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 . 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

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

22 2 2 2  ∂∂1 ∂ ()1 ∂ ∂   22+−22()1+χ • − −  ∂∂y z c ∂t ∂xy∂ ∂xz∂  2222 2  ∂∂∂1 ∂()1 ∂ L =− 22+−22()1+χ • −   ∂∂xy ∂x ∂z c ∂t ∂y∂z   22222 ∂∂∂∂1 ∂()1  −−22+−22()1+χ •  ∂∂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 11−+()kk '' kk '2 ∂T 2  Formal Derivation of NLS IV

Solvability condition at order µ yields linear dispersion relation

Ok()µ : = k(ω ) Solvability condition at order µ2 ∂B ∂B Ok()µ 2 :'+ = 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 ∂∂BiB∂B k'' ∂B n2 2 Oi()µ :|=+22−2+ikB|B ∂∂Zk2 22X∂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 ∂∂∂ZT22kX∂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 2 ω =Ψω (k ,| | ) G Expand in a multi-variable Taylor series about (,k0 0)

d ∂∂ωω ω =+ω  ()kk− + ||Ψ2 00∑ ∂∂k jj||Ψ2 j=1 j ()k ,,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- 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 can cause a rapid increase in 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

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. 1. J.V. Moloney, Proc. Roy. Soc., A313, 429 (1984) 2. J.V. Moloney, J.Q.E. IEEE, 21 , 1393 (1985). Filamentation of Intense Laser Pulse in Air

NRL Laser Parameters

Pulse energy = 1.425 J Pulse length = 400 fsec Peak power = 3.56 TW Propagation distance = 10 m Arizona Center for Mathematical Sciences Center for Mathematical Arizona Optical Breakdown

•Avalanche Photo-Ionization - free impurity or seed (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 to the light intensity I N ≡ | A | 2 N where N is the number of quanta needed to reach the ionization continuum. Mathematical Model

Diffraction GVD Kerr Nonlinearity Plasma Absorption/ refraction '' 2 ∂A i 2 k ∂ A 2 σ = ∇T A − i + i()1− fR kn2 | A | A − ()1+ iωτ ρA ∂z 2k 2 ∂t 2 2 (N ) β 2N −2 t ' ' 2 ' − | A | A + ifRkn2 ∫ R(t ) | A(t − t ) | dt A 2 −∞ Multi-photon absorption Delayed Raman Response

∂ρ 1 σ β (N ) | A |2N = ρ | A |2 + − aρ 2 2 - Plasma Drude Model ∂t nb Eg N=ω

Avalanche generation Multi-photon generation Plasma recombination Nonlinear Spatial Replenishment - multiple recurrence of collapse within pulse

M. Mlejnek, E.M. Wright and J.V. Moloney, Opt. Letts., 32, 382 (1998) Single Filament - Radial symmetry 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 285-290, December (1999). 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. 29, p51 1997.

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. 23, p628, 1973]

Strong Turbulence Scenario! 4D Strong Turbulence Simulation

Computational Challenges: Simulation - full (x,y,z,t) resolution •Severe compression of multiple - field intensity plots interacting light filaments in space and time requires adaptive mesh Single Time Slice refinement and parallel computation.

• 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. 2938-2941 (1999). Adaptive Mesh Parallel Solver

Dynamic regriding in space and time Turbulent Interaction and Nucleation of Light Filaments Iso-surface 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 Z=9.0 m random. Collapse singularity arrested by strong defocusing which conserves energy. Dissipation is very weak in air. Optical Turbulence

• Chaotic filament creation within propagating pulse

- 3D box contains main part of pulse

Leading Trailing Edge Edge Summary of NLSE Model

• Captures qualitative behavior of light string creation and plasma generation

• 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 Arizona Center for Mathematical Sciences Center for Mathematical Arizona