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

-100 -80 -60 -40 -20 0 20 40 60 80 1000

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