文档介绍：§ Effective SHG Coefficient
SHG coefficient matrix
when the Kleinman's symmetry condition is valid, introduce a SHG coefficient in nonlinear crystal
Then, the nonlinear polarization
Assuming that dijk is symmetric in its last two indices jk
whenever Kleinman's symmetry condition is valid.
it is valid in general for SHG, since in this case n and m are equal.
simplify the notation by introducing a contracted matrix dil according to the prescription:
(jk )
11
22
33
23
32
13
31
12
21
l
1
2
3
4
5
6
The nonlinear susceptibility tensor is 3 6 matrix
the Kleinman permutation symmetry condition is the indices of dijk can be freely permuted, thus
dil have only 10 independent elements
As an example, the spatial symmetry ( 4 m) of the KDP crystal,
The matrix equation of the nonlinear polarization
How to calculate the effective NLO coefficient deff
Under phase matching condition, NLO crystal has definite orientation in the space even for uniaxial crystal
The direction of the crystal and the interacting beams
Eo must stay in xy-plane
electric ponents
where aj is the direction cosine of ordinary field with respect to each axis.
The extraordinary polarization is perpendicular to both the k vector and the ordinary polarization
in negative uniaxial crystal for type-I phase matching
The ponents of fundamental beam, with o polarization, would contribute to the induced polarization, i. e.
Only the projector P on the e polarization is useful
the effective NLO susceptibility is defined by
For KDP, its effective NLO susceptibility is
the maximum deff occurs at =90 and =45. In practice, under phase matching condition, in order to get the maximum conversion coefficient, the angles are selected as =m and =45.