Wave eqn for 2nd Order nonlinearity ( Ch. 3 of notes ) Apr. 14, 2014

+ Nonlinear wave equation (paraxial):
where correspond to the indices for the three waves. Note: left hand side is the linear wave equation right hand is the nonlinear term which couples the other beams to the current one, e.g.
and the phase of is .

+ Phase matching:
To have max. coupling -- phase matching.
Measured by coherent length .
Require long interaction length, i.e. all beams parallel.

+ 2nd harmonic generation:

where , and

+ Nondepleted pumping, i.e. :
Good for phase mismatch and nonparallel cases
where
.
-- small signal power conversion efficiency
Finite beam size, consider diffraction .

+ Depleted pumping:
Good for perfect phase match or parallel beams

where and

+ Three wave mixing:

+ Nondepleted pumping(s):
, ; sum freq generation
, ; up conversion
, down conversion

+ Phase matching with birefringence:
1. Choose the output polarization with lower refractive index at .
2. Type I phase matching -- two inputs with same polarizations; type II phase matching -- two inputs with different polarizations.
3. Determine propagation direction, e.g. 2nd harmonic
4. find

+ Sum freq generation:

+ Up-conversion ():
; where

+ Down-conversion (parametric amplification):
; where

3rd order nonlinearity ( Ch4 of notes ) Apr. 14, 2014

+ 3 order nonlinearity:

1 input freq 3rd harmonic; nonlinear refractive index.
2 input freq -- DC Kerr, DC 2nd harmonic generation, Raman scattering
3 input freq -- sum and difference freq generation

+ One input freq:
-- optical Kerr effect; automatically phase match.
-- 3rd harmonic generation; require phase match.

+ Nonlinear refractive index :

where and are the nonlinear refractive index coef.

+ Self-phase modulation (time):
Phase modulation is introduced to laser beam according to its intensity

+ Self-focusing (space):
n increases as I increases
Gaussian beam (2D transverse plane) converges in space, i.e. variable focusing lens. What happen at focus?
Slab beam (1D transverse plane) self-trapped inside a Kerr medium spatial soliton, i.e. forming an induced wave guide which has the most stable self-guiding mode in the shape of sech.
Recap -- ; slab beam stabilized to form soliton; Gaussian collapses catastrophically and media break down; require threshold power (const for Gaussian and proportional to 1/width for slab)

+ Self-bending (space):
Angle of deflection depends of the sign of .

+ Self-defocusing (space):
Laser beam diverges in additional to diffraction.

+ Spatial dark soliton & vortex soliton:
-- phase ramp and dark core at center
No threshold intensity; background intensity control size of vortex; Soliton (self-trapping) on 2D transverse plane; guidance for another beam.

+ Modeling equation:
where ;
Paraxial approx nonlinear Schrodinger equation (NSE)

+ 1+1D NSE:
Fundamental soliton where .
Threshold power
For 2D transverse dimension,

+ Split Step FFT method:

where

+ Raman gain:

Compare
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HW #10 due 4/22/14
1. Problem 19.2-2 (p. 796, 1st Ed; Problem 21.2-5, p. 934 2nd Ed)
2. Ex. 19.4-3 on p. 765 ( Ex. 21.4-2, p. 909 2nd Ed) (Note: what is )
3. An 8-cm-long ADP crystal (n=1.5, ) is used to ampli fy He-Ne laser light of wavelength 633nm. The pump is an argon laser of wavelength 334nm and intensity . Determine the gain of the amplifier. (Ex. 19.4-6 on p. 773)
4. Problem 19.4-1 on p. 797 (Problem 21.4-7, p. 935 2nd Ed)
Extra-Credit
Find the d matrix for 3m class crystal. Show that for two input waves in ordinary polarization will give nonlinear polarization with for . Use the procedure in Section 2.4 of the notes.