shock waves & potentials in nonlinear optics
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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?. - PowerPoint PPT PresentationTRANSCRIPT
Shock Waves & Potentials In Nonlinear Optics
Laura Ingalls HuntleyProf. 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.
Conduction Band
Valence Band
Eex
e-
impurity levels
Band Transport Model, cont.
• When an Sr impurity is ionized by incoming light, the emitted electron is promoted to the conduction band.
Conduction Band
Valence Band
hν
Eex
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.
Conduction Band
Valence Band
Eex
Band Transport Model, cont.
• Once out of the area of illumination, the electron relaxes back into holes in the band gap.
Conduction Band
Valence Band
Eex
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.
Valence Band
Eex
Esc
+++
---
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.
Etot
x-axis of crystal
n0
n
Eex
2
2
2
1E
Eb
En
Focusing & Defocusing Nonlinearities
Linear
Linear Case:Diffraction
Top view
Nonlinear
Δn = γI
Focusing Case: Spatial Soliton
Defocusing Case: Enhanced Diffraction
Nonlinear
Nonlinear
-100 -80 -60 -40 -20 0 20 40 60 80 1000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Defocusing Case & Background: Dispersive Waves
Shock wave = Gaussian + Plane Wave
Input Linear Diffraction Nonlinear Shock Wave
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0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
-100 -80 -60 -40 -20 0 20 40 60 80 1000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
-100 -80 -60 -40 -20 0 20 40 60 80 1000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Simulation:
Experiment:
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
Optical Shock WavesBEC Shock Waves
The Wave Equation
2
2
2
22
t
E
c
nE
∂∂
=∇
)(),,(),,,( tkziezyxtzyxE ωψ −=
02
12
2
=∂
Ψ∂+
∂Ψ∂
xkzi
Slowly-varying amplitude
Rapid phase
The Linear Wave Equation:
For a beam propagating along the z-axis:
We derive the Schrödinger equation:
Assuming that the propagation length in z is much larger than the wavelength ofthe light. I.e.:
kn
c⎟⎠
⎞⎜⎝
⎛=ω
Linear
Top view
zzz LL λψψ
<<2zzL λ>>z
kz z ∂
∂<<
∂∂ ψψ
2
2
The Wave Equation, Cont.
2
2
22 1
t
D
cE
The Nonlinear Wave Equation:
EnnnEnnEnED 222
Where the electric displacement operator is approximated by:
02
1 2
22
ψψψψnn
k
kzi
We derive the nonlinear Schrödinger equation: Kerr coefficient
DiffractionIntensity
Propagation Nonlinearity
Focusing
Defocusing
Nonlinear Schrödinger Equation
02
1 2
0
022
0
ψψψψn
kn
kzi
Nonlinear Optical SystemNonlinear Schrödinger equation
Coherent |ψ|2 = INTENSITY
• Propagation in space
• Diffraction
• Nonlinear interaction term: Kerr focusing or defocusing
SAME EQUATION SAME PHYSICS
02
222
ψψψψg
mti
Cold Atom SystemGross-Pitaevskii equation
Coherent |ψ|2 = PROBABILITY DENSITY • Evolution in time
• Kinetic energy spreading
• Nonlinear interaction term: mean-field attraction or repulsion
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 intensity-dependent; 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 split-step beam propagation method in MatLab.
Linear Part:
Nonlinear Part:
2
2
2
1
xkzi
ψψ
ψψψ 2
2nn
k
zi
Shock Wave & Potential
Experimental Set-up
Mirror
Beam Splitter
Lenses (Circular, Cylindrical)
Spatial Filter
Pincher
Attenuator
Laser Beam
Potential
Plane Wave
Shock
Laser (532
nm
)
SBN:75 (Defocusing Nonlinearity) Top Beam Steerer
The output face of the crystal, before the nonlinearizing voltage is applied across the extraordinary axis of the crystal.
Experimental Results
y
x
Experimental Results, cont.
After a defocusing voltage (-1500 v) has been applied to the extraordinary axis of the crystal for 5 minutes.
x
y