From second order Maxwell to first order with

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From second order Maxwell to first order with nonlinear index} No time dependence Normalization:

From second order Maxwell to first order with nonlinear index} No time dependence Normalization: and 2 D NONLINEAR SCHROEDINGER EQUATION

BEAM PROPAGATION MAXWELL Normalization: and solution of an eigenvalue equation (dimensionless form): Townes Soliton:

BEAM PROPAGATION MAXWELL Normalization: and solution of an eigenvalue equation (dimensionless form): Townes Soliton: 2 D NONLINEAR SCHROEDINGER EQUATION

Scaling parameters: Radius: Peak Field: o Such that o 2 2 o o critical

Scaling parameters: Radius: Peak Field: o Such that o 2 2 o o critical power SOLUTION: TOWNES SOLITON An infinite family of solutions because of The Townes soliton explains why a filamenting beam, as it starts collapsing, gets a profile independent of initial conditions

w Self trapping critical power: z What is the profile that is self-maintaining? For

w Self trapping critical power: z What is the profile that is self-maintaining? For P > Pcr , the beam collapses to a point.

Scaling parameters: Radius: r 0 Field: E 0 SOLUTION: TOWNES SOLITON Such that E

Scaling parameters: Radius: r 0 Field: E 0 SOLUTION: TOWNES SOLITON Such that E 02 r 02 critical power Alexander L. Gaeta Cornell University

Simulations with Elliptical Input Beam z=0 y x Alexander L. Gaeta Cornell University 3.

Simulations with Elliptical Input Beam z=0 y x Alexander L. Gaeta Cornell University 3. 5 Pcr z = 0. 8 z = 1. 8 4 X

K. D. Moll, G. Fibich, and A. L. Gaeta, Phys. Rev. Lett. 90, 203902

K. D. Moll, G. Fibich, and A. L. Gaeta, Phys. Rev. Lett. 90, 203902 (2003). intensity • Experiments performed in glass aberrated input power position output

We have seen the 2 D soliton in space. Time –space analogy: there is

We have seen the 2 D soliton in space. Time –space analogy: there is a 1 D soliton in time The derivation is another illustration of Fourier Transforms. We start with Propagation in the time domain where n = n(t) which produces phase modulation (for instance: n = n 0 + n 2 I) Next step: Propagation in the frequency domain where n = n(W) which implies k = n W/c = k(W) In frequency we have (redefining the retarded frame)

Study of linear propagation Expansion orders in k(W)--- Material property

Study of linear propagation Expansion orders in k(W)--- Material property

Propagation in the time domain PHASE MODULATION e E(t) = (t)eiwt-kz e(t, 0) n(t)

Propagation in the time domain PHASE MODULATION e E(t) = (t)eiwt-kz e(t, 0) n(t) or k(t) e(t, 0) eik(t)d

Propagation in the frequency domain DISPERSION e(DW, 0) n(W) or k(W) e(DW, 0)e-ik(DW)z Retarded

Propagation in the frequency domain DISPERSION e(DW, 0) n(W) or k(W) e(DW, 0)e-ik(DW)z Retarded frame and taking the inverse FT:

PHASE MODULATION e E(t) = (t)eiwt-kz e(t, 0) n(t) or k(t) e(t, 0) eik(t)d

PHASE MODULATION e E(t) = (t)eiwt-kz e(t, 0) n(t) or k(t) e(t, 0) eik(t)d d DISPERSION e(DW, 0) n(W) or k(W) e(DW, 0)e-ik(DW)z

PHASE MODULATION DISPERSION W

PHASE MODULATION DISPERSION W

Equation in the retarded frame Characteristic field: Normalized distance: Solitons: solutions of the eigenvalue

Equation in the retarded frame Characteristic field: Normalized distance: Solitons: solutions of the eigenvalue equation Characteristic time:

Solution: try first order soliton: In these normalized units, there is an infinite family

Solution: try first order soliton: In these normalized units, there is an infinite family of solutions of ``order 1'', defined as the product of peak amplitude A by pulse duration ts being unity. Correct pulse area:

Why non-linear Schroedinger equation? ] Multiply by z t z This is the “pure”

Why non-linear Schroedinger equation? ] Multiply by z t z This is the “pure” soliton equation. And that is where I get totally upset, because the space variable is now called t, and the time variable z, and the physics is for the birds!