Post on 16-Feb-2018
Ultrashort Optical Pulse Propagators
Jerome V MoloneyArizona 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
Maxwell Equations I
Maxwell’s Equations
;
; 0
BE Dt
DH j Bt
ρ∂∇× = − ∇• =
∂∂
∇× = + ∇• =∂
Constitutive Relations
0 0;B H D E Pµ ε= = +
Polarization couples light to matter.( ),P r t
Maxwell Equations II
Vector Wave Equation
( )2 2
22 2 2 2
0
1 1E PE Ec t c tε
∂ ∂∇ − −∇ ∇• =
∂ ∂
Note: Most pulse propagation models derived from Maxwellassume that . Coupling between field componentscan only occur via nonlinear polarization i.e. nonlinear birefringence.
( ) 0E∇ ∇• =
( ) ( ) ( ), , ,L NLP r t P r t P r t= +
Maxwell Equations IIILinear and Nonlinear Polarization Models
( ) ( ) ( ) ( ) ( ) ( ) ( )10 0, , , , , ,LP r t r t E r d P r r E rε χ τ τ τ ω ε χ ω ω+∞
−∞
= − ⇔ =∫
( ) ( ) ( ) ( ) ( ) ( )
( )( ) ( ) ( ) ( ) ( )( )
30 1 2 3 1 2 3 1 2 3
31 2 3 1 2 3
01 2 3 1 2 3
, , , , , , ,
, , , , , ,,
NL
NL
P r t r t t t E r E r E r d d d
r E r E r E rP r
d d d
ε χ τ τ τ τ τ τ τ τ τ
χ ω ω ω ω ω ωω ε
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!
Maxwell Equations IIISome nonlinear polarization models:
Induced nonlinear refractive index
( ) ( ) ( )0 20
2 1NL bP E n n f I f R t I t dε χ τ τ∞
= ∆ = − + −
∫
Rotational/vibrational Raman responseCoupling to Plasma
( )
( ) ( ) ( )
2
2
/
t
t ce
aI b I c
ej t E t j tm
ρ ρ
ρ τ
∂ = + −
= −
Formal Derivation of NLS I
( ) ( ) ( ) ( )2 2
122 2 2 2
2
2 20
1 1
1 NL
ELE E E t E dc t c t
Pc t
χ τ τ τ
ε
∂ ∂≡ ∇ −∇ ∇• − − −
∂ ∂∂
=∂
∫
- asymptotic expansion in a small parameter µ
Seek solutions in the form of a perturbation expansion2 3 4
0 1 2 3E E E E Eµ µ µ µ= + + + +
where
( ) ( ) ( )( )0 , , , *i kz tE e B X x Y y Z z T t e ωµ µ µ µ −= = = = = +
is a wavepacket linearly polarized perpendicular to the direction of propagation z.
Formal Derivation of NLS II
( )( )( )( )
( )( )
2 2 2 2 21
2 2 2 2
2 2 2 2 21
2 2 2 2
2 2 2 2 21
2 2 2 2
1 1
1 1
1 1
y z c t x y x z
Lx y x z c t y z
x z y z x y c t
χ
χ
χ
∂ ∂ ∂ ∂ ∂+ − + • − − ∂ ∂ ∂ ∂ ∂ ∂ ∂
∂ ∂ ∂ ∂ ∂= − + − + • − ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
− − + − + • ∂ ∂ ∂ ∂ ∂ ∂ ∂
Assume slow variation of B
1 2
B B BZ Z Z
µ∂ ∂ ∂= + +
∂ ∂ ∂to remove secular terms proportional to z in , etc. 1 2,E E
Formal Derivation of NLS IIICan write ( ) ( ) ( )2 3
0 0 1 2 *i kz tLE e L Be L Be L Beω µ µ µ−= + + + +
( )( )
( )
2 2
2 20
2
0 00 00 0
k kL k k
k
ωω
ω
− + = − +
1
11
2 2 ' 0
0 2 2 ' 0
2 '
ik ikk ikZ T X
L ik ikkZ T
ik ik ikkX T T
∂ ∂ ∂ + − ∂ ∂ ∂ ∂ ∂ = + ∂ ∂
∂ ∂ ∂ − − ∂ ∂ ∂
( )
( )
( )
2
2 2 22
2 212
2 21
2
22 22
2 2 212
2 21
2 2
2 2 2 2
21 1 2
2
2
'' '
2
'' '
'' '
ikX Z
X Y X Zkk kk
Z T
ikX Z
LX Y Y Z
kk kkZ T
X YX Z Y Z
kk kkT
∂ ∂+ ∂ ∂ ∂ ∂ − −
∂ ∂ ∂ ∂∂ ∂+ − + ∂ ∂
∂ ∂ +∂ ∂∂ ∂ = − − ∂ ∂ ∂ ∂∂ ∂ + − +∂ ∂
∂ ∂ +
∂ ∂ ∂ ∂− − ∂ ∂ ∂ ∂ ∂ − + ∂
Formal Derivation of NLS IV
Solvability condition at order µ yields linear dispersion relation
( ) ( ):O k kµ ω=
Solvability condition at order µ2
( )2
1
: ' 0B BO kZ T
µ ∂ ∂+ =
∂ ∂
reflecting the fact that, to leading order, the wavepacket travelswith the group velocity.
Formal Derivation of NLS V
Solvability condition at order µ3 yields NLSE
( )2 2 2
3 222 2 2
2
'': | |2 2
nB i B B k BO i ik B BZ k X Y T n
µ ∂ ∂ ∂ ∂
= + − + ∂ ∂ ∂ ∂
Adding terms at O(µ2 ) and O(µ3)
2 2 222
2 2 2
''' | |2 2
nB B i B B k Bik i ik B BZ T k X Y T n
∂ ∂ ∂ ∂ ∂= − + + − + ∂ ∂ ∂ ∂ ∂
Higher order correction terms such as 3rd GVD, Raman and Envelope shock terms appear at next order!
Heuristic Derivation of NLSENLSE is most easily derived from the general dispersion relation:
( )2,| |kω ω= Ψ
Expand in a multi-variable Taylor series about ( ,0 0)k
( )( )
( )
( ) ( )( )
0 00 0
0 0
20 0 2
1 , ,0, ,0
2
0 0, ,0, 1
| || |
12
jj
j
d
j jj j kk
d
j j l lkj l j l
k kk
k k k k hotk k
ωω
ω
ω ωω ω
ω
=
=
∂ ∂= + − + Ψ ∂ ∂ Ψ
∂+ − − +
∂ ∂
∑
∑
Generic description of weakly nonlinear dispersive waves invert Fourier transform
D-Dimensional Nonlinear Schrödinger Equation
Canonical description of weakly nonlinear dispersive waves
∑=
∑=
=ΨΨ
Ψ∂
∂+
∂∂Ψ∂
∂∂∂
+∂Ψ∂
∂∂
+∂Ψ∂ D
lj ljlj
D
j jj xxkkxkti
1,2
2
22
10||
||ωωω
Group Velocity Group Velocity Dispersion Nonlinearity
D=1 - Integrable D=2,3 - Blowup singularity in finite time
MoloneyMoloney and Newell, Nonlinear Optics (2004): 1and Newell, Nonlinear Optics (2004): 1stst Edition (1992)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 spread over characteristic length scales 2
02 wLD λπ≈
0w
Dispersion: Short laser pulses of duration spread as they propagate over distances
where is the group velocitydispersion.
pτ||/ ''2 kL pDisp τ= ''k
Self-Phase ModulationSPM (Kerr) SPM (Delayed Nonlinearity)
SPM with Normal GVD
Dispersive shock(Wave breaking)
StokesAnti-Stokes
Plasma-Induced Spectral Blue ShiftRae 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
2 30
2 308i
e
e N L dZm c dtλλ
π ε∆ = −
Ray Optics Estimation of Critical Power
Circular Aperture
Self-TrappedWaveguiden0+n2<E2>
0
0261.0anDλθ =
2a θc
n0
Incident PlaneWave
n0
0
202
nEn
c =θ Diffraction Angle:Critical Angle:
2
20
2
128)22.1(n
cPcλ
=Critical Power: (Air Pc = 3-4 GW )
Role of Dimension in NLS0||2 =+∇+ AAAiA s
Dz
s
••
•
•2 4
CBA
D
Critical collapseboundary (sD = 4)
Region of supercriticalcollapse (sD >4)
Summarys D NLO Manifestation
A: 1 0+1 cw spatial soliton1+0 temporal soliton
B: 2 0+2 cw beam critical focus1+1 simultaneous compression
in space and timeC: 3 1+2 3D collapse in bulk
Note: Supercritical collapse can occur in optics in anomalous GVD regime- when Laplacian operator is positive definite.0'' <k
8
D4
2
Normal GVD Arrests CollapseAAAkAaiA ttTz
2''2 ||+−∇=3D NLS:
• Anomalous GVD -- 3D Collapse (simultaneous space-time compression)
• Normal GVD -- 2D Collapse with pulse splitting in time
0'' <k
0'' >k
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
Self-similar solution: where
τζα
ζχτζτii
egtA+−
=2
4)()(),,()()(1)( 0 tztzg −τ
)4()(
42
2
βαεββ
αβα
α
τ
τ
τ
−−−−=
−=
=
aav
gg Small parameter:)("
0" tzk=ε
Phase Portraits
Fixed point
∝
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
( )2 2| | 0z TiA A pN A Aγ+ ∇ + =
Where is the complex slowly-varying envelope of the electric field ( , )A x z
Key Idea: Linearize about an infinite plane wave solution
( ) *( ), ipN gg zA x z g e=
i.e N(|A|2) is an exact integral of NLSE for a plane wave
Modulational Instability II
( ) ( )( ) ( ), , ipN I zA x z g y x z e= +Assume
where is a small perturbation on the plane wave.( ),y x z
Substitute into NLSE
( ) ( )( ) ( ) ( )( )
( ) ( )( ) ( )( )
20 0
* *
0
, , ( ) ,
, , , 0
z Tiy pN I g y x z y x z pN I g y x z
Np g y x z gy x z g y x zI
γ− + + ∇ + +
∂+ + + =
∂
Canceling second and fourth terms and retaining terms first-order in y2 2 *
0
* 2 * * *20
' '
' 'z T
z T
iy y pI N y pg N y
iy y pI N y pg N y
γ
γ
+ ∇ + = −
− + ∇ + = −
Modulational Instability IIITo determine stability, seek perturbations with transverse spatial structurethat grow with z i.e
( ) ( ) ( )( ) ( ) ( )* ** * * * *
,
,
z iK x iK x z iK x iK x
z iK x iK x z 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
σ σ
σ σ
• − • − • − •
− • • − − • •
= + + +
= + + +
After straightforward algebra coupled system of linear pde’s reduce to the solution of a matrix problem:
For σ real:2 *2
0*2 2
0
' '0
' 'ai K pI N pg Nbpg N i K pI N
σ γσ γ
− + − = − − − +
For σ=iν, pure imaginary2 *2
0*2 2
0
' '0
' 'aK pI N pg Nbpg N K pI N
υ γυ γ
− − + − = − − +
Modulational Instability IIISummary:
1. Transverse perturbations will grow if Re(σ)>02. Instability growth determined by the condition
3. Critical wavenumber: fastest growing mode
2 2 2 402 ' 0pI N K Kσ γ γ= − >
Reσ
KKc
Periodic Box
Continous system
Kn
02 ' 2c
c
pI NK πγ λ
= =
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 JPulse length = 400 fsecPeak power = 3.56 TWPropagation distance = 10 m
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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 proportionalto the light intensity where N is the number of quanta needed to reach the ionizationcontinuum.
NN AI 2||≡
Mathematical Model
( ) ( )
∫∞−
− −+−
+−−+∂∂
−∇=∂∂
tR
NN
RT
AdtttAtRknifAA
AiAAknfitAkiA
ki
zA
'2''2
22)(
222
2''2
|)(|)(||2
12
||122
β
ρωτσ
GVD Plasma Absorption/refraction
Multi-photon absorption Delayed Raman Response
Diffraction Kerr Nonlinearity
22)(
22
||||1 ρω
βρσρ aN
AAEnt
NN
gb−+=
∂∂
Avalanche generation Multi-photon generation Plasma recombination
- Plasma Drude Model
Nonlinear Spatial Replenishment- multiple recurrence of collapse within pulse
M. Mlejnek, E.M. Wright and J.V. Moloney, Opt. Letts., 32, 382 (1998)
Experimental Verification
Detected radiation indicating recurrence of collapse.
A. Talebpour, S. Petit and S.L. Chin,“Re-focusing during propagation of afocused 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 profilebreaks up into multiple collapsing light filaments.[Campillo et al., Appl. Phys. Lett., Vol. 23, p628, 1973]
Strong Turbulence Scenario!
4D Strong Turbulence SimulationComputational Challenges:
•Severe compression of multiple interacting light filaments in space and time requires adaptive mesh refinement and parallel computation.
• Data Representation: - surface plot of filaments as slider
moves from front to back of pulsefixed propagation distance
- isosurface representation of filamentdistribution over pulse as a function of z.
Simulation- full (x,y,z,t) resolution- field intensity plots
Single Time Slice
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).
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.
Z=9.0 m
Iso-surface
Optical Turbulence• Chaotic filament creation within propagating pulse
- 3D box contains main part of pulse
LeadingEdge
TrailingEdge
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 modelplanned with Becker and Faisal.
• Beyond NLSE – resolve optical carrier while propagating over meter distances: UPPE model
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