Unidirectional Pulse Propagation Eqn (UPPE)

Unidirectional Pulse Propagation Eqn (UPPE)

General Philosophy:

1. Retain the rigor of Maxwell ( carrier resolved) but enable propagation over macroscopic distances

2. Provides a unifying theoretical framework for seamlesslyderiving the many nonllinear propagation equations in the literature

Two Implementations:

1. Z-propagated UPPE – connects to envelope description2. T-propagated UPPE – links directly to Maxwell equations

Maxwell Equations

Non-magnetic medium (µ=µ0) with linear permittivity ε(ω,x,y).

0

0

t t

t

j P E H

H E

ε ε

µ

+ ∂ + ∂ ∗ = ∇×

− ∂ = ∇×where

( ) ( )0

( )P t E E t dε ε τ τ τ∞

= ∗ ≡ −∫is the linear optical response.

Note: We return to the nonlinear optical response later.

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Maxwell Numerical Solvers

Vector Maxwell Simulators

Maxwell’s Equations:

0,0

;

=•∇=•∇∂

∂=×∇

∂∂

−=×∇

→→

→→

BDtDH

tBE

Constitutive relations:→→→→

+== PEDHB 00 , εµ

FDTD Method Yee scheme

Simulation size grows as N3 making memory and CPU prohibitive!

Note: Evolve the solution in time!

Example of AMR FDTD discretization for TM-mode in two space dimensions

- A.R.Zakharian, M. Brio

, yx z z

y xz

HH E Et y t x

H HEt x y

∂∂ ∂ ∂= − =

∂ ∂ ∂ ∂∂ ∂∂

= −∂ ∂ ∂

( )

( )

1/ 2 1/ 2, 1/ 2, , 1/ 2, , 1/ 2, 1/ 2 , 1/ 2, 1/ 2

1/ 2 1/ 2, 1/ 2, , 1/ 2, , 1/ 2, 1/ 2 , 1/ 2, 1/ 2

n n n nx i j x i j z i j z i j

n n n ny i j y i j z i j z i j

tH H E EytH H E Ex

+ −+ + + + + −

+ −+ + + + − +

∆− = − −

∆∆

− = −∆

AMR refines the computational domain locally using nested rectangular grid patches. A standard FDTD update is applied to each patch.

At the coarse/fine grid interfacesthe solution is interpolated. Dashed lines denote boundaries of the ghost cells around the fine region. Arrows show a sample interpolation from coarse to fine values of the electric field.

Object-Oriented Implementation

• AMR object coordinates the recursive time-stepping of each refinement level• Level object performs memory management for a collection of subregions • Interlevel objects handle interpolation, averaging and circulation consistency• Material objects encapsulate an algorithm for a particular material model.

Nested Grids on a 3D PBG Structure- A. Zakharian, C. Dineen

- confined defect mode of a 3D PBG lattice

Resolving a QD in a PBG Lattice4-20 nm

-Jens Foerstner - experiment Gibbs et al.

QD Wavefunction

AMR Mesh

230 240 250 260 270 280 290 300

emis

sion

frequency [THz]

Coupling QD to High Q Defect Mode

Energy level splitting

Scattering from a metal Nanosphere – 3 nested AMR Levels

Maxwell Solvers are too Expensive!!

Need EM pulse propagators that bridge the considerablegap between Maxwell FDTD and slowly-varying envelope

For extreme NLO studies, such propagators must resolvethe underlying optical carrier wave while propagating overmacroscopic distances.

Carrier-wave shocking can occur for extreme electric fields

R.G. Flesch, A. Pushkarev and J.V. Moloney, “Carrier wave shocking of femtosecond optical pulses,” Phys. Rev. Let., 76(14), 2488-2491 (1996).

z-Propagated UPPE I- more suitable for long propagation distances and waveguides

Standard Textbook Approach

Expand transverse fields in terms of modes (free space or waveguide)

( )

,

( )

,

( , , , ) ( , ) ( , , )

( , , , ) ( , ) ( , , )

m

m

i z i tm m

m

i z i tm m

m

E x y z t A z E x y e

H x y z t A z H x y e

β ω ω

ω

β ω ω

ω

ω ω

ω ω

−

−

=

=

∑

∑Shorthand notation:

( )

( )

( , , )

( , , )

m

m

i z i tm m

i z i tm m

E E x y e

H H x y e

β ω ω

β ω ω

ω

ω

−

−

≡

≡

z-Propagated UPPE II

Scalar multiply Maxwell eqns by complex conjugate modal fields

*0

*0

( ) ( )m t m t m

m t m

E j P E E E H

H H H E

ε ε

µ

∗ ∗

∗

• + ∂ + • ∂ ∗ = • ∇ ×

− • ∂ = • ∇×

( ) ( ) ( )b a a b a b• ∇ × = ∇ • × + • ∇ ×Use formula

( )( )

0

*0

( ) ( )m t m t m m

m t m m

E j P E E H E H E

H H E H E H

ε ε

µ

∗ ∗ ∗ ∗

∗ ∗

• + ∂ + • ∂ ∗ = ∇ • × + • ∇×

− • ∂ = ∇ • × + • ∇ ×

z-Propagated UPPE IIISince modal fields themselves satisy Maxwell’s eqns

0

0

m t m

m t m

E H

H E

µ

ε ε

∗ ∗

∗ ∗

∇× = − ∂

∇× = − ∂ ∗

Substitute in previous eqns:

( )( )

0 0

*0 0

( ) ( )m t m t m t m

m t m t m

E j P E E H E H H

H H E H E E

ε ε µ

µ ε ε

∗ ∗ ∗ ∗

∗ ∗

• + ∂ + • ∂ ∗ = ∇ • × − • ∂

− • ∂ = ∇ • × + • ∂ ∗

Subtract two equations and collect terms involving full time derivatives

( )( )

0

*0

( ) [ ]

[ ]

m t t m m

t m m

E j P E E H E

H H E H

ε ε

µ

∗ ∗ ∗

∗

• + ∂ + ∂ • ∗ = ∇ • ×

−∂ • − ∇ • ×

z-Propagated UPPE IVNow integrate over whole xyt domain – Note that all terms except the first and ∂z (implicit in ∇) are derivatives that give rise to surface terms that vanish for localized pulses

( ) [ ] [ ]m t z m z mE j P dxdydt z H E dxdydt z E H dxdydt∗ ∗ ∗• + ∂ = ∂ • × −∂ • ×∫ ∫ ∫

Because only transverse fields enter above eqns, we use modal expansions

( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ),

,

[ , ]

[ , ]

n m

n m

m t

i z i t i z i tz n n mn

i z i t i z i tz n n mn

E j P dxdydt

z A z H E e e dxdydt

z A z E H e e dxdydt

β β ω ω

β β ω ω

ω

ω

∗

Ω − Ω − +∗Ω

Ω − Ω − +∗Ω

• + ∂ =

∂ • Ω Ω × −

∂ • Ω Ω ×

∫∑∫∑∫

Integration over time reduces to Kronecker delta between angular frequencies, removing sum over Ω

z-Propagated UPPE VCollecting like terms in the previous equation

( ) ( ) ( ) ( )

( ) ( ) ( )

,

[ , , , , ( , , ) , , ]

n mi z i zm t z nn

n m n m

E j P dxdydt A z e e

z H x y E x y E x y H x y dxdy

β ω β ω

ω ω ω ω

−∗

∗ ∗

• + ∂ = ∂ Ω ×

• × − ×

∑∫∫

This equation can be reduced further by using the general orthogonality relation:

( ),[ ] 2m n m n m n mz E H H E dxdy Nδ ω∗ ∗• × − × =∫To yield

( ) ( ) ( ) ( ) ( ),, 2n mi z i zm t z n m n mn

E j P dxdydt A z e e Nβ ω β ω δ ω−∗ • + ∂ = −∂ Ω∑∫Evolution equation for the expansion coefficients follows immediately

z-Propagated UPPE VI

( ) ( )

( ) ( )

2 2 2

2 2 2

1,2

, ,

m

T Y X

i z i tz m

T Y Xm

m t

A z dt dy dx eN XYT

E x y j P

β ω ωω

ω

+ + +

− +

− − −

∗

∂ = − ×

• + ∂

∫ ∫ ∫

Specializing to a homogeneous medium (plane waves), the frequency and wavenumber propagation constant is

( ) ( ) ( )2 2 2 2, , , , , /

x yk k s z x y x yk k k c k kβ ω ω ω ε ω± ≡ = − −

( ), , ,

, , , , , ,0

exp , ,

1x y

x y x y

k k s s x y z x y

k k s k k s

E e ik x ik y k k k

H k E

ω

µ ω

±

± ±

= + ±

= ×

where es=1,2 are unit polarization vectors normal to , ,x y zk k k k=

and the modal field amplitudes are

z-Propagated UPPE VIIFor this plane wave basis, the normalization constant is easily calculated to be

( ), , ,

0

, ,x y

z x yk k s

k k kN

ω

µ ω± = ±

Substituting this expression into eqn …

( )

( )( )

2 2 2( )0

2 2 2

,2

, , , ( , , , )

x yz

T Y X

i t k x k yik zz m

T Y Xz

s t

A z e dt dy dx ek XYT

e j x y z t P x y z t

ωωµω+ + +

− −−

− − −

∂ = − ×

• + ∂

∫ ∫ ∫

Above integral is nothing other than the spatial and temporal Fourier integral transform and can be written in the spectral domain as

z-Propagated UPPE VIII

( ) ( )( ), ,20

, ( , ) ,2

z

x y x y

ik zz m s k k k k

z

A z e e i P z j zc kωω ω ω ω

ε−∂ = − • −

This version is implemented numerically in the spectral domain!

For completeness, we give the spectral representation for evolving the total transverse field:

( ) ( ) ( ), ,, , , ,

1,2

, , z x y

x y x y

ik k k zk k s k k s

s

E z e A z e ωω ω⊥ ⊥+

=

= ∑

( ) ( )

( )

, ,

2

, ,2 20 0

, ,

( , ) ,2 2

x y x y

x y x y

z k k z k k

s s k k k kz z

E z ik E z

ie e P z j zc k c k

ω ω

ω ωω ωε ε

⊥ ⊥

⊥

∂ = +

• −

NumericsExample: Scalar version of z-UPPE

( ) ( )

( )

, ,

2

, ,2 20 0

, ,

( , ) ,2 2

x y x y

x y x y

z k k z k k

k k k kz z

E z ik E z

i P z j zc k c k

ω ω

ω ωω ωε ε

⊥ ⊥∂ = +

−

Actually solve for “plane-wave/waveguide” expansion coefficients

( ) ( )( ), ,20

, ( , ) ,2

z

x y x y

ik zz m k k k k

z

A z e i P z j zc kωω ω ω ω

ε−∂ = − −

Large system of coupled ode’s. Linear propagator part in spectral domain is trivially parallelizable. However, nonlinearpolarization results in global coupling of all field variables!- require shared memory machine

t-Propagated UPPE I

This version of UPPE is most closely aligned with Maxwell eqns.Starting point is different from previous case.Based on a projection operator technique – Kolesik,Moloney and MlejnekPRL, 89, p283902, (2002); Kolesik and Moloney, PRE, 70, 036604 (2004).

Involves derivation of a unidirectional propagation for the electric displacementvector ( ),D r t

Let denote the spatial Fourier transforms of the fields e.g

( ) ( ) ( ), ,E k H k D k

( ) ( ) ( ) ( ) ( ) ( )1D k D r k D r D k r−= ℑ = ℑ

t-Propagated UPPE IIDefine the following projection operations

( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )2

1sgn12

sgn

z

z

D k k k H kD k kP

kH k H k k k D kk

ω

ω±

× = ± ×

∓

These operators provide a unity decomposition

1P P+ −+ =

They behave as projectors as long as they act over divergence-free subspaces i.e

2P P± ±=

t-Propagated UPPE IIIP+ leaves invariant any plane wave solution to Maxwell’s equations that propagate with kz component > 0 i.e it annihilates all plane wave components propagating in the negative z-direction. To show this use

( )( )

( ) ( )( ) ( )2

1 k H kD k k

kH k k D kk

ω

ω

− × = ×

To obtain

( )( )

( ) ( )( ) ( )

( ) ( )( )

2

11 sgn( )12

1 sgn( )

1 sgn2

z

z

z

k k H kD k kP

kH kk k D k

k

D kk

H k

ω

ω±

− ×

= ± × + ± =

∓

t-Propagated UPPE IVKey steps in the derivation:

Constitutive Relation ( ) ( ) ( ) ( )0, , , ,NLD r t r t E r t P r tε ε= ∗ +

( ) ( )( ) ( ) ( )0 NLD k k E k P kε ε ω= +Fourier Representation:

where implicitly defines ( ) ( )( ) 2k k kω ε = ω2

( )kω

Introduce compact notation:

( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( )0 0

1 1;L NL L NLE k D k E k E k E k P kk kε ε ω ε ε ω

= = − =

t-Propagated UPPE VUse this formal splitting in Maxwell’s equations written in Fourier (plane wave basis)

( )( )

( )( )

( )( ) ( )

00 0

0

tNLL

ik H k ik H kD kii i k E kk E k k E kH k µµ µ

× × ∂ = ≡ + − × − × − ×

Acting on this equation with P+, projects out forward propagating component.

( )( )

( )( )

( )( )

( )( ) ( )

0

00

0

f

t t

f

NLL

ik H kD k D kP P i k E kH k H k

ik H kP P ii k E kk E k

µ

µµ

+ +

+ +

× ∂ ≡ ∂ = − × × ≡ + − ×− ×

t-Propagated UPPE VIBecause the projector is diagonal in the plane wave basis

( )( ) ( ) ( )

( )0

f

L f

ik H k D kP i ki k E k H k

ω

µ

+

× = − − ×

Second term on RHS is evaluated using definition of projector

( )( ) ( )

( )

( ) ( )

( )( ) ( )

( ) ( ) ( ) ( )

( )( ) ( )

0

00

22

22

0 2

2

2 22

22

NL

NLNL

NL NLNL

NLNL

i k k E kk

P i k E k i k E k

k i i kkk P k k P ki k k P k kkcc i k P ki k P k

kk

µ ω

µµ

ωω ω

ε ωε ω

+

× × = − × − ×

− • × × = =

− × − ×

Unidirectional Pulse Propagation Equation Unidirectional Pulse Propagation Equation

tt--propagated UPPEpropagated UPPE

Equation written in spectral domain.Not a PDE in real-space representation

Linear propagation,Contains space-time focusing “terms”

Nonlinear response couplingContribution from self-steepening

div .E - related termNonzero due to gradients.

Nonlinear polarization of medium calculatedfrom material equation (Kerr effect,plasma,...)

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This equation is exact as long as D is full field

To close the system of equations for numerical simulations, we approximate:

Horizontal scale equal to shortest integration stepduring the whole simulation

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Only eight points per wavelength, still correctdispersion properties

Shock regularization in femtosecond pulse propagation

Continuum generation starts in the steep trailing edge of the pulse.Situation after 0.55 m propagation.

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Continuum generation “regularizes” the shock.Situation after 0.57 m propagation. The high-frequencytrailing part of the pulse carries the continuum light.

Supercontinuum generation•White light continuum is generated in the back of the pulse over a

short propagation distance of a few centimeters.

•After explosive spectral broadening and “shock” regularization the

spectral content of the pulse remains essentially unchanged

•Off-axis directions exhibit relatively higher supercontinuum intensities

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What measures extent and BandgapDependence of supercontinuum?

• Interplay between optical Kerr effect, plasma generation and chromatic dispersion

Calculation for Water

M. Kolesik, J.V. Moloney, G. Katona and E.M. Wright , “Physical Factors Limiting the Spectral Extent and Band Gap Dependence of Supercontinuum Generation”, Physical Review Letters, 91, No. 4, (2003)

Conventional wisdom: Interplay between self-focusing and mechanismthat arrests collapse i.e plasma generation – chromatic dispersion not a player

Carrier frequency

real water dispersion

Artificial medium dispersion

Conclusion: chromatic dispersion controls bandwidth of generated supercontinuum

Transition from UPPE to Various Envelope Models

Write z-propagated scalar version of UPPE in compact form

( ) ( ) ( ), , ,, , ,x y x y x yz k k k k k kE z iKE z iQP zω ω ω∂ = +

( ) ( )2 2 2 2, , /x y x yK k k c k kω ω ε ω= − −where

is the linear propagator in the spectral representation, and

( )( )

2

2 2 2 2 20

, ,2 /

x y

x y

Q k kc c k k

ωωε ω ε ω

=− −

will be called the nonlinear coupling term

Idea: Replace K and Q by suitable Taylor approximations!

Transition from UPPE … II

To obtain envelope equations, we need to express the total field interms of an envelope function and a reference carrier frequency ωRwith corresponding wavenumber kR = K(0,0, ωR).

( ) ( ) ( ), , , , , , R Ri k z tE x y z t A x y z t e ω−=

Similar factorization applied to P(x,y,z,t)

Key Identifications:

( ) ( )( )R t x x y y R z zi ik ik i k kω ω ω− ↔ ∂ ↔ ∂ ↔ ∂ − ↔ ∂

Using these convert from spectral domain to real space variables.

Derivation of Nonlinear Schrödinger Equation

Taylor expand linear spectral propagator (paraxial approximation)

( ) ( )

( ) ( ) ( )

2 2 2 2

21 2 2

, , /

'' 12 2

x y x y

R g R R x yR

K k k c k k

kk v k kk

ω ω ε ω

ω ω ω ω−

= − −

≈ + − + − − +

In nonlinear coupling term ignore all variable dependence and set ω = ωR

( )( ) ( )

2

2 2 2 2 200

, ,22 /

Rx y

Rx y

Q k kn cc c k kωωω

ε ωε ω ε ω= ≈

− −

Substitute these truncated expressions into z-UPPE

Nonlinear Schrödinger Equation (NLSE)

For simplicity assume an instantaneous Kerr effect

( ) 20 22 | |RP n n A Aε ω=

Inserting the above into z-UPPE

( ) ( ) ( )21 2 2 22

'' | |2 2

Rz g R R x y

R

k iA iv A i A k k A i n A Ak c

ωω ω ω ω−∂ = − + − − + +

Final step – use Fourier –real space identities above

( )1 2 22

'' | |2 2

Rz g t T tt

R

i kv A A i A i n A Ak c

ω−∂ + ∂ = ∇ − ∂ +

This derivation shows explicitly the approximations made in deriving NLSE in a physically self-consistent manner!

Nonlinear Envelope Equation (NEE)Brabec and Krausz, PRL (1997)

Now Taylor expand the linear spectral propagator in wavenumber (paraxial approx) but retain exact frequencydispersion.

( ) ( )

( ) ( ) ( )

2 2 2 2

2 2

, , /

2

x y x y

x yR

K k k c k k

ck k kn

ω ω ε ω

ωω ω

= − −

≈ − +

where k(ω) is the full frequency dispersion and can be written

( ) ( ) ( ) ( )1R g R Rk k v Dω ω ω ω ω ω−= + − + −

and

( ) ( )2 !

R

nnR

R nn

kDnω ω

ω ωω ω

ω

∞

= =

− ∂− = ∂

∑

Nonlinear Envelope Equation (NEE) IIUnlike the NLSE case, we partially retain the frequency dependencein the nonlinear coupling term but neglect transverse wavenumberdependence

( )( ) ( )

2

2 2 2 2 200

( ), ,22 /

R Rx y

Rx y

Q k kn cc c k k

ω ω ωωωε ωε ω ε ω

− += ≈

− −

Inserting these approximations into z-UPPE

( ) ( )

( )( ) ( )

( )( )

1

1

2 2

0

12

12

z g R R

Rx y

R R R

RR

R R

A iv A iD A

ic k k An

i Pcn

ω ω ω ω

ω ωω ω ω

ω ωωε ω ω

−

−

∂ = − + −

−− + +

−

+ +

Nonlinear Envelope Equation (NEE)

The final step is to convert from spectral to real space using previous identities to yield NEE

( ) ( )

11 2

20

1 12 2

Rz g t t t T t

R R b R R

iki i iv A iD i A A Pk nω ε ω ω

−

− ∂ + ∂ = ∂ + + ∂ ∇ + + ∂

This equation was originally derived in a rather ad hoc fashionby Brabec and Krausz (1997).

Here approximations are explicit.

Practically, the dispersion operator D should be evaluated exactlyin the spectral domain!

Partially Corrected NLS (PC-NLS)This equation is essentially a further but dangerous approximationto NEE!!Essentially, the inverse frequency term in the free propagator isapproximated as: 1

1 1R R

R R

ω ω ω ωω ω

− − −

+ ≈ −

The only potential justification of this is that the generated spectralbandwidth during propagation is much less in magnitude thanthe carrier reference frequency – yet used in supercontinuum studies!

PC-NLS Model

( ) ( )1 2

20

1 12 2

Rz g t t t T t

R R b R R

iki i iv A iD i A A Pk nω ε ω ω

− ∂ + ∂ = ∂ + − ∂ ∇ + + ∂

Relation to envelope equations Relation to envelope equations –– summary:summary:

identifying physical meaning of underlying approximationsidentifying physical meaning of underlying approximations

UPPE (exact): Almost always very small deviationsfrom exact relation

Brabec&Krausz:NEE

Paraxial (NLSE):

Partially corrected

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Locally better than in NLSE,but globally this partial correction is worse than none correction at all!

Truncated Envelope Models Yield Spurious Artifacts in SC

• Graphs of coefficient of transverse wavenumber for various approximations to UPPE.

Coefficient of k⊥ for different equationsSupercontinuum for different models

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extremely dangerous pcNLS

Spatial power spectrum comparison: UPPE vs. partially corrected NLSAr

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emati

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UPPE pcNLS

First Order Propagation Equation (FOP)Geissler et al. Phys.Rev. Letts, 83, p2930 (1999)

While this equation resolves the carrier wave (non-envelope), it neglects entirely linear chromatic dispersion i.e approximate K by

( ) ( ) ( )2 2 2 2 2 2, , /2x y x y x ycK k k c k k k k

cωω ω ε ω

ω= − − ≈ − +

Similarly for Q

( )( )

2

2 2 2 2 200

, ,22 /

x y

x y

Q k kcc c k k

ω ωωεε ω ε ω

= ≈− −

i.e same as NEE except with vacuum in role of linear medium!

Here we retain the total field rather than an envelope.

First Order Propagation Equation (FOP)Substituting the approximations for K and Q into the scalar z-UPPE

( ) ( ) ( ), , ,, , ,x y x y x yz k k k k k kE z iKE z iQP zω ω ω∂ = +

we obtain

( ) ( ) ( ) ( )

( )

2 2, , ,

,0

, , ,2

,2

x y x y x y

x y

z k k k k x y k k

k k

cE z i E z i k k E zc

i P zc

ωω ω ωω

ω ωε

∂ = − +

+

When transforming to real space the term ω-1 gives rise to an integralover time.FOP Equation

( ) ( ) ( )2

0

1( ) , , ,2 2

t

z t T tc iE r t d E r P r

c cτ τ τ

ε⊥ ⊥ ⊥−∞

∂ + ∂ = ∇ − ∂∫

Forward Maxwell Equation (FME)Hasakou and Hermann, Phys. Rev. Letts, 87, p203901 (2001)

Another non-envelope equation – although written in vector form, individual components decouple due to neglect of ∇∇•ELinear propagator and nonlinear coupling closely parallel NEE

( ) ( )

( ) ( ) ( )

2 2 2 2

2 2

, , /

2

x y x y

x yR

K k k c k k

ck k kn

ω ω ε ω

ωω ω

= − −

≈ − +

( )( ) ( )

2

2 2 2 2 200

, ,22 /

x ybx y

Q k kn cc c k k

ω ωωε ωε ω ε ω

= ≈− −

Only difference with NEE is that the chromatic dispersion ofthe index of refraction is preserved!

Forward Maxwell Equation (FME) II

Substituting previous approximations for K and Q into z-UPPE

( ) ( ) ( ), , ,, , ,x y x y x yz k k k k k kE z iKE z iQP zω ω ω∂ = +

yields

( ) ( ) ( ) ( ) ( )

( ) ( )

2

0

, , , , , , , , ,2

, , ,2

z T

b

iE x y z ik E x y z E x y zk

ci P x y zn

ω ω ω ωω

µ ω ωω

∂ = + ∇

+

This is essentially FME except that we have not transformed to aframe moving with the vacuum light velocity. The reason that wedo not do this here is that, in a strongly dispersive medium, weneed a reference frame moving with the group velocity.

Extreme Focusing - ∇•Ε≠0

Initial tightly focused 170 fs pulse with 2 µm initial waist simulated with full UPPE model

Radial Symmetry

Equivalent “Paraxial” scalar NLSE description: Stops criticalfocusing

Summary of Propagation Equations

1. The many propagation equations are mostly easily derived from UPPE

2. Both NEE and FME are very closely related. The most efficient way to solve NEE/FME is via a split-operatormethod where the dispersion operator is computed exactlyin the spectral domain.

3. It is necessary to define a reference carrier frequency/wavenumber pair ( ωR,kR). In principle, these can be chosenarbitrarily although the best choice is at then central pulse wavelength. Envelope models are in general sensitive to thechoice of this pair and care has to be exercised in numerics.

Atmospheric Femtosecond ProbeWöste et al., Laser und Optoelektronik, Vol. 29, p51 1997.

White Light Continuum

The Physics

• White light continuum spectroscopic probe

• Optical breakdown creates narrow plasma filaments - RF emission/lightning control.

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Light String Physics

• Nonlinear self-focusing in air – Pth=3 GW

• Extreme self-phase modulation - remote white light supercontinuum spectroscopic source.

• Dilute plasma channel generation – THz source,remote LIBS spectroscopy.

• Light string diameters below inner turbulence scale – obscurant penetration

Energy Fluence – 3D + 1 Simulation- Turbulent Atmospheric Light strings - M. Mlejnek et al. PRL 83, 2938 (1999)

- 6 meter light string propagation distance- 5 mm x 5 mm patch in center of pulse

Missing Physics?O2 and N2 are key constituents of air – other species H2O, Ar

Collaboration with A. Becker Dresden

• Saturation of MPI rates

• Saturation and delay of nonlinear self-focusing

• Nonequilibrium carriers in “hot” plasma

• Explanation for explosive filamentation followed by quiescent regime?

Supercontinuum generation in air• White light continuum is generated in the back of the pulse over a

short propagation distance of few centimeters.

• After explosive spectral broadening and “shock” regularization the

spectral content of the pulse remains essentially unchanged

• Off-axis directions exhibit relatively higher supercontinuum intensities

Femtosecond White Light Lidar- Remote ultra-broadband spectroscopic source - from Teramobile group

Water Spectrum

828 829 830 831 832

0.4

0.5

0.6

0.7

0.8

0.9

1.0

m easurem ent

wavelength [nm ]

calculation(after HITRAN)

920 nm

680 nm

Remote LIBS Spectroscopy-Teramobile group – K. Stelmaszczyk et al. APL, 85, 3977 (2004)- Potential to extend to Kilometer ranges.- 250 mJ 80 fs chirped pulses at 800nm. Beam diameter - 3 cm

Penetration through Obscurants- femtosecond light string self-healing

Measured Energy in Finite Aperture- Courvoisier et al, APL (2003)

Energy loses are minimal even for large (100 micron) droplets

- opaque screen (droplet) insensitive to location of pulse during nonlinear replenishment cycle – background dynamically restores pulse

propagation distance

Self-focusing filament

Screen modeling a dropletFilament recovers

Before droplet After hitting droplet 10cm after droplet

All filament’s evolution stages are robust: timing of the collision is unimportant

M. Kolesik et al., Optics Letters, 29, 590 (2004)

Incoherent THz Emission from Light Strings

- fully microscopic theory – Hoyer et al. PRL, 94, p115004, (2005)

ω-independent

T = 3000 K

T = 2000 K

Emission Spectrum THz Polarization Dependence

Experiment Tzortzakis et al.OL 27, 1944 ‘02

String Axis

ω < ω PL

ω ≥ ω PL

Calculation:Microscopic Theory String Axis

Nonlinear X-Waves- normal GVD + collapseResearch Highlighted in Physics Today, October 2004

M. Kolesik et al., PRL, 92, 253901-2 (2004)M. Kolesik et al., PRL, 91, 043905 (2003)

Lake Como Billboard

Induced Nonlinear Dynamical Grating- dynamical 3 wave interaction

Supercontinuum Generation in Sub-Micron Diameter Tapered Fibers

• Unidirectional Pulse Propagation Equation (UPPE)GOAL: Generate spectrally flat SC

• Sub-micron core or microstructured fiber design is critical to generating flat SC.

Evolution of Pulse and SC in Sub-Micron Diameter fiber

• Application of z-UPPE using waveguide modal basis- M.Kolesik, E.M.Wright, J.V. Moloney, Simulation of femtosecond pulse propagation in sub-micron diameter tapered fibers, Apply. Phys. B 79,293 (2004).

Extreme Focusing - ∇•Ε≠0

Equivalent “Paraxial” scalar NLSE description:

Initial tightly focused 170 fs pulse with 2 µm initial waist simulated with full UPPE model

Radial Symmetry

Stops criticalfocusing

Extreme SC Spectral Shift

• Pancake pulse 5 mm x 5 mm patch

• Input peak intensity = 4x1016 W m-2

• Pulse duration = 7 fs

• λ = 800 nm

Implication: Nonlinear dispersion of Kerr and MPI crucial!

Single Focused 5 Fs Filament

770 nm

900 nm

Approach to Self-focusing collapse After collapse – onset of red-shift

Red-shift of central carrier wavelength

Summary• UPPE equation rigorously describes ultrashort pulse propagation

under extreme conditions.

• Critical need to understand nonlinear dispersion of n2 and MPI cross-sectionby careful low power diagnostic experiments of air constituents.

• Applications include:

- Atmospheric light string propagation – fs LIDAR

- Penetration through obscurants

- Remote LIBs spectroscopy over multi-Km distances

- THz generation by collapsing light strings

- Supercontinuum shaping in sub-micron core and photonic crystal fibers