Crystal optics
the crystal optics is usually occupied with the reciprocal effect of electromagnetic radiation, in the visible wavelength coverage, with crystalline or otherwise anisotropic solids, but generalizing also with optically active liquids. It is a subsection of the optics, solid-state physics and the mineralogy.
general
the optical characteristics of the crystals, which are responsible among other things for reflection , refraction and absorption of the light, are by their regular internal structure certain. Differently than with the optically isotropic Gläsern one usually finds the phenomenon with crystalsthe anisotropy: Important characteristics as for example the refractive index depend on the direction of propagation of the light in the crystal and its polarization.
More exactly said this applies to all crystals, which do not exhibit the cubic crystal system. For illustration one carries in a three-dimensional diagram for each possible direction of propagationof light in the crystal the value refractive index in this direction. Thus always an ellipsoid with usually three results unequally is enough for centerlines standing perpendicularly one on the other, which one calls also Indikatrix.
- If the crystal is cubic, the ellipsoid reduces to thatSpecial case of a ball, since all three centerlines have the same length. The light propagation is isotropic in this case.
- In case of the hexagonal, trigonalen and tetragonal crystal system only two of the centerlines are of equal length, one speak then of optically crystals with one axle or university-axial. In the designationaddressed axle stands perpendicularly on the two centerlines of equal length. At beam of light parallel to this axle no birefringence takes place .
- Three centerlines of different lengths are for the orthorhombische, monoclinic and triclinic crystal system, the crystal are called now optically fourwheel or biaxial. These two axles do not fallwith centerlines of the ellipsoid together, them it is defined rather clearly by the fact that they stand perpendicularly on the only two circles, which can be produced by cut of one level by the center of the ellipsoid with the Indikatrix (all other cuts result in ellipses and no circles). ThatRadius of these circles corresponds the three centerlines middle from the length.
An important consequence of the anisotropy of crystals is the birefringence, i.e. the fragmentation of light hitting the crystal into a tidy and an extraordinary jet, the one different polarizationexhibit.
Also the optical activity of crystals can be attributed to its anisotropy: The polarization plane linear polarized light turned at an angle proportional to the distance put back in the crystal. One differentiates between depending on whether the level the hand or counterclockwise is turned, if oneexactly against the direction of propagation of the light looks, right and anti-clockwise rotating crystals, which are called also optical modifications. As examples link quartz and Rechtsquartz are mentioned.
A third specifically optical feature applicable on crystals is the Pleochroismus in such a way specified. That means that light depending uponPropagation and polarization direction are differently strongly absorbed. Since the absorption depends additionally still on the wavelength, the Pleochronismus shows up in a direction-controlled color change of the through-radiated light, which can be recognized already with the naked eye in extreme cases.
The optical characteristics of a crystal let themselves throughexpresses electrical and magnetic fields, in addition, by mechanical load, in first case speaks one affects of the electrooptical effect, in the second case of the magentooptischen effect. Turned around they can be consulted for the diagnosis of these external influences.
mathematical formalism
basis of themathematical formalism is the fact that the electrical field strength E and the electrical shift density D are no longer directly arranged. With it the dielectric function can <math> \ varepsilon< /math>, which links the two formula sizes, no more than scalar be understood, but must than tensor of second stage be treated.The relationship between D and E is written now:
<math> \ begin {pmatrix} D_x \ \ D_y \ \ D_z \ end {pmatrix} = \ to 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} to E_x \ \ E_y \ \ E_z \ end {pmatrix},< /math>
whereby <math> \ epsilon_0< /math> the dielectric constant of the vacuum represents.
How an electromagnetic wave in the anisotropic medium spreads, can be computed by release of the wave equation for anisotropic bodies:
- <math> \ varepsilon \ \ vec {E cdot} = n^2 (\ vec {E} - \ vec {k} (\ vec {E} \ cdot \ vec {k}))</math>.
Here k represents a unit vector, to that in direction of propagation thatWave shows, n is the refractive index.
The wave equation a system from three coupled equations is algebraic, from which the two refractive indices for the two different polarization directions can be derived. The set of equations is not however generally clear regarding the polarization direction. Therefore becomesProcedure uses, in order to reduce the three equations to two. First one designs a system from three vectors standing perpendicularly in pairs one on the other. Two of it are the direction of propagation k and shift density D, third are the magnetic field strength H. There k no longer howin the isotropic solid body in the 90-Grad-Winkel to E to be located, is not suitable the wave equation must, in order to determine the polarization character of the waves.
Now it is used that D stands perpendicularly on the direction of propagation k. It is
- <math> \ vec {E} = \ varepsilon^ {- 1} \ cdot \ vec {D}< /math>,
whereby <math> \ epsilon^ {- 1}< /math> too <math> \ epsilon< /math> inverse tensor is.By choice of a new coordinate system with the coordinates A, b, C, which is so selected that the C-direction lies parallel to k, can one the set of equations of three to two equations reduce:
- <math> D_a = n^2 \ varepsilon_ {aa} ^ {- 1} D_a + n^2 \ varepsilon_ {off} ^ {- 1} D_b< /math>
- <math> D_b = n^2 \ varepsilon_ {ba} ^ {- 1}D_a + n^2 \ varepsilon_ {bb} ^ {- 1} D_b< /math>
By release of this set of equations one receives the two refraction indices and the polarization character for any direction.
