Macroscopic polarization, equation

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). 

An expression similar to equation 8 can be written for the macroscopic polarization of a medium or ensemble of molecules as... 

The macroscopic polarization of the phase is given by equations 1 and 2, where Di is the number density of the ith conformation, jlj is the component of the molecular dipole normal to the tilt plane when the ith conformation of the molecule is oriented in the rotational minimum in the binding site, ROFj is the "rotational orientation factor", a number from zero to one reflecting the degree of rotational order for the ith conformation, and e is a complex and unmeasured dielectric constant of the medium (local field correction). 

Equations (1.3) and (1.4) describe the mean properties of the dielectric. This macroscopic point of view does not consider the microscopic origin of the polarization [3], The macroscopic polarization P is the sum of all the individual dipole moments pj of the material with the density Nj. 

In the above derivation of the direct fifth-order response, the five field-matter interactions used to create the macroscopic induced polarization, [Equation (14)] were restricted to the five external fields. In other words, internal fields generated by lower-order induced polarizations were completely ignored in the field-matter interactions. This approximation is very good for the lowest-order NLO processes, such as four-wave mixing spectroscopies. However, the internal field produced by the lower-order induced nonlinear polarization can be of crucial importance in higher-order NLO processes such as the fifth-order processes considered here. 

We consider now the NLO response of a molecule to an electric field. The resulting equations will be found to be analogous to the ones derived for a bulk medium. Instead of bulk susceptibilities however, molecular polarizabilities of nth order appear. For the latter, by convention, the lower-case Greek letters in ascending order (a, 3, y,. ..) are used. Again, an electric field of the form defined in (16) is used. Similar to the macroscopic polarization (17), the expression (33) for the molecular dipole moment p t) contains linear and non-linear terms. 

These definitions for P, Pi, and aN may now be substituted into Equation 31 to give Equation 22. The roles of the phenomenological relaxation times Tj and T2 are now clear. The macroscopic polarization P, and hence the components P - and Pi, relax to their equilibrium values of zero with a relaxation time T2. The population difference N relaxes to its equilibrium value aNq with a relaxation time Tj. 

Here /Tq is the intrinsic dipole moment, o is the linear polarizability, /3o the hyperpolarizability, yo the secmid hyperpolarizability, etc. Equation (41.10) is a molecular level analog of Eq. (41.9) for the macroscopic polarization. Thus, we are for instance interested in the second harmonic... 

The province of conventional dielectric measurements is here taken to be the determination of the relations of the polarization E and current density J. to the electric field in the macroscopic Maxwell equations. Proper theory should account for these relations in condensed phases as a function of state variables time dependence of applied fields and molecular parameters by appropriate statistical averaging over molecular displacements determined by the equations of motion in terms of molecular forces and fields. Simplifying assumptions and approximations are of course necessary. One kind often made and debated is use of an effective or mean local field at a molecule rather than the sum of microscopic... 

With these developments the macroscopic Maxwell equations relate to fields to macroscopic charge current and moment densities with further statistical averaging over appropriate ensembles giving expectation values. Before going on to these operations two points which sometimes cause difficulty should be mentioned. The first is that only total current density s 1 dP/dt is related to fields and hence that only can be determined by purely electromagnetic measurements but not conduction current 2 if significant separately from polarization current This is evident in equation (4). It may not seem so for f = 4ir (p—V f) but combining this with = 0 for... 

Before going on to discuss recent developments in theory of static or equilibrium permittivity for specific models in the next section mention should be made of different kinds of formalism developed by Fulton (18) and by Felderhof and Titulacr (19) The reader may have noticed that in the theories described so far the macroscopic polarization is evaluated as a statistical mechanical average consistent with the basic definition for the Maxwell equations discussed inl.lp while the macroscopic and thence Cg is not so computed but is instead introduced by electrostatic or cavity arguments. Both Fulton and Felderhof-Tltulaer have dispensed with cavities in their treatments relating the permittivity to polarization fluctuations (as expressed by <(Zi ) > for... 

In hie most recent papers (20) Fulton has put his case in the statement e take the view that if nature does not endow herself with cavities we should not have to introduce them. . He avoids such introduction by taking an external charge and current density as sources of electromagnetic field in the medium and by using methods of quantum electrodynamics obtains solutions of microscopic and macroscopic field equations for the polarizations and fields with susceptibility and permittivity obtained as functional derivatives of polarization with respect to source field and macroscopic . 

Macroscopic polarization, 52 Macroscopic polarization field, 53, 54 Macroscopic susceptibility, 33, 35, 53-55 Magnetic dipole moment, 14 Maxwell electric field, 53, 54 Maxwell equation, 53 Maxwell field, 27, 33, 34 Mixed electric magnetic hyper-magnetizabili-ty, 28... 

In order to provide a more concrete image, some equations arc developed here [54]. The three-dimensioaal displacement vector of an atom b denoted by Ut(r). The macroscopic polarization and electric field are denoted by P(r) and E(r), respectively. Under the harmonic, idisbatic. and electnKtatic approximations, these quantities are coupled together in the following manner. 

The microscopic origin of the nonlinear response is the distortion induced in the molecular charge distribution due to the electrical field. The presence of a microscopic dipole produces a macroscopic polarization in the unit volume P = N r) where N is the number density of polarizable units and (er) the expectation value of the dipole moment induced in each unit. In order to evaluate (sr) we will use the density matrix formalism, because it is the easiest way to relate microscopic properties to macroscopic ones and to cope with macroscopic coherence effects. In the absence of fields, the medium is supposed to be described by an unperturbed Hamiltonian Hq and to be at equilibrium. When the fields are applied, the field-matter interaction contributes a time-dependent term V(t) =-E(t)P(t) to the global energy. The evolution of the system under this perturbation can be described through the equation of motion of the density operator ... 

Nondiagonal terms in Equation [14] produce a macroscopic polarization Pjj(t) departs from zero at equilibrium and includes an oscillatory term at the frequency co y. The existence of a nondiagonal term Pij(t) 0 is linked to the creation of a coherent superposition of states i and / due to the perturbation. This coherence is broken through the interaction with the molecules of the bath, with a rate constant of the order of F y= I/T2. Usually this dephasing is much faster than the energy relaxation and T2 Ti. 

In the case where the transition layer is represented by adsorbed atoms or molecules, or by a perturbed surface layer, its thickness d is of the order of interatomic distances. Such a situation is beyond the applicability limit of the macroscopic Maxwell equations and therefore the above consideration is no longer valid. Instead, one has to take into account the microscopic structure of the transition layer. The first approach to this problem is to consider the polarization of the atomic dipoles, both in the medium and in the surface transition layer, averaged over an infinitesimal volume, as a source of radiation resulting in reflected light. Such an average is carried out to evaluate the radiation field far from the surface and therefore is reasonable for a microscopic layer. 

Equation 6.86 provides us with an expression for the macroscopic polarization and the following remarks are worthy of note ... 

In the previous sections we have described the interaction of the electromagnetic field with matter, that is, tlie way the material is affected by the presence of the field. But there is a second, reciprocal perspective the excitation of the material by the electromagnetic field generates a dipole (polarization) where none existed previously. Over a sample of finite size this dipole is macroscopic, and serves as a new source tenu in Maxwell s equations. For weak fields, the source tenu, P, is linear in the field strength. Thus,... 

The continuum electrostatic approximation is based on the assumption that the solvent polarization density of the solvent at a position r in space is linearly related to the total local electric field at that position. The Poisson equation for macroscopic continuum media... 

In reference to Figure 5 for MNA crystals, the polar axes of the individual molecular sites are aligned with one another along the crystal polar axis. The microscopic co mo nents gx add resulting in the large macroscopic x i i i following equation 7. 

In this subsection, the connection is made between the molecular polarizability, a, and the macroscopic dielectric constant, e, or refractive index, n. This relationship, referred to as the Lorentz-Lorenz equation, is derived by considering the immersion of a dielectric material within an electric field, and calculating the resulting polarization from both a macroscopic and molecular point of view. Figure 7.1 shows the two equivalent problems that are analyzed. 

In the study of the KE for a selection of pure liquids [30] the concept of effective polarizabilities was extended to introduce the contribution of the output wave. Radiation at a frequency to induces a macroscopic nonlinear polarization density (pm)NL at the same frequency, the output wave, generating an additional perturbing field. The molecules of the liquid respond with an additional effective polarizability a(—to to), whereby Equation (2.242) becomes... 

Increase of the solvent dielectric constant caused by the compression of the solvent layer, if the concept of dielectric constant has still a meaning with such thin layers. The Microscopic Maxwell equations formalism would be more appropriate. ( Macroscopic Maxwell s equations are applied to macroscopic averages of the fields, which vary wildly on a microscopic scale closed to individual atoms. It is only in this averaged sense that one can define quantities such as the permittivity, and permeability of a material, as well as the polarization and induction field). 

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