Cartesian polarization components

X lT + E r Xij ki, which refer in their order of appearance to the Cartesian polarization components of the CRS, pump, probe, and Stokes fields in the four-wave mixing process [31]. In transparent and optically inactive media, where the input frequencies are away from any electronic transition frequencies, and only the molecular ground state is populated, the selection rules of both resonant coherent and spontaneous Raman scattering are identical... 

Fig. 13.4 Profiles of the intensities l J2, Ey 2, and EZ 2 of the Cartesian coordinate components of the electric field in a fundamental mode with quasi linear polarization. The insets show the inner parts of the profiles, which correspond to the field inside the fiber. The parameters used a 200 nm, X 1,300 nm, 1.4469, n2 1. Reprinted from Ref. 61 with permission. 2008 Elsevier...

Table 4.5. Cartesian dipole components of the H2-He system, for three inter-nuclear spacings r after [279]. The center-of-mass of the H2 molecule is at the origin and the He atom is at z = R. A positive induced dipole moment p2 corresponds to the polarity H He+ (py — 0).

Details of the extended triple zeta basis set used can be found in previous papers [7,8]. It contains 86 cartesian Gaussian functions with several d- and f-type polarisation functions and s,p diffuse functions. All cartesian components of the d- and f-type polarization functions were used. Cl wave functions were obtained with the MELDF suite of programs [9]. Second order perturbation theory was employed to select the most energetically double excitations, since these are typically too numerous to otherwise handle. All single excitations, which are known to be important for describing certain one-electron properties, were automatically included. Excitations were permitted among all electrons and the full range of virtuals. 

In order to find the vertical component of the field we can apply the same approach as before, namely, the integration over the volume of the spheroid, only in this case the polar axis of the spherical system should be directed along the z-axis. However, we solve this problem differently and will proceed from the second equation of the gravitational field. In Cartesian system of coordinates we have... 

It has two components, (x, y) in cartesian coordinates, and these can be transformed into plane polar coordinates to give... 

Polarization functions may be optionally chosen to be of pure or Cartesian form (by another keyword). In the former case, one includes the expected number of angular-momentum components (i.e., five d orbitals, seven f orbitals, etc.), whereas in the latter case some additional component(s) of lower angular momentum are included (e.g., a Cartesian d set includes five d orbitals plus one s orbital, a Cartesian f set includes seven f orbitals plus three p orbitals, and so forth). 

Cartesian coordinates is a product of N — 1 Jacobians for the local transformations from polar to Cartesian components for each bond vector Q for j < — 1, times the Jacobian det[A ]p = 1 for the transformation of Q... 

The incident plane wave has only field components perpendicular to the direction of propagation. In contrast, the evanescent field has components along all directions X, y, and z of a Cartesian coordinate system attached to the IRE, as shown in Fig. 2. The direction of the incident field vector can be selected by use of a polarizer. The symbols II and denote electric field vectors parallel and perpendicular to the... 

Hence there exists a complete set of common eigenfunctions for L2 and any one of its components. The eigenvalue equations for L2 and Lz are found to be separable in spherical polar coordinates (but not in Cartesian coordinates). Using the chain rule to transform the derivatives, we can find... 

Our task here is to determine whether any of the three Cartesian components is nonzero. Since, in DAh the z vector transforms according to the AZu representation and a and y jointly according to the representation, we need to know whether either of the direct products, Blg x AZu x BZu or BiK x Eu x BZu contains the Aljt representation. It is a simple matter to show that the first one is equal to AlK while the second is equal to Eg. Thus, the <5— S transition is electric-dipole allowed with z polarization and forbidden for radiation with its electric vector in the xy plane. 

Figure H3.1.2 The oscillatory response of a food material that possesses both elastic and viscous properties can be represented by a complex variable G. This variable has two components that can be expressed as either the Cartesian coordinates G and G or the polar coordinates IG I and 8. IG I is the magnitude of the imaginary G and is measured as the ratio of the amplitudes of stress and strain.

The bond polarization model gives the chemical shift of an atom a as the sum over the Na bonds of this bond. The bond contributions are formed of a component for the unpolarized bond (which also includes the inner shell contributions to the magnetic shielding) and a polarization term. The bond contributions are represented by a tensor with its principal axes along the basis vectors of the bond coordinate system. The transformation from the bond coordinate system into a common cartesian system is given by the transformation matrix Z). ... 

In order to illustrate the mixed state, an example with five sample wavelets will be discussed in detail. Each wavelet is represented by its components ax and ay in the Cartesian basis (optical definition, see Section 9.2.2). If the polarization vector is described by a polarization ellipse with major and minor axes a = cos y and b = sin y, by a tilt angle X of this ellipse against a fixed coordinate frame (see Fig. 1.15), and by the direction of rotation of the electric field vector indicated by the sign of y, the components ax and cty follow from... 

In the discussion of light polarization so far the Cartesian basis and spherical basis have been considered. Because the linear polarization might be tilted with respect to the (ex, e -basis, a third basis system has to be introduced against which such a tilted polarization state can be measured via its non-vanishing components. This coordinate system is called (e e and its axes are rotated by +45° with respect to the previous ones. This leads to a third representation of the arbitrary vector b ... 

We now transform the governing equations in cylindrical coordinates into polar coordinates. Since the motion is axisymmetric, the transformation from (r, z) to (R, 6), as shown in Fig. 3.1, is analogous to the transformation from Cartesian coordinates (x, y) to cylindrical coordinates (r, 0) in a two-dimensional domain. The stream function is related to the velocity components in polar coordinates by... 

Equation (11) is written in the form of Newton s second law and states that the mass times acceleration of a fluid particle is equal to the sum of the forces causing that acceleration. In flow problems that are accelerationless (Dx/Dt = 0) it is sometimes possible to solve Eq. (11) for the stress distribution independently of any knowledge of the velocity field in the system. One special case where this useful feature of these equations occurs is the case of rectilinear pipe flow. In this special case the solution of complex fluid flow problems is greatly simplified because the stress distribution can be discovered before the constitutive relation must be introduced. This means that only a first-order differential equation must be solved rather than a second-order (and often nonlinear) one. The following are the components of Eq. (11) in rectangular Cartesian, cylindrical polar, and spherical polar coordinates ... 

Figure 2.12 Definition of the components of angular momentum in cartesian and in spherical polar coordinates.

It has the same form as the cartesian components and the solution, = ke tmf describes rotation about the polar axis in terms of the orbital angular momentum vector LZ1 specified by the eigenvalue equation... 

Within the polarization scheme used here where p-polarized light is polarized in the plane of incidence, and s-polarized light is polarized perpendicular to the plane, the Cartesian components of , (a>, ) are given by the projection of the s and p components of the incident wave... 

The hrst suffix of each tensor component gives the row and the second the column in which the component appears. The xis term, for example, measures the component of the polarization parallel to X2 (usually the y direction in a Cartesian coordinate system) when a field is applied parallel to X3 (the z direction). The susceptibility tensor must conform to any restrictions imposed by crystal symmetry, see Eqs. 6.5-6.9. 

The induced polarization in a piezoelectric, Pj, is a first-rank tensor (vector), and mechanical stress, is a second-rank tensor (nine components), which is represented in a Cartesian coordinate system with axes x, y, and z, as ... 

As expected for a guided mode, the electric field in the substrate and cover regions decays exponentially, and it shows an oscillatory behavior inside the waveguide film. For the TM polarization, one obtains similar, although not identical, relations for the magnetic field Hy, which then allows the calculation of the Cartesian components and E, of the electric field using the generalized Ampere s law in the Maxwell s equations [7]. 

For the case of cubic zone-center optical phonons, the Raman tensor is of rank three, with a second-rank tensor for phonon components polarized along X, y, or z Cartesian coordinate directions ... 

---