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
2 MATERIAL SLOW LIGHT 13
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
2 MATERIAL SLOW LIGHT 14
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.
2 MATERIAL SLOW LIGHT 15
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.
2 MATERIAL SLOW LIGHT 16
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.
2 MATERIAL SLOW LIGHT 17
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.
2 MATERIAL SLOW LIGHT 18
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.
2 MATERIAL SLOW LIGHT 19
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
2 MATERIAL SLOW LIGHT 20
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.
2 MATERIAL SLOW LIGHT 21
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)
3 STRUCTURAL SLOW LIGHT 23
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
3 STRUCTURAL SLOW LIGHT 24
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)
3 STRUCTURAL SLOW LIGHT 25
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% $r cos #( )
&'(
)*+
(3.11)
Figure 3.2: Theoretically predicted net transmission, phase shift, and group delay for an ideal ring resonator with single-pass transmissivity τ = 0.95 and varying values of the finesse. The normalized detuning is the angular frequency difference between the light field and the nearest cavity resonance multiplied by the ring circulation time.
Near each resonance, the phase undergoes a rapid variation with respect to the round-
trip phase shift. This round-trip phase shift may be interpreted as a frequency detuning
3 STRUCTURAL SLOW LIGHT 26
normalized through multiplication by the round-trip time. The radian-frequency
derivative of the phase shift is the group delay which is proportional to both the finesse
F ! " / 1# r *$( )( ) 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 mea- sure 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.
3 STRUCTURAL SLOW LIGHT 27
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
3 STRUCTURAL SLOW LIGHT 28
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.
3 STRUCTURAL SLOW LIGHT 29
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].
3 STRUCTURAL SLOW LIGHT 30
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
3 STRUCTURAL SLOW LIGHT 31
!!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
3 STRUCTURAL SLOW LIGHT 32
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
3 STRUCTURAL SLOW LIGHT 33
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
3 STRUCTURAL SLOW LIGHT 34
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
4 SCATTERING SLOW LIGHT 36
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.
4 SCATTERING SLOW LIGHT 37
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
4 SCATTERING SLOW LIGHT 38
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
4 SCATTERING SLOW LIGHT 39
!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
4 SCATTERING SLOW LIGHT 40
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
4 SCATTERING SLOW LIGHT 41
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.
4 SCATTERING SLOW LIGHT 42
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 SCATTERING SLOW LIGHT 43
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
4 SCATTERING SLOW LIGHT 44
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
4 SCATTERING SLOW LIGHT 45
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
4 SCATTERING SLOW LIGHT 46
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)
4 SCATTERING SLOW LIGHT 47
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.
4 SCATTERING SLOW LIGHT 48
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.
4 SCATTERING SLOW LIGHT 49
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.
4 SCATTERING SLOW LIGHT 50
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-
4 SCATTERING SLOW LIGHT 51
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 PULSE DELAY IN SATURABLE MEDIA 53
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,
5 PULSE DELAY IN SATURABLE MEDIA 54
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!
$%&
'()*s Is "
n!+ s Is "
n!+ pI p (5.1)
where ρ is the Er3+ ion density, τ is the metastable level lifetime (10.5 ms), Ip is the pump
intensity in units of photons/area/time, Is is the signal intensity, βs is the signal emission
coefficient, and αp and αs are the pump and signal absorption coefficients [68]. The
5 PULSE DELAY IN SATURABLE MEDIA 55
steady-state solution for n to eq. (1) is given by n0 = (ρ/τ +βsIs)/ωc, where we have
defined the “center frequency” as
! c =1"+# pI p$
+# 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
$%&
'()Img . (5.6)
The phase shift of the modulation is described by
5 PULSE DELAY IN SATURABLE MEDIA 56
d!dz
=I0Im
"2 Im , I p , I0( ) (5.7)
where
!2 Im , I p , I0( ) = ! s + "s
#$%&
'()
*+ 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
5 PULSE DELAY IN SATURABLE MEDIA 57
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.
5 PULSE DELAY IN SATURABLE MEDIA 58
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
5 PULSE DELAY IN SATURABLE MEDIA 59
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.
5 PULSE DELAY IN SATURABLE MEDIA 60
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
5 PULSE DELAY IN SATURABLE MEDIA 61
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
5 PULSE DELAY IN SATURABLE MEDIA 62
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
5 PULSE DELAY IN SATURABLE MEDIA 63
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
5 PULSE DELAY IN SATURABLE MEDIA 64
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
5 PULSE DELAY IN SATURABLE MEDIA 65
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
5 PULSE DELAY IN SATURABLE MEDIA 66
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.
5 PULSE DELAY IN SATURABLE MEDIA 67
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
5 PULSE DELAY IN SATURABLE MEDIA 68
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
5 PULSE DELAY IN SATURABLE MEDIA 69
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
5 PULSE DELAY IN SATURABLE MEDIA 70
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,
5 PULSE DELAY IN SATURABLE MEDIA 71
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.
5 PULSE DELAY IN SATURABLE MEDIA 72
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
5 PULSE DELAY IN SATURABLE MEDIA 73
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
5 PULSE DELAY IN SATURABLE MEDIA 74
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,
5 PULSE DELAY IN SATURABLE MEDIA 75
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
5 PULSE DELAY IN SATURABLE MEDIA 76
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.
5 PULSE DELAY IN SATURABLE MEDIA 77
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.
5 PULSE DELAY IN SATURABLE MEDIA 78
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
5 PULSE DELAY IN SATURABLE MEDIA 79
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 81
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 82
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 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 83
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 84
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 85
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 86
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 87
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 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 88
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.
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 89
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 90
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 91
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.
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 92
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 93
(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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 94
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 95
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 96
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.
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 97
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.
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 98
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.
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 99
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 100
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.
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 101
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.
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 102
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.
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 103
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.
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 104
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.
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 105
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 106
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 107
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,
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 108
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 109
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 110
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
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 111
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.
6 SLOW LIGHT LASER DETECTION AND RANGING (SLIDAR) 112
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.
BIBLIOGRAPHY 116
Bibliography
1. D. J. Blumenthal, P. R. Pruncal, and J. R. Sauer, Proc. IEEE 82, 1650–1667,
(1994).
2. Z. Shi, R. W. Boyd, R.M. Camacho, P.K. Vudyasetu, and J.C. Howell, Phys. Rev.
Lett. 99, 240801 (2007).
3. Z. Shi, R. W. Boyd, D.J. Gauthier, and C.C. Dudley, Opt. Lett. 32, 915–917 (2007).
4. J. S. Toll, Phys. Rev. 104, 1760–1770 (1956).
5. G. M. Gehring, R. W. Boyd, A. L. Gaeta, D. J. Gauthier, and A. E. Willner, J.
Lightwave Technol. 26, 3752–3762 (2008).
6. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594–598
(1999).
7. A. V. Turukhin, V. S. Sudarshanam, M. S. Shahriar, B. S. Ham, and P. R. Hemmer,
Phys. Rev. Lett. 88, 023602 (2002).
8. R. W. Boyd, J. Opt. Soc. Am. B 28, A38–A44 (2011).
9. P. K. Vudyasetu, R. M. Camacho, and J. C. Howell, Phys. Rev. A 82, 053807
(2010).
10. J. E. Heebner and R. W. Boyd, J. Mod. Opt. 49, 036619 (2002).
11. J. E. Heebner, R. W. Boyd, and Q. Park, Phys. Rev. E 65, 2629–2636 (2002).
12. J. E. Heebner, R. W. Boyd, and Q. Park, J. Opt. Soc. Am. B 19, 722–731 (2002).
BIBLIOGRAPHY 117
13. D. N. Urquidez, S. Stepanov, H. S. Ortiz, N. Togunov, I. Ilichev, and A. Shamray,
Appl. Phys. B 106, 51–56 (2012).
14. K. Totsuka and M. Tomita, J. Opt. Soc. Am. B 23, 2194–2199 (2006).
15. Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNabb, Nature 438, 65–69,
(2005).
16. J. E. Sharping, Y. Okawachi, and A. L. Gaeta., Opt. Express 13, 6092–6098 (2005).
17. Y. Okawachi, R. Salem, and A.L. Gaeta, J. Lightwave Technol. 25, 3710–3715
(2007).
18. R. Tucker, P. Ku, and C. Chang-Hasnain, Electronics Lett. 41, 208–209 (2005).
19. A. E. Willner, B. Zhang, L. Zhang, L. S. Yan, and I. Fazal, IEEE J. Sel. Topics
Quantum Electron. 14, 691–705 (2008).
20. M. D. Lukin and A. Imamoglu, Phys. Rev. Lett. 84, 1419 (2000).
21. C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau., Nature 409, 490–493 (2001).
22. S. T. Johns, D. D. Norton, C. W. Keefer, R. Erdmann, and R. A. Soref, Electronics
Lett. 29, 555–556 (1993).
23. R. D. Esman, M. Y, Frankel, J. L. Dexter, L. Goldberg, M. G. Parent, D. Stilwell,
and D. G. Cooper, IEEE Photon. Tech. Lett. 5, 1347–1349 (1993).
24. R.Y. Chiao, Phys. Rev. A 48, R34–R37 (1993).
25. L. J. Wang, A. Kuzmich, and A. Dogariu, Nature 406, 277–279 (2000).
26. A. Dogariu, A. Kuzmich, and L. J. Wang, Phys. Rev. A 63, 053806 (2001).
27. M. D. Stenner, D. J. Gauthier, and M. A. Neifeld, Nature 425, 695–698 (2003).
28. B. Macke and B. Segard, Eur. Phys. J. D 23, 125–141 (2003).
BIBLIOGRAPHY 118
29. G. S. Agarwal and S. Dasgupta, Phys. Rev. A 70, 023802 (2004).
30. M. D. Stenner, M. A. Neifeld, and Z. M. Zhu, Opt. Express 13, 9995–10002 (2005).
31. D. Grischkowsky, Phys. Rev. A 7, 2096–2102 (1973).
32. H. Tanaka, H. Niwa, K. Hayami, S. Furue, K. Nakayama, T. Kohmoto, M.
Kunitomo, and Y. Fukuda, Phys. Rev. A 68, 053801 (2003).
33. B. Macke and B. Segard, Phys. Rev. A 73, 043802 (2006).
34. Z. Zhu and D. J. Gauthier, Opt. Express 14, 7238–7245 (2006).
35. R. M. Camacho, C. J. Broadbent, I. Ali-Khan, and J. C. Howell, Phys. Rev. Lett. 98,
043902 (2007).
36. A. Kasapi, M. Jain, and G. Y. Yin, Phys. Rev. Lett. 74, 2447–2450 (1995).
37. K.Y. Song, M. G. Herraez, and L. Thevanez, Opt. Lett. 30, 1782–1784 (2005).
38. R. M. Camacho, M. V. Pack, and J. C. Howell, Phys. Rev. A 74, 033801 (2006).
39. R. M. Camacho, M. V. Pack, and J. C. Howell, Phys. Rev. A 73, 063812 (2006).
40. R. W. Boyd and D. J. Gauthier, Prog. Opt. 43, 497–530 (2002).
41. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, IEEE J. Quantum
Electron. 37, 525–532 (2001).
42. H. Winful, Physics Reports 436, 1–69 (2006).
43. J. E. Heebner and R. W. Boyd, Opt. Lett. 24, 847–849, 1999.
44. R. W. Boyd and J. E. Heebner, Appl. Opt. 40, 5742–5747, 2001.
45. A. Kasapi, M. Jain, G. Y. Yin, and S. E. Harris, Phys. Rev. Lett. 74, 2447 (1995).
46. O. Kocharovskaya, Y. Rostovtsev, and M. O. Scully, Phys. Rev. Lett. 86, 628–631
(2001).
BIBLIOGRAPHY 119
47. A. K. Patnaik, J. Q. Liang, and K. Hakuta, Phys. Rev. A 66, 063808 (2002).
48. D. Budker, V. Yashchuk, and M. Zolotorev, Phys. Rev. Lett. 81, 5788–5791 (1998).
49. H. A. Lorentz, The Theory of Electrons and Its Applications to the Phenomena of
Light and Radiant Heat, Teubner, Leipzig, Germany, 1909.
50. W. E. Lamb and R. C. Retherford, Phys. Rev. 81, 222–232 (1951).
51. P. R. Hemmer and M. G. J. Prentiss, J. Opt. Soc. Am. B 5, 1613–1623 (1988).
52. G. L. Garrido Alzar, M. A. G. Martinez, and P. Nussenzveig, Am. J. Phys. 70, 37–
41 (2001).
53. T. Opatrny and D. G. Welsch, Phys. Rev. A 64, 23805 (2001).
54. D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R.W. Boyd, Phys.
Rev. A 69, 63804 (2004).
55. A. M. Akulshin, S. Barreiro, and A. Lezama, Phys. Rev. A 57, 2996–3002 (1998).
56. S. E. Harris, J. E. Field, and A. Imamoglu, Phys. Rev. Lett. 64, 1107–1110 (1990).
57. K. J. Boller, A. Imamoglu, and S. E. Harris, Phys. Rev. Lett. 66, 2593–2596 (1991).
58. A. Naweed, G. Farca, S. I. Shipova, and A.T. Rosenberger, Phys. Rev. A 71, 043804
(2005).
59. Y. Dumeige, T. K. N. Nguyen, L. Ghisa, S. Trebaol, and P. Feron, Phys. Rev. A 78,
013818 (2008).
60. K. Totsuka, N. Kobayashi, and M. Tomita, Phys. Rev. Lett. 98, 213904 (2007).
61. Y. D. Zhang, N. Wang, H. Tian, H. Wang, W. Qiu, J. F. Wang, and P. Yuan, Phys.
Lett. A 372, 5848–5852 (2008).
62. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic Press, New York, 2007).
BIBLIOGRAPHY 120
63. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, New York, 2008).
64. R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, Phys. Rev. A 71,
023801 (2005).
65. A. L. Gaeta and R. W. Boyd, Int. J. Nonlinear Opt. Phys. 1, 581–594 (1992).
66. A. Loayssa, R. Hernandez, D. Benito, and S. Galech, Opt. Lett. 29, 638–640 (2004).
67. P. Ku, C. Chang-Hasnain, and S. Chuang, Electron. Lett. 38, 1581–1583 (2002).
68. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science 301, 200–202 (2003).
69. E. Shumakher, A. Willinger, R. Blit, D. Dahan, and G. Eisenstein, Opt. Express 14,
8540–8545 (2006).
70. Y. Okawachi, J. E. Sharping, C. Xu, and A. L. Gaeta, Opt. Express 14, 12022–
12027 (2006).
71. Z. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, J. Lightwave
Technol. 25, 201–206 (2007).
72. K. Y. Song and K. Hotate, Opt. Lett. 32, 217–219 (2007).
73. T. Schneider, M. Junker, and K.-U. Lauterbach, Opt. Express 14, 11082–11087
(2006).
74. A. Zadok, A. Eyal, and M. Tur, J. Lightwave Technol. 25, 2168–2174 (2007).
75. A. Minardo, R. Bernini, and L. Zeni, Opt. Express 14, 5866–5876 (2006).
76. Z. Shi, R. Pant, Z. Zhu, M. D. Stenner, M. A. Neifeld, D. J. Gauthier, and R. W.
Boyd, Opt. Lett. 32 1986–1988. (2007)
77. Z. Lu, Y. Dong, and Q. Li, Opt. Express 15, 1871–1877 (2007).
BIBLIOGRAPHY 121
78. T. Sakamoto, T. Yamamoto, K. Shiraki, and T. Kurashima, Opt. Express 16, 8026–
8032 (2008).
79. M. Lee, R. Pant, M. D. Stenner, and M. A. Neifeld, Opt. Comm. 281, 2975–2984
(2008).
80. K. Y. Song, M. G. Herraez, and L. Thevenaz, Opt. Express 13, 9758–9765 (2005).
81. S. Chin, M. G. Herraez, and L. Thevenaz, Opt. Express 9, 10814–10821 (2007).
82. S. Chin, M. G. Herraez, and L. Thevenaz, Opt. Express 16, 12181–12189 (2008).
83. V. P. Kalosha, L. Chen, and X. Bao, Opt. Express 14, 12693–12703 (2006).
84. R. W. Boyd and P. Narum, J. Mod. Opt. 54, 2403–2411 (2007).
85. R. Pant, M. D. Stenner, M. A. Neifeld, Z. Shi, R. W. Boyd, and D. J. Gauthier,
Appl. Opt. 46, 6513–6519 (2007).
86. E. Cabrera-Granado, O. G. Calderon, S. Melle, and D. J. Gauthier, Opt. Express 16,
16032–16042 (2008).
87. J. Freeman and J. Conradi, IEEE Photonics Tech. Lett. 5, 224–226 (1993).
88. S. Novak and A. Moesle, J. Lightwave Technol. 20, 975–985 (2002).
89. S. Jarabo, J. Opt. Soc. Am. B 14, 1846–1849 (1997).
90. S. E. Schwartz and T. Y. Tan, Appl. Phys. Lett. 10, 4–7 (1967).
91. N. G. Basov, R. V. Ambartsumyan, V. S. Zuev, P. G. Kryukov. and V. S. Letokhov,
Sov. Phys. Dokl. 10, 1039–1040 (1966).
92. A. C. Selden, Brit. J. Appl. Phys. 18, 743–748 (1967).
93. A. C. Selden, J. Phys. D 3, 1935–1943 (1970).
BIBLIOGRAPHY 122
94. M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Phys. Rev. Lett. 90, 113903
(2003).
95. G. Piredda and R. W. Boyd, J. Eur. Opt. Soc–Rapid 2, 07004 (2007).
96. R. Y. Chiao, Phys. Rev. A 48, R34–R37 (1993).
97. M. Ware, S. A. Glasgow, and J. Peatross, Opt. Express 9, 506–518 (2001).
98. M. Ware, S. A. Glasgow, and J. Peatross, Opt. Express 9, 519–532 (2001).
99. S. Chu and S. Wong, Phys. Rev. Lett. 48, 738–741 (1982).
100. B. Segard and B. Macke, Phys. Lett. 109, 213–216 (1985).
101. M. A. I. Talukder, Y. Amagishi, and M. Tomita, Phys. Rev. Lett. 86, 3546–3549
(2001).
102. S. Chu and S. Wong, Phys. Rev. Lett. 49, 1293–1293 (1982).
103. L. J. Wang, A. Kuzmich, and A. Dogariu, Nature 406, 227–279 (2000).
104. R. L. Smith, Am. J. Phys. 38, 978–984 (1970).
105. G. P. Agrawal and N. A. Olsson, IEEE J. Quantum Electron. 25, 2297–2306
(1989).
106. D. Eliyahu, R. A. Salvatore, J. Rosen, A. Yariv, and J. Drolet, Opt. Lett. 20, 1412–
1414 (1995).
107. M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, J. Phys. Condens.
Matter 18, 3117–3126 (2006).
108. H. Cao, A. Dogariu, and L. J. Wang, IEEE J. Sel. Top. Quantum Electron. 9, 52–58
(2003).
BIBLIOGRAPHY 123
109. U. Keller, K. J. Weingarten, and F. X. Kartner, IEEE J. Sel. Top. Quantum
Electron. 2, 435–453 (1996).
110. A. Dementjev, V. Gulbinas, L. Valkunas, I. Motchalov, H. Raaben, and A.
Michailovas, Appl. Phys. B 77, 595–599 (2003).
111. M. van der Poel, J. Mørk and J. M. Hvam, Opt. Express 13, 8032–8037 (2005).
112. H. Su and S. L. Chuang, Opt. Lett. 31, 271–273 (2006).
113. P.S. Argall and R.J. Sica, “Lidar,” in Encyclopedia of Imaging Science and
Technology, J.P. Hornak, ed. (Wiley, New York, 2002).
114. P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Friedman, D. S. Hobbs,
M. Holz, S. Liberman, H. Q. Nguyen, D. P. Resler, R. C. Sharp, and E. A. Watson,
Proc. IEEE 84, 268–298 (1996).
115. P. F. McManamon, J. Shi, and P. K. Bos, Opt. Eng. 44, 128004 (2005).
116. S. J. Augst, T. Y. Fan, and A. Sanchez, Opt. Lett. 29, 474–476 (2004).
117. S. Serati, H. Masterson, and A. Linnenberger, IEEE Aerospace Conference
Proceedings 3, 1046–1052 (2004).
118. T. M. Shay, Opt. Express 14, 12188–12195 (2006).
119. T. M. Shay, V. Benham, J. T. Baker, B. Ward, A. D. Sanchez, M. A. Culpepper, D.
Pilkington, J. Spring, D. J. Nelson, and C. A. Lu, Opt. Express 14, 12015–12021
(2006).
120. I. Frigyes, IEEE Trans. Microw. Theory 43, 2378–2386 (1995).
121. S. T. Johns, D. A. Norton, C. W. Keefer, R. Erdman, and R. A. Soref, Electron.
Lett. 29, 555–556 (1993).
BIBLIOGRAPHY 124
122. M. Muszkowski and E. Sędek, J. Telecomun. Inform. Tech. 25, C61–C64 (2008).
123. M. Bashkansky, Z. Dutton, A. Gulian, D. Walker, F. Fatemi, and M. Steiner, Proc.
SPIE 7226, 72260A (2009).
124. M. Bashkansky, D. Walker, A. Gulian, and M. Steiner, Proc. SPIE 7949, 794918
(2011).
125. D. R. Walker, M. Bashkansky, A. Gulian, F. K. Fatemi, and M. Steiner, J. Opt. Soc.
Am. B 25, C61–C64 (2008).
126. F. Vasey, F. K. Reinhart, R. Houdré, and J. M. Stauffer, Appl. Opt. 32, 3220–3232
(1993).
127. F. Xiao, W. Hu, and A. Xu, Appl. Opt. 44, 5429–5433 (2005).
128. F. Xiao, G. Li, Y. Li, and A. Xu, Opt. Eng. Lett. 47, 040503 (2008).
129. N. J. Miller, M. P. Dierking, and B. D. Duncan, “Optical sparse aperture imaging,”
Appl. Opt. 46, 5933–5943 (2007).
130. C. R. DeHainaut, D. C. Duneman, R. C. Dymale, J. P. Blea, B. D. O'Neil, and C. E.
Hines, Opt. Eng. 34, 876–880 (1995).
131. T. Y. Fan, IEEE J. Sel. Top. Quant. Electron. 11, 567–577 (2005).
132. S. J. Augst, J. K. Ranka, and T. Y. Fan, J. Opt. Soc. Am. B 24. 1707–1715 (2007).
133. P. Zhou, Z. Liu, X. Wang, Y. Ma, X. Xu, S. Guo, IEEE J. Sel. Top. Quant.
Electron. 15, 248–256 (2009).
134. J. H. Abeles and R. J. Deri, Appl. Phys. Lett. 53, 1375–1377 (1988).
135. J. A. Overbeck, M. S. Salisbury, M. B. Mark, and E. A. Watson, Appl. Opt. 34,
7724–7730 (1995).
BIBLIOGRAPHY 125
136. M. Y. Frankel, P. J. Matthews, and R. D. Esman, Opt. Quant. Electron. 30, 1033–
1050 (1998).
137. Z. Shi, A. Schweinsberg, J. E. Vornehm Jr., M. A. M. Gámez, and R. W. Boyd,
Phys. Lett. A 374, 4071–4074 (2010).