Table 1
Possible Interactions for THGa
Phase-Matching Type Interaction SHG SFG THG Type I ss → s ss → f ss → f sf → f sss → f Type II ss → s fs → f sf → s ss → f ss → f ff → f sf → f sf → f ssf → f Type III sf → s fs → f ff → s ss → f sf → f ff → f ff → f sf → f sff → f
a The polarization directions
f / s are given in order of longest wavelength to shortest.
Table 2
Effective Nonlinear Coefficient for Second-Order Interactions, Neglecting Dispersion (sss Interactions)
Crystal Class
d eff
3
- d 11 sin 3 ϕ - d 22 cos 3 ϕ
3 m
- d 22 cos 3 ϕ
6 ¯
- d 11 sin 3 ϕ - d 22 cos 3 ϕ
6 and 4 0
6 mm
and 4 mm
0 622 and 422 0
6 ¯ m 2
- d 22 cos 3 ϕ
4 ¯
0 32
- d 11 sin 3 ϕ
4 ¯ 2 m ,
4 ¯ 3 m ,
and 23 0
Table 3
Effective Nonlinear Coefficient for Second-Order Interactions, Neglecting Dispersion (ssf Interactions)
Crystal Class
d eff
3
cos θ m ( d 11 cos 3 ϕ - d 22 sin 3 ϕ ) + d 31 sin θ m
3 m
- d 22 cos θ m sin 3 ϕ + d 31 sin θ m
6 ¯
cos θ m ( d 11 cos 3 ϕ - d 22 sin 3 ϕ )
6 and 4
d 31 sin θ m
6 mm
and 4 mm
d 31 sin θ m
622 and 422 0
6 ¯ m 2
- d 22 cos θ m sin 3 ϕ
4 ¯
- sin θ m ( d 31 cos 2 ϕ + d 36 sin 2 ϕ )
32
d 11 sin θ m cos 3 ϕ
4 ¯ 2 m ,
4 ¯ 3 m ,
and 23
- d 36 sin θ m sin 2 ϕ
Table 4
Effective Nonlinear Coefficient for Second-Order Interactions, Neglecting Dispersion (sff Interactions)
Crystal Class
d eff
3
cos 2 θ m ( d 11 sin 3 ϕ + d 22 cos 3 ϕ )
3 m
d 22 cos 2 θ m cos 3 ϕ
6 ¯
cos 2 θ m ( d 11 sin 3 ϕ + d 22 cos 3 ϕ )
6 and 4 0
6 mm
and 4 mm
0 622 and 422 0
6 ¯ m 2
d 22 cos 2 θ m cos 3 ϕ
4 ¯
sin 2 θ m ( d 36 cos 2 ϕ - d 31 sin 2 ϕ )
32
d 11 cos 2 θ m sin 3 ϕ
4 ¯ 2 m ,
4 ¯ 3 m ,
and 23
sin 2 θ m d 36 cos 2 ϕ
Table 5
Effective Nonlinear Coefficient for Second-Order Interactions, Neglecting Dispersion (fff Interactions)
Crystal Class
d eff
3
cos 3 θ m ( - d 11 cos 3 ϕ + d 22 sin 3 ϕ ) + 3 d 31 sin θ m cos 2 θ m + d 33 sin 3 θ m
3 m
d 22 cos 3 θ m sin 3 ϕ + 3 d 31 sin θ m cos 2 θ m + d 33 sin 3 θ m
6 ¯
cos 3 θ m ( - d 11 cos 3 ϕ + d 22 sin 3 ϕ )
6 and 4
3 d 31 sin θ m cos 2 θ m + d 33 sin 3 θ m
6 mm
and 4 mm
3 d 31 sin θ m cos 2 θ m + d 33 sin 3 θ m
622 and 422 0
6 ¯ m 2
d 22 cos 3 θ m sin 3 ϕ
4 ¯
3 sin θ m cos 2 θ m ( d 36 sin 2 ϕ + d 31 cos 2 ϕ )
32
- d 11 cos 3 θ m cos 3 ϕ
4 ¯ 2 m ,
4 ¯ 3 m ,
and 23
3 d 36 sin θ m cos 2 θ m sin 2 ϕ
Table 6
Effective Nonlinear Coefficient for Third-Order Interactions, Neglecting Dispersion (ssss Interactions)
Crystal Class
C eff
Trigonal (3 and 3 ¯ )
C 11
Trigonal (3 m ,
3 m ¯ ,
and 32)
C 11
Hexagonal (6, 6 ¯ ,
6 / m ,
622,
6 mm ,
6 / mmm ,
and 6 ¯ m 2 )
C 11
Tetragonal (4, 4 ¯ ,
and 4 / m )
1 4 [ C 11 ( 3 + cos 4 ϕ ) + 6 C 18 sin 2 2 ϕ ] + C 21 sin 4 ϕ
Tetragonal (4 ¯ 2 m ,
422, 4 mm ,
and 4 / mmm )
1 4 [ C 11 ( 3 + cos 4 ϕ ) + 6 C 18 sin 2 2 ϕ ]
Cubic (23, m 3 ,
432, 4 ¯ 3 m ,
and m 3 m )
1 4 [ C 11 ( 3 + cos 4 ϕ ) + 6 C 16 sin 2 2 ϕ ] Isotropic
C 11
Table 7
Effective Nonlinear Coefficient for Third-Order Interactions, Neglecting Dispersion (sssf Interactions)
Crystal Class
C eff
Trigonal (3 and 3 ¯ )
sin θ m ( C 10 cos 3 ϕ + C 15 sin 3 ϕ )
Trigonal (3 m ,
3 m ¯ ,
and 32)
C 10 cos 3 ϕ sin θ m
Hexagonal (6, 6 ¯ ,
6 / m ,
622,
6 mm ,
6 / mmm ,
and 6 ¯ m 2 ) 0 Tetragonal (4, 4 ¯ ,
and 4 / m )
- cos θ m [ C 21 cos 4 ϕ + 1 4 sin 4 ϕ ( 3 C 18 - C 11 ) ]
Tetragonal (4 ¯ 2 m ,
422,
4 mm
and 4 / mmm )
1 4 cos θ m sin 4 ϕ ( C 11 - 3 C 18 )
Cubic (23, m 3 ,
432,
4 ¯ 3 m ,
and m 3 m )
1 4 cos θ m sin 4 ϕ ( C 11 - 3 C 16 )
Isotropic 0
Table 8
Effective Nonlinear Coefficient for Third-Order Interactions, Neglecting Dispersion (ssff Interactions)
Crystal Class
C eff
Trigonal (3 and 3 ¯ )
1 3 C 11 cos 2 θ m + C 16 sin 2 θ m + sin 2 θ m × ( C 10 sin 3 ϕ
- C 15 cos 3 ϕ )
Trigonal (3 m ,
3 m ¯ ,
and 32)
1 3 C 11 cos 2 θ m + C 16 sin 2 θ m + C 10 sin 2 θ m sin 3 ϕ
Hexgonal (6, 6 ¯ ,
6 / m ,
622,
6 mm ,
6 / mmm ,
and 6 m 2 )
1 3 C 11 cos 2 θ m + C 16 sin 2 θ m
Tetragonal (4, 4 ¯ ,
and 4 / m )
1 2 cos 2 θ m [ C 11 sin 2 2 ϕ + C 18 ( 3 cos 2 2 ϕ - 1 )
- 2 C 21 sin 4 ϕ ] + C 16 sin 2 θ m
Tetragonal (4 ¯ 2 m ,
422, 4 mm ,
and 4 / mmm )
1 2 cos 2 θ m [ C 11 sin 2 2 ϕ + C 18 ( 3 cos 2 2 ϕ - 1 ) ]
+ C 16 sin 2 θ m
Cubic (23, m 3 ,
432, 4 ¯ 3 m ,
and m 3 m )
1 2 cos 2 θ m [ C 11 sin 2 2 ϕ + C 16 ( 3 cos 2 2 ϕ - 1 ) ]
+ C 16 sin 2 θ m
Isotropic
1 3 C 11
Table 9
Effective Nonlinear Coefficient for Third-Order Interactions, Neglecting Dispersion (sfff Interactions)
Crystal Class
C eff
Trigonal (3 and 3 ¯ )
- 3 cos 2 θ m sin θ m ( C 10 cos 3 ϕ + C 15 sin 3 ϕ )
Trigonal (3 m ,
3 m ¯ ,
and 32)
- C 10 cos 2 θ m sin θ m cos 3 ϕ
Hexagonal (6, 6 ¯ ,
6 / m ,
622,
6 mm ,
6 / mmm ,
and 6 ¯ m 2 ) 0 Tetragonal (4, 4 ¯ ,
and 4 / m )
cos 3 θ m [ C 21 cos 4 ϕ + 1 4 sin 4 ϕ ( 3 C 18 - C 1 ) ]
Tetragonal (4 ¯ 2 m ,
422, 4 mm ,
and 4 / mmm )
1 4 ( 3 C 18 - C 11 ) cos 3 θ m sin 4 ϕ
Cubic (23, m 3 ,
432, 4 ¯ 3 m ,
and m 3 m )
1 4 ( 3 C 16 - C 11 ) cos 3 θ m sin 4 ϕ
Isotropic 0
Table 10
Effective Nonlinear Coefficient for Third-Order Interactions, Neglecting Dispersion (ffff Interactions)
Crystal Class
C eff
Trigonal (3 and 3 ¯ )
4 sin θ m cos 3 θ m ( C 15 cos 3 ϕ - C 10 sin 3 ϕ ) + C 11 cos 4 θ m
+ 3 2 C 16 sin 2 2 θ m + C 33 sin 4 θ m
Trigonal (3 m ,
3 m ¯ ,
and 32)
- 4 C 10 sin θ m cos 3 θ m sin 3 ϕ + C 11 cos 4 θ m
+ 3 2 C 16 sin 2 2 θ m + C 33 sin 4 θ m
Hexagonal (6, 6 ¯ ,
6 / m ,
622,
6 mm ,
6 / mmm ,
and 6 ¯ m 2 )
C 11 cos 4 θ m + 3 2 C 16 sin 2 2 θ m + C 33 sin 4 θ m
Tetragonal (4, 4 ¯ ,
and 4 / m )
cos 4 θ m [ 1 4 C 11 ( 3 + cos 4 ϕ ) + 3 2 C 18 sin 2 2 ϕ + 2 C 21 sin 4 ϕ ]
+ 3 2 C 16 sin 2 2 θ m + C 33 sin 4 θ m
Tetragonal (4 ¯ 2 m ,
422, 4 mm ,
and 4 / mmm )
1 4 C 11 cos 4 θ m ( 3 + cos 4 ϕ ) + 3 2 C 16 sin 2 2 θ m
+ 3 2 C 18 cos 4 θ m sin 2 2 ϕ + C 33 sin 4 θ m
Cubic (23, m 3 ,
432, 4 ¯ 3 m ,
and m 3 m )
1 4 C 11 [ cos 4 θ m ( 3 + cos 4 ϕ ) + sin 4 θ m ]
+ 3 2 C 16 [ sin 2 2 θ m + cos 4 θ m sin 2 2 ϕ ]
Isotropic
C 11
Table 11
Nonzero Third-Order Tensor Elements for Crystal Class 3 m
zzzz
xxyy = yyxx
xxxx = yyyy = xxyy + xyyx + xyxy
xyyx = yxxy
xyxy = yxyx
yyzz = xxzz
zzyy = zzxx
yyyz = - yxxz = - xyxz = - xxyz
zyyz = zxxz
yyzy = - yxzx = - xyzx = - xxzy
yzzy = xzzx
yzyy = - yzxx = - xzyx = - xzxy
yzyz = xzxz
zyyy = - zyxx = - zxyx = - zxxy
zyzy = zxzx