Crystal opticsCrystal optics is the branch of optics that describes the behaviour of light in anisotropic media, that is, media (such as crystals) in which light behaves differently depending on which direction the light is propagating. Crystals are often naturally anisotropic, and in some media (such as liquid crystals it is possible to induce anisotropy by applying e.g. an external electric field.
Typical transparent media such as glasses are isotropic, which means that light behaves the same way no matter which direction it is travelling in the medium. In terms of Maxwell's equations in a dielectric, this gives a relationship between the electric displacement field D and the electric field E:
- D = ε0E + P ,
- P = ε0χE ,
- D = ε0E + ε0χE = ε0(1+χ)E .
- n = (1+χ)1/2 = (εr)1/2 .
- P = ε0 χ×E .
or using the summation convention:
- Pi = ε0 χij Ej .
From thermodynamics arguments it can be shown that χij = χji, i.e. the χ tensor is symmetric. In accordance with the spectral theorem, it is thus possible to diagonalise the tensor by choosing the appropriate set of cooridinate axes, zeroing all components of the tensor except χxx, χyy and χzz. This gives the set of relations:
- Px = ε0 χxx Ex
- Py = ε0 χyy Ey
- Pz = ε0 χzz Ez
It follows that D and E are also related by a tensor:
- D = ε0E + P = ε0E + ε0 χ×E = ε0 (1+χ)×E = ε0 ε×E .
- nxx = (1 + χxx)1/2 = (εxx)1/2 .
- nyy = (1 + χyy)1/2 = (εyy)1/2 .
If χxx = χyy ≠ χzz, the crystal is known as uniaxial. If χxx ≠ χyy and χxx ≠ χzz the crystal is called biaxial. A uniaxial crystal exhibits two refractive indicies, an "ordinary" index (no) for light polarised in the x or y directions, and an "extraordinary" index (ne) for polarisation in the z direction. Light polarised 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.
Certain nonlinear optical phenomena such as the electro-optic effect cause a variation of a medium's permittivity tensor when an external electric field is applied, proportional (to lowest order) to the strength of the field. This causes a rotation of the principal axes of the medium and alters the behaviour of light travelling through it; the effect can be used to produce light modulators.
In response to a magnetic field, some materials can have a dielectric tensor that is complex-Hermitian; this is called a gyro-magnetic or magneto-optic effect. In this case, the principle axes are complex-valued vectors, corresponding to elliptically polarized light, and time-reversal symmetry can be broken. This can be used to design optical isolators, for example.
(A dielectric tensor that is not Hermitian gives rise to complex eigenvalues, which corresponds to a material with gain or absorption at a particular frequency.)