Материал: Weber H., Herziger G., Poprawe R. (eds.) Laser Fundamentals. Part 1 (Springer 2005)(263s) PEo

Fig. 4.1.3. Dependence of refractive index on light propagation direction and polarization (index surface) in biaxial crystals: (a) nX < nY < nZ , (b) nX > nY > nZ .

For a positive biaxial crystal the bisectrix of the acute angle between optical axes coincides with nmax and for a negative one the bisectrix coincides with nmin .

Equations for calculating phase-matching angles upon propagation in principal planes of biaxial crystals are given in Table 4.1.2 [87Nik, 99Dmi].

4.1.2.4 E ective nonlinearity

has small dispersion and is almost constant for a wide range of crystals.

For anisotropic media the coe cients χ(1) and χ(2) are, in general, the secondand third-rank tensors, respectively. In practice, the tensor

is used instead of χijk . Usually, the “plane” representation of dijk in the form dil is used, the relation between l and jk is:

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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

(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 )

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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

can be determined from the wave equation derived from Maxwell’s equations [64Akh, 65Blo, 73Zer, 99Dmi], see also (1.1.4)–(1.1.7),

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:

where τ is the radiation pulse width and ν is the mismatch of reverse group velocities. For SHG

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