Uniaxial crystals

Boulanger, B.; Zyss, J.

doi:10.1107/97809553602060000634

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

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International Tables for Crystallography (2006). Vol. D. ch. 1.7, p. 189

Section 1.7.3.2.2.2. Uniaxial crystals

B. Boulanger^a ^* and J. Zyss^b

^a Laboratoire de Spectrométrie Physique, Université Joseph Fourier, 140 avenue de la Physique, BP 87, 38 402 Saint-Martin-d'Hères, France, and ^bLaboratoire de Photonique Quantique et Moléculaire, Ecole Normale Supérieure de Cachan, 61 Avenue du Président Wilson, 94235 Cachan, France
Correspondence e-mail: benoitb@satie-bourgogne.fr

1.7.3.2.2.2. Uniaxial crystals

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The configurations of polarization in terms of ordinary and extraordinary waves depend on the optic sign of the phase-matching direction with the convention given in Section 1.7.3.1: Tables 1.7.3.1 and 1.7.3.2 must be read by substituting (+, −) by (e, o) for a positive crystal and by (o, e) for a negative one.

Because of the symmetry of the index surface, all the phase-matching directions for a given type describe a cone with the optic axis as a revolution axis. Note that the previous comment on the anaxial class is valid for a propagation along the optic axis ( [n_o=n_e] ).

Fig. 1.7.3.4 shows the example of negative uniaxial crystals ( [n_o>n_e] ) like β-BaB₂O₄(BBO) and KH₂PO₄ (KDP).

Figure 1.7.3.4 | top | pdf |

Index surface sections in a plane containing the optic axis z of a negative uniaxial crystal allowing collinear type-I phase matching for SFG ( $[\omega_3=\omega_1+\omega_2]$ ), $[\gamma = 3]$ , or for SFG ( $[\omega_4=\omega_1+\omega_2+\omega_3]$ ), $[\gamma = 4]$ . $[{\bf u}^{\rm I}_{\rm PM}]$ is the corresponding phase-matching direction.

From Fig. 1.7.3.4, it clearly appears that the intersection of the sheets is possible only if $[(n_{e_\gamma })/(\lambda _\gamma)\,\lt\, \textstyle\sum_{i = 1}^{\gamma - 1}(n_{o_i })/(\lambda _i) ]$ $[[\lt\,(n_{o_\gamma })/(\lambda _\gamma)]]$ with $[\gamma = 3]$ for a three-wave process and $[\gamma = 4]$ for a four-wave one. The same considerations can be made for the positive sign and for all the other types of phase matching. There are different situations of inequalities allowing zero, one or several types: Table 1.7.3.3 gives the five possible situations for the three-wave interactions and Table 1.7.3.4 the 19 situations for the four-wave processes.

Table 1.7.3.3 | top | pdf |
Classes of refractive-index inequalities for collinear phase matching of three-wave interactions in positive and negative uniaxial crystals

Types I, II and III refer to SFG; the types of the corresponding DFG are given in Table 1.7.3.1 (Fève et al., 1993).

Positive sign ()	Negative sign ()	Types of SFG
$[{n_{o3}\over \lambda_3}\,\lt\,{n_{o1}\over \lambda_1}+{n_{e2}\over \lambda_2};{n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}]$	$[{n_{o1}\over\lambda_1}+{n_{e2}\over\lambda_2},{n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}\,\lt\,{n_{e3}\over\lambda_3}]$	I, II, III
$[{n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}\,\lt\,{n_{o3}\over\lambda_3}\,\lt\,{n_{o1}\over \lambda_1}+{n_{e2}\over \lambda_2}]$	$[{n_{o1}\over \lambda_1}+{n_{e2}\over \lambda_2}\,\lt\,{n_{e3}\over\lambda_3}\,\lt\,{n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}]$	I, II
$[{n_{o1}\over \lambda_1}+{n_{e2}\over \lambda_2}\,\lt\,{n_{o3}\over \lambda_3}\,\lt\,{n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}]$	$[{n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}\,\lt\,{n_{e3}\over\lambda_3}\,\lt\,{n_{o1}\over \lambda_1}+{n_{e2}\over \lambda_2}]$	I, III
$[{n_{o1}\over \lambda_1}+{n_{e2}\over \lambda_2}, {n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}\,\lt\, {n_{o3}\over \lambda_3}\,\lt\, {n_{e_1}\over\lambda_1}+{n_{e2}\over\lambda_2}]$	$[{n_{o1}\over \lambda_1}+{n_{e2}\over \lambda_2}, {n_{e1}\over\lambda_1}+{n_{o2}\over\lambda_2}\,\lt\, {n_{e3}\over \lambda_3}\,\lt\, {n_{o_1}\over\lambda_1}+{n_{o2}\over\lambda_2}]$	I
$[{n_{e_1}\over\lambda_1}+{n_{e2}\over\lambda_2}\,\lt\,{n_{o3}\over \lambda_3}]$	$[{n_{o_1}\over\lambda_1}+{n_{o2}\over\lambda_2}\,\lt\,{n_{e3}\over \lambda_3}]$	None

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 (Fève, 1994).

Positive sign ()	Negative sign ()	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