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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? • 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 2nn 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