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

International Tables for Crystallography (2013). Vol. D, ch. 1.6, pp. 169-175

Section 1.6.5. Optical rotation

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:  glazer@physics.ox.ac.uk

1.6.5. Optical rotation

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1.6.5.1. Introduction

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Optical rotation or gyration, as it is sometimes known, was first recorded by Arago in 1811. Since then, a great deal of work has been done to try to explain this phenomenon, and at the present time it is one of the few physical properties of a crystal that can be successfully understood in terms of the underlying crystal structure (Glazer & Stadnicka, 1986[link]). Lowry (1935[link]) has given a good historical account of the subject.

Optical rotation is the phenomenon observed in some crystals (and in some solutions of, usually, organic compounds) of the rotation of plane-polarized light on passing through the crystal. If the rotation, as seen by the observer, is to the left or counterclockwise it is known as laevorotation; if to the right or clockwise it is known as dextrorotation. A crystal that shows this effect is sometimes called optically active or gyrotropic. It is therefore clear that, from a symmetry point of view, optical rotation can only occur in a crystal in which one direction is not equivalent to its opposite, i.e. there are no inversion symmetry operations that can change chirality.

One of the earliest theories to explain the origin of optical rotation was given by Fresnel. This was based on the idea that a plane-polarized wave can be equally described by two opposing circularly polarized waves propagating along the same direction. In Fig. 1.6.5.1[link](a), the polarization vector [\bf OA] is simply given by the vector sum of [\bf OB] and [\bf OC], polarization vectors at some instant of time belonging to a right-circular and a left-circular wave, respectively. Here it is assumed that the light propagates in a direction towards the viewer. If the light now passes through a gyrotropic material, one of the circularly polarized waves will be slowed down with respect to the other (this corresponds to having a small difference in refractive indices for the two waves). Thus on emerging from the crystal, there will be a phase difference between the circular waves, so that on recombining, the resulting plane of linear polarization OA will have rotated through an angle [\varphi]. This is shown in Fig. 1.6.5.1[link](b), where the right-circular (clockwise) component travels faster than the left-circular (anticlockwise) component. Because of this, in a given time, the electric vector for the right-circular wave, observed in a fixed plane perpendicular to the line of sight, will have rotated through a larger angle clockwise than the electric vector for the left-circular wave will have rotated anticlockwise. The result is a net rotation to the right, i.e. dextrorotation, of the vector OA. The left and right-circular refractive indices, [n_L] and [n_R], are inversely related to the velocities of the waves, and so Fig. 1.6.5.1[link](b) corresponds to the case where [n_L\,\gt\, n_R]. Thus this theory shows that a positive value of [n_L - n_R] defines dextrorotation and that optical rotation is in fact a form of circular birefringence.

[Figure 1.6.5.1]

Figure 1.6.5.1 | top | pdf |

Fresnel's explanation of optical rotation.

Fig. 1.6.5.2[link] illustrates in a more generalized way the nature of the birefringences possible in a crystal. Formally, the refractive index of a medium can be written in terms of real and imaginary components: [ n = n^{\prime} + in^{\prime\prime}, \eqno(1.6.5.1)]the imaginary component referring to the absorption of the light (the real and imaginary terms are related as usual by the Kramers–Kronig relationship). Similarly, the linear and circular birefringences can also be written in real and imaginary terms: [\eqalignno{\Delta n_{\rm linear} & = (n_1 - n_2)^{\prime} + i(n_1 - n_2)^{\prime\prime} &\cr \Delta n_{\rm circular} & = (n_L - n_R)^{\prime} + i(n_L - n_R)^{\prime\prime}. &(1.6.5.2)}]The real linear birefringence (a) shows a resonance, changing sign, at a particular wavelength. Its variation with wavelength is known as birefringent dispersion. The imaginary component (b) peaks at this wavelength and corresponds to the difference in absorption between linear polarization states. This is called linear dichroism (LD) and is determined by quantum-mechanical selection rules resulting from matrix elements of the type [ \langle\Psi_1|{\hat p}|\Psi_0\rangle], where [{\hat p}] is the electric dipole operator. The circular components follow similar behaviour. The real circular birefringence (c) corresponds to the optical rotation and its change with wavelength is known as optical rotatory dispersion (ORD). The imaginary circular birefringence (d) corresponds to the difference in absorption of opposite circularly polarized states. This is called circular dichroism (CD) and is determined by the matrix elements of the type [\langle\Psi_1|{\hat \alpha}|\Psi_0\rangle], where [{\hat \alpha}] is the polarizability operator. Since [{\hat \alpha}] is an even function, whereas [{\hat p}] is an odd function, the selection rules for LD and CD are different, and each type of spectrum gives different information.

[Figure 1.6.5.2]

Figure 1.6.5.2 | top | pdf |

Real and imaginary birefringence as a function of wavelength: (a) real linear; (b) imaginary linear; (c) real circular; (d) imaginary circular.

The link between optical rotation and crystal structure has been the subject of a great deal of work from the 18th century to date. Experimental methods for determining absolute chirality (Glazer & Stadnicka, 1989[link]) of crystals only became routine in the late 1940s with the use of the X-ray anomalous dispersion effect, which made it possible to determine the absolute sense of the chiral nature of a crystal and then to link this to the sense of optical rotation. However, despite the preponderance of a great many complex theories, until recently, experimental evidence to support these theories has been fragmentary. It was shown by Glazer & Stadnicka (1986[link]) that, at least for inorganic crystals, the problem lay in mistakes in the experiments or publications describing the experiments, rather than in the underlying theories. Once these errors are taken into account, it is possible to find a direct link between the chiral nature of the structure and the optical rotation, which appears to work in almost all cases. It has even been possible to produce a computer program OPTACT (Devarajan & Glazer, 1986[link]; Glazer, 2002[link]), using a polarizability theory based on earlier work by Born (1933[link]), that is capable of calculating both the sign and magnitude of the effect with reasonable precision in many cases. OPTACT also calculates the refractive indices with good reliability.

1.6.5.2. The dielectric tensor and spatial dispersion

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The relevant polarization term to consider here is [ P_i^\omega = \omega_0\chi_{ij\ell}\nabla_\ell E_j^\omega. \eqno (1.6.5.3)]

The important part of this expression is the use of the field gradient, which implies a variation of the electric field across the unit cell of the crystal rather than the assumption that [\bf E] is everywhere constant. This variation in [\bf E] is known as spatial dispersion (Agranovich & Ginzburg, 1984[link]).

Assume propagation of a plane wave given by [{\bf E}=] [{\bf E_o}\exp (i{\bf k}\cdot {\bf r})] through an optically active crystal. Substituting into the expression for the polarization gives [ P_i^\omega = i\omega_0\chi_{ij\ell}E_j^\omega k_\ell .\eqno (1.6.5.4)]This term can now be treated as a perturbation to the dielectric tensor [\varepsilon_{ij}(\omega)] to form the effective dielectric tensor [\varepsilon_{ij}(\omega, {\bf k})]: [\varepsilon_{ij}(\omega, {\bf k}) = \varepsilon_{ij}(\omega) + i\gamma_{ij\ell}k_\ell , \eqno (1.6.5.5)]where [\gamma_{ij\ell}] has been written for the susceptibility [\chi_{ij\ell}] in order to distinguish it from the use of [\chi] elsewhere. Note that this can be expressed more generally as a power-series expansion in the vector [\bf k] (Agranovich & Ginzburg, 1984[link]) to allow for a generalization to include all possible spatial dispersion effects: [\varepsilon_{ij}(\omega, {\bf k}) = \varepsilon_{ij}(\omega) + i\gamma_{ij\ell}(\omega)k_\ell + \alpha_{ij\ell m}(\omega)k_\ell k_m, \eqno (1.6.5.6)]where the susceptibilities are in general themselves dependent on frequency.

1.6.5.3. Symmetry of effective dielectric tensor

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To determine the symmetry constraints on the effective dielectric tensor, it should be recognized that the application of a real electric field [\bf E] must lead to a real dielectric displacement [\bf D]. This therefore implies that one can write [\varepsilon_{ij}(\omega, {\bf k}) = \varepsilon_{ij}^*(-\omega^*, -{\bf k}^*)] or [\varepsilon_{ij}(\omega^*, {\bf k}^*)= \varepsilon_{ij}^*(-\omega, -{\bf k}). \eqno(1.6.5.7)]Furthermore, in the absence of any absorptive processes, [\varepsilon_{ij}] must be Hermitian, that is [\varepsilon_{ij}(\omega, {\bf k}) = \varepsilon_{ji}(\omega, -{\bf k}). \eqno(1.6.5.8)]This is fulfilled by the following symmetry constraints for the leading terms in the effective dielectric tensor: [\varepsilon_{ij}(\omega) = \varepsilon_{ji}(\omega); \quad \gamma_{ij\ell}(\omega) = -\gamma_{ji\ell}(\omega); \quad \alpha_{ij\ell m}(\omega) = \alpha_{ji\ell m}(\omega). \eqno(1.6.5.9)]In a gyrotropic crystal, there must be at least one direction that is not equivalent to its opposite, and so such a crystal cannot have a centre of symmetry. (Only a noncentrosymmetric crystal can be gyrotropic. However, it is true to say that all non-gyrotropic crystals must be centrosymmetric.) Therefore, for a non-gyrotropic crystal, [\varepsilon_{ij}(\omega, {\bf k}) = \varepsilon_{ij}(\omega, -{\bf k}). \eqno (1.6.5.10)]It follows therefore in such a case that [\gamma_{ij\ell}(\omega) = 0 .\eqno (1.6.5.11)]It is obvious then that the susceptibility [\gamma_{ij\ell}(\omega)] has the required symmetry for gyration and that it forms an antisymmetric tensor of rank 3 (see Section 1.1.4.5.3[link] for the properties of antisymmetric tensors).

1.6.5.4. Gyration tensor

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It is convenient to rewrite the effective dielectric expression in the following way (just keeping the first two terms): [\varepsilon_{ij}(\omega, {\bf k}) = \varepsilon_{ij}(\omega) + i{\hat e}_{ijm}g_{m\ell}k_l ,\eqno(1.6.5.12)]where [{\hat e}_{ijm}] is a unit antisymmetric pseudotensor of rank 3 or permutation tensor ([{\hat e}_{123}= 1], [{\hat e}_{213}= -1], [{\hat e}_{112}= 0] etc.; [{\hat e}_{ijm}] is not affected by mirror reflection) and [g_{m\ell}] represents a pseudotensor (i.e. axial tensor) of rank 2. One can then write further [\varepsilon_{ij}(\omega, {\bf k}) = \varepsilon_{ij}(\omega) + i{\hat e}_{ijm}G_{m} , \eqno(1.6.5.13)]where [G_m= g_{m\ell}k_\ell] is a component of a pseudovector (i.e. axial vector), known as the gyration vector. The formula for the dielectric displacement can then be expressed in the form [ D_i = \varepsilon_o\varepsilon_{ij}(\omega, {\bf k}) E_j = \varepsilon_o\left[\varepsilon_{ij}(\omega) E_j - i\left({\bf G \times E}\right)_i\right]. \eqno(1.6.5.14)]The operation [{\bf G \times E}] can also be represented by the product of an antisymmetric tensor [[G]] with the vector [\bf E]: [\eqalignno{ \pmatrix{ 0&G_{12}&G_{13}\cr -G_{12}&0&G_{23}\cr -G_{13}&-G_{23}&0 }\pmatrix {E_1\cr E_2\cr E_3} &= \pmatrix{G_{12}E_2+G_{13}E_3\cr G_{23}E_3-G_{12}E_1\cr -G_{13}E_1-G_{23}E_2 }&\cr &= \pmatrix{-G_{3}E_2+G_{2}E_3\cr -G_{1}E_3+G_{3}E_1\cr -G_{2}E_1+G_{1}E_2 }, &\cr &&(1.6.5.15)}]where [-G_1 =] [G_{23} =] [-G_{32}], [-G_2 =] [-G_{13} =] [G_{31}] and [-G_3 =] [G_{12} =] [-G_{21}]. If [{\hat {\bf s}}] is the unit vector in the propagation direction, then [ {\bf G} = G{\hat {\bf s}}\eqno(1.6.5.16)] and G represents the magnitude of the gyration vector. Thus, [{\hat {\bf s}}\cdot {\bf G} = {\hat {\bf s}}\cdot G {\hat {\bf s}} = G {\hat {\bf s}}\cdot {\hat {\bf s}} = G. \eqno(1.6.5.17)]Consequently, if one knows the direction of propagation inside the crystal, the coefficient G can be calculated via [G = {\hat s}_1G_1 + {\hat s}_2G_2 + {\hat s}_3G_3 \eqno (1.6.5.18)]and then the optical rotatory power is defined as [ \rho = {\pi G \over \lambda n}, \eqno (1.6.5.19)]where [\rho] is the angle of rotation in degrees per millimetre. According to the way in which the sign of G has been defined here, a positive value of [\rho] means dextrorotation and a negative value means laevorotation.

There is a possibility of confusion here in terminology with respect to the term `gyration tensor'. It is seen that there exists an antisymmetric tensor [[G]]: this is sometimes referred to as the gyration tensor. However, returning to equation (1.6.5.12)[link], it is the pseudotensor given by [g_{m\ell}] that is more often described as the gyration tensor. The difference between them is really one of emphasis. The [[G]] tensor refers to the polarization directions of the wave, whereas the g tensor is referred to the direction of wave propagation. Thus, for example, [G_{23}] refers to a wave whose polarization lies in the [x_2x_3] plane, and so propagates in the [x_1] direction, according to the axial gyration vector component [-G_1]. The gyration is equally described by the tensor component [g_{11}]. Being an axial tensor, [g_{11}] corresponds to a wave travelling along [x_1], along which direction one observes a rotation in the [x_2x_3] plane.

Although g is in general an antisymmetric tensor, since it is only the scalar products [g_{m\ell}{\hat s}_m{\hat s}_\ell] that are important in determining the rotation of the plane of polarization, the components of [\bf D], and hence the refractive index behaviour, are independent of the antisymmetric part of [g_{m\ell}]. It is thus possible to construct a table of tensor invariances for the symmetric part of the gyration tensor with regard to the possible symmetry classes of a crystal (Table 1.6.5.1[link]). For a discussion of the importance of the antisymmetric terms, see Agranovich & Ginzburg (1984[link]).

Table 1.6.5.1| top | pdf |
Symmetry constraints (see Section 1.1.4.10[link] ) on the gyration tensor [g_{ij}]

All [g_{ij}] components are zero for the centrosymmetric point groups plus [4mm], [\bar 43m], [3m], [6mm], [\bar 6] and [\bar 6m2].

TriclinicMonoclinicOrthorhombic
Point group 1 Point group 2 ([2\parallel x_2]) Point group m ([m \perp x_2]) Point group 222
[\,\,\,\pmatrix{ g_{11}& g_{12}& g_{13}\cr g_{12}& g_{22}& g_{23}\cr g_{13}& g_{23}& g_{33}\cr}] [\pmatrix{ g_{11}& 0& g_{13}\cr 0& g_{22}& 0\cr g_{13}& 0& g_{33}\cr}] [\pmatrix{0& g_{12}& 0\cr g_{12}& 0& g_{23}\cr 0& g_{23}& 0\cr}] [\pmatrix{ g_{11}& 0& 0\cr 0& g_{22}& 0\cr 0& 0& g_{33}\cr}]
  Point group 2 ([2\parallel x_3]) Point group m ([m \perp x_3]) Point group [mm2]
  [\pmatrix{ g_{11}& g_{12}& 0\cr g_{12}& g_{22}& 0\cr 0& 0& g_{33}\cr}] [\pmatrix{ 0&& g_{13}\cr&0& g_{23}\cr g_{13}& g_{23}& 0\cr}] [\pmatrix{0& g_{12}& 0\cr g_{12}& 0& 0\cr 0& 0& 0\cr}]

TetragonalTrigonal and hexagonalCubic and isotropic
Point groups 4, 422 Point group [\bar 4] Point groups 3, 32, 6, 622 Point groups 432, 23
[\pmatrix{g_{11}& 0& 0\cr 0& g_{11}& 0\cr 0& 0& g_{33}\cr}] [\pmatrix{ g_{11}& g_{12}& 0\cr g_{12}& -g_{11}& 0\cr 0& 0& 0\cr}] [\pmatrix{g_{11}& 0& 0\cr 0& g_{11}& 0\cr 0& 0& g_{33}\cr}] [\pmatrix{ g_{11}& 0& 0\cr 0& g_{11}& 0\cr 0& 0& g_{11}\cr}]
Point group [\bar 42m]   Isotropic, no centre of symmetry
[\pmatrix{g_{11}& 0& 0\cr 0& -g_{11}& 0\cr 0& 0& 0\cr}]   [\pmatrix{ g_{11}& 0& 0\cr 0& g_{11}& 0\cr 0& 0& g_{11}\cr}]

1.6.5.5. Optical rotation along the optic axis of a uniaxial crystal

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Consider a uniaxial crystal such as quartz, crystallizing in point group 32. In this case, the only dielectric tensor terms (for the effect of symmetry, see Section 1.1.4.10[link] ) are [\varepsilon_{11} = \varepsilon_{22} \ne \varepsilon_{33}], with the off-diagonal terms equal to zero. The equations for the dielectric displacements along the three coordinate axes [x_1], [x_2] and [x_3] are then given, according to equations (1.6.5.14)[link] and (1.6.5.15)[link], by [\eqalignno{D_1& = \varepsilon_{o}\varepsilon_{11}E_1 - i\varepsilon_{o}\left[G_{12}E_2 + G_{13}E_3\right]&\cr D_2& = \varepsilon_{o}\varepsilon_{22}E_2 - i\varepsilon_{o}\left[G_{23}E_3 - G_{21}E_1\right]&\cr D_3 &= \varepsilon_{o}\varepsilon_{33}E_3 - i\varepsilon_{o}\left[G_{31}E_1 - G_{32}E_2\right]. & (1.6.5.20)}]If the light is taken to propagate along [x_3], the optic axis, the fundamental optics equation (1.6.3.14)[link] is expressed as [\displaylines{ \pmatrix{ \varepsilon_{11}& - iG_{12}& -iG_{13}\cr iG_{12}&\varepsilon_{11}& -iG_{23}\cr iG_{13}&iG_{23}&\varepsilon_{33}+n^2} \pmatrix{E_1\cr E_2\cr E_3 } \cr\hfill\quad= \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(1.6.5.21)}]As usual, the longitudinal solution given by [iG_{13}E_1 + iG_{23}E_2 + \varepsilon_{33}E_3 = 0 \eqno(1.6.5.22)]can be ignored, as one normally deals with a transverse electric field in the normal case of propagating light. For a non-trivial solution, then, [ \left| \matrix{ \varepsilon_{11}-n^2&-iG_{12}\cr iG_{12}&\varepsilon_{11}-n^2} \right| = 0, \eqno (1.6.5.23)]which gives [ n^2 = \varepsilon_{11} \pm G_{12}. \eqno (1.6.5.24)]This results in two eigenvalue solutions, [n_1] and [n_2], from which one has [ (n_1 - n_2)(n_1 + n_2) = 2G_{12} = -2G_3 \eqno (1.6.5.25)]and thus [ n_1 - n_2 = {G_{12}\over {\bar n}} = - {G_{3}\over {\bar n}}. \eqno (1.6.5.26)]The optical rotatory power (1.6.5.19)[link] is then given by [\rho = {\pi G_{3}\over \lambda {\bar n}} = {\pi (n_2 - n_1)\over \lambda}. \eqno(1.6.5.27)]Note that in order to be consistent with the definition of rotatory power used here, [n_2\,\gt\, n_1] for a dextrorotatory solution. This implies that [n_2] should be identified with [n_L] and [n_1] with [n_R]. To check this, find the eigenvectors corresponding to the two solutions (1.6.5.24)[link].

For [n_1^2 = \varepsilon_{11} - G_{3} ], the following matrix is found from (1.6.5.21)[link]: [ \pmatrix{G_3&iG_{3}\cr-iG_{3}&G_3} = 0, \eqno (1.6.5.28)] giving the Jones matrix [(1/2^{1/2})\pmatrix{ 1\cr -i} = 0. \eqno (1.6.5.29)]This corresponds to a right-circularly polarized wave. It should be noted that there is confusion in the optics textbooks over the Jones matrices for circular polarizations. Jones (1948[link]) writes a right-circular wave as [\pmatrix{ 1\cr i},]but this is for a definition of right-circularly polarized light as that for which an instantaneous picture of the space distribution of its electric vector describes a right spiral. The modern usage is to define the sense of circular polarization through the time variation of the electric vector in a given plane as seen by an observer looking towards the source of the light. This reverses the definition given by Jones.

For [n_2^2 = \varepsilon_{11} + G_{3} ], the following matrix is found: [ \pmatrix{-G_3&iG_{3}\cr -iG_{3}&-G_3} = 0, \eqno (1.6.5.30)] giving [ (1/{2}^{1/2})\pmatrix{ 1\cr i} = 0. \eqno (1.6.5.31)]

This corresponds to a left-circularly polarized wave. Therefore it is proved that the optical rotation arises from a competition between two circularly polarized waves and that in equation (1.6.5.26)[link] [n_1 = n_R] and [n_2 = n_L], the refractive indices for right- and left-circularly polarized light, respectively. Note that Fresnel's original idea of counter-rotating circular polarizations fits nicely with the eigenvectors rigorously determined in (1.6.5.29)[link] and (1.6.5.31)[link]. Thus [\rho = {\pi (n_L - n_R)\over \lambda}. \eqno(1.6.5.32)]Finally, for light propagating along [x_3] in quartz, one can write the direction of the wave normal as [ {\hat s}_1 = 0; \quad {\hat s}_2 = 0;\quad {\hat s}_3 = 1 \eqno(1.6.5.33)] and then the gyration vector is given by [ G = g_{i3} {\hat s}_i {\hat s}_3 = g_{13} + g_{23} + g_{33} = g_{33} \eqno(1.6.5.34)]as [g_{12}= g_{23} = 0] in point group 32. Thus from (1.6.5.26)[link] it is seen that [ g_{33} = G_3 = -G_{12}, \eqno(1.6.5.35)]thus linking one of the components of the gyration tensor [g_{m\ell}] with a component of the gyration vector [\bf G] and a tensor component of [[G]].

1.6.5.6. 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 (1.6.3.14)[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(1.6.5.36)}] 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 (1.6.5.37)] and then [ n^4 - n^2(\varepsilon_{11} + \varepsilon_{33}) + \varepsilon_{11}\varepsilon_{33} - G_{23}^2 = 0. \eqno (1.6.5.38)] 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 (1.6.5.39)]The eigenvectors for the two solutions [n_1] and [n_2] can easily be shown to correspond to elliptical polarizations. Notice that in equation (1.6.5.39)[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 (1.6.5.40)] 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, & (1.6.5.41)}]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(1.6.5.42)][\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. (1.6.5.41)[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. 1.6.5.3[link]. From this, the ellipticity [\kappa] of the polarization is given by [\kappa = \tan({\textstyle{1\over 2}} \theta), \eqno (1.6.5.43)] where [ \tan \theta = {2\rho \over \delta}. \eqno (1.6.5.44)] Thus [ \tan \theta = {G_1 \over {\bar n}(n_o - n_e)} = {g_{11} \over {\bar n}(n_o - n_e)} .\eqno(1.6.5.45)]

[Figure 1.6.5.3]

Figure 1.6.5.3 | top | pdf |

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 (1.6.5.44)[link] and (1.6.5.43)[link]]. This value of κ = 1 means that the two waves are circular (see Section 1.6.5.5[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 (1.6.5.27)[link]]. In a direction normal to the optic axis, from equation (1.6.5.40)[link] one finds ρ = −13.3° mm−1. However, in this direction, the crystal appears linearly birefringent with δ = 110.88 mm−1 [equation (1.6.5.42)[link]]. Thus the ellipticity κ = −0.00209, as calculated from equations (1.6.5.44)[link] and (1.6.5.43)[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.

References

Agranovich, V. M. & Ginzburg, V. C. (1984). Crystal optics with spatial dispersion, and excitons. Berlin: Springer.
Born, M. (1933). Dynamische Gittertheorie der Kristalle. In Handbuch der Physik, 24, 623–794.
Devarajan, V. & Glazer, A. M. (1986). Theory and computation of optical rotatory power in inorganic crystals. Acta Cryst. A42, 560–569.
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