced to

adopt a hybrid approach [131]. Instead of having a huge number of emitters with width

on the order of a wavelength, we have several large apertures, with “sub-aperture”

control of the phase ramp in each element provided by a spatial light modulator, a

mechanical element like a Risley prism pair or, in the case of our demonstration, simply a

mirror.

Furthermore, the simultaneous requirements for good far-field transverse resolution

(large D), good longitudinal resolution (short pulse duration τ), and significant steering


angle θ, create a problem in that for wide-angle scanning, simultaneously emitted signals

will arrive at the target at slightly different times and will not overlap to interfere

properly. We provide an all-optical solution for this problem by incorporating slow-light

tunable delay elements into each channel. A diagram illustrating the central concept of

such a system for Slow-Light Detection and Ranging (SLIDAR) is presented in figure

6.2. Pulses are carved from the output of a laser, which is then split into an array of

channels. The relative path difference between the two farthest separated emitters is

given by D sin !( ) . One sees that if this distance is a substantial fraction of the pulse

length c/τ it becomes essential to use slow light delay elements to compensate.

Figure 6.2: Conceptual diagram of a SLIDAR system. Each channel must contain independent phase control, and group-delay control as well, if the desired steering angle is too large for a given pulse duration.

6.3 Setup

A partial schematic of our fiber-based SLIDAR experimental setup is shown in figure

6.3. A single optical reference channel is split off, and the remaining signal is pulse-

carved by an intensity-modulating elecrtro-optic modulator (EOM) and divided into a

number of channels leading to phased emitters. Each channel contains 1.1 km of optical

fiber with different proportions of standard single-mode fiber (SMF) vs. dispersion


compensating fiber (DCF) for group delay control, a phase modulating EOM with

feedback to provide phase control, and an erbium-doped fiber amplifier (EDFA) running

in constant-output mode before the output collimator to increase the emitted power and

ensure a stable output level. Polarization controllers are needed before every electro-optic

modulator to ensure that the input is linearly polarized and properly aligned relative to the

EOM’s fast axis.

Figure 6.3: Partial SLIDAR schematic, showing one channel with the reference and a block diagram of the electronic feedback path. EDFA: Erbium-doped fiber amplifier. EOM: electro-optic modulator, POL: fiber polarizer, φ: electro-optic phase modulator, AOM: acousto-optic modulator. Red lines connecting elements in the diagram represent optical fiber; black lines represent electrical connections. Polarization control is also needed to produce high-visibility interference patterns

between the signal and reference. Half-wave plates after the output couplers of the three

channels are used to ensure high-visibility interference of the free-space emission.

Our goal is to use 1–4 ns pulses in the near infrared (1550 nm) to imitate a system

that observes targets at a range of over a kilometer with a full effective aperture of 1

meter. Because of space and power restrictions, our experimental target is approximately


6 meters away, with 2.1 mm diameter apertures (FWHM) and a full effective aperture of

~6.6 mm. This setup makes the ratio between distance to the target and the effective

aperture ~910:1, on the order of the ratio of 1000:1 that we are considering for a

hypothetical full-scale system.

6.4 Phase control

As with all phased-array systems, it is necessary to control the phase of each emitter

in order to maintain the desired phase relation among different apertures. Because of the

thermal and vibrational noise that our signals were exposed to when traveling through

long fiber-optic channels, phase locking proved to be a significant challenge.

In our first attempt at phase locking, we locked two combined beams at a fixed

intensity output. Two channels were combined in a fiber coupler and the output of one

channel sent to a detector. This was fed into the inverting input of an op-amp which had

its output voltage sent directly to an EOM in one of the channels. Negative feedback was

provided by the op-amp as it adjusted its output signal to equalize the signal coming from

the reference detector with a control voltage sent to the non-inverting input. This simple

circuit is shown in figure 6.4, and the results of the experiment are shown in figure 6.5.

Figure 6.4: Circuit diagram for “intensity” phase lock. The capacitor at the output was added as a low-pass filter to reduce op-amp’s tendency to overcompensate and go into a “ringing” mode.


We can see that even simple phase locking based on the intensity can be moderately

effective, but the technique has several drawbacks. Slow time dependent variations in

signal attenuation in different channels would require the control voltages to be

frequently re-adjusted. This circuit lacks both a mechanism for maintaining phase lock

when the op-amp is driven to one of its ±15 Volt rails, and a purely electronic feedback

path that would create a more stable operation.

Figure 6.5: Results of the “intensity-based” phase lock. The oscillating blue trace is the optical signal in the absence of phase locking. The two constant traces red (upper) and green (lower) display the combined signal with an activated phase lock for two different values of the control voltage, 3.8 volts and 8.6 volts respectively. (The three traces are plotted together, though not actually taken concurrently.)

Instead, we have developed a heterodyne-locking scheme to phase lock every emitter

to the reference that is split off before the slow-light elements. The scheme is illustrated

in figure 6.5, where only a single signal channel is shown for simplicity. Each channel is

independently phase-locked to the reference optical field, whose frequency is shifted by

55 MHz using an acousto-optic modulator (AOM). The reference field is split with a star


coupler so that a portion of the reference can be interfered with each output channel. This

optical beat signal is detected, mixed with a 55 MHz local oscillator in an RF mixer, and

passed through a 1 MHz low-pass filter to give a low-frequency error signal representing

the phase difference between the optical beating and the local oscillator. The error signal

is input to a proportional-integral controller circuit that regulates the phase-modulating

EOM and keeps the phase of the channel locked to the reference. Fine control of the

relative phases between the output channels is obtained by modulating low-frequency

voltage signals sent to electronic phase shifters in the control circuits. The circuit also

incorporates a “snap-back” mechanism that suddenly resets the voltage to the EOM by

2Vπ when necessary to prevent a continuous ramp of phase noise from outrunning the

voltage range of the phase-locking circuit.

The operation of the snapback circuit is as follows: The half-wave voltage (Vπ ) of our

EOM is nominally 4 V, and the loop filter uses ±15 V supply voltages. When the loop

filter’s output VEOM exceeds 3Vπ (nominally 12 V), an analog switch engages and strongly

drives VEOM toward negative voltages. Once VEOM is below Vπ (nominally 4 V), the analog

switch disengages, and the loop filter resumes tracking, having shifted its operating point

by 2π radians of phase error.

Several issues of detail are illustrated in figure 6.6. In our design, the loop filter is an

inverting circuit (negative voltage gain), so to drive VEOM negative, the analog switch

connects the positive voltage supply to the loop filter’s input summing junction. The

analog switch disengages the snapback by connecting signal ground to the input summing

junction; adding 0 V to the summing junction in this way is conceptually the same as


disconnecting the snapback circuit from the loop filter, but the signal inputs are not left

floating. The logic of engaging and disengaging the analog switch is handled by two

analog comparators and a set-reset (S-R) latch. Careful attention must be paid to the logic

sense of each input or output (normal or inverted). A complementary circuit (not shown)

handles negative-voltage snapbacks, triggering when VEOM goes below −3Vπ and

disengaging when VEOM returns above −Vπ . Allowing the phase error to wander over a

range of ±3π before activating the snapback reduces the frequency of snapback events,

giving a lower residual phase error than a naive approach that simply tracks the phase

error modulo 2π.


Figure 6.6: Block diagram of the proportional-integral (P-I) controller, including the loop filter and the fast 2π-phase snapback circuit. EOM: electro-optic phase modulator; LPF: low pass filter; Comp.: analog comparator (output is logical 1 when voltage at positive input exceeds voltage at negative input). Bars over input names indicate inverted-sense (active low) logic inputs. Note that only the positive-voltage arm of the snapback circuit is shown here; a complementary circuit provides negative-voltage snapback functionality.

The entire snapback process takes about 1 µs. Since this is longer than the response

time of the loop filter, the circuit parameters must be adjusted to produce an optimal

snapback effect. Specifically, the DC voltage levels used by the analog comparators and

the value of the resistor that determines the snapback circuit drive strength were tuned

empirically. These values may also be chosen to cause a phase snapback of integer

multiples of 2π radians. In our system, the snapback circuit was tuned to provide a 4π

snapback. We found that the phase error tended to drift continuously in one direction


(either positive or negative), presumably due to environmental temperature changes, and

the 4π snapback allowed a longer time between snapback events.

The operation of the snapback circuit can be seen in figure 6.7. The EOM output

voltages of two channels locked to a reference are plotted simultaneously with their

combined optical signal as the lock is turned on.

Figure 6.7: SLIDAR phase locking of two independent channels, shown concurrently with the control signal sent to the two EOMs. Two snap-back events can be seen, but there is no significant discontinuity in the interference signal.

Using the setup diagrammed in figure 6.3, we have successfully locked the phases of

three channels, each containing a few kilometers of single-mode fiber for producing

tunable slow light. Figure 6.8 shows the phase locking performance when the three


output channels are combined in a fiber and allowed to interfere. The system begins with

only one channel locked to the reference (i.e. no phase lock between any output channels)

and the full intensity fluctuations of the phase noise can be seen.

Figure 6.8: SLIDAR phase locking. Clearly visible are the regions with no channels locked (left), two of the three channels are locked together (center), and all three channels locked (right). Note that this signal is measured off of a reference detector that is effectively conjugate to the emitted signal, so the system output is high when this signal is low.

Roughly 12 seconds into data collection, a second channel is locked to the reference, and

much of the intensity fluctuation vanishes. At approximately 17 seconds, the third


channel is also locked, and while some jitter remains, the relative stability of the

combined signal is evident. With averages taken over about 20 s, we measure the RMS

residual error to be approximately π/5 radians (1/10 wave), comparable to results

obtained in several other optical phase locking systems [132,133]. This error remains

roughly constant regardless of how the slow light is tuned in the system, holding steady

for tests using dispersive delay as well as SBS.

6.5 Slow light demonstration and system tests

To control the relative group delay, we use dispersive slow light, which utilizes the

difference in frequency-dependent refractive index of different types of optical fiber to

produce tunable relative delay by controlling the signal wavelength. The differential

delay is given simply by

!T =Lcng "1( ) # ng "2( )( ) (6.3)

or ΔT = L D Δλ, where L is the fiber length and D is the fiber dispersion. In our

experiment, we use a tunable diode laser (Koshin Kogaku LS-601A) as the light source.

We use standard single mode fiber as a reference medium and highly dispersive optical

fiber, similar to the fiber used in telecom dispersion-compensating modules, as the slow-

light medium. We have measured that a 550 m length of this fiber has a dispersion of −73

ps/nm. Each of the three channels in our SLIDAR system has approximately the same

total length of fiber in each arm, but with different amounts of DCF. The relative group


delay experienced by signal pulses propagating through different channels depends on the

wavelength of the signal field.

Figure 6.9: Demonstration of variable dispersive delay of three SLIDAR channels. In this figure, the relative delay for each channel is set to zero at 1550 nm.

Figure 6.9 shows the relative pulse delays as functions of signal wavelength for the

three channels with varying ratios of DCF to SMF. It is clear that by tuning the

wavelength, we can control the relative delays of the three channels.

Figure 6.10: Comparison of dispersive delay in two channels. Each line is a time trace of a short signal pulse taken in wavelength increments of 1 nm.


Figure 6.10 shows multiple time traces taken at 1 nm wavelength intervals of short

pulses traveling through comparable lengths of DCF and SMF. It can be seen that the

pulses experience significant delays with negligible distortion using this method.

We have tested the SLIDAR system to show the synchronization of pulses from three

channels traveling in free space, simultaneously phase-locked to a common reference. A

diagram of the setup is shown in figure 6.11. The optical signal pulses are put on a DC

background so that the phase-control circuitry has a continuous signal to lock with the

reference. The output pulses of the three channels are sent approximately 6 meters away

where they overlap and interfere to form a far field pattern. The overlapped beam is split

with a beam splitter. One part is directed onto a CCD camera to observe the transverse

beam profile, while the other portion reaches the target. Because we do not have enough

emitted power in our model system, we use a retro-reflecting target, rather than a

scattering one.


Figure 6.11: Setup for SLIDAR system tests. Diagram is not to scale; ~6 meters separate the emitters and the target. (b) Illustration of how translating apertures simulate side-steering of a larger scale system. The top two images represent our setup; the bottom two represent the equivalent pictures for a steered array. The green and blue bars indicate the differences in path length of light from the farther apertures.

For distance resolution measurements, a delay element was placed in the target and

scanned mechanically on a translation stage. For measurements of spatial resolution, a

0.44 mm slit was scanned across the beam at the target. Light retro-reflected by the target

is directed back to a point near the emitters, where is collected with a lens and the signal

is recorded with a 3 GHz amplified InGaAs photodetector.

Laboratory space and power constraints required an innovative approach to

demonstrate the ability to phase-lock and synchronize a SLIDAR system with an

effective aperture of one meter. In order to mimic beam steering of a full-scale system

with a large aperture and long detection range, two of the output collimators were placed

on translation stages.


Figure 6.12: Eye diagrams showing phase locking and pulse resynchronization through dispersive delay. (a) represents a recorded signal at 1535 nm without phase locking and with the emitters in the straight-ahead position. In going to (b), we have turned on the phase locking. When we shift the emitters to imitate an angular sweep, we obtain the signal seen in (c). Adjusting the wavelength to 1542 nm gives us the recombined signal of (d), showing proper synchronization and phase locking. Positioning the stages close to the fixed collimator is equivalent to the situation when the

beam beams are directed straight ahead, while moving the stages back is equivalent to

steering the beam off-axis to one side, as illustrated in figure 6.11b. We imitate an

angular sweep of a full-scale laser radar system by moving the stages and then showing

that we can resynchronize the pulses for every steering angle by controlling the

wavelength of our signal field, with the beams always being phase-locked properly in

each case. The results of the experiment are shown in figure 6.12.

In 6.12(a), the carved pulse is shown without the phase-locking mechanism engaged.

As much of the light is blocked by the slit at the target, the rapidly shifting far-field

interference fringes lead to intensity fluctuations in the return signal. Here, the signal


wavelength is 1535 nm. In 6.12 (b), the locking circuits are switched on, and the intensity

of the return pulse is seen to stabilize. 6.12 (c) shows what happens when we shift the

emitters, showing the pulse-breakup effect that would occur when the output from a

system with a one-meter full-aperture is steered 20 degrees to one side. In 6.12(d), the

laser wavelength has been adjusted to 1542 nm which, as accurately predicted by the

dispersion measurements shown in 6.9, was the amount needed to adjust the relative

pulse delays by 1.14 ns and resynchronize the output. Despite the relative positions of the

emitters, the signal can be resynchronized with a single wavelength shift because of the

differing amounts of DCF in each channel. The channel with the aperture that was moved

the furthest had the most dispersion (−145.6 ps/nm), while the fixed channel had the least

(18.2 ps/nm).

We have also performed an experiment to measure the range-measurement accuracy

of our time-of-flight SLIDAR system. As mentioned above, we mounted a retroreflector

on a scanning translation stage containing at the target position. We calculated the

distance to the target by measuring the return time of the emitted pulses as the stage was

scanned in a series of 1 cm steps. The results are shown figure 6.13. Fitting a line with a

slope of 1.00 cm/step to the data, we find the error to have a second-order moment of

only 0.38 mm.


Figure 6.13: Data showing tracking of an object moving away from the emitter. The inset shows the time traces of the return signal corresponding to the closest (blue) and furthest (red) positions.

Since the stage positioning should be accurate to well under this value, this experiment

demonstrates that the SLIDAR system can measure the distance to the target with very

high precision.

By monitoring the return signal as a narrow object (in this case a slit) is moved

through the beam at the target, we are able to trace out the vertically-integrated intensity

pattern in the far field and estimate the spatial resolution of the SLIDAR in the scanning

dimension. Traces for two different emitter configurations are shown in figure 6.14.

Trace (a) shows the signal produced by two emitters having a 3.3 mm center-to-center

spacing. Trace (b) shows the result if a third emitter is added, keeping the same emitter

spacing and therefore increasing the total aperture to 6.6 mm. It should be possible to

localize a narrow target object to within the lateral width of the SLIDAR system’s central

lobe of power in the far field.


Figure 6.14: Spatial resolution measurements for different beam output configurations. A slit was scanned across the far-field interference pattern of the operational SLIDAR. Trace (a) shows the return signal with two output channels where the emitters were widely spaced. In trace (b), two closely spaced emitters are used. For trace (c), we use three emitters with the same spacing as in (b), and therefore a larger full aperture. The insets beside each data curve show the CCD image of the interference pattern. It can be seen that the addition of a third emitter allows for a narrower central lobe that still contains a high fraction of the beam’s power.

Comparing the results with three and two emitters, we can see that increasing the number

of emitters enables us to narrow the central lobe while keeping a large fraction of the

output power contained within it. In devices with more channels, it is possible to use a

non-uniform spacing between emitters to reduce the peak power of the side-lobes [134].

By blurring the side-lobes into a more uniform pattern, interpretation of the return signal

is greatly facilitated, as suggested in figure 6.1.


We have proposed and demonstrated a proof-of-concept version of a coherently-

combined multi-aperture slow-light laser radar, operating in the 1550 nm wavelength

region. We have also shown that we can use dispersive delay slow-light to address the

problem of mismatched pulse arrival times due to large-angle steering in one dimension.

Phase control of each emitter in our system is successfully maintained with an active-

feedback phase-locking circuit. Tests of our system show good resolution in the

longitudinal and transverse dimensions and suggest the feasibility of a system designed on

a larger scale with many more and larger emitters. The realization of a full-scale system

would require more channels, and larger, more complicated emitter structures. Combining

estimates for the necessary lidar emitter pulse energy for a 20-km slant range through the

atmosphere with well-known scaling laws suggests that required energies required for 1-

km and 5-km remote sensing could be as low as 3.3 µJ and 81 µJ respectively under clear

atmospheric conditions. Hazy atmospheric conditions would increase the required power

by roughly a factor of 10 [135]. To meet these energy requirements, a full scale system

would require more powerful EDFAs at the end of each channel, as compared to the 21

dBm models in our experiment.

6.6 A two-dimensional SLIDAR

In this section, we demonstrate a proof-of-concept version of a two-dimensionally-

scanning coherently-combined pulsed laser radar system incorporating tunable slow light

delay elements.


In a one-dimensional system, it is possible to use only dispersive delay as a means of

group delay control. Using system channels constructed partly with standard single-mode

fiber (SMF) and partly with highly-dispersive, dispersion compensating fiber (DCF), one

can build a linear array, sometimes referred to as a fiber-optic prism, where each channel

has a different dispersion, and tuning the input wavelength smoothly changes their

relative pulse delays [120,121]. In the above section, we tested one-dimensional SLIDAR

system showing that dispersive delay elements could be used to compensate for the

steering-induced path length differences. This technique is very convenient, yet because

tuning of the source wavelength provides only a single free parameter, its extension to a

two-dimensional system is complex [136]. We choose an approach wherein we combine

two independent slow light mechanisms, using both dispersive slow light and stimulated

Brillouin scattering (SBS) based slow light to perform the path-length compensation in

two orthogonal dimensions. The system also requires accurate optical phase control

among all apertures. It therefore contains all of the conceptual difficulties of a

multichannel 2D system.


Figure 6.15: Schematic diagrams of the 2-D SLIDAR system, including illustrations of the front view (a) and top view (b) of the emitter layout. The top view shows two of the output-couplers are mounted on translation stages that are moved to simulate the degree of delay that would be required in a full-scale system. Emitters are arranged in a “right-angle” pattern to create a rectangular interference pattern in the far-field. (c) shows a diagram of the remainder of the system. EDFA: Erbium-doped fiber amplifier. OPL: optical phase lock circut, POL: fiber polarizer, φ: electro-optic phase modulator, AOM: acousto-optic modulator, VOA: variable optical attenuator. Numerous polarization controllers have been omitted for clarity.

To simulate a system with a full aperture of roughly a meter, targeting objects over a

kilometer away with 6 ns pulses, we have built a model that is roughly 1:1000 in scale.

We have three apertures each with an output beam FWHM of 2.1 mm, and a center-to-

center spacing of 3.3 mm. These are shown in a front and top view in Figs. 6.15(a) and

6.15(b) respectively. If the beam is steered in the x direction, we could need to change the


delay of channel 3 relative to channels 1 and 2, which we do with dispersive delay. When

steering in the y direction, we need to change the delay of channel 1 relative to channels

2, and 3, which we do with SBS slow light.

A diagram of the remainder of the 2-D SLIDAR system is shown in figure 6.15(c).

Because we are using SBS-based slow light in addition to dispersive delay, it is necessary

to have a signal that maintains a fixed frequency difference between itself and the SBS

pump, even as the wavelength is tuned to steer the beam in the second dimension.

Specifically, this frequency difference must be equal to the ~10.6 GHz Brillouin

frequency ΩB of the fiber we use in our SBS delay stages. Therefore, the 1550-nm field

emitted from our tunable diode laser is first split into two parts, one part is used for the

SBS pump and the other part is converted into the signal. The signal is modulated at ΩB

in an electro-optic modulator (EOM) aligned for minimum DC transmission. The

modulation frequency must be reduced by 7 MHz/nm as the wavelength of the diode

laser is increased in order to account for the wavelength dependence of ΩB. Sending this

field through a 3.3-km length of counter-pumped fiber amplifies the field at the down-

shifted SBS Stokes frequency while attenuating the part of the field at the anti-Stokes

frequency. After further amplification, this field is then split with part of it being set aside

for use as a continuous reference, and the rest sent to the pulse-carving EOM before

being divided into the individual signal channels. The EOM carves 6 ns pulses and leaves

part of the optical background for use in phase locking.

Each channel has elements for group delay control and phase control. For phase

control, each channel at the output is split and mixed with the split-off reference, which is


shifted in frequency by 55 MHz using an acousto-optic modulator (AOM). The resulting

beat signal for each channel is sent to a phase locking circuit, which feeds back to a

phase-controlling EOM in each channel. Further details of the phase locking setup will be

published elsewhere. For group delay elements, our 3-channel proof-of-concept system

uses one type of delay in each channel. Channel 2 has a 2.2 km length of dispersion-

shifted fiber (DSF). We have measured this fiber to have a dispersion of -0.69 ps/(nm

km) in the vicinity of 1550 nm. Outputs of the other emitters are tilted around this one,

which changes its delay only very minimally with a shift in wavelength. Channel 3 uses

2.2 km of DCF, which possesses in total a dispersion of approximately −291 ps/nm.

Channel 1 also contains 2.2 km of DSF, but it is designed to be strongly counter-pumped

to produce SBS gain and pulse delay. Our pulse train signal has a FWHM frequency

bandwidth of approximately 75 MHz, which is larger than the 20 MHz intrinsic SBS

linewidth of the DSF. To minimize the signal distortion, we here adopt a two-stage pump

modulation method to achieve a flat-top broadened SBS gain profile, as in the

experiments of section 4.2. Our 3-channel proof-of-concept system uses one type of delay

in each channel. Channel 2 has a 2.2 km length of dispersion-shifted fiber (DSF). Outputs

of the other emitters are tilted around this one, which changes its delay only very

minimally with a shift in wavelength. Channel 3 uses 2.2 km of DCF, which possesses in

total a dispersion of approximately −291 ps/nm. Channel 1 also contains 2.2 km of DSF,

but it is designed to be strongly counter-pumped to produce SBS gain and pulse delay.

Since we wish to apply delay to a signal that will have a higher bandwidth than the

natural linewidth of the SBS stokes line, we can modulate the pump to broaden the gain,


minimizing pulse distortion. We use two intensity-modulating EOMs in series to

modulate the pump field. The biases of the modulators are set to make the gain profile as

flat as possible. Once the desired shape of the gain profile is obtained, tunable delay is

achieved by controlling the power of the pump field. The standard deviation of the

measured SBS delay due to pump power fluctuation is typically less than 2% of the

pulse width [137]. A measured frequency trace of the broadened SBS gain profile for

positive time delay is shown plotted as the blue solid line in figure 6.16.

Figure 6.16: Measured SBS gain coefficient as a function of frequency detuning. The red dotted line shows the spectrum of Gaussian pulses with FWHM of 6 ns.

The group delay accumulated in transmission through a SBS gain stage is

approximately G/Γ, where G is the gain parameter and Γ is the resonance linewidth [10].

Once the gain profile is fixed, this enables us to tune the delay by controlling the output

level of the EDFA.

At the end of each channel an EDFA set to constant output power mode is used to

maintain steady output from each emitter. After the emitters, half-wave plates are added

and tuned to maximize the interference visibility. The output of each channel is directed


to the target with adjustable mirrors, as control of the phase ramp across each sub-

aperture is necessary when using near-IR or visible wavelengths.

Figure 6.17: Far field intensity pattern of the SLIDAR (a) and tests showing simulated beam steering and compensation via slow light delay mechanisms (b-f). Curves are traces of the normalized return signal plotted against time (ns). The dotted red, thin solid teal, and dashed green traces correspond to signals from channels 1, 2, and 3 respectively. The bold blue trace is the combined signal.

We performed a series of tests to demonstrate the effectiveness of our slow light

modules in compensating for group delay mismatch. For these tests, we use a reflecting

target approximately 6 meters distant, in keeping with our 1:1000 scale. Figure 6.17(a) is

an image of the interference pattern at this position. The contrast ratio of the pattern is

quite high, indicating robust phase locking among all three channels. The RMS phase jitter

of the locked signal is approximately λ⁄10 whether the SBS delay module is turned on or


off. As a receiver, we use a collection lens to focus the returning light onto a 3 GHz

amplified InGaAs photodetector.

Shown also in figure 6.16 are time traces of the detected return signal from each

channel, and also of the combined signal. In figure 6.17(b), the wavelength is set to 1530

and the translation stages are positioned to simulate “straight ahead” targeting. In (c), a

tilt in the y direction is simulated using the translation stage in channel 3, creating a delay

mismatch, resulting in significant pulse distortion and broadening and leading to

deterioration of the longitudinal resolution. This is corrected in figure (d) by shifting the

wavelength to 1549.5 nm. In (e), we have simulated an x-steering arrangement, causing

an advance in the signal from the SBS channel. Such a mismatch can be compensated

using SBS slow light with a pump power of 240 mW, and the result is plotted in (f). One

sees that using both slow-light techniques, the group delay mismatch in both x and y

directions can be corrected without significant pulse broadening in the returned signal.

We will close with some comments on scaling. For a more complicated and larger

system with more than three apertures, one can design a 2D grid of delay modules. For

example, a single multitapped SBS delay line can provide horizontal delays, with each

tap feeding a tapped dispersive delay line that provides the appropriate vertical delays to

each emitter. A diagram showing how this might be implemented is given in figure

(6.18). The operation of the SBS and dispersive delay modules remains the same, with

SBS pump power and laser wavelength controlling the x and y delay compensation

simultaneously for all apertures. Note also that for dense arrays, each delay channel can


be used to feed multiple neighboring emitters, since it is not necessary to control the

group delay as precisely as the phase.

Figure 6.18: (a) Schematic showing a possible method for integrating SBS and conversion / dispersion delay (C/D) into a full 2-D system with many emitters. Note specifically how the pump field shown in blue is cycled so that parts of the signal field split off at different junctions will see a different amount of SBS gain. Figure (b) shows some ideas for different possible layouts of the emitters.


Finally, let us note that while much of our focus has been on the possibility of scaling

up our laboratory-sized system. There are possibilities for a miniaturized version of a

SLIDAR as well, possibly in smaller devices where moving mechanical elements are

inconvenient or difficult to incorporate.

In conclusion, we have designed a three-channel SLIDAR system with a 2D aperture

array layout. Using two independent slow-light mechanisms, namely dispersive delay and

SBS, we have dynamically compensated the group delay mismatch among different

apertures during beam steering in both x and y directions, while simultaneously

maintaining control over the relative optical phases of the three channels.

7 CONCLUSION 113

7 Conclusions

We have now reached the end of our tour through the phenomenon of slow light.

Rather than use this space to discuss the details of all the work presented prior, I would

prefer to make some remarks about the larger picture and how these individual projects

relate to each other and to the whole.

Armed with knowledge of the Kramers-Kronig relations and the importance of the

dispersive term in the group index, we began with a study of slow light in a system with

material resonances. However, instead of attempting to find fast light by tuning directly

on an absorption resonance or slow light by tuning to a gain resonance, we observed that

we could solve several problems by working in between the two absorption resonances of

the cesium D2 lines. The strong absorption of atomic transitions allows for very large

group indices, despite spacing of over a GHz, while the fact of working in a region where

absorption is low and the GVD is a local minimum further enhances the ability to obtain

long total delays. We were able to obtain delays of 6.8 ns for 275 ps pulses and 59 ns for

740 ps pulses.

We then turned to structural slow light. Structural slow light is appealing because our

ability to design structures and artificial materials for slow light offers the promise of not

only imitating what nature gives us, but creating what it does not. Tunable resonators and

microfabricated waveguides allow for control over the transmission and phase response

7 CONCLUSION 114

of a system. We showed that if we tuned light to the resonance of a low-finesse fiber ring

resonator, we could delay a 51-ns pulse by 27 ns, but the true usefulness of the idea is not

in that delay, but in the thought that such a resonator could be one element in a larger

system. We also saw how structural resonances in coupled fiber resonators could be used

to imitate the behavior of EIT, one of the common systems for creating slow light in

gases (both hot and cold).

In our studies of stimulated Brillouin scattering, we recall the spirit of the preceding

two sections. From chapter 2, we have learned the effectiveness of double-resonance

systems and from chapter 3, we carry the idea of tuning a medium’s response with

reflections and interference. With SBS, we can use a strong pump beam that interacts

with the probe to produce a moving grating within a fiber. Since light is scattered from

the pump beam into the probe when they differ by the Stokes frequency, we have a

system where we can not only slow down light of any wavelength, but one where we

have control over the shape of the gain resonances, which we exert by adding sidebands

to the pump beam with electro-optic modulators. This flexibility is likely why SBS has

become such a popular system for slow light.

In chapter 5, we explored the properties of pulses propagating in saturable media.

While their behavior can be thought of as providing a time-dependent gain or absorption

envelope on a pulse, we find that this behavior, especially when a background is provided

to the signal, imitates the slow light performance of conventional material resonances to a

surprising degree. In erbium-doped fiber, we observe the appearance of a “spectral hole”

in the modulation gain with a linewidth comparable to the metastable state lifetime. We

7 CONCLUSION 115

even observe the appearance of a backwards traveling pulse peak inside the medium

when it is exhibiting negative pulse delays. We also show that we can actually exploit the

different behavior of saturable slow light media to limit pulse distortion, using the gain

recovery of the medium to compensate for the effects of pulse spectrum broadening that

would be exhibited in a linear resonant medium. The closing results demonstrating high-

bandwidth pulse delay with PbS quantum dots reminds us that while the bandwidth of

any given saturable medium might only be moderately tunable with a pump field, the

concept as a whole has great flexibility, given the wide range of materials that exhibit

saturable gain or absorption with different response times.

Finally, we show one way that slow light can be made useful by applying it to the

problem of group delay mismatch in optical phased-array beam-steering. We develop a

system that uses dispersive delay and an SBS setup much like that developed in section

4.2 to dynamically compensate for this mismatch, even as the beam is steered in two

dimensions.

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