ABSTRACT
Title of dissertation: HIGH-INTENSITY LASER-MATTER
INTERACTIONS
PHYSICS AND APPLICATIONS
Zachary Epstein
Doctor of Philosophy, 2019
Dissertation directed by: Professor Phillip Sprangle
Department of Physics
This dissertation consists of three distinct research topics which involve high-intensity
laser-matter interactions. The topics are (1) High-Power Supercontinuum IR Generation, (2)
Spectral Broadening of the NIKE KrF Laser in a Negative Nonlinear Index Medium, and (3)
Remote Optical Magnetometry.
(1) High-average power, ultra-broadband, mid-IR radiation can be generated by illuminat-
ing a nonlinear medium with a multi-line laser radiation. Propagation of a multi-line, pulsed
CO2 laser beam in a nonlinear medium, e.g. gallium arsenide or chalcogenide, can generate
directed, broadband, IR radiation in the atmospheric window (2 – 13 µm). A 3-D laser code
for propagation in a nonlinear medium has been developed to incorporate extreme spectral
broadening resulting from the beating of several wavelengths. The code has the capability to
treat coupled forward and backward propagating waves, as well as transverse and full linear
dispersion effects. Methods for enhancing the spectral broadening are proposed and analyzed.
Grading the refractive index radially or using a cladding will tend to guide the CO2 radiation
and extend the interaction distance, allowing for enhanced spectral broadening. Nonlinear
coupling of the CO2 laser beam to a backwards-propagating, reflected beam can increase the
rate of spectral broadening in the anomalous dispersion regime of a medium. Laser phase
noise associated with the finite CO2 linewidths can significantly enhance the spectral broad-
ening as well. In a dispersive medium laser phase noise results in laser intensity fluctuations.
These intensity fluctuations result in spectral broadening due to the self-phase modulation
mechanism. Finally, we present propagation through a chalcogenide fiber as an alternative for
extreme spectral broadening of a frequency-doubled CO2 multi-line laser beam.
(2) In inertial confinement (ICF) experiments at the NIKE [1] laser facility, the high-power
krypton fluoride (KrF) laser output beams propagate through long (∼75m) air paths to achieve
angular multiplexing. This is required because the KrF medium does not store energy for a
sufficiently long time. Recent experiments and simulations have shown that, via stimulated
rotational Raman scattering (SRRS), this propagation can spectrally broaden the laser beam
well beyond the ∼1 THz laser linewidth normally achieved by the induced spatial incoherence
(ISI) technique used in NIKE [2]. These enhanced bandwidths may be enough to suppress the
laser-plasma instabilities which limit the maximum intensity that can be incident on the ICF
target. We investigate an alternative technique that achieves spectral broadening by self-phase
modulation in Xe gas, which has a large, negative nonlinear refractive index ∼ 248 nm [3],
and thus completely avoids transverse filamentation issues. The collective, nonlinear atomic
response to the chaotic, non-steady state ISI light is modeled using a two-photon vector model.
The effect of near-resonant behavior on the spectral broadening is also studied.
(3) Here we analyze a mechanism for remote optical measurements of magnetic field vari-
ations above the surface of seawater. This magnetometry mechanism is based on the polar-
ization rotation of reflected polarized laser light, in the presence of the earth’s magnetic field.
Here the laser light is reflected off the surface of the water and off an underwater object. Two
mechanisms responsible for the polarization rotation are the Surface Magneto-Optical Kerr
Effect (SMOKE) and the Faraday effect. In both mechanisms the degree of polarization rota-
tion is proportional to the earth’s local magnetic field. Variations in the earth’s magnetic field
due to an underwater object will result in variations in the polarization rotation of the laser
light reflected off the water’s surface (SMOKE) and off the underwater object (Faraday effect).
An analytical expression is obtained for the polarization-rotated field when the incident plane
wave is at arbitrary angle and polarization with respect to the water’s surface. We find that
the polarization rotated field due to SMOKE is small compared to that due to the Faraday
effect.
HIGH-INTENSITY LASER-MATTER INTERACTIONS
PHYSICS AND APPLICATIONS
by
Zachary Epstein
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
2019
Advisory Committee:
Professor Phillip Sprangle1,2, Chair/Advisor
Professor Thomas Antonsen1
Professor Howard Milchberg1
Dr. Joseph Peñano2
Dr. Robert Lehmberg2
Professor Rajarshi Roy1 (Dean’s Representative)
1University of Maryland, College Park, MD 20742-4111
2Plasma Physics Division, Naval Research Laboratory, Washington DC 20375
©c Copyright by
Zachary Epstein
2019
Acknowledgments
My thesis was made possible by the guidance and contributions of many individuals.
(i) I’d first like to thank Professor Phillip Sprangle for being an extraordinary advisor and
mentor. He granted me the opportunity to work on interesting, challenging, and applicable
projects, made himself very available, and offered advice and guidance throughout the process.
It has been a pleasure to work with and learn from him, from the day five years ago when I
arranged a meeting with him to chat about his career and which strategies he used to make
such large contributions to science and to the country.
(ii) I would also like to thank Dr. Robert Lehmberg, Professor Antonio Ting, Dr. Luke
Johnson, Dr. Joseph Penano, and Dr. Bahman Hafizi for many fruitful discussions and
for their guidance. I am grateful to Professors Howard Milchberg and Thomas Antonsen
for agreeing to serve on my thesis committee and for making it a pleasure to work in the
department.
(iii) My office mates have enriched my graduate life in many ways and deserve a special
mention - Joshua Isaacs, Luke Johnson, Chenlong Miao, Thomas Rensink, Evan Dowling, and
Paul Gradney. It has been a lot of fun sharing an office with you all.
(iv) I would like to acknowledge help and support from the IREAP staff members. In
particular, Ed Condon’s computer hardware help was highly appreciated, and Leslie Delabar
made it a breeze to navigate involved technical issues.
(v) I would also like to acknowledge the Office of Naval Research for funding the work
contained in this dissertation and in particular Quentin Saulter ONR USPL PM.
ii
(vi) Thank you to my wife Nadine for her patience, love, warmth, energy, support, and
mesiras nefesh, and for being a true tzadekkes throughout the entire PhD.
(vii) Thank you to my parents for instilling within the me strength to make it through
graduate school while balancing a wide range of priorities, and for supporting me through
every step of my education.
(viii) I would like to acknowledge financial support from ONR, NEEC, and NRL for the
various projects discussed in this dissertation.
iii
Contents
1 Acknowledgements ii
2 List of Figures viii
3 List of Tables xiii
I Introduction 1
II High-Power Supercontinuum IR Generation 3
1 Introduction 4
2 Propagation and Interaction Model 6
2.1 Nonlinear Propagation Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Self-Guiding in GaAs Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Longitudinal Modulation Instability of Counterpropagating Beams . . . . . . . 12
3 Simulation Results 13
3.1 Spectral Broadening in GaAs . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.2 Spectral Broadening in an Index Tapered Medium . . . . . . . . . . . . . . . . 16
3.3 Effect of Laser Phase Noise on Spectral Broadening . . . . . . . . . . . . . . . 17
3.4 Enhancement of Spectral Broadening via Coupling to a Backward-Propagating,
Reflected Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.5 Second-Harmonic Spectral Broadening in As2S3 Fiber . . . . . . . . . . . . . . 21
iv
4 Discussion 23
4.1 Raman Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Laser Damage Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Conclusion 24
6 References 25
7 Supplement 28
7.1 Supplement: Evaluation of Nonlinear Polarization Term . . . . . . . . . . . . . 28
7.2 Supplement: Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . . . 29
III Spectral Broadening of the NIKE KrF Laser in a Negative
Nonlinear Index Medium 32
1 Introduction 33
2 Model 36
2.1 Laser Pulse Propagation - Generalized Nonlinear Schrodinger Equation . . . . 36
2.2 Spatial and Temporal Incoherence . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 Nonlinear Response of Xe Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.1 Density matrix equations . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.2 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3.3 Separation of density matrix elements into harmonics of the laser frequency 41
2.3.4 Two-photon vector model . . . . . . . . . . . . . . . . . . . . . . . . . 42
v
2.3.5 Two-photon adiabatic following approximation . . . . . . . . . . . . . . 42
2.3.6 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4 Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3 Numerical Results 49
4 Discussion 54
5 Conclusions 54
6 References 55
7 Supplement 57
IV Remote Optical Magnetometry 63
1 Introduction 63
2 Model 65
2.1 Polarization Field of Magnetized Water . . . . . . . . . . . . . . . . . . . . . . 66
2.2 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3 Polarization Rotation 70
4 Discussion 75
5 References 75
6 Supplement 76
vi
V List of Publications and Presentations 82
VI Complete Reference List 84
vii
List of Figures
1 A schematic of the concept showing the output spectrum extending from 2 –
13 µm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Self-focusing and spectral broadening of a beat wave propagating in GaAs, for
the parameters of Table 1. The initial laser line separation is, respectively, (a)
∆λ/λ0 = 0.01 , (b) ∆λ/λ0 = 0.04 . In the latter case, the wavelength 10.6
µm has been used. The spot-size is determined by fitting the transverse laser
profile to a Gaussian profile. The spectrum is defined here as the range over
which there are individual frequency components containing at least 0.005% of
the laser beam’s total power, see Fig. 3. . . . . . . . . . . . . . . . . . . . . . 15
3 Spectrum of a beat wave propagating in GaAs, for the parameters in Table
1. The spectrum is shown at a distance when the on-axis intensity nears
the breakdown threshold. The input laser line separation is, respectively, (a)
∆λ/λ0 = 0.01 , (b) ∆λ/λ0 = 0.04 , (c) ∆λ/λ0 = 0.08 . These lines are com-
prised of 8 individual lines in the range from 9.15 µm – 9.95 µm , together with
eight lines in the range 4.575 µm – 4.975 µm, where each line contains 0.08 MW. 15
4 Self-focusing and spectral broadening of a beat wave near the focusing power
in GaAs. The peak power here is P = 0.93 PK . . . . . . . . . . . . . . . . . 16
viii
5 The total power spectrum, at z = 11.2 cm, is shown in (a) for a beat wave near
the focusing power. The spectrum (b) on-axis and (c) at the spot size R0 are
shown for comparison. The spectrum is broader along the axis because of the
higher intensity on axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6 Self-focusing and spectral broadening of a Gaussian-profile laser beat wave prop-
agating through a GaAs waveguide with a graded index. The laser line separa-
tion is ∆λ/λ0 = 0.04 , and the peak power is given by P/PK = 5 . . . . . . . . 17
7 Power spectrum for ∆λ/λ0 = 0.04 after propagating 1.5 cm in a GaAs rod with
a graded index characterized by RC = 70 R0 , see Fig. 6 . . . . . . . . . . . . 17
8 Power per wavelength-regime, given a varying laser linewidth due to phase
noise. The total power in spectrum (a), (b), and (c) is the same. The spectrum
is shown after propagating 2 cm in GaAs. Initially the CO2 radiation consisted
of two lines each with linewidth (a) 0.00∆λ (no noise) (b) 0.01∆λ (c) 0.10∆λ
due to phase noise. The initial laser line separation is ∆λ/λ0 = 0.01. In
frequency space this corresponds to (a) 0 GHz (no noise) (b) 3 GHz (c) 30 GHz. 18
ix
9 The power spectrum is shown at z = 3 cm for a forward-propagating beat-
wave in GaAs with the parameters in Table 1, with coupling to a backward-
propagation reflected wave. This simulation has been carried out in 1D for
an initial laser line separation of ∆λ/λ0 = 0.02 and demonstrates the useful-
ness of BTUL’s bi-directional capability as well as the ability of a backward-
propagating reflected wave to enhance spectral broadening in the anomalous
dispersion regime of a nonlinear medium. (a) The black dotted curve repre-
sents propagation given 29% reflectivity at the end of the medium (i.e., at
z = 3 cm), corresponding to reflection off a GaAs - Air boundary. The red,
solid curve corresponds to 0% reflectivity, i.e., unidirectional propagation. (b)
The black dotted curve represents propagation given 100% reflectivity at the
end of the medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10 1D simulation of CO2 laser beat-wave propagating through (a) GaAs (b) As2S3
at various intensities and wavelengths. The fractional bandwi(d∫th is estima)ted−1
h(∫ere as the third mom)ent in frequency space, i.e. µ3(z) = P (ω, z)dω
(ω − 3 1/3ω0) P (ω, z)dω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
11 Optical guiding, self-focusing, and spectral broadening of a Gaussian-profile
frequency-doubled CO2 beat-wave with 200 µm initial spot size, laser wave-
length λ0 = 4.7 µm, and laser line separation ∆λ/λ0 = 0.005 propagating
through an As2S3 fiber. Although the laser operates below the nonlinear focus-
ing power PK ∝ λ2, extreme spectral broadening yields high frequency compo-
nents which are indeed above their respective nonlinear focusing power. . . . . 22
x
12 Power spectrum for ∆λ/λ0 = 0.005, where λ0 = 4.7 µm , after propagating 8
meters in an As2S3 fiber, see Fig. 10. . . . . . . . . . . . . . . . . . . . . . . . 22
13 (a) The effective spot size of a Gaussian-profile, monochromatic laser beam of
initial spot size R0 = 200 µm is plotted as a function of propagation distance
through GaAs for various input powers ‘P’ which are above the nonlinear fo-
cusing power, i.e. P > PK = 2π c
2/ (ω20 n0 n2) = 0.26 MW. The collapse
distance is extrapolated and (b) plotted as a function of input power in order
to show consistency with an approximate numerical formula [9] for the collapse
distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
14 The nonlinear optical response of Xe near its two-photon 5p6 → 6p[1/2]0 reso-
nance is that of a three-level system with multiple intermediate states |2〉, and
allowed electronic transitions from |1〉 → |2〉 and |2〉 → |3〉. . . . . . . . . . . . 34
15 Wavelength-dependence of the nonlinear refractive index n2 of 200mbar Xe in
the regime of the two-photon resonance. The curve is continuous but extends
beyond the range shown here. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
16 Propagation of a 30 M2x (times-diffraction-limit) temporally and spatially inco-
herent KrF laser beam through 50m of 200mbar Xe gas, see Table 2: (a) Total
power spectrum (b) Axial lineout of near-field beam profile (c) Axial lineout
of far-field beam profile , modeled via the Two-Photon Vector Model (TPVM)
using either Eq. (9) or Eq. (10), the Kerr response n2(φ(˙t)) see Eq. (6-7), and
the narrow-bandwidth, steady-state Kerr response n2(ω0). Properties of the
incident light (z=0) are indicated in black. . . . . . . . . . . . . . . . . . . . . 52
xi
17 Spectral Broadening Factor, Far-field Broadening Factor, Beam Spreading Fac-
tor are plotted as a function of the times-diffraction-limit (M2x), for propaga-
tion of a temporally and spatially incoherent KrF laser beam through 50m of
200mbar Xe gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
18 Polarized laser light is propagated to the surface of the water and exhibits
polarization rotation upon reflection due to Faraday rotation and SMOKE.
The solid arrows in the figure denote wave vectors. . . . . . . . . . . . . . . . 65
19 A simplified configuration of reflected laser light undergoing polarization rota-
tion due to Faraday rotation and SMOKE. The purpose of this configuration is
to obtain estimates for the polarization rotation angle ∆θ. . . . . . . . . . . . 70
20 Ratio of the polarization-rotated reflected field to the incident field as a function
of object depth h, according to Eq. (10). The polarization rotation results from
a combination of Faraday rotation and SMOKE; however, the contribution due
to SMOKE is small. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
21 nR = 1.34 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
22 The function |δER,⊥/E0| plotted above represents the dependence of the polarization-
rotated field on the incident wave vector’s orientation for a specific geometry
(θB = 0,Ey = 0), according to Eq. (15). Here δER,⊥ is the polarization-
rotated component of the reflected laser field and E0 is the incident laser field.
This plot is specific to water with refractive index nR = 1.34. . . . . . . . . . . 73
23 Coordinate systems used in the calculation of the polarization-rotated reflected
field associated with SMOKE for an arbitrary geometric configuration. . . . . 77
xii
24 The functions above, within the context of Eq. (15), describe the angular depen-
dence of the polarization-rotated field on the incident wave vector’s orientation. 80
List of Tables
1 Parameters used in the spectral broadening examples . . . . . . . . . . 13
2 Simulation parameters for Fig. 16-17 . . . . . . . . . . . . . . . . . . . . 50
xiii
Part I
Introduction
This dissertation consists of three topics which involve high-intensity laser-matter interactions.
Involvement in several projects has enabled a broader understanding of the fields. The topics
are (1) High-Power Supercontinuum IR Generation, (2) Spectral Broadening of the NIKE KrF
Laser in a Negative Nonlinear Index Medium, and (3) Remote Optical Magnetometry. The
main two topics - (1) and (2) - both have the goal of spectrally broadening a laser, albeit
under very different circumstances for different applications.
Topics (1) and (3) were published as first-author peer-reviewed publications in JOSA B
and IEEE Quantum Electronics, respectively, and the third topic will be submitted to Physical
Review E this month for peer review. In addition, numerous presentations have been made
involving content of this dissertation (see List of Publications and Presentations).
The first research topic involves the generation of high-power ultra-broadband mid-IR
radiation, which is relevant to the remote detection of chemicals and to IR countermeasures,
and can be achieved by beating together two CO2 laser lines and propagating the resulting
beam through a GaAs crystal. Previously, supercontinuum IR radiation up in the 0.1-1 kW
average IR power regime had been analytically and numerically studied only in 1D (see Ref.
[5-6] of Part 1). A 3D bi-directional code (BTUL) was developed with the capability to treat
extreme spectral bandwidths and coupling between forward and backward waves. It also
includes all χ(1) effects (group velocity dispersion, third-order dispersion, etc) and χ(3) effects
(self-phase modulation, four-wave mixing, cross-phase modulation, third harmonic generation,
1
etc), as well as spatial variations in the linear refractive index. Three methods are proposed
and analyzed to enhance spectral broadening: i) radially grading the linear refractive index, ii)
coupling to a backwards-propagating, reflected wave, and iii) taking advantage of laser phase
noise. Spectral broadening in a chalcogenide fiber is considered as well.
The second research topic involves the spectral broadening of KrF laser light in Xe, which
has a large, negative nonlinear index at the 248.4 nm laser wavelength due to a two-photon
resonance at 249.6 nm. The research is relevant to inertial confinement fusion, in which a
large bandwidth can mitigate laser plasma instabilities and cross beam transfer. In particular,
the Naval Research Laboratory’s NIKE KrF laser beams propagate across distances of tens
of meters, over which spectral broadening in a tube of Xe gas is a possibility. The negative
value of the nonlinear index is an ideal property because it precludes the occurrence of laser
beam breakup. Spectral broadening and beam profile degradation in Xe are quantified and
simulated, and the effect of the initial times-diffraction-limited value (M2X) on these quantities
is found to be significant.
The third research topic involves two mechanisms for detecting variations in the Earth’s
magnetic field due to underwater objects. It is found that reflecting polarized light off the
surface of seawater results in two contributions of polarization-rotated light. The first is a
purely surface effect called the Surface Magneto-Optical Kerr Effect (SMOKE) which turns out
to be small. The latter contribution is a volume effect - the Faraday effect. The contribution
from the Faraday effect can be significant but is only relevant to shallow (∼ 10m depth)
objects. The contribution of SMOKE is analyzed for arbitrary polarization direction, incident
angle, and magnetic field direction.
2
Part II
High-Power Supercontinuum IR Generation
High-average power, ultra-broadband, mid-IR radiation can be generated by illuminating a
nonlinear medium with a multi-line laser radiation. Propagation of a multi-line, pulsed CO2
laser beam in a nonlinear medium, e.g. gallium arsenide or chalcogenide, can generate di-
rected, broadband, IR radiation in the atmospheric window (2 – 13 µm). A 3-D laser code
for propagation in a nonlinear medium has been developed to incorporate extreme spectral
broadening resulting from the beating of several wavelengths. The code has the capability to
treat coupled forward and backward propagating waves, as well as transverse and full linear
dispersion effects. Methods for enhancing the spectral broadening are proposed and analyzed.
Grading the refractive index radially or using a cladding will tend to guide the CO2 radiation
and extend the interaction distance, allowing for enhanced spectral broadening. Nonlinear
coupling of the CO2 laser beam to a backwards-propagating, reflected beam can increase the
rate of spectral broadening in the anomalous dispersion regime of a medium. Laser phase
noise associated with the finite CO2 linewidths can significantly enhance the spectral broad-
ening as well. In a dispersive medium laser phase noise results in laser intensity fluctuations.
These intensity fluctuations result in spectral broadening due to the self-phase modulation
mechanism. Finally, we present propagation through a chalcogenide fiber as an alternative for
extreme spectral broadening of a frequency-doubled CO2 multi-line laser beam.
3
1 Introduction
High-average power and high peak power, ultra-broadband radiation sources in the long-wave
(8 – 13 µm) and mid-wave (2 – 5 µm) infrared regime have applications for remote sensing and
IR countermeasures. The generation of a mid-IR ultra-broadband (supercontinuum) source
was demonstrated using a 3ps CO2 laser pulse with GW peak power propagating through
GaAs [1]. For higher average powers (up to ∼100 mW), there are ongoing experiments and
modeling efforts at UCLA. In one experiment, a train of seven interspersed 3 ps pulses with
GW peak power was propagated through GaAs just below the breakdown intensity, i.e., ∼10
GW/cm2 [2]. Another experiment [3] employed a 200 ps , CO2 pulse train operating at a peak
power of 150 MW and consisting of consecutive, ∼10 ps micropulses just below the breakdown
intensity of GaAs, i.e., ∼1 GW/cm2 for pulses far exceeding 1 ps. In these experiments it
was observed that second order nonlinear processes restrictively affected the supercontinuum
generation. Elimination of the second order nonlinear effect via selecting the appropriate
orientation of GaAs may enable the generation of an efficient, pulsed radiation source in the
given regime [2 – 4].
Supercontinuum IR radiation in the 0.1 – 1 kW average IR power regime in GaAs has been
analytically and numerically studied in 1D [5 – 6]. In these early theoretical studies a spectrum
spanning the mid-IR regime was generated by the beating of two lines of CO2 radiation in
GaAs. The intensity of each line was 250 MW/cm2, and the resulting spectrum covered ∼ 5
– 16 µm after propagating 7.8 cm in GaAs. However, transverse effects and surface damage
thresholds were not considered. The simulations were based on the 1D Nonlinear Schrodinger
Equation (NLSE) model which is only valid when the fractional frequency spread is somewhat
4
less than unity.
Our formulation of the spectral broadening process allows for fractional frequency spreads
approaching unity and coupling to a backwards-propagating beam. In addition, we include
transverse and full linear dispersion effects, and methods for enhancing the spectral broadening
are proposed and analyzed.
Ultra-broad band radiation can be generated by a number of coupled nonlinear optical
processes [1, 7] which occur in certain material, e.g., GaAs, when illuminated by multi-lined
CO2 radiation. To cover the entire IR atmospheric transmission windows (8 – 13 µm) and (2
– 5 µm) the CO2 laser would operate at both the fundamental and second harmonic.
The dominant interrelated nonlinear processes include: i) self-phase modulation, ii) 4-
wave mixing, iii) cross-phase modulation, iv) stimulated Raman scattering and v) stimulated
Brillouin scattering. These processes can lead to wide spectral broadening of the lines around
the fundamental (∼ 10 µm) and second harmonic (∼ 5 µm) with high conversion efficiencies.
These mechanisms have the flexibility of adjustable spectral widths. A schematic of the
concept is shown in Fig. 1.
Figure 1: A schematic of the concept showing the output spectrum extending from 2 – 13 µm.
5
2 Propagation and Interaction Model
The wave equations which form the basis of the Bi-directional, Three-dimensional, Ultra-
broadband Laser (BTUL) code are obtained in this section. This code will be used to simulate
supercontinuum generation and self-focusing.
2.1 Nonlinear Propagation Code
By taking the electric field to be linearly polarized along the x-axis, and neglecting free charges
and coupling to polarization-rotated fields (arising from the divergence term), the electric field
in a spatially inhomogenous medium is given by the wave equation
1 ∂2 ∂2 2∇2 ∂E(r, t)− E(r, t) = µ0 (P (r, t) + P (r, t)) + E(r, t) , (1)
c2 ∂t2 ∂t2
L NL
∂x2
( )
where the last term emerges from the divergence term ∇~ ∇~ · E(r, t) , and where
E(r, t) = (1/2)Ê(r, t) exp (i(k0z − ω0t)) + c.c. is the electric field, Ê(r, t) is the complex am-
plitude, ω0 is the carrier frequency, k0 = nL(ω0)ω0/c is the carrier wavenumber, nL(ω) is the
linear refractive index, and PL(r, t) and PNL(r, t() are respe)ctively, the linear and nonlinear
polarization fields. In Eq. (1), The term (∂/∂y) ∇~ · E(r, t) is absent in Eq. (1) because it
drives the field along the y-axis.
Using the relation ∇ · (n2(r, t)E(r, t)) = ρf , where ρf = 0 is the free charge density, the
divergence term in Eq. (1) can be written as,
6
( ) ( )
∇~ · ~ − ~ · ∇n
2(r, t) ∂ ( )
E(r, t) = E(r, t) = −E(r, t) ln n2(r, t)
n2(r, t) ∂x
(2)
≈ −2E(r, t) ∂ (δnL(r) + δnNL(r, t))
nL0 ∂x
where n(r, t) = nL0 + δnL(r) + δnNL(r, t), nL0 is the linear refractive index of the medium,
and δnL(r) and δnNL(r, t) are respectively, linear and nonlinear variations in the refrac-
tive index of the (spatially-varying) medium. We express the latter term as δnNL(r, t) =
n2nL0ε0c|E(r, t)|2/2 , where n2 represents the Kerr effect. Terms of the form (∂/∂y) do not
appear in Eq. (2) since the electric field is x-polarized. Because the beam is smooth with
spot size R0  λ0, the divergence term is small for a linear refractive index that is constant
or weakly varying in space. Its treatment will be discussed at the end of this section.
Taking the Fourier transform (neglecting the divergence term) of Eq. (1) in time and
transverse space yields the following equation,
( )
∂2
+ k2z Ẽ(kx, ky, z, ω) = −µ 22 0ω P̃NL(kx, ky, z, ω), (3)∂z
where k2 2z = k (ω) − k2 2x − ky , k(ω) = nL(ω)ω/c and kx, ky denote transverse spa∫t∫ia∫l frequen-
cies. In obtaining Eq. (3) the Fourier transforms are defined as Q̃(kx, ky, z, ω) = Q(r, t)×
exp (i(ωt− kxx− kyy)) dtdxdy and the tr√ansform of the linear polarization field is P̃L(kx, ky, z, ω)
= ε0χL(ω)Ẽ(kx, ky, z, ω) where nL(ω) = 1 + χL(ω) and χL is the linear susceptibility.
In the case of an instantaneous nonlinear Kerr response, the nonlinear polarization field in
7
Eq. (3) can be written as
∫∫∫
P̃NL(kx, ky, z, ω) = ε0χNL(ω) E(r, t)E(r, t)E(r, t) exp (i(ωt− kxx− kyy)) dtdxdy (4)
where χNL is the nonlinear susceptibility, which in general is frequency dependent. The fields
in Eq. (4) can be written as the sum of a forward and backward propagating wave, i.e.,
Ẽ = Ẽ+ + Ẽ− and P̃ = P̃+ −NL NL + P̃NL. Using this form, Eq. (3) becomes (∂/∂z+ ikz)(∂/∂z−
ikz)Ẽ
±(kx, ky, z, ω) = −µ0ω2P̃±NL(kx, ky, z, ω). If the backward propagating wave is small
compared to the forward propagating wave, |E−|  |E+| , the transform of the nonlinear
polarization field is
∫∫∫ ( )
+ 3P̃NL(kx, ky, z, ω) = ε0 χNL dtdxdy E
+(r, t) exp (i(ωt− kxx− kyy)) (5a)
∫∫∫ [
− ( ) ]+ 2P̃NL(kx, ky, z, ω) = 3 ε0 χNL dtdxdy E (r, t) E−(r, t) exp (i(ωt− kxx− kyy)) .−
(5b)
where the subscript “ − ” refers to the backwards-propagating component. The general case,
in which the backward propagating wave can be significant, is treated in the Supplement.
Using the approximation (∂/∂z ± ik )Ẽ±(k , k , z, ω) ≈ ±2ik Ẽ±z x y z (kx, ky, z, ω), results in the
8
final expression
∂
Ẽ±(kx, ky, z, ω) =
∂z
(6)
2
± iµ0ωik ±zẼ (k ±x, ky, z, ω)± P̃NL(kx, ky, z, ω).2kz
The bi-directional capability of BTUL will enable, for example, study of how coupling to
a reflected backwards-propagating wave affects extreme spectral broadening in a nonlinear
medium. In the absence of a backward-propagating wave, the BTUL equations reduce to
the Unidirectional Pulse Propagation Equation (UPPE) derived in Ref. [8 – 9]. The UPPE
allows for simulations of ultra-broadband radiation with dispersion. We note the existence
of a generalized UPPE (gUPPE) as given in Ref. [10], which allows for the treatment of a
strongly-varying spatial dependence in the linear refractive index, though it does so in the
absence of a backwards propagating wave.
A weakly-varying spatial depende∫nce in the linear refractive index may be incorporated
by the addition of the term ±(i/2kz) δk2(x, y, z, ω)E±(x, y, z, ω) exp(−i(kxx+ kyy))dxdy to
the right-hand side of Eq. (6), where the linear dispersion relation is written as k2(x, y, z, ω) =
k2(ω)+δk2(x, y, z, ω) , δk2(x, y, z, ω) k2(ω)∫. The(divergence term in Eq.)(2) can be incorpo-∂ ∂δnL(r)
rated by the addi∫tion o(f the terms±(i/nL0kz) ) E
±(x, y, z, ω) exp (−i(kxx+ kyy)) dxdy
∂x ∂x
∂
and ±(i/n k ) E± ∂δnNL(r, t)L0 z (x, y, z, t) × exp (−i(kxx+ kyy) + iωt) dxdydt to the
∂x ∂x
right-hand side of Eq. (6) , where nNL(r, t) = n2I(r, t) and the laser intensity time-averaged
over several optical cycles is given by I(r, t) = nL0ε0c 〈E(r, t)2〉.
9
2.2 Self-Guiding in GaAs Waveguide
In the case of a monochromatic field, the 3D generalization of the nonlinear para-axial envelope
equation is
∂Ê(r) i ∣ ∣2
= ∇2 i ∣ ∣⊥Ê(r) + δk2(r)Ê(r) + iγ0∣Ê(r)∣ Ê(r), (7)∂z 2k0 2k0
where the nonlinear coefficient is γ0 = (3ω0/8nL0c)χNL(ω0) = n2ω0nL0ε0/2 , n2 represents
the Kerr e∣ffec∣t, the total refractive index is n = n + δn + δn , δn = n I , and∣∣ ∣
L0 L NL NL 2
2
I = nL0ε0c Ê∣ /2 is the intensity. The term proportional to δk2(r) = n2L(r)ω2/c2 − k20 allows
for a transverse spatial variation in the linear refractive index, i.e. δnL. In the case of a GaAs
medium, appropriate orientation has been assumed so that second order effects are small.
Representing the complex amplitude of the field in the form Ê(r) = A(z) exp (iθ(z))×
exp (−(1 + iα(z))r2 2 1/2/R (z)), where r = (x2 + y2) , α is related to the radius of curvature of
the wavefront, R is the spot size and A and θ are real, Eq. (7) can be written as
( )
∂ lnA ∂θ − ∂a r
2 r2
2ik0 + i i + 2(1 + iα)( ∂z )(∂z ∂z R2 ) R3
− 1 + iα − (1 + iα)r
2
4 1 (8)
R2 R2
≈ −δk(r)− 2k0γ0A2(1− 2r2/R2),
where on the right hand side of Eq. (8) the Kerr term has been expanded in r. The radial
variation in the linear refractive index is taken to be quadratic in r, i.e., n2L(r, ω) = n
2
L0 +
(n2L0−1)r2/R2C , where RC is the characteristic radial scale length. Equating powers of r up to
r2 terms and real and imaginary terms in Eq. (8) results in four first order coupled differential
10
equations for A, θ, α, R: . Combining these equations we find that the laser spot size is given
by ( )
∂2R 4 − P k
2
0(n
2
L0 − 1) R4= 1 + , (9)
∂z2 k2R3 P n2 R20 K L0 C
where PK = λ
2
0/(8πnL0n2) is the nonlinear focusing power and λ0 is the wavelength in vacuum.
A derivation of the laser spot size based on the Source Dependent Expansion method [11 – 12]
yields a value for PK which is 4 times larger than that obtained using the expansion approach
in Eq. (8). In the following, we use the correct value of PK [12]. Simulations over multiple
Rayleigh lengths indicate that Eq. (9) is satisfied at low powers for the guiding condition,
i.e., ∂2R/∂z2 = 0 . It can be shown that the guiding condition is stable with respect to
radial perturbations in the spot size. The osci√llations of the spot-size perturbations have a√
longitudinal period given by λosc = 2πnL0RC/ n2L0 − 1 . For typical parameters considered
here, λosc is on the order of cm. In the absence of radial variation in the refractive index, we
1/2
see from Eq. (9) that the spot size is given by R(z) = R (1 + z20 (1− P/P 2K)/ZR) , where
ZR = n
2
L0πR0/λ0 is the Rayleigh length.
To lowest ord∑er in the nonlinearity, the field of a beat wave can be written as E(r, z, t) =
E exp(−r2/R20 0) ± cos ((k0 ±∆k/2)z − (ω0 ±∆ω/2)t) where ∆k = nL0∆ω/c and ∆ω ≈
2πc∆λ/λ20. The power in the beat wave is PB(z, t) = P cos
2(∆kz/2 − ∆ωt/2) where P is
the peak power. The field is now generalized to include the nonlinearity, i.e. E(r, z, t) ∼
exp (−r2/R2(z)) exp (i(kz − ω0t)) , where k(z, t) = (nL0 + nNL)ω/c = (nL0 + n2I(z, t))ω/c
and the on-axis beat wave intensity is I(z, t) = 2PB(z, t)/ (πR
2(z)) . The instantaneous
11
frequency spread is δω ≈ −∂ (k(z, t)z) /∂t and the fractional frequency spread is given by
δω(z, t) ≈ δωmax(z) sin (∆kz −∆ωt) . (10)
ω0 ω0
where ( )( )( )( )
δωmax(z) ≈ 1 ∆λ z P R
2
0 . (11)
ω0 2 λ 20 ZR PK R (z)
Inserting the expression for R(z) into Eq. (11), we find that a laser operating above the
nonlinear focusing power and below the breakdown intensity will spectrally broaden and self-
focus until the breakdown intensity is reached. For injection below the focusing power and the
breakdown intensity, the spectral broadening takes place for the entire length of the medium.
2.3 Longitudinal Modulation Instability of Counterpropagating Beams
Counterpropagating beams can affect one another’s modulation instability in the anomalous
dispersion regime of a Kerr medium. We have found that the wave vector K(ωm) associated
with longitudinal instability in the forward-going wave is given by the following dispersion
relation:
2
(K2 − ωK2 )(K2−K2 )− 4 mK2+ − v2
(12)
ωm
= 4β2ω42 mΩNL−ΩNL+ + 2K (K
2
+ −K2v ∣ ∣−
) ,
∣ ∣2
where K2 = β ω2± 2 m(β
2
2ωm/4 + ΩNL±) − ω2 2 2m/v and ΩNL± = χNLk0 ∣ʱ∣ /nL , where χNL is
the nonlinear susceptibility, nL is the linear refractive index, k0 = nLω0/c is the wave vector
associated with the pump frequency ω0 , β2 is the group velocity coefficient in the medium at
12
ω0 , v is the group velocity , and Êp and Êm are respectively the forward- and backward-going
electric field amplitude.
As will be seen in Section 3.4, coupling to a backward propagating, reflected wave can sig-
nificantly affect beam propagation in the anomalous dispersion regime of a nonlinear medium.
3 Simulation Results
3.1 Spectral Broadening in GaAs
The parameters used in this study are shown in Table 1. Numerical techniques are discussed
in the Supplement.
Table 1: Parameters used in the spectral broadening examples
Parameter Value
Laser wavelength λ0 = 9.55 µm
Laser line separation ∆λ/λ0 = 0.01
Initial spot size R0 = 0.02 cm
Peak laser power P = 1.28 MW
Pulse duration T  1 ps
Linear refractive index (GaAs) nL0 = 3.3
Nonlinear refractive index (GaAs) n = 1.7× 10−132 cm2/W
Self-focusing power at λ0 PK = 0.26 MW
Breakdown intensity IBD ∼ 0.4 – 1.5 GW/cm2
The parameters in Table 1 are applied to all simulations in the paper, unless otherwise
specified; in particular, the laser line separation is four times larger in Figures 2b, 3b, 3c,
6, 7, and multiple laser lines are present in Fig. 3c. Figures 6-7 incorporate the n2L(r, ω) =
n2 2 2 2L0 + (nL0− 1)r /RC gradient described in Section 2.2, and Figures 11-12 incorporate a lower
13
peak laser power, i.e., ∼ 0.1 MW (in order to remain below the nonlinear focusing power of
As2S3 for frequency doubled-light). These figures (Fig. 11-12) also incorporate a (smooth)
radial dip in the refractive linear index outside of a 500 µm radius which enables the beam to
maintain its spot size over many Rayleigh lengths. Figures 8 and 10 are simulated according to
the standard one-dimensional Nonlinear Schrodinger Equation with second order dispersion.
Figures 2-7, 9, and 11-12 are simulated according to the BTUL code, i.e. Eq. (6), although
Figure 9 is a 1D simulation (kz becomes k, and the kx and ky arguments are ignored). For
Figures 2-7 and 11-12, the backward-propagating field and backward-propagating polarization
have been neglected; these simulations are unidirectional, and the nonlinear polarization term
is the standard Kerr term, i.e. Eq. (5a). For Figure 9, which is bi-directional, the full
polarization is calculated according to the two lined equations in Section 7.1 of the Supplement.
Our 3D simulations reproduce the spectrums in Ref. [5] in the 1D limit, i.e., R0 →
∞ . However, when transverse effects are incorporated, self-focusing limits the propagation
distance, thus reducing the degree of spectral broadening. We terminate the propagation
when the on-axis intensity reaches roughly the breakdown intensity (∼ 1 GW/cm2). When an
initially diverging beam with the parameters in Table 1 propagates in GaAs and self focuses
until the intensity nears the breakdown threshold, a spectrum of 9 – 10 µm can be achieved, see
Fig. 2a, 3a. To obtain an 8 – 13 µm spectral bandwidth requires a line spacing of ∆λ/λ ∼ 0.04,
see Fig. 2b, 3b. Spectra presented here are spatially averaged, unless otherwise specified.
The spectrum obtained for a number of input conditions is shown in Fig. 3. The degree
of spectral broadening in the long-wavelength regime is maximized for ∆λ/λ0 = 0.04 . For
large line separations the fractional bandwidth δω/ω0 is no longer proportional to ∆λ/λ0 due
14
Figure 2: Self-focusing and spectral broadening of a beat wave propagating in GaAs, for the parame-
ters of Table 1. The initial laser line separation is, respectively, (a) ∆λ/λ0 = 0.01 , (b) ∆λ/λ0 = 0.04 .
In the latter case, the wavelength 10.6 µm has been used. The spot-size is determined by fitting the
transverse laser profile to a Gaussian profile. The spectrum is defined here as the range over which
there are individual frequency components containing at least 0.005% of the laser beam’s total power,
see Fig. 3.
to phase mismatch. In the normal dispersion regime (∂2(ωn(ω))/∂ω2 < 0 ), e.g., λ < 7 µm for
GaAs, dispersion effects broaden the laser pulse duration, thus reducing the rate of spectral
broadening. Spectral broadening in the normal dispersion regime is shown in Fig. 3c.
Figure 3: Spectrum of a beat wave propagating in GaAs, for the parameters in Table 1. The spectrum
is shown at a distance when the on-axis intensity nears the breakdown threshold. The input laser
line separation is, respectively, (a) ∆λ/λ0 = 0.01 , (b) ∆λ/λ0 = 0.04 , (c) ∆λ/λ0 = 0.08 . These
lines are comprised of 8 individual lines in the range from 9.15 µm – 9.95 µm , together with eight
lines in the range 4.575 µm – 4.975 µm, where each line contains 0.08 MW.
The propagation distance can be extended by operating near the nonlinear focusing power
PK . Propagation over several Rayleigh lengths (ZR = 4 cm ) is shown in Fig. 4. The resulting
spectral broadening is in good agreement with Eq. (11), see Fig. 5. The spectral broadening
in this case is larger than that obtained when operating above the nonlinear focusing power
15
(see Fig. 2a, 3a) because the pulse propagates longer distances before radial collapse.
Figure 4: Self-focusing and spectral broadening of a beat wave near the focusing power in GaAs.
The peak power here is P = 0.93 PK .
Figure 5: The total power spectrum, at z = 11.2 cm, is shown in (a) for a beat wave near the
focusing power. The spectrum (b) on-axis and (c) at the spot size R0 are shown for comparison. The
spectrum is broader along the axis because of the higher intensity on axis.
3.2 Spectral Broadening in an Index Tapered Medium
An alternative way to overcome the limitations due to self-focusing when P > PK is to operate
in a GaAs rod with a radially-graded refractive index. In the following example, a graded-
index rod in which the refractive index increases as a function of r2 is used to defocus the
beam, thereby mitigating the nonlinear self-focusing and extending the propagation distance,
see Fig. 6. A supercontinuum which extends beyond 5 – 14 µm can be obtained. We note
16
that each pulse is compressed tenfold, see Discussion. The spectral broadening in a graded
GaAs rod exceeds that in a uniform GaAs rod, see Fig. 7 and Fig. 3b.
Figure 6: Self-focusing and spectral broadening of a Gaussian-profile laser beat wave propagating
through a GaAs waveguide with a graded index. The laser line separation is ∆λ/λ0 = 0.04 , and the
peak power is given by P/PK = 5 .
Figure 7: Power spectrum for ∆λ/λ0 = 0.04 after propagating 1.5 cm in a GaAs rod with a graded
index characterized by RC = 70 R0 , see Fig. 6
3.3 Effect of Laser Phase Noise on Spectral Broadening
Given that high power CO2 lasers can be ’noisy’, i.e., have a substantial laser line width, see
Ref. [13], the effect of laser noise on spectral broadening is an important consideration for
enhancing spectral broadening in a dispersive, nonlinear medium.
We model the phase noise of the CO2 laser according to a phase diffusion model [14].
Laser noise effects are typically dominated by phase noise and not intensity noise [15]. In
17
the following, intensity laser noise is neglected. In a dispersive medium the phase noise can
result in intensity fluctuations [16-17]. In addition, if the medium has a Kerr nonlinearity
the fluctuations lead to additional spectral broadening via self–phase modulation and other
nonlinear processes, see Fig. 8.
Figure 8: Power per wavelength-regime, given a varying laser linewidth due to phase noise. The
total power in spectrum (a), (b), and (c) is the same. The spectrum is shown after propagating 2
cm in GaAs. Initially the CO2 radiation consisted of two lines each with linewidth (a) 0.00∆λ (no
noise) (b) 0.01∆λ (c) 0.10∆λ due to phase noise. The initial laser line separation is ∆λ/λ0 = 0.01.
In frequency space this corresponds to (a) 0 GHz (no noise) (b) 3 GHz (c) 30 GHz.
The area under the curves in each plot in Fig. 8 is equal. In Fig. 8a, 1 MW of power is
contained between the wavelengths 9.0 and 10.1 µm. In Fig. 8b, the same amount of power
is contained between 8.0 and 11 µm. Finally in Fig. 8c, 1 MW of power is contained between
7.7 to 11.5 µm. Note that in Fig. 8c the spectrum approaches a continuum.
For a high-power CO2 laser with laser linewidth ∼ 500 MHz, phase noise does not signif-
icantly affect the laser propagation, given the parameters in Table 1. The same is true for
intensity noise, which can be incorporated by instead modeling the laser noise as a collection
18
of N  1 single-frequency, uncorrelated emitters distributed according to the spectral line
shape, see Ref. [18]. Increasing the laser linewidth, however, can significantly enhance the
spectral broadening, see Fig. 8.
We note that when two substantial frequency lines amplify a third line from the background
via four wave mixing (FWM), the total amplification exp(g z) (where g is the FWM gain) is
so large that the magnitude and structure of the background noise have little effect on the
resulting spectral bandwidth. In the case of white noise distributed from ω0/2 to 2ω0 , for
example, this can be true for noise field magnitudes up to |δE/E| ∼ 1/10.
3.4 Enhancement of Spectral Broadening via Coupling to a Backward-
Propagating, Reflected Wave
A backward-propagating wave nonlinearly induces temporal fluctuations in the refractive index
of the medium. These fluctuations induce a phase shift in the forward-propagating wave which
can result in spectral broadening.
The BTUL bidirectional simulations are in agreement with the dispersion relation in Eq.
(12) for the modulation instability, and predict that coupling to a backward-propagating,
reflective wave can enhance the spectral broadening over substantial propagation distances
specifically in the anomalous dispersion regime of a nonlinear medium, see Fig. 9.
We note that the backwards-propagating wave cannot directly induce new frequencies in
a forward-propagating wave via four wave mixing due to a phase mismatch.
19
Figure 9: The power spectrum is shown at z = 3 cm for a forward-propagating beat-wave in GaAs
with the parameters in Table 1, with coupling to a backward-propagation reflected wave. This
simulation has been carried out in 1D for an initial laser line separation of ∆λ/λ0 = 0.02 and
demonstrates the usefulness of BTUL’s bi-directional capability as well as the ability of a backward-
propagating reflected wave to enhance spectral broadening in the anomalous dispersion regime of a
nonlinear medium. (a) The black dotted curve represents propagation given 29% reflectivity at the
end of the medium (i.e., at z = 3 cm), corresponding to reflection off a GaAs - Air boundary. The
red, solid curve corresponds to 0% reflectivity, i.e., unidirectional propagation. (b) The black dotted
curve represents propagation given 100% reflectivity at the end of the medium.
20
3.5 Second-Harmonic Spectral Broadening in As2S3 Fiber
Chalcogenide (As2S3) is a promising material for spectral broadening in the long-wave IR
regime. The benefits are (i) the significant nonlinear refractive index (∼1/6 that of GaAs), (ii)
the long propagation distances (∼8 m) attainable via low-loss As2S3 optical fibers [19], see Eq.
(11), and (iii) the small group velocity dispersion coefficient in the vicinity of its zero dispersion
wavelength at 4.8 µm. These benefits are apparent in 1D simulations which incorporate the
full dispersion relation of GaAs and As2S3, see Fig. 10. The fractional bandwidth δω/ω0 as
a function of propagation d(∫istance is est)im(a∫ted here via the third)moment in frequency space
µ3(z) = 〈(ω − ω )30 〉1/3
−1 1/3
= P (ω, z)dω (ω − ω )30 P (ω, z)dω , where ω0 = 2πc/λ0 is
the laser frequency and P (ω, z) is the power spectrum as a function of the propagation distance
z of a CO2 laser-beat wave through GaAs, As2S3, respectively.
Figure 10: 1D simulation of CO2 laser beat-wave propagating through (a) GaAs (b) As2S3 at various
intensities and wavelengths. (T∫he fraction)al b(an∫dwidth is estimated)here as the third moment in−1 1/3
frequency space, i.e. µ3(z) = P (ω, z)dω (ω − ω 30) P (ω, z)dω .
To simulate laser propagation through a chalcogenide (As2S3) fiber, we represent the
21
cladding as a ∆nL = 1.0 dip in the refractive index with a radius 2.5 R0 with its ramp
occurring over a transverse distance of 30 λ0. The divergence term of the field is incorporated,
see Eq. (2). We see that a 4 – 6 µm bandwidth can be achieved via the propagation of a
multi-line frequency-doubled CO2 beat-wave through a chalcogenide fiber at just one tenth of
the intensity and power given in Table 1, see Fig. 11 – 12.
Figure 11: Optical guiding, self-focusing, and spectral broadening of a Gaussian-profile frequency-
doubled CO2 beat-wave with 200 µm initial spot size, laser wavelength λ0 = 4.7 µm, and laser
line separation ∆λ/λ0 = 0.005 propagating through an As2S3 fiber. Although the laser operates
below the nonlinear focusing power P ∝ λ2K , extreme spectral broadening yields high frequency
components which are indeed above their respective nonlinear focusing power.
Figure 12: Power spectrum for ∆λ/λ0 = 0.005, where λ0 = 4.7 µm , after propagating 8 meters in
an As2S3 fiber, see Fig. 10.
22
4 Discussion
Carbon Dioxide lasers have tunable lines in the 9 – 11 µm range and can be frequency doubled.
Transversely excited atmospheric (TEA) CO2 lasers can provide average output powers of
tens of kilowatts together with repetition rates from kHz to as high as 100 kHz, and have long
operating lifetimes in excess of 104 hours [20]. Commercially available TEA CO2 lasers are
compact and relatively low cost per watt. They can have average power levels in the multi
tens of kW range and operating efficiencies as high as 30%.
4.1 Raman Scattering
Raman amplification of input noise by a CO2 laser will impede the spectral broadening in a
GaAs waveguide at propagation distances beyond a characteristic Raman amplification length
[21]. For a CO2 laser of average power P = 2.5 PK and spot size R0 = 0.02 cm we find that
the characteristic Raman amplification length is LCR ≈ 3 cm [22].
In the case of second-harmonic spectral broadening in a chalcogenide fiber, Brillouin scat-
tering may play an important role, depending on the laser bandwidth and pulse length. Further
studies are required.
The energy backscattered via other nonlinear and linear processes has been neglected here
in Part II, given the homogeneity of GaAs crystals, the availability of custom anti-reflection
coatings, and the relatively modest peak powers.
23
4.2 Laser Damage Threshold
Laser damage to the GaAs and the optics is an important consideration in the applications
of ultra-broadband IR radiation. A 3 – 8 MW/mm2 intensity threshold was measured at the
Hirst Research Centre for propagation through GaAs of a 50 mJ CO2 laser pulse with a third
of the pulse energy in a 60 ns pulse width [23]. In the thermal damage regime, the intensity
threshold is approximately independent of the pulse duration and inversely proportional to
the spot size [24]. The parameters of Table 1 indicate an intensity threshold of 0.4 – 1.5
GW/cm2. This is consistent with the finding in Ref. [2] that 0.7 GW/cm2 is safely below the
intensity threshold. The simulations here in Part II remain below 1 GW/cm2 on the timescale
of multiple micropulses (¿10 ps), and each micropulse is restricted to 10 GW/cm2.
Surface damage sets an energy density threshold, thereby limiting the allowed pulse du-
ration. The parameters given in Ref. [5] exceed this damage threshold for the proposed 200
ns CO2 TEA laser pulse, which has an energy density of 200 J/cm
2 and higher on-axis. An
energy density limit of 1 – 2 J/cm2 in the optics and coating was reported in Ref. [3]. For
high average power applications, it may be useful to experiment with higher energy densities
using an uncoated GaAs crystal and Cu/Ni/Au mirrors, see Fig. 9 and Ref. [25].
5 Conclusion
We have developed a three-dimensional ultra-broadband laser propagation code (BTUL) which
can treat coupled forward and backward propagating waves in a nonlinear, dispersive medium.
We have also demonstrated that 3D effects play an important role in the spectral broadening
24
process. One dimensional models of spectral broadening can overestimate the degree of broad-
ening by almost an order of magnitude. Our 3D simulations show that when operating near or
above the nonlinear focusing power, the fractional line width separation of ∆λ/λ0 ≈ 0.04 leads
to substantial broadening in GaAs. This linewidth separation appears to be near optimal given
the parameter range considered. We have also demonstrated that the use of a graded-index
or cladding around the nonlinear medium can extend the propagation range and therefore
the degree of spectral broadening. The beat-wave formed by two CO2 lines can generate a
spectrum extending beyond 5 – 14 µm via propagation through a graded-index GaAs rod,
and the beat-wave formed by two frequency-doubled CO2 lines can generate a spectrum of 4
– 6 µm via propagation through a As2S3 fiber. In addition, we have shown enhancement of
spectral broadening via coupling to a backward-propagating reflected wave in the anomalous
dispersion regime of a nonlinear medium. Finally, we have shown that laser phase noise in a
dispersive medium can be a significant contributor to spectral broadening.
6 References
[1] Corkum, P.B., Ho, P.P., Alfano, R.R. and Manassah, J.T., 1985. Generation of infrared
supercontinuum covering 3–14 µm in dielectrics and semiconductors. Optics letters, 10(12),
pp.624-626.
[2] Pigeon, J. J., S. Ya Tochitsky, C. Gong, and C. Joshi. ”Supercontinuum generation from 2
to 20 µm in GaAs pumped by picosecond CO 2 laser pulses.” Optics letters 39, no. 11 (2014):
3246-3249.
[3] Pigeon, J.J., Tochitsky, S.Y. and Joshi, C., 2015. High-power, mid-infrared, picosecond
25
pulses generated by compression of a CO 2 laser beat-wave in GaAs. Optics letters, 40(24),
pp.5730-5733.
[4] Pigeon, J., Tochitsky, S. and Joshi, C., 2016, March. Generation of broadband 10 µm
pulses using four-wave mixing compression in GaAs. In Mid-Infrared Coherent Sources (pp.
MT2C-7). Optical Society of America.
[5] Kapetanakos, C.A., Hafizi, B., Milchberg, H.M., Sprangle, P., Hubbard, R.F. and Ting,
A., 1999. Generation of high-average-power ultrabroad-band infrared pulses. IEEE journal of
quantum electronics, 35(4), pp.565-576.
[6] Kapetanakos, C.A., Hafizi, B., Sprangle, P., Hubbard, R.F. and Ting, A., 2001. Progress
in the development of a high average power ultra-broadband infrared radiation source. IEEE
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tion equation. Physical review letters, 89(28), p.283902.
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From Maxwell’s to unidirectional equations. Physical Review E, 70(3), p.036604.
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036706.
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laser pulses in ionizing gases and plasmas. IEEE Journal of Quantum Electronics, 33(11),
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pp.1879-1914.
[12] Sprangle, P., Penano, J.R. and Hafizi, B., 2002. Propagation of intense short laser pulses
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pp. 34-37.
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[17] Isaacs, J. Sprangle, P., “The Effect of Laser Noise on the Propagation of Laser Radiation
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27
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mirror-laser.html
7 Supplement
7.1 Supplement: Evaluation of Nonlinear Polarization Term
When a forward propagating field is coupled to a non-perturbative backward propagating field,
Eq. (4) must be generalized as follows: The nonlinear polarization field is given by
∫∫∫
P̃±NL(k , k( x y
, z, ω) = ε0χNL exp(iωt− ikxx− ikyy)dtdxdy×
( ± ) {( ) ( ) } )3 2 2 ±E (r, t) + 3 E+(r, t) E−(r, t) + E−(r, t) E+(r, t)
Evaluation of the right-hand term requires knowledge of the directionality of each com-
ponent of (E(r, t))3 . To directly determine the directionality would require appending the
28
( )
variable k = ± k2(ω)− k2 − 2 1/2z x ky to the (kx, ky, ω) vector space. This is inefficient and
generally exceeds the computational memory limit. Rather, it is appropriate to divide the for-
ward and backward going field into frequency regimes as a means of resolving directionality.
For example, in the case where δω/ω0 < 2/3 , we can divide the forward and backward going
field into two frequency regimes, i.e., ω > 0, ω < 0 . Denoting these regimes by a subscript,
i.e., E± ±ω<0, Eω>0 , the nonlinear polarization field P̃
±
NL can be written as:
P̃±
∫∫∫
NL(kx, ky, z, ω)= ε0χNL exp(iωt− ikxx− ikyy)dtdxdy×  (E± 3 ( (r, t)) + 6E
∓
ω>0(r, t)E
∓
ω<0(r,)t)E
±(r, t)+
( )2 ( ) 2
3 E± (r, t) + E± (r, t) E∓ω>0 ω<0 (r, t)
To derive this relation, each field component Cω′,k′ exp(ik
′z − iω′t) is represented as the
v∫e∫ctor (ω
′, k′) in ω−k space, where the field is general and can decomposed as E(r, t) =
C ′ ′ ′ ′ ′ω ,k′ exp(ik z − iω t)dω dk . For example, a field propagating in the forward direction
contains components in the (+,+) and/or (−,−) quadrant of ω−k space. Each component
of (E(r, t))3 represents the combination of 3 fields, i.e. the sum of 3 vectors in ω−k space.
7.2 Supplement: Numerical Techniques
With the exception of Section 3.4, the simulations only involve unidirectional propagation;
BTUL then reduces to a 3D UPPE which takes the form of an ODE. It is solved in 3D Fourier
space via iterative determination of the field’s slope along the longitudinal axis. The nonlinear
polarization density at each step along the longitudinal axis is calculated in the time domain.
29
Negative frequency components are included in the analysis, and the sign of kz in Section 2.1
is defined accordingly, i.e., kz(kx, ky, z,−ω) = −kz(kx, ky, z, ω) for ω > 0 .
Regarding the additional terms which are absent in a traditional UPPE: The divergence
term can be numerically integrated beside the nonlinear polarization density term. In the case
of a spatially varying refractive index, the additional term below Eq.(6) is incorporated at each
step ∆z in spatial coordinates via the operator exp (i δk2(x, y, z, ω)E(x, y, z, ω) ∆z / 2k0) .
In all simulations with the exception of Section 3.3, periodic temporal boundary conditions
are employed. Hence, a pulse duration has been specified in Table 2 which well exceeds the
dispersion-induced walk-off time.
For bi-directional propagation, in summary - iteration from forward propagation to back-
ward propagation to forward propagation etc. converges to a stable solution, which represents
the middle of a long pulse which contains within it periodic behavior. For each iteration, the
electric field at each point along the propagation axis is stored in an array in order to be
incorporated into the nonlinear polarization expression given in the Supplement.
With the exception of Fig. 8-10, all simulations are carried out in three spatial di-
mensions. To analyze the transverse beam profile of the propagating beam, we define an
‘effective spot size’ REff(z) and ‘effective on-axis intensity’ IEff(z) as the parameters deter-
mined by fitting t(he transverse beam)profile I(x, y) to a Gaussian surface G(x, y) given by
G(x, y) = I exp −2x2/R2 − 2y2/R2 , R = (R2 +R2)1/2Eff x y Eff x y . Consistency between BTUL
and the 3D Nonlinear Schrodinger Equation (NLSE) has been benchmarked.
Guard bands are placed in transverse space and in transverse spatial frequency space. The
guard bands absorb all energy beyond some spatial bound and spatial frequency threshold,
30
thereby avoiding aliasing across the periodic boundary imposed by the Fast Fourier Transform
operation. Grid resolution requirements in both vector spaces are also imposed. The dotted
line in Fig. 13a indicates the resolution limit REff(z) = 3∆xgrid below which the beam may
be insufficiently resolved in transverse space, where ∆xgrid denotes the transverse distance
between grid points.
As a laser beam self-focuses towards collapse, transverse spatial frequencies above a spec-
ified threshold will be generated and then absorbed by the guard bands in order to avoid
aliasing. When significant guard band absorption occurs, the BTUL code terminates the
propagation in order to avoid nonphysical results associated with the exclusion of sharply
converging field components.
As a simple benchmark, BTUL is used to reproduce the power law for a monochromatic,
perfect Gaussian beam , see Fig. 13a and 13b, where the self-focusing distance is defined
as the collapse distance, i.e., limz→z REff(z)/R0 = 0 . The extrapolated collapse distance isSF
compared, respectively, to the numerical result zSF(P,PK, R0) [9].
31
Figure 13: (a) The effective spot size of a Gaussian-profile, monochromatic laser beam of initial spot
size R0 = 200 µm is plotted as a function of propagation distance through GaAs for various input
powers ‘P’ which are above the nonlinear focusing power, i.e. P > PK = 2π c
2/ (ω20 n0 n2) =
0.26 MW. The collapse distance is extrapolated and (b) plotted as a function of input power in order
to show consistency with an approximate numerical formula [9] for the collapse distance.
Part III
Spectral Broadening of the NIKE KrF Laser
in a Negative Nonlinear Index Medium
In inertial confinement (ICF) experiments at the NIKE [1] laser facility, the high-power kryp-
ton fluoride (KrF) laser output beams propagate through long (∼75m) air paths to achieve
angular multiplexing. This is required because the KrF medium does not store energy for a
sufficiently long time. Recent experiments and simulations have shown that, via stimulated
rotational Raman scattering (SRRS), this propagation can spectrally broaden the laser beam
well beyond the ∼1 THz laser linewidth normally achieved by the induced spatial incoherence
(ISI) technique used in NIKE [2]. These enhanced bandwidths may be enough to suppress the
laser-plasma instabilities which limit the maximum intensity that can be incident on the ICF
32
target. We investigate an alternative technique that achieves spectral broadening by self-phase
modulation in Xe gas, which has a large, negative nonlinear refractive index ∼ 248 nm [3],
and thus completely avoids transverse filamentation issues. The collective, nonlinear atomic
response to the chaotic, non-steady state ISI light is modeled using a two-photon vector model.
The effect of near-resonant behavior on the spectral broadening is also studied.
1 Introduction
A primary challenge in inertial confinement fusion is the growth of hydrodynamic instabili-
ties and laser plasma instabilities (LPI) [4]. Increasing the laser spectral bandwidth can be
an effective way of reducing the growth rates of LPI instabilities [1]. Recent experiments on
the NIKE KrF laser, which uses a 75 m propagation bay for beam multiplexing, have shown
that stimulated rotational Raman scattering (SRRS) in the air paths can spectrally broaden
the bandwidth of the chaotic, incoherent light well beyond the ∼ 1 THz normally allowed by
NIKE’s induced spatial incoherence (ISI) technique. While SRRS is small in normal NIKE
operation at moderate beam intensities (∼ 50 MW/cm2), these experiments generated signif-
icant SRRS by imposing a ∼ 150 MW/cm2, 400 ps spike on the pulse and folding the beam
to lengthen the optical air path. The amount of spectral broadening that was achieved, how-
ever, was limited to several THz. Alternatively, to propagate in an inert gas requires a large
nonlinear refractive index n2 ∼ 60 × 10−19cm2/W - five times that of air - for comparable
spectral broadening. Though this would mean self-filamentation for a Gaussian beam (where
the nonlinear focusing power at 248nm in air is 100 MW), the important parameter for ISI
light is the power within a single coherence zone; for a 75 times diffraction limited beam, the
33
nonlinear focusing power in air becomes ∼ 500 GW.
For linearly polarized light, there exists a two-photon resonance in atomic xenon (Xe) at
249.6 nm which has been discussed in the literature [3,5]. As a result of this resonance, Xe
has a large, negative nonlinear index n2 at the 248.4nm KrF laser wavelength whose doubled
frequency lies only 11.9 THz above the resonance, see Fig. 14-15. Hence, propagation through
Xe gas has the potential to substantially increase the spectral bandwidth of the KrF laser
without the problem of self-filamentation, which is a positive-n2 effect. The negative nonlinear
index near-resonance has been observed and accurately calculated from known Xe electric
dipole matrix elements for narrowband KrF light operating under steady-state conditions
[3,6].
Figure 14: The nonlinear optical response of Xe near its two-photon 5p6 → 6p[1/2]0 resonance is
that of a three-level system with multiple intermediate states |2〉, and allowed electronic transitions
from |1〉 → |2〉 and |2〉 → |3〉.
At the NIKE facility, however, the KrF laser uses incoherent beam smoothing techniques
that produce chaotic variations on a ps timescale. The resulting multi THz spectral band-
widths of interest can thus include frequencies that are much closer to the two-photon reso-
nance than the 11.9 THz detuning of the narrowband KrF light. As a result, the nonlinear
Kerr response can experience time delays and the upper level may become partially populated
34
Figure 15: Wavelength-dependence of the nonlinear refractive index n2 of 200mbar Xe in the regime
of the two-photon resonance. The curve is continuous but extends beyond the range shown here.
during the pulse.
We model the nonlinear response of Xe gas to chaotic non-steady-state, near-resonance
light using a two-photon vector model. Although this model includes small collisional damping
coefficients, they play a negligible role in spectral broadening at Xe pressures up to at least
200mbar. At 200mbar, nonlinear effects and linear attenuation due to the small (0.02%)
dimerization of the Xe molecules are expected to be minimal, but further investigation is
needed, see Ref. [3,15]. Earlier work on two-photon vector models can be found in Ref. [7-8].
Though the laser propagation in a nonlinear medium yields spectral broadening (which is
useful for the reduction of hydrodynamic and laser plasma instabilities in inertial confinement
fusion) the propagation also results in far-field broadening. The resulting degradation of the
laser beam profile is studied.
A detailed overview of the NIKE laser facility and its KrF laser can be found in Ref.
[1]. The NIKE laser facility uses echelon-free induced spatial incoherence (ISI) to produce a
laser beam that is spatially and temporally incoherent [9-10]. The beam’s speckle fluctua-
35
tions occur over rapid (picosecond) timescales such that they average out over hydrodynamic
timescales to yield a spatially and temporally uniform illumination. This, in turn, minimizes
the hydrodynamic instabilities that would otherwise be strongly seeded by laser beam spatial
nonuniformities.
In Section 2, we present the Nonlinear Schrodinger Equation that is used to model laser
propagation through a nonlinear medium, discuss how to model the directed spatially and
temporally incoherent KrF radiation, derive a model for the nonlinear response of the Xe
gas, and describe the numerical techniques. In Section 3, we present simulation results. In
Section 4, we discuss the results and propose a new direction for the inertial confinement
fusion experiments at the NIKE laser facility. In Section 5, we summarize our results.
2 Model
2.1 Laser Pulse Propagation - Generalized Nonlinear Schrodinger
Equation
In the absence of free charges, the electric field is given by the wave equation with a nonlinear
source term, i.e.
∇2E(r, t) = (∂/∂t)2E(r, t)/c2 + µ0(∂/∂t)2 (PL(r, t) + PNL(r, t)) ,
where PL(r, t) and PNL(r, t) are respectively the linear and nonlinear polarization of the prop-
agation medium. The electric field is taken to be linearly polarized along the x-axis, and the
36
divergence term has been neglected since the beam’s spatial fluctuations occur over distances
much larger than the wavelength.
Because (a) the nonlinear changes in the refractive index are small relative to the linear
changes in the refractive index and (b) the KrF laser’s fractional bandwidth ∆ω/ω0  1, we
can make a slowly varying envelope approximation and express the wave equation as a three-
dimensional generalization of the Nonlinear Schrodinger Equation (NLSE) [11]. The envelope
equation is given by
( )
(∂/∂η)Ep(r, τ) =i∇2⊥Ep(r, τ)/2k0 + −iβ2(∂/∂τ)
2Ep(r, τ)/2 + ...
( ) (1)
+ i ω2 20/2ε0c k0 (1 + i(∂/∂τ)/ω
2
0) PNL,p(r, τ) ,
where η = z and τ = t − z/v are the longitudinal and temporal variables in a frame moving
at the group velocity v, k0 is the wavenumber at the carrier frequency ω0 = 2πc/λ0, and
β2 is the group velocity dispersion coefficient at ω0. The electric field is given by E(r, t) =
(1/2)Ep(r, t) exp(ik0z − iω0t)+(1/2)E∗p(r, t) exp(−ik0z + iω0t), and the nonlinear polarization
field (excluding higher harmonics) is given by
PNL(r, t) = (1/2)P
∗
NL,p(r, t) exp(ik0z − iω0t) + (1/2)PNL,p(r, t) exp(−ik0z + iω0t). The wave
equation given in Eq. (1) can be solved numerically via the split-step method, see Ref. [12],
and is driven by the nonlinear polarization of the Xe gas, which is derived in Section 2.3 .
2.2 Spatial and Temporal Incoherence
The front-end of the NIKE KrF light source is a spatially and temporally incoherent ASE
(amplified spontaneous emission) oscillator, which can be represented by a collection of many
37
independent oscillators emitting at frequencies in the range of the carrier frequency, and which
are scattered throughout space in such a way that the spatial frequency distribution is roughly
symmetric about zero. Numerically, this is represented by a 3D function determined according
to the average output of the ASE, which we model here as Gaussian in frequency and spatial
frequency space and centered about (ω0, 0, 0). Each amplitude Aω,kx,ky ( corresponding to
the sum of a large collection of independent oscillators at (ω, kx, ky)) is thus a G∣ aussian∣-2
distributed complex random number with random phase. The amplitude squared ∣A ∣ω,kx,ky
is then multiplied by the incident power spectrum. The resulting function Aω,kx,ky is Fourier
transformed into time and space, resulting in the function G(t, x, y). The function G has been
normalized according to the power of the KrF beam and multiplied by a top-hat shaped filter
function, which corresponds to the beam aperture.
The ASE light, which is modeled as a quasi-collimated multimode beam, traverses an
aperture whose width D is ∼ 75 times larger than the spatial coherence zones determined
by the (kx, ky) angular divergence. It thus takes on the structure of a 75x75 transverse array
of uncorrelated beams, each of which is mostly uncorrelated with itself after every coherence
time τc = 2π/∆ω, where ∆ω is the spectral bandwidth. Each uncorrelated beam is produced
by a varying number of random oscillators, so that the structure can be viewed as a three-
dimensional, checkered array of random Gaussian numbers with random phase (labeled as
F (t, x, y)) which when integrated over many coherence times yields a flat-top beam profile.
Nonetheless, it will suffice to model the field envelope as E0 × G(t, x, y), see Ref. [2]. A
measurement of the initial beam profile or the far field profile can be incorporated for more
accurate results.
38
2.3 Nonlinear Response of Xe Gas
In this section, we derive the nonlinear polarization of the Xe gas, which is the source term of
the wave propagation equation, see Eq. (1).
2.3.1 Density matrix equations
The electric response of the Xe atom is modeled by the Hamiltonian Ĥ(r, t) = Ĥ0(r, t)+V̂ (r, t),
where Ĥ0(r, t) represents the unperturbed atom, V̂ (r, t) = −µ(r) · E(r, t) = −qr · E(r, t)
represents the electric dipole interaction, µ is the dipole moment operator, and E(r, t) =
|Ep(r, t)| cos(φ(r, t)) = (1/2)Ep(r, t) exp(ik0z − iω0t)+c.c. is the classical field associated with
the KrF laser light. Although the following equations can produce a nonzero third harmonic
polarization, the third harmonic electric field is neglected here. This is justified because the
large phase mismatch created by dispersion and cross-phase modulation strongly suppresses
the convective growth of the third-harmonic field at this far UV wavelength, which is also
strongly absorbed by photoionization. The slowly-varying amplitude |Ep(r, t)| and the phase
φ(r, t) are real, with the instantaneous frequency given by ω(t) = φ̇(t) = ω0 + δω(t), where
ω0 = 2πc/λ0 and λ0 = 248.4nm. If the KrF light is linearly polarized, a two-photon near-
resonance with the Xe 6p[1/2]0 state at 80119cm−1 (2 / 249.63nm) result∑s in a negative
nonlinear refractive index around λ0. We write the wave function as Ψ = cn(r, t)un(R)
n
where un(R) is the eigenfunction corresponding to the eigenstate |n〉 of H0 and R is the position
of the outer electron with respect to the nucleus. Although multiple odd-parity intermediate
eigenstates contribute to the two-photon transition, we model a three-level system here for
simplicity, where |1〉 is the 5p6 ground state, |2〉 represents the odd-parity intermediate states,
39
and |3〉 is the Xe 6p[1/2]0 two-photon near-resonant state. For the closed, three-level system,
the von Neumann equation dρ/dt = [Ĥ, ρ]/i~ yields the following density matrix equations:
∂ρ11/∂t =i (Ω12ρ21 − Ω21ρ12) + ΓI(ρ22 − ρeq22) +RΓc(ρ33 − ρ
eq
33),
∂ρ22/∂t =− ΓI(ρ22 − ρeq22) + i (Ω
eq
21ρ12 − Ω12ρ21) + i (Ω23ρ32 − Ω32ρ23) + (1−R)Γc(ρ33 − ρ33),
∂ρ33/∂t =− Γc(ρ eq33 − ρ33) + i (Ω32ρ23 − Ω23ρ32) ,
∂ρ12/∂t =− γNρ12 + i ω21ρ12 + iΩ12 (ρ22 − ρ11)− iΩ̄32ρ13,
∂ρ13/∂t =− γcρ13 + i ω31ρ13 + i (Ω12ρ23 − Ω23ρ12),
∂ρ23/∂t =− γLρ23 + i ω32ρ23 + iΩ23 (ρ33 − ρ22) + iΩ21ρ13 ,
(2)
where the density matrix elements are defined as ρnm(r, t) = 〈n|ρ(r, t)|m〉 = c∗m(r, t)cn(r, t)
and where ωmn = ωm − ωn . The transition frequencies are defined as
Ωmn(r, t) = 〈m| µ̂ |n〉 · E(r, t)/~ = Ω+ (r, t) exp(iφ(r, t)) + Ω−mn mn(r, t) exp(−iφ(r, t)), where
Ω±mn(r, t) = µmn · |Ep(r, t)|/2~ varies slowly in time and space and ± is a carrier frequency
marker; for linear polarization we take µnm to be real and express Ω
+
mn(r, t) → Ωmn(r, t) =
µmn|Ep(r, t)|/2~. Equation (2) applies to a general three-level system with ground state |1〉,
intermediate state |2〉, and excited state |3〉. The spatial symmetry of the Hamiltonian requires
that Ωnn = 0 for each eigenstate |n〉, and we have specified the allowed transitions to be
|1〉 → |2〉 → |3〉. The dynamics of inelastic collisions and spontaneous emission in the system
are general, where R satisfies 0 ≤ R ≤ 1 and describes the fraction of the energy emitted from
40
|3〉 that transfers directly to |1〉.
2.3.2 Parameters
For a 248.4nm laser pulse propagating through an atomic Xe gas, the two-photon excited state
is the Xe 6p[1/2]0 state. For a KrF NIKE pulse, we have 600ps short pulses and 4ns long
pulses, which yield a frequency scale 1.7 GHz and 0.3 GHz, respectively. The elastic collision
rate at 200mbar is 0.5 GHz, and the radiative decay rate of the excited Xe state is 0.03 GHz,
see Supplement.
2.3.3 Separation of density matrix elements into harmonics of the laser frequency
To simplify the notation, the spatial dependence of the field and matrix elements will be
dropped here; however, it can be reincorporated in a straightforward manner. We write each
element as the sum of components at roughly the various harmonics 0, ± φ̇(t), ±2φ̇(t), ±3φ̇(t)
of the laser frequency:
ρij = σij(0) + σij(−φ) exp(−iφ) + σij(φ) exp(iφ) + σij(−2φ) exp(−2iφ) + σij(2φ) exp(2iφ)
+σij(−3φ) exp(−3iφ) + σij(3φ) exp(3iφ),
where σij(−sφ) = [σji(sφ)]∗. The density matrix equations are then rewritten according to
Eq. (S1) for integer s ∈ (−3,−2, ..., 3). We calculate the full nonlinear response to the first-
harmonic field here, but will ultimately only include the first-harmonic polarization component
in the analysis.
41
2.3.4 Two-photon vector model
For an ISI beam with temporal fluctuations that are much longer than the timescale associated
with the detuning, e.g. 2π/∆ ∼ 0.2ps where ∆ = (2ω0−ω31)/2, the off-diagonal density matrix
elements are only driven significantly near the various harmonics of the laser frequency, see
Supplement and Eq. (S3). In the case where damping terms can be neglected, we define a
real vector r = (r1, r2, r3) according to r1 = σ13(2φ) + [σ13(2φ)]
∗, r2 = i(σ13(2φ)− [σ13(2φ)]∗),
r3 = σ33(0φ) − σ11(0φ) , and obtain the familiar (see Ref. [7-8]) two-photon vector model
shown below. See Eq. (9) for the formulation that is used for our simulations.
ṙ1 = −γ3r2 ,
ṙ2 = γ (3)3r1 − γ1(r3 +O(∆/ω0)) ≈ γ3r1 − γ1r3 ,
ṙ3 = γ1r2 ,
where ( ) ( )
1 1 2 Ω Ω µ µ |E |212 23 12 23 p
γ1 = Ω12Ω23 − ≈ = , γ3 = −(ω( − − ) (− ~ − ) 31
− 2φ̇+ δω31),
ω21 φ̇ ω 232 φ̇ ω21 φ̇ 2 (ω21 φ̇)
1 1 1 1
δω 2 231 = Ω12 ( + + Ω− − 23 +φ̇ ω32 3φ̇ ω32 ) φ̇− ω21 3φ̇−(ω21 )
| |2 1 1 | 1 1= µ12Ep/2~ + + µ23Ep/2~|2 + .
φ̇− ω32 3φ̇− ω32 φ̇− ω21 3φ̇− ω21
2.3.5 Two-photon adiabatic following approximation
When the phase and amplitude of the field envelope vary slowly with respect to the magnitude
of the two-photon rotation vector γ, e.g., |∂v(t)/∂t|/|v(t)|  |γ3(t)|, where v(t) = E2p(t)r3(t),
Eq. (2) can be solved approximately via an adiabatic following approximation [13], and the
42
collective state of the Xe atoms can be expressed as:
r1 = ±γ /(γ2 2 1/21 1 + γ3)
r 2 2 1/2 (4)3 = ±γ3/(γ1 + γ3)
γ1γ̇3 − γ3γ̇1
r2 = ṙ3/γ1 = ± 3/2
(γ2 21 + γ3)
Given the initial condition r = (0, 0,−1) and positive γ3 , the lower sign is appropriate.
The temporal incoherence of the NIKE laser induces large enough field fluctuations that
the adiabatic following approximation condition is violated; however, the approximation is
nonetheless a useful comparison. We further note that the adiabatic following approximation
yields an effective modulation-instability gain that is asymmetric about the pump frequency.
This is not, however, the cause of the small blue shift which appears in the n2(φ̇) nonlinear
response simulations, see Section 3.
2.3.6 Polarization
The total polarization is given by P(r, t) = N 〈µ̂〉 = NTr(ρµ̂) = N (〈2| µ̂ |1〉 ρ12 + 〈3| µ̂ |2〉 ρ23 + c.c.)
whose component along the linearly-polarized real field of magnitudeE(r, t) = |Ep(r, t)| cos(φ(r, t))
can be written as
P = N (µ12σ12(φ) exp(iφ) + µ12σ12(3φ) exp(3iφ) + µ23σ23(φ) exp(iφ) + µ23σ23(3φ) exp(3iφ)) +c.c.
The third-harmonic polarization component is included for completeness in this expression,
but is neglected in the remainder of the discussion and in the simulations.
43
The envelopes σ12(nφ) and σ23(nφ) are w∣ ell appro∣ximat∣ed by the∣ir adiabatic solutions in∣ ∣ ∣ ∣
Eq. (S2b) because the respective detunings ∣ω21 − nφ̇∣ and ∣ω32 − nφ̇∣ that appear in σ̇12(nφ)
and σ̇23(nφ) (see Eq. (S1)) far exceed the damping terms γN and γL or any spectral components
in σ12(nφ) and σ23(nφ). Applying these results, ignoring the small damping terms γN,L, using
Ω±mn = µmn |Ep| /2~, then writing σ33(0φ) ' (1 + r3)/2, σ11(0φ) ' (1 − r3)/2, σ13(2φ) =
(r1 − ir2)/2, we obtain the polarization associated with the two-photon vector model:
[ ]
P/N ' −µ21(φ̇− ω )−121 Ω+12σ11(0φ) + Ω−32σ13(2φ) exp(iφ)
, (5a)
+µ (φ̇− ω )−
[
1 Ω+ σ −
]
32 32 23 33(0φ) + Ω21σ13(2φ) exp(iφ) + c. c. ,
which becomes
( )
P/N ' −(2~)−1 (µ
2
12(φ̇− ω21)−1(1− r )− µ23 )23(φ̇− ω
−1
32) (1 + r3) |Ep| cos(φ)
(5b)
−(2~)−1µ12µ23 (φ̇− ω −121) − (φ̇− ω −132) |Ep| (r1 cos(φ) + r2 sin(φ)) ,
where we have defined the matrix elements µ12 = µ21 and µ23 = µ32 to be real, and ignored all
even harmonic terms except σ13(2φ). Using the two-photon adiabatic following approximation
of Eq. (4) and neglecting the small damping term r2 ∝ γ̇1 , γ̇3 , we obtain
P/N ' −(2~)−1(γ2 2 −1/21 + γ3) × µ2 (φ̇− ω )−1((γ2 + γ2)1/2 12 21( 1 3 + γ3)− µ
2
2)3(φ̇− ω )
−1((γ232 1 + γ
2 1/2
3) − γ3) |Ep| cos(φ) .
− − −

1
µ12µ23γ1 (φ̇ ω21) −
−1
(φ̇− ω32)
(5c)
44
In the small field limit where γ3  γ1, this reduces to
( ( )2( ))
µ2 2 2
P/N ≈ − 12 − | 2µE 12µ23 1 1 1
− ~ p
| × − |Ep| cos(φ) .
(φ̇ ω ) 8~321 φ̇− ω21 φ̇− ω32 2φ̇− ω31
(5d)
The polarization has the usual linear, DC terms as well as the nonlinear terms at the laser
frequency. We can obtain Eq. (5d) for the nonlinear polarization alternatively by expanding
the density matrix elements in powers of the field amplitude, see Ref. [6].
From (5d), we obtain the following expression for the nonlinear refractive index:
Nµ212µ
2
ñ 232 (ω0) = (6)
4ε0n 30~ (ω31 − 2ω 20) (ω21 − ω0)
and the nonlinear polarization is given by
( ( ) )
PNL,p(t) = 2ε0n0ñ2 φ̇(t) |Ep(t)|2 Ep(t), (7)
where E(t) = (1/2)Ep(t) exp(−iω0t) + c.c. = |Ep(t)| cos(φ(t)), and
PNL(t) = (1/2)PNL,p(t) exp(−iω0t) + c.c. .
This result is obtained from the first-harmonic term of the nonlinear polarization and
is consistent with the nonlinear index derived in Ref(. [3,17], ac〈cordi〉n)g to ñ2(= (1/2)ñ2,alt =2 )2
(1/8π)ñ 2 22,alt,cgs where ñ2,alt is defined accordin〈g to ε〉= n0 + ñ2,alt Ep −1 = n0 + 2ñ2 |Ep| −
1 where the time-averaged field magnitude E 2p = |Ep|2 /2 and the subscript ’cgs’ refers to
cgs units. We note that it has been derived in the small field limit under the two-photon
adiabatic following approximation.
45
In the more general case, i.e. Eq. (5b), the nonlinear polarization
PNL(t) = (1/2)PNL,p(t) exp(−iω0t) + c.c. can be expressed as shown below (see Supplement
for a discussion of the additional intermediate levels):
 µ12(ω21 − φ̇− iγ )
−1
N (µ23ρZ − µ12(1 + r3)) 
PNL,p(t)/Ep(t) = (N/2~

)  ,
−µ23(ω32 − φ̇− i γ −1L) (µ23(1 + r3) + µ12ρZ)
where ρZ = 2σ13(2φ)
∗ = (r1 + ir2) , and Eq. (3) gives
dρZ/dt = iγ3ρZ − iγ1r3,
dr3/dt = γ1r2 = iγ1(ρ
∗
Z − ρZ)/2 = iγ ∗1ρZ/2 + cc,
1 1 µ12µ23
γ1 = Ω12Ω23( − ) = |Ep/2|2
1 1
( − ) , and
ω21 − φ̇ ω32 − φ̇ ~2 ω21 − φ̇ ω32 − φ̇
γ3 = −(ω31 − 2φ̇+ δω31) ≈ −(ω31 − 2φ̇).
We note that the (1− r3) term has been rewritten as 2− (1 + r3). The first term represents
the contribution to the linear index of refraction (approximately δn0 ∼ 5×10−4) and has been
eliminated in order to yield the expression for PNL,p(t) .
46
Alternatively, we can write the nonlinear polarization as
 
µ12(ω21 − φ̇− iγN)
−1(µ23ρL)  E∗p(t)
PNL,p(t)/Ep(t) = (N/2~)  Ep(t)
 −µ −123(ω32 − φ̇− i γL) (µ 
12ρL)
(8)
µ12(ω21 − φ̇− iγ )
−1
N (−µ12(1 + r3)) 
+(N/2~)  ,
−µ23(ω32 − φ̇− i γ )−1L (µ23(1 + r3))
where ρL = (r1 + ir2) exp(2iω0t− 2iφ(t)), and
dρ /dt = iγ′L 3ρL − iγ′1r3,
dr /dt = iγ′ ρ∗3 1 L/2 + c.c. = Im[(γ
′ ∗
1) ρL]
(9)
′ µ12µ23γ = (E /2)2
1 1
1 p ( − ) , and~2 ω21 − ω0 ω32 − ω0
γ′3 = −(ω31 − 2ω0 + δω31) ≈ −(ω31 − 2ω0).
This latter form is preferable from a numerical standpoint because the instantaneous fre-
quency φ̇ of a temporally incoherent laser tends to fluctuate wildly over short timescales, i.e.,
a much higher temporal resolution is required to resolve ρZ than to resolve ρL.
In the case of negligible population redistribution, Eq. (9) can be expressed by a single,
driven harmonic oscillator equation given by
(∂/∂t− iγ′3) ρL = iγ′1 (10)
47
and solved exactly, i.e.,
( ∫ ) ( )t ∫t t
ρ ′ ′ ′ ′′ ′ ′′
∫ ′
L(t) = ρL(t0) exp i γ3(t )dt dt iγ1(t ) exp i γ3(t
′)dt′ .
t0 t0 t′′
This solution can be evaluated more simply in frequency space, i.e., ρ̃ ′ ′L = iγ̃1/(iω − iγ3).
2.4 Numerical Techniques
The generalized Nonlinear Schrodinger Equation, see Eq. (1), is integrated along the propa-
gation axis via a split-step method, where the diffraction term has been incorporated into the
dispersive propagator and where calculation of the nonlinear propagator is parallelizable.
The nonlinear polarization of Xe gas according to the two-photon vector model can be
calculated most generally via numerical time integration of the coupled first order equations
dr3/dt and dρL/dt in Eq. (9) according to a straightforward iterative scheme - or in the case of
negligible population redistribution Eq. (10). In the latter case, we opt for the straightforward
frequency domain solution discussed above, which is an order of magnitude faster. Doing so
requires that a small imaginary component be added to γ′3 to prevent divergent behavior.
The instantaneous frequency at time t is not required in our formulation of the two-photon
vector model, see Eq. (8), but it can be calculated by taking the derivative of the phase of the
electric field envelope having accounted for 2π phase jumps. Alternatively, it can be calculated
by locating the spectral peak of the Fourier Transform of the product of the field envelope and
a narrow Gaussian centered about time t. The instantaneous frequency is used to calculate
the nonlinear source term for the n2(φ̇) response model given in Eq. (6-7).
48
The input condition for the generalized Nonlinear Schrodinger Equation is discussed in
Section 2.2. The filter function generating the top-hat beam shape is constructed from two
hyperbolic tangent functions.
Guard bands are employed to absorb all energy beyond some frequency threshold, thereby
avoiding aliasing across the periodic boundary imposed by the Fast Fourier Transform oper-
ation as well as eliminating any high-frequency artifacts which arise from the numerical inte-
gration of the coherence term. Any error (especially for a longe∫r pu∣ lse) in th∣e numerical inte-T/2 ∣ ∣2
gration can slightly violate the energy conservation condition ∣Ê(t, x, y)∣ r2(t, x, y)dt = 0
−T/2
which is valid in the limit that ΓcT  1 . To address this concern, r2 is shifted accord-
ingly upward or downward by a very small, smooth plateau of the pulse width T after each
integration step. The necessary resolution along the propagation axis can be reduced by split-
ting the growth∑/de(c√ay part nonline)ar propagator into smaller, energy-conserving steps, i.e.2n
exp(αr2 ∆z) ≈ 1 + αr2 ∆z/n . Grid resolution requirements in both vector spaces
n
are imposed as well.
3 Numerical Results
The propagation of a temporally and spatially incoherent KrF laser beam through a chamber
of 200mbar Xe gas is simulated in this section. The simulation parameters are listed in Table
3 and correspond to the incident beam in Ref. [2]. The times diffraction limit is varied in
Figure 17.
Fig. 16-17 are simulated according to Eq. (1). The nonlinear polarization for the TPVM in
Fig. 16 is determined by solving Eq. (10) in frequency space; to determine it according to Eq.
49
Table 2: Simulation parameters for Fig. 16-17
KrF Beam/Xe Property Value
Laser wavelength λ0 = 9.55 µm
Peak power 33 GW
Beam size 15 cm × 15 cm
Transverse coherence length 0.5 cm (i.e. times-diffraction-limit M2x = 30)
Pulse length >10ps
Pulse temporal shape Gaussian
Pulse transverse spatial shape Flat-top beam with smoothed edges
Linewidth 1 THz
Detuning of 2ω0 from two-photon resonance 11.9 THz
aNonlinear refractive index at ω0 -190 × 10−19 cm2/W
Group velocity dispersion coefficient 8.2 × 10−6 ps2/cm
aRef. [5]. This value may be a 15-20% overestimate, see Ref. [3]
(9) yields the same result because population inversion is small. The nonlinear polarization
in the second model in Fig. 16 is determined according to Eq. (6-7), and the nonlinear
polarization in the third model in Fig. 16 is the instantaneous Kerr nonlinearity, also referred
to here as the steady-state solution. Each data point in Fig. 17 is simulated also according
to Eq. (1), for its own, noise-generated incident beam with the specified times-diffraction-
limit. To clarify the physical behavior, the nonlinear polarization used in Fig. 17 has been
determined according to the steady-state as opposed to Eq. (6-7); the trends and conclusions,
however, are the same.
We obtain the fo∫llowing results, in which the coherence time is estimated according to
the expressi∫on tc = (P (ν))
2dν where P (ν) is the normalized total power spectrum of the
beam, i.e., P (ν)dν = 1. This enables the calculation of a spectral broadening factor de-
fined as SBF = tc(0)/tc(z) , where z is the propagation distance. Similarly, to quantify the
50
broadening of the far-field profile (whose shape is identical to the near-field transverse spa-
tial frequency spectrum), we substitute it into the 2D spatial version of the tc expression to
calculate the near-field transverse coherence zone radius rc; this gives the far-field broadening
factor FFBF = rc(0)/rc(z). This factor is essentially unchanged if we use the 1D (along the
x-axis) spatial version of the tc expression. The small broadening of the near-field profile itself
can be estimated by the beam spread factor BSF = 1 + (k⊥/k0)z/D, where k⊥/k0 is the beam
spreading angle, k⊥ is the spatial frequency version of rc, and D = 15 cm is the beam width.
The small beam spread arises in part due to the ISI beam divergence and in part due to the
effects of the nonlinear far field broadening.
The amount of spectral broadening seen in Fig. 16 for 50m of 200mbar Xe is comparable
to that seen in Ref. [2] for beam propagation through a 100m air path.
The transverse spatial behavior in Fig. 16 is independent of whether the nonlinearity
is calculated according to the TPVM, Kerr response, or instantaneous Kerr response. The
spectral behavior, however, is model-dependent. The TPVM displays a reduction in high
frequencies; its difference with respect to the delayed Kerr response can be attributed to the
large field fluctuations, which render inaccurate the step used to obtain Eq. (5c) (in which
the adiabatic following approximation was applied and the γ̇i term was neglected).
Additionally, we find that the population redistribution and collisional damping are too
small for any decrease in the amount of spectral broadening to be observed.
Since the simulations are only for a 30 times-diffraction-limited beam (due to compu-
tational constraints), it is useful to consider how the propagation properties of a 75 times-
diffraction-limited beam will differ, see Fig. 17. The spectral broadening factor (SBF) is
51
Figure 16: Propagation of a 30 M2x (times-diffraction-limit) temporally and spatially incoherent KrF
laser beam through 50m of 200mbar Xe gas, see Table 2: (a) Total power spectrum (b) Axial lineout
of near-field beam profile (c) Axial lineout of far-field beam profile , modeled via the Two-Photon
Vector Model (TPVM) using either Eq. (9) or Eq. (10), the Kerr response n ˙2(φ(t)) see Eq. (6-7),
and the narrow-bandwidth, steady-state Kerr response n2(ω0). Properties of the incident light (z=0)
are indicated in black.
52
Figure 17: Spectral Broadening Factor, Far-field Broadening Factor, Beam Spreading Factor are
plotted as a function of the times-diffraction-limit (M2x), for propagation of a temporally and spatially
incoherent KrF laser beam through 50m of 200mbar Xe gas.
reduced as M2x increases, as is the far-field broadening factor (FFBF). These effects can be
attributed to the phase mismatch of the 3D wave vectors involved in four wave mixing, or equiv-
alently, the interruption of amplification paths. For a smaller spot size, the beam spreading
factor (BSF) can become significant and increase with M2x , such that the SBF and FFBF de-
crease more significantly with M2x than in Fig. 17 and roughly in proportion to (1 +M
2/a)−2x ,
where a is some constant. The BSF in Fig. 17 is due to the effects of far-field broadening,
roughly an order of magnitude larger than it would be due to ISI beam divergence alone. Its
slight decrease with M2x appears not to be a statistical anomaly, but rather a nonlinear effect
occurring in the regime where the nonlinear index and the group velocity dispersion coefficient
have opposite signs.
It is expected (for the parameters in Table 2) that at 75 times-diffraction-limit the SBF
will be roughly 2, and that the FFBF will be less than that, see Fig. 17. It must be noted,
however, that the nonlinearity may be overestimated by 15-20%, see Table 2. The SBF is
larger than the FFBF in Fig. 17 which we have seen in simulations is in part due to the four
53
wave mixing phase mismatch mentioned above, but can be mostly attributed to the positive
group velocity dispersion coefficient, which in a negative-n2 medium sharpens the pulse in
time, thereby increasing the rate of spectral broadening.
4 Discussion
We have shown that propagation of the KrF NIKE laser output beams through Xe gas may
be an effective way of increasing the laser bandwidth beyond that which can be achieved
via propagation through air. The beam spreading factor is sufficiently small. The far-field
broadening factor is significant, though it is at least smaller than the spectral broadening
factor which has been enhanced by group velocity dispersion. Far-field broadening may limit
the propagation distance and thus the spectral broadening that can be achieved. The best
way to maximize the spectral broadening while limiting the degradation of the far-field beam
profile is likely propagation through a chamber containing both air and Xe.
5 Conclusions
The nonlinear response of Xe gas to chaotic non-steady-state, near-resonance light has been
modeled using a two-photon vector model. The propagation of the KrF NIKE laser light
through 200mbar Xe has been simulated, and a dependence of the spectral broadening and
beam profile degradation on the times-diffraction-limit is observed and discussed. The results
of the TPVM are compared with those of a delayed response model and an instantaneous
response model. We conclude that propagation of the KrF NIKE laser output beams through
54
Xe may be an effective way of increasing the laser bandwidth, thereby suppressing the laser-
plasma instabilities.
6 References
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[2] Weaver, J., Lehmberg, R., Obenschain, S., Kehne, D. and Wolford, M., 2017. Spectral
and far-field broadening due to stimulated rotational Raman scattering driven by the Nike
krypton fluoride laser. Applied optics, 56(31), pp.8618-8631.
[3] Lehmberg, R. H., C. J. Pawley, A. V. Deniz, M. Klapisch, and Y. Leng. ”Two-photon
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[5] Nibbering, E. T. J., G. Grillon, M. A. Franco, B. S. Prade, and André Mysyrowicz. ”Deter-
mination of the inertial contribution to the nonlinear refractive index of air, N 2, and O 2 by
use of unfocused high-intensity femtosecond laser pulses.” JOSA B14, no. 3 (1997): 650-660.
[6] Lehmberg, R. H., J. Reintjes, and R. C. Eckardt. ”Negative nonlinear susceptibility of
cesium vapor around 1.06 µm.” Physical Review A 13, no. 3 (1976): 1095.
[7] Grischkowsky, D., M. M. T. Loy, and P. F. Liao. ”Adiabatic following model for two-photon
transitions: nonlinear mixing and pulse propagation.” Physical Review A 12, no. 6 (1975):
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2514.
[8] Grischkowsky, D. ”Coherent excitation, incoherent excitation, and adiabatic states.” Phys-
ical Review A 14, no. 2 (1976): 802.
[9] Lehmberg, R. H., and S. P. Obenschain. Use of Induced Spatial Incoherence for Uniform
Illumination on Laser Fusion Targets. No. NRL-MR-5029. Naval Research Lab, Washington
DC, 1983.
[10] Lehmberg, R. H., A. J. Schmitt, and S. E. Bodner. ”Theory of induced spatial incoher-
ence.” Journal of applied physics 62, no. 7 (1987): 2680-2701.
[11] Brabec, Thomas, and Ferenc Krausz. ”Nonlinear optical pulse propagation in the single-
cycle regime.” Physical Review Letters 78, no. 17 (1997): 3282.
[12] Taha, Thiab R., and Mark I. Ablowitz. ”Analytical and numerical aspects of certain
nonlinear evolution equations. II. Numerical, nonlinear Schrodinger equation.” Journal of
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[13] Crisp, M.D. ”Adiabatic-Following Approximation.” Physical Review A 8, no. 4 (1973):
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[14] Junnarkar, Mahesh R., and Naoshi Uesugi. ”Near-two-photon-resonance short-pulse prop-
agation in atomic xenon.” Optical Pulse and Beam Propagation. Vol. 3609. International
Society for Optics and Photonics, 1999.
[15] Gornik, W., et al. ”Two-photon excitation of xenon atoms and dimers in the energy
region of the 5 p 56 p configuration.” The Journal of Chemical Physics 75.1 (1981): 68-74.
[16] Bruce, M. R., et al. ”Radiative lifetimes and collisional deactivation of two-photon ex-
cited xenon in argon and xenon.” The Journal of chemical physics 92.5 (1990): 2917-2926. [17]
56
Grischkowsky, D. and Armstrong, J.A., 1972. Self-defocusing of light by adiabatic following
in rubidium vapor. Physical Review A, 6(4), p.1566.
7 Supplement
Parameters
For a 248.4nm laser pulse propagating through an atomic Xe gas, the two-photon excited
state is the Xe 6p[1/2]0 √state, and a classical collisional model predicts an elastic colli-
sion rate of γ 2c ∼ 4PπrXe 2/mXekT = 2.4GHz × (P/bar) where P is the pressure of the
Xe gas, T is the gas temperature, and mXe and rXe are the mass and van der Waals ra-
dius of an individual Xe molecule. We estimate the spontaneous emission rates as A32 =
|µ |2ω3 /(3πε ~c3) ≈ 40MHz and A = |µ |2ω3 /(3πε ~c332 32 0 21 21 21 0 ) ≈ 5GHz (see Ref. [14]). The
FWHM of the two-photon absorption spectrum giv√en a laser linewidth ∆ωL = 2.9GHz is
measured to be ∆νFWHM = 4.6 + 3.5(P/bar)GHz ≈ ∆ν2L + ∆ν2D + γc (see Ref. [15]), where
the Doppler broadening lin√ewidth ∆νD = 2ν0(2kBT ln 2/m
1/2
Xe) /c ≈ 1.3GHz and the elastic
collision rate γc ∼ 4Pπr2Xe 2/mXekT = 2.4GHz× (P/bar) are in agreement with the exper-
imental result. The radiative rate of the excited Xe state is measured to be ∼ 30 MHz (see
Ref. [16]), which is consistent with the limiting spontaneous emission rate of 40 MHz. For a
KrF NIKE pulse, the 600ps short pulses and 4ns long pulses correspond to a frequency scale
1.7 GHz and 0.3 GHz, respectively. The elastic collision rate at 200mbar is 0.5 GHz, and the
inelastic collision rate is presumably at least an order of magnitude smaller.
57
Density Matrix Equations
The density matrix equations given in Eq. (1) can be rewritten for integer s ∈ (−3,−2, ..., 3).
σ̇11(sφ) = (−isφ̇− Γo)σ11(sφ) + δ Γ ρeqs 0 11 + aΓI(σ
eq
22(sφ)− δsρ22) + abΓ (σ (sφ)− δ ρ
eq
c 33 s )
∑ [ ± ± ]
33
+ i Ω12σ21(φ(s∓ 1))− Ω21σ12(φ(s∓ 1)) ,
±
σ̇22(sφ) = (−isφ̇− ΓI)σ22(sφ) + δ Γ eqs Iρ22 + a(1− b)Γc(σ
eq
33(sφ)− δsρ33)
∑ ( )
+ i Ω± ± ±21σ12(φ(s∓ 1))− Ω12σ21(φ(s∓ 1)) + Ω23σ32(φ(s∓ 1))− Ω±32σ23(φ(s∓ 1) ,
± ∑ [ ]
σ̇33(sφ) = (−isφ̇− Γc)σ33(sφ) + δ Γ ρeq ± ±s c 33 + i Ω32σ23(φ(s∓ 1))− Ω23σ32(φ(s∓ 1)) ,
∑± ( )
σ̇12(sφ) = (−isφ̇+ i ω21 − γN)σ12(sφ) + i Ω±12 [σ22(φ(s∓ 1))− σ11(φ(s∓ 1))]− Ω±32σ13(φ(s∓ 1)) ,
∑± [ ± ± ]σ̇13(sφ) = (−isφ̇+ i ω31 − γc)σ13(sφ) + i Ω12σ23(φ(s∓ 1))− Ω23σ12(φ(s∓ 1)) ,
∑± ( )
σ̇23(sφ) = (−isφ̇+ i ω32 − γL)σ23(sφ) + i Ω±23 (σ33(φ(s∓ 1))− σ22(φ(s∓ 1))) + Ω±21σ13(φ(s∓ 1)) .
±
(S1)
Two-Photon Vector Model
On a timescale that is much longer than the inverse of the detuning 2π/∆ ∼ 0.2ps where
∆ = (2ω0 − ω31)/2, the off-diagonal density matrix elements will be significantly driven
only near the various harmonics of the laser frequency. This means the com∣ ponents σ∣mn(sφ)∣ ∣
can be taken to be slowly varying in time, i.e., |σ̇mn(sφ)| / |ω0σmn(sφ)| ∼ ∣(ω31 − 2φ̇)∣ /ω0 =
|∆| /ω0  1. In the given case where each one-pho∣ton transition is off-resonanc∣e, this allows∣ ∣
us to take, to zeroth order in ∆/ω0 , |σ̇12(sφ)|  ∣(−isφ̇+ i ω21 − γN)σ12(sφ)∣ , |σ̇23(sφ)| 
58
∣∣∣ ∣∣(−isφ̇+ i ω32 − γL)σ23(sφ)∣. In this case it is also reasonable to neglect the population of
the intermediate state σ22 because it is driven off-resonance and is to second order in the field.
Since the timescale of the pulse may be comparable to or shorter than the average elastic colli-
sion time, there are non-negligible transient components of ρ12 and ρ23 as well. However, these
components are non-resonant with the harmonics of the laser frequency and thus represent
only a contribution to the linear response.
The near-resonance components σ11(0), σ33(0), σ13(2φ) can be identified as the ones which
drive other elements and contribute significantly to the polarization field. Simplifying the
expression and redefining the eigenstates such the dipole moments µ12 = µ21 and µ23 = µ32
are real, we obtain, for linear polarization:
+ ∗ [ ]Ω12 = Ω− + +12 = Ω12 → Ω12, Ω23 → Ω23 , and (S2a)
∑
σ12(sφ) ≈ (sφ̇− ω21 − iγ )−1N [Ω12 (σ11(φ(s∓ 1))) + Ω32σ13(φ(s∓ 1))] ,
±∑
σ23(sφ) ≈ (−sφ̇+ ω + iγ )−132 L [Ω23σ33(φ(s∓ 1)) + Ω21σ13(φ(s∓ 1))] ,
±
σ̇13(2φ) = Aσ13(2φ) +Bσ11(0φ) + Cσ
(S2b)
33(0φ) ,
σ̇11(0φ) = Eσ13(2φ) + c.c.+ Fσ11(0φ) + c.c.+Gσ33(0φ) + c.c.+H ,
σ̇33(0φ) = Jσ13(2φ) + c.c.+ Lσ33(0φ) + c.c.+M,
59
where E = B and J = C from Eq. (S2a), and
( )
1 1
A = (i(ω31(− 2φ̇)− γc)− Ω
2
12 +
iφ̇− i ω32 + γL) 3iφ̇− i ω32 + γL
− 2 1 1Ω23 + ,
iφ̇− i ω21 + γN 3iφ̇− i ω21 + γN
B = −Ω12Ω23(iφ̇− i ω21 + γN)−1 ,
C = −Ω Ω (iφ̇− i ω + γ )−112 23 32 L ,
E = −Ω12Ω23(iφ̇− i ω −121 + γN) = B ,
F = −Γo/2− Ω212(iφ̇− i ω −1 221 + γN) − Ω12(−iφ̇− i ω21 + γ )−1N ,
G = +abΓc/2 ,
H = Γ eq eq0ρ11 − abΓcρ33 ,
J = −Ω12Ω23(iφ̇− i ω32 + γ )−1L = C ,
L = −Γc/2− Ω223(iφ̇− i ω32 + γ )−1L − Ω223(−iφ̇− i ω + γ )−132 L ,
M = Γcρ
eq
33 .
Defining a real vector r = (r1, r2, r3) where r1 = σ13(2φ) + [σ13(2φ)]
∗, r2 = i(σ13(2φ) −
[σ13(2φ)]
∗), r3 = σ33(0φ)− σ11(0φ), we can rewrite Eq. (S2b) to zeroth order in ∆/ω0 as
60
[ ]
ṙ1 = A(r1 −
γN
ir2)/2 + c.c.− Ω12Ω23 [ 2 (1−
γL
r3) + 2 (1 + r3) ,
(φ̇− ω ) + γ2 (φ̇− ω ) + γ221 N 32 L ]
φ̇− ω21 φ̇− ω32
ṙ2 = i (A(r1 − ir2)/2) + c.c.− Ω12Ω23 2 (1− r( 3
) + 2 (1 + r3) ,
(φ̇− ω 221) )+ γN (φ̇− ω32) + γ2L
− 1 1ṙ3 = (iΩ12Ω23 ( − (r1 − i− − )r2))/2 + c.c.ω32 φ̇+ iγL ω21 φ̇+ iγN
1 1
+(Γ0/2 + γ
2
NΩ12 2 + 2 (1− r3) + (Γc(1 + ab)ρ
eq
33 − Γ0ρ
eq
11)
(φ̇− ω2(1) + γ2N (φ̇+ ω21) + γ2N ))
1 1
+ −(1 + ab)Γc/2− γLΩ223
− 2 2
(1 + r3) ,
(φ̇ ω32) + γ2L (φ̇+ ω
2
32) + γL
(S3)
where we have made the replacements: σ13(2φ) = (r1− ir2)/2, σ11(0φ) ' (1−r3)/2, σ33(0φ) '
(1 + r3)/2.
Polarization
Treatment of Intermediate States
In the case of multiple intermediate levels, we can sum over states |2〉 to obtain the total
polarization. According to energy of each intermediate state and its transition dipole moments,
one can determine the fractional contribution of each state to the nonlinear index given in
Eq. (7). However, treatment of multiple intermediate levels in the two-photon vector model
shouldn’t be important for comparing simulations with future experiments (except in the case
when population redistribution is significant). This is because the additive contribution of
each intermediate state to the nonlinear index ∼ µ12µ23/(ω21 − ω0) is of the same form as its
contribution to the more general expression for the nonlinear polarization, see Eq. (9) and
61
Eq. (5a), thereby producing a response that is representative of a two-photon-transition with
only one intermediate state.
Two-Photon Absorption Rate
Solving Eq. (3) given the initial condition r(t = 0) = (0, 0,−1) and a constant (or slowly
varying in time() field am(pl√itude and)p)h/ase, i.e. , γ̇/γ
2  1 , we obtain
r 23 = −1 + γ1 1− cos t γ2 + ∆21 (γ21 + ∆2) . (W√e can then)e/xpress the two-photon
excited population ρ33 as ρ33 ≈ (1 + r3)/2 = γ2 2 2 21sin t γ1 + ∆ /2 (γ2 + ∆21 ) ∼ 10−9 for
the average field intensity and detuning given in Table 2. For long times, in the low-intensity
limit, we express ρ33 as a function of frequency[and obtain t]he following function which peaks
γ2sin2 (t∆/2) πγ2≈ 1 1t sin
2(∆t/2)
at ∆ = 0: ρ33(∆) = . Applying the relation for the
∆2 2 π(t/2)∆2
long time limit, we can recover t∫h[e standard two-]photon absorption rat∫e R2γ. The transition
probability is given by R2γt = lim ρ (∆) g(∆)d∆ = (πγ
2
33 1t/2) δ(∆)g(∆)d∆ , from
(t/2)→∞
which we obtain: R2γ = πγ
2g(0)/2 = 2πg(0)(Ω Ω )2/(ω − φ̇)21 12 23 21 , where g(∆) is the energy
density. After 50m propagation according to the parameters in Table 2, the energy density
on-resonance is given by g(0) ∼ 10−3 THz−1. Hence a transition rate R2γ ∼ kHz is obtained
for a sufficiently slowly varying field amplitude and phase, i.e. γ̇/γ2  1.
62
Part IV
Remote Optical Magnetometry
Here we analyze a mechanism for remote optical measurements of magnetic field variations
above the surface of seawater. This magnetometry mechanism is based on the polarization
rotation of reflected polarized laser light, in the presence of the earth’s magnetic field. Here
the laser light is reflected off the surface of the water and off an underwater object. Two
mechanisms responsible for the polarization rotation are the Surface Magneto-Optical Kerr
Effect (SMOKE) and the Faraday effect. In both mechanisms the degree of polarization
rotation is proportional to the earth’s local magnetic field. Variations in the earth’s magnetic
field due to an underwater object will result in variations in the polarization rotation of the
laser light reflected off the water’s surface (SMOKE) and off the underwater object (Faraday
effect). An analytical expression is obtained for the polarization-rotated field when the incident
plane wave is at arbitrary angle and polarization with respect to the water’s surface. We find
that the polarization rotated field due to SMOKE is small compared to that due to the Faraday
effect.
1 Introduction
Optical magnetometry can be a highly sensitive method for measuring small variations in
magnetic fields [1-3]. The development of a remote optical magnetometry system can have
important applications for the detection of underwater and underground objects that perturb
63
the local ambient magnetic field. For magnetic anomaly detection (MAD) applications, mag-
netic field variations must be detected at standoff distances greater than from the detector
[4-5]. The remote atmospheric optical magnetometry mechanism considered here is based on
polarization changes in reflected laser light from the sea water surface.
The dielectric properties of water in the presence of a magnetic field will cause the polariza-
tion of an optical field to rotate. The polarization rotation occurs via two distinct mechanisms:
the Surface Magneto-Optical Kerr Effect (SMOKE) and the Faraday rotation effect. SMOKE
is purely a surface phenomenon in which the degree of polarization rotation of the reflected
light is proportional to the magnetic field and independent of the propagation distance in the
dielectric medium [6]. The SMOKE mechanism is used in material science research, for ex-
ample, the Kerr microscope. The Faraday effect is due to the difference in propagation speed
of right-hand and left-hand polarized light, and the resulting degree of polarization rotation
is proportional to both, the magnetic field and the propagation distance.
Here in Part III, we analyze whether these effects can be used for the remote detection
of underwater objects. An underwater object can produce appreciable magnetic field per-
turbations above the water’s surface. The analyzed magnetometry detection configuration is
shown schematically in Fig. 18. Here, a linearly-polarized blue-green laser beam is propa-
gated to and reflects off the water’s surface near the detection site. The degree of polarization
rotation in the reflected field is proportional to the local magnetic field. The magnitude of
the polarization-rotated component of the reflected field is measured at various locations near
the detection site. An irregularity in the measurement at one location would correspond to
a variation in the local magnetic field. The local field variation indicate the presence of an
64
underwater object.
Figure 18: Polarized laser light is propagated to the surface of the water and exhibits polarization
rotation upon reflection due to Faraday rotation and SMOKE. The solid arrows in the figure denote
wave vectors.
The purpose is to estimate the degree of polarization rotation in the reflected field. If the
polarization rotation is large enough, the mechanism schematically outlined in Fig. 18 may
enable the measurement of variations in the earth’s magnetic field. This may indicate the
presence of an underwater object.
2 Model
To analyze the Surface Magneto-Optical Kerr and the Faraday rotation effects, we first obtain
the dielectric properties of magnetized, pure water. The polarization field associated with
magnetized water at the laser frequency is obtained using an extended form of the Lorentz
model [7]. The polarization field is used to derive a wave equation for the reflected laser
field. By appropriately matching boundary conditions at the various interfaces, we obtain the
reflected field which has its polarization modified by both the SMOKE and Faraday effect.
The polarization-rotated component of the field is proportional to the local magnetic field. In
65
applying the boundary conditions, the laser propagation direction and polarization as well as
the Earth’s magnetic field are taken to be in an arbitrary direction.
2.1 Polarization Field of Magnetized Water
In the Lorentz model, the dielectric properties of the molecule are represented by the motion
of the forced displacement of the electron distribution from its equilibrium position. The
displacement of the electron distribution δr(r, t) is modeled by the following driven harmonic
oscillator equation,
( ) ( )
∂2 2 ∂ q ∂δr+ ΩB + γ δr = E + × B2 o/c , (1)∂ t ∂ t m ∂ t
where ΩB is the electron binding frequency, γ is the effective damping rate of water in
the optical regime, q is the electronic charge, m is the electronic mass, and Bo is the lo-
cal(magnetic field, i.e)., Earth’s magnetic field ∼ 0.5G. The laser electric field is E(r, t) =
Re Ê(r) exp(− i ω t) whereω is the frequency. The magnetic field associated with the laser
is neglected on the right hand side of Eq. (1) since it produces forces at frequencies of 2ω and
0.
The polarization field associated with the magnetized water molecules is given by P̂ (r) =
qNδr̂, where δr̂(r) represents the electronic displacement from equilibrium of the water molecule,
N is the effective bound electron density of the water molecules, and pha(sor notation is use)d
throughout to represent the various field quantities, i.e., Q (r, t) = Re Q̂ (r) exp (− i ω t) .
66
The displacement δr̂(r) is given by
( )
(−ω2 + Ω2B − i γ ω) δr̂(r) = q Ê(r) − i ω δr̂(r)×Bo/c /m. (2)
The polarization field, correct to first order in the magnetic field, is
ω2 ( )pN
P̂ (r) = q N δr̂ ≈ × Ê(r) + i ω Ωo × Ê(r)/D (ω) , (3)
4 πD (ω)
1/2
where D (ω) = Ω2 2B − ω − i γ ω, ωpN = (4πq2N/m) , and Ω0 = qBo/mc is the cyclotron
frequency. The laser frequency is chosen to be in the blue-green range where the penetration
depth is maximum. In the blue-green regime, the real part of the refractive index is Re[n0] ≈
1.34. In the following examples, the laser frequency is taken to be ω = 3.8 × 1015rad/s
(ω = 2.48 eV/~) where pure water’s absorption coefficient is minimum, i.e., α = 5×10−4cm−1;
the corresponding penetration depth (e-folding length) is he ≈ 20 m. The lowest energy
electron-binding frequency in water is ΩB ≈ 23.5eV/~, ωpN ≈ 21eV/~, and the effective
damping rate of water in the optical regime is γ ≈ 3 × 109s−1. The cyclotron frequency
corresponding to Bo ∼ 0.5G is Ω ∼ 107 s−1o .
In general, there are many resonances above ΩB which contribute to the refractive index
of water. We neglect these binding frequencies since they correspond to substantially higher
energies and have little effect on the dielectric properties of water in the optical regime. In
addition, the oscillator strength parameter associated with the ΩB resonance is taken to be
unity [8]. Nonlinear effects in water are neglected since the laser intensity is well below the
level where these become important.
67
2.2 Wave Equation
In the absence of free charges and currents, Maxwell’s equations for the fields in phasor notation
are given by ∇× Ĥ = − i (ω/c) D̂ , ∇ · D̂ = 0 , ∇× Ê = i (ω/c) B̂ , and ∇ · B̂ = 0 .
These equations can be combined to give
∇2Ê + (ω2/c2) D̂ = ∇(∇ · Ê). (4)
To obtain a wave equation for Ê, we note that the electric flux density D̂ = (Ê + 4π P̂)
is given by D̂ = (εo + i (εo − 1) (ω/D(ω)) Ωo× ) Ê, where the electric polarization field
is given in Eq. (3) and where the dielectric constant of water, in the absence of the earth’s
magnetic field, is εo(ω) = 1 + ω
2
PN/D(ω). Since ∇ · D̂ = 0, we find that
∇ · − (εo − 1) ωÊ = i ∇ · (Ωo × Ê) =
εo D(ω) (5)
(εo − 1) ω
i (Ωo · (∇× Ê)),
εo D(ω)
where the final form for ∇ · Ê is valid since Ωo is spatially uniform. Using Eqs. (4) and (5),
the wave equation takes the form
2
∇2 ω − ω
2 ω
Ê + εo Ê = i (εo − 1) (Ω2 2 o × Ê)c c D(ω) (6)
(εo − 1) ω
+ i ∇(Ωo · (∇× Ê)).
εo D(ω)
The right-hand side of Eq. (6) can be simplified by noting that ∇(Ωo · (∇ × Ê)) = Ωo ×
(∇ × ∇ × Ê)) + (Ωo · ∇)(∇ × Ê) and, to lowest order in Ωo, ∇ × ∇ × Ê = (ω2/c2)εoÊ.
68
The final form for the wave equation, correct to first order in Ωo, is
( )
2
∇2 ω (εo − 1) ω+ εo Ê = i (Ωo · ∇)(∇× Ê). (7)
c2 εo D(ω)
Since the right-hand side of Eq. (7) is small, we can write the total field as Ê(r) = Êo + δÊ,
where Êo here denotes the laser field in the absence of the perturbation (i.e. the Earth’s
magnetic field) and δÊ is the small polarization-modified component of the laser field due to
the Earth’s magnetic field and induced by the laser field Êo . Separate wave equations can be
written for these fields:
( )
2
∇2 ω+ εo Êo = 0, (8a)
( c2 )
2
∇2 ω (εo − 1) ω+ εo δÊ = i (Ωo · ∇)(∇× Êo). (8b)
c2 εo D(ω)
In the special case where the laser field Eo is transverse to the magnetic field Bo, one can
recover from Eq. (8b) the usual Faraday rotation angle, i.e. ∆θF ≈ ω2 2pN ω Ω0 L/(2 cΩ4B) ∼
(10−4 G−1m−1rad)LBo , where L is the propagation distance [9]. These wave equations, Eq.
(8a,b), are numerically solved for arbitrary laser field propagation direction, polarization and
magnetic field direction.
69
3 Polarization Rotation
The magnetometry method is based on the polarization rotation of reflected laser radiation.
The reflection occurs both off the surface of the water (SMOKE effect) and off the underwater
object (Faraday effect). The purpose of the analysis is thus to calculate the degree of polar-
ization rotation as denoted by ∆θ = (1/R)|δE|/|E0| , where δE is the polarization-rotated
component of the reflected laser field, E0 is the incident laser field and the reflection coefficient
is R = (no − 1)/(no + 1) ≈ 0.15.
We first consider a simplified configuration as shown in Fig. 19 to obtain estimates for the
amount of polarization rotation, i.e., ∆θ. In this simplified configuration, a linearly-polarized,
monochromatic laser beam is propagated parallel to Earth’s magnetic field which is taken to
be perpendicular to the surface of the water. An optically reflective underwater object at
depth h below the water’s surface is represented by a perfectly conducting plate parallel to
the surface. Two distinct contributions will emerge from this simplified configuration to the
polarization rotation, i.e., the Faraday effect and SMOKE.
Figure 19: A simplified configuration of reflected laser light undergoing polarization rotation due
to Faraday rotation and SMOKE. The purpose of this configuration is to obtain estimates for the
polarization rotation angle ∆θ.
70
The solid arrows in Fig. 19 represent the wave vectors while the dashed arrows de-
note the electric fields corresponding to the incident wave Eo, the transmitted wave ET , the
object-reflected wave ER2 , and the surface-reflected wave ER1 which is modified by a small
polarization-rotated component δE. The electric field for z < 0 (air) can be written as,
Ê(z) = E0 exp(ikz)êx + ER1 exp(−ikz)êx + δE exp(−ikz)ê (9a)y,
and for 0 < z < h (water),
Ê(z) = E+T exp(ik+z)ê+ + E
−
T exp(ik−z)ê− + E
+
R2 exp(−ik+z)ê + E
−
+ R2 exp(−ik−z)ê−,
(9b)
√
where ê± = (êx ± i êy)/ 2, êxand êy are unit vectors in the x and y direction respectively,
√
k = ω/c is the wavenumber in air, n± = ck±/ω = εo ± ε1 is the complex index of refraction
in magnetized water, and ε1 = (εo − 1)ωΩ/D(ω). Application of the electromagnetic
boundary conditions yields for the configuration shown in Fig. 19 the following ratio for the
polarization-rotated field,
( )
δE − n+ cos (k+h) − n− cos (k−h)= i , (10)
E0 n+ cos (k+h) − i sin (k+h) n− cos (k−h) − i sin (k−h)
where n± ≈ no (1 ± (n2o − 1)ωΩo/(2n2oD(ω))) is the refracted index corresponding to right
and left-handed polarized waves. In the limit that h → ∞, the reflected Faraday rotation
vanishes and the polarization rotation is due solely to SMOKE. In the SMOKE limit Eq. (10)
71
reduces to
| n+ − n−δE/E0|h→∞ = = ξ, (11)(1 + n+) (1 + n−)
where ξ ≈ (n0 − 1)(ωΩ /Ω20 B)/(n20 + n0) ∼ 3× 10−12 . The ratio of the fields given in Eq.
(10) is shown in Fig. 20 as a function of the depth of the conducting plate h.
Figure 20: Ratio of the polarization-rotated reflected field to the incident field as a function of object
depth h, according to Eq. (10). The polarization rotation results from a combination of Faraday
rotation and SMOKE; however, the contribution due to SMOKE is small.
The Faraday effect yields the ratio |δE/E0| ∼ 12π n0 h ξ/λ, whereas SMOKE yields
|δE/E0| ≈ ξ ∼ 3× 10−12 . The Faraday effect is larger by the ratio h/λ 1 .
We now consider the general SMOKE contribution to the polarization rotation by analyzing
the process for arbitrary field directions. The polarization-rotated field due to SMOKE as a
function of the field components of an incident plane wave, for arbitrary polarizations and
angle of incidence as well as arbitrary orientation of the magnetic field, is obtained in the
Supplement. The general expression for the rotated polarization component of the field is
−iK√β(Ex, Ey, θ, θB, φB)δER,⊥ = , (12)
|Ex|2 + P 2|E |2y
72
where K = Ω 2 −120ω(nR − 1)/ΩBnR(nR + 1) ∼ 2.5 × 10 , β = cos θ (u |E |
2
B 1 x + u2P |E 2y| ) +
sin θ ∗B cosφB(u3ExEy)+sin θB sinφB(u4|Ex|
2+u 25P |Ey| ), and P (θ), ui(θ) ∼ 1 . In Eq. (12), Ex
and Ey are the components of the incident field, and θ is the angle of the incident wave vector
with respect to the water’s surface normal vector. The angles θB and φB are, respectively, the
polar and azimuthal angle of Earth’s magnetic field with respect to the incident wave vector,
see Supplement.
Equation (12) is written with the explicit θ dependence the Supplement, see Eq. (15). The
angular function u1(θ) is used to construct the following plot of SMOKE’s angular dependence,
for which the local magnetic field is taken to be parallel to the incident wave vector i.e.
θB = φB = 0 , Ey = 0 .
Figure 21: nR = 1.34 .
Figure 22: The function |δER,⊥/E0| plotted above represents the dependence of the polarization-
rotated field on the incident wave vector’s orientation for a specific geometry (θB = 0,Ey = 0),
according to Eq. (15). Here δER,⊥ is the polarization-rotated component of the reflected laser field
and E0 is the incident laser field. This plot is specific to water with refractive index nR = 1.34.
The angular dependence of Eq. (12) is further analyzed in the Supplement, where it is
shown that an alternative geometry can increase |δER,⊥/E0| to 5.1×10−12. We find, however,
that no choice of polarization or angle of incidence will significantly increase the SMOKE
contribution to the polarization rotation, see Fig. 21. The SMOKE contribution thus is too
small to serve as a feasible mechanism for the measurement of the local magnetic field or for
73
the measurement of small variations in this field.
Regarding the Faraday effect contribution, small variations in the magnetic field are not
of interest. Even so, measurement of the field ratio, |δE/E0| ∼ 12 π n0 h ξ/λ ∼ 10−3 (see Fig.
20), may be complicated by a number of factors which we address in the remainder of this
section.
The underwater object’s surface may not be conducting; in the case of rubber (nR = 1.52),
we find that 98% of the energy is lost to transmission, which would require |δE/E0| ∼ 10−4.
Alternatively, the object may not be in the desired depth range; a sub-meter or above-80m
depth will require |δE/E0| < 10−4, see Fig. 20.
With regard to variations in temperature, even if the laser beam propagates through water
that is fairly hot, the Faraday contribution will be increased by only a small amount, according
to |δE/E0| ∝ (n0(T )−1)/Ω2B(n0(T )+1), where the change in the parameter Ω2B with respect
to temperature yields, approximately, |δE/E0| ∝ (n 2 20(T )− 1) /(n0(T ) + 1) .
Regarding orientations in space, a percentage of the signal will be lost if the laser beam’s
path somewhat bypasses the underwater object or if the object’s geometry is such that it
mostly reflects the beam away from the detector’s surface. A non-vertical magnetic field at
angle θ with respect to the water’s surface will reduce the signal by a factor of sec θ.
At the water’s surface, bubbles and ripples on a smaller spatial scale than the laser beam
radius will likely refract away a significant amount of the available signal, and waves on a larger
spatial scale than the laser beam radius may refract away the full signal, requiring multiple
measurement attempts in each location.
74
4 Discussion
We have analyzed mechanisms for the possible remote optical measurement of magnetic field
variations above the surface of seawater. In this process, a polarized blue-green laser beam is
reflected off the water’s surface and the reflected, polarization-rotated electric field recorded.
A variation in the measurements at nearby locations may indicate the presence of an under-
water conducting object. The two mechanisms responsible for the polarization rotation are
the Surface Magneto-Optical Kerr Effect (SMOKE) and the Faraday effect. An analytical
expression is obtained for the polarization-rotated field when the incident plane wave is at
arbitrary angle and polarization with respect to the water’s surface. We conclude that the
polarization rotated field due to SMOKE is small and would challenging to implement as a
magnetometry mechanism. The Faraday effect is feasible for shallow (∼ 1-10 m) conducting
objects. Such an object would be too light to detect via gravitational field measurements, and
though sonar is a viable detection method, it is a more local method.
Underwater objects may be detected directly by measuring the un-rotated component of
the reflected laser field. By time gating the incident laser pulses and measuring the total
reflected field, the object’s depth can be determined.
5 References
[1] D. Budker, W. Gawlik, D.F. Kimball, S.M. Rochwester, V.V. Yashchuk and A. Weis, Rev.
Mod. Phys. 74, 1154 (2002).
[2] Optical Magnetometry, D. Budker and D.F.J. Kimball (eds.) (Cambridge University Press,
75
Cambridge, UK, 2013).
[3] G. Bison, R. Wynands, and A. Weis, Appl. Phys. B Lasers Opt. 76, 325 (2003).
[4] J.P. Davis, M.B. Rankin, L.C. Bobb, C. Giranda, M.J. Squicciarini, “REMAS Source
Book,” Mission and Avionics Tech. Dept., Naval Air Development Center (1989).
[5] L. A. Johnson, P. Sprangle, B. Hafizi, and A. Ting. “Remote atmospheric optical magne-
tometry,” Journal of Applied Physics, 116(6), (2014).
[6] Z. Qiu and S. Bader, “Surface magneto-optical Kerr effect,” Review of Scientific Instru-
ments 71 (3), 1243-1255 (2000).
[7] K. Oughstun and R. Albanese, ”Magnetic field contribution to the Lorentz model,” J. Opt.
Soc. Am. A 23, 1751-1756 (2006).
[8] J. D. Jackson, Classi cal electrodynamics. (Wiley, New York, 1999).
[9] Andrei, E. 2015. Faraday Rotation [pdf]. Retrieved from:
http://www.physics.rutgers.edu/grad/506
6 Supplement
A perturbative analysis on the wave equations in Eqs.(8a,b) is performed to obtain the polar-
ization rotated reflected field associated with SMOKE. The treatment allows for the incident
field to have an arbitrary angle and polarization with respect to the water’s surface and allows
for arbitrary magnetic field orientation. The coordinate system used in this analysis is shown
in Fig. 23.
The incident wave vector is denoted by the arrow in the z′ < 0 (air) regime; its direction
defines the z axis. The plane’s normal vector to the water’s surface plane defines the z′ axis.
76
Figure 23: Coordinate systems used in the calculation of the polarization-rotated reflected field
associated with SMOKE for an arbitrary geometric configuration.
The arrow in the water region corresponds to the transmitted wave vector in the absence of a
magnetic field. The boundary conditions ensure that it is coplanar with the z and z’ axis and
that k0 sinα = (ω/c) sin θ, where k0 is the magnitude of the transmitted wave vector. The
coordinate systems are defined such that x = x′ = x′′ coincide.
The fields Ê(r) , D̂(r) , and B̂(r) are expanded to first order in the small parameter
∆ = ωΩ0/D(ω) ∼ 10−11. The field is written in the form Ê(r) = Ê0(r)+δÊ(r), where E0
is the field in the absence of a magnetic field and δÊ(r) is the first order correction to the
field due to the magnetic field. The wave equations for E0 and δE have been derived, i.e. Eq.
(8a,b), and are repeated below for convenience,
( )
2
∇2 ω+ ε Ê = 0 (13a)o 2 o
( c )
2
∇2 ω (εo − 1) ω+ εo δÊ = i (Ωo · ∇)(∇× Ê (13b)2 o)c εo D(ω)
77
The fields can be written in the form
δÊ(r) = δÂ(r) exp(ik z′′) and Ê (r) = A exp(ik z′′0 0 T,0 0 ),
in order to employ the paraxial approximation,∇2 → − k 20 + 2ik0∂/∂z′′, which implies that
the field varies little over a wavelength. Applying the paraxial approximation to Eq. (13b)
yields
′′ i (ε0 − 1)ω (Ω · ik ′′0 0ẑ ) (i k ẑ′′0 × AT,0)∂δÂ(r) /∂z = , (14)
2 i k0 ε0D(ω)
where z′ > 0 (water). The right-hand side of Eq. (14) is independent of spatial variables and
′ i (ε0 − 1)ω (Ω0 · ik ′′0ẑ ) (ik ′′0ẑ × AT,0)yields δÂ(r) = a + z b, where b = . We have used the
2ik0ε0D(ω) cosα
fact that the boundary conditions equate the spatial dependence of δÂ(r) and ∇× δÂ(r) at
z′ = 0. Having solved the wave equation in each regime, we now apply the boundary conditions.
The electric field boundary conditions allow us to express a in terms of the components of δÊR
which denotes the first order correction to the reflected field ÊR. Three linearly independent
equations for δÊR are obtained by equating the tangential magnetic fields at the boundary.
Solving these equations yields a complicated expression for δÊR. This expression is not, in
general, perpendicular to the zeroth order reflected field. Thus, the desired expression for the
polarization rotated field δER,⊥ is given by
∗/
δER,⊥ = δÊR · (ẑref × Ê0,R) |Ê0,R|,
where Ê ′0,R is the zeroth order reflected field at z = 0 and ẑref is the direction along which it
78
propagates. The polarization rotated field δER,⊥ is given by
−iKβ(Ex, Ey, θ, θB, φB)
δER,⊥ = √ , (15)
|E |2x + P 2|Ey|2
where K = Ω0ω(nR − 1)/Ω2BnR(nR + 1) ∼ 2.5× 10−12 and
β = cos θB(u1|Ex|2 + u 22P |Ey| ) + sin θB cosφ ∗B(u3ExEy) + sin θB sinφB(u4|E |
2
x + u
2
5P |Ey| ). .
The angular functions P and ui are given by:
( √ )( ( ))( √cos θ +√n2 − sin2θ sin2θ + n2R R −1 + cos θP (θ) = )( ( √n2R − sin2θ)) ,
cos θ − n2R − sin2θ −sin2θ + n2R 1 + cos θ n2R − sin2θ
ui(θ) = vi(θ)(nR + 1)
2 c(os θ, ) ( √ )
cos θ n2R − sin2θ + sin2θ −2nR + n2R − sin2θ
v1(θ) = − ( √ )( ( √ )) ,
cos θ + n2 − sin2θ√ −sin2θ + n2 2 2R R 1 + cos θ nR − sin θ( √ sin2θ + co)s θ( n2R − sin2θv2(θ) = √ ) ,
cos θ + n2R − sin2θ n2R cos θ + n2R − sin2θ
v3(θ) = −
2nR sin θ √ ,
n2 + n4 c(os2θ − sin2R R θ + 2n2R(cos θ n2R√− sin2θ ))( sin√θ n2R − sin2)θ(− cos θ −2nR(+ n2 2R −v4(θ) = √sin θ )) ,
cos θ + n2 −(sin2θ √−sin2θ + n2 )1 + cos θ n2 − sin2R R R θ
v5(θ) = ( √sin θ cos θ)−( n2 − sin2R √θ ) .
cos θ + n2 2 2R − sin θ nR cos θ + n2R − sin2θ
In Eq. (15), θ is the angle of the incident wave vector with respect to the water’s surface
normal vector. The angles θB and φB are, respectively, the polar and azimuthal angle of
79
earth’s magnetic field with respect to the incident wave vector where θB = π/2, φB = 0,
corresponds to an orientation along the x axis (Fig. 23). The fields Ex and Ey correspond
to the component of the incident electric field Ê(r) along the x and y axis, respectively. The
angular functions used in this result are plotted in Fig. 24.
Figure 24: The functions above, within the context of Eq. (15), describe the angular dependence of
the polarization-rotated field on the incident wave vector’s orientation.
In the first plot of Fig. 24, the zero in P (θ) corresponds to the Brewster angle. From this
plot, we see that |P (θ)| ≤ 1 , which we will see is useful in minimizing the denominator of Eq.
(15) and maximizing the amount of polarization rotation. From the second plot of Fig. 24,
we see that when the incident wave propagates perpendicularly to the water’s surface (as in
the case of Fig. 19), i.e. , θ = 0 , the SMOKE contribution is proportional to the magnetic
field component in the direction of propagation. The geometry of Fig. 19 is not optimal.
The SMOKE contribution for water is largest when all of the following apply: i) the incident
light comes in at the Brewster’s angle (with respect to the water’s surface normal vector), ii)
the electric field of the incident wave is almost entirely polarized in the plane which contains
the incident wave vector, the transmitted wave vector, and the water’s surface normal vector,
i.e. 0 < |Ex|  |Ey|, and iii) the earth’s magnetic field is perpendicular to this plane, i.e.
θB = π/2, φB = 0. This specific geometry increases the polarization rotation by a factor of
80
≈ 2 (corresponding to the peak of u3 in Fig. 24) with respect to the geometry of Fig. 19. We
find, however, that no choice of polarization or angle of incidence will significantly increase
the SMOKE contribution to the polarization rotation.
81
Part V
List of Publications and Presentations
Publications:
[1] Epstein, Z., Lehmberg, R., and Sprangle, P., 2019. ”Spectral Broadening of the NIKE KrF
Laser via Propagation through Xenon in the Negative Nonlinear Index Regime.” (Manuscript
to be submitted April 2019).
[2] Epstein, Z., Hafizi, B., Peñano, J. and Sprangle, P., 2018. Generation of high-average-
power ultra-broadband infrared radiation. JOSA B, 35(11), pp.2718-2726.
[3] Epstein, Z., Hafizi, B., Peñano, J. and Sprangle, P., 2018, May. High-average power, ultra-
broadband, infrared radiation generation. In Ultrafast Bandgap Photonics III (Vol. 10638, p.
106381V). International Society for Optics and Photonics.
[4] Epstein, Z. and Sprangle, P., 2016. An Optical Magnetometry Mechanism Above the Sur-
face of Seawater. IEEE Journal of Quantum Electronics, 52(6), pp.1-6.
Presentations:
[3] Epstein, Z., Hafizi B., Penano J., Sprangle P. ”High Power Supercontinuum IR Genera-
tion.” Invited Talk. Defense and Commerical Sensing, SPIE, Orlando, FL, Apr. 18, 2018.
[2] Epstein, Z., Sprangle, P., Penano, J. ”High Power Supercontinuum Generation in the
Atmospheric IR Window.” Division Seminar. Plasma Physics Division (Code 6700), Naval
Research Laboratory, Washington, DC, Oct. 13, 2016.
[3] Epstein, Z., Sprangle, P., Penano, J. ”Generation of an IR-CM source using CO2 Beams.”
Directed Energy Scholar’s Summer Intern Presentation, Naval Research Laboratory, Wash-
ington, DC, Sep. 12, 2016.
[4] Epstein, Z., Sprangle, P., Penano, J. ”Generation of an IR-CM source using CO2 Beams.”
Directed Energy Scholar’s Summer Kickoff, Naval Research Laboratory, Washington, DC, Jun.
10, 2016.
[5] Epstein, Z., Sprangle, P., Lehmberg, R. “Effect of the Delayed Nonlinear Response on
Propagation of NIKE Laser through Xenon.” Laser Plasma Branch (Code 6730), Naval Re-
search Laboratory, Washington, DC, May 12, 2016.
82
[6] Epstein, Z., Sprangle, P. ”Maximization of PV Cell Efficiency for the Remote Recharging
of a UAV.” Beam Physics Branch (Code 6790), Naval Research Laboratory, Washington, DC,
Aug. 31, 2015.
[7] Epstein, Z., Sprangle, P., Ting, A. ”Propagation of a ‘Budding Flat-top’ beam through
Atmospheric Turbulence.” Directed Energy Scholar’s Summer Intern Presentation, Naval Re-
search Laboratory, Washington, DC, Aug. 19, 2015.
[8] Epstein, Z., Sprangle, P., Ting, A. ”Propagation through Atmospheric Turbulence.” Di-
rected Energy Scholar’s Summer Kickoff, Naval Research Laboratory, Washington, DC, Jun.
19, 2015.
[9] Epstein, Z., Sprangle, P. ”Optical Magnetometry Above the Surface of Seawater.” Naval
Engineering Education Center’s Annual Meeting, Carderock, VA, Apr. 07, 2015. Poster.
[10] Epstein, Z., Sprangle, P.”Remote Optical Magnetometry.” IREAP Graduate Student Sem-
inar, University of Maryland, College Park, MD, Mar. 13, 2015.
[11] Epstein, Z., Sprangle, P. ”Optical Magnetometry Above the Surface of Seawater.” Amer-
ican Society of Naval Engineers Day, Arlington, VA, Mar. 05, 2015. Poster.
[12] Epstein, Z., Johnson, L., Sprangle. P. “Laser Remote Sensing of Magnetic Fields in
the Atmosphere by Optical Pumping Techniques,” IREAP Laser-Matter Interactions Group
Meeting, College Park, MD, Aug. 06, 2014.
83
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