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Interaction of a Shock Wave with a Refractive Index Potential in Nonlinear Optics
Laura I. Huntley and Prof. Jason Fleischer
Princeton University, PCCM/PRISM REU Program
10 August 2007
INTRODUCTION
The correspondence between the physics of nonlinear optics (NLO) and BoseEinstein condensates (BEC) allows for the study of superfluidity without need for a high
vacuum or ultracold temperatures. This equivalence arises from the coherent and
nonlinear properties shared by both systems.1 In BEC, shocks and waves are described
by a nonlinear Schrödinger equation known as the Gross-Pitaevskii equation.2 Given a
quasi-coherent beam of light propagating along the z- axis of a nonlinear Kerr-like
medium, the slowly-varying amplitude  of the light can also be described by the
nonlinear Schrödinger equation
i
 1 2
2
    n    0 ,
z 2k
 
(1)
where k  2n0
 is the wave number of light with wavelength  in a material with
 initial refractive index n0 . The change in refractive index caused by the nonlinearity is

given by

k
2
n   n2  ,
n0
(2)
where n2 is the Kerr coefficient of the medium and  is the intensity of the beam.1 A
2
 photorefractive crystal will generally have one axis along which the magnitude of n2 is
maximized – this is the extraordinary axis. The mechanisms by which a photorefractive


material such as a Kerr medium form index potentials in response to light has been well
described.3,4
By applying the Madelung Transformation  (x,z)  (x,z)e iS(x,z) (  is the
intensity of the light, S is its coherent phase, and x is the extraordinary axis) to equation
(1), fluid dynamic equations are derived that further illuminate the correlation between


coherent light in a nonlinear medium and a superfluid.1,2 Notably, intensity in NLO is
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analogous to density in BEC, and the transverse Laplacian of the phase,   S , is
analogous to fluid velocity.1
Previous experiment has shown that the evolution of a shock wave (e.g. an

intensity hump with a plane wave background) propagating through a nonlinear medium
is characterized by transverse diffraction and symmetric splitting of the initial beam, with
local maxima repelling one another.1 The current research probes the effects of a
refractive index potential on an intersecting shock wave propagating through a material
with a defocusing ( n2,n  0) nonlinearity.

Matlab Simulation:
EXPERIMENTAL
The initial shock amplitude was modeled as a gaussian profile in x plus a small
constant representing the plane wave. Equation (1) was solved iteratively using a spiltstep method to propagate the shock wave in a material with index response (2) and
through an initial refractive index potential in the form of a negatively-valued Gaussian
in y. The intensity at various z-values was outputted as plots of the xy-plane with color
modulation indicating differences in intensity.
Experimental Set-up:
An 8x8x6 mm SBN:75 (SrxBa1-xNb2O6, x=0.75) photorefractive crystal was the
nonlinear medium used in this experiment. Light (532 nm) from a laser was split into
three separate beams. Cylindrical lenses were used to create an intensity hump along the
(extraordinary) x-axis of the crystal (this will create the index potential) and a separate
intensity hump along the (ordinary) y-axis, while the third piece remained a plane wave.
The beams were then recombined and focused on the input face of the crystal (See Figure
1). A defocusing voltage of -500 V was applied across the x-axis for approximately 5
min., after which the intensity hump along the x-axis was blocked and the output face of
the crystal was imaged using a charged-coupled device camera (See Figure 2).
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x
y
Figure 1. Image of the output face of the crystal before the
defocusing voltage was applied. The vertical stripe is an
intensity hump in y and the horizontal stripe is an intensity
hump in x.
Figure 2. Schematic of optical set-up and equipment used
for this experiment. All optical components shown are
mounted on a tuned-damping optics bench.
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RESULTS
x
y
Figure 3. Plots generated by Matlab split-step beam
propagation program. First plot shows the input face of the
“crystal” with the initial gaussian intensity hump focused
along the y-axis. Successive plots show slices of the xyplane at increasing values of z.
x
y
Figure 4. Output face of crystal after defocusing voltage (500 V) has been applied for 5 min. The potential-creating
intensity hump along the y-axis has been blocked
momentarily.
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DISCUSSION
Matlab Simulation:
Intensity modulations far from the initial index potential are consistent with
previous experiment on a lone shock wave.1 Light is attracted to areas of higher n; thus,
the negatively-valued potential appears as a dark strip along the x-axis with a growing
diffraction pattern surrounding it (see Figure 3).
Experiment:
Figure 4 shows signs of the dark vertical stripe with accompanying diffraction
pattern predict by the simulation. However, the quality and alignment of the beams was
not perfect enough to make an definitive claims about the outcome of the experiment.
Further work must be done to attain a clearer result.
REFERENCES
1. Wan, W.; Jia, S.; Fleischer, J.W. Dispersive superfluid-like shock waves in
nonlinear optics. Nature Physics (2007), 3, 46-51.
2. Yeh, P. Introduction to Photorefractive Nonlinear Optics. John Wiley & Sons,
Inc., 1993.
3. Günter, P.; Huignard, J. (eds.). Photorefractive Materials and Their Applications:
1-3. Springer, Inc., 2006.
4. Roberts, P. H.; Berloff, N. G. The nonlinear Schrödinger equation as a model
of superfluidity. Lecture Notes in Physics (2001), 571, 236-257.