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

International Tables for Crystallography (2013). Vol. D, ch. 1.7, pp. 189-190

Section Propagation in the principal planes

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: Propagation in the principal planes

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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, &(}]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 ([link], where [\theta] is replaced by [\varphi], no by ny and ne by nx:[e_x^e=0\quad e_y^e=0\quad e_z^e=1.\eqno(]

  • (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 ([link] and ([link] with [\varphi=\pi/2]. The ordinary walk-off is nil and the extraordinary one is given by ([link] with [n_o=n_y] and [n_e=n_z].

  • (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, eo and ee are given by ([link] and ([link] with [\varphi = 0]. The walk-off is relative to the extraordinary wave and is calculated from ([link] with [n_o=n_x] and [n_e = n_z].

    • (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 ([link] 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 ([link], that is[e_x^o=0\quad e_y^o=-1\quad e_z^o=0.\eqno(]The extraordinary walk-off angle is given by ([link] with [n_o = n_x] and [n_e = n_z].

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

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


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

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