Ultrashort Optical Pulse Propagators

Transcription

1 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

2 An Artists Rendering - New Scientist, February 19, 000

3 From Linear to Extreme NLO Light string propagation in air.

4 Maxwell Equations I Maxwell s Equations B E = ; D= ρ t D H = + j; B= 0 t Constitutive Relations B = µ 0H ; D = ε0e + P Polarization P rt, couples light to matter. ( )

5 Vector Wave Equation Maxwell Equations II 1 E 1 E ( E) = c t ε0c t Note: Most pulse propagation models derived from Maxwell assume that ( E) = 0. Coupling between field components can only occur via nonlinear polarization i.e. nonlinear birefringence. P rt, = P rt, + P rt, ( ) ( ) ( ) L NL P

6 Maxwell Equations III Linear and Nonlinear Polarization Models + P r, t = ε χ r, t τ E r, τ dτ P r, ω = ε χ r, ω E r, ω L ( ) ( ) () 1 ( ) ( ) ( ) ( ) ( ) 0 0 P r, t = ε χ r, t τ, t τ, t τ E r, τ E r, τ E r, τ dτ dτ dτ NL ( 3 ) ( ) ( ) ( ) ( ) P NL ( r, ω) = ε 0 χ ω ω ω ω ω ω ( 3 ) ( r, 1,, 3) E( r, 1) E( r, ) E( r, 3) ( + + ) d d d δ ω ω ω ω ω ω ω 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!

7 Maxwell Equations III Some nonlinear polarization models: Induced nonlinear refractive index PNL = ε0 χe = nbn ( 1 f ) I + f R( t) I( t τ) dτ 0 Coupling to Plasma Rotational/vibrational Raman response t e ( ) ρ = ai ρ + b I c e jt = ρ t E t j t m () () () / τ c

8 Formal Derivation of NLS I - asymptotic expansion in a small parameter µ 1 E 1 1 NL 0c t ( ) () 1 ( ) ( ) LE E E χ t τ E τ dτ c t c t P = ε Seek solutions in the form of a perturbation expansion 3 4 E = µ E + µ E + µ E + µ E where ikz ( ωt) E0 = e B( X = µ x, Y = µ y, Z = µ z, T = µ t) e + * ( ( )) is a wavepacket linearly polarized perpendicular to the direction of propagation z.

9 Formal Derivation of NLS II L () 1 ( 1 χ ) y z c t xy xz 1 () 1 = + ( 1+ χ ) xy x z c t yz xz yz x y c t () 1 ( 1 χ ) Assume slow variation of B B B B = + µ + Z Z1 Z to remove secular terms proportional to z in E, E, etc. 1

10 Can write Formal Derivation of NLS III k + k 0 0 L0 = k + k ( ω ) 0 0 k L LE e L Be L Be L Be ( ωt ) ( 3 µ µ µ ) ( ) ikz 0 = * ( ω ) ik + ikk ' 0 ik Z1 T X 0 0 L1 = 0 ik + ikk' 0 Z1 T ( ω ) ik ik ikk ' X T T + ik X Z XY XZ 1 + ( kk '' + kk ' ) Z1 T + ik X Z = XY YZ 1 + ( kk '' + kk ' ) Z1 T + X Y X Z1 Y Z 1 ( kk '' + kk ' ) T

11 Formal Derivation of NLS IV Solvability condition at order µ yields linear dispersion relation ( µ ): = ( ω ) O k k Solvability condition at order µ O ( µ ) B B : + k' = 0 Z T 1 reflecting the fact that, to leading order, the wavepacket travels with the group velocity.

12 Formal Derivation of NLS V Solvability condition at order µ 3 yields NLSE ( 3 ) B i B B k'' B n O µ : = + i ik B B + Z k X Y T n Adding terms at O(µ ) and O(µ 3 ) n Z T k X Y T n B B i B B k'' B = ik ' + + i + ik B B Higher order correction terms such as 3 rd GVD, Raman and Envelope shock terms appear at next order!

13 Heuristic Derivation of NLSE NLSE is most easily derived from the general dispersion relation: ω = ω k, Ψ ( ) Expand in a multi-variable Taylor series about ( k 0, 0) ω d ω ( ) ω = ω0 + kj k0 j + j 1 k j Ψ = ( k ) Ψ ( 0, 0,0) 0 j, ω0,0 k j ω d 1 ω + ( ) ( k ) 0 ( 0 ) k0 j, 0,0 j k j kl k l + hot jl, 1 kj k ω = l Generic description of weakly nonlinear dispersive waves invert Fourier transform

14 D-Dimensional Nonlinear Schrödinger Equation Canonical description of weakly nonlinear dispersive waves = = Ψ = Ψ Ψ + Ψ + Ψ + Ψ D l j l j l j D j j j x x k k x k t i 1, 1 0 ω ω ω Group Velocity Group Velocity Dispersion Nonlinearity D=1 - Integrable D=,3 - Blowup singularity in finite time Moloney Moloney and Newell, Nonlinear Optics (004): 1 and Newell, Nonlinear Optics (004): 1 st st Edition (199) Edition (199)

15 Strong Turbulence Theory and Applications: Higher Dimensional NLS 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)

16 Types of Dispersion in NLO Diffraction: Narrow waist beams of initial width w 0 spread over characteristic length scales π LD w0 λ Dispersion: Short laser pulses of duration spread as they propagate over distances '' '' LDisp = τ p / k where k is the group velocity dispersion. τ p

17 Self-Phase Modulation SPM (Kerr) SPM (Delayed Nonlinearity) Anti-Stokes Stokes SPM with Normal GVD Dispersive shock (Wave breaking)

18 Plasma-Induced Spectral Blue Shift Rae and Burnett, PRA 46, p1084 (199) 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 enλ L dz λ = 8πε dt 3 i 0 3 0mc e

19 Ray Optics Estimation of Critical Power Circular Aperture n 0 Incident Plane Wave a θ c n 0 +n <E > Self-Trapped Waveguide n 0 Critical Angle: θc = ne n 0 0 P c Diffraction Angle: 0 (1.) λ c = 18n 0.61λ θ 0 D = an Critical Power: (Air P c = 3-4 GW ) 0

20 Role of Dimension in NLS ia z D + A+ A A s = 0 8 D 4 C B A 4 Critical collapse boundary (sd = 4) Region of supercritical collapse (sd >4) D s Summary s D NLO Manifestation A: cw spatial soliton 1+0 temporal soliton B: 0+ cw beam critical focus 1+1 simultaneous compression in space and time C: D collapse in bulk Note: Supercritical collapse can occur in optics in anomalous GVD regime '' - when k < 0 Laplacian operator is positive definite.

21 Normal GVD Arrests Collapse 3D NLS: ia z = a T A k '' A tt + A A Anomalous GVD - k '' < 0-3D Collapse (simultaneous space-time compression) Normal GVD - k '' > 0 - D Collapse with pulse splitting in time Review Articles: F. Fibich and G. Papanicolaou, SIAM J. Appl. Math. 60, pp183-40, (000) L. Berge, Physics Reports, 303, pp59-37, (1998)

22 ODE Description near Collapse Manifold Self-similar solution: where τ ζ α ζ χ τ ζ τ i i e g t A + = 4 ) ( ) ( ),, ( ) ( ) ( 1 ) ( 0 t z t z g τ ) 4 ( ) ( 4 β α ε β β α β α α τ τ τ = = = a av g g Small parameter: ) ( " 0 " t z = k ε Phase Portraits Fixed point Luther, Newell and Moloney, Physica D, Vol. 74, p59 (1994)

23 Modulational Instability I A modulational instability across the wide beam front causes growth of collapsing filaments D 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 ( ) iaz + γ TA + pn A A = 0 A( xz, ) Where is the complex slowly-varying envelope of the electric field Key Idea: Linearize about an infinite plane wave solution A x z (, ) = g e * ipn ( gg ) z i.e N( A ) is an exact integral of NLSE for a plane wave

24 Assume Modulational Instability II A x z g y x z e (, ) = + (, ) (, ) y x z ( ) ( ) ipn I z where is a small perturbation on the plane wave. Substitute into NLSE iy pn I g+ y x z + y x z + pn I g+ y x z z (, ) γ T (, ) ( )( (, )) ( ) ( ) 0 0 * * N + p( g y( x, z) + gy ( x, z) )( g+ y( x, z) ) = 0 I Canceling second and fourth terms and retaining terms first-order in y iy + γ y + pi N ' y = pg N ' y z * T 0 iy + γ y + pi N ' y = pg N ' y * * * * z T 0 0

25 Modulational Instability III To determine stability, seek perturbations with transverse spatial structure that grow with z i.e σz ik x ik x σz ik x ik x y x, z = e ae + be + e ce + de ( ) ( ) ( ) * * * * * * * y x z e a e b e e c e d e ( ) ( z ik x ik x, ) z ( σ σ ik x ik x = ) After straightforward algebra coupled system of linear pde s reduce to the solution of a matrix problem: For σ real: * iσ γk + pi0n' pg N' a * = pg N ' iσ γk + pi0n ' b 0 For σ=iν, pure imaginary * υ γk + pi0n' pg N' a * = pg N ' υ γk + pi0n ' b 0

26 Modulational Instability III Summary: 1. Transverse perturbations will grow if Re(σ)>0. Instability growth determined by the condition σ = pi N ' γk γ K > Critical wavenumber: fastest growing mode Continous system Reσ K c Periodic Box K K c pi N ' π γ λ 0 = = c K n

27 Saturable Forced-Damped D NLS - a paradigm for optical turbulence Movie first shown at Optical Bistability Meeting, Aussois, J.V. Moloney, Proc. Roy. Soc., A313, 49 (1984). J.V. Moloney, J.Q.E. IEEE, 1, 1393 (1985). Made with 16 mm camera of Willis Lamb Jnr.

28 Filamentation of Intense Laser Pulse in Air Arizona Center for Mathematical Sciences NRL Laser Parameters Pulse energy = 1.45 J Pulse length = 400 fsec Peak power = 3.56 TW Propagation distance = 10 m

29 Optical Breakdown Avalanche Photo-Ionization - free impurity or seed electrons (produced by multi-photon ionization) are accelerated by light field and multiply exponentially. Multi-Photon Ionization - high intensity laser field ionizes electrons essentially instantaneously at a rate proportional N N to the light intensity I A where N is the number of quanta needed to reach the ionization continuum.

30 Mathematical Model ( ) ( ) = t R N N R T A dt t t A t R kn if A A A i A A kn f i t A k i A k i z A ' ' ' ) ( '' ) ( ) ( 1 1 β ρ ωτ σ GVD Plasma Absorption/ refraction Multi-photon absorption Delayed Raman Response Diffraction Kerr Nonlinearity ) ( 1 ρ ω β ρ σ ρ a N A A E n t N N g b + = Avalanche generation Multi-photon generation Plasma recombination - Plasma Drude Model

31 Nonlinear Spatial Replenishment - multiple recurrence of collapse within pulse M. Mlejnek, E.M. Wright and J.V. Moloney, Opt. Letts., 3, 38 (1998)

32 Single Filament - Radial symmetry

33 Experimental Verification Detected radiation indicating recurrence of collapse. A. Talebpour, S. Petit and S.L. Chin, Re-focusing during propagation of a focused femtosecond pulse in air, Opt. Commun., Vol. 171, pp 85-90, December (1999).

34 Optically Turbulent Atmospheric Light Guide What happens when the femtosecond laser pulse is wide and contains tens to hundreds of critical powers? Woeste et al., Laser und Optoelektronik, Vol. 9, p A wide laser beam can undergo a modulational instability whereby an initially smooth beam profile breaks up into multiple collapsing light filaments. [Campillo et al., Appl. Phys. Lett., Vol. 3, p68, 1973] Strong Turbulence Scenario!

35 4D Strong Turbulence Simulation Computational Challenges: Severe compression of multiple interacting light filaments in space and time requires adaptive mesh refinement and parallel computation. Simulation - full (x,y,z,t) resolution - field intensity plots Single Time Slice Data Representation: - surface plot of filaments as slider moves from front to back of pulse fixed propagation distance - isosurface representation of filament distribution over pulse as a function of z. M. Mlejnek, M. Kolesik, J.V. Moloney, and E.M. Wright, "Optically Turbulent Femtosecond Light Guide in Air." Phys. Review Letters, 83(15) pp (1999).

36 Adaptive Mesh Parallel Solver Dynamic regriding in space and time

37 Turbulent Interaction and Nucleation of Light Filaments Movie sweeps from front to back of laser pulse at a fixed propagation distance z=9.0 m. Filaments collapse on front end, decay and recur at random. Collapse singularity arrested by strong defocusing which conserves energy. Dissipation is very weak in air. Iso-surface Z=9.0 m

38 Optical Turbulence Chaotic filament creation within propagating pulse - 3D box contains main part of pulse Leading Edge Trailing Edge

39 Summary of NLSE Model Captures qualitative behavior of light string creation and plasma generation Arizona Center for Mathematical Sciences Severe time (and space) compression within violently focusing pulse may invalidate NLSE description Simple Drüde plasma model with multiphoton ionization source captures lowest order plasma effects improved model planned with Becker and Faisal. Beyond NLSE resolve optical carrier while propagating over meter distances: UPPE model