150 4.1.2 Fundamentals [Ref. p. 187
Kleinman symmetry conditions [62Kle]: d21 = d16 , d24 = d32 , d31 = d15 , d13 = d35 , d14 = d36 , d25 = d12 = d26 , d32 = d24 are valid in the case of non-dispersion of electron nonlinear polarizability. The equations for calculating the conversion e ciency include the e ective nonlinear coe cients de , which comprise all summation operations along the polarization directions of the interacting waves and thus reduce the calculation to one dimension. E ective nonlinearities de for di erent crystal point groups under valid Kleinman symmetry conditions are presented in Table 4.1.3.
The conversion factors for SI and CGS-esu systems are given in Table 4.1.4.
Table 4.1.3. Expressions for de in nonlinear crystals when Kleinman symmetry relations are valid.
(a) Uniaxial crystals
Point group |
Type of interaction |
|
|
|
|
|
|
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|
|
|
ooe, oeo, eoo |
|
|
|
|
|
eeo, eoe, oee |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
4, 4mm |
d15 sin θ |
|
|
|
|
|
0 |
|
|
|
|
|
|
6, 6mm |
d15 sin θ |
|
|
|
|
|
0 |
|
|
|
|
|
|
¯ |
d 22 |
|
cos θ sin (3ϕ) |
|
|
d22 cos |
2 |
θ cos ϕ |
|
|
|
6m2 |
|
|
|
|
|
|
|
3m |
d 15 |
|
sin θ − d22 cos θ sin (3ϕ) |
|
d22 cos2 θ cos (3ϕ) |
2 |
|
¯ |
(d11 cos (3ϕ) − d22 sin (3ϕ)) cos θ |
|
(d11 sin (3ϕ) + d22 cos (3ϕ)) cos2 |
θ |
6 |
sin θ |
3 |
(d |
11 |
cos (3ϕ) |
− |
d |
22 |
sin (3ϕ)) cos θ + d |
(d11 |
sin (3ϕ) + d |
22 |
cos (3ϕ)) cos |
θ |
32 |
|
|
|
|
15 |
|
2 |
|
|
|
d 11 |
|
cos θ cos (3ϕ) |
|
|
d11 cos |
|
θ sin (3ϕ) |
|
|
¯ |
(d14 sin (2ϕ) + d15 cos (2ϕ)) sin θ |
|
(d14 cos (2ϕ) − d15 sin (2ϕ)) sin (2θ) |
4 |
|
¯ |
d 36 |
|
sin θ sin (2ϕ) |
|
|
|
d36 sin (2θ) cos (2ϕ) |
|
42m |
|
|
|
|
|
(b) Biaxial crystals (assignments of crystallophysical and crystallographic axes: for mm2 and 222 point groups: X, Y, Z → a, b, c ; for 2 and m point groups: Y → b )
Point |
Principal |
Type of interaction |
|
group |
plane |
ooe, oeo, eoo |
eeo, eoe, oee |
|
|
|
|
2 |
XY |
d23 cos ϕ |
d36 sin (2ϕ) |
|
Y Z |
d21 cos θ |
d36 sin (2θ) |
|
XZ |
0 |
d21 cos2 θ + d23 sin2 θ − d36 sin (2θ) |
m |
XY |
d13 sin ϕ |
d31 sin2 ϕ + d32 cos2 ϕ |
|
Y Z |
d31 sin θ |
d13 sin2 θ + d12 cos2 θ |
|
XZ |
d12 cos θ − d32 sin θ |
0 |
mm2 |
XY |
0 |
d31 sin2 ϕ + d32 cos2 ϕ |
|
Y Z |
d31 sin θ |
0 |
|
XZ |
d32 sin θ |
0 |
222 |
XY |
0 |
d36 sin (2ϕ) |
|
Y Z |
0 |
d36 sin (2θ) |
|
XZ |
0 |
d36 sin (2θ) |
|
|
|
|
Landolt-B¨ornstein
New Series VIII/1A1
Ref. p. 187] |
|
4.1 Frequency conversion in crystals |
151 |
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Table 4.1.4. Units and conversion factors. |
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Nonlinear coe cient |
MKS or SI units |
|
CGS or electrostatic units |
|
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|
|
χ(1) |
1 |
(SI, dimensionless) |
= |
|
1 |
(esu, dimensionless) |
ij |
|
|
|
|
4 π |
|
|
|
|
|
|
|
dij or χ(2) |
1 |
V−1m |
= |
|
3 × 104 |
(erg−1 cm3) 21 |
ijk |
|
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|
4 π |
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|
4 π |
1 |
δij |
1 |
C−1m2 |
= |
|
|
(erg−1 cm3) 2 |
3 × 105 |
Note that in SI units P (n) |
= |
ε0 χ(n)E n (with P (n) |
expressed in C m−2 ), whereas in CGS or esu |
units P (n) = χ(n)E n (with P (n) |
expressed in esu). |
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|
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|
|
4.1.2.5 Frequency conversion e ciency
4.1.2.5.1 General approach
The conversion e ciency of a three-wave interaction process for the case of square nonlinearity
P nl = ε0 χ(2)E2 |
(4.1.14) |
can be determined from the wave equation derived from Maxwell’s equations [64Akh, 65Blo, 73Zer, 99Dmi], see also (1.1.4)–(1.1.7),
× × E + |
(1 + χ(1)) ∂2E |
= − |
1 |
∂2P nl |
(4.1.15) |
c2 |
|
∂t2 |
ε0 c2 |
|
∂t2 |
with the initial and boundary conditions for the electric field E .
An exact calculation of the nonlinear conversion e ciency for SHG, SFG, and DFG generally requires a numerical calculation. In some simple cases analytical expressions are available. In order to choose the proper method, the contribution of di erent e ects in the nonlinear mixing process should be determined. For this purpose the following approach is introduced [99Dmi]:
–Consider the e ective lengths of the interaction process:
1.Aperture length La:
La = d0 ρ−1 , |
(4.1.16) |
where d0 is the beam diameter. |
|
2. Quasistatic interaction length Lqs: |
|
Lqs = τ ν−1 , |
(4.1.17) |
where τ is the radiation pulse width and ν is the mismatch of reverse group velocities. For SHG
|
ν = uω−1 − u2−ω1 , |
(4.1.18) |
|
where uω and u2ω are the group velocities of the corresponding waves ω and 2ω . |
|
3. |
Di raction length Ldif : |
|
|
L |
= k d2 . |
(4.1.19) |
|
dif |
0 |
|
4. |
Dispersion-spreading length Lds: |
|
|
Lds = τ 2g−1 , |
(4.1.20) |
Landolt-B¨ornstein
New Series VIII/1A1