In electrostatics, the polarization is the vector field that results from permanent or induced electric dipole moments in a dielectric material. The polarization vector P is defined as the dipole moment per unit volume. Electrostatics is the branch of physics that deals with the force exerted by a static (i. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... A dipole (Greek: dyo = two and polos = pivot) is a pair of electric charges or magnetic poles of equal magnitude but opposite polarity (opposite electronic charges), separated by some (usually small) distance. ... The electrons in the molecules shift toward the positively charged left plate. ...

In a homogeneous linear and isotropic dielectric medium, the polarization is aligned with and proportional to the electric field E. In an anisotropic material, the polarization and the field are not necessarily in the same direction. Then, the i^th component of the polarization is related to the j^th component of the electric field according to: Homogeneous is an adjective that has several meanings. ... Isotropic means independent of direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. ... The electrons in the molecules shift toward the positively charged left plate. ... The word proportionality may have one of a number of meanings: In mathematics, proportionality is a mathematical relation between two quantities. ...

Pi =	∑	ε0χijEj,
	j

where ε₀ is the permittivity of free space, and χ is the electric susceptibility tensor of the medium. The permittivity of a medium is an intensive physical quantity that describes how an electric field affects and is affected by the medium. ... The electric susceptibility χe is essentially the constant of proportionality (which may be a tensor) relating an applied electric field E to the induced dielectric polarisation P such that: Where ε0 is the electric permittivity of free space. ...

If the polarization P is not proportional to the electric field E, the medium is termed nonlinear and is described by the field of nonlinear optics. If the direction of P is not aligned with E, as in many crystals, the medium is anisotropic and is described by crystal optics. Nonlinear optics is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization P responds nonlinearly to the electric field E of the light. ... Quartz crystal A crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. ... Crystal 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. ...