The hyperpolarizability, a nonlinear-optical property of a molecule, is the second order electric susceptibility per unit volume.
[1] The hyperpolarizability can be calculated using quantum chemical calculations developed in several software packages.
[2][3][4] See nonlinear optics.
The linear electric polarizability
in isotropic media is defined as the ratio of the induced dipole moment
of an atom to the electric field
that produces this dipole moment.
[5] Therefore, the dipole moment is: In an isotropic medium
In an anisotropic medium
can be in different directions and the polarisability is now a tensor.
The total density of induced polarization is the product of the number density of molecules multiplied by the dipole moment of each molecule, i.e.: where
In a nonlinear optical medium, the polarization density is written as a series expansion in powers of the applied electric field, and the coefficients are termed the non-linear susceptibility: where the coefficients χ(n) are the n-th-order susceptibilities of the medium, and the presence of such a term is generally referred to as an n-th-order nonlinearity.
In isotropic media
is zero for even n, and is a scalar for odd n. In general, χ(n) is an (n + 1)-th-rank tensor.
It is natural to perform the same expansion for the non-linear molecular dipole moment: i.e. the n-th-order susceptibility for an ensemble of molecules is simply related to the n-th-order hyperpolarizability for a single molecule by: With this definition
defined above for the linear polarizability.
However, care is needed because some authors[6] take out the factor
, which is convenient because then the (hyper-)polarizability may be accurately called the (nonlinear-)susceptibility per molecule, but at the same time inconvenient because of the inconsistency with the usual linear polarisability definition above.
This optics-related article is a stub.