hyperpolarizability

The linear electric polarizability $\backslash alpha$ in isotropic media is defined as the ratio of the induced dipole moment $\backslash mathbf\{p\}$ of an atom to the electric field $\backslash mathbf\{E\}$ that produces this dipole moment.Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, {{ISBN|81-7758-293-3}}

Therefore, the dipole moment is:

:$\backslash mathbf\{p\}=\backslash alpha\; \backslash mathbf\{E\}$

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

:$\backslash mathbf\{P\}\; =\; \backslash rho\; \backslash mathbf\{p\}\; =\; \backslash rho\; \backslash alpha\; \backslash mathbf\{E\}\; =\; \backslash varepsilon\_0\; \backslash chi\; \backslash mathbf\{E\},$

where $\backslash rho$ is the concentration, $\backslash varepsilon\_0$ is the vacuum permittivity, and $\backslash chi$ 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:

:$\backslash mathbf\{P\}(t)\; =\; \backslash varepsilon\_0\; \backslash left(\; \backslash chi^\{(1)\}\; \backslash mathbf\{E\}(t)\; +\; \backslash chi^\{(2)\}\; \backslash mathbf\{E\}^2(t)\; +\; \backslash chi^\{(3)\}\; \backslash mathbf\{E\}^3(t)\; +\; \backslash ldots\; \backslash right),$

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 $\backslash chi^\{(n)\}$ 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:

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

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:

:$\backslash alpha^\{(n)\}=\backslash frac\{\backslash varepsilon\_0\}\{\backslash rho\}\; \backslash chi^\{(n)\}\; .$

With this definition $\backslash alpha^\{(1)\}$ is equal to $\backslash alpha$ defined above for the linear polarizability. Often $\backslash alpha^\{(2)\}$ is given the symbol $\backslash beta$ and $\backslash alpha^\{(3)\}$ is given the symbol $\backslash gamma$. However, care is needed because some authors{{cite book |last1=Boyd |first1=Robert |title=Nonlinear Optics |publisher=Elsevier |isbn=978-81-312-2292-8 |edition=3rd}} take out the factor $\backslash varepsilon\_0$ from $\backslash alpha^\{(n)\}$, so that $\backslash mathbf\{p\}=\backslash varepsilon\_0\backslash sum\_n\backslash alpha^\{(n)\}\; \backslash mathbf\{E\}^n$ and hence $\backslash alpha^\{(n)\}=\backslash chi^\{(n)\}/\backslash rho$, 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

{{Reflist}}

• [http://nlosource.com/Polarizability.html The Nonlinear Optics Web Site]

Category:Nonlinear optics