Hyperpolarizability

The linear electric polarizability ${\displaystyle \alpha }$ in isotropic media is defined as the ratio of the induced dipole moment ${\displaystyle \mathbf {p} }$ of an atom to the electric field ${\displaystyle \mathbf {E} }$ that produces this dipole moment.[5]

Therefore the dipole moment is

${\displaystyle \mathbf {p} =\alpha \mathbf {E} }$

In an isotropic medium ${\displaystyle \mathbf {p} }$ is in the same direction as ${\displaystyle \mathbf {E} }$, i.e. ${\displaystyle \alpha }$ is a scalar. In an anisotropic medium ${\displaystyle \mathbf {p} }$ and ${\displaystyle \mathbf {E} }$ 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.

${\displaystyle \mathbf {P} =\rho \mathbf {p} =\rho \alpha \mathbf {E} =\varepsilon _{0}\chi \mathbf {E} ,}$

where ${\displaystyle \rho }$ is the concentration, ${\displaystyle \varepsilon _{0}}$ is the Vacuum permittivity, and ${\displaystyle \chi }$ 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.

${\displaystyle \mathbf {P} (t)=\varepsilon _{0}\left(\chi ^{(1)}\mathbf {E} (t)+\chi ^{(2)}\mathbf {E} ^{2}(t)+\chi ^{(3)}\mathbf {E} ^{3}(t)+\ldots \right),}$

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 ${\displaystyle \chi ^{(n)}}$ 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

${\displaystyle \mathbf {p} (t)=\alpha ^{(1)}\mathbf {E} (t)+\alpha ^{(2)}\mathbf {E} ^{2}(t)+\alpha ^{(3)}\mathbf {E} ^{3}(t)+\ldots ,}$

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

${\displaystyle \alpha ^{(n)}={\frac {\varepsilon _{0}}{\rho }}\chi ^{(n)}.}$

With this definition ${\displaystyle \alpha ^{(1)}}$ is equal to ${\displaystyle \alpha }$ defined above for the linear polarizability. Often ${\displaystyle \alpha ^{(2)}}$ is given the symbol ${\displaystyle \beta }$ and ${\displaystyle \alpha ^{(3)}}$ is given the symbol ${\displaystyle \gamma }$. However, care is needed because some authors[6] take out the factor ${\displaystyle \varepsilon _{0}}$ from ${\displaystyle \alpha ^{(n)}}$, so that ${\displaystyle \mathbf {p} =\varepsilon _{0}\sum _{n}\alpha ^{(n)}\mathbf {E} ^{n}}$ and hence ${\displaystyle \alpha ^{(n)}=\chi ^{(n)}/\rho }$, 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.

• Intrinsic hyperpolarizability

• The Nonlinear Optics Web Site