Macroscopic electric field power series

Further subclassification of nonlinear optical materials can be explained by the foUowiag two equations of microscopic, ie, atomic or molecular, polarization,, and macroscopic polarization, P, as power series ia the appHed electric field, E (disregarding quadmpolar terms which are unimportant for device appHcations) ... 

The fundamental equation (1) describes the change in dipole moment between the ground state and an excited state jte expressed as a power series of the electric field E which occurs upon interaction of such a field, as in the electric component of electromagnetic radiation, with a single molecule. The coefficient a is the familiar linear polarizability, ft and y are the quadratic and cubic hyperpolarizabilities, respectively. The coefficients for these hyperpolarizabilities are tensor quantities and therefore highly symmetry dependent odd order coefficients are nonvanishing for all molecules but even order coefficients such as J3 (responsible for SHG) are zero for centrosymmetric molecules. Equation (2) is identical with (1) except that it describes a macroscopic polarization, such as that arising from an array of molecules in a crystal (10). 

Before proceeding, it is necessary to introduce the nonlinear polarizabilities and susceptibilities that we will be dealing with. The polarization of a molecule or a macroscopic material subject to an applied electric field is expanded as a power series in the applied field. The molecular polarization is given by ... 

In this review, the focus is on electro-optic materials. Such materials are members of the more general class of second-order nonlinear optical materials, which also includes materials used for second harmonic generation (frequency doubling). The term second-order derives from the fact that the magnitude of these effects is defined by the second term of the power series expansion of optical polarization as a function of applied electric fields. The power series expansion of polarization with electric field can be expressed either in terms of molecular polarization (p Eq. 1) or macroscopic polarization (P Eq. 2)... 

The power-series expansion of p(f) can be used with Eq. (10.14) to find the expectation value of the macroscopic electric dipole for an ensemble of systems exposed to an electromagnetic field ... 

A(u = complex pulse area D = number of distinguishable permutations of the pairs (o, a) (er) = expectation value of the molecular dipole operator E = energy of level i E t) = electric field vector E t) = component along ju of the electric field vector E cd) = amplitude of the Fourier component of E at frequency (o E = amplitude of the monochromatic wave at frequency a> E = component along ju of E E t) = envelope of the quasi-monochromatic field at central frequency a> H(t) = time-dependent total Hamiltonian Hq = unperturbed Hamiltonian in the absence of electromagnetic fields = intensity of the monochromatic wave at frequency w N = number density of molecules = refractive index at frequency macroscopic polarization vector P< ) = w-th order term in the series expansion of P in powers of... 

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