1
Crystal Optics

1.1 Introduction

Crystal optics is the branch of optics that describes the behavior of electromagnetic waves in anisotropic media, that is, media (such as crystals) in which light behaves differently depending on the direction in which the light is propagating. The characteristic phenomena of crystals that are studied in crystal optics include double refraction (birefringence), polarization of light, rotation of the plane of polarization, etc.

The phenomenon of double refraction was first observed in crystals of Iceland spar by the Danish scientist E. Bartholin in 1669. This date is considered the beginning of crystal optics. Problems of the absorption and emission of light by crystals are studied in crystal spectroscopy. The effect of electric and magnetic fields, mechanical stress, and ultrasound waves on the optical properties of crystals are studied in electro-optics, magneto-optics, photoelasticity, acousto-optics, and photorefractivity, which are based on the fundamental laws of crystal optics.

Since the lattice constant (of the order of 10?Å) is much smaller than the wavelength of visible light (4000-7000?Å), a crystal may be regarded as a homogeneous but anisotropic medium. The optical anisotropy of crystals is caused by the anisotropy of the force field of particle interaction. The nature of the field is related to crystal symmetry. All crystals, except crystals of the cubic system, are optically anisotropic.

1.2 Index Ellipsoid or Optical Indicatrix

In isotropic materials, the electric field displacement vector D is parallel to the electric field vector E, related by D =??0?rE =??0E + P, where ?0 is the permittivity of free space, ?r is the unitless relative dielectric constant, and P is the material polarization vector.

The optical anisotropy of transparent crystals is due to the anisotropy of the dielectric constant. In an anisotropic dielectric medium (a crystal, for example), the vectors D and E are no longer parallel; each component of the electric flux density D is a linear combination of the three components of the electric field E.

where i, j = 1, 2, 3 indicate the x, y, and z components, respectively. The dielectric properties of the medium are therefore characterized by a 3 × 3 array of nine coefficients {?ij} forming a tensor of second rank known as the electric permittivity tensor and denoted by the symbol ?. Equation (1.1) is usually written in the symbolic form D = ?E. The electric permittivity tensor is symmetrical, ?ij = ?ji, and is therefore characterized by only six independent numbers. For crystals of certain symmetries, some of these six coefficients vanish and some are related, so that even fewer coefficients are necessary.

Elements of the permittivity tensor depend on the choice of the coordinate system relative to the crystal structure. A coordinate system can always be found for which the off-diagonal elements of ?ij vanish, so that

where ?1 = ?11, ?2 = ?22, and ?3 = ?33. These are the directions for which E and D are parallel. For example, if E points in the x-direction, D must also point in the x-direction. This coordinate system defines the principal axes and principal planes of the crystal. The permittivities ?1, ?2, and ?3 correspond to refractive indices.

are known as the principal refractive indices and ?0 is the permittivity of free space.

In crystals with certain symmetries two of the refractive indices are equal (nl =?n2 ??n3) and the crystals are called uniaxial crystals. The indices are usually denoted n1 =?n2 =?no and n3 =?ne. The uniaxial crystal exhibits two refractive indices, an "ordinary" index (no) for light polarized in the x- or y-direction, and an "extraordinary" index (ne) for polarization in the z-direction. The crystal is said to be positive uniaxial if ne >?no and negative uniaxial if ne <?no. The z-axis of a uniaxial crystal is called the optic axis. In other crystals (those with cubic unit cells, for example) the three indices are equal and the medium is optically isotropic. Media for which the three principal indices are different (i.e. n1 ??n2 ??n3) are called biaxial. Light polarized at some angle to the axes will experience a different phase velocity for different polarization components and cannot be described by a single index of refraction. This is often depicted as an index ellipsoid.

The optical properties of crystals are described by the index ellipsoid or optical indicatrix. It is generated by the equation

where x1, x2, and x3 are the principal axes of the dielectric constant tensor and n1, n2, and n3 are the principal dielectric constants, respectively. Figure 1.1 shows the optical indicatrix of a biaxial crystal. It is a general ellipsoid with n1 ??n2 ??n3 representative of the optical properties of triclinic, monoclinic, and orthorhombic crystals.

Figure 1.1 The index ellipsoid. The coordinates (x, y, z) are the principal axes and (n1, n2, n3) are the principal refractive indices of the crystal.

1.3 Effect of Crystal Symmetry

In the case of cubic crystals, which are optically isotropic, ? is independent of direction and the optical indicatrix becomes a sphere with radius n. In crystals of intermediate systems (trigonal, tetragonal, and hexagonal), the indicatrix is necessarily an ellipsoid of revolution about the principal symmetry axis (Figure 1.2). The central section is perpendicular to the principal axis, and only this central section is a circle of radius n0. Hence, only for a wave normal along the principal axis is there no double refraction. The principal axis is called the optic axis and the crystals are said to be uniaxial. A uniaxial crystal is called optically positive (+) when ne >?n0 and negative (-) when ne <?n0.

Figure 1.2 The indicatrix for a (positive) uniaxial crystal.

For crystals of the lower systems (orthorhombic, monoclinic, and triclinic), the indicatrix is a triaxial ellipsoid. There are two circular sections (Figure 1.3) and hence two privileged wave normal directions for which there is no double refraction. These two directions are called the primary optic axes or simply the optic axes, and the crystals are said to be biaxial.

Figure 1.3 The two circular sections of the indicatrix and the two primary optic axes OP1, OP2 for a biaxial crystal.

1.4 Wave Surface

If a point source of light is situated within a crystal the wave front emitted at any instant forms a continuously expanding surface. The geometric locus of points at a distance v from a point O is called the ray surface or wave surface. Actually, the wave surface is a wave front (or pair of wave fronts) completely surrounding a point source of monochromatic light. This is also a double-sheeted surface. In most crystalline substances, however, two wave surfaces are formed; one is called the ordinary wave surface and the other is called the extraordinary wave surface. In both positive and negative crystals, the ordinary wave surface is a sphere and the extraordinary wave surface is an ellipsoid of revolution.

1.4.1 Uniaxial Crystal

In uniaxial crystals, one surface is a sphere and the other is an ellipsoid of revolution touching one another along the optical axis - OZ as shown in Figure 1.4a,b. In positive (+) crystals (ne >?n0) the ellipsoid is inscribed within the sphere (Figure 1.4a), whereas in negative (-) crystals (ne <?no) the sphere is inscribed within the ellipsoid (Figure 1.4b).

Figure 1.4 Ray surfaces of uniaxial crystals: (a) positive, (b) negative, (OZ) optical axis of the crystal, (v0) and (ve) phase velocities of ordinary and extraordinary waves propagating in the crystals.

The dependence of the ray velocity of a plane wave propagating in a crystal on the direction of propagation and the nature of polarization of the wave leads to the splitting of light rays in crystals. In a uniaxial crystal, one of the...