# Crystal optics

The crystal optics is concerned with the interaction of electromagnetic radiation, typically in the visible wavelength range, with crystalline or otherwise anisotropic solids, but also generalized with optically active liquids. It is a branch of optics, solid state physics and mineralogy.

## Background

The optical characteristics of crystals, which are responsible, inter alia, for reflection, refraction and absorption of the light are determined by its regular internal structure. Unlike the optically isotropic crystals glasses can be found in generally the phenomenon of the anisotropy: Important properties such as refractive index are dependent on the direction of light propagation in the crystal and its polarization.

More specifically, this applies to all the crystals, which do not have the cubic crystal system. To illustrate, carries a three dimensional diagram for any wave normal in the crystal direction of the light the value of the refractive indices in the two vibration directions perpendicular to this direction. This always results in an ellipsoid with three unequal generally mutually perpendicular principal axes, which are also called index ellipsoid, ellipsoid or indicatrix Fletcher.

- The crystal is cubic, the ellipsoid is reduced to the special case of a ball, since all three main axes have the same length. The propagation of light is isotropic in this case.
- In the case of the hexagonal, trigonal and tetragonal crystal system, only two of the principal axes are the same length, then one speaks of optically uniaxial or uniaxial crystals. Mentioned in the description, the axis is perpendicular to the two equally long major axes. When light is incident parallel to the axis of birefringence does not take place.
- Three different lengths of the principal axes can be found for the orthorhombic, monoclinic and triclinic crystal system, the crystal is now called optically biaxial or biaxially. These two axes do not coincide with the principal axes of the ellipsoid, rather they are clearly defined by the fact that they are perpendicular to the only two circles, which can be generated by intersection of a plane through the center of the ellipsoid with the indicatrix (all other sections can be derived ellipses and not circles ). The radius of this circle corresponds to the middle in length to the three main axes.

An important consequence of anisotropy of crystals is the birefringence, i.e. the splitting of incident light onto the crystal in an ordinary ray and an extraordinary ray having different polarization.

The optical activity of crystals can be attributed to their anisotropy: The plane of polarization of linearly polarized light is rotated by a proportional to the distance traveled in the crystal angle. It differs depending on whether the plane is rotated clockwise or counterclockwise if you look closely to the direction of light propagation, the right - and left-handed crystals, which are also referred to as visual modifications. Examples Links quartz and quartz rights are mentioned.

A third specific appropriate crystals to optical phenomenon is the so -called pleochroism. This means that light is absorbed to different extents, depending on the propagation and polarization direction. Additionally because the absorption depends on the wavelength, the Pleochronismus shows a direction-dependent color change of the irradiated light that is detectable in extreme cases, even with the naked eye.

The optical properties of the crystal can be influenced by external electric and magnetic fields, but also due to mechanical stress, in the former case one speaks of the electro-optic effect, in the second case of the magneto-optical effect. Conversely, they can be used for diagnosis of these external influences.

## Mathematical formalism

The basis of the mathematical formalism is the fact that the electric field strength and the electric displacement field are not the same direction. Thus, the dielectric function that combines the two formula sizes, not to be construed as a scalar, it must be treated as a second order tensor. The relationship between, and writes now:

Wherein the dielectric constant of vacuum is.

As an electromagnetic wave in an anisotropic medium spreads, can be calculated by solving the wave equation for anisotropic body:

Here represents a unit vector pointing in the propagation direction of the shaft, N is the refractive index.

Algebraically, the wave equation is a system of three coupled equations from which can be derived two refractive indices for the two different polarization directions. However, the equation system is not clearly shown in general in relation to the direction of polarization. Therefore, a method is used to reduce the three equations to two. First, we construct a system of three pairwise orthogonal vectors. Two of them are the direction of propagation and dislocation density, and the third is the magnetic field strength. Since no longer needs to stand as in isotropic solids at 90- degree angle, the wave equation is not suitable to determine the polarization character of the waves.

Is now utilized that is perpendicular to the propagation direction. It is

Said to inverse tensor. By selection of a new coordinate system with the coordinates A, B, C, which is selected such that the c- direction is parallel to, the system of equations can be reduced from three to two equations:

By solving this system of equations yields the two refractive indices and the polarization character for any direction.