The linear electric
polarizability \alpha in
isotropic media is defined as the ratio of the induced
dipole moment \mathbf{p} of an atom to the electric field \mathbf{E} that produces this dipole moment. Therefore, the dipole moment is: \mathbf{p}=\alpha \mathbf{E} In an isotropic medium \mathbf{p} is in the same direction as \mathbf{E}, i.e. \alpha is a scalar. In an anisotropic medium \mathbf{p} and \mathbf{E} 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.: \mathbf{P} = \rho \mathbf{p} = \rho \alpha \mathbf{E} = \varepsilon_0 \chi \mathbf{E}, where \rho is the concentration, \varepsilon_0 is the
vacuum permittivity, and \chi is the
electric susceptibility. 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: \mathbf{P}(t) = \varepsilon_0 \left( \chi^{(1)} \mathbf{E}(t) + \chi^{(2)} \mathbf{E}^2(t) + \chi^{(3)} \mathbf{E}^3(t) + \ldots \right), 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 \chi^{(n)} 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: \mathbf{p}(t) = \alpha^{(1)} \mathbf{E}(t) + \alpha^{(2)} \mathbf{E}^2(t) + \alpha^{(3)} \mathbf{E}^3(t) + \cdots , 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: \alpha^{(n)} = \frac{\varepsilon_0}{\rho} \chi^{(n)} . With this definition \alpha^{(1)} is equal to \alpha defined above for the linear polarizability. Often \alpha^{(2)} is given the symbol \beta and \alpha^{(3)} is given the symbol \gamma. However, care is needed because some authors take out the factor \varepsilon_0 from \alpha^{(n)}, so that \mathbf{p} = \varepsilon_0 \sum_n \alpha^{(n)} \mathbf{E}^n and hence \alpha^{(n)}=\chi^{(n)}/\rho, 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. ==See also==