Michael Scalora U.S. Army Research, Development, and Engineering Center Redstone Arsenal, Alabama, & Universita' di Roma "La Sapienza" Dipartimento di Energetica OPTICS BY THE NUMBERS L’Ottica Attraverso i Numeri Rome, April-May 2004

BPM:Propagation in Planar Waveguides Retarded Coordinate trasformation: time dependence, Raman scattering, self-phase modulation in PCFs

air core 14  m 5  m fig.(4) Study the transmissive properties of guided modes. Propagation into the page

Assuming steady state conditions…

Wave front does not distort: Plane Wave propagation Diffraction is very important

This equation is of the form: Where: Using the split-step BPM algorithm

a=1.4  m b=1  m Example: Incident angle is 5 degrees Assume   =0

x The cross section along x renders the problem one-dimensional in nature

Transmissive properties in the linear (low intensity) regime For two different fibers. We set   =0

Field tuning corresponds to High transmission state. Direction of propagation

Same as previous figure.

For the example discussed: 5-mm guide ~ 8 minutes on this laptop 3.2GHz, 1Gbts RAM

If  (3) is non-zero, the refractive index is a function of the local intensity. Solutions are obtained using the same algorithm but with a nonlinear potential.

air core 14  m 5  m fig.(4) Optical Switch

The band shifts because the location and the width of each gap depends on the exact values of n 2 and n 1, and on their local difference.

fig.(5a)

fig.(5b) Optical Switch on off

fig.(6) off on

Retarded coordinate Transformation

N.B.:An implicit and important assumption we have made is that one can go to a retarded coordinate provided the grating is shallow so that a group velocity can be defined unumbiuosly and uniquely. In other words, the effect of the grating on the group velocity is scaled away into an effective group velocity v. It is obvious that care should be excercised at every step when reaching conclusions, in order to properly account for both material index and modal dispersion, if the index discontinuity is large.

Symplifying and Dropping all Higher order Derivatives…

Now we look at the linear regime, by injecting a beam inside the guide from the left and then from the right.

Input Spectrum Output Spectrum Propagating from left to right the pulse is tuned on the red curve, igniting self-phase modulation, and the spectral shifts indicated on the graph. A good portion of the input energy is transmitted. Spectra are to scale.   Fig. 4 I  W/cm 2 n 2  5  cm 2 /W L  8 cm   100 fs ON-AXIS

Propagation from right to left does not induce nonlinearities because the light quickly dissipates. The pulse is tuned with respect to the blue curve. Spectra are to scale. Fig. 4

Initial pulse profile Final profile

Spectrum of the pulse as it propagates. Note splitting. Initial profile

Self-phase modulation A process whereby new frequencies (or wavelengths) are generated such that: Example: input 100fs pulse at 800nm is broadened by ~30nm

Stimulated Raman Scattering

The simplest case Raman Soliton: A sudden relative phase shift between the pump and the Stokes at the input field generates a “phase wave”, or soliton, a temporary repletion of the pump at the expense of the Stokes intensity

The simplest case The gain changes sign temporarily, For times of order 1/  The soliton is the phase wave The Input Stokes field undergoes a  -phase shift

The Pump signal is temporarily repleted The Stokes minimum is referred to as a Dark Soliton

PUMP FIELD z  z=0 z,  =L,0

PUMP FIELD

STOKES FIELD

PUMP FIELD

STOKES FIELD

The onset of diffraction causes the soliton to decay… …almost as expected. Except that…

… the Stokes field undergoes significant replenishement on its axis, as a result of nonlinear self focusing

PUMP FIELD

STOKES FIELD

PUMP FIELD

STOKES FIELD Poisson Spot like effect

Examples: single slit

Direction of Propagation