International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 192-195

Section 1.7.3.2.2.3. Biaxial crystals

B. Boulangera* and J. Zyssb

aInstitut Néel CNRS Université Joseph Fourier, 25 rue des Martyrs, BP 166, 38042 Grenoble Cedex 9, France, and bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail:  benoit.boulanger@grenoble.cnrs.fr

1.7.3.2.2.3. Biaxial crystals

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The situation of biaxial crystals is more complicated, because the two sheets that must intersect are both elliptical in several cases. For a given interaction, all the phase-matching directions generate a complicated cone which joins two directions in the principal planes; the possible loci a, b, c, d are shown on the stereographic projection given in Fig. 1.7.3.5[link].

[Figure 1.7.3.5]

Figure 1.7.3.5 | top | pdf |

Stereographic projection on the optical frame of the possible loci of phase-matching directions in the principal planes of a biaxial crystal.

The basic inequalities of normal dispersion (1.7.3.7)[link] forbid collinear phase matching for all the directions of propagation located between two optic axes at the two frequencies concerned.

Tables 1.7.3.5[link] and 1.7.3.6[link] give, respectively, the inequalities that determine collinear phase matching in the principal planes for the three types of three-wave SFG and for the seven types of four-wave SFG.

Table 1.7.3.5| top | pdf |
Refractive-index conditions that determine collinear phase-matching loci in the principal planes of positive and negative biaxial crystals for three-wave SFG

a, b, c, d refer to the areas given in Fig. 1.7.3.5[link]. The types corresponding to the different DFGs are given in Table 1.7.3.1[link] (Fève et al., 1993[link]).

Types of SFGPhase-matching loci in the principal planesInequalities determining three-wave collinear phase matching in biaxial crystals
Positive biaxial crystalNegative biaxial crystal
[n_x(\omega_i) \,\lt\, n_y(\omega_i)\,\lt\, n_z(\omega_i)][n_x(\omega_i)> n_y(\omega_i)> n_z(\omega_i)]
Type I a [{n_{x3}\over\lambda_{3}} \,\lt\, {n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} \,\lt\, {n_{z3}\over\lambda_{3}}] [{n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{y3}\over\lambda_{3}}> {n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}]
b [{n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} \,\lt\, {n_{y3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}] [{n_{x3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}]
c [{n_{x3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} \,\lt\, {n_{y3}\over\lambda_{3}}] [{n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}]
d [{n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} \,\lt\, {n_{x3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}] [{n_{y3}\over\lambda_{3}}> {n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}]
Type II a [{n_{x3}\over\lambda_{3}} \,\lt\, {n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}};{n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} \,\lt\, {n_{z3}\over\lambda_{3}}] [{n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{y3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}]
b [{n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{y3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}] [{n_{x3}\over\lambda_{3}}> {n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}};{n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}]
c [{n_{x3}\over\lambda_{3}} \,\lt\, {n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}};{n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} \,\lt\, {n_{y3}\over\lambda_{3}}] [{n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}> {n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}]
c* [{n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} \,\lt\, {n_{x3}\over\lambda_{3}};{n_{y3}\over\lambda_{3}} \,\lt\, {n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}] [{n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}> {n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}]
d [{n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} \,\lt\, {n_{x3}\over\lambda_{3}} \,\lt\, {n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}] [{n_{y3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}};{n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}]
d* [{n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} \,\lt\, {n_{x3}\over\lambda_{3}} \,\lt\, {n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}] [{n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}> {n_{y3}\over\lambda_{3}};{n_{z3}\over\lambda_{3}}> {n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}]
Type III a [{n_{x3}\over\lambda_{3}} \,\lt\, {n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}};{n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} \,\lt\, {n_{z3}\over\lambda_{3}}] [{n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}> {n_{y3}\over\lambda_{3}}> {n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}]
b [{n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} \,\lt\, {n_{y3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}] [{n_{x3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}};{n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}]
c [{n_{x3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}};{n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} \,\lt\, {n_{y3}\over\lambda_{3}}] [{n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}]
c* [{n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} \,\lt\, {n_{x3}\over\lambda_{3}};{n_{y3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}] [{n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}> {n_{y1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}]
d [{n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} \,\lt\, {n_{x3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}] [{n_{y3}\over\lambda_{3}}> {n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}};{n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}> {n_{z3}\over\lambda_{3}}]
d* [{n_{y1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} \,\lt\, {n_{x3}\over\lambda_{3}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}}] [{n_{x1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}}> {n_{y3}\over\lambda_{3}};{n_{z3}\over\lambda_{3}}> {n_{x1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}}]
Conditions c, d are applied if [{n_{y1}\over\lambda_{1}} - {n_{x1}\over\lambda_{1}}, {n_{y2}\over\lambda_{2}} - {n_{x2}\over\lambda_{2}} \,\lt\, {n_{y3}\over\lambda_{3}} - {n_{x3}\over\lambda_{3}}] [{n_{y1}\over\lambda_{1}} - {n_{z1}\over\lambda_{1}}, {n_{y2}\over\lambda_{2}} - {n_{z2}\over\lambda_{2}} \,\lt\, {n_{y3}\over\lambda_{3}} - {n_{z3}\over\lambda_{3}}]
Conditions c*, d* are applied if [{n_{y3}\over\lambda_{3}} - {n_{x3}\over\lambda_{3}} \,\lt\, {n_{y1}\over\lambda_{1}} - {n_{x1}\over\lambda_{1}}, {n_{y2}\over\lambda_{2}} - {n_{x2}\over\lambda_{2}}] [{n_{y3}\over\lambda_{3}} - {n_{z3}\over\lambda_{3}} \,\lt\, {n_{y1}\over\lambda_{1}} - {n_{z1}\over\lambda_{1}}, {n_{y2}\over\lambda_{2}} - {n_{z2}\over\lambda_{2}}]

Table 1.7.3.6| top | pdf |
Refractive-index conditions that determine collinear phase-matching loci in the principal planes of positive and negative biaxial crystals for four-wave SFG

The types corresponding to the different DFGs are given in Table 1.7.3.2[link] (Boulanger et al., 1993[link]).

(a) SFG type I.

Phase-matching loci in the principal planesInequalities determining four-wave collinear phase matching in biaxial crystals
Positive signNegative sign
a [{n_{x4}\over\lambda_{4}} \,\lt\, {n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} + {n_{y3}\over\lambda_{3}} \,\lt\, {n_{z4}\over\lambda_{4}}] [{n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} + {n_{z3}\over\lambda_{3}} \,\lt\, {n_{y4}\over\lambda_{4}} \,\lt\, {n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} + {n_{x3}\over\lambda_{3}}]
b [{n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} + {n_{x3}\over\lambda_{3}} \,\lt\, {n_{y4}\over\lambda_{4}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} + {n_{z3}\over\lambda_{3}}] [{n_{z4}\over\lambda_{4}} \,\lt\, {n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} + {n_{y3}\over\lambda_{3}} \,\lt\, {n_{x4}\over\lambda_{4}}]
c [{n_{x4}\over\lambda_{4}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} + {n_{z3}\over\lambda_{3}} \,\lt\, {n_{y4}\over\lambda_{4}}] [{n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} + {n_{y3}\over\lambda_{3}} \,\lt\, {n_{z4}\over\lambda_{4}} \,\lt\, {n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} + {n_{x3}\over\lambda_{3}}]
d [{n_{y1}\over\lambda_{1}} + {n_{y2}\over\lambda_{2}} + {n_{y3}\over\lambda_{3}} \,\lt\, {n_{x4}\over\lambda_{4}} \,\lt\, {n_{z1}\over\lambda_{1}} + {n_{z2}\over\lambda_{2}} + {n_{z3}\over\lambda_{3}}] [{n_{z4}\over\lambda_{4}} \,\lt\, {n_{x1}\over\lambda_{1}} + {n_{x2}\over\lambda_{2}} + {n_{x3}\over\lambda_{3}} \,\lt\, {n_{y4}\over\lambda_{4}}]

(b) SFG type II ([i=1,j=2,k=3]), SFG type III ([i=3,j=1,k=2]), SFG type IV ([i=2,j=3,k=1]).

Phase-matching loci in the principal planesInequalities determining four-wave collinear phase matching in biaxial crystals
Positive signNegative sign
a [{n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}}; {n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}}] [{n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
b [{n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}] [{n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}}; {n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}}]
c [{n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}; {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}}] [{n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
c* [{n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}}; {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}} ] [{n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
d [{n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}] [{n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}; {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}}]
d* [{n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}] [{n_{zi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}}; {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
SFG type II [ (i,j) = (1,2)]; SFG type III [(i,j) = (1,3)]; SFG type IV [(i,j) = (2,3)]
Conditions c, d are applied if [{n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} - {n_{xi}\over\lambda_{i}} - {n_{xj}\over\lambda_{j}} \,\lt\, {n_{y4}\over\lambda_{4}} - {n_{x4}\over\lambda_{4}}] [{n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} - {n_{zi}\over\lambda_{i}} - {n_{zj}\over\lambda_{j}} \,\lt\, {n_{y4}\over\lambda_{4}} - {n_{z4}\over\lambda_{4}}]
Conditions c*, d* are applied if [{n_{y4}\over\lambda_{4}} - {n_{x4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} - {n_{xi}\over\lambda_{i}} - {n_{xj}\over\lambda_{j}}] [{n_{y4}\over\lambda_{4}} - {n_{z4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} - {n_{zi}\over\lambda_{i}} - {n_{zj}\over\lambda_{j}}]

(c) SFG type V4 ([i=1,j=2,k=3]), SFG type VI4 ([i=2,j=3,k=1]), SFG type VII4 ([i=3,j=1,k=2]).

Phase-matching loci in the principal planesInequalities determining four-wave collinear phase matching in biaxial crystals
Positive signNegative sign
a [{n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}}; {n_{zi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}}] [{n_{yi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
b [{n_{yi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}] [{n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}}; {n_{xi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}}]
c [{n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}; {n_{yi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}}] [{n_{zi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
c** [{n_{xi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}}; {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}] [{n_{zi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
d [{n_{xi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}] [{n_{z4}\over\lambda_{4}} \,\lt\, {n_{zi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}; {n_{yi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}} \,\lt\, {n_{y4}\over\lambda_{4}}]
d** [{n_{xi}\over\lambda_{i}} + {n_{yj}\over\lambda_{j}} + {n_{yk}\over\lambda_{k}} \,\lt\, {n_{x4}\over\lambda_{4}} \,\lt\, {n_{xi}\over\lambda_{i}} + {n_{zj}\over\lambda_{j}} + {n_{zk}\over\lambda_{k}}] [{n_{zi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}} \,\lt\, {n_{z4}\over\lambda_{4}}; {n_{y4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} + {n_{xj}\over\lambda_{j}} + {n_{xk}\over\lambda_{k}}]
SFG type V4, ([i = 1]); SFG type VI4 ([i = 2]) ; SFG type VII4 ([i = 3])
Conditions c′, d′ are applied if [{n_{yi}\over\lambda_{i}} - {n_{xi}\over\lambda_{i}} \,\lt\, {n_{y4}\over\lambda_{4}} - {n_{x4}\over\lambda_{4}} ] [{n_{yi}\over\lambda_{i}} - {n_{zi}\over\lambda_{i}} \,\lt\, {n_{y4}\over\lambda_{4}} - {n_{z4}\over\lambda_{4}}]
Conditions c**, d** are applied if [{n_{y4}\over\lambda_{4}} - {n_{x4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} - {n_{xi}\over\lambda_{i}}] [{n_{y4}\over\lambda_{4}} - {n_{z4}\over\lambda_{4}} \,\lt\, {n_{yi}\over\lambda_{i}} - {n_{zi}\over\lambda_{i}}]

The inequalities in Table 1.7.3.5[link] show that a phase-matching cone which would join the directions a and d is not possible for any type of interaction, because the corresponding inequalities have an opposite sense. It is the same for a hypothetical cone joining b and c.

The existence of type-II or type-III SFG phase matching imposes the existence of type I, because the inequalities relative to type I are always satisfied whenever type II or type III exists. However, type I can exist even if type II or type III is not allowed. A type-I phase-matched SFG in area c forbids phase-matching directions in area b for type-II and type-III SFG. The exclusion is the same between d and a. The consideration of all the possible combinations of the inequalities of Table 1.7.3.5[link] leads to 84 possible classes of phase-matching cones for both positive and negative biaxial crystals (Fève et al., 1993[link]; Fève, 1994[link]). There are 14 classes for second harmonic generation (SHG) which correspond to the degenerated case ([\omega_1=\omega_2]) (Hobden, 1967[link]).

The coexistence of the different types of four-wave phase matching is limited as for the three-wave case: a cone joining a and d or b and c is impossible for type-I SFG. Type I in area d forbids the six other types in a. The same restriction exists between c and b. Types II, III, IV, V4, VI4 and VII4 cannot exist without type I; other restrictions concern the relations between types II, III, IV and types V4, VI4, VII4 (Fève, 1994[link]). The counting of the classes of four-wave phase-matching cones obtained from all the possible combinations of the inequalities of Table 1.7.3.6[link] is complex and it has not yet been done.

For reasons explained later, it can be interesting to consider a non-collinear interaction. In this case, the projection of the vectorial phase-matching relation (1.7.3.26)[link] on the wavevector [{\bf k}(\omega_\gamma,\theta_\gamma,\varphi_\gamma)] of highest frequency [\omega_\gamma] leads to[\textstyle\sum\limits_{i = 1}^{\gamma - 1}{\omega _i }n(\omega _i, \theta _i, \varphi _i)\cos \alpha _{i\gamma } = \omega _\gamma n(\omega _\gamma, \theta _\gamma, \varphi _\gamma),\eqno(1.7.3.29)]where [\alpha_{i\gamma}] is the angle between [{\bf k}(\omega_i,\theta_i,\varphi_i)] and [{\bf k}(\omega_\gamma,\theta_\gamma,\varphi_\gamma)], with [\gamma = 3] for a three-wave interaction and [\gamma = 4] for a four-wave interaction. The phase-matching angles ([\theta_\gamma,\varphi_\gamma]) can be expressed as a function of the different ([\theta_i,\varphi_i]) by the projection of (1.7.3.26)[link] on the three principal axes of the optical frame.

The configurations of polarization allowing non-collinear phase matching are the same as for collinear phase matching. Furthermore, non-collinear phase matching exists only if collinear phase matching is allowed; the converse is not true (Fève, 1994[link]). Note that collinear or non-collinear phase-matching conditions are rarely satisfied over the entire transparency range of the crystal.

References

Fève, J. P. (1994). Existence et symétrie des interactions à 3 et 4 photons dans les cristaux anisotropes. Méthodes de mesure des paramètres affectant les couplages à 3 ondes: étude de KTP et isotypes. PhD Dissertation, Université de Nancy I, France.
Fève, J. P., Boulanger, B. & Marnier, G. (1993). Calculation and classification of the direction loci for collinear types I, II and III phase-matching of three-wave non linear optical parametric interactions in uniaxial and biaxial acentric crystals. Optics Comm. 99, 284–302.
Hobden, M. V. (1967). Phase-matched second harmonic generation in biaxial crystals. J. Appl. Phys. 38, 4365–4372.








































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