Nonlinear Optics (Ch. 19 1st Ed, Ch. 21 2nd Ed) Apr. 7, 2014

+ Magnitude of nonlinear coef.: [equation] (MKS) or ([equation]) and [equation] (MKS).

+ Method of solution for nonlinear wave eqn:
1. Born approx. -- [equation] , [equation] , [equation] and so on.
Iteration continues until error, [equation], is less than a small value.
1st Born approx [equation] stop at the 1st step if depletion of [equation] by nonlinear process is small
2. Coupled wave theory -- use paraxial wave eqn with [equation]

+ 2nd order nonlinearity: [equation]
1. 2nd harmonic and rectification -- [equation], e.g. IR [equation] visible light or IR to microwave or dc field
[equation] and [equation]
2. Electro-optic effect -- [equation] is dc and [equation] is optical.
3. 3-wave mixing -- [equation] and [equation]
Result in dc ( [equation] ), 2nd harmonics ( [equation] ), sum freq ( [equation] ) and difference freq ( [equation] ).
Conditions for conversion --
a. Conservation of energy [equation]
b. Phase matching [equation]; small angle between [equation] and [equation] is allowed
Optimum conditions --
High beam intensity and long interaction length (overlapping of beams)

+ Where is the 3rd wave for 3 wave mixing?
Interactions between 2 inputs and 1 output
a. Parametric interaction ([equation])
Up conversion -- inputs [equation], [equation].
Down conversion -- inputs [equation], [equation] [equation].
b. Parametric amplifier -- input pump ([equation]) and signal ([equation]) [equation] increase power at [equation] while decrease power at [equation] after certain distance; by-product idler at [equation]
c. Parametric oscillator -- put the amplifier into a cavity and noise as signal; with feedback the system oscillates
Input pump at [equation] and output power at [equation] and/or [equation] depending on the phase matching condition

+ Quantum picture: 2 low freq photons combine to form 1 high freq photon based on conservation of energy
Lead to Manley-Rowe Relations
[equation] where [equation].

2nd Order Nonlinearity (Ch2 of notes) Apr. 7, 2014

+ Linear susceptibility (Lorentz Model):
Oscillator model -- [equation]
Nonlinear oscillator -- [equation]

+ Tensor of 2nd order nonlinearity:
[equation] -- second harmonic generation
[equation] -- sum and difference frequency generations; parametric amplification
[equation] -- dc linear electro-optic effect
[equation] -- optical rectifications ([equation] electro-optic effect)


where [equation] is the Miller's index.

+ Nonlinear polarization: 1 input beam


2 input beams


+ Off-axis (optic axis) nonlinear polarization:
[equation], sum-frequency generation
[equation], second-harmonic generation.
Steps --
1) Decompose [equation] in term of principal axis directions
2) Find nonlinear polarization
3) Transform nonlinear polarization from principal axes coordinates in terms of the transverse plane.
HW #9 due 4/15/14
1. Problem 18.2-3 (p. 736 in 1st Ed; problem 20.2-5 on p. 872 in 2nd Ed)
2. Exercise 19.1-1 (p. 741 1st Ed; Exercise 21.1-1, p. 878 2nd Ed)
3. Prove Eqs. (19-2-11) (p. 747 1st Ed; Eqs. (21.2-13) p. 883) for 2nd order nonlinearity with inputs of [equation] and [equation].
Based on Lorentz model with anharmonicity, we see the separation of charge in E field (transverse) direction leading to nonlinear polarization of 2nd order (see supplementary notes sec.2.1-2.2). In this process, we have not considered magnetic force in the Lorentz equation. Show that if we consider effect of magnetic force, we will see DC rectification and 2nd harmonic effect along the longitudinal direction. (see Physical Review A, vol. 82, 013802, 2010)

Last Modified: Apr 06, 2014
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