Crystal optics
Encyclopedia
Crystal optics is the branch of optics
that describes the behaviour of light
in anisotropic media, that is, media (such as crystal
s) in which light behaves differently depending on which direction the light is propagating. The index of refraction depends on both composition and crystal structure and can be calculated using the Gladstone-Dale relation
. Crystals are often naturally anisotropic, and in some media (such as liquid crystal
s) it is possible to induce anisotropy by applying an external electric field.
es 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:
where ε0 is the permittivity
of free space and P is the electric polarization (the vector field
corresponding to electric dipole moments
present in the medium). Physically, the polarization field can be regarded as the response of the medium to the electric field of the light.
medium, this polarisation field P is proportional to and parallel to the electric field E:
where χ is the electric susceptibility
of the medium. The relation between D and E is thus:
where
is the dielectric constant
of the medium. The value 1+χ is called the relative permittivity of the medium, and is related to the refractive index
n, for non-magnetic media, by
Here χ is not a number as before but a tensor
of rank 2, the electric susceptibility tensor. In terms of components in 3 dimensions:
or using the summation convention:
Since χ is a tensor, P is not necessarily colinear with E.
From thermodynamic
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 coordinate axes, zeroing all components of the tensor except χxx, χyy and χzz. This gives the set of relations:
The directions x, y and z are in this case known as the principal axes of the medium. Note that these axes will be orthogonal if all entries in the χ tensor are real, corresponding to a case in which the refractive index is real in all directions.
It follows that D and E are also related by a tensor:
Here ε is known as the relative permittivity tensor or dielectric tensor. Consequently, the refractive index
of the medium must also be a tensor. Consider a light wave propagating along the z principal axis polarised such the electric field of the wave is parallel to the x-axis. The wave experiences a susceptibility χxx and a permittivity εxx. The refractive index is thus:
For a wave polarised in the y direction:
Thus these waves will see two different refractive indices and travel at different speeds. This phenomenon is known as birefringence
and occurs in some common crystals such as calcite
and quartz
.
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 indices, an "ordinary" index (no) for light polarised in the x or y directions, and an "extraordinary" index (ne) for polarisation in the z direction. A uniaxial crystal is "positive" if ne > no and "negative" if ne < no. 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
.
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 principal axes are complex-valued vectors, corresponding to elliptically polarized light, and time-reversal symmetry can be broken. This can be used to design optical isolator
s, 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.)
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
that describes the behaviour of light
Light
Light or visible light is electromagnetic radiation that is visible to the human eye, and is responsible for the sense of sight. Visible light has wavelength in a range from about 380 nanometres to about 740 nm, with a frequency range of about 405 THz to 790 THz...
in anisotropic media, that is, media (such as crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...
s) in which light behaves differently depending on which direction the light is propagating. The index of refraction depends on both composition and crystal structure and can be calculated using the Gladstone-Dale relation
Gladstone-Dale relation
The Gladstone–Dale relation is used for optical analysis , or to calculate the density of a liquid for use in fluid dynamics .The relation has also been used to calculate refractive index of glass and minerals in optical mineralogy The Gladstone–Dale relation (J. H. Gladstone and T. P. Dale, 1864)...
. Crystals are often naturally anisotropic, and in some media (such as liquid crystal
Liquid crystal
Liquid crystals are a state of matter that have properties between those of a conventional liquid and those of a solid crystal. For instance, an LC may flow like a liquid, but its molecules may be oriented in a crystal-like way. There are many different types of LC phases, which can be...
s) it is possible to induce anisotropy by applying an external electric field.
Isotropic media
Typical transparent media such as glassGlass
Glass is an amorphous solid material. Glasses are typically brittle and optically transparent.The most familiar type of glass, used for centuries in windows and drinking vessels, is soda-lime glass, composed of about 75% silica plus Na2O, CaO, and several minor additives...
es 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
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
in a dielectric
Dielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...
, this gives a relationship between the electric displacement field D and the electric field
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
E:
where ε0 is the permittivity
Permittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
of free space and P is the electric polarization (the vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
corresponding to electric dipole moments
Electric dipole moment
In physics, the electric dipole moment is a measure of the separation of positive and negative electrical charges in a system of charges, that is, a measure of the charge system's overall polarity with SI units of Coulomb-meter...
present in the medium). Physically, the polarization field can be regarded as the response of the medium to the electric field of the light.
Electric susceptibility
In an isotropic and linearLinear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
medium, this polarisation field P is proportional to and parallel to the electric field E:
where χ is the electric susceptibility
Electric susceptibility
In electromagnetism, the electric susceptibility \chi_e is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applied electric field...
of the medium. The relation between D and E is thus:
where
is the dielectric constant
Dielectric constant
The relative permittivity of a material under given conditions reflects the extent to which it concentrates electrostatic lines of flux. In technical terms, it is the ratio of the amount of electrical energy stored in a material by an applied voltage, relative to that stored in a vacuum...
of the medium. The value 1+χ is called the relative permittivity of the medium, and is related to the refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
n, for non-magnetic media, by
Anisotropic media
In an anisotropic medium, such as a crystal, the polarisation field P is not necessarily aligned with the electric field of the light E. In a physical picture, this can be thought of as the dipoles induced in the medium by the electric field having certain preferred directions, related to the physical structure of the crystal. This can be written as:Here χ is not a number as before but a tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
of rank 2, the electric susceptibility tensor. In terms of components in 3 dimensions:
or using the summation convention:
Since χ is a tensor, P is not necessarily colinear with E.
From thermodynamic
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
arguments it can be shown that χij = χji, i.e. the χ tensor is symmetric. In accordance with the spectral theorem
Spectral theorem
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
, it is thus possible to diagonalise
Diagonalization
In mathematics, diagonalization may refer to:* Diagonal matrix, which is in a form with nonzero entries only on the main diagonal* Diagonalizable matrix, which can be put into a form with nonzero entries only on the main diagonal...
the tensor by choosing the appropriate set of coordinate axes, zeroing all components of the tensor except χxx, χyy and χzz. This gives the set of relations:
The directions x, y and z are in this case known as the principal axes of the medium. Note that these axes will be orthogonal if all entries in the χ tensor are real, corresponding to a case in which the refractive index is real in all directions.
It follows that D and E are also related by a tensor:
Here ε is known as the relative permittivity tensor or dielectric tensor. Consequently, the refractive index
Refractive index
In optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....
of the medium must also be a tensor. Consider a light wave propagating along the z principal axis polarised such the electric field of the wave is parallel to the x-axis. The wave experiences a susceptibility χxx and a permittivity εxx. The refractive index is thus:
For a wave polarised in the y direction:
Thus these waves will see two different refractive indices and travel at different speeds. This phenomenon is known as birefringence
Birefringence
Birefringence, or double refraction, is the decomposition of a ray of light into two rays when it passes through certain anisotropic materials, such as crystals of calcite or boron nitride. The effect was first described by the Danish scientist Rasmus Bartholin in 1669, who saw it in calcite...
and occurs in some common crystals such as calcite
Calcite
Calcite is a carbonate mineral and the most stable polymorph of calcium carbonate . The other polymorphs are the minerals aragonite and vaterite. Aragonite will change to calcite at 380-470°C, and vaterite is even less stable.-Properties:...
and quartz
Quartz
Quartz is the second-most-abundant mineral in the Earth's continental crust, after feldspar. It is made up of a continuous framework of SiO4 silicon–oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall formula SiO2. There are many different varieties of quartz,...
.
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 indices, an "ordinary" index (no) for light polarised in the x or y directions, and an "extraordinary" index (ne) for polarisation in the z direction. A uniaxial crystal is "positive" if ne > no and "negative" if ne < no. 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
Index ellipsoid
In optics, an index ellipsoid is a diagram of an ellipsoid that depicts the orientation and relative magnitude of refractive indices in a crystal....
.
Other effects
Certain nonlinear opticalNonlinear optics
Nonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...
phenomena such as the electro-optic effect
Electro-optic effect
An electro-optic effect is a change in the optical properties of a material in response to an electric field that varies slowly compared with the frequency of light...
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
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
, some materials can have a dielectric tensor that is complex-Hermitian
Hermitian
A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite:*Hermitian adjoint*Hermitian connection, the unique connection on a Hermitian manifold that satisfies specific conditions...
; this is called a gyro-magnetic or magneto-optic effect
Magneto-optic effect
A magneto-optic effect is any one of a number of phenomena in which an electromagnetic wave propagates through a medium that has been altered by the presence of a quasistatic magnetic field...
. In this case, the principal axes are complex-valued vectors, corresponding to elliptically polarized light, and time-reversal symmetry can be broken. This can be used to design optical isolator
Optical isolator
An optical isolator, or optical diode, is an optical component which allows the transmission of light in only one direction. It is typically used to prevent unwanted feedback into an optical oscillator, such as a laser cavity...
s, 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.)