Electric polarization general case

In the most general case, the above two equations mean that the electric field vector traces an ellipse in the j>z plane. There are two special cases of note in this general situation. If the phase difference between the two components of the field (5K -5.) is zero or some integral multiple of 7r, the ellipse flattens to a line. If the phase difference is t/2 or any odd integral multiple of tt/2 and the amplitudes of the two components are equal, the ellipse is rounded to a circle. In the former case we speak of the radiation as being plane polarized, and in the latter case as being circularly polarized. 

In an isotropic dielectric such as a normal glass, the induced electrical polarization is always parallel to the applied electric field and therefore the susceptibility is a scalar. In general this is not the case in anisotropic dielectrics when the polarization depends on both the direction and magnitude of the applied field. The three components of the polarization are written... 

The QA solutions developed in the previous section were based on the assumption that the electrical reflectivity tensor was a scalar. This assumption reduces the areas of practical applications of the QA approximations because in this case the anomalous (scattered) field is polarized in a direction parallel to the background field within the inhomogeneity. However, in general cases, the anomalous field may be polarized in a direction different from that of the background field, which could generate additional errors in the scalar QA approximation. To overcome this difficulty, we introduce in this section a tensor quasi-analytical (TQA) approximation. The TQA approximation uses a tensor A, which permits different polarizations for the background and anomalous (scattered) fields. 

The constitution of ordered layers of water at interfaces with carbonaceous adsorbents in aqueous suspensions is governed by three major factors, namely, hydrophilic properties of the surface, porosity of the material, and the feasibility of polarization of the surface at the expense of the formation of regions carrying electric charges of opposite signs. In the general case, the thickness of an adsorbed water layer on the surface is detennined by the action radius of surface forces in whose field the orientation of electric dipoles of water molecules occurs and the formation of its surface clusters takes place. 

X = e - 1 is the linear susceptibility and x is the p -ortlcr nonlinear susceptibility which is a tensor of rank p - -1. The actual situation is, in fact, more complicated, as the preceding expression holds for isotropic and homogeneous media with nonlinear susceptibilities being real scalar quantities. In the general case, the susceptibilities are complex tensors, defined through the relationship between the spatiotemporal Fourier components of both the polarization and the electric field [68, 69],... 

In the general case, polarizability is anisotropic it depends on the position of the molecnle with respect to the orientation of the external electric field. To consider the orientation, a is expressed by a fnnction — atom polarizability tensor — that defines the indnced dipole moment for each possible direction of the electric field. This atom polarizability tensor a describes the distortion in the nuclear arrangement in a molecnle (i.e., the tendency of polarization in three dimensions). The polarizability determined in an experiment is an average polarizability it is the sum of polarizabilities in three principal directions that are collinear with the external field [59],... 

As previously indicated (p. 43), the moment induced in a particle by a unit incident electric vector is—except with spherically symmetrical (isotropically polarizable) bodies—a function of the orientation of the particle relatively to the polarizing field E. In the general case, therefore, molecular polarizability must be regarded as a tensor of the second order. Adopting rectangular space-fixed co-ordinates, each component of the induced moment m is a linear vector function of the three components of E ... 

In nonlinear optics the theoretical framework falls naturally into two parts. First, one must calculate a nonlinear susceptibility (x) which describes the microscopic (i.e., atomic contribution to the polarization induced by the propagation of the laser beam at frequency w,). For the general case, this polarization is written as a power series in the electric field (w,) so that... 

All atoms and molecules can be polarized by an electric field. The polarization (induced dipole of a unit volume) is P = aE where a is molecular polarizability. For spherically symmetric atoms or molecules (like C60 fullerenes) the polarizability is a scalar quantity (tensor of zero rank) and P E. In general case of lath-like molecules, QLij is a second rank tensor (9 components) and Py = a,yE/. By a proper choice of the reference frame the tensor can be diagonalized... 

An important experimental criterion to discriminate the various spectroscopic processes arising in Eq. (28) is the emission direction k of the polarization. Because the incident electric fields Ei, E2 radiate along their wave vectors ki, k2, the emission directions k of the thus induced polarization are given as linear combinations of ki and k2- In the most general case (i.e. when none of the assumptions above hold) it is clear that we get an enormous number of contributions and corresponding emission directions k in the expansion (28). Employing the RWA, however, it has been shown in Ref. 25 that the polarization of 2N + l)th order only radiates into the directions... 

Antennas create a state of electromagnetic polarization generally described as linear, circular or elliptical. Figure 13.4(a) depicts a linear polarized electric held (vertical polarization in this case). This would result from one of the dipoles shown in Fig. 13.1(c). The polarization of the wave is related to the orientation of the antenna wire. The electric held in Fig. 13.4(a) oscillates up and down at the angular frequency co. 

The definition of piezoelectricity extended considerably in time. Originally it was used only for crystals in connection with compressions, but later it was generalized to polymers and other materials for any strains and stresses, including shear. For historical reasons, in most cases direct and converse piezoelectric effects are distinguished. When electric polarization is produced by... 

Consider first the quenching of spin-polarized 2s ions in the absence of hyperfine structure or magnetic field effect . Then the 2sj state can decay by allowed Ml transitions, or by electric-field induced El or M2 transitions. An emitted photon is characterized by a propagation vector k and a polarization vector e, and the initial atomic state by a spin-polarization vector P. We choose a coordinate system such that k lies along the z-axis, and write e for the general case of elliptical polarization in the form... 

When 8j and 82 have different phases, the tip of the electric vector traces out an ellipse in any fixed plane perpendicular to k and the wave is said to be elliptically polarized. This general case is roost easily discussed if we choose a coordinate system in which the z-axis coincides with the direction of propagation k and 2 te in the directions Ox, Oy respectively. If the complex amplitudes are... 

The polarization of the electric field is determined by the motion of the elementary charge producing the time-dependent dipole moment. If the electron is performing linear oscillations, equations (2.69) and (2.70) show that the radiation fields are also linearly polarized. In the more general case of elliptical or circular motion, the... 

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