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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, p. 194

Table 1.7.3.5 

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 1.7.3.5| 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}}]