Electric field, second-order effects

In a strong electric field second-order and even third-order effects may be present ... 

Two of the most important nonlinear optical (NLO) processess, electro-optic switching and second harmonic generation, are second order effects. As such, they occur in materials consisting of noncentrosymmetrically arranged molecular subunits whose polarizability contains a second order dependence on electric fields. Excluding the special cases of noncentrosymmetric but nonpolar crystals, which would be nearly impossible to design from first principles, the rational fabrication of an optimal material would result from the simultaneous maximization of the molecular second order coefficients (first hyperpolarizabilities, p) and the polar order parameters of the assembly of subunits. (1)... 

Many of the different susceptibilities in Equations (2.165)-(2.167) correspond to important experiments in linear and nonlinear optics. x<(>> describes a possible zero-order (permanent) polarization of the medium j(1)(0 0) is the first-order static susceptibility which is related to the permittivity at zero frequency, e(0), while ft> o>) is the linear optical susceptibility related to the refractive index n" at frequency to. Turning to nonlinear effects, the Pockels susceptibility j(2)(- to, 0) and the Kerr susceptibility X(3 —to to, 0,0) describe the change of the refractive index induced by an externally applied static field. The susceptibility j(2)(—2to to, to) describes frequency doubling usually called second harmonic generation (SHG) and j(3)(-2 to, to, 0) describes the influence of an external field on the SHG process which is of great importance for the characterization of second-order NLO properties in solution in electric field second harmonic generation (EFISHG). 

Just as second-order effects involve the interaction of two electric fields with the electrons of a material, third-order effects involve three electric fields, 1 2 and 3. In the special case where all these three fields have the same frequency and where y(21 is zero (which is necessarily the case in centrosymmetric materials -see Section 11.1.2), we see that... 

The dipole polarizability also expresses the second-order effect of an electric field on the energy levels of an atom or molecule. We can write, for an atom (which has no dipole moment and therefore, in general, no first-order effect) ... 

Through cascade second-order effects, the second-order optical nonlinearities result on a third-order optical effect in a multistep or cascade process. This process is a due to the existence of microscopic electric fields that are generated by second-order nonlinear aligning of molecular dipoles. 

Wave mixing of two electric fields can give rise to second-order effects of nonlinear optics [4]. One of these is the harmonic generation that converts the fundamental wavelength of a laser into its half (see Section 12.2.2). But, if electric fields at different frequencies are used, the response of a medium with sufficient second-order dielectric susceptibility can be frequency shifted to the sum and the difference of the two laser frequencies [4]. In particular, sum frequency generation (SFG) is often used to study surfaces and has found applications to examine catalytic combustion [9,36]... 

Here, o is the conductivity, Vthemioeiectric is the thermovoltage, tj the Seebeck coefficient, T the temperature, and k is the thermal conductivity for a vanishing electrical field. The second equation is here set to zero, since Joule heating as a second order effect does not play a significant role. More details can be found in Rettig (2008), or Nagy and Nayfeh (2000). 

Most discrete MTP implementations are similar in many respects, e.g., limited expansion up to order 2-4, spherical harmonic description, interaction calculation in the atoms local frames. Hence, what distinguishes these force fields and implementations from each other is primarily in how they treat the other interaction terms. Most importantly, static multipoles only consist of a first-order perturbation of the electrostatic operator. Describing second-order effects leads to polarizability—the charge density s ability to respond to an external electric field—a critical aspect of certain systems (e.g., dielectric changes) [62-64]. Here, possible implementations are ordered in terms of increased overall accuracy (and thus computational investment and larger parametrization effort). Given the heavy requirements of such refined force fields, it is important to point out that "more is not always better," and each system of interest will call for a fine balance of accuracy and statistical sampling. 

Let us consider another, so-called, second-order effect of an external electric field E on a given molecule M2 This field influences the molecular charges, electTOTis, and nuclei, causing their displacements, and as a result, there appears the induced dipole moment d . ... 

Second Order Non-Linear Effects. In order to probe second-order effects, materials normally must be subjected to electric fields of high enough intensity to polarise the material substantially, so that the induced polarisation becomes a non-linear function of the field strength. The relationship between the two common parameters which characterise bulk materials, viz. dielectric constant e and refractive index r, is shown in the following relationships ... 

Calculation of the electric birefiingence is slightly more complicated because it is a second-order effect in the electric field. Let us consider the case that a time-dependent, weak electric field E t) is applied in ffie z direction. Straightforward perturbation calculation gives ... 

The above equation is called the polarization equation. In order for the materials to exhibit second-order effects, the n electrons must de-localize in response to applied electric fields. This results in spatial asymmetry, as already stated above. Also, for the second-order effects to be large, fliere must be large differences in die dipole moments between the ground state and the excited states. 

When a polymer is subject to an intense sinusoidal electric field such as that due to an intense laser pulse, Fourier analysis of the polarization response can be shown to contain not only terms in the original frequency co, but also terms in 2(0 and 3nonlinear response depends on the square of the intensity of the incident beam for 2co, and the third power for 3 . For the second-order effects, the system must have some asymmetry, as discussed previously. For poling, this means both high voltage and a chemical organization that will retain the resulting polarization for extended periods of time. Polymeric systems investigated have been of three basic types ... 

If the net interaction between solvent and solute is small, their molecules will remain far apart since the van der Waals radii of molecules are relatively large. Their MOs will not therefore overlap significantly, so the corresponding terms P, in equation (5.80) will be small. The main second-order effect therefore corresponds to a polarization of solvent and/or solute by the electric fields due to the other. Thus if the solvent is nonpolar and the solute is polar, the solute will remain unperturbed (since the solvent molecules have no net fields), but the solvent will be polarized by the fields of the polar solute molecules. The polarization is such as to lead to an attraction between solute... 

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