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. 193

Table 1.7.3.4 

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 1.7.3.4| top | pdf |
Classes of refractive-index inequalities for collinear phase matching of four-wave interactions in positive ([n_a=n_e, n_b=n_o]) and negative ([n_a=n_o, n_b=n_e]) uniaxial crystals with [(n_{b4}/\lambda_4)\,\lt\,(n_{a1}/\lambda_1)+(n_{a2}/\lambda_2)+(n_{a3}/\lambda_3)]

If this inequality is not verified, no phase matching is allowed. The types of phase matching refer to SFG; the types of the corresponding DFG are given in Table 1.7.3.2[link] (Fève, 1994[link]).

Positive sign ([n_e> n_o])Negative sign ([n_o> n_e])Types of SFG
[{n_{a1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}]   I
[{n_{a1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}, {n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}]   I, V4
[{n_{a1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}, {n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}]   I, VI4
[{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}} \,\lt\, {n_{a1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}]   I, VII4
[{n_{a1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] [{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}} \,\lt\, {n_{b4}\over\lambda_{4}}] I, V4, VI4
[{n_{b4}\over\lambda_{4}} \,\lt\, {n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] I, II, V4, VI4
[{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] [{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}} \,\lt\, {n_{b4}\over\lambda_{4}}] I, V4, VII4
[{n_{b4}\over\lambda_{4}} \,\lt\, {n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}] I, III, V4, VII4
[{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}},{n_{a1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}] [{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}} \,\lt\, {n_{b4}\over\lambda_{4}}] I, VI4, VII4
  [{n_{b4}\over\lambda_{4}} \,\lt\, {n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}] I, IV, VI4, VII4
[{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}}+{n_{a2}\over\lambda_{2}}+{n_{b3}\over\lambda_{3}},{n_{a1}\over\lambda_{1}}+{n_{b2}\over\lambda_{2}}+{n_{a3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] [{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}] I, V4, VI4, VII4
[{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] I, II, V4, VI4, VII4
[{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}] I, III, V4, VI4, VII4
[{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}] I, IV, V4, VI4, VII4
[{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] I, II, III, V4, VI4, VII4
[{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] I, II, IV, V4, VI4, VII4
[{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}\,\lt\,{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}}] I, III, IV, V4, VI4, VII4
[{n_{b4}\over\lambda_{4}}\,\lt\,{n_{a1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{a2}\over\lambda_{2}} + {n_{b3}\over\lambda_{3}},{n_{b1}\over\lambda_{1}} + {n_{b2}\over\lambda_{2}} + {n_{a3}\over\lambda_{3}}] All