Shock Waves & Potentials In Nonlinear Optics

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 e- impurity 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

Eex Etot n0 n x-axis of crystal 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.

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

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

Optical Shock Waves Nonlinear Optics & BEC BEC Shock Waves

Slowly-varying amplitude Rapid phase Linear Top view The Wave Equation 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 of the light. I.e.:

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 Nonlinearity Diffraction

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

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. Fluid Dynamics

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: Shock Wave & Potential

Laser (532 nm) Mirror Beam Splitter Lenses (Circular, Cylindrical) Spatial Filter Pincher Attenuator Laser Beam Potential Plane Wave Shock SBN:75 (Defocusing Nonlinearity) Top Beam Steerer Experimental Set-up

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

Shock Waves & Potentials In Nonlinear Optics