Third-order permittivity

The symbol A has the same meaning as before and stands for either constant finite strain V or stress t. It describes the field dependence of the conventional linear permittivity. Usually, the factor sq is extracted, and the third-order permittivity written as sqkkim but this would involve yet additional symbols, also for the impermittivities, and really makes only limited sense as it fails to render kklm dimensionless. Similar considerations also hold for the notation of all sorts of electrostriction coefficients. 

The third-order permittivity is described by the third rank tensor Its symmetry properties are similar to those of the piezoelectric tensors. In particular all tensor coordinates vanish for all centro-symmetric crystal classes and the class 432. However, all three indices are interchangeable. As a consequence of this either the first two or the last two indices may be contracted to a single matrix index which allows representation in a 3 by 6 matrix. This offers advantages for tabulation of data but has httle merit otherwise since there is no corresponding matrix index for the respective variables. 

The above technique has successfully been applied to the ferroelectric copolymer P(VDF-TrFE) by Heiler and Floss (1994), who studied the linear permittivity 8i along with the second- and third-order permittivities 82 and 83 for poled and unpoled samples (cf Fig. 8). While 81 reveals the ferroelectric to paraelecfric transition including the typical Curie-Weiss behavior for T > Tq, the second-order permittivity appears to be sensitive to the state of poling. Combining 81 and 83 gives even... 

Fig. 8 First-, second-, and third-order permittivity of 56/44 mol-% P(VDF-TrFE) as a function of temperature. The nonlinearities were measured after poling the sample with E ox = 130 V/pm ( ), after thermal depolarization (O), and after poling with poi = —130 V/mm (A). Bottom right the remanent polarization reconstructed from the first- and second-order permittivity (Reprinted with permission from Heiler and Ploss (1994))...

Many of the different susceptibilities in (18)-(21) correspond to important experiments in linear and non-linear optics. The argument in parentheses again describes the kind of interacting waves. TWo waves interact in a first-order process as described above in (9), three waves in a second-order process, and four in a third-order process. x ° describes a possible zeroth-order (permanent) polarization of the medium t- (0 0) is the first-order static susceptibility which is related to the relative permittivity (dielectric constant) at zero frequency, e,.(0), by (22). 

Mackay, T.G., Lakhtakia A.. Weiglhofer, W.S. Homogeneisation of isotropic, cubically nonlinear, composite mediums by the strong-permittivity-fluctuation theory third-order considerations. Opt. Commun. 204. 219-228 (2002)... 

Here, f, Fx and are local field factors, a is a molecular parameter including the dipole moment, second order and third order hyperpolarizabilities. Under condition F = Fx = the formula does not agree with the experiment for 8CB. Indeed, it follows that rji should increase at the transition from the nematic to smectic A phase according to the increase in (F2) However, in the experiment, Fn markedly decreases. The similar temperature behavior was earlier observed for dielectric permittivity of 8CB. In the latter case the decrease in is due to the antiparallel correlation of molecular dipoles in the smectic A phase, which results in a decrease in the effective dipole moment /x Thus, the decrease... 

Hruska CK (1993) Conflicting values of the third-order nonlinear permittivity of a-quartz. J AppI... 

D is the dielectric displacement, Co is the permittivity of free space, Xi >> the dielectric susceptibility of ith order, is the applied electric field), one gets the values of Ihble 2. It should be pointed out that the component should vanish in unpoled samples for thermodynamic reasons. Blending obviously reduces the third-order susceptibility. 

In these equations, eq and a are respectively the vacuum permittivity and the linear absorption coefficient. E a>) represents the apphed electric field at the frequeney co [ (complex conjugate of ( )], whilea>, —m) and/( >,< , —< ), which are fourth-order tensors, are respectively the third-order susceptibUity and hyperpolarisabUity. The real part of these tensors represents the induced refractive index change and is a function of the laser intensity (m) oc (imaginary part describes the 2PA process [40]. 

The three phase dielectric system backbone-waterlayer-air of a real RF aerogel is reduced to a two layer system. The third phase (air) is neglected because of its relative low influence (compared to the other two phases) on the compound dielectric permittivity according to its own material parameters e and k. In order to explain the measured spectra by Maxwell-Wagner polarization processes due to the absorbed water we propose the following model. ... 

The first term, eq, represents the unperturbed part of the permittivity the second term describes the coupling of e to u, with b an elasto-optic coefficient] and the third term describes higher-order terms in E, with / the nonlinear dielectric susceptibility. Substituting (7.15) into (7.14) and retaining only those terms containing u, one gets... 

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