Cartesian field components

In each medium the Vt In e term vanishes and each cartesian field component satisfies a simplified wave equation ... 

For waveguides with a step-index profile, i.e. n = in the core and n — 0 1 in the cladding, all terms involving Vtlnn in Eqs. (11-40) and (11-44) vanish within the core and cladding but not on the core-cladding interface. Consequently each cartesian field component of Eq. (11-40) satisfies a simplified equation within the core and cladding, e.g. 

If we work with the cartesian field components of Eq. (30-16) and the separable fields of Eq. (30-4), then in source-free regions Eq. (30-17) reduces to the homogeneous vector wave equations with transverse and longitudinal components given by... 

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...

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... 

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... 

Equations (D.6) and (D.7) are Gilbert s equation in spherical polar coordinates. To obtain the Gilbert-Langevin equation in such coordinates we augment the field components //, and with random field terms and h. By graphical comparison of the Cartesian and spherical, polar coordinate systems, we find these to be... 

Let us consider the general form of the wave equation that governs the evolution of any electric or magnetic field component f. Its homogeneous version in a Cartesian coordinate system (x, y, z) is given by... 

Both eqs. 1.34 and 1.37 are valid for electromagnetic fields regardless of the rate of change of the field with time. Equation 1.37 is the third Maxwell equation in differential form. We must stress that there is a fundamental difference between the two forms presented above for the third Maxwell equation. While the integral form can be appUed everywhere, it is necessary to be careful in the use of the differential form. This caution must be exercised because the function div E might not be defined at certain points, lines or surfaces. As a matter of fact, div E is expressed in terms of the first spatial derivatives of the field components. In Cartesian coordinates for example, we have ... 

FIGURE 37.4 The relation between the spherical coordinate system and the Cartesian coordinate system in which the field components in every point were calculated. R is the radius vector to the point inside the sphere where the field is computed, and r is the vector to the differential coil element on which the integration is performed. 

When a molecule is located in a non-homogeneous electric field, the perturbation operator has the form = — fi-qPq — 3 qq qq qq where Eq for q — X, y, z denote the electric field components along the corresponding axes of a Cartesian coordinate system, Eqqi stands for the component of the gradient of Eq, while [Lq, qq Stand for the operators of the corresponding components of the dipole and quadrupole moments. In a homogeneous electric field Eqqi = 0), this reduces to = — Y2q P-q q-... 

In a molecule which has tetrahedral symmetry (CCI4) or octahedral symmetry (SFg) an external electric field will generate an induced dipole moment whose direction is the same as that of the field regardless of the orientation of the molecule (see Fig. 1.33). Such a molecule is said to be isotropic. In such a case when the electric field vector is resolved into cartesian coordinate components, Eq. (1.110) becomes... 

This equation is sometimes referred to as the scalar wave equation. However, in this book we describe a field component as a solution of the scalar wave equation only if (i) it satisfies Eq. (11-45) for all values of x and y, including the interface, and consequently (ii) is continuous and has continuous first derivatives everywhere - see Section 33-1. This is not the case for any cartesian component of on a step-profile waveguide of arbitrary cross-sectional shape, since the Vj In n terms are nonzero at the interface. Thus, to solve for Cgj everywhere, we impose the boundary conditions of Maxwell s equations on the solutions of Eq. (11-45) derived in each homogeneous region. The component h j is derived similarly. Alternatively we can solve Eq. (11-44) with V lnn terms retained. The transverse components then follow from Eq. (11-43). 

As emphasized above, the vector operator couples the field components in an arbitrary coordinate system. However, if the field vectors have components referred to fixed cartesian directions, this coupling does not occur and the vector operator is replaced by the scalar Laplacian V. Thus, if we set... 

Fig. 1. The time evolution (top) and average cumulative difference (bottom) associated with the central dihedral angle of butane r (defined by the four carbon atoms), for trajectories differing initially in 10 , 10 , and 10 Angstoms of the Cartesian coordinates from a reference trajectory. The leap-frog/Verlet scheme at the timestep At = 1 fs is used in all cases, with an all-atom model comprised of bond-stretch, bond-angle, dihedral-angle, van der Waals, and electrostatic components, a.s specified by the AMBER force field within the INSIGHT/Discover program.

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

The creation terms embody the changes in momentum arising from external forces in accordance with Newton s second law (F = ma). The body forces arise from gravitational, electrostatic, and magnetic fields. The surface forces are the shear and normal forces acting on the fluid diffusion of momentum, as manifested in viscosity, is included in these terms. In practice the vector equation is usually resolved into its Cartesian components and the normal stresses are set equal to the pressures over those surfaces through which fluid is flowing. 

H now differentiate these expressions with respect to some parameter a that (jould be a Cartesian coordinate, the component of an applied electric field, or whatever. We then have... 

To illustrate Equation (1.8), consider a solution of the forward and inverse problems in the simplest possible case, when the field is caused by an elementary mass. Suppose that a particle with mass m q) is situated at the origin of a Cartesian system of coordinates. Fig. 1.2a, and the field is observed on the plane z — h. Then, as follows from Equation (1.8), the components of the attraction field at the point p(x,y,h) are... 

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