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

International Tables for Crystallography (2013). Vol. D, ch. 1.6, pp. 173-175

Section Optical rotation perpendicular to the optic axis of a uniaxial crystal

A. M. Glazera* and K. G. Coxb

aDepartment of Physics, University of Oxford, Parks Roads, Oxford OX1 3PU, England, and bDepartment of Earth Sciences, University of Oxford, Parks Roads, Oxford OX1 3PR, England
Correspondence e-mail: Optical rotation perpendicular to the optic axis of a uniaxial crystal

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The magnitude of circular birefringence is typically about 10−4 times smaller than that of linear birefringence. For this reason, optical rotation has usually been observed only in directions where the linear birefringence is absent, such as in optic axial directions. However, it has been clear for some time that optical rotation also exists in other directions and that with specialized techniques it is even possible to measure it. The techniques (Moxon & Renshaw, 1990[link]; Moxon et al., 1991[link]) are complex and require very precise measuring capabilities, and therefore are generally not commonly available.

Probably the best known case where optical rotation has been measured in a linearly birefringent section is that of quartz. It has been seen that it is easy to measure the rotation along the optic axial direction, since this is the direction along which the crystal looks isotropic. Szivessy & Münster (1934[link]) measured the rotation in a direction perpendicular to the optic axis and found that its magnitude was smaller and opposite in sign to that along the optic axis of the same crystal.

To see the relationship between the linear and circular birefringences, consider light travelling along [x_1] in quartz. The fundamental equation ([link] then becomes [\displaylines{ \pmatrix{ \varepsilon_{11}-n^2& - iG_{12}& -iG_{13}\cr iG_{12}&\varepsilon_{11}& -iG_{23}\cr iG_{13}&iG_{23}&\varepsilon_{33}} \pmatrix{ E_1\cr E_2\cr E_3 } \cr\quad\hfill= \pmatrix{ n^2&0&0\cr 0&n^2&0\cr 0&0&n^2 }\pmatrix{ E_1\cr E_2\cr E_3 }. \hfill(}] Solving for the non-trivial transverse solutions [ \left| \matrix{ \varepsilon_{11}-n^2&-iG_{23}\cr iG_{23}&\varepsilon_{33}-n^2} \right| = 0 \eqno (] and then [ n^4 - n^2(\varepsilon_{11} + \varepsilon_{33}) + \varepsilon_{11}\varepsilon_{33} - G_{23}^2 = 0. \eqno (] Finding the roots of this equation considered as a quadratic in [n^2], the following birefringence is obtained:[ n_1 - n_2 = \left[\left(n_o-n_e\right)^2 +(G_{23}^2/{\bar n}^2)\right]^{1/2}. \eqno (]The eigenvectors for the two solutions [n_1] and [n_2] can easily be shown to correspond to elliptical polarizations. Notice that in equation ([link], two refractive-index solutions are obtained whose difference depends on two terms, one with respect to the linear birefringence [n_o - n_e] and the other to the circular birefringence represented by the gyration component [G_{23} = -G_1]. The refractive-index difference [n_1 - n_2] gives rise to a phase shift between the two elliptically polarized components of the light, given by [\Delta = (2\pi/\lambda) (n_1 - n_2) ,\eqno (] from which [\eqalignno{\Delta^2 &= (4\pi^2/\lambda^2)\left[(n_o - n_e)^2 +(G_{23}^2 /{\bar n}^2)\right]&\cr & = \delta^2 + (2\rho)^2, & (}]where [\delta = {2\pi\over \lambda}(n_o - n_e)\; \hbox{ and } \; \rho ={-\pi G_{23} \over \lambda {\bar n}} = {-\pi G_{1} \over \lambda {\bar n}}. \eqno(][\delta] is the phase difference when there is no optical rotation and [2\rho] is the phase difference corresponding to a normal optical rotation [\rho] when there is no linear birefringence. ([link] shows that the linear and circular terms simply add, and this is known as the principle of superposition.

This reveals that an elliptical polarization is created by the simple vector addition of a linearly polarized wave to a circularly polarized wave, as indicated in Fig.[link]. From this, the ellipticity [\kappa] of the polarization is given by [\kappa = \tan({\textstyle{1\over 2}} \theta), \eqno (] where [ \tan \theta = {2\rho \over \delta}. \eqno (] Thus [ \tan \theta = {G_1 \over {\bar n}(n_o - n_e)} = {g_{11} \over {\bar n}(n_o - n_e)} .\eqno(]


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The principle of superposition.

Generally speaking, the ellipticity is extremely small and difficult to measure (Moxon & Renshaw, 1990[link]). In right-handed quartz (right-handed with respect to optical rotation observed along the c axis), no = 1.544, ne = 1.553, g11 = g22 = −5.82 × 10−5 and g33 = 12.96 × 10−5 measured at λ = 5100 Å. Since the c axis is also the optic axis, δ = 0 for the (0001) plane, and thus κ = 1 for this section [equations ([link] and ([link]]. This value of κ = 1 means that the two waves are circular (see Section[link]), i.e. there is no linear birefringence, only a pure rotation. In this direction, the gyration g33 means a rotation of ρ = 29.5° mm−1 [using equation ([link]]. In a direction normal to the optic axis, from equation ([link] one finds ρ = −13.3° mm−1. However, in this direction, the crystal appears linearly birefringent with δ = 110.88 mm−1 [equation ([link]]. Thus the ellipticity κ = −0.00209, as calculated from equations ([link] and ([link]. In other words, the two waves are very slightly elliptical, and the sense of rotation of the two ellipses is reversed. Because of the change in sign of the gyration coefficients, it is found that at an angle of 56° 10′ down from the optic axis κ = 0, meaning that waves travelling along this direction show no optical rotation, only linear birefringence.


Moxon, J. R. L. & Renshaw, A. R. (1990). The simultaneous measurement of optical activity and circular dichroism in birefringent linearly dichroic crystal sections: I. Introduction and description of the method. J. Phys. Condens. Matter, 2, 6807–6836.Google Scholar
Moxon, J. R. L., Renshaw, A. R. & Tebbutt, I. J. (1991). The simultaneous measurement of optical activity and circular dichroism in birefringent linearly dichroic crystal sections: II. Description of the apparatus and results for quartz, nickel sulphate hexahydrate and benzil. J. Phys. D Appl. Phys. 24, 1187–1192.Google Scholar
Szivessy, G. & Münster, C. (1934). Über die Prüfung der Gitteroptik bei aktiven Kristallen. Ann. Phys. (Leipzig), 20, 703–736.Google Scholar

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