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

4.1.2 Fundamentals

4.1.2.1 Three-wave interactions

Dielectric polarization P (dipole moment of unit volume of the substance) is related to the field E by the material equation of the medium [64Akh, 65Blo] (Chap. 1.1):

with

ε0 = 8.854 × 10−12 CV−1m−1 : dielectric permittivity of free space,

χ(1) = n2 − 1 : the linear, and χ(2) , χ(3) etc.: the nonlinear dielectric susceptibilities.

In the present chapter, Chap. 4.1, we consider only three-wave interactions in crystals with square nonlinearity (χ(2) = 0). The following nonlinear frequency conversion processes are considered:

Second Harmonic Generation (SHG):

Sum-Frequency Generation (SFG) or up-conversion:

Di erence-Frequency Generation (DFG) or down-conversion:

ki : the wave vectors for ω1 , ω2 , ω3 , respectively.

Two types of phase matching are introduced:

type I: o + o → e or e + e → o , type II: o + e → e or o + e → o ,

or with shortened notations:

ooe: o + o → e or e → o + o , eeo: e + e → o or o → e + e , eoe: e + o → e or e → e + o , oeo: o + e → o or o → e + o .

In the shortened notation (ooe, eoe, . . . ) applies: ω1 < ω2 < ω3, i.e. the first symbol refers to the longest-wavelength radiation, and the latter to the shortest-wavelength radiation. Here, o-beam, or ordinary beam, is the beam with polarization normal to the principal plane of the crystal, i.e.

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the plane containing the wave vector k and crystallophysical axis Z (or optical axis, for uniaxial crystals). The e-beam, or extraordinary beam, is the beam with polarization in the principal plane.

The methods of angular and temperature phase-matching tuning are used in frequency converters. Angular tuning is rather simple and more rapid than temperature tuning. Temperature tuning is generally used in the case of 90 ◦ phase matching, i.e., when the birefringence angle is zero. This method is mainly used in crystals with a strong temperature dependence of phase matching: LiNbO3, LBO, KNbO3, and Ba2NaNb5O15.

4.1.2.2 Uniaxial crystals

For uniaxial crystals the di erence between the refractive indices of the ordinary and extraordinary beams, birefringence ∆n , is zero along the optical axis (crystallophysical axis Z) and maximum in the normal direction. The refractive index of the ordinary beam does not depend on the direction of propagation, however, the refractive index of the extraordinary beam ne(θ) is a function of the polar angle θ between the Z axis and the vector k (but not of the azimuthal angle ϕ) (Fig. 4.1.1):

where no and ne are the refractive indices of the ordinary and extraordinary beams in the plane normal to the Z axis and termed as corresponding principal values. Note that if no > ne , the crystal is negative, and if no < ne , it is positive. For an o-beam the indicatrix of the refractive indices is a sphere with radius no , and an ellipsoid of rotation with semiaxes no and ne for an e-beam (Fig. 4.1.2). In the crystal the beam, in general, is divided into two beams with orthogonal polarizations; the angle between these beams ρ is the birefringence (or walk-o ) angle.

Equations for calculating phase-matching angles in uniaxial crystals are given in Table 4.1.1 [86Nik, 99Dmi].

Z

k

Y

X

Fig. 4.1.1. Polar coordinate system for description of refraction properties of uniaxial crystals (k is the light propagation direction, Z is the optic axis, θ and ϕ are the coordinate angles).

4.1.2.3 Biaxial crystals

For biaxial crystals the optical indicatrix has a bilayer surface with four points of interlayer contact which correspond to the directions of two optical axis. In the simple case of light propagation in

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Fig. 4.1.2. Dependence of refractive index on light propagation direction and polarization (index surface) in uniaxial crystals: (a) negative: no > ne and (b) positive: ne > no .

Table 4.1.1. Equations for calculating phase-matching angles in uniaxial crystals [86Nik, 99Dmi].

the phase-matching angle θpmooe is determined from the above presented equation using the new coe cients U and W :

U = (A2 + B2 + 2AB cos γ)/C2 , W = (A2 + B2 + 2AB cos γ)/F 2 ,

where γ is the angle between the wave vectors k1 and k2 .

the principal planes XY , Y Z, and XZ the dependencies of refractive indices on the direction of light propagation represent a combination of an ellipse and a circle (Fig. 4.1.3). Thus in the principal planes a biaxial crystal can be considered as a uniaxial crystal, e.g. a biaxial crystal with nZ > nY > nX in the XY plane is similar to a negative uniaxial crystal with no = nZ and

The angle VZ between the optical axis and Z axis for the case nZ > nY > nX can be found from:

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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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