Nonlinear dielectric polarization

D. A. Kleinman, Nonlinear dielectric polarization in optical media, Phys. Rev. 126 1977-1979 (1962). 

Assuming that the amplitude of the light electric field changes during propagation in the x direction, then the electric field of the light, E, emitted by the forced oscillation of the nonlinear dielectric polarization is given by... 

When two laser beams (Eu nonlinear optical medium, the nonlinear dielectric polarization, PNL, exhibits the following forced oscillations,... 

Thus irradiation of nonlinear optical media with coherent light induces nonlinear dielectric polarization and emission of frequency-converted light. Second harmonic generation (SHG) and third harmonic generation (THG) are typical examples of this effect. They are used for example to generate laser light emitting in the ultraviolet and visible spectral domains. 

Polarization which can be induced in nonconducting materials by means of an externally appHed electric field is one of the most important parameters in the theory of insulators, which are called dielectrics when their polarizabiUty is under consideration (1). Experimental investigations have shown that these materials can be divided into linear and nonlinear dielectrics in accordance with their behavior in a realizable range of the electric field. The electric polarization PI of linear dielectrics depends linearly on the electric field E, whereas that of nonlinear dielectrics is a nonlinear function of the electric field (2). The polarization values which can be measured in linear (normal) dielectrics upon appHcation of experimentally attainable electric fields are usually small. However, a certain group of nonlinear dielectrics exhibit polarization values which are several orders of magnitude larger than those observed in normal dielectrics (3). Consequentiy, a number of useful physical properties related to the polarization of the materials, such as elastic, thermal, optical, electromechanical, etc, are observed in these groups of nonlinear dielectrics (4). 

The most important materials among nonlinear dielectrics are ferroelectrics which can exhibit a spontaneous polarization PI in the absence of an external electric field and which can spHt into spontaneously polarized regions known as domains (5). It is evident that in the ferroelectric the domain states differ in orientation of spontaneous electric polarization, which are in equiUbrium thermodynamically, and that the ferroelectric character is estabUshed when one domain state can be transformed to another by a suitably directed external electric field (6). It is the reorientabiUty of the domain state polarizations that distinguishes ferroelectrics as a subgroup of materials from the 10-polar-point symmetry group of pyroelectric crystals (7—9). 

When a strong static electric field is applied across a medium, its dielectric and optical properties become anisotropic. When a low frequency analyzing electric field is used to probe the anisotropy, it is called the nonlinear dielectric effect (NLDE) or dielectric saturation (17). It is the low frequency analogue of the Kerr effect. The interactions which cause the NLDE are similar to those of EFLS. For a single flexible polar molecule, the external field will influence the molecule in two ways firstly, it will interact with the total dipole moment and orient it, secondly, it will perturb the equilibrium conformation of the molecule to favor the conformations with the larger dipole moment. Thus, the orientation by the field will cause a decrease while the polarization of the molecule will cause an... 

We have shown in this paper the relationships between the fundamental electrical parameters, such as the dipole moment, polarizability and hyperpolarizability, and the conformations of flexible polymers which are manifested in a number of their electrooptic and dielectric properties. These include the Kerr effect, dielectric polarization and saturation, electric field induced light scattering and second harmonic generation. Our experimental and theoretical studies of the Kerr effect show that it is very useful for the characterization of polymer microstructure. Our theoretical studies of the NLDE, EFLS and EFSHG also show that these effects are potentially useful, but there are very few experimental results reported in the literature with which to test the calculations. More experimental studies are needed to further our understanding of the nonlinear electrooptic and dielectric properties of flexible polymers. 

We consider a nonlinear dielectric medium in one dimension. The relation between the polarization P and the electric field E is given by ... 

Figure 3.1 shows a simplified picture of an interface. It consists of a multilayer geometry where the surface layer of thickness d lies between two centrosymmetric media (1 and 2) which have two different linear dielectric constants e, and e2, respectively. When a monochromatic plane wave at frequency co is incident from medium 1, it induces a nonlinear source polarization in the surface layer and in the bulk of medium 2. This source polarization then radiates, and harmonic waves at 2 to emanate from the boundary in both the reflected and transmitted directions. In this model, medium 1 is assumed to be linear. 

A linear dielectric medium is characterized by a linear relation between the polarization density and the electric field, P = e0xE, where eo is the permittivity of free space and x is the electric susceptibility of the medium. A nonlinear dielectric medium, on the other hand, is characterized by a nonlinear relation between P and E, as illustrated in Figure 4.19. 

In order to compare calculated and experimentally observed phase portraits it is necessary to know very exactly all the coefficients of the describing nonlinear differential Equation 14.3. Therefore, different methods of determination of the nonlinear coefficient in the Duffing equation have been compared. In the paraelectric phase the value of the nonlinear dielectric coefficient B is determined by measuring the shift of the resonance frequency in dependence on the amplitude of the excitation ( [1], [5]). In the ferroelectric phase three different methods are used in order to determine B. Firstly, the coefficient B is calculated in the framework of the Landau theory from the coefficient of the high temperature phase (e.g. [4]). This means B = const, and B has the same values above and below the phase transition. Secondly, the shift of the resonance frequency of the resonator in the ferroelectric phase as a function of the driving field is used in order to determine the coefficient B. The amplitude of the exciting field is smaller than the coercive field and does not produce polarization reversal during the measurements of the shift of the resonance frequency. In the third method the coefficient B was determined by the values of the spontaneous polarization... 

With this background, we have proposed and developed a new purely electrical method for imaging the state of the polarizations in ferroelectric and piezoelectric material and their crystal anisotropy. It involves the measurement of point-to-point variations of the nonlinear dielectric constant of a specimen and is termed scanning nonlinear dielectric microscopy (sndm) [1-7]. This is the first successful purely electrical method for observing the ferroelectric polarization distribution without the influence of the screening effect from free charges. To date, the resolution of this microscope has been improved down to the subnanometer order. 

Here we describe the theory for detecting polarization and the technique for nonlinear dielectric response and report the results of the imaging of the ferroelectric domains in single crystals and thin films using sndm. Especially in a measurement of pzt thin film, it was confirmed that the resolution was sub-nanometer order. We also describe the theoretical res-... 

Here, 33, 333, 3333, and 33333 correspond to linear and nonlinear dielectric constants and are tensors of rank 2nd, 3rd, 4th and 5th, respectively. Even-ranked tensors including linear dielectric constant 33 do not change with polarization inversion, whereas the sign of the odd-ranked tensors reverses. Therefore, information regarding polarization can be elucidated by measuring odd-ranked nonlinear dielectric constants such as 333 and 33333. 

Nonlinear polarization characteristics of centrosymmetric molecules modified by the introduction of substituent groups are expressed in the following manner. The dielectric polarization consists of odd-number-order terms as in Eq. (5.23). 

Displacements that occur between several equilibrium sites for which the probability of occupancy of each site depends on the strength of the external field. This mechanism is also known as dipolar or ion jump polarization and is depicted schematically in Fig. 14.10. Another definition of ion jump polarization is the preferential occupation of equivalent or near-equivalent lattice sites as a result of the applied field biasing one site with respect to the other. If the alignment occurs spontaneously and cooperatively, nonlinear polarization results and the material is termed ferroelectric. Because of the relatively large displacements, relative dielectric constants on the order of 5000 can be attained in these materials. Nonlinear dielectrics are dealt with separately in Chap. 15. But if the polarization is simply due to the motion of ions from one adjacent site to another, the polarization behavior is linear with voltage. These solids are discussed below. 

Nonlinear optics (NLO) is the branch of optics that describes the behavior of light in nonlinear media, in which the dielectric polarization P responds nonlinearly to the electric field E of the light. This nonlinearity is typically observed only at very high light intensities such as those provided by pulsed lasers. 

Nonlinear optical (NLO) properties of organic polymers can be viewed as dielectric phenomena, and their origins can conveniently be explained by considering a planar wave propagation through a nonlinear dielectric medium [1-4]. In a dielectric medium the polarization P induced by the incident field E can be written as a power series of the field strength as follows ... 

The dielectric polarization P of a medium with nonlinear susceptibility x. subject to an electric field E, can be written as an expansion in powers of the applied field... 

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