Propagation in the principal planes

Boulanger, B.; Zyss, J.

doi:10.1107/97809553602060000634

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. D. ch. 1.7, pp. 186-187

Section 1.7.3.1.4.1. Propagation in the principal planes

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

1.7.3.1.4.1. Propagation in the principal planes

| top | pdf |

It is possible to define ordinary and extraordinary waves, but only in the principal planes of the biaxial crystal: the ordinary electric field vector is perpendicular to the z axis and to the extraordinary one. The walk-off properties of the waves are not the same in the [xy] plane as in the [xz] and [yz] planes.

(1) In the xy plane, the extraordinary wave has no walk-off, in contrast to the ordinary wave. The components of the electric field vectors can be established easily with the same considerations as for the uniaxial class: $[\eqalignno{e_x^o&=-\sin[\varphi\pm\rho^\mp(\varphi,\omega)]&\cr e_y^o&=\cos[\varphi\pm\rho^\mp(\varphi,\omega)]&\cr e_z^o&=0, &(1.7.3.15)}]$ with $[+\rho^-(\varphi,\omega)]$ for the positive class and $[-\rho^+(\varphi,\omega)]$ for the negative class . $[\rho^\pm(\varphi,\omega)]$ is the walk-off angle given by (1.7.3.13), where $[\theta]$ is replaced by $[\varphi]$ , n_o by n_y and n_e by n_x: $[e_x^e=0\quad e_y^e=0\quad e_z^e=1.\eqno(1.7.3.16)]$
(2) The yz plane of a biaxial crystal has exactly the same characteristics as any plane containing the optic axis of a uniaxial crystal. The electric field vector components are given by (1.7.3.11) and (1.7.3.12) with $[\varphi=\pi/2]$ . The ordinary walk-off is nil and the extraordinary one is given by (1.7.3.13) with and .

(3) In the xz plane, the optic axes create a discontinuity of the shape of the internal and external sheets of the index surface leading to a discontinuity of the optic sign and of the electric field vector. The birefringence, [n_e-n_o] , is nil along the optic axis, and its sign changes on either side. Then the yz plane, xy plane and xz plane from the x axis to the optic axis have the same optic sign, the opposite of the optic sign from the optic axis to the z axis. Thus a positive biaxial crystal is negative from the optic axis to the z axis. The situation is inverted for a negative biaxial crystal . It implies the following configuration of polarization:

(i) From the x axis to the optic axis, e^o and e^e are given by (1.7.3.11) and (1.7.3.12) with $[\varphi = 0]$ . The walk-off is relative to the extraordinary wave and is calculated from (1.7.3.13) with and .
(ii) From the optic axis to the z axis, the vibration plane of the ordinary and extraordinary waves corresponds respectively to a rotation of π/2 of the vibration plane of the extraordinary and ordinary waves for a propagation in the areas of the principal planes of opposite sign; the extraordinary electric field vector is given by (1.7.3.12) with $[\varphi = 0]$ , $[-\rho^-(\varphi,\omega)]$ for the positive class and $[+\rho^+(\varphi,\omega)]$ for the negative class, and the ordinary electric field vector is out of phase by π in relation to (1.7.3.11), that is $[e_x^o=0\quad e_y^o=-1\quad e_z^o=0.\eqno(1.7.3.17)]$ The extraordinary walk-off angle is given by (1.7.3.13) with and .

The π/2 rotation on either side of the optic axes is well observed during internal conical refraction (Fève et al., 1994).

Note that for a biaxial crystal, the walk-off angles are all nil only for a propagation along the principal axes.

References

Fève, J. P., Boulanger, B. & Marnier, G. (1994). Experimental study of internal and external conical refractions in KTP. Optics Comm. 105, 243–252.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.7, pp. 186-187