Crystal optics

The crystal optics deals with the interaction of electromagnetic radiation , usually in the visible wavelength range , with crystalline or otherwise anisotropic solids , but in general terms with optically active liquids. It is a branch of optics , solid-state physics and mineralogy .

background

The optical properties of crystals, which u. a. responsible for reflection , refraction and absorption of light are determined by their regular internal structure. In contrast to optically isotropic glasses , the phenomenon of anisotropy is usually found in crystals: important properties such as e.g. B. the refractive index depend on the direction of propagation of the light in the crystal and its polarization (more precisely this applies to all crystals that do not have the cubic crystal system).

To illustrate this, enter the value of the refractive indices in the two oscillation directions perpendicular to this direction in a three-dimensional diagram for each possible wave normal direction of light in the crystal. This always results in an ellipsoid with generally three unequal, mutually perpendicular main axes , which is also referred to as an index ellipsoid , Fletcher ellipsoid or indicatrix :

• If the crystal is cubic, the ellipsoid is reduced to the special case of a sphere , since all three main axes have the same length. The propagation of light is isotropic in this case.

Birefringence on a calcite crystal:
ordinary and extraordinary rays are visible through red fluorescence in the crystal

• In the case of the hexagonal , trigonal and tetragonal crystal system, only two of the main axes are of the same length ( ellipsoid of revolution ), one then speaks of optically uniaxial or uniaxial crystals. The optical axis mentioned in the designation is perpendicular to the two main axes of equal length. There is no birefringence only when light is incident parallel to this axis .
• There are three main axes of different lengths ( three-axis ellipsoid ) for the orthorhombic , monoclinic and triclinic crystal system, the crystal is now optically called biaxial or biaxial. These two axes do not coincide with the main axes of the ellipsoid; rather, they are clearly defined by the fact that they are perpendicular to the only two circles that can be created by intersecting a plane with the ellipsoid through its center point (all other cuts do not result in circles, but ellipses ). The radius of these circles corresponds to the middle of the length of the three main axes.

An important consequence of the anisotropy of crystals is birefringence; H. the splitting of light that hits the crystal into an ordinary and an extraordinary ray, which have a different polarization.

The optical activity of crystals can also be traced back to their anisotropy: The plane of polarization of linearly polarized light is rotated by an angle that is proportional to the distance covered in the crystal. Depending on whether the plane is rotated clockwise or counterclockwise, if one looks exactly against the direction of propagation of the light, a distinction is made between right and left rotating crystals, which are also referred to as optical modifications . Examples are left and right quartz .

A third optical phenomenon that applies specifically to crystals is pleochroism . This means that light is absorbed to different degrees depending on the direction of propagation and polarization. Since the absorption also depends on the wavelength , the pleochronism shows itself in a direction-dependent color change of the transmitted light, which in extreme cases can be seen with the naked eye.

The optical properties of a crystal can be influenced by external electrical and magnetic fields , but also by mechanical stress. In the first case one speaks of the electro-optical effect , in the second of the magneto-optical effect . Conversely, these effects can be used to diagnose external influences.

Mathematical formalism

The basis of the mathematical formalism is the fact that the electric field strength and the electric displacement density are no longer directed in the same way. This means that the dielectric function , which links the two formula quantities, can no longer be understood as a scalar , but has to be treated as a second order tensor . The relationship between and is now written: ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {D}}}$ ${\displaystyle \varepsilon }$ ${\displaystyle {\vec {D}}}$ ${\displaystyle {\vec {E}}}$

${\displaystyle {\begin{pmatrix}D_{x}\\D_{y}\\D_{z}\end{pmatrix}}=\varepsilon _{0}{\begin{pmatrix}\varepsilon _{xx}&\varepsilon _{xy}&\varepsilon _{xz}\\\varepsilon _{yx}&\varepsilon _{yy}&\varepsilon _{yz}\\\varepsilon _{zx}&\varepsilon _{zy}&\varepsilon _{zz}\end{pmatrix}}\cdot {\begin{pmatrix}E_{x}\\E_{y}\\E_{z}\end{pmatrix}},}$

where represents the dielectric constant of the vacuum . ${\displaystyle \varepsilon _{0}}$

How an electromagnetic wave propagates in anisotropic medium can be calculated by solving the wave equation for anisotropic bodies:

${\displaystyle \varepsilon \cdot {\vec {E}}=n^{2}({\vec {E}}-{\vec {k}}({\vec {E}}\cdot {\vec {k}}))}$.

Here represents a unit vector pointing in the direction of propagation of the wave, n is the refractive index. ${\displaystyle {\vec {k}}}$

Algebraically, the wave equation is a system of three coupled equations from which the two refractive indices for the two different directions of polarization can be derived. However, the system of equations is generally ambiguous with regard to the direction of polarization. Therefore, a method is used to reduce the three equations to two. First, a system is constructed from three vectors that are perpendicular to each other in pairs. Two of them are the direction of propagation and displacement density , the third is the magnetic field strength . Since there is no longer a need to stand at a 90-degree angle, as in an isotropic solid , the wave equation is not suitable for determining the polarization character of the waves. ${\displaystyle {\vec {k}}}$ ${\displaystyle {\vec {D}}}$ ${\displaystyle {\vec {H}}}$ ${\displaystyle {\vec {k}}}$ ${\displaystyle {\vec {E}}}$

Use is now made of the fact that it is perpendicular to the direction of propagation . It is ${\displaystyle {\vec {D}}}$ ${\displaystyle {\vec {k}}}$

${\displaystyle {\vec {E}}=(\varepsilon )^{-1}\cdot {\vec {D}}}$,

where is the inverse tensor. By choosing a new coordinate system with the coordinates a, b, c, which is chosen so that the c-direction is parallel to , the system of equations can be reduced from three to two equations: ${\displaystyle (\varepsilon )^{-1}}$ ${\displaystyle (\varepsilon )}$ ${\displaystyle {\vec {k}}}$

${\displaystyle D_{a}=n^{2}\varepsilon _{aa}^{-1}D_{a}+n^{2}\varepsilon _{ab}^{-1}D_{b}}$

${\displaystyle D_{b}=n^{2}\varepsilon _{ba}^{-1}D_{a}+n^{2}\varepsilon _{bb}^{-1}D_{b}}$

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

literature

• Werner Döring: Introduction to Theoretical Physics, Volume III (Optics). Göschen Collection, Berlin 1957.

Web links

• H.-G. Stosch: Script for Crystal Optics II - Mineral Microscopy. (PDF 21789 kB) In: agw.kit.edu. University of Freiburg, October 14, 2009, accessed on September 23, 2019 . 
• Interference figures on birefringent, optically active crystal plates