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

International Tables for Crystallography (2006). Vol. D, ch. 1.6, pp. 152-154

Section 1.6.3. Linear optics

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

1.6.3. Linear optics

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1.6.3.1. The fundamental equation of crystal optics

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It is necessary, in order to understand fully the propagation of light through a general anisotropic crystal, to address the question of the way in which an electromagnetic wave is affected by its passage through a regular array of atoms or molecules. A full analysis of this problem at a microscopical level is complicated and was treated, for example, by Ewald (1916[link]), who showed through consideration of a `half-crystal' how to link the electromagnetic field outside the crystal to that inside (a good description of Ewald's work on this can be read in the book P. P. Ewald and his Dynamical Theory of X-ray Diffraction, published by the International Union of Crystallography, Oxford Science Publications, 1992). For the purposes needed here, it is sufficient to apply Maxwell's equations to a bulk anisotropic continuum crystal, thus taking a macroscopic approach. The treatment here follows that given by Nussbaum & Phillips (1976[link]).

Consider the relationship between the dielectric displacement [\bf D] and an electric field [\bf E] which in tensor terms is given by [D_i = \varepsilon_o\varepsilon_{ij}E_j, \eqno (1.6.3.1)]where [\varepsilon_o] is the vacuum dielectric permittivity and [\varepsilon_{ij}] is a second-rank tensor, the relative dielectric tensor. Correspondingly, there is an induced polarization [\bf P] related to [\bf E] via [P_k = \varepsilon_o\chi_{k\ell}E_\ell ,\eqno (1.6.3.2)]where [\chi_{k\ell}] is another second-rank tensor, called the dielectric susceptibility tensor. Note that the restriction to a linear relationship between [\bf D] and [\bf E] (or [\bf P] and [\bf E]) confines the theory to the region of linear optics. Addition of higher-order terms (see above) gives nonlinear optics. (Nonlinear optics is discussed in Chapter 1.7[link] .)[\eqalignno{{\rm curl}\,{\bf H} &= \partial {\bf D}/\partial t &(1.6.3.3)\cr {\rm curl}\,{\bf E} &= - \partial {\bf B}/\partial t , &(1.6.3.4)}%fd1.6.3.4]where [\bf B] and [\bf H] are the magnetic induction and magnetic field intensity, respectively. It is customary at this point to assume that the crystal is non-magnetic, so that [{\bf B} = \mu_o {\bf H}], where [\mu_o] is the vacuum magnetic permeability. If plane-wave solutions of the form [\eqalignno{{\bf E}& = {\bf E}_o\exp[ i({\bf k\cdot r} - \omega t)] &(1.6.3.5)\cr {\bf H}& = {\bf H}_o\exp [i({\bf k\cdot r} - \omega t)] & (1.6.3.6)\cr {\bf D}& = {\bf D}_o\exp[ i({\bf k\cdot r} - \omega t) ]& (1.6.3.7)}%fd1.6.3.7]are substituted into equations (1.6.3.3)[link] and (1.6.3.4)[link], the following results are obtained: [\eqalignno{{\bf k \times H} &= \omega {\bf D} & (1.6.3.8)\cr {\bf k \times E} &= -\omega {\bf B}. & (1.6.3.9)}%fd1.6.3.9]These equations taken together imply that [\bf D], [\bf H] and [\bf k] are vectors that are mutually orthogonal to one another: note that in general [\bf E] and [\bf D] need not be parallel. Similarly [\bf B] (and hence [\bf H]), [\bf E] and [\bf k] are mutually orthogonal. Now, on substituting (1.6.3.9)[link] into (1.6.3.8)[link], [{1\over\mu_o\omega^2}{\bf k \times (k \times E)} = - {\bf D}. \eqno(1.6.3.10)]Defining the propagation vector (or wave normal) [\bf s] by [{\bf s} = {c\over \omega}{\bf k} = n {\hat {\bf s}}, \eqno (1.6.3.11)]where [{\hat {\bf s}}] is the unit vector in the direction of [\bf s] and n is the refractive index for light propagating in this direction, equation (1.6.3.10)[link] then becomes [{1\over \mu_oc^2}{\bf s \times (s \times E)} = - {\bf D}. \eqno(1.6.3.12)]Via the vector identity [{\bf A}\times({\bf B} \times {\bf C}) = ({\bf A} \cdot {\bf C}){\bf B} - ({\bf A} \cdot {\bf B}){\bf C}], this result can be transformed to [-({\bf s} \cdot {\bf s}){\bf E} + ({\bf s} \cdot {\bf E}){\bf s} = -\mu_oc^2 {\bf D}. \eqno (1.6.3.13)][{\bf s} \cdot {\bf s}] is equal to [n^2] and [{\bf s} \cdot {\bf E}] can be expressed simply in tensor form as [\textstyle\sum_j s_jE_j]. Now, with equation (1.6.3.1)[link], the fundamental equation of linear crystal optics is found: [ \textstyle\sum\limits_j (\varepsilon_{ihj}+ s_is_j)E_j = n^2 I E_i, \eqno (1.6.3.14)] where I is the unit matrix.

1.6.3.2. The optical indicatrix

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Equation (1.6.3.14)[link] is the relevant starting point for the derivation of the way in which light propagates in an anisotropic medium. To solve it in a particular case, treat it as an eigenvector–eigenvalue problem: the [E_i] are the eigenvectors and [n^2] the eigenvalues. For example, take the case of a uniaxial crystal. The dielectric tensor is then given by [ \pmatrix{ \varepsilon_{11}&0&0\cr 0&\varepsilon_{11}&0\cr 0&0&\varepsilon_{33}}.\eqno (1.6.3.15) ] Assume that light propagates along a direction in the [x_2x_3] plane, at an angle [\theta] to the [x_3] axis. Then, using (1.6.3.11)[link], it is seen that [\eqalignno{s_1 &= 0 &\cr s_2 &= n \sin \theta &\cr s_3 &= n \cos \theta & (1.6.3.16)}]and [ s_is_j = \pmatrix{ 0&0&0\cr 0&n^2\sin^2\theta&n^2\sin\theta\cos\theta\cr 0&n^2\sin\theta\cos\theta&n^2\cos\theta}.\eqno (1.6.3.17) ]Substituting into equation (1.6.3.14)[link] yields [\displaylines{\pmatrix{\varepsilon_{11}&0&0\cr 0&\varepsilon_{11}+n^2\sin^2\theta&n^2\sin\theta\cos\theta\cr 0&n^2\sin\theta\cos\theta&\varepsilon_{33}+n^2\cos\theta} \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 (1.6.3.18) }]Solving this for the eigenvalues n gives [n_1^2= \varepsilon_{11} \eqno(1.6.3.19)] as one solution and [{ 1\over n_{2}^2} = {\cos ^2\theta \over \varepsilon_{11}} + {\sin ^2\theta \over \varepsilon_{33}} \eqno(1.6.3.20)] as the other. This latter solution can be rewritten as [{ 1\over n_{2}^2} = {\cos ^2\theta \over n_o^2} + {\sin ^2\theta \over n_e^2}, \eqno(1.6.3.21)]showing how the observed refractive index [n_2] varies between the limits set by [n_o] and [n_e], called the ordinary and extraordinary refractive index, respectively (sometimes these are denoted by [\omega] and [\varepsilon], respectively). Equation (1.6.3.21)[link] can be thought of as the equation of a uniaxial ellipsoid (circular cross section) with the lengths of the semi-axes given by [n_o] and [n_e]. This is illustrated in Fig. 1.6.3.1[link], where [\bf OZ] is the direction of propagation of the light ray at an angle [\theta] to [x_3]. Perpendicular to [\bf OZ], an elliptical cross section is cut from the uniaxial ellipsoid with semi-axes [OA] equal to [n_o] and [OB] given by (1.6.3.21)[link]: the directions [\bf OA] and [\bf OB] also correspond to the eigenvectors of equation (1.6.3.4)[link].

[Figure 1.6.3.1]

Figure 1.6.3.1 | top | pdf |

The optical indicatrix.

Direction [\bf OA] is therefore the direction of the electric polarization transverse to the propagation direction, so that the refractive index measured for light polarized along [\bf OA] is given by the value [n_o]. For light polarized along [\bf OB], the refractive index would be given by equation (1.6.3.7)[link]. When [\bf OZ] is aligned along [x_3], a circular cross section of radius [n_o] is obtained, indicating that for light travelling along [\bf OZ] and with any polarization the crystal would appear to be optically isotropic. The ellipsoid described here is commonly known as the optical indicatrix, in this case a uniaxial indicatrix.

Two cases are recognized (Fig. 1.6.3.2[link]). When [n_o \,\lt\, n_e], the indicatrix is a prolate ellipsoid and is defined to be positive; when [n_o\,\gt\, n_e], it is oblate and defined to be negative. Note that when [n_o= n_e] the indicatrix is a sphere, indicating that the refractive index is the same for light travelling in any direction, i.e. the crystal is optically isotropic. The quantity [\Delta n = n_e-n_o] is called the linear birefringence (often simply called birefringence). In general, then, for light travelling in any direction through a uniaxial crystal, there will be two rays, the ordinary and the extraordinary, polarized perpendicular to each other and travelling with different velocities. This splitting of a ray of light into two rays in the crystal is also known as double refraction.

[Figure 1.6.3.2]

Figure 1.6.3.2 | top | pdf |

Positive and negative uniaxial indicatrix.

The origin of the birefringence in terms of the underlying crystal structure has been the subject of many investigations. It is obvious that birefringence is a form of optical anisotropy (the indicatrix is not spherical) and so it must be linked to anisotropy in the crystal structure. Perhaps the most famous early study of this link, which is still worth reading, is that of Bragg (1924[link]), who showed that it was possible to calculate rough values for the refractive indices, and hence birefringence, of calcite and aragonite. His theory relied upon the summation of polarizability contributions from the Ca2+ and O2− ions.

Returning now to the theory of the indicatrix, more general solution of the fundamental equation (1.6.3.14)[link] leads to a triaxial ellipsoid, i.e. one in which all three semi-axes are different from one another (Fig. 1.6.3.3[link]).

[Figure 1.6.3.3]

Figure 1.6.3.3 | top | pdf |

Biaxial indicatrix, showing the two optic axes and corresponding circular cross sections.

It is conventional to label the three axes according to the size of the refractive index by [n_\gamma\,\gt\, n_\beta\,\gt\, n_\alpha] (or simply [\gamma\,\gt\, \beta\,\gt\, \alpha]). In such an ellipsoid, there are always two special directions lying in the γ–α plane, known as the optic axial plane, and perpendicular to which there are circular cross sections (shown shaded) of radius β. Thus these two directions are optic axes down which the crystal appears to be optically isotropic, with a measured refractive index β for light of any polarization. For this reason, crystals with this type of indicatrix are known as biaxial. When the angle between the optic axes, denoted conventionally as 2V, is acute about the γ axis, the crystal is positive biaxial, and when it is acute about α the crystal is negative biaxial. Note that as 2V becomes smaller, the biaxial indicatrix becomes closer to a uniaxial indicatrix (positive or negative). In all general directions the crystal is optically anisotropic. Thus, for light along [x_3], the measured refractive indices will be α and β for light polarized along [x_1] and [x_2], respectively; for light along [x_2], α and γ are measured for light polarized along [ x_1] and [x_3], respectively; and along [x_1], β and γ are measured for light polarized along [x_2] and [x_3], respectively. There are therefore three different linear birefringences to measure: γ–β, β–α and γ–α.

The different indicatrices are oriented in the crystal according to symmetry considerations (Table 1.6.3.1[link]), and so their observation can form valuable and reliable indicators of the crystal system.

Table 1.6.3.1| top | pdf |
Symmetry constraints on the optical indicatrix

Crystal systemIndicatrix Orientation constraints
Cubic Isotropic (sphere) None
Tetragonal Uniaxial Circular cross section perpendicular to c
Trigonal
Hexagonal
Orthorhombic Biaxial All indicatrix axes aligned along a, b and c
Monoclinic Biaxial One indicatrix axis aligned along b (second setting)
Triclinic Biaxial None

1.6.3.3. The dielectric impermeability tensor

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It has been seen how the refractive indices can be described in a crystal in terms of an ellipsoid, known as the indicatrix. Thus for orthogonal axes chosen to coincide with the ellipsoid axes, one can write [ {x_1^2 \over n_1^2} + {x_2^2 \over n_2^2} + {x_3^2 \over n_3^2} = 1, \eqno(1.6.3.22)]where [n_1 = (\varepsilon_{11})^{1/2}], [n_2 =(\varepsilon_{22})^{1/2}] and [n_3 = (\varepsilon_{33})^{1/2}]. One can write this equation alternatively as [\eta_{11}x_1^2 + \eta_{22}x_2^2 + \eta_{33}x_3^2 = 1, \eqno (1.6.3.23)]where the [\eta_{ii} = 1/\varepsilon_{ii}] are the relative dielectric impermeabilities. For the indicatrix in any general orientation with respect to the coordinate axes[\eta_{11}x_1^2 + \eta_{22}x_2^2 + \eta_{33}x_3^2 + 2 \eta_{12}x_1x_2 + 2 \eta_{23}x_2x_3 + 2 \eta_{31}x_3x_1= 1. \eqno (1.6.3.24)]Thus the dielectric impermeability tensor is described by a second-rank tensor, related inversely to the dielectric tensor.

References

Bragg, W. L. (1924). The refractive indices of calcite and aragonite. Proc. R. Soc. London Ser. A, 105, 370.
Ewald, P. P. (1916). Zur Begründung der Kristalloptik. Ann. Phys. (Leipzig), 49, 1–38, 117–143.
Nussbaum, A. & Phillips, R. A. (1976). Contemporary optics for scientists and engineers. New Jersey: Prentice Hall.








































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