ABSTRACT
Title of dissertation: NONLINEAR SELF-CHANNELING
OF HIGH-POWER LASERS
THROUGH TURBULENT ATMOSPHERES
Gregory P. DiComo, Doctor of Philosophy, 2018
Dissertation directed by: Professor Thomas Antonsen
Department of Electrical and Computer Engineering
A variety of laser applications have been considered which depend on long-
distance atmospheric propagation of the beam to attain practical utility. The effec-
tiveness of these applications is limited to some extent by beam distortions caused
by atmospheric optical turbulence. Often the limiting factor is the instantaneous
beam spreading due to turbulence, which makes it impossible to create a small laser
spot at the receiver.
In the absence of turbulence, laser beams of sufficient peak power propagating
in atmosphere have been shown to undergo nonlinear self-guiding, in which the
beam size remains constant over multiple Rayleigh lengths. Recent research suggests
that self-guiding beams of sufficiently small diameter might exhibit resistance to
turbulent spreading, in a propagation mode known as nonlinear self-channeling.
Presented here is an experimental demonstration of such self-channeling through
an artificially controlled turbulent atmosphere, with investigation into the region of
parameter space over which it can occur. This research makes use of a distributed-
volume turbulence generator and long propagation ranges at the Naval Research
Laboratory and the Air Force Research Laboratory in order to produce a controlled
propagation environment suitable for the study of high-power beams.
Nonlinear self-channeling is found to resist the diffractive effects of turbu-
lence, with its effectiveness decreasing significantly as the inner scale of turbulence
decreases below the size of the beam.
NONLINEAR SELF-CHANNELING
OF HIGH-POWER LASERS
THROUGH TURBULENT ATMOSPHERES
by
Gregory P. DiComo
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2018
Advisory Committee:
Professor Thomas Antonsen, Chair/Advisor
Professor Christopher Davis
Professor Phillip Sprangle
Professor Antonio Ting
Professor Rajarshi Roy, Dean’s Representative
At your disturbing power is this instrument come; whence will you go?
Four directions, then one. Alive must I be brought to Embelyon.
Jack Vance, The Dying Earth
ii
Acknowledgments
I owe my gratitude to all the people who have made this thesis possible and
because of whom my graduate experience has been one that I will cherish forever.
It is impossible to remember all of the people who have contributed to this work. I
apologize to anyone whom I have neglected to mention by name.
First I must thank my advisor, Professor Thomas Antonsen, for his keen in-
sight, tremendous teaching skill, and great patience.
Thanks are due as well to Doctors Phillip Sprangle, Christopher Davis, Anto-
nio Ting, and Rajarshi Roy, for agreeing to serve on my thesis committee.
I would also like to thank Doctors Richard Fischer and Antonio Ting, of the
Naval Research Laboratory, for giving me the opportunity to work in the field of
experimental physics. They had confidence in me when I had no confidence in
myself. My career in physics, as opposed to any other field, is due to their faith in
my abilities.
I would like to acknowledge the support of my other colleagues at the Naval
Research Laboratory, especially Doctors Joseph Peñano and Michael Helle, who
suggested the line of research which makes up this thesis. In addition, my current
and past coworkers: Doctors Daniel Gordon, Theodore Jones, Dmitri Kaganovich,
Yu-Hsin Chen, Melissa Hornstein, and John Palastro.
I want to thank Zachary Wilkes and Sanjay Varma, who showed me that the
life of a graduate student, while hard, contained room for friendship and happiness.
Paul Jaffe, Iyad Lababidi, Bipin Gyawali, Bisrat Addissie, Clayton Crocker, and all
iii
of the many other graduate students with whom I spent so many sleepless nights
solving problem sets. I would also like to thank Gregory Flynn, Richard Winters,
Michael DiComo, Emily Bannister, Gramm and Sarah Richardson, Cassandra Wire,
Brian Hildebrandt, and all the other good friends who have helped keep me sane.
I acknowledge the financial support of the Office of Naval Research, the High
Energy Laser Joint Technology Office, and the Naval Research Laboratory Base
Program, who provided the stable, ongoing project funding which made this research
possible.
To my parents, Paul DiComo and Susan Putnam, I owe a lifelong love of
science, engineering, and the arts.
I would like to acknowledge the motivating influence of my son, Alexander
Joseph DiComo, against whose arrival I raced to finish this work.
Finally, and most importantly, I would like to acknowledge the unflagging love
and support provided by my wife. Dawn, without you none of this would have been
possible.
iv
Table of Contents
List of Figures viii
List of Abbreviations xiii
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Atmospheric Turbulence 4
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Atmospheric Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 The Approach of Kolmogorov . . . . . . . . . . . . . . . . . . 6
2.2.2 Correlation and Structure of Turbulence . . . . . . . . . . . . 7
2.2.3 Velocity and Temperature Turbulence . . . . . . . . . . . . . . 9
2.2.4 Optical Turbulence . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Turbulence Characterization . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Hot Wire Anemometry . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1.1 Effect of Temperature Fluctuations . . . . . . . . . . 14
2.3.1.2 Benchmark . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Scintillometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.3 LISSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Linear Laser Beam Propagation 21
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Without Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.1 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.2 Gaussian Beam Propagation . . . . . . . . . . . . . . . . . . . 23
3.3 With Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 Weak Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1.1 Rytov Variance and Scintillation Index . . . . . . . . 27
3.3.1.2 Beam Wander . . . . . . . . . . . . . . . . . . . . . . 28
3.3.1.3 Long-Term Beam Size . . . . . . . . . . . . . . . . . 29
v
3.3.1.4 Short-Term Beam Size . . . . . . . . . . . . . . . . . 30
3.3.2 Strong Turbulence . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3.2.1 Coherence Length . . . . . . . . . . . . . . . . . . . 32
3.3.2.2 Scintillation Index . . . . . . . . . . . . . . . . . . . 32
3.3.2.3 Long-Term Beam Size . . . . . . . . . . . . . . . . . 34
3.3.2.4 Beam Wander . . . . . . . . . . . . . . . . . . . . . . 35
3.3.2.5 Short-Term Beam Size . . . . . . . . . . . . . . . . . 35
4 Artificial Turbulence Generation 36
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 Turbulence Experiments Outdoors . . . . . . . . . . . . . . . . . . . . 36
4.3 Previous Turbulence Generators . . . . . . . . . . . . . . . . . . . . . 40
4.4 This Work’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.4.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . 43
4.4.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.5 Characterization of the Turbulence Generator . . . . . . . . . . . . . 48
4.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.5.2 Anemometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.5.3 Optical Techniques . . . . . . . . . . . . . . . . . . . . . . . . 53
4.6 Modeling and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 58
5 Nonlinear Beam Propagation 61
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Without Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2.1 Nonlinear Self-Focusing . . . . . . . . . . . . . . . . . . . . . . 62
5.2.2 Filamentation and Self-Guiding . . . . . . . . . . . . . . . . . 64
5.2.3 Group Velocity Dispersion . . . . . . . . . . . . . . . . . . . . 66
5.3 With Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6 Experiments at NRL 71
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2 Outline of Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2.1 Propagation Range . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2.2 Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2.3 Turbulence Generation . . . . . . . . . . . . . . . . . . . . . . 75
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.3.1 First Observation of Nonlinear Self-Channeling . . . . . . . . 75
7 Experiments at AFRL 81
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.2 Outline of Methodology . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.2.1 Propagation Range . . . . . . . . . . . . . . . . . . . . . . . . 81
7.2.2 Laser System . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.2.3 Turbulence Generation and Characterization . . . . . . . . . . 82
7.2.4 Optical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
vi
7.2.5 Pulse Length and Power . . . . . . . . . . . . . . . . . . . . . 86
7.2.6 Data Organization . . . . . . . . . . . . . . . . . . . . . . . . 90
7.2.7 Plotting and Theory Methodology . . . . . . . . . . . . . . . . 94
7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.3.1 Turbulence Generation . . . . . . . . . . . . . . . . . . . . . . 95
7.3.2 Single-Pass Experiments . . . . . . . . . . . . . . . . . . . . . 96
7.3.3 Triple Pass Data . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.3.4 Both Single and Triple Pass . . . . . . . . . . . . . . . . . . . 116
7.3.5 Effects of Power and Beam Clipping . . . . . . . . . . . . . . 118
8 Conclusion 119
8.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
8.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A Image Analysis 121
A.1 Approach to Analysis of Beam Profile Images . . . . . . . . . . . . . 121
A.1.1 Blob Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A.1.2 Beam Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A.1.3 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Bibliography 136
vii
List of Figures
2.1 The Great Wave off Kanagawa, woodcut by Hokusai. Multiple scale
lengths of turbulence are visible in this image, from the outer scale
at the size of the largest wave, through the inner scale at the size of
the smallest eddies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 The Kolmogorov and Von Karman turbulence spectra. The Von Kar-
man spectrum includes inner and outer scale effects. . . . . . . . . . . 12
2.3 One-dimensional velocity turbulence power spectrum Vu(k) for a nat-
urally turbulent atmosphere. A −5/3-slope line is displayed in black.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 Laser beam images taken before (a) and after (b) propagation through
more than 1 km of natural turbulence at Starfire Optical Range. Note
the increase in the spot size, as well as the significant distortion of
the intensity profile. The instantaneous 1/e2 beam radius is 2.1 cm
before propagation, and 7.8 cm after. The propagation range is 3.2
km; C2 −14 −2/3 2n = 1.4 × 10 m ; the Rytov parameter σR = 3.6. This is
considered to be “strong” turbulence. . . . . . . . . . . . . . . . . . 26
4.1 A typical day of C2n variation taken at Kirtland AFB, Albuquerque,
NM. Note the pronounced evening null event, and the accompanying
rapid variation of turbulence across several orders of magnitude. . . . 38
4.2 Schematic arrangement of Davis’ 1998 experiment incorporating the
hot-liquid turbulence tube. . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 C. Nelson’s flow-driven hot air turbulence generator. Panel a: exterior
view with the directions and relative temperatures of flow marked, as
well as the positions of the five turbulence sections. Figure b: interior
view with the direction of laser propagation marked. . . . . . . . . . . 41
4.4 Schematic of W. Nelson’s convection-driven turbulence generator.
Panel a: monostatic beam propagation path with outgoing and in-
coming beams experiencing identical turbulence. Panel b: bistatic
path with outgoing and incoming beams experiencing different tur-
bulence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
viii
4.5 A symbolic diagram of the GDFACT. Heater wires strung on an ex-
tended framework running below the beam propagation path impart
thermal and kinetic energy into the air, which naturally produces re-
alistic Kolmogorov turbulence. Turbulence diagnostics co-propagate
with the laser beam, characterizing the turbulence as the experiment
takes place. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.6 One-dimensional velocity turbulence power spectra Vu(k) for a nat-
urally turbulent atmosphere (NRL parking lot, red), and four power
density settings on the GDFACT (orange through blue). A −5/3-
slope line is displayed in black. High-wavenumber peaks on each
data set represent high-frequency instrumental noise related to the
oscilloscope sampling frequency. . . . . . . . . . . . . . . . . . . . . . 50
4.7 One-dimensional acceleration turbulence power spectra Vu(k) for the
GDFACT. The effects of the inner and outer scales are plainly visible. 52
4.8 Optical characterization of the NRL installation of GDFACT via
LISSD. The laboratory HVAC system remained on for this test. . . . 54
4.9 Optical characterization of the GDFACT via LISSD. The solid line
represents the structure parameter C2n, as determined from LISSD
scintillation data under the weak turbulence approximation, equation
(2.17). The dotted line represents the C2n output from the commercial
scintillometer. The laboratory HVAC system remained off for this test. 55
4.10 Optical characterization of the GDFACT via LISSD. a) Scintillation
index. b) Standard deviation of the angle of arrival. c) Inner scale,
determined according to the method of Consortini, equation (2.18). . 57
4.11 Qualitative comparison between experimental and simulation results.
a) and b) Still frames from video of laser beam scattered off target
board in front of LISSD. c) and d) Still frames of animation produced
in HELCAP. The spot size and fine beam structure observed in ex-
periment are qualitatively reproduced in simulation. Low dynamic
range of the camera prevents quantitative comparison of these images. 60
5.1 Computer simulations of sub-inner-scale low-power and high-power
beam profiles after propagating through deep turbulence. Three real-
izations of turbulence are presented for both high and low power cases.
Top row: low-power propagation with linear focusing to target. Bot-
tom row: high-power propagation with nonlinear self-focusing. Sim-
ulation parameters: 7 km propagation distance, C2 = 10−15m−2/3n ,
σ2R = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.1 The moderate distance indoor laser propagation range at the NRL
Plasma Physics Division. . . . . . . . . . . . . . . . . . . . . . . . . 72
6.2 The kilohertz Titanium Femtosecond Laser at the NRL Plasma Physics
Division. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
ix
6.3 Cumulative distribution functions for the spot size of low-power and
high-power beams propagating through turbulence at the NRL prop-
agation range. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.4 The standard deviation of the low-power beam size increases as a
function of turbulence intensity. However, the high-power beam’s
statistics remain nearly unchanged as turbulence increases. . . . . . . 79
6.5 Increase of beam wander as a function of turbulence intensity. Both
the low-power and high-power beam exhibit identical increases in
beam wander, with the exception of a constant offset. . . . . . . . . . 80
7.1 Photograph of indoor turbulence range located at AFRL. The range
is 180 meters long, and its ambient turbulence conditions have been
characterized by LISSD and scintillometer. This view is looking back-
wards over the turbulence generator, from the receiver towards the
transmitter. Forward-scattering light from the HeNe beam used for
the LISSD can be seen due to aerosol scattering over the long path
length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.2 Diagram of experimental setup . . . . . . . . . . . . . . . . . . . . . 85
7.3 Autocorrelation trace of the short-pulse PHEENIX laser beam im-
mediately prior to propagation down range. The nominal grating
position is 0.418 inches. . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.4 Laser burns taken after the final iris on the transmitter table. Left:
beam used for single-pass experiments. Right: beam used for triple-
pass experiments. The scale is the same for both images. . . . . . . . 89
7.5 Instantaneous beam images taken at the receiver after one pass over
the propagation range. Left side: high power propagation with non-
linear self focusing. Right side: low power linear propagation. Top
panel: typical beam fluence profile at the receiver. Bottom panel:
horizontal lineout across beam center. The presence of lower-power
pre- and post-pulses can be seen in the “skirt” surrounding the self-
channeling high power beam. . . . . . . . . . . . . . . . . . . . . . . . 92
7.6 Instantaneous beam images taken at the receiver after three passes
over the propagation range. Left side: high power propagation with
nonlinear self focusing. Right side: low power linear propagation.
Top panel: typical beam fluence profile at the receiver. Bottom panel:
horizontal lineout across beam center. The distinction between the
self-channeling portion of the beam and the lower-power pre- and
post-pulses is even clearer. . . . . . . . . . . . . . . . . . . . . . . . . 93
7.7 Inner scale of turbulence as a function of structure parameter C2n. A
line of best fit and its generating equation are shown. . . . . . . . . . 95
7.8 Beam wander (standard deviation of distance from time-averaged cen-
ter) as a function of turbulence. Single-pass experiment (180 m). . . . 97
7.9 Violin plot for the short-term radius of the whole beam as a func-
tion of Rytov variance. Solid and dashed curves represent theoretical
values for strong and weak turbulence, respectively. . . . . . . . . . . 99
x
7.10 Violin plot for the short-term radius of the hotspot of the beam as a
function of Rytov variance. Solid and dashed curves represent theo-
retical values for strong and weak turbulence, respectively. . . . . . . 101
7.11 Violin plot for the maximum fluence on target as a function of tur-
bulence intensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.12 Probability of improvement as a function of Rytov variance. . . . . . 105
7.13 Probability of improvement as a function of the ratio of the initial
beam radius to the inner scale. . . . . . . . . . . . . . . . . . . . . . . 107
7.14 Beam wander (standard deviation of distance from time-averaged cen-
ter) as a function of turbulence. Triple-pass experiment (540 m). . . . 109
7.15 Violin plot for the radius of the whole beam as a function of Rytov
variance. The dashed curve represents the weak-turbulence theory
result; the solid curve the strong-turbulence result. . . . . . . . . . . 111
7.16 Violin plot for the radius of the instantaneous beam hotspot as a
function of Rytov variance. For low turbulence, the high power beam
has a lump of probability mass at low radius which the low-power
beam does not possess. However, as turbulence increases, the chances
that a lucky shot will form a small region of high fluence on the target
increase, and the low-power beam begins to develop a similar feature.
By σ2R ∼ 3 the distributions are very similar. . . . . . . . . . . . . . . 112
7.17 Violin plot for the maximum fluence on target as a function of Rytov
variance. At all turbulence intensities, the high-power beam is capa-
ble of producing higher fluence on target than the low-power beam.
The strength of this effect is diminished somewhat as turbulence in-
creases, but never goes away. . . . . . . . . . . . . . . . . . . . . . . . 114
7.18 Probability of improvement for the triple-pass experiment. For Rytov
variances near 10, all three metrics are found to approach parity with
the low-power case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.19 The probability that the NSC beam will produce a spot with radius
smaller than the average radius of a low-power beam focused to the
target from an infinitely large beam director. The performance of
the infinite-radius linear beam was estimated using Equation 3.29.
Nonlinear self-channeling starts to outperform the ideal linear case
around σ2R ≈ 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.20 Average beam size as a function of power. . . . . . . . . . . . . . . . 118
A.1 An example unmodified image. Note the presence of significant “static-
like” background noise. . . . . . . . . . . . . . . . . . . . . . . . . . . 122
A.2 The image modified by perspective transformation. The 512x512 im-
age has been projected onto a larger canvas which matches the di-
mensions of the entire imaging surface. This allows images of any size
to be adjusted correctly with the same perspective transformation. . . 123
A.3 The image after cropping and background subtraction. . . . . . . . . 125
A.4 The image after the bandpass filter. Notice the significantly reduced
level of background noise. . . . . . . . . . . . . . . . . . . . . . . . . 126
xi
A.5 The image after thresholding. . . . . . . . . . . . . . . . . . . . . . . 127
A.6 The image after thresholding, contrast-enhanced in order to draw
attention to low-amplitude noise. . . . . . . . . . . . . . . . . . . . . 128
A.7 Location and size of the blobs detected by the DoG blob-detection
algorithm. The underlying image is contrast-enhanced. . . . . . . . . 130
A.8 Output of the DoG blob-detection algorithm. The image has been
contrast-enhanced for visibility. . . . . . . . . . . . . . . . . . . . . . 131
A.9 Final image. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
xii
List of Abbreviations
σx The variance of any quantity x
x̄ The average of any quantity x
E~ Electric field
~r Arbitrary vector (x, y, z)
r Radial coordinate
R0 Beam radius at the transmitter
R Beam radius at the receiver in the absence of turbulence
RLT Long-term beam radius at the receiver in the presence of turbulence
RST Short-term beam radius at the receiver in the presence of turbulence
σ2c Beam Wander
AFRL Air Force Research Laboratory
CTA Constant Temperature Anemometer
DEPS Directed Energy Professional Society
GDFACT Generator of Distributed Free Atmospheric Convective Turbulence
GVD Group Velocity Dispersion
IREAP Institute for Research in Electronics and Applied Physics
LED Light-Emitting Diode
LISSD Laser Inner Scale and Scintillation Diagnostic
NASA National Aeronautics and Space Administration
NRL Naval Research Laboratory
NSC Nonlinear Self-Channeling
NSF Nonlinear Self-Focusing
PDF Probability Density Functions
RSI Research Support Instruments
UMD University of Maryland
USPL Ultrashort Pulse Laser
xiii
Chapter 1: Introduction
1.1 Introduction
A variety of laser applications have been considered which depend on long-
distance atmospheric propagation of the beam to attain practical utility. Such ap-
plications include point-to-point laser communications [1], laser power beaming [2],
and directed energy [3]. In each case, the effectiveness of the application is limited to
some extent by beam distortions caused by atmospheric turbulence. Extensive effort
has been devoted to modeling and experimentally investigating this phenomenon for
both low-power linear [4–8] and high-power nonlinear beams [9, 10].
Recently, a new paradigm for long-range propagation of high-power laser pulses
has been predicted [11]. Nonlinear self-channeling (NSC) is a regime of propagation
in which the laser power is comparable to the self-focusing power, and the radius
of the beam is smaller than the transverse coherence radius of turbulence. Such
a laser pulse is expected to remain highly collimated over many Rayleigh lengths,
without experiencing either turbulence or diffraction-induced expansion. However,
the NSC phenomenon has not yet been investigated experimentally. This work
reports on experiments which establish the statistical behavior of nonlinear self-
channeling beams.
1
In order to facilitate this work, an artificial turbulence generator suitable for
use with high-power lasers has been built and characterized. This device allows
controlled, accurate reproduction of the linear and nonlinear properties of naturally-
occurring turbulence. The turbulence generated by this generator has been shown
to be similar to that observed in an outdoor environment.
Using this generator as a source of turbulence, experiments have been con-
ducted which demonstrate and characterize the statistics of NSC beams propagating
through atmospheric turbulence. This work establishes that NSC is robust against
increase in turbulence strength into the strong turbulence regime. It exhibits grace-
ful degradation, with the probability of generating a small spot after propagation
decreasing smoothly with increasing turbulence. Even under strong turbulence, an
NSC beam is likely to produce a smaller, more intense spot at the end of propagation
than a linearly propagating beam.
1.2 Organization
This work begins with an brief introduction to the theory of atmospheric
turbulence in Chapter 2. For a fuller treatment of this deep and complex subject,
see, for example, the books by Tatarskii or Andrews & Phillips [7, 12]. Thereafter,
Chapter 3 introduces the theory of linear laser propagation through a turbulent
atmosphere, insofar as it is necessary to present the equations used later in the thesis
to compare theory to experiment. Chapter 4 then outlines the theory and practice
of generating artificial turbulence in the laboratory, and describes the design and
2
characterization of the turbulence generator used throughout this work. Chapter 5
then gives a brief overview of recent work concerning nonlinear laser propagation
through turbulence.
With the groundwork thus laid, Chapter 6 presents experiments performed
over a moderate range at the U.S. Naval Research Laboratory (NRL), which pro-
vided the first experimental demonstration of NSC. Chapter 7 presents a detailed
analysis of experiments performed over a long range at the U.S. Air Force Research
Laboratory (AFRL), comparing the propagation statistics of high-power NSC beams
to those of low-power beams. These experiments establish the robust resistance of
NSC beams to atmospheric turbulence into the strong-turbulence regime. Finally,
Chapter 8 contains concluding remarks and plans for future research in this area.
3
Chapter 2: Atmospheric Turbulence
2.1 Overview
Atmospheric turbulence is the fluctuation of the fluid-dynamical properties
of air, over a range of scale lengths, in response to large-scale flow. An artistic
rendering of turbulent motion at a variety of scale lengths is presented in Figure
2.1. Credit for this clever example of turbulent flow is due to Ball [13].
Following the foundational work of Kolmogorov, turbulent fluctuations in air
properties are typically modeled as randomly-fluctuating fields, without reference to
the deterministic flows which drive those fluctuations. In spite of this simplification,
many statistical properties of turbulence have been calculated, with good agreement
with experiment. This analysis has led to ways of experimentally characterizing the
statistical properties of turbulence without being forced to measure the entire flow
field.
4
Figure 2.1: The Great Wave off Kanagawa, woodcut by Hokusai. Mul-
tiple scale lengths of turbulence are visible in this image, from the outer
scale at the size of the largest wave, through the inner scale at the size
of the smallest eddies.
5
2.2 Atmospheric Turbulence
2.2.1 The Approach of Kolmogorov
Although velocity turbulence has been known as a phenomenon of fluid motion
for generations, the foundational statistical treatment of such turbulence is generally
held to be that of Kolmogorov [4]. Kolmogorov developed a power spectrum for ho-
mogenous, isotropic turbulence for a range of scales between the large deterministic
flows (the “outer scale,” denoted L0) and the small vortices whose kinetic energy is
dissipated by viscosity (the “inner scale,” denoted l0). Between these scales lies the
“inertial subrange.”
Kolmogorov made two major hypotheses: the first, presented as an extended
footnote, was that turbulent motion in the inertial subrange consists of a succession
of increasingly tiny self-similar vortices, in which energy is passed from large scale
sizes to small, with the energy dissipation rate per unit mass constant over all scales.
The second, originally formulated as two separate statements, was that the motion
within the inertial subrange is independent of both the nature of the driving flow at
the outer scale, as well as the viscous dissipation occurring at the inner scale. Under
these circumstances, Kolmogorov developed an expression describing the statistical
properties of turbulent velocity fluctuations.
6
2.2.2 Correlation and Structure of Turbulence
Kolmogorov treats turbulence quantities as random variables. Here, and
throughout this work, “turbulence quantities” refers to any physical quantity which
fluctuates as a function of position and time within the turbulent fluid; most often
the velocity, temperature, or index of refraction. As random variables, turbulence
quantities can be characterized by their statistical moments, such as mean, correla-
tion, et cetera.
Consider a randomly varying scalar field x(~r, t). The correlation function of
random field x is defined as:
Bx(~r
∗
1, ~r2, t) = 〈[x(~r1, t)− x̄(~r1, t)][x (~r2, t)− x̄∗(~r2, t)]〉 (2.1)
where angle brackets indicate expectation, an asterisk indicates the complex conju-
gate, x̄(~r1, t) is the expected value of x at position ~r1 and time t, and all expectations
are performed over probability space; that is, averaging over all possible realizations
of the random field. For a statistically stationary, homogenous, and isotropic field,
x̄ is independent of space and time, and the correlation function reduces to
Bx(R) = 〈x(~r ∗1)x (~r2)〉 − x̄2 (2.2)
where R is the separation |~r2 − ~r1| for any choice of ~r1,2. Bx(R) is now a function
only of the separation between the measurement points. When considering tur-
bulent quantities, the assumption of ergodicity is often made in addition to that of
stationarity; that is, that expectations over time can be substituted for expectations
7
over realizations. This assumption is necessary for experimental results, which are
necessarily performed over time within a single realization of the universe, to be
compared to theoretical results, which are free to consider multiple realizations.
The assumptions of homogeneity and isotropy drive Kolmogorov’s argument; it
is particularly important that the mean value of each turbulent quantity be constant
over all space. This is not frequently the case; however, the difference between
turbulent field quantities at two locations almost always behaves like a statistically
homogenous random field [4, 7]. Therefore, we introduce the structure function in
lieu of the correlation [14].
The structure function for a time-varying random field x(~r, t) is defined as:
〈 〉
Dx(~r1, ~r2, t) = |x(~r1, t)− x(~r2, t)|2 (2.3)
If the random field is stationary, locally homogenous (that is, it consists of a
zero-mean statistically homogenous fluctuation plus a locally-varying mean, x(~r) =
x1(~r)+ x̄(~r) with x̄1 = 0 statistically homogenous), and isotropic, then the structure
function depends only on the separation R between the two points, and not on which
particular points you pick. It can then be written as
〈 〉
Dx(R) = |x(~r )− x(~r +R~ )|21 1 (2.4)
for any choice of ~r1 and R~ .
If the field is indeed statistically homogenous, not merely locally homogenous,
then we have lost nothing by choosing to use the structure function; it is related to
8
the correlation function by
Dx(R) = 2[Bx(0)−Bx(R)] (2.5)
In the first of his 1941 papers, Kolmogorov performed a coordinate transform
to a reference frame which includes subtraction of velocities at one point from those
at another; in effect, every correlation function in [4] is in fact a structure function.
Another method often used to characterize turbulence quantities is the power
spectrum of turbulence. This can be calculated from the Fourier transform of the
structure function [15]. The one-dimensional power spectrum is given by [7, 16]
∫ ∞ [ ]1 sinκR d
Φ (κ) = R2
d
x Dx(R) dR (2.6)
4π2κ2 0 κR dR dR
where κ is the spatial wavenumber associated with scale size l, κ = 2π/l in units of
radians per meter.
2.2.3 Velocity and Temperature Turbulence
Discussion now returns to Kolmogorov’s approach to turbulence. Kolmogorov
showed that within the inertial subrange, when the hypotheses of self-similarity and
scale independence hold, the structure function for the component of fluid velocity
along a path between two measurement points must be given by [4,7, 17]
DV (R) = C
2R2/3V (2.7)
where C2V is the velocity structure parameter in units of m
4/3/s2 and R is the
9
separation between the measurement points. Velocity turbulence of this form is
modeled as a stationary, locally homogenous, and isotropic random field.
C2V is related to the energy dissipation rate by C
2
V = m
2/3, where m is a
dimensionless constant experimentally determined to be approximately 2, and  is
the rate of energy dissipation per unit mass, in units of m2/s3. DV (R) has units
of m4/3/s2. Equation 2.7 describes mathematically the “Kolmogorov cascade” in
which energy enters the system at the largest spatial scale and “cascades” to ever
smaller scales, where it is eventually dissipated by viscosity. However, it does not
describe the driving or dissipative mechanisms.
2.2.4 Optical Turbulence
Optical turbulence in the atmosphere is generated when turbulent velocity
fluctuations act to mix parcels of air of different temperatures together. The re-
sulting turbulent fluctuations in air temperature lead to turbulent fluctuations in
density, and thus to fluctuations in the index of refraction. With this in mind, the
method of Kolmogorov was extended to index fluctuations. An outer scale, inner
scale, and inertial subrange are defined for index fluctuations, analogous to the cor-
responding quantities for velocity. For optical atmospheric turbulence (henceforth
simply turbulence), the structure function in the inertial subrange is given by
Dn(R) = C
2
nR
2/3 (2.8)
where C2n is the index of refraction structure parameter in m
−2/3. Near the ground,
10
C2n typically ranges from as little as 10
−16 for very weak turbulence, through 10−12
for very strong turbulence.
Based on this form for the structure function, the spatial power spectrum of
turbulence can be shown to be [7]
Φn(κ) = 0.033C
2κ−11/3n (2.9)
where κ is the spatial wavenumber in rad/m. This is the Kolmogorov spectrum
for turbulence. It neglects inner and outer scale effects, and is therefore only valid
within the inertial subrange, l0  1/κ  L0. However, it is still used for many
cases in which the scale of interest is far from the inner and outer scales.
When inner and outer scale effects cannot be neglected, modifications to the
Kolmogorov spectrum must be made in order to truncate the cascade at high and
low wavenumbers [12, 18, 19]. Without truncation at high wavenumber, the energy
dissipation into viscosity is not described; without truncation at low wavenumber,
the spectrum contains infinite energy. The modified von Kármán spectrum is a
commonly used spectrum which includes both of these effects in a simple manner;
it is given by [7]
Φ (κ) = 0.033C2n n(κ
2 + κ2 −11/60) exp(−κ2/κ2m) (2.10)
Φ (κ) = 0.033C2(κ2 + κ2)−11/6n n 0 exp(−κ2R(z)2) (2.11)
where κ0 = 2π/L0 and κm = 5.92/l0 define the outer and inner scale cutoff wavenum-
11
bers, respectively.
The outer and inner scale sizes are determined by the geometry of the turbu-
lence. For turbulence in the open atmosphere, in the 100 meters nearest the ground,
the outer scale is equal to approximately 1/2 the distance above the ground [7,12].
The inner scale of turbulence is related to the energy dissipation rate by [12]
l = aν3/4−1/40 (2.12)
where ν is the fluid viscosity,  is the energy dissipation rate, and a is a medium-
dependent dimensionless constant, equal to 7.4 for air [19].
104
100
10-4
10-8
KolmogorovSpectrum
10-12 von KarmanSpectrum
10-1 100 101
Wavenumber ( / ) 102 1031 m
Figure 2.2: The Kolmogorov and Von Karman turbulence spectra. The
Von Karman spectrum includes inner and outer scale effects.
In conditions when the average wind speed is greater than the typical eddy
flow speed, the turbulence can be conceived of as a more-or-less static random
12
Φn(κ) / 0.033C 2n
field of index variations, which is swept through a volume of interest (such as a
laser propagation path) by a transverse wind [20]. This is the Taylor frozen-flow
hypothesis. The required conditions often hold; typical time scales for transverse
wind motion are on the order of 1 second, while the time scale for eddy movement
is on the order of 10 seconds [7]. This hypothesis implies that spatial and temporal
statistics of turbulence should be the same.
2.3 Turbulence Characterization
Multiple methods of experimentally characterizing atmospheric turbulence ex-
ist. Methods used in this work include hot wire anemometry, commercial scintil-
lometer, and a combination measurement of beam intensity fluctuation and angle-
of-arrival variation. Each of these methods is described below.
2.3.1 Hot Wire Anemometry
Velocity turbulence can be characterized by measuring fluctuations in the local
air velocity within the turbulent volume. A hot wire anemometer is an instrument
capable of measuring the wind velocity past a small probe at high speed and with
great accuracy.
Hot wire anemometers operate on the principle that the resistance of a con-
ductor is a function of its temperature. A small probe wire (∼ 5 µ m diameter and 1
mm long) is heated to a high temperature (∼ 150 deg C); since the wire has very low
heat capacity, air flow past the probe rapidly cools the wire by forced convection.
13
Ohmic heating and convective cooling dominate the heat transfer equation; thus the
wire temperature can be related to the flow velocity [15].
In a constant current anemometer, the current through the wire is kept con-
stant and the resistance is measured to determine the temperature, and thereby
the air flow velocity. In a constant temperature anemometer (CTA) the resistance
of the wire (and thus its temperature) is kept constant by a feedback circuit; the
voltage applied to accomplish this is then measured to determine the velocity. In
either case, careful calibration is necessary in order to determine the relationship
between the air velocity and measured electrical quantity. Commercial anemometers
are typically calibrated at the factory. Since the thermal mass of the wire is small,
fluctuations in velocity can be measured at frequencies on the order of one to ten
kilohertz [15]. In this work, a CTA has been used to characterize turbulent velocity
fluctuations.
2.3.1.1 Effect of Temperature Fluctuations
For a CTA, the voltage across the wire will be proportional to the rate of heat
loss due to convection. The following analysis demonstrates that fluctuations in
the rate of heat loss are dominated by fluctuations in the velocity of the fluid, and
not significantly influenced by fluctuations in the temperature. That is, a CTA is
a constant-temperature anemometer, and not a constant-temperature thermometer.
Smolyakov has written that air temperature does not affect the measurements of
velocity fluctuations relative to the mean; a calculation to support this assertion is
14
found below [15].
In the limit of forced convection, the rate of heat loss for a hot cylinder im-
mersed in a fluid flow is given by
Q̇ = hA(Tc − Tf ) (2.13)
where A is the surface area of the cylinder, given by πDL, D and L are the diameter
and length of the cylinder, Tc and Tf are the temperatures of the cylinder and the
fluid respectively, and h is the geometry-dependent heat transfer coefficient. For a
cylinder experiencing forced convection, the heat transfer coefficient is given by [21]
k Tc
h = (0.24 + 0.56Re0.45D )( )
−0.17 (2.14)
D Tf
where k is the thermal conductivity of the fluid, Re is the Reynolds number for the
flow due to the influence of the cylinder, and the constants have been determined
experimentally. The Reynolds number is a nondimensional number which quantifies
the ratio of inertial to viscous forces in the fluid flow. The Reynolds number for
flow at velocity v around an object of characteristic dimension x is given by
ρvx
Rex = (2.15)
µ
where ρ is the fluid mass density and µ is the fluid’s dynamic viscosity.
The constants in equation 2.14 have been determined under the conditions
that Reynolds number Re < 77. For the case of a CTA measuring atmospheric
turbulence, the characteristic dimension is the wire’s diameter (on the order of
15
microns) and the characteristic velocity is on the order of 1 m/s. The resulting
Reynolds number is thus less than one.
Returning to the analysis of Q̇ fluctuations, typical temperature fluctuations
are on the order of 0.1 C, and the velocity fluctuates on the order of 0.05 m/s. In
this case the temperature contribution to the heat transfer fluctuations (and thus
the measured voltage) is approximately 5% of the total fluctuation.
2.3.1.2 Benchmark
In this work, a commercial CTA manufactured by TSI was used to characterize
velocity turbulence. The CTA was benchmarked by measuring the turbulent velocity
fluctuations one meter above a concrete parking lot. The velocity structure function
was generated by using the Taylor frozen-flow hypothesis to convert time-resolved
data into space-resolved data. The average wind speed past the probe during the
measurement was 0.26 m/s. The one-dimensional power spectrum of the structure
function demonstrates a slope of approximately −5/3, as expected for Kolmogorov
turbulence. These results are summarized in Figure 2.3.
16
100
10-2
10-4
10-6
10-8
10-10 NRL Parking Lot
10-2 10-1 100 101 102
Wavenumber k H1êmL
Figure 2.3: One-dimensional velocity turbulence power spectrum Vu(k)
for a naturally turbulent atmosphere. A −5/3-slope line is displayed in
black.
17
VuHkL H
3
m ê 2s L
2.3.2 Scintillometer
Optical turbulence can be characterized by passing light through a volume of
uniform turbulence and observing the resulting changes in the light. Under certain
assumptions, measured statistical properties of the light can be related to statistical
properties of the turbulence.
A scintillometer is a diagnostic designed to measure optical turbulence by
projecting a broad beam of incoherent light through the turbulent volume to a
remote receiver with a large aperture. The large aperture acts to increase the
effective range at which the device can determine C2n [22]. The fluctuating intensity
of the received light, or scintillation, is recorded in order to back out C2n according
to theoretical equations. For such a device, the structure parameter is given by
C2 = 1.12σ2 D7/3L−3n lnI (2.16)
where σ2lnI is the normalized variance of the log of the intensity fluctuations, and D
is the diameter of the transmitter and receiver apertures [23]. Scintillometer theory
and practice are well-developed, and the technique has been benchmarked against
independent measurements of turbulence [22]. Scintillometers are widely available
as an industry-standard method of measuring C2n. A Kipp & Zonen LAS MkII
scintillometer was used throughout this work as an industrial standard to charac-
terize C2n and benchmark other diagnostics. However, the increased aperture size
of the scintillometer also increases the minimum scintillation necessary to produce
a response. The scintillometer is therefore inappropriate for short-range or very
18
low-turbulence environments.
2.3.3 LISSD
Another optical diagnostic for measuring turbulence parameters is the Laser
Inner Scale and Scintillation Diagnostic (LISSD). The LISSD is capable of measur-
ing the structure parameter C2n, as well as the inner scale of turbulence l0. This
diagnostic was first developed and implemented by Consortini et al. in 2003 [24].
The LISSD uses a low-power helium-neon laser (λ = 632.8 nm) which propa-
gates through the turbulence. The laser must have a Rayleigh range much shorter
than the propagation distance, or a pinhole must be placed in front of the laser, in
order to ensure that the wavefront at the detector is a spherical wave. The detector
is a digital video camera positioned a fixed distance behind a pinhole. Scintilla-
tion and beam wander cause the spot formed on the camera sensor to fluctuate in
both intensity and position. The intensity fluctuations are used to determine the
scintillation index σ2I , the variance of the intensity normalized to the square of the
mean [7]. The position wander, coupled with the distance between the detector and
the pinhole, gives the angle-of-arrival fluctuation 〈α2〉 . These two measurements
give the path-averaged C2n and l0 according to
7/6
2 0.5k L
11/6
Cn = (2.17)σ2I
19
and
( )
〈 2〉 1/2α
l0 = 1.08L (2.18)
σ2I
The implementation of the LISSD used in this work has been benchmarked
against the Kipp & Zonen commercial scintillometer over a distance of 100 meters
in a typical outdoor environment at NRL. Under these conditions the C2n values
reported by the LISSD matched those reported by the commercial instrument. The
scintillometer’s operating wavelength of 850 nm is taken into account by its internal
software when calculating C2n, so the wavelength difference between the LISSD and
the scintillometer does not affect the measurement. Because of its superior ability
to make measurements over short distances or under low turbulence conditions, as
well as its ability to measure l0, the LISSD is the primary turbulence diagnostic
used in this work.
20
Chapter 3: Linear Laser Beam Propagation
3.1 Overview
When a laser beam propagates through a turbulent medium, it acquires ran-
dom phase perturbations caused by the spatially-varying refractive index of the
medium. As turbulent flows move, these phase perturbations fluctuate randomly.
The index of refraction fluctuations can be thought of as as weak, distributed lenses
throughout the turbulent volume. Those fluctuations much larger than the beam
tend to steer it as a whole; those much smaller act as scattering centers; those in
the intermediate range act to alter the intensity profile of the beam. Fluctuations
of all sizes act on the beam simultaneously. The resulting fluctuations of the beam
profile, size, and position have been extensively studied, and theoretical expressions
exist to describe their statistics.
The treatment of laser beam propagation in turbulence in this chapter is due
to Andrews and Phillips; for a more detailed look at the subtleties of this topic see
their 2005 book on the subject [7]. The presentation here is limited to only those
equations necessary to reproduce the figures in Chapters 6 and 7.
21
3.2 Without Turbulence
3.2.1 The Wave Equation
The propagation of light in the absence of medium-dependent effects is gov-
erned by the wave equation
2 ~
∇2 1 ∂ EE~ − = 0 (3.1)
c2 ∂t2
where ∇2 is the Laplacian operator, E~ (x, y, z, t) is the complex electric field, and c
is the speed of light. Under the assumption of a linearly polarized monochromatic
wave (that is, E~ (x, y, z, t) = x̂E (x, y, z)e−iωt0 ), and orienting the coordinate system
so that the z-axis points in the direction of propagation, the equation becomes
2
∇2 ∂ E0 2⊥E0 + + k E0 = 0 (3.2)∂z2
where k is the optical wavenumber of the laser field, given by k = ω/c = 2π/λ.
Assuming(radial symmet)ry (E0(x, y, z) = E0(r, z)) and taking the paraxial
2
approximation ∇2 E  ∂ E0⊥ 0 2 , the wave equation becomes∂z
∇2rE0 + k2E0 = 0 (3.3)
22
3.2.2 Gaussian Beam Propagation
The lowest order transverse mode of a laser resonator is a Gaussian beam [25].
Therefore, we look for solutions to Equation 3.3 of the form [7].
[ ( )]
2
E0(r, z) = A(z) exp −
1 α0kr
(3.4)
p(z) 2
where α = 2 + i 10 2 , R0 is the initial radius of the beam, and FkR F 0 is the radius0 0
of curvature of the phase front. A(z) is the propagation distance-dependent peak
field, and p(z) contains the phase. The exact form of Equation 3.4 has been cho-
sen to simplify later analysis, and maintain consistency with [7]. By solving the
wave equation with appropriate boundary conditions, the remaining parameters in
Equation 3.4 can be shown to be
z 2z
p(z) = 1 + iα0z = 1− + i
F kR20 0 (3.5)
1
A(z) =
p(z)
These parameters in turn lend themselves to expression in terms of the follow-
ing parameters [7].
z
Θ0 = 1−
F0 (3.6)
2z
Λ0 = ,
kR20
known as input beam parameters since they characterize the beam at the beginning
of its propagation. The focusing parameter, Θ0, characterizes the initial divergence
of the beam: it is equal to 1 for collimated beams, less than 1 for converging beams,
23
and greater than 1 for diverging beams. The Fresnel ratio, Λ0, characterizes the
distance from the beam relative to its initial size.
The input beam parameters can in turn be used to define the output beam
parameters
Θ0 z
Θ = = 1 +
Θ20 + Λ
2
0 F (3.7)
Λ0 2z
Λ = = ,
Θ20 + Λ
2
0 kR
2
which characterize the beam after a distance of propagation z. Here R and F are
the beam’s radius and phase front radius of curvature after propagation. The beam
size changes as a function of distance, according to
( ) 2
R(z)2 = R2 2 2
R
Θ + Λ = 00 0 0 (3.8)Θ2 + Λ2
For the case of a collimated beam (Θ0 = 1), the beam propagates as
( ( ) )2
z
R(z)2 = R20 1 + (3.9)zR
where zR = πR
2
0/λ is the Rayleigh range, the distance over which the radius of the
√
beam increases by a factor of 2.
As the beam propagates, conservation of energy requires that the peak inten-
sity change as the radius does, decreasing as R increases. The resulting intensity
profile can be shown to be
( ) ( ) ( )
1 2r2 2r2
I(r) = exp − = (Θ2 + Λ2) exp − (3.10)
Θ20 + Λ
2 R20 R
2
24
with on-axis intensity
R20 1I(0) = = = Θ2 + Λ2 (3.11)
R2 Θ2 + Λ20 0
3.3 With Turbulence
When a laser beam propagates through turbulence, the intensity profile re-
ceived at the target is observed to vary in time. As the beam propagates, it acquires
random phase perturbations due to turbulence. The beam wanders, its profile varies,
and its intensity fluctuates. An illustrative example of the effect turbulence can have
on a beam profile is displayed in Figure 3.1.
3.3.1 Weak Turbulence
As a laser beam propagates through a medium with spatially varying index,
different transverse locations in the beam will acquire random phase errors and
decohere from one another. The transverse coherence length of such a beam, the
distance in the transverse dimension over which it remains coherent, will naturally
decrease with distance. The weak turbulence limit applies when this change in
transverse coherence length can be neglected [26]. Under these conditions the electric
field of the wave can be expressed as E(x, y, z) = E0 exp[ψ(x, y, z)], where E0 is the
electric field without turbulence and ψ contains all complex phase perturbations due
to turbulence. This analysis is known as the Rytov method [12].
25
Figure 3.1: Laser beam images taken before (a) and after (b) propagation
through more than 1 km of natural turbulence at Starfire Optical Range.
Note the increase in the spot size, as well as the significant distortion
of the intensity profile. The instantaneous 1/e2 beam radius is 2.1 cm
before propagation, and 7.8 cm after. The propagation range is 3.2 km;
C2 = 1.4×10−14m−2/3n ; the Rytov parameter σ2R = 3.6. This is considered
to be “strong” turbulence.
26
3.3.1.1 Rytov Variance and Scintillation Index
The fluctuation of the intensity of a laser beam is known as scintillation.
Its strength is characterized by the scintillation index, equal to the variance of
the on-axis intensity normalized to the square of the mean. For an infinite plane
wave of wavenumber k, propagating a distance L through Kolmogorov turbulence,
and keeping perturbations on the phase to second order, the scintillation index is
predicted to be
σ2R = 1.23C
2k7/6L11/6n (3.12)
also known as the Rytov variance, or Rytov parameter. The Rytov parameter is
often used to characterize the strength of turbulence. For σ2R  1, the turbulence
is said to be weak; for σ2R  1, the turbulence is said to be strong. For weak
turbulence, the scintillation index of the plane wave is equal to the Rytov parameter.
As turbulence becomes strong, the scintillation index increases more slowly than the
Rytov parameter. The Rytov parameter continues to be used to characterize the
strength of turbulence, even beyond the point where it represents the scintillation
index of any real beam.
For a Gaussian beam propagating in weak turbulence with a von Karman
spectrum, the scintillation index is given by
27
[( ) ]1/6
2 2 5/6 ΛQ
2
m
σ = 3.93σ Λ − 1.29(ΛQ )1/6 rI R { 01 + 0.52ΛQm R2( )
2
2 [(1 + 2Θ) + (2Λ + 3/Q )
2
m ]
11/12 11
+ 3.86σR 0.40 sin ϕ0 + ϕm[(1 + 2Θ)2 + 4Λ2]1/2 ( 6) }5/6
− 6Λ − 11 1 + 0.31ΛQm (3.13)
11/6
[ Q] [(1 + 2Θ)2m +[ 4Λ2] ]6 Qm
where ϕ = tan−1 2Λ , ϕ = tan−1 (1+2Θ)Qm0 m , Q = Lκ
2 /k represents the
1+2Θ 3+2ΛQ mm m
effect of the inner scale, Q = Lκ20 0/k represents the effect of the inner scale, and
recalling that the dependence on propagation distance L is contained in parameters
Λ and Θ.
3.3.1.2 Beam Wander
Beam wander is defined as the variance of the distance of the beam centroid
from the long-term averaged centroid location. It is due to perturbations to the
phase with scale size larger than the radius of the beam. An expression for the
wander can be derived by multiplying the turbulence spectrum Φ by a longpass
filter function which causes Φ to fall off exponentially at κ larger than 1/R(z). The
result of this procedure is
∫ {1 [ ] }2 2 1/6
2 11/3 1 κ Rσc = 7.25C
2
nL
3R 2 0 00 ξ − dξ
0 |Θ0 + Θ̄ 1/3 2 2 20ξ| 1 + κ0R0(Θ0 + Θ̄0ξ)
(3.14)
where Θ̄0 = 1−Θ0. The absence of an inner scale term is due to the filter function.
In the absence of outer scale effects (that is, using a Kolmogorov spectrum), the
28
beam wander of a collimated Gaussian beam reduces to
2 2 3 −1/3σc = 2.42CnL R0 (3.15)
3.3.1.3 Long-Term Beam Size
Although the instantaneous intensity profile of an initially Gaussian beam is
more or less distorted after propagation through turbulence, the long-term average
intensity profile nevertheless remains approximately Gaussian [7]. This long-term
average profile has a characteristic spot size, which is increased relative to the vac-
uum case by turbulence. For a Gaussian beam propagating through weak turbu-
lence, and assuming a von Karman turbulence spectrum (equation 2.10), it can be
written
( ( ))
2 2 2 5/6 (1 + 0.31ΛQ )
5/6
m − 1
R = R 1 + 3.54σ Λ − 0.36(ΛQ 1/6LT R 5/6 0) (3.16)(ΛQm)
where R is the radius of the beam in the absence of turbulence.
In the absence of inner and outer scale effects, and assuming a collimated
beam, the Kolmogorov spectrum takes the place of the von Karman spectrum and
this expression simplifies to
( )( ( ) )
2 5/6z z zR
R2LT = R
2 1 + 1 + 1.33σ2R (3.17)z2 2R z + z
2
R
The long-term spot size includes independent contributions from both instan-
taneous spreading and breakup of the beam (detailed in the next section), as well
29
as the random wander of the beam centroid over time [27]. The contributions sum
in quadrature, as
R2 = R2 + σ2LT ST c (3.18)
where RST is the instantaneous beam radius due to spreading and breakup, and σ
2
c
is the variance of the beam centroid.
3.3.1.4 Short-Term Beam Size
Combining equations 3.17, 3.18, and 3.15 gives an expression for the expected
instantaneous spot size,
( )( ( ) )
2 5/6z z z ( )2 2 R −1/3RST = R0 1 + 1 + 1.33σ2R − 2.42C2L3R (3.19)z2R z2 + z2 n 0R
The instantaneous spot size is expected to match this formula on average,
not for each individual realization of turbulence. An analogous expression for the
more general case of a non-collimated Gaussian beam in von Karman turbulence can
be calculated by subtracting Equations 3.16 and 3.14 in accordance with Equation
3.18. However, it has no algebraic closed form and will not be presented here. This
expression will be evaluated numerically in Chapters 6 and 7 to compare theory to
experiment.
30
3.3.2 Strong Turbulence
As a beam propagates through turbulence, its transverse coherence length is
decreased as turbulence introduces random phase errors into the beam. In weak
turbulence theory, we neglect this turbulence-induced decrease in coherence length,
and treat the coherence length as always infinite. This is a reasonable approxima-
tion because weak turbulence is not capable of altering the coherence length much.
However, in strong turbulence, the beam’s coherence length decreases significantly
with propagation distance. This decreases the effect of turbulence relative to what
would be expected from weak turbulence theory. Turbulent fluctuations larger than
the coherence length of the beam no longer have any power to spoil the wavefront
of the beam, as the wavefront is already fully randomized at those length scales. In
order to account for this effect, strong turbulence theory introduces filter functions,
which act to kill the contribution of turbulence at scales larger than the coherence
length.
Equations comparable to those for weak turbulence exist for the case of strong
turbulence. Some involve integrals which do not easily simplify to algebraic expres-
sions; those integrals are presented unevaluated here. For the portions of this work
which took place under conditions of strong turbulence, the appropriate equations
were evaluated numerically in order to compare theory to experiment.
31
3.3.2.1 Coherence Length
Assuming the beam is much smaller than the outer scale of turbulence, as is
the case for this work, the coherence length of a Gaussian beam propagating through
strong turbulence with a Von Karman spectrum is given by
( )1/2 ( )−1/23 2 2 −1/3
[
2 2 ] 1.64C1+Θe+Θ +Λ nk Ll0 ρ0  l0e eρ0 =  (3.20)3/58 2 2 3/5
11/6 (1.46C k L) l0  ρ0  L0
3(a ne+0.62Λe )
where the effective strong-turbulence receiver beam parameters are given by
Θ− 0.81(σ2 )6/5R ΛΘe =
1 + 1.63(σ2 )6/5R Λ
Λ
Λe = (3.21)
1 + 1.63(σ2 )6/5R Λ
and the parameter ae is given by
1−Θ8/3 Θ ≥ 0 1−Θae =  (3.22)1+|Θ|8/3 Θ < 0
1−Θ
3.3.2.2 Scintillation Index
For a Gaussian beam propagating in strong turbulence with a Modified At-
mospheric spectrum, the scintillation index is given by
 2
σ2 = expσ2 2 0.51σG I lnX(l0)− σlnX(L0) + ( ) − 1 (3.23)5/6
12/5
1 + 0.69σG
32
where the Gaussian-beam Rytov variance with nonzero inner scale is given by
{ [ ( )
[(1 + 2Θ)22 2 + (2Λ + 3/Q )
2]11/12l 11
σG = 3.86σR 0.40 sin ϕ0 + ϕl[(1 + 2Θ)2 + 4Λ2]1/2 ( 6)
2.61 4
+ sin(ϕ0 + ϕ2 2 2 1/4 l[(1 + 2Θ) Ql + (3 + 2ΛQl) ] 3 )]
− 0.52 5sin ϕ0 + ϕl
[(1 + 2Θ)2Q2l + (3 + 2ΛQ )
2
l ]7[/2(4 4 )5/6
− 13.40Λ − 11 1 + 0.31ΛQl
11/6
Ql [(1 + 2Θ)
2 + 4Λ2] 6 Ql ]}
1.10(1 + 0.27ΛQ )1/3l − 0.19(1 + 0.24ΛQl)
1/4
+ (3.24)
5/6 5/6
Ql Ql
and the large-scale log-irradiance variance is
( )( )7/6
2 1 − 1 1 ηXQlσlnX(l0) = 0.49σR [ Θ̄ + Θ̄
2
3 2 5 ηX +Ql ]
ηX
(1 + 1.75(
1/2
))( −
ηX
0.25( )7/12
ηX +Ql )ηX +Ql (3.25)7/6
1 1 1 ηX0Ql
σlnX(L0) = 0.49σ
2
R [ − Θ̄ + Θ̄
2
3 2 5 ηX0 +Ql ]
ηX0
1 + 1.75( )1/2 − ηX00.25( )7/12
ηX0 +Ql ηX0 +Ql
and the nondimensional cutoff frequencies for the inner and outer scales are given
by
[ ]−1
0.38 1 1 12 1/6 − Θ̄ + Θ̄
2
η = + 0.47σ Q ( 3 2 5 )6/7X
1− 3.21Θ̄ + 5.29Θ̄2 R l 1 + 2.20Θ̄
(3.26)
ηXQ0
ηX0 =
ηX +Q0
where the outer scale parameter Q0 = Lκ
2
0/k, inner scale parameter Ql = Lκ
2
l /k,
ϕ = tan−1( 2Λ0 ), and ϕl = tan
−1( (1+2Θ)Ql ).
1+2Θ 3+2ΛQl
33
3.3.2.3 Long-Term Beam Size
The details of the long-term spot size depend on the relationship between the
propagation distance L and the distance at which the transverse coherence length
of the beam is equal to the inner scale.
The distance at which the transverse coherence radius of the Gaussian beam
becomes equal to the inner scale is given by:
5/3
zi = 1/(C
2k2n l0 ) (3.27)
Under the limiting cases of propagation over a distance much greater than, or
much less than zi, and in the absence of outer-scale effects, the long-term spot size
for a Gaussian beam is given by:
( )
R2 2LT = R (1 + 0.982σ
2 ΛQ1/6R m ) , for L zi (3.28)
= R2 1 + 1.630(σ2 6/5R) Λ , for L zi
where R is the radius of the beam in the absence of turbulence. The long-term
spot size includes independent contributions from both instantaneous spreading and
breakup of the beam, as well as the random wander of the beam centroid over
time [27]. The contributions sum in quadrature, as before:
W 2 = W 2 + σ2LT ST cen (3.29)
34
3.3.2.4 Beam Wander
Under strong fluctuation conditions, the beam wander is given by
∫ [1
σ2
−1/3 1
c = 7.25C
2L3n R
2
0 ξ 12/5
0 (Θ0 + Θ̄0ξ)2 + 1.63σ Λ 16/5R 0(1− ξ) ]
{ [ (κ R )1/3− 0 0 ]} dξ (3.30)1/6
1 + κ2R2
12/5
0 0 (Θ0 + Θ̄0ξ)
2 + 1.63σR Λ0(1− ξ)16/5
This expression has no closed-form analytic solution. It is the general expres-
sion for beam wander under all conditions. For σ2R  1 it reduces to Equation 3.14;
therefore, it will be evaluated numerically to model beam wander under both weak
and strong turbulence.
3.3.2.5 Short-Term Beam Size
As is the case for weak turbulence, the short-term beam size under strong
turbulence conditions can be constructed by combining Equations 3.30 and 3.28
in accordance with Equation 3.29. Combining these equations yields no algebraic
simplification or additional insight; therefore, the full expression is not presented
explicitly here.
35
Chapter 4: Artificial Turbulence Generation
4.1 Overview
Outdoor field experiments to study the effects of turbulence in a natural set-
ting are complicated by natural uncontrollable variation in turbulence strength.
Laboratory turbulence simulators using spatial light modulators or phase plates do
not correctly capture the nonlinear dynamics of atmospheric propagation, and are
not robust against interaction with high-power laser pulses. Fluid-based artificial
turbulence generators offer a way to study turbulence under controlled laboratory
conditions, without being subject to the limitations of turbulence simulators. The
turbulence generator used in this work operates by heating a long, narrow volume
of air and allowing free convection to create turbulence similar to that observed in
nature. This turbulence generator has been demonstrated to generate turbulence
with statistical properties similar to those seen in a natural outdoor environment.
4.2 Turbulence Experiments Outdoors
Experimental turbulence research can often be performed in the field, simply
by allowing the laser to propagate across a long distance. The strength of optical
36
turbulence is characterized by C2n, the coefficient of the structure function for the
index of refraction. In the surface layer of the atmosphere, less than ∼100 meters
above the ground, C2n experiences natural daily variation over several orders of mag-
nitude as the ground heats and cools with the rising and setting of the sun. A typical
example of daily C2n variation is presented in Figure 4.1. This data was collected via
differential image motion monitor (DIMM) [28], operating at a wavelength of 632.8
nm, on a clear day in early September 2008. This record includes a pronounced
“null event,” a rapid decrease in C2n associated with sunrise and sunset. A large
range of turbulence conditions can be sampled by observing the laser propagation
throughout the day.
Many outdoor ranges suitable for long-distance laser propagation exist, in-
cluding at Starfire Optical Range [29], the U.S. Naval Academy [30], across the
Chesapeake Bay [31] and at the former Space Shuttle runway [32], among others.
However, long-range outdoor experiments have the disadvantage of relying on un-
controllable natural variations, and the desired turbulence conditions may not occur
during the experimental window. Many hours of experiment are required in order
to secure a reasonable volume of data, which in any case will necessarily be under
conditions which are not entirely known. In addition, laser safety considerations
preclude the propagation of high-power beams at many outdoor ranges.
For these reasons, many techniques have been developed in order to generate
or simulate turbulence in a more compact, more controllable manner in a laboratory
setting [33]. These include a variety of fixed and variable phase screen methods, as
well as turbulent fluid methods.
37
Figure 4.1: A typical day of C2n variation taken at Kirtland AFB, Albu-
querque, NM. Note the pronounced evening null event, and the accom-
panying rapid variation of turbulence across several orders of magnitude.
38
Fluid-based turbulence generators adopt perhaps the most direct approach to
generating turbulence short of performing field experiments. A volume of air, water,
or some other fluid is stirred via some combination of mechanical action (pumps,
fans, or jets) and thermal action (heat guns, hot plates, heater wires). Gradients in
temperature result in turbulent fluctuations of the index of refraction. This method
has several advantages: realistic Kolmogorov turbulence is naturally obtained; the
strength of the turbulence is easily tuned by varying the strength of the heating; rare
events such as intense scintillations naturally occur with the appropriate frequency;
the turbulence is distributed along the path rather than being tightly concentrated at
one or a few longitudinal positions; appropriate construction materials are typically
far cheaper than custom phase screens or deformable optics. The disadvantages are
as follows: the inner and outer scales cannot be varied independently of C2n; the
turbulence is truly random and cannot be reproduced except in a statistical sense;
the generator occupies a large volume relative to phase screens; the heat used to
generate the turbulence must be dissipated in some manner [33]. Turbulent fluid
methods are well-suited for the study of high-power propagation because there are
no physical optics, such as phase screens or spatial light modulators, which could be
damaged by the high-power beam. Additionally, if the fluid used is air, it correctly
captures the nonlinear properties of actual atmospheric laser propagation.
39
4.3 Previous Turbulence Generators
Davis has developed a hot-liquid turbulence generator for creating turbulence
on laboratory scales [34]. A diagram of the experimental apparatus is shown in
Figure 4.2. This device uses a heating wire at the bottom of a liquid-filled acrylic
tube to generate a temperature differential. The tube is generally filled with water,
although other liquids can be used. The liquid can be allowed to convect naturally,
or a pump can be used to provide additional stirring. The pump must be run from
time to time, or the liquid will reach a uniform temperature and cease to exhibit
turbulence. This method allows a large phase difference to be produced in a short
distance. However, the turbulence evolves much more slowly than in air, and the
inner scale is substantially larger than in air, owing to water’s higher viscosity.
Figure 4.2: Schematic arrangement of Davis’ 1998 experiment incorpo-
rating the hot-liquid turbulence tube.
Majumdar, C. Nelson, and Keskin have all developed flow-driven hot-air tur-
bulence generators [30, 35, 36]. These similar devices all operate by directing one
40
Figure 4.3: C. Nelson’s flow-driven hot air turbulence generator. Panel
a: exterior view with the directions and relative temperatures of flow
marked, as well as the positions of the five turbulence sections. Figure
b: interior view with the direction of laser propagation marked.
or more pairs of hot and cold air flows into a semi-enclosed chamber. Photographs
of C. Nelson’s device are displayed in Figure 4.3 as an example of the type. The
air flows enter the chamber perpendicular to the beam propagation path, with each
hot nozzle more or less opposed by a cold nozzle. The air leaves the chamber via
the laser entrance and exit ports, as well as additional exhausts. The strength of
turbulence is adjusted by varying the temperature difference between the air jets,
as well as the speed of the flow. This method produces a more rapidly evolving
turbulence. However, no obvious relationship exists between temperature / flow
speed and turbulence strength. The energy dissipation rate, useful for estimating
the inner scale of turbulence (Equation 2.12), is unclear.
Finally, convection-driven turbulence generators have been used by Salamé
and W. Nelson [37,38]. Salamé used a series of candles placed below his beam path
41
rather than an evenly distributed turbulent volume, in effect producing a series of
randomly varying gaseous phase screens. W. Nelson has used a series of hot plates
to generate distributed turbulence over a short distance, approximately 3 meters. A
diagram of W. Nelson’s experimental setup is displayed in Figure 4.4. By arranging
the beam path over the hot plates, W. Nelson could produce a situation in which
the turbulence experienced by the light returning from the target to the imaging
camera experienced turbulence which was either identical to, or different from, the
turbulence experienced by the outgoing laser beam.
Figure 4.4: Schematic of W. Nelson’s convection-driven turbulence gen-
erator. Panel a: monostatic beam propagation path with outgoing and
incoming beams experiencing identical turbulence. Panel b: bistatic
path with outgoing and incoming beams experiencing different turbu-
lence.
42
4.4 This Work’s Approach
4.4.1 Theoretical Background
In the lowest layers of the atmosphere near the surface, atmospheric turbu-
lence is driven primarily by natural convection of hot air rising from the sun-warmed
ground. A natural approach to generating artificial turbulence is therefore to heat
some object placed below the propagation range and allow convection to drive tur-
bulence. A long cylindrical heating element, such as a heating wire, allows the
generation of turbulence along the beam’s entire propagation path while consuming
only a modest amount of power - on the order of 10 watts per meter.
The behavior of flow due to natural convection is characterized by the nondi-
mensional local Nusselt number Nu, the ratio of convective to conductive heat trans-
fer rates (qconv and qcond, respectively). For large values of Nu convection dominates
over conduction; for small values the reverse is true. Consider a layer of air of thick-
ness x above a heated surface; the Nusselt number at the top of that layer is given
by
qconv h∆T hx
Nux = = = (4.1)
q ∆Tcond k κx
where h is the geometry-dependent convective heat transfer coefficient in W/m2 ·K,
∆T is the temperature difference between the heated surface and the ambient fluid,
and k is the thermal conductivity of the fluid in W/m · K. The local Nusselt
number increases with height above the source, as the air layer becomes thicker and
43
conduction becomes less important compared to convection. Below a critical Nusselt
number, between 100 and 1000, the flow remains laminar. Beyond this point, heat
transfer is dominated by fluid motion and the flow is considered to be turbulent. For
fully developed turbulence with infinite Nusselt number, no deterministic behavior
resulting from large-scale laminar flows remains.
Some difficulty lies in determining the convective heat transfer coefficient h,
which is influenced by the geometry of the problem and the ability of air to flow
around the heated surface. Experimentally determined correlating equations, re-
lating the Nusselt number for the flow around a horizontal cylinder to other fluid
mechanical parameters, have been developed by Churchill et al [39]. From their
analysis, the convective heat transfer coefficient is given by
( )
1/6 2
k 0.387Ra
h = 0.6 + Dcyl (4.2)
D (1 + (0.559/Pr)9/16)8/27
where Pr is the Prandtl number for air (0.713 at 20 ◦C), RaD is the Rayleigh number
for scale size D, and D is the diameter of the cylinder. The numerical factors have
been determined experimentally through curve fitting over a wide range of Rayleigh
numbers, 10−5 < RaD < 10
12.
The nondimensional Rayleigh number, Ra, characterizes whether the flow
around a heated object will be predominantly convective or conductive. The Rayleigh
number for scale size x is given by
ρgβ∆Tx3
Rax = (4.3)
αµ
44
where ρ is the density, g is the acceleration due to gravity, β is the coefficient of
thermal expansion, α is the thermal diffusivity, and µ is the fluid’s dynamic viscosity.
4.4.2 Implementation
This section reports on the construction and characterization of a hot-air con-
tinuously variable distributed turbulence generator. Two copies of this generator
were constructed: one, over a distance of 30 meters, at NRL; and one, over a prop-
agation range of 180 meters, at an indoor laser propagation range at AFRL. At the
time, the latter device constituted to the author’s knowledge the world’s longest
continuously variable optical turbulence generator.
The generator, depicted symbolically in Figure 4.5, consists of a cluster of
industrial heating wires strung on steel cables running below the beam propaga-
tion path. Each wire is 73 meters long. Tension on the steel cables and a series
of quadrupods positioned periodically along the path support the wires at a fixed
distance below the beam. As the wires are heated, buoyant hot air from the wire
convects upwards to mix with the ambient air of the lab. Laboratory HVAC sys-
tems remove the excess heat and prevent the average temperature of the lab from
increasing. Since the turbulence is driven by natural convective mixing, this method
is more akin to a hot water chamber or a series of hot plates than it is to typical flow-
driven hot-air tubulence generators. The strength of turbulence can be varied by
adjusting the amount of power applied to the heater wires. The power consumed by
the wires gives the rate of energy dissipation ; this will enable comparison between
45
the turbulence spectrum produced by the generator and theoretical predictions (for
instance, equations 2.7 and 2.12). Because it generates a continuously-distributed
turbulent volume in air by the process of free convection, this hot-wire turbulence
generation method will henceforth be referred to as the Generator of Distributed
Free Atmospheric Convective Turbulence (GDFACT).
Figure 4.5: A symbolic diagram of the GDFACT. Heater wires strung
on an extended framework running below the beam propagation path
impart thermal and kinetic energy into the air, which naturally produces
realistic Kolmogorov turbulence. Turbulence diagnostics co-propagate
with the laser beam, characterizing the turbulence as the experiment
takes place.
GDFACT shares the advantages of all turbulent fluid-based generators. It
naturally produces tunable, continuously-distributed Kolmogorov turbulence at a
low cost. More importantly, it is compatible with high-power laser propagation, in
that it contains no physical optics to be damaged by the beam. It also produces
natural variation in both the linear and nonlinear indices of refraction.
There are two downsides to this method. First, it is based on a line source of
turbulence, rather than a plane source as in propagation over land. As the beam
46
propagates, care must be taken that the beam not wander or spread beyond the
narrow corridor of turbulent air. This disadvantage can be alleviated by stringing
several heater wires side-by-side, creating a broader propagation path. This work
was performed using a line source turbulence generator; future work in this field
would benefit from the use of a broad multiple-wire device capable of accommodating
a larger beam with more wander. Second, since the strength of the turbulence is
comparable to that found in nature, a long indoor propagation range is necessary
in order to reach deep turbulence (σ2R ≥ 1). This can be alleviated by placing
high-power dielectric mirrors at the ends of the propagation range and performing
multiple passes over the turbulence generator. The use of such mirrors, however, is
limited by their optical damage threshold, and care must be taken to ensure that
filamentation does not occur.
For the first set of experiments, the wires were strung 20 cm below the beam
propagation path; this separation can be adjusted from an arbitrarily small distance
up to as much as 1 meter. The 20 cm separation was chosen to provide a scintillation
index of 1 for a helium-neon laser (He-Ne, λ = 632.8 nm) at the highest heater
setting, 22.3 W/m per wire. Under these conditions, buoyancy carries the air flow
up across the beam; this is like having a weak transverse wind, in which case the
Taylor Frozen Flow hypothesis is expected to apply [7].
The heater wires were measured to reach a temperature of 65 C above ambient.
The corresponding Nusselt number 20 cm above the wires is approximately 100. This
is marginally within the range of Nusselt numbers considered to be characteristic
of turbulent flow; therefore, the turbulence thus generated may not exhibit fully
47
Kolmogorov statistics.
4.5 Characterization of the Turbulence Generator
4.5.1 Overview
A constant temperature anemometer was used to characterize the spectrum of
velocity fluctuations produced by the GDFACT. The turbulence spectrum was found
to be Kolmogorov within the inertial subrange, with an inner scale of approximately
1 cm at full power. In addition, the optical turbulence was characterized at a
variety of power settings, using the scintillometer and LISSD diagnostics described
in Chapter 2. The scintillometer and LISSD were found to agree with one another
except for the lowest power settings. The inner scale determined by LISSD was
found to agree with that determined by CTA.
4.5.2 Anemometer
At NRL, the turbulence produced by GDFACT was measured with the CTA
to verify that its statistics match those of natural atmospheric turbulence. These
spectra were compared to naturally-occurring turbulent velocity fluctuations one
meter above a concrete parking lot. In each case, the structure function was gener-
ated by using the Taylor frozen-flow hypothesis to convert time-resolved data into
space-resolved data [6]. The structure function takes a long time to compute (a
naive calculation requires a number of comparisons O(n2)), so instead it is calcu-
lated from the correlation, using equation 2.5 and the Wiener-Khintchine Fourier
48
transform relation [7], as follows:
Analysis begins with V (t), the voltage as a function of time. This can be
converted to v(t), the velocity as a function of time, via the CTA manufacturer-
supplied calibration formula
v(t) = c + c V (t) + c V (t)20 1 2 + c3V (t)
3 (4.4)
where c0 = 0.546 m/s, c1 = 3.29202 m/s per V, c2 = 3.03385 m/s per V
2, and
c3 = 34.1096 m/s per V
3.
The Taylor frozen-flow hypothesis is used to convert the time-series to a space-
series according to R = tv̄, where v̄ is the average velocity in the velocity-time series.
Next the Fourier transform of velocity is calculated, and squared to get the
power spectral density of velocity,
∣∣∣∣ ∫ ∞ ∣∣2Sv(κ) = 2 v(R) cos(κR)dR∣∣ (4.5)
0
From the power spectral density, the correlation function is calculated according to:
∫ ∞
Bx(R) = 2 Sx(κ) cos(κR)dκ (4.6)
0
Given the correlation function Bx(R), the structure function Dx(R) can be
calculated from equation 2.5. This approach is less straightforward than calculating
the structure function directly from the velocity fluctuations, but is significantly
more computationally efficient, executing in O(n log n) rather than O(n2).
The one-dimensional power spectrum of the structure function is then calcu-
49
lated according to 2.6, in order to facilitate comparison between theory and experi-
ment.
This spectrum demonstrates a slope of approximately −5/3, as expected for
Kolmogorov turbulence. These results are presented in Figure 4.6. The −5/3 slope
predicted by Kolmogorov and observed in the natural outdoor environment is re-
produced in the experiment.
100
10-4
10-8
NRL Parking Lot
15 Wêm
10-12
31 Wêm
46 Wêm
10-16 77 Wêm
10-2 10-1 100 101 102 103 104 105
Wavenumber k H1êmL
Figure 4.6: One-dimensional velocity turbulence power spectra Vu(k)
for a naturally turbulent atmosphere (NRL parking lot, red), and four
power density settings on the GDFACT (orange through blue). A −5/3-
slope line is displayed in black. High-wavenumber peaks on each data set
represent high-frequency instrumental noise related to the oscilloscope
sampling frequency.
Fiorino et al have shown that the coefficient of the velocity power spectrum,
C2v , can be related to C
2
n, the gradient of the index of refraction, and the verti-
cal gradient of the wind velocity [40]. Unfortunately, an insufficient number of
50
VuHkL H
3
m ê 2s L
anemometers were available to collect wind velocity gradient information in order
to apply this technique. In addition, the inner scale of turbulence cannot be re-
solved. However, these results provide evidence that within the inertial subrange,
the GDFACT is creating Kolmogorov turbulence similar to that encountered in a
natural environment.
A separate experiment was recently performed in order to measure the inner
and outer scales. This experiment was performed on a second-generation GDFACT,
consisting of four heater wires in parallel with one inch spacing between them.
Velocity fluctuations were measured via CTA, as before. Following the method of
Champagne [41], the power spectrum of the derivative of the velocity was calculated.
This quantity, the square of the Fourier transform of the derivative of the velocity,
is presented in Figure 4.7. Each point is the average over a wavenumber bin with
equal width on a log scale. The plotted data is the average of ten such binned
measurements, with error bars representing the standard deviation for each point.
The expected slope of 1/3 is observed within the inertial subrange. The inner
scale, the high-wavenumber point at which the inertial subrange ends, is observed
to be approximately 1 cm; the outer scale, the equivalent low-wavenumber point, is
observed to be approximately 30 cm. This corresponds to the CTA’s height above
the GDFACT when this data was taken.
51
Figure 4.7: One-dimensional acceleration turbulence power spectra Vu(k)
for the GDFACT. The effects of the inner and outer scales are plainly
visible.
52
4.5.3 Optical Techniques
Optical characterization of the GDFACT was first performed at the NRL ex-
perimental site. At NRL, the propagation path runs between a short-pulse laser
and the wall of the lab; it is too narrow for the scintillometer to fit. Therefore, only
LISSD data was collected.
The turbulence produced by GDFACT at the NRL range was measured via
LISSD, as a function of electrical power applied to the wires; these results are
presented in Figure 4.8. The baseline value of ∼ 10−14 is somewhat higher than
might be preferred; this is likely due to the presence of fans and chillers necessary
for the operation of the short-pulse laser. Nevertheless, GDFACT is able to produce
an increase of two orders of magnitude in C2n.
At AFRL, concurrent measurements of the structure function were made con-
currently by scintillometer and LISSD, as a function of electrical power applied to
the wires. However, the baseline turbulence at AFRL is so low that it was some-
times below the minimum detection threshold of the scintillometer. Therefore, in the
low-turbulence regime the results of the LISSD are considered to take precedence.
Measurements of the optical turbulence produced by GDFACT were performed
at a variety of power levels, as well as with the facility HVAC system turned both
on and off. The results are displayed in Figures 4.10 and 4.9. At the maximum
power level of 45 W/m, the intensity fluctuations caused dynamic range saturation
in the LISSD detector; therefore, this data has been omitted. This dynamic range
saturation is a result of the 8-bit color camera used for the LISSD; this limited the
53
10-11
10-12
10-13
10-14
10-15
0 20 40 ( 6/0 ) 80Power Density W m
Figure 4.8: Optical characterization of the NRL installation of GDFACT
via LISSD. The laboratory HVAC system remained on for this test.
size of the intensity fluctuations to a factor of 255. This deficiency was corrected in
the laser propagation experiments which were later performed at AFRL (presented
in Chapter 7).
With the HVAC system turned off the turbulence in the propagation range
is extremely weak, with C2 as low as 10−16m−2/3n leading to a scintillation index as
low as 10−4. Activating the HVAC system increases this baseline level of turbulence
by approximately an order of magnitude. At heater settings above 5 W/m, the
turbulence produced by GDFACT dominates the natural turbulence of the facility,
and the effect of the HVAC system becomes insignificant.
The C2n values displayed in Figure 4.9 are calculated under the assumption
of weak Kolmogorov turbulence, and are therefore subject to the same conditions.
54
C 2n (m-2 / 3)
Figure 4.9: Optical characterization of the GDFACT via LISSD. The
solid line represents the structure parameter C2n, as determined from
LISSD scintillation data under the weak turbulence approximation,
equation (2.17). The dotted line represents the C2n output from the
commercial scintillometer. The laboratory HVAC system remained off
for this test.
55
The agreement between the LISSD-derived values and those obtained from the com-
mercial scintillometer is observed to be excellent, except at the lowest turbulence
intensities. The scintillometer performs a significant amount of space averaging by
focusing a large collection aperture (10 cm diameter) onto its internal detector; this
renders the device resistant to large-amplitude scintillation saturation [22], but also
averages out small fluctuations. This places an effective lower limit on the strength
of the scintillations which the scintillometer can observe. The LISSD, which uses a
small sampling aperture, does not have this limitation.
The scintillation index and angle-of-arrival fluctuations displayed in Figure
4.10 are based directly on observation. The inner scale plotted in Figure 4.10b has
been calculated using equation 2.18, and hold under the conditions given by [24];
namely, that L l20/λ, and that the turbulence is Kolmogorov. Of these conditions,
the first holds for all but the highest power settings. For a He-Ne laser and an
inner scale of 1 cm, the limiting length is 158 meters. This is on the order of our
propagation distance; therefore, for the highest powers the actual inner scale may
be somewhat smaller than depicted. The calculation of the Nusselt number (∼100
at full power) gives us reason to believe that the second condition may not hold for
lower power levels. However, anemometer measurements indicate that GDFACT
produces a power spectrum similar to that of natural turbulence; see Figure 2.3.
The angle-of-arrival fluctuations and inner scales reported in Figure 4.10 are
averages of the x- and y-components determined by Consortini’s method. A slight
anisotropy exists: the y-component is more often larger than the x-component (23
out of 31 runs). On average, the y-component of the angle-of-arrival fluctuations,
56
101 (a)
100
10-1
10-2
10-3 HVACoff
- HVACon10 4
50 (b)
40
30
20
10
0
50 (c)
40
30
20
10
0
0 10 20 30
Power Density (W /m)
Figure 4.10: Optical characterization of the GDFACT via LISSD. a)
Scintillation index. b) Standard deviation of the angle of arrival. c)
Inner scale, determined according to the method of Consortini, equation
(2.18).
57
ℓ0 ( σ 2Imm) σα (μ rad)
and thus the inner scale in the y-dimension, is 9.75% larger than the x-component.
This difference is too small to be visible in Figure 4.10; therefore, only the average
has been presented. The anisotropy may result from the average flow of air upward
from the hot wires of the turbulence generator.
4.6 Modeling and Simulation
To benchmark propagation models, simulations were performed of the prop-
agation of a low-power He-Ne laser beam through the turbulence generated at the
AFRL range. Simulations were performed using the High Energy Laser Code for
Atmospheric Propagation (HELCAP), a 4D (3D space + time) computer simulation
code developed at NRL [42,43]. With the turbulence generator turned off, the initial
divergence of the He-Ne laser beam was determined by measuring the laser spot size
in the transmitter and receiver planes. The ambient turbulence was not sufficient
to affect the beam divergence angle.
With the initial divergence angle and spot size of the He-Ne beam established,
C2n and the inner scale were varied in the simulations according to the measured
values. The simulated beam was propagated 180 meters through the turbulence
(assumed to be homogenous). HELCAP does not natively include buoyant con-
vection; to mimic the effect of heat convection from the wires, a vertical wind was
used in the simulation with speed equal to the buoyancy speed associated with the
temperature of the wires.
Simulated beam images were compared with observed beam images in the re-
58
ceiver plane for various turbulence conditions (Figure 4.11). Over the majority of
the experiments, which produced a wide range of turbulence intensities (including
both weak and strong turbulence regimes), it was observed that turbulence did not
significantly increase the spot size of the beam in the receiver plane. Weak turbu-
lence was observed to produce large-scale structures that convect across the beam,
while strong turbulence produced smaller-scale convecting filamentary structures.
The same behavior was observed in the simulations, which were in good qualitative
agreement with the experiments over the entire range of turbulence measured. This
behavior is to be expected in the parameter regime for “case 3” of Fante’s seminal pa-
per [5], in which there is very little beam spreading but high incoherence can occur.
A example comparison of beam intensity in the receiver plane between experiments
and simulations is shown in Figure 4.11 for weak turbulence (C2 −14 −2/3n = 6 ∗ 10 m ,
σ2I = 0.15, frames a and c) and strong turbulence (C
2 −13 −2/3
n = 5 ∗ 10 m , σ2I = 1.2,
frames b and d) cases.
59
(a) (b)
(c) (d)
Figure 4.11: Qualitative comparison between experimental and simula-
tion results. a) and b) Still frames from video of laser beam scattered
off target board in front of LISSD. c) and d) Still frames of animation
produced in HELCAP. The spot size and fine beam structure observed
in experiment are qualitatively reproduced in simulation. Low dynamic
range of the camera prevents quantitative comparison of these images.
60
Chapter 5: Nonlinear Beam Propagation
5.1 Overview
The propagation of a laser beam is changed fundamentally once the intensity
of the beam becomes great enough to alter the optical properties of the medium
through which is propagates. A variety of nonlinear processes are capable of tak-
ing place. For the purposes of the present work, the most important is nonlinear
self-focusing, in which the index of refraction of air increases with increasing laser
intensity, resulting in transverse focusing of a Gaussian beam proportional to the
laser power. The nonlinear self-focusing effect allows high-power beams to propa-
gate many Rayleigh lengths while remaining below the inner scale of turbulence,
thus enabling the nonlinear self-channeling which is the focus of this work.
5.2 Without Turbulence
Even in the absence of turbulence, a high peak power laser pulse propagates
through the atmosphere in a manner both quantitatively and qualitatively different
from that of an ordinary low power laser. This is due to nonlinear interaction with
the matter of the atmosphere through which the pulse propagates [44,45].
61
In addition to the usual linear atmospheric effects of diffraction, dispersion,
absorption, and scattering, high power laser pulses are susceptible to a variety of
nonlinear effects. These include nonlinear self-focusing and self-phase modulation.
Numerical simulation codes exist to model these complex effects [43].
For the nonlinear case, equation 3.1 is modified by the inclusion of a source
term:
1 ∂2E~∇2E~ − = S~ (5.1)
c2 ∂t2
where S~ is a source term, or terms, which are nonlinear in the laser electric field [42].
As before, we look for solutions of the form
E~ (x, y, z, t) = A(x, y, z, t) exp [iψ(z, t)] x̂/2 + c.c. (5.2)
5.2.1 Nonlinear Self-Focusing
In nonlinear self-focusing, the index of refraction of the atmosphere is influ-
enced by the intense electric field of the laser pulse. The index of refraction in any
physical medium derives from the polarization of the medium in response to the
optical electric field. For small optical fields, the polarization can be modeled by
electrons moving in a harmonic potential. The polarization is linearly proportional
to the applied field. This results in the ordinary, linear index of refraction.
However, for sufficiently intense optical fields the electrons are transported
beyond the harmonic portion of the atomic potential. For centrosymmetric media
62
such as air, the lowest-order correction to the potential is quarti[c, which leads t]o
a third-order nonlinearity in the polarization. That is, P =  χ(1)E + χ(3)E30 ,
where 0 is the permittivity of free space, and χ
(1) and χ(3) are the first and third-
order susceptibilities. This manifests as a local change in the index of refraction
proportionate to the intensity of the laser beam:
n = n0 + 2n2I (5.3)
(3)
where n0 is the linear index of refraction, I is the beam intensity, and n2 =
3χ is the
4n0
nonlinear index of refraction. The majority of materials, including air, have positive
n2 at optical wavelengths; that is, they exhibit increased index with increasing
optical intensity. Therefore, beams with on-axis intensity maxima effectively create
a traveling positive lens, which leads the beam to converge on itself. This process
is known as nonlinear self-focusing [44]. For the laser pulses used in this work, a
typical n value for air is on the order of 5× 10−192 cm2/W.
The effect of nonlinear self-focusing can be included in the nonlinear wave
equation by the use of the source term
ω2n2n2
SNSF =
0 0 |A(x, y, z, t)|2A(x, y, z, t) (5.4)
4πc
A “critical power” at which nonlinear self-focusing becomes experimentally
significant can be defined by comparing the strength of nonlinear self-focusing to
the defocusing strength of diffraction. Since the diffractive spreading angle θdiff and
the nonlinear self-focusing angle θsf both scale as R
2, the ratio depends not on the
63
intensity, but on the total beam power. The critical power, Pcrit, is the power at
which the ratio of diffraction to nonlinear self-focusing is 1. For a Gaussian beam,
the critical power is given by [44]
π(0.61)2 λ2
Pcrit = (5.5)
8 n0n2
Beams containing much less than one critical power experience diffraction and
linear propagation. Beams which contain much greater than one critical power
experience rapid self-focusing. Using n2 = 5 × 10−19 cm2/W and a wavelength of
800 nm, the theoretical critical power in air is found to be approximately 1.9 GW.
In practice, the power required to induce nonlinear self-focusing is often higher
than the estimate provided by Equation 5.5. For instance, Braun et al. required 10
GW of power to initiate self-focusing, rather than the 1.8 GW predicted for their
wavelength [46]. Similarly, Liu and Chin found the critical power for 800 nm laser
pulses to vary from 10 GW for a 42 fs pulse, to 5 GW for pulses 200 fs and longer.
[47] This is due to time-dependent contributions to the nonlinear refractive index
resulting from the difference between atomic and molecular polarization times [44].
5.2.2 Filamentation and Self-Guiding
Laser pulses with very high intensities, on the order of 1013 W/cm2, can ionize
the air to form plasma filaments, in which nonlinear self-focusing is balanced by the
combination of diffractive spreading and plasma defocusing [48]. These filaments
can propagate many Rayleigh lengths in air at a spot size on the order of 100
64
microns, without appreciable change in the size of the beam [49,50]. Much research
on the propagation of high power laser pulses has concentrated on the filamentation
regime [51–53].
Even below the ionization threshold, the action of nonlinear self-focusing can
overcome beam spreading due to diffraction [54]. For laser pulses with power equal
to Pcrit, self-focusing and diffraction balance one another. This persists for many
Rayleigh lengths, until the delicate balance is upset, devolving into either eventual
beam collapse or diffraction. In this self-guiding regime, the beam undergoes long-
distance propagation without filamentation [44].
Although both plasma filaments and self-guiding laser beams can propagate
many Rayleigh lengths, because the diameter of a filament is so much smaller than
that of a self-guiding beam, the filament’s Rayleigh length is correspondingly shorter.
Thus, the absolute propagation distance which can be attained with self-guiding
is longer than that of a filament [11]. Long distance propagation is necessary to
reach the strong turbulence regime. For this reason, catastrophic beam collapse and
plasma filamentation are to be avoided in order to explore the interplay of nonlinear
self-focusing and strong turbulence. Plasma filamentation is often accompanied by
the conversion of laser photons to supercontinuum light, spanning from the UV to
the infrared [55]; the presence of this light can be used to diagnose and avoid the
presence of plasma filaments during self-guiding propagation experiments.
65
5.2.3 Group Velocity Dispersion
Light propagating through a medium is subject to group velocity dispersion
(GVD). Since the index of refraction of a physical medium in general varies with
the frequency of light, different wavelengths of light propagate at different velocities.
Consider a laser pulse with significant bandwidth; the duration of such a pulse
changes as it propagates, as different frequency components lag behind others. This
effect is GVD. The effects of GVD is especially significant for short laser pulses,
such as those produced by CPA laser systems. The broad bandwidth of such pulses
increases the absolute magnitude of the change in pulse length, while the short pulse
length increases the relative significance of that change.
The change in pulse length due to group velocity dispersion is given by
∆TGVD = d×D(λ0)×∆Λ0 (5.6)
where d is the propagation distance, D is the group velocity dispersion coefficient
(proportional to ∂n ), and ∆Λ0 is the bandwidth of the laser. The value of D in air∂λ0
at 800 nm is −0.062493ps/(nm× km) [56].
In order to estimate the effect of GVD on the experiments presented in this
work, the expected change in pulse length due to GVD has been calculated for each
case. The two experiments discussed in this work were performed at propagation
ranges at NRL (Chapter 6) and AFRL (Chapter 7).
For the NRL facility, the maximum propagation distance is 90 m and the
bandwidth is 27 nm, based on a 35 fs transform-limited short pulse. The change in
66
pulse length ∆T is therefore ∼ 151 fs. Since the shortest pulse length used in that
experiment is 1 ps, the fractional change due to GVD is at most 15%.
For the AFRL facility, the maximum propagation distance is 540 meters and
the bandwidth is still 27 nm, based on a 35 fs transform-limited short pulse. The
resulting value of ∆T is about 906 fs. Since the shortest pulse used at AFRL is
about 900 fs long, the fractional change in pulse length is on the order of 100%.
This amount of GVD could cause the short pulse to compress to its transform-
limited pulse length of 35 fs as it propagates through the atmosphere, thereby greatly
exceeding the critical power near the target. This effect is likely to be significant;
however, at the time that experiments were performed, the diagnostics necessary to
characterize the size of this effect were not in place. Future simulations of these or
similar future experiments should include the effects of GVD in order to accurately
model the propagation.
5.3 With Turbulence
As with low-power linear laser beams, the propagation of high-power laser
pulses is complicated by the presence of turbulence. The generation and wander of
plasma filaments in turbulence have been studied extensively [49, 57–59]. However,
there has been relatively little study of nonlinear propagation through turbulence
in the sub-filamentation regime (P ∼ Pcrit). In a series of simulations, Hafizi et al
found the effect of turbulence to be similar to reducing the beam quality of the laser
pulse [60].
67
It is plausible to imagine that nonlinear self-focusing might be employed to
resist turbulent spreading, just as it resists diffraction. In the case of w∣eak∣tur∣bulen∣ce
∣an eff∣ective turbulent spreading angle θturb can be defined [27]. When ∣θ ∣ = ∣ ∣sf θturb +∣θ ∣diff , the strength of nonlinear self-focusing should balance the combined effects of
turbulence and diffraction. Indeed, self-focusing has been observed in simulation to
overcome turbulent spreading for sufficiently weak turbulence [59].
In strong turbulence the formation of self-reinforcing hot spots, seeded by
turbulence-induced phase distortions and driven by nonlinear self-focusing, can lead
to beam breakup. Turbulent fluctuations smaller than the size of the beam leads to
the random formation of hot spots. These hot spots become self-reinforcing through
nonlinear self-focusing and eventually lead to filamentation. The beam breaks up
into a number of filaments approximately equal to the number of critical powers
within the beam. The propagation of sub-ionization high power laser pulses in strong
turbulence has been studied using Monte Carlo simulation to predict the probability
of beam collapse as a function of turbulence strength and laser propagation distance
[9].
Under certain conditions, however, beam breakup might be averted. Turbulent
eddies smaller than the inner scale of turbulence are strongly dissipated by viscosity,
leading index of refraction variations to experience fast falloff at sizes below the inner
scale. For beams much smaller than the smallest turbulent eddy, index-induced
phase errors should remain first-order across the beam. The effect of turbulence for
such a scenario would be restricted to pointing fluctuations. In the case of linear
laser propagation, any beam with initial spot size smaller than the inner scale will
68
rapidly diffract to a larger size, allowing turbulence to introduce higher-order phase
errors. However, this may not be true in the case of a nonlinear self-channeling
beam, which can maintain its size over long distances.
In fact, recent simulations performed at NRL have indicated that a high power
laser pulse operating in the self-guiding regime might be capable of propagating
through turbulence without undergoing turbulent spreading, so long as the beam
size remains smaller than the inner scale of the turbulence [10, 11, 61, 62]. Beam
images demonstrating several realizations of the simulated propagation of nonlinear
self-channeling beams are presented in Figure 5.1. All beams begin the simulation
at a radius smaller than the inner scale; those beams on the bottom row undergo
nonlinear self-focusing, maintaining a small spot size even after propagation over a
long distance.
69
Figure 5.1: Computer simulations of sub-inner-scale low-power and high-
power beam profiles after propagating through deep turbulence. Three
realizations of turbulence are presented for both high and low power
cases. Top row: low-power propagation with linear focusing to tar-
get. Bottom row: high-power propagation with nonlinear self-focusing.
Simulation parameters: 7 km propagation distance, C2 = 10−15m−2/3n ,
σ2R = 2
70
Chapter 6: Experiments at NRL
6.1 Overview
Moderate-range nonlinear propagation experiments were performed at the U.S.
Naval Research Laboratory (NRL), Plasma Physics Division (PPD). These experi-
ments demonstrated NSC for the first time, over moderate range (> 10 m) and in
moderate turbulence (Rytov ∼ 0.3).
6.2 Outline of Methodology
6.2.1 Propagation Range
The Plasma Physics division at NRL is home to a moderate-distance indoor
propagation range. This 30-meter range runs alongside a short-pulse laser system
(described below) and through three labs, to a target table at the end of the build-
ing. High-power dielectric optics are available to reflect the beam across this range
multiple times, reaching a total maximum propagation range of approximately 90
meters. Only the 30-meter path was used in these experiments. A schematic and
photograph of the laboratory space are presented in Figure 6.1.
71
Figure 6.1: The moderate distance indoor laser propagation range at the
NRL Plasma Physics Division.
72
6.2.2 Laser System
The laser used for this portion of the work is the kilohertz Titanium Femtosec-
ond Laser (kTFL) at NRL PPD. kTFL is a cryogenically cooled 800 nm Titanium-
doped sapphire laser (Ti:sapphire), delivering 20 mJ of energy per pulse at a pulse
rate of 1 kHz. It uses chirped pulse amplification to reach a minimum pulse length
of 35 fs, and thus a maximum peak power of 570 GW.
Chirped pulse amplification (CPA) is a technology for amplifying ultrashort,
high peak power laser pulses [63, 64]. This technique has enabled the development
of lasers with peak powers on the order of petawatts [65]. In this technique, the high
bandwidth (10s of nm), low energy (hundreds of µJ), transform limited short pulse
(as short as tens of fs) output of a mode-locked laser oscillator is used as a seed.
If this pulse were amplified directly, nonlinear self-focusing of the resulting high
power pulse would destroy the amplifier gain medium. Instead, the seed pulse is
stretched temporally. This is done by introducing a frequency chirp which disperses
the spectral content of the pulse in time, creating a long (100s of ps) chirped pulse.
This chirped pulse is amplified, and then the chirp is removed, typically with a pair
of diffraction gratings, to recompress the pulse. Because the final intense pulse is
formed on the surface of a diffraction grating rather than within the bulk material
of the gain medium, two main advantages are realized. First, the beam can be
increased in size, reducing the peak laser intensity without the expense or pumping
and cooling challenges of a large gain medium; second, the path-integrated n2L is
much reduced. Thus the risk of catastrophic self-focusing is reduced relative to
73
simply amplifying the original short pulse directly.
In addition to providing an ultrashort, high-power beam, CPA allows the
pulse length of the beam to be varied. By adjusting the separation of the diffrac-
tion gratings which compress the pulse, the amplified beam can be recompressed
incompletely, leading to a chirped high-power pulse with increased pulse length. By
increasing the pulse length, the peak power of the beam is reduced.
Figure 6.2: The kilohertz Titanium Femtosecond Laser at the NRL
Plasma Physics Division.
The peak power output of CPA lasers such as kTFL is far in excess of what
is needed for this research, since P/Pcrit need only be on the order of 1. The CPA
system allows kTFL’s peak power to be tuned down from as much as 570 GW in
order to access the range of interest (∼2 GW).
74
6.2.3 Turbulence Generation
The GDFACT turbulence generator has been implemented over the 30 meter
propagation range at NRL. The generator can be seen in the right-hand panel of
Figure 6.1. The turbulence generated at NRL is similar to that produced at the
AFRL site, with the exception of a higher baseline when the turbulence generator
is turned off. This difference can be seen by comparing the lowest turbulence points
of Figures 4.8 and 4.9; the baseline is approximately 10−14 at NRL, two orders
of magnitude higher than the baseline of 10−16 measured at AFRL. This elevated
turbulence background is due to the NRL propagation range’s location along one
side of an active laser lab, rather than in a dedicated hallway as at AFRL. The laser
cooling systems and laboratory HVAC produce a good deal of air flow, which must
remain running in order to keep the lasers in working order.
6.3 Results
6.3.1 First Observation of Nonlinear Self-Channeling
Preliminary high-power propagation experiments have been performed at the
NRL Plasma Physics Division experimental site. These experiments consisted of a
single pass over the 30 meter long turbulence generator. Both a high-power beam
and a low-power beam were propagated through the same turbulence in order to
compare the wander and spreading of the linear and nonlinear propagation.
For the nonlinear beam, the compressor was detuned to produce a high-power
75
pulse with P ∼ Pcrit. This was determined by slowly adjusting the compressor,
decreasing the pulse length until the beam at the receiver was the same size as the
beam at the transmitter. This indicated that nonlinear self-focusing was balancing
diffraction. The linear beam was produced by blocking the seed beam before it
enters kTFL’s regenerative amplifier. The amplifier’s self-lasing output has the
same energy as the seeded beam; however, it is a much longer pulse.
At each turbulence generator setting, data was taken with both the high-
power and the low-power beam. A sampling of pulses was captured by a Blackfly
camera, and the resulting beam images were analyzed to yield spot size and beam
wander statistics. Simultaneously, another Blackfly camera recorded data for a
LISSD diagnostic based on a HeNe laser beam co-propagating with the infrared
beam from kTFL.
Figure 6.3 plots the cumulative distribution functions (CDFs) for beam radius
for both high- and low-power beams. The CDF for the low-power beam at the lowest
turbulence setting is a nearly vertical line. This indicates that each pulse undergoes
essentially the same propagation and produces essentially the same spot every time.
As the turbulence increases, the random process of turbulence causes the beam to
fluctuate. The low-power CDF becomes more sloped.
In contrast, the high-power beam has significant fluctuation in spot size even
at the lowest values of turbulence. Nonlinear processes are sensitive to small fluc-
tuations in initial conditions, so small changes in pulse energy or the small natural
amount of turbulence are enough to cause the nonlinear beam to vary in spot size.
However, this behavior is insensitive to the increase in turbulence. The CDF traces
76
remain essentially the same, even as turbulence increases to the maximum extent
reached in these experiments.
Figure 6.4 summarizes the properties of the CDFs in Figure 6.3. For each
CDF, the mean and standard deviation in spot size is plotted. The linear beam
spot size fluctuations increase, while the statistics of the nonlinear beam remain
constant.
This preliminary set of experiments also investigated beam wander. Figure
6.5 shows that the nonlinear and linear beams undergo essentially the same increase
in wander as a function of turbulence. The nonlinear beam begins with a larger
amount of wander than the linear beam does, once again due to the sensitivity of
nonlinear propagation to small variations in initial conditions. However, both beams
respond to turbulence by experiencing the same increase in wander.
These experiments act as a proof of concept for nonlinear guiding in turbulence,
and motivated the work described in Chapter 7.
77
Figure 6.3: Cumulative distribution functions for the spot size of low-
power and high-power beams propagating through turbulence at the
NRL propagation range.
78
Figure 6.4: The standard deviation of the low-power beam size increases
as a function of turbulence intensity. However, the high-power beam’s
statistics remain nearly unchanged as turbulence increases.
79
Figure 6.5: Increase of beam wander as a function of turbulence intensity.
Both the low-power and high-power beam exhibit identical increases in
beam wander, with the exception of a constant offset.
80
Chapter 7: Experiments at AFRL
7.1 Overview
Following the initial proof-of-concept work at NRL, long-range nonlinear prop-
agation experiments were performed at the U.S. Air Force Research Laboratory
(AFRL) in Albuquerque, New Mexico. These experiments demonstrated NSC over
long range (> 100 m) and in deep turbulence (Rytov > 10).
7.2 Outline of Methodology
7.2.1 Propagation Range
AFRL is home to a long-distance indoor laser propagation range. This facility
offers an uninterrupted indoor propagation length of up to 180 meters, with the
option to perform multiple passes through the turbulent volume using dielectric
mirrors. In these experiments, data was taken using either one or three passes
down the range. The AFRL indoor range has excellent temperature stability, low
humidity, and tightly controlled access for laser safety.
81
7.2.2 Laser System
Experiments were performed using the AFRL PHEENIX laser system. PHEENIX
is a chirped-pulse amplification laser system operating at 800 nm, capable of pro-
ducing up to 1.6 Joules per pulse at 10 Hz. The minimum pulse length is as short as
35 fs, for a peak power output of approximately 45 TW. This is far in excess of the
power necessary for this experiment. Therefore, the final amplifiers were left turned
off, reducing the output energy to approximately 20 mJ.
7.2.3 Turbulence Generation and Characterization
Artificial turbulence was generated using the convective flow artificial turbu-
lence generator described in Chapter 3. A LISSD, as described in Chapter 2, was
used to characterize the structure parameter C2n and inner scale l0 of turbulence
produced by the generator. The Helium-Neon laser for the LISSD was aligned in
order to provide the maximum overlap between the LISSD sample volume and the
USPL propagation path without permitting cross-contamination between the two
systems. A photograph of the HeNe beam propagating down the AFRL range is
depicted in Figure 7.1. Implementation of the LISSD was improved relative to our
prior efforts [66] by the use of a 12-bit monochrome camera as the imaging device.
82
Figure 7.1: Photograph of indoor turbulence range located at AFRL.
The range is 180 meters long, and its ambient turbulence conditions have
been characterized by LISSD and scintillometer. This view is looking
backwards over the turbulence generator, from the receiver towards the
transmitter. Forward-scattering light from the HeNe beam used for the
LISSD can be seen due to aerosol scattering over the long path length.
7.2.4 Optical Setup
The PHEENIX laser system is located in a separate room, offset from the main
propagation range. In order to produce a well-characterized and well-controlled
beam at the entrance to the propagation range, the ∼20 mJ uncompressed pulse
from the PHEENIX laser was transmitted through a series of beam tubes to a
“transmitter end” optical table immediately prior to the turbulence generator. This
prevented undesired nonlinear self-focusing from occurring outside the region of
controlled turbulence. The transmitter end table contained a spatial filter, pulse
compressor, beam imaging system, spectrometer, LISSD laser, and alignment optics.
The beam quality of the PHEENIX laser is imperfect, with Strehl ratio on the order
83
of 0.6. Therefore, a spatial filter on the transmitter table was used to clean the beam
prior to compression. After the compressor, a variable aperture was placed to allow
the higher-order modes at the perimeter of the beam to be clipped off, improving
the beam quality at the cost of some energy.
The total amount of energy available after the spatial filter and compressor was
as much as 13 mJ. For the single-pass runs, the amount of energy allowed through
the aperture was 5 mJ. For the triple-pass runs, the aperture was opened to allow
11 mJ through.
The receiver table contained the LISSD diagnostic pinhole and camera, a Ma-
cor ceramic target board for the USPL beam, a USPL imaging camera directed at
the Macor, and a photodiode to allow synchronization of the camera triggering with
the laser pulse. The LISSD receiver and the USPL receiver were offset from one
another by approximately 30 cm on center, with their placement reversed relative
to the transmitter end; in this way the beams crossed midway down the propaga-
tion path to maximize the overlap. A dielectric mirror, transparent at 632.8 nm and
highly reflective at 800 nm, was placed in front of the LISSD detector to prevent
the nonlinear beam from affecting the statistics of the LISSD. A schematic diagram
of the setup is presented in Figure 7.2.
84
Figure 7.2: Diagram of experimental setup
85
7.2.5 Pulse Length and Power
Unlike at NRL, the PHEENIX laser cannot produce a longer pulse by re-
lying on amplifier self-lasing. Therefore, two settings for the compressor grating
separation were used, corresponding to a “high power” self-guiding case and a “low
power” linear propagation case. The compressor spacing for the high power case was
chosen by aligning the beam to the receiver table and adjusting the spacing until
the smallest spot size was reached at the receiver. This corresponds to the desired
self-channeling mode. For some compressor settings, catastrophic self-focusing and
generation of plasma or supercontinuum light was visible at the target as a “hollow”
beam profile or a pattern of rainbow light; these cases were intentionally avoided.
For the low power case, the compressor was detuned to the limit of its travel.
Laser pulse length measurements were made with an autocorrelator at nominal
grating positions of 0.408, 0.418, and 0.443 inches as measured on a translation stage.
The autocorrelation traces, a typical example of which is displayed in Figure 7.3,
indicate the presence of a short pulse containing very little energy, overlaid on a
much longer pulse. The full-width-half-maximum pulse duration of the primary,
higher energy pulse was measured to be 929 fs at a setting of 0.418 inches. However,
the condition which resulted in the best self-channeling propagation was a nominal
position of 0.429 inches. The pulse length at this grating position was not measured.
However, it can be estimated.
86
Figure 7.3: Autocorrelation trace of the short-pulse PHEENIX laser
beam immediately prior to propagation down range. The nominal grat-
ing position is 0.418 inches.
87
For a first-order grating pair compressor detuned by a distance x from its ideal
spacing, the pulse length is given by [45]
√ ( )2
x
τG = τG0 1 + (7.1)
Ld
where τG0 is the minimum pulse length, Ld is the effective dispersion length Ld =
πc2d2 cos2 β 2
3 τG0, c is the speed of light, d is the grating line spacing, β is the centralλ
wavelength first-order diffraction angle, and λ is the laser central wavelength.
The PHEENIX compressor had a grating line density of 1500 lines per mm and
a central wavelength first-order diffraction angle of 54 degrees. The minimum pulse
length which can be produced by the PHEENIX laser is 35 fs. Solving Equation 7.1
with these parameters, the expected compressor position corresponding to a 35-fs
pulse would be 0.526 inches.
The expected pulse length for the self-channeling condition is then found to be
835 fs. The low-power case was performed at a nominal grating position of 0 inches,
for an expected pulse length of 4.51 ps. This pulse is too long to be measured on
the autocorrelator.
A simple estimate of the peak power for the high power case (5 mJ / 835 fs)
yields approximately 6 GW, approximately three times the nominal value of 1.9 GW
cited in Chapter 5, and somewhat in excess of the typical experimental value of 5
GW [46,47]. One possible explanation for the increased energy necessary to obtain
self-focusing is the presence of some energy outside the self-channeling mode, in the
shorter pulse visible in Figure 7.3.
88
Additionally, the critical power for self-focusing is calculated under the as-
sumption of a Gaussian beam profile; any deviation from that profile will result
in a greater diffractive spreading angle, and therefore a greater amount of power
necessary to maintain self-focusing in balance with diffraction. The beam quality of
the PHEENIX laser is non-ideal, as can be seen in Figure 7.4. For the single-pass
data, the best beam at the receiver was produced by a beam whose wings had been
significantly clipped by a final iris at the transmitter, resulting in a beam with a
profile more similar to a top-hat profile than a Gaussian. For the triple-pass data,
this iris was opened to increase the energy at the cost of allowing the asymmet-
ric, nonuniform wings of the beam to propagate. Laser burns of these profiles are
presented in Figure 7.4.
Figure 7.4: Laser burns taken after the final iris on the transmitter
table. Left: beam used for single-pass experiments. Right: beam used
for triple-pass experiments. The scale is the same for both images.
89
7.2.6 Data Organization
In order to capture the statistical fluctuations of turbulence, many images
must be taken under each set of experimental conditions. For this experiment,
a single “data shot” consists of a sequence of five hundred images of the USPL
pulse on the Macor imaging surface, coupled with five hundred LISSD images of
light from the HeNe, at a single setting of the turbulence generator. Although the
number of images is the same for both beams and both diagnostics, the LISSD and
USPL cameras are not synchronized. The LISSD camera runs at 30 fps, while the
PHEENIX laser repetition rate is only 10 Hz. The LISSD images were taken while
the USPL camera was running, so both sets of images should represent, statistically
speaking, the same turbulence conditions.
The LISSD images were analyzed in order to yield statistics representative
of the turbulence strength and inner scale. Since only the centroid location and
total energy falling on the camera is necessary for this diagnostic, image analysis
was simple. The background level of the image was determined by averaging over a
region of the image far from the beam center, and this level was subtracted from the
entire image. After that, the beam centroid and total pixel value integrated over
the camera aperture were recorded. The fluctuations in these quantities were used
to determine turbulence parameters according to the method laid out by Consortini
[24].
Images taken by the USPL imaging camera were analyzed to yield statistics
representative of the IR beam’s behavior. In order to determine beam radius ac-
90
curately, significant image processing must be performed in order to minimize the
effect of noise. Image analysis was carried out according to the procedure laid out
in Appendix A. This procedure yields two ways of looking at the beam: whole-beam
analysis, and hotspot analysis.
As outlined in the previous section, for each notional output pulse the PHEENIX
laser actually produces a train of at least two pulses, only one of which is properly
chirped in order to be compressed by the compressor gratings. Therefore, beam
images taken at the target typically look like Figures 7.5 and 7.6, containing a cen-
tral high-intensity spot from the pulse which undergoes nonlinear self-channeling,
surrounded by lower-intensity light from the pulse or pulses which do not. For this
reason, the hotspot data more accurately captures the effect of ideal nonlinear self-
channeling than the whole-beam data does. However, the whole-beam analysis still
represents the actual achieved results with this laser system. Therefore, results of
applying both metrics to the image data have been retained, and are presented in
the following sections.
91
Figure 7.5: Instantaneous beam images taken at the receiver after one
pass over the propagation range. Left side: high power propagation
with nonlinear self focusing. Right side: low power linear propagation.
Top panel: typical beam fluence profile at the receiver. Bottom panel:
horizontal lineout across beam center. The presence of lower-power pre-
and post-pulses can be seen in the “skirt” surrounding the self-channeling
high power beam.
92
Figure 7.6: Instantaneous beam images taken at the receiver after three
passes over the propagation range. Left side: high power propagation
with nonlinear self focusing. Right side: low power linear propagation.
Top panel: typical beam fluence profile at the receiver. Bottom panel:
horizontal lineout across beam center. The distinction between the self-
channeling portion of the beam and the lower-power pre- and post-pulses
is even clearer.
93
Turbulence statistics collected for each shot included the structure parameter
C2n and inner scale length l0. IR beam statistics included the beam centroid wander,
√
the radius (given by 2 times the RMS radius), and the peak fluence. For each
data shot, these statistics were compiled into a probability density function (PDF)
describing the likelihood of any given outcome for a single laser shot under the given
experimental conditions. Parameters varied included the laser power, turbulence
intensity, and propagation distance.
7.2.7 Plotting and Theory Methodology
For each shot, the LISSD measures scintillation and angle-of-arrival fluctua-
tions for the HeNe beam. The scintillation index is used to calculate C2n according
to Equation 3.23. The scintillation index and angle-of-arrival variation are com-
bined according to Equation 2.18 to yield the inner scale. Next, the C2n measured
with the HeNe is used to calculate the Rytov variance for the IR beam according to
Equation 3.12. Although this equation was developed under the assumption of weak
turbulence, it remains the definition of the Rytov variance under all conditions, even
though it no longer represents the scintillation index of any real beam. All other
beam statistics are plotted as a function of the Rytov variance.
94
7.3 Results
7.3.1 Turbulence Generation
Turbulence was generated with the GDFACT, as outlined in previous sections.
C2n varied from 5× 10−16 to 5× 10−12. LISSD measurements indicate that the inner
scale of turbulence varied as a function of C2n, from a maximum of 2 cm with the
GDFACT turned off, to a minimum of 5 mm at full power. The relationship between
inner scale and C2n is observed to be nearly linear on a log scale.
Figure 7.7: Inner scale of turbulence as a function of structure parameter
C2n. A line of best fit and its generating equation are shown.
95
This relationship between C2n and the inner scale is carried through the follow-
ing analysis. Whenever the inner scale is called for in an equation, it is determined
by reference to the line of best fit displayed in Figure 7.7.
7.3.2 Single-Pass Experiments
The results of the image processing are presented below. Figure 7.8 presents
the beam wander as a function of turbulence. Wander is calculated as the standard
deviation of the distance of the instantaneous beam centroid from its time-averaged
location. This statistic is relatively insensitive to the method used to process the raw
beam images: both hotspot and whole-beam processing result in the same centroid
behavior. As expected, the beam wander is insensitive to changes in pulse length,
and increases with increasing turbulence.
The black curve is the expected beam wander based on the expression for
beam wander under conditions of strong turbulence, Equation 3.30. Nearly identical
results are produced when using the weak-turbulence expression, Equation 3.14;
this is represented by the dashed line. The free parameter for this curve was the
intrinsic wander of the beam due to laser and platform fluctuations; it was set at
2.5 milliradians. The beam wander terms due to intrinsic wander and turbulence
are statistically independent, and thus sum in quadrature. The resulting agreement
between theory and experiment is observed to be excellent.
96
Figure 7.8: Beam wander (standard deviation of distance from time-
averaged center) as a function of turbulence. Single-pass experiment
(180 m).
97
From here on, analysis is presented in the following order: whole beam,
hotspot, peak fluence. These represent successively increasing amounts of abstrac-
tion from what is actually observed in the experiment. Different readers of this
thesis may prioritize different metrics. Those who are interested in the most di-
rect report of what I observed with this laser in this experiment will wish to direct
their attention to the whole beam data. Those readers who want to know how a
more ideal beam (single short pulse) would behave, and those who are interested in
creating a small spot after propagating a long distance, should look at the hotspot
analysis. Those readers whose priority is the maximum fluence or intensity created
on target should take note of the maximum fluence plots.
Figure 7.9 presents PDFs for the radius of the beam as a function of Rytov
variance, in the form of a violin plot. A violin plot is similar to a box-and-whisker
plot. However, instead of plotting summary statistics, such as the mean and in-
terquartile range, the violin plot shows the entire PDF, oriented vertically, in a
violin-like shape. There is one violin for each data shot, with the violin’s horizontal
position within the overall graph representing the turbulence conditions when the
data was taken. Each violin is split into two halves: one for low-power, long-pulse
data (blue), and one for high-power, short-pulse data (red). The width of the violin
is proportional to the probability density, with each pair of violins (red and blue)
in scale to one another, so that the integrated area under each violin is normalized
to one. Each pair of violins is not in scale to the other pairs; all of them are set
so that the violins have, as nearly as possible, the same width. This maintains the
mutual legibility of all the violins, even if some PDFs are narrowly distributed and
98
others are very broad.
The solid black theory curve is based on the expression for the short-term
beam radius under strong fluctuation conditions, Equation 3.29. The dashed curve
is based on the equivalent weak-turbulence expression, Equation 3.19. There is one
free parameter, set to the same value for both equations: the initial divergence of the
laser beam, represented by quantity F0. For Figure 7.9, F0 was set to −17 meters.
This likely represents a combination of imperfect adjustment of the collimating lens
and imperfect laser beam quality.
Figure 7.9: Violin plot for the short-term radius of the whole beam as a
function of Rytov variance. Solid and dashed curves represent theoretical
values for strong and weak turbulence, respectively.
For low turbulence, the low power beam is narrowly distributed about the ana-
99
lytical result for the expected beam size in the absence of turbulence. As turbulence
increases, the variance of both the low-power and high-power distributions begins
to increase. The mean radius for the low-power beam remains relatively constant,
while the mean radius for the high-power beam begins to increase as self-channeling
begins to fail. However, the high-power beam never grows as large as the low-power
beam, even for Rytov over 10. As turbulence increases, the self-guiding beam statis-
tics begin to resemble the low-power statistics more and more. However, under all
conditions, the high-power beam remains more likely to produce a smaller spot than
the low-power beam.
Analysis of the hotspot radius is presented in Figure 7.10. For low turbulence,
the low power beam is narrowly distributed about the analytical result for its ex-
pected beam size in the absence of turbulence. As turbulence increases, the variance
of both distributions begins to increase, while the mean remains relatively constant.
As its variance increases, the low-power beam begins to have some probability of
generating a small hotspot on target due to “lucky shots,” in which turbulence some-
times acts as a positive lens over a sub-region of the beam [67]. However, the high
power beam maintains a significant lump of probability mass at low radius which
the low-power beam never matches.
100
Figure 7.10: Violin plot for the short-term radius of the hotspot of the
beam as a function of Rytov variance. Solid and dashed curves represent
theoretical values for strong and weak turbulence, respectively.
101
The theory curves are once again based on Equations 3.29 (solid, strong) and
3.18 (dashed, weak). However, this time the free parameter F0 was set to −80
meters. Since the hot spot represents the propagation of a distinct mode of the
beam, it comes as no surprise that it should have a different divergence from the
low-quality beam as a whole.√Additionally, the spot size predicted by theory has
been reduced by a factor of log 2 . This is the difference between the 12 radius2 e
of a Gaussian beam and its radius at half maximum fluence. Since the hotspot
size is calculated after performing a 50% threshold, this measure of beam size more
accurately represents the quantity being measured.
Theory fails to predict the decrease in the mean radius of the hotspot of the
low-power beam. This is likely a result of our definition of a “hot spot” and resulting
image processing techniques. As turbulence increases, the scintillation index of the
beam increases, leading to some pulses with increasingly intense hotspots. As the
hotspot grows more intense, more of the wings of the beam fall below the fluence
threshold necessary to count as part of the hotspot, and are therefore eliminated.
This reduces the apparent radius.
The peak fluence on target was determined in arbitrary units by taking the
maximum pixel value in the image, adjusted for background noise, camera gain, and
the presence of attenuating filters in front of the camera. This statistic does not
vary based on whether the whole beam or the hotspot is considered, since the entire
difference between those two methods lies in the low-value pixels.
Under this metric, increasing turbulence actually increases the likelihood that
the beam will produce high fluence. As turbulence increases, lucky shots focus part
102
of the beam and allow it to produce fluences that would not otherwise be possible.
At the highest turbulence levels, the low-power and high-power beams exhibit nearly
identical peak fluence statistics. Nevertheless, for any given peak fluence (in units
of J/m2), the peak intensity (in W/cm2) is still far higher for the high-power beam,
since its pulse length is much shorter than for the low-power beam.
Figure 7.11: Violin plot for the maximum fluence on target as a function
of turbulence intensity.
For each of these metrics, a convenient way to summarize the data is to ask
the question: how much statistical benefit does the high power beam provide? If
we were to randomly select one realization from both the high-power and low-power
probability distributions, what is the probability that the high-power beam would
103
outperform the low-power beam? This probability can be computed by
∑N ∑M
n=0 m=0 H (Rm(h)−Rn(l))Pr[R(h) < R(l)] = (7.2)
M ×N
where R(h) and R(l) are the radii of beams randomly selected from the samples of
high-power and low-power data sets, Rm(h) and Rn(l) are the n
th and mth samples in
the high-power and low-power beam radius data sets, M and N are the total number
of samples in the high-power and low-power data sets, and H is the Heaviside step
function, equal to zero when its argument is negative and one when its argument is
positive.
If both distributions were the same, the probability of improvement would be
50%. A similar statistic can be calculated for the probability that the high power
beam will produce a higher peak fluence than the low-power beam, by replacing the
radius with the peak fluence and interchanging h and l. These probabilities have
been calculated for all three metrics; they are plotted in Figure 7.12.
104
Figure 7.12: Probability of improvement as a function of Rytov variance.
105
Interestingly, the behavior of all three metrics is very similar. With the ex-
ception of a few outliers, each data set exhibits a gradual decrease in the benefit
of nonlinear self-channeling as the Rytov variance increases. The roll-off begins at
σ2R ∼ 1. For the strongest turbulence tested, σ2R ∼ 10, the probability of improve-
ment due to nonlinear self-channeling is reduced to approximately 75%.
Another interesting way to look at this is in light of the inner scale. The data
points in Figure 7.13 have the same y-coordinates as Figure 7.12, but the x-axis
has been replaced with the ratio of the beam diameter to the inner scale. The
qualitative behavior remains the same: all of the metrics exhibit similar roll-off as
turbulence increases. However, now the decrease in improvement is very modest
until the ratio of beam diameter to inner scale approaches one, at which point it
drops precipitously. This validates the expectation that nonlinear self-channeling
would cease to be effective for beams larger than the inner scale of turbulence.
106
Figure 7.13: Probability of improvement as a function of the ratio of the
initial beam radius to the inner scale.
107
7.3.3 Triple Pass Data
The experiments reported in the previous section were repeated, this time with
three passes over the experimental range. High-reflectance mirrors were set up at
both ends of the range, so that the IR beam could bounce back and forth. This
enabled a total range of 540 meters. In order to reduce the risk of saturating the
LISSD detector, the LISSD was left set up over only one pass. Although the optical
turbulence (in terms of σ2R) is much stronger over a longer distance, the atmospheric
turbulence (in terms of C2n) is the same. Therefore, measurements taken with the
LISSD over one pass can be used to determine the Rytov variance over three passes,
by scaling according to Equation 3.12.
Figure 7.14 presents the beam wander as a function of turbulence, as in Figure
7.8. Again, the beam wander is observed to be insensitive to pulse length. The
theory curve is once again calculated from Equation 3.30, with intrinsic wander set
at 2.5 milliradians. However, over this distance the agreement between theory and
experiment is very poor.
108
Figure 7.14: Beam wander (standard deviation of distance from time-
averaged center) as a function of turbulence. Triple-pass experiment
(540 m).
109
Even allowing the intrinsic wander to vary from the value used over 180 meters,
the theory still reliably predicts an amount of wander far greater than observed. A
possible explanation for this is that the IR beam, in being reflected down the prop-
agation path multiple times, may have strayed outside the bounds of the turbulent
region created by the GDFACT. In this case, the increased wander (and, as we shall
see, beam size) would result almost entirely from the additional diffractive propa-
gation through non-turbulent air. This would explain a persistent overestimate of
wander and beam size by theory.
In this section analysis is presented in the same order as last time: whole
beam, hotspot, peak fluence.
Figure 7.15 presents PDFs for the radius of the beam as a function of Rytov
variance. The violin plots in this section represent a subset of the data actually
collected: the full data set is too dense to read at low Rytov variance and has
therefore been downsampled for clarity. As in Figure 7.9, the solid and dashed curves
represent theoretical calculations for the expected short-term beam radius based on
strong and weak turbulence theory, respectively. The free focusing parameter F0
was set to 540 m to produce these curves. The low-turbulence curve temporarily
decreases below the vacuum result; this behavior illustrates the danger of using
low-turbulence theory outside its regime of applicability.
110
Figure 7.15: Violin plot for the radius of the whole beam as a function
of Rytov variance. The dashed curve represents the weak-turbulence
theory result; the solid curve the strong-turbulence result.
111
At low turbulence, the low power beam is distributed about the analytical re-
sult for its expected beam size. As turbulence increases, both distributions broaden
and the mean spot size of both distributions increases. However, the mean spot size
of the high-power beam increases faster, until by σ2R ∼ 1 the distributions are very
similar.
The behavior of the hotspot radius is given in Figure√7.16. As before, the
beam radii for the theory curves are reduced by a factor of log 2 to represent the
2
effect of thresholding. The focal length for this case was again 540 meters.
Figure 7.16: Violin plot for the radius of the instantaneous beam hotspot
as a function of Rytov variance. For low turbulence, the high power beam
has a lump of probability mass at low radius which the low-power beam
does not possess. However, as turbulence increases, the chances that a
lucky shot will form a small region of high fluence on the target increase,
and the low-power beam begins to develop a similar feature. By σ2R ∼ 3
the distributions are very similar.
112
Examining Figure 7.16, the statistics of the high-power beam are seen to be
insensitive to the effects of turbulence. However, high turbulence increasingly pro-
duces lucky shots which allow the low-power beam to produce statistics similar to
that of the high-power beam. By a Rytov variance of approximately 10, the distri-
butions appear nearly identical.
Examining the maximum fluence, rather than the spot size, the results look
somewhat more favorable for the high-power beam. Although the low-power beam
experiences lucky shots which increase its maximum possible fluence, the high-power
beam nevertheless continues to produce fluences which the low-power beam cannot
achieve. When the increased peak intensity due to the high-power beam’s shorter
pulse length is taken into account, this gap increases further. These results are
presented in Figure 7.17
113
Figure 7.17: Violin plot for the maximum fluence on target as a function
of Rytov variance. At all turbulence intensities, the high-power beam is
capable of producing higher fluence on target than the low-power beam.
The strength of this effect is diminished somewhat as turbulence in-
creases, but never goes away.
114
Once again, it is instructive to examine the probability of improvement in
each of these metrics realized by moving from the linear propagation regime of a
low-power beam to the NSC regime. If both statistical distributions were the same,
the probability of improvement would be 50%. A probability less than 50% indicates
that the high-power beam underperformed the low-power beam. The results for the
triple-pass experiment are summarized in Figure 7.18.
Figure 7.18: Probability of improvement for the triple-pass experiment.
For Rytov variances near 10, all three metrics are found to approach
parity with the low-power case.
The probability of improvement for all three metrics decreases approximately
linearly with increasing log Rytov variance. Unlike the case for the single-pass, there
115
is a quantitative difference between the metrics: the probability of improvement for
the hotspot radius is lower than that for the whole beam or the peak fluence for all
values of turbulence. Additionally, nearly the entire benefit of the high-power beam
is eroded by the highest turbulence values.
7.3.4 Both Single and Triple Pass
In applications which require a small spot on target, the baseline competition
to a nonlinear self-channeling beam is a linear beam which begins with a very large
radius and focuses linearly to the target. The nonlinear beam starts small, and
uses nonlinear self-focusing to fight diffraction and stay small. It might be argued
that a sufficiently large linear beam would not need to fight diffraction, and might
outperform NSC so long as turbulence is not too intense. That argument is addressed
in Figure 7.19. The short-term spot size statistics for the whole beam, in both single-
and triple-pass experiments, are compared to the theoretical result from Equation
3.29 for the case of an infinitely large beam with focal length equal to the propagation
distance.
116
Figure 7.19: The probability that the NSC beam will produce a spot with
radius smaller than the average radius of a low-power beam focused to
the target from an infinitely large beam director. The performance of the
infinite-radius linear beam was estimated using Equation 3.29. Nonlinear
self-channeling starts to outperform the ideal linear case around σ2R ≈
0.5.
117
7.3.5 Effects of Power and Beam Clipping
Additionally, experiments were performed to examine the statistics of the high-
power beam at the target (single pass) as the initial beam power is allowed to vary.
By fixing the turbulence at its highest strength and varying the compressor grating
separation, the beam behavior as a function of power can be investigated with
greater fidelity than simply comparing the low-power and high-power cases. The
results of this experiment are plotted in 7.20
Figure 7.20: Average beam size as a function of power.
118
Chapter 8: Conclusion
8.1 Future Work
The present work provides experimental confirmation of the ability of NSC
to produce small laser spots at long ranges, even in deep turbulence. Additional
work to model and simulate the experiments performed at AFRL would enhance
understanding of these processes. Special attention should be paid to the effect of
GVD, since it has been shown to have an effect on the order of 100% of the pulse
length over a range of 580 meters.
Future experimental work should be performed at longer range, in order to
allow atmospheric scattering and absorption to come into play. These are expected
to be important effects which will have to be balanced in order to maintain NSC over
long distances. For instance, Peñano et al use a chirped pulse which compresses due
to GVD to balance the loss of energy due to attenuation and maintain NSC [11].
Future experiments should use a new, improved version of the GDFACT, which
was developed after the AFRL experiments took place. The “Lemmy” design is
effectively multiple GDFACT devices in parallel with one another, suspended side-
by-side with small (∼ 1 inch) gaps between the wires. This design provides a wider
zone of turbulence in which the beam can propagate, potentially allowing multiple
119
passes over the same range without the beam exiting the turbulent volume, as it may
have done at AFRL. Additionally, as the linear beam propagates longer distances it
will naturally diffract to larger size; the turbulent volume needs to remain at least
as wide as the beam in order to correctly emulate natural atmospheric turbulence.
8.2 Conclusion
Nonlinear self-channeling, a novel mode of propagation for high power laser
pulses, has been experimentally demonstrated in the presence of artificially gen-
erated atmospheric turbulence. In nonlinear self-channeling, the laser spot size
remains approximately constant as a function of propagation distance, at a size
somewhat smaller than the inner scale of turbulence. Beams propagating in this
regime are resistant to the beam-spreading effects of turbulence, although their
wander still increases in accord with linear theory. Examining Figures 7.12 and
7.13, the benefits of nonlinear self-channeling begin to decline as turbulence enters
the strong turbulence regime (σ2R > 1), and are largely eliminated once the inner
scale becomes smaller than the beam radius. Under these extreme conditions, the
high-power beam still has some likelihood of producing a small beam on target, but
it is no more likely to do so than a linear beam of the same diameter. Neverthe-
less, it is still expected to out-perform an extremely large beam which is focused to
the target (Figure 7.19). With these caveats, nonlinear self-channeling beams have
potential value in any application which demands the propagation of laser light
through the atmosphere to produce a high fluence or high intensity at long range.
120
Appendix A: Image Analysis
A.1 Approach to Analysis of Beam Profile Images
Images taken during the experimental portion of this work were analyzed in
order to extract statistical quantities such as laser spot size, position, and peak
intensity. A custom script written in Python was used for this purpose. Python was
chosen for its execution speed, array of applicable pre-programmed modules, and
extensive online tutorials.
Analysis begins with one of the roughly 500 images of the IR beam taken for
each set of experimental conditions. These images range from 512x512 to 2560x2048
pixels in size, with 12-bit grayscale depth padded to a 16-bit image. The total
volume of data is approximately 2.5 TB. Each image shows the beam profile of a
single laser pulse at the end of propagation. Depending on experimental conditions,
the signal-to-noise ratio of the image could be as high as several thousand, or as
low as two. The beam could be large or small, tightly localized or diffuse, or spread
out into multiple sub-beams. A single image analysis routine is needed which can
convert all of these images into summary statistics, such as beam size and position,
for comparison to theory and simulation.
An example of a typical beam image is shown in Figure A.1. The results of
121
applying successive steps of the analysis routine to this same example image will be
presented in successive figures.
Figure A.1: An example unmodified image. Note the presence of signif-
icant “static-like” background noise.
The perspective of the image is corrected by means of a perspective transfor-
mation. During data collection, the camera is offset from the surface normal of the
imaging plane in order to allow the laser beam to pass by, causing the flat rectangle
of the imaging surface to appear as a trapezium. Collection of a wide-angle “setup”
122
image establishes the coordinates of the four corners of the imaging surface in both
real space and image space. A perspective transform uses these known points to
map every point in image space to the appropriate point in real space, producing a
rectangular image showing the imaging surface as it would have appeared were the
camera centered. The results of this procedure are shown in Figure A.2.
Figure A.2: The image modified by perspective transformation. The
512x512 image has been projected onto a larger canvas which matches
the dimensions of the entire imaging surface. This allows images of any
size to be adjusted correctly with the same perspective transformation.
123
The image is cropped to a region of interest which corresponds to an area
on the target surrounding the beam. This region of interest remains constant for
all images within a single day’s data. The resulting reduction in image size speeds
processing of all following steps. A “background” region of the image far from the
beam is also defined. The pixel values in this region are averaged to yield an estimate
of the average background level of the image. This value is subtracted from every
pixel in the image. Typical results of cropping and background subtraction can be
seen in Figure A.3.
A Fourier transform is now applied to the entire image. A Gaussian filter in
Fourier space rejects the highest frequencies in the image. The filter radius is 1⁄4
the radius of the entire image; this value was chosen empirically to reduce high-
frequency noise while leaving beam features largely intact. Additionally, a second
Gaussian filter, typically of radius 0.5, rejects the lowest frequencies in the image.
This eliminates vignetting and reduces the noise level near the beam. The resulting
Fourier image is inverse transformed, and the real part is returned as the cleaned
image (Figure A.4).
A thresholding step now takes place in order to reduce noise further. There
are two typical operational modes: thresholding to a multiple of the noise standard
deviation in order to reduce noise while leaving the beam largely intact, and thresh-
olding to 50% of the image maximum in order to examine the behavior of the beam
hot-spot. In either case, pixels with value below the threshold are not simply set
to zero: to do so would yield a background of value zero, containing intermittent
intense noise pixels of value slightly greater than the threshold value. This type of
124
Figure A.3: The image after cropping and background subtraction.
125
Figure A.4: The image after the bandpass filter. Notice the significantly
reduced level of background noise.
126
noise is extremely disruptive to beam size algorithms. Instead, the threshold value
is subtracted from the entire image, and then any pixels with negative value are set
to zero. In this case the remaining noise pixels have low values and do not con-
tribute disproportionately to apparent beam size. Typical results of this procedure
are displayed in Figure A.5.
Figure A.5: The image after thresholding.
Finally, in the case of noise-deviation thresholding, there is often still some
127
remaining low-value noise far from the beam center (see Figure A.6). Since the
RMS spot size metric is extremely sensitive to small amounts of noise far from the
beam centroid, it is important to eliminate these pixels. In this case, a Difference
of Gaussians (DoG) blob-detection algorithm is applied to the image.
Figure A.6: The image after thresholding, contrast-enhanced in order to
draw attention to low-amplitude noise.
128
A.1.1 Blob Detection
The DoG algorithm locates and estimates the size of bright spots on a dark
background by convolving the image with successively larger Gaussian blur kernels,
and taking the difference between successive blurred images. In regions of constant
or near-constant amplitude, the amount of blur applied does not change the image
much, so the difference is small. The difference image is dark and uniform in these
regions. However, in regions where the image has edges, the amount of blurring
strongly affects the size of the blurred spot. In these regions the difference image
is bright. Blob location is determined by the location of a local maximum in the
difference image; blob size is determined by the size of the Gaussian blur kernels
which were used to detect the blob. The array of possible blob sizes is determined
by the set of Gaussian kernels used to convolve the image: more discrete sizes leads
to longer processing times and smaller blob size quantization error. For this reason,
blob detection is not sufficiently precise to produce a true measurement of spot size.
Nevertheless, blob detection suffices to determine a reasonably-sized region of
interest which excludes noise while including the beam. Every blob detected by the
algorithm defines a circle with known position and radius, as visualized in Figure
A.7. This radius is multiplied by three (empirically chosen), and the set of all pixels
within the resulting circles forms the new region of interest. Pixels outside this
region are set to zero, as in Figure A.8. For each image, the radius of the largest
blob is reported for completeness. Typical results of this image cleaning procedure
can be seen in Figure A.9.
129
Figure A.7: Location and size of the blobs detected by the DoG blob-
detection algorithm. The underlying image is contrast-enhanced.
130
Figure A.8: Output of the DoG blob-detection algorithm. The image
has been contrast-enhanced for visibility.
131
Figure A.9: Final image.
132
A.1.2 Beam Metrics
Proper beam analysis can begin now that image cleaning is done. The beam
centroid is determined by calculating the first moment of the image in the x- and
y-dimensions, as follows:
∑
Px,yα
α0 = ∑ (A.1)
Px,y
where α denotes either x or y.
The beam radius is then calculated in the x-, y-, and r-dimensions by calcu-
lating the second central moment of the image, as in the following equations:
√∑
P∑ 2x,y(α− α0)Rα = 2 (A.2)
Px,y
√
√ ∑P∑~r|~r − r~ 20|Rr = 2 (A.3)
P~r
where ~r = ~x + ~y and r~0 = x̂x0 + ŷy0. The primary measurement is the radius in
the r-dimension; the radius in the x- and y-dimensions are reported separately in
√
order to detect beam asymmetry. The r-radius which is reported is 2 times the
RMS radius of the beam. This metric was chosen because for a perfect Gaussian
beam, it matches the 1/e2 intensity radius. It is also a commonly used definition of
beam width throughout the literature, including the work of Siegman [68], the ISO
standard for measuring laser beam quality (M-squared) [69], and the calculations of
Andrews and Phillips [7].
133
Additionally, the 86.5% Power-In-The-Bucket (PIB) radius is reported. This
is the radius of a circle centered on the beam centroid which contains 86.5% of the
pixel value integrated over the entire image. Because it lacks the |~r|2 factor, the PIB
radius is not so sensitive to the presence of low-value pixels far from the centroid.
It concentrates on the size of the central lobe of the beam while giving lower weight
to the distribution of the wings. For a perfect Gaussian, the RMS matches the 1/e2
radius; for other beams, it is typically somewhat smaller.
A.1.3 Error Analysis
Error analysis has been performed for these procedures, by applying them to a
series of noiseless images produced by simulation, to which artificial noise matching
the spectrum and intensity distribution of real image noise has been added.
Table A.1: From left to right: the simulated beam without noise, with
noise, and after image cleaning. Some low-amplitude portions of the
beam are lost in exchange for eliminating the noise.
In this manner, the true r-radius can be measured under ideal conditions, and
the error in determining the radius due to noise and beam cleaning can be assessed.
134
The results of this procedure are summarized in the following table:
Without Cleaning With Cleaning
Without Noise 0 % -2.0 %
With Noise 634 % -9.6 %
Table A.2: Percentage error in r-radius spot size determination rel-
ative to the noiseless simulated beam. The application of the beam
measurement algorithm to the noiseless image produces zero error by
definition. The presence of noise, in the absence of image cleaning,
causes the beam size to be overestimated by a factor of more than 6.
The image cleaning routines discussed above restore the accuracy of
the beam size to within 10%.
The beam cleaning significantly reduces the noise-related error in beam size,
while having only a small effect on the apparent beam size of an image without
noise. There is a general tendency to underestimate the size of the beam. This is
preferable to an overestimate, since small excesses of noise far from beam center can
greatly inflate the apparent beam size.
135
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