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

+
Magnitude of nonlinear coef.:
(MKS) or ()
and (MKS).` `

+
Method of solution for nonlinear wave eqn:

1. Born approx. -- ,
,
and so on.` `

Iteration continues until error, , is less
than a small value.` `

1st Born approx
stop at the 1st step if depletion of by nonlinear process is
small

2. Coupled wave theory -- use paraxial wave eqn with

+
2nd order nonlinearity:

1. 2nd harmonic and rectification -- , e.g.` `
IR visible light or IR to microwave or dc field

and

2. Electro-optic effect -- is dc and is optical.` `

3. 3-wave mixing -- and

Result in dc ( ), 2nd harmonics
( ),
sum freq ( )
and difference
freq ( ).` `

Conditions for conversion --

a. Conservation of energy

b. Phase matching ; small
angle between and 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 ()

Up conversion -- inputs , .` `

Down conversion -- inputs , .` `

b. Parametric amplifier -- input pump () and signal
() increase power at while decrease power at
after certain distance; by-product idler at

c. Parametric oscillator -- put the amplifier into a cavity and noise as
signal; with feedback the system oscillates

Input pump at and output power at and/or
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

where .` `

**2nd Order Nonlinearity
()** Apr. 7, 2014

+
Linear susceptibility (Lorentz Model):

Oscillator model --

Nonlinear oscillator --

+
Tensor of 2nd order nonlinearity:

--
second harmonic generation

--
sum and difference frequency generations;
parametric amplification

--
dc linear electro-optic effect

--
optical rectifications ( electro-optic effect)

where is the Miller's index.` `

+ Nonlinear polarization: 1 input beam

2 input beams

+
Off-axis (optic axis) nonlinear polarization:

,
sum-frequency generation

,
second-harmonic generation.` `

Steps --

1) Decompose 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
and .` `

Extra-Credit

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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