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Ultrashort Optical Pulse Propagators Ultrashort Optical Pulse Propagators Jerome V Moloney Arizona Center for Mathematical Sciences University of Arizona, Tucson Lecture I. From Maxwell to Nonlinear Schrödinger Lecture II. A Unidirectional Maxwell Propagator Lecture III. Relation to other Propagators and Applications An Artists Rendering - New Scientist, February 19, 2000 From Linear to Extreme NLO Light string propagation in air. Maxwell Equations I Maxwell’s Equations G GG∂B ∇×ED=− ; ∇• = ρ ∂t G GG∂D G ∇×Hj= + ;0∇•B= ∂t Constitutive Relations G GGGG B = µε00HD; =+EP G G Polarization P () rt , couples light to matter. Maxwell Equations II Vector Wave Equation G G GG22 2 11∂ EP∂ ∇−EE22−∇()∇• = 22 ct∂ ε0c∂t Note: Most pulse propagationG models derived from Maxwell assume that ∇∇ () • E = 0 . Coupling between field components can only occur via nonlinear polarization i.e. nonlinear birefringence. GGG GGG P(rt,,) =+PLN(rt) PL(rt,) Maxwell Equations III Linear and Nonlinear Polarization Models GGGG+∞ GGGGGG Pr,,t=−ε χτ()1 rt Er,τdτ⇔Pr,ω=εχr,ωEr,ω L () 00∫ ()() ()()() −∞ GGGG GGGGG Pr,,t=−ε χτ(3) rt ,t−τ,t−τEr,τEr,τEr,τdτdτdτ NL () 01∫∫∫ ( 23) ( 1) ( 2) ( 3) 123 8 3 GGGGGGG G G χω()()rE,,ω,ω()r,ωE(r,ω)E()r,ω Pr,ωε= 123 1 2 3 NL ()0 ∫∫∫ δω()12++ω ω3−ωddω1ω2dω3 Practically, the linear polarization response is treated at some approximate level – nonlinear polarization is assumed to be instantaneous (Kerr) with a possible delayed (Raman) term. For ultra wide bandwidth pulses even linear dispersion is nontrivial to describe! Maxwell Equations III Some nonlinear polarization models: Induced nonlinear refractive index GG ∞ P =∆ε χτEn=21n −fI+fRtIt−dτ NL 02b () ∫ ()() 0 Rotational/vibrational Raman response Coupling to Plasma 2 ∂=t ρ aI ρ +b(I ) −c GGe2 G jttc=−ρ ()E()tj()t/τ me Formal Derivation of NLS I - asymptotic expansion in a small parameter µ G 22 GG G11∂∂E 1 G LE ≡∇2 E −∇ ∇•E − − χ ()()t −ττE ()dτ ()ct22∂∂c2t2∫ G 2 1 ∂ PNL = 22 ε0ct∂ Seek solutions in the form of a perturbation expansion GG G G G 23 4 EE=µµ01+E+++µE2µE3" where G ik( z−ωt) Ee0 ==()B()Xµµx,,Y=yZ=µz,T=µte +(*) is a wavepacket linearly polarized perpendicular to the direction of propagation z. Formal Derivation of NLS II ∂∂221 ∂2 ∂2 ∂2 +− 1+χ ()1 • − − 2222() ∂∂y z c ∂t ∂xy∂ ∂xz∂ ∂∂22∂21 ∂2 ∂2 L =− +−1+χ ()1 • − 2222() ∂∂xy ∂x ∂z c ∂t ∂y∂z ∂∂22∂2∂21 ∂2 −−+−1+χ ()1 • 2222() ∂∂xz ∂y∂z ∂x ∂y c ∂t Assume slow variation of B ∂B ∂∂BB = ++µ " ∂∂ZZ ∂Z 12 G G to remove secular terms proportional to z in EE 12 , , etc. Formal Derivation of NLS III G ik( z−ωt) 23 Can write LE00= e (µµL Be ++L1Be µL2Be +") +(*) ∂ ∂∂ 22ik +−ikk ' 0 ik 22 ∂∂Z TX∂ −+kk()ω 00 1 22 ∂∂ Lk0 =−00+k()ω Li=+02k2ikk'0 1 ∂∂ZT 2 1 00k ()ω ∂∂∂ −−ik ik 2'ikk ∂∂XT∂T ∂∂2 2 + 2ik ∂∂XZ ∂∂22 2 −− 22 ∂∂2 ∂∂XY ∂∂XZ1 +−22()kk ''+kk ' ∂∂ZT1 ∂∂2 + 2ik 2 2 2 ∂∂∂∂XZ2 L =− − 2 ∂∂XY 22∂Y∂Z ∂∂2 1 +−22()kk ''+kk ' ∂∂ZT1 ∂∂22 + ∂∂22∂∂XY22 −− ∂∂XZ ∂Y∂Z 2 112 ∂ −+()kk '' kk ' 2 ∂T Formal Derivation of NLS IV Solvability condition at order µ yields linear dispersion relation Ok()µ : = k(ω ) Solvability condition at order µ2 2 ∂B ∂B Ok()µ :'+ = 0 ∂∂ZT1 reflecting the fact that, to leading order, the wavepacket travels with the group velocity. Formal Derivation of NLS V Solvability condition at order µ3 yields NLSE 22 2 3 ∂∂BiB∂B k'' ∂B n2 2 Oi()µ :|=+22−2+ikB|B ∂∂Zk2 22X∂Y ∂T n Adding terms at O(µ2 ) and O(µ3) 22 2 ∂∂BBi∂B∂Bk'' ∂Bn2 2 =−ik '|+ 22+ −i 2+ik B |B ∂∂∂ZT22kX∂Y ∂Tn Higher order correction terms such as 3rd GVD, Raman and Envelope shock terms appear at next order! Heuristic Derivation of NLSE NLSE is most easily derived from the general dispersion relation: G ω =Ψω k ,| |2 ( ) G Expand in a multi-variable Taylor series about (,k0 0) d ∂∂ωω ω =+ω kk− + ||Ψ2 00∑ ()jj2 j=1 ∂∂k j ||Ψk ,,ω 0 ()k00j ,,ω 0 ()00j 1 d ∂2ω +−∑ ()k jjk00()kl−k l+hot ()k00j ,,ω 0 2 jl,1= ∂∂kkjl Generic description of weakly nonlinear dispersive waves invert Fourier transform D-Dimensional Nonlinear Schrödinger Equation Canonical description of weakly nonlinear dispersive waves 2 2 ∂Ψ D ∂ω ∂Ψ D ∂ ω ∂ Ψ ∂ω 2 i + ∑ + ∑ + | Ψ | Ψ = 0 2 ∂t j=1 ∂k j ∂x j j,l=1 ∂k j∂kl ∂x j∂xl ∂ | Ψ | Group Velocity Group Velocity Dispersion Nonlinearity D=1 - Integrable D=2,3 - Blowup singularity in finite time Moloney and Newell, Nonlinear Optics (2004): 1st Edition (1992) Strong Turbulence Theory and Higher Dimensional NLS Applications: • Deep Water Waves • Langmuir Turbulence in Plasmas • Relativistic beam plasma turbulence • Bose Einstein Condensates Reference: Robinson, Reviews of Modern Physics, “Nonlinear wave collapse and strong turbulence”, Vol. 69, p507 (1997) Types of Dispersion in NLO Diffraction: Narrow waist beams of initial width w spread over characteristic length 0 2π scales L ≈ w2 D λ 0 Dispersion: Short laser pulses of duration τ p spread as they propagate over distances 2 '' '' LDisp = τ p / | k | where k is the group velocity dispersion. Self-Phase Modulation SPM (Kerr) SPM (Delayed Nonlinearity) Anti-Stokes Stokes SPM with Normal GVD Dispersive shock (Wave breaking) Plasma-Induced Spectral Blue Shift Rae and Burnett, PRA 46, p1084 (1992) Optical field induced ionization can cause a rapid increase in electron density and, hence a decrease in refractive index - end result is a strong blue shift in the generated spectrum Spectral Blue Shift 23 eNiλ0 L dZ ∆=λ − 23 8πε0mce dt Ray Optics Estimation of Critical Power Circular Aperture n0 Incident Plane θ Self-Trapped Wave 2a c 2 Waveguide n0+n2<E > n0 n E 2 0.61λ0 Critical Angle: θ = 2 0 Diffraction Angle: θ D = c 2an0 n0 2 2 (1.22) λ0c Critical Power: Pc = (Air Pc = 3-4 GW ) 128n2 Role of Dimension in NLS 2 s iAz + ∇ D A+ | A | A = 0 8 Critical collapse Summary boundary (sD = 4) s D NLO Manifestation D Region of supercritical A: 1 0+1 cw spatial soliton 4 collapse (sD >4) 1+0 temporal soliton •C B: 2 0+2 cw beam critical focus B 1+1 simultaneous compression 2 • D A in space and time • • C: 3 1+2 3D collapse in bulk 24 s Note: Supercritical collapse can occur in optics in anomalous GVD regime '' - when k < 0 Laplacian operator is positive definite. Normal GVD Arrests Collapse 2 '' 2 3D NLS: iAz = a∇T A − k Att + | A | A • Anomalous GVD - k '' < 0 - 3D Collapse (simultaneous space-time compression) • Normal GVD - k '' > 0 - 2D Collapse with pulse splitting in time Review Articles: F. Fibich and G. Papanicolaou, SIAM J. Appl. Math. 60, pp183-240, (2000) L. Berge, Physics Reports, 303, pp259-372, (1998) ODE Description near Collapse Manifold α −i ζ 2 +iτ Self-similar solution: A(τ,ζ ,t) = g(τ )χ(ζ )e 4 where g(τ )∝1 z(t) − z0 (t) Phase Portraits Fixed point Small parameter: gτ = αg " " 2 ε = k z0 (t) ατ = 4β −α 2 βτ = −av(β ) − ε(a −α 4 − β ) Luther, Newell and Moloney, Physica D, Vol. 74, p59 (1994) Modulational Instability I • A modulational instability across the wide beam front causes growth of collapsing filaments – 2D analog of 1D MI in a fiber • When the instability growth spatial wavelength is much less than the transverse dimension of the beam, one can analyze it as a perturbation on a plane wave Starting Point is the NLSE 22 iAzT+∇γ A +pN (||A ) A =0 G Where A (, xz ) is the complex slowly-varying envelope of the electric field Key Idea: Linearize about an infinite plane wave solution G * Ax( , z) = geipN ()gg z i.e N(|A|2) is an exact integral of NLSE for a plane wave Modulational Instability II GGipN(I )z Assume Ax(),,z=+( g y( xz))e G where yx () , z is a small perturbation on the plane wave. Substitute into NLSE G 2 GG iyzT−+pN()I00()g yx( ,,z) +γ∇y( xz) +pN(I)( g+yx( ,z)) **GG G∂N + pg()y()x,,z++gy()xz ()g y()x,z =0 ∂I 0 Canceling second and fourth terms and retaining terms first-order in y 22* iyzT+∇γ y +pI0 N ''y =−pg N y *2* * *2 −+iyzTγ∇y +pI0 N ''y =−pg N y Modulational Instability III To determine stability, seek perturbations with transverse spatial structure that grow with z i.e G G GGGGGG G σσz iK•−x iK•x −z iK•x −iK•x yx(), z=+e (ae be ) +e (ce +de ) GGG GGG GG **G σσ**z −•iK x *iK•x −z *−•iK x *iK•x yx(), z=+e ()ae be +e (ce +de ) After straightforward algebra coupled system of linear pde’s reduce to the solution of a matrix problem: For σ real: iKσγ−+2*pIN''−pg2N a 0 = 0 22* −−pg N ''iσγ−K +pI0 N b For σ=iν, pure imaginary −−υγKp2*+IN''−pg2N a 0 = 0 22 * −−pg N ''υγK +pI0 N b Modulational Instability III Summary: 1. Transverse perturbations will grow if Re(σ)>0 2. Instability growth determined by the condition Continous system Reσ 2224 σ =−2'pI0 N γγK K >0 K Kc 3. Critical wavenumber: fastest growing mode Periodic Box 2'pI N 2π K ==0 c Kn γ λc Saturable Forced-Damped 2D NLS - a paradigm for optical turbulence Movie first shown at Optical Bistability Meeting, Aussois, 1984 Made with 16 mm camera of Willis Lamb Jnr.
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