Transcript PrincetonPresentation

Shock Waves & Potentials
In Nonlinear Optics
Laura Ingalls Huntley
Prof. Jason Fleischer
Princeton University, EE Dept.
PCCM/PRISM REU Program
9 August 2007
What is Nonlinear Optics?
• Nonlinear (NL)
optics is the regime
in which the
refractive index of a
material is
dependant on the
intensity of the light
illuminating it.
Photorefractive Materials
• Examples: BaTiO3, GaAs, LiNbO3
• Large single crystal (~1 cm3) with single electric domain required for
experiment
– Single domain attained by poling
• Exhibit ferroelectricity:
– Spontaneous dipole moment
– Extraordinary axis is along dipole moment
• SBN:75
– Strontium Barium Niobate
– SrxBa(1-x)Nb2O6 where x=0.75
Band Transport Model
• Describes the
mechanism by which
the illuminated SBN
crystal experiences an
index change.
• Sr impurities have
energy levels in the
band gap.
• An external field is
useful, but not
necessary.
Eex
Conduction Band
eimpurity levels
Valence Band
Band Transport Model, cont.
• When an Sr impurity
is ionized by incoming
light, the emitted
electron is promoted
to the conduction
band.
Eex
Conduction Band
hν
Valence Band
Band Transport Model, cont.
• Once in the conduction
band, the electron
moves according to the
external electric field.
• If no external field is
present, diffusion will
cause the electrons to
travel away from the
area of illumination.
Eex
Conduction Band
Valence Band
Band Transport Model, cont.
• Once out of the area
of illumination, the
electron relaxes back
into holes in the band
gap.
Eex
Conduction Band
Valence Band
Band Transport Model, cont.
• In time, a charge
gradient arises, as
shown.
• The screening electric
field is contrary to the
external field.
• The screening field
grows until its
magnitude equals that
of the external field.
Eex
Esc
-
+
+
+
Valence Band
The Electro-optic/Kerr Effect
• Where the electric field is
non-zero, the index of
refraction is diminished.
• Snell’s Law dictates that light
is attracted to materials with
higher index, n.
• In the case shown, the index
change is focusing.
• The defocusing case occurs
when Eex is negative, and the
illuminated part of the crystal
develops a lower index.
n 
E
Etot
n0
n
x-axis of crystal
2
1 b E
Eex
2
 E
2
Focusing & Defocusing
Nonlinearities
Linear Case:
Diffraction
Linear
Defocusing Case:
Enhanced Diffraction
Nonlinear
Top view
Focusing Case:
Spatial Soliton
Defocusing Case &
Background: Dispersive Waves
Nonlinear
Nonlinear
0.1
0.09
0.08
Δn = γI
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
-100
-80
-60
-40
-20
0
20
40
60
80
100
Shock wave =
Gaussian + Plane Wave
Experiment:
Simulation:
0.5
0.45
0.1
0.1
0.09
0.09
0.4
0.08
0.08
0.35
0.07
0.07
0.3
0.06
0.06
0.25
0.05
0.05
0.2
0.04
0.04
0.15
0.03
0.03
0.1
0.02
0.02
0.05
0.01
0.01
0
-100
-80
-60
-40
-20
0
20
40
Input
60
80
100
0
-100
-80
-60
-40
-20
0
20
40
60
80
100
Linear Diffraction
0
-100
-80
-60
-40
-20
0
20
40
60
80
100
Nonlinear Shock Wave
Nonlinear Optics & Superfluidity
• The same equations
govern the physics of
waves in nonlinear optics
and cold atom physics
(BEC).
• Thus, the behavior of a
superfluid may be probed
using simple optical
equipment, thus
alleviating the need for
vacuum isolation and
ultracold temperatures.
Nonlinear Optics & BEC
BEC Shock Waves
Optical Shock Waves
The Wave Equation
The Linear Wave Equation:
2
2
n

E
2
 E 2 2
c t
For a beam propagating along the z-axis:
E ( x, y, z, t )   ( x, y, z )ei ( kz t )
We derive the Schrödinger equation:
 1  

0
2
z 2k x
2
i
c
n
   k
Linear
Slowly-varying Rapid
amplitude
phase
Top view
Assuming that the propagation length in z is much larger than the wavelength of
the light. I.e.:
2
Lz  z



2
Lz
z Lz


 k z
2
z
z
The Wave Equation, Cont.
1  D
 E 2 2
c t
2
The Nonlinear Wave Equation:
2
Where the electric displacement operator is approximated by:
D  E  n
2
2


E  n  n E  n  2nn E
We derive the nonlinear Schrödinger equation:
Defocusing
Kerr coefficient

1 2
k
2
i

   n2    0
z 2k
n
Focusing
Propagation
Diffraction
Nonlinearity
Intensity
Nonlinear Schrödinger Equation
Nonlinear Optical System
Cold Atom System
Nonlinear Schrödinger equation
Gross-Pitaevskii equation
n2 k0 2

1 2
i

  
  0
z 2k0
n0

2 2
2
i

   g    0
t
2m
Coherent |ψ|2 = INTENSITY
Coherent |ψ|2 = PROBABILITY
DENSITY
• Evolution in time
• Propagation in space
• Diffraction
• Kinetic energy spreading
• Nonlinear interaction term:
Kerr focusing or defocusing
• Nonlinear interaction term:
mean-field attraction or
repulsion
SAME EQUATION
SAME PHYSICS
Fluid Dynamics
• The Madelung
transformation allows us
to write fluid dynamic-like
equations from the
nonlinear Schrödinger
equation.
• Intensity is analogous to
density.
• Shock speed is intensitydependent; thus, a more
intense beam in a
defocusing nonlinearity
with a plane wave
background will diffract
faster.
A Shock Wave & A Potential
Step 1:
A gaussian shock focused along the
extraordinary (y) axis of the crystal creates an
index change in the crystal, but does not feel it.
Step 2:
A gaussian shock focused along the ordinary
(x) axis with a plane wave background feels both
the index potential created by the first beam and
its own index change.
MatLab Simulation
The nonlinear
Schrödinger equation
is solved using a splitstep beam propagation
method in MatLab.
Linear Part:

1  2
i

z
2k x 2
Nonlinear Part:

k
2
i
  n2  
z
n
Shock Wave & Potential
Experimental Set-up
Attenuator
Beam Splitter
Laser Beam
Lenses (Circular, Cylindrical)
Potential
Spatial Filter
Plane Wave
Pincher
Shock
SBN:75 (Defocusing Nonlinearity)
Top Beam Steerer
Laser (532 nm)
Mirror
Experimental Results
The output face of
the crystal, before
the nonlinearizing
voltage is applied
across the
extraordinary axis
of the crystal.
y
x
Experimental Results, cont.
After a defocusing
voltage (-1500 v)
has been applied
to the extraordinary
axis of the crystal
for 5 minutes.
y
x