1. Shock Waves & Potentials
In Nonlinear Optics
Laura Ingalls Huntley
Prof. Jason Fleischer
Princeton University, EE Dept.
PCCM/PRISM REU Program
9 August 2007

2. 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.

3. 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
Strontium barium niobium

4. 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 e-
impurity levels
Valence Band

5. 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

6. 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

7. 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

8. 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

9. 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

10. Focusing & Defocusing Nonlinearities
Linear Case:
Diffraction
Top view
Defocusing Case: Enhanced Diffraction
Nonlinear
Nonlinear
Defocusing Case & Background: Dispersive Waves
Nonlinear
Δn = γI
Focusing Case: Spatial Soliton

11. Shock wave = Gaussian + Plane Wave
Input
Linear Diffraction
Nonlinear Shock Wave
Simulation:
Experiment:

12. 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.

13. Nonlinear Optics & BEC
BEC Shock Waves
Optical Shock Waves

14. Slowly-varying amplitude
The Wave Equation
The Linear Wave Equation:
For a beam propagating along the z-axis:
Slowly-varying amplitude
Rapid
phase
We derive the Schrödinger equation:
Linear
Top view
Assuming that the propagation length in z is much larger than the wavelength of
the light. I.e.:

15. The Wave Equation, Cont. The Nonlinear Wave Equation:
Where the electric displacement operator is approximated by:
We derive the nonlinear Schrödinger equation:
Kerr coefficient
Defocusing
Focusing
Intensity
Propagation
Diffraction
Nonlinearity

16. Nonlinear Schrödinger Equation
Nonlinear Optical System
Cold Atom System
Nonlinear Schrödinger equation
Gross-Pitaevskii equation
Coherent |ψ|2 = INTENSITY
• Propagation in space
• Diffraction
• Nonlinear interaction term:
Kerr focusing or defocusing
Coherent |ψ|2 = PROBABILITY
DENSITY
• Evolution in time
• Kinetic energy spreading
• Nonlinear interaction term:
mean-field attraction or
repulsion
SAME EQUATION
SAME PHYSICS

17. 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.

18. 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.

19. MatLab Simulation Shock Wave & Potential
The nonlinear Schrödinger equation is solved using a split-step beam propagation method in MatLab.
Linear Part:
Nonlinear Part:
Shock Wave & Potential

20. SBN:75 (Defocusing Nonlinearity)
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

22. Experimental Results
The output face of the crystal, before the nonlinearizing voltage is applied across the extraordinary axis of the crystal. y x

23. Experimental Results, cont.
After a defocusing voltage (-1500 v) has been applied to the extraordinary axis of the crystal for 5 minutes. y x