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 Biaxial class

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: Biaxial class

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In a biaxial crystal, the three principal refractive indices are all different. The graphical representations of the index surfaces are given in Fig.[link] for the positive biaxial class ([n_x\,\lt\,n_y\,\lt\, n_z]) and for the negative one ([n_x>n_y>n_z]), both with the usual conventional orientation of the optical frame. If this is not the case, the appropriate permutation of the principal refractive indices is required.


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Index surfaces of the negative and positive biaxial classes. [{\bf E}_{o.e}^{\pm}] are the ordinary (o) and extraordinary (e) electric field vectors relative to the external (+) or internal (−) sheets for a propagation in the principal planes. OA is the optic axis.

In the orthorhombic system, the three principal axes are fixed by the symmetry; one is fixed in the monoclinic system; and none are fixed in the triclinic system. The index surface of the biaxial class has two umbilici contained in the xz plane, making an angle V with the z axis:[\sin^2V(\omega)={n^{-2}_y(\omega)-n^{-2}_x(\omega)\over n^{-2}_z(\omega)-n_x^{-2}(\omega)}.\eqno(]The propagation along the optic axes leads to the internal conical refraction effect (Schell & Bloembergen, 1978[link]; Fève et al., 1994[link]). 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. Propagation out of the principal planes

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It is impossible to define ordinary and extraordinary waves out of the principal planes of a biaxial crystal: according to ([link] and ([link], e+ and e have a nonzero projection on the z axis. According to these relations, it appears that e+ and e are not perpendicular, so relation ([link] is never verified. The walk-off angles ρ+ and ρ are nonzero, different, and can be calculated from the electric field vectors:[\rho^\pm(\theta,\varphi,\omega)=\varepsilon\arccos[{\bf e}^\pm(\theta,\varphi,\omega)\cdot{\bf u}(\theta,\varphi,\omega)]-\varepsilon\pi/2.\eqno(][\varepsilon = +1] or [-1] for a positive or a negative optic sign, respectively.


Fève, J. P., Boulanger, B. & Marnier, G. (1994). Experimental study of internal and external conical refractions in KTP. Optics Comm. 105, 243–252.
Schell, A. J. & Bloembergen, N. (1978). Laser studies of internal conical refraction. I. Quantitative comparison of experimental and theoretical conical intensity distribution in aragonite. J. Opt. Soc. Am. 68, 1093–1106.

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