文档介绍：Chapter 2 Theory of Nonlinear Optical Susceptibility
Symmetry of the Susceptibility
Intrinsic Permutation Symmetry
a field (1, j) oscillating in j direction with frequency 1
another field (2, k) oscillating in k direction with frequency 2,
generates a field named (1+ 2, i) oscillating in i direction with frequency (1+ 2)
takes role where generates
Then
It is also valid for third-order and higher-order susceptibilities.
Full Permutation Symmetry
if the relaxation effects can be neglected, as in the lossless medium, the frequencies of the incident fields and produced fields are far from resonance ng of the medium. Eq. (-53),
a full permutation
In the second-order NLO susceptibility, the factors (-1-2, i), ((1, j), and (2, k) can be interchanged freely for both input and output fields,
For general lossless medium, as the relaxation effect can be neglected,
the susceptibilities are real, hence
All of the frequency arguments of the nonlinear susceptibility can be freely interchanged, as long as the corresponding Cartesian indices are interchanged simultaneously.
let
two relations hold :
the sum frequency generation and the different frequency generation are related through
Kleinman's Symmetry
If optical waves whose frequencies i are much smaller than the lowest resonance frequency (lowest transition frequency) of the medium, the susceptibility will not strongly depend on the frequency i of the fields.
The magnitude of (n) measured in one wavelength can be used at other wavelengths, or for sum frequency generation and difference frequency generation will have the same value.
Kleinman's Symmetry condition
Spatial Symmetry
Axial symmetry, mirror symmetry and anti-centrosymmetry have special symmetry, which will affect the symmetry of the susceptibility tensor
utilizing spatial symmetry of the materials/crystals would greatly simplify the NLO susceptibility tensor and reduce the number of the nonzero ele