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. 195-196

Table 1.7.3.6 

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

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}}]