Z-dependent nonuniformities,

A fiber with z-dependent nonuniformities is no longer cylindrically symmetric and therefore cannot support individual modes of the unperturbed fiber. 

The unperturbed fiber has refractive-index profile h(x, y) and the perturbed fiber has z-dependent nonuniformities described by n(x, y, z). Both fibers are weakly guiding and we assume h = n s n j everywhere. For reasons given in the previous section, we ignore all polarization effects and work solely with solutions of the scalar wave equation. For convenience we assume the electric field is x-polarized and set... 

By definition, the area-average term, il/0(z), depends only on the variable, z, and by assumption is considerably larger in magnitude than the nonuniform term, p(r), which depends fully on the three spatial coordinates (x, y, z). 

When the nonuniform fiber has a step profile, the core index n z) depends only on z, and the upper limit of integration r,p (z) in Eq. (5-41) is replaced by the core radius p(z). If the path makes angle 0 (z) with the axial direction, then with the help of Eq. (5-2) we have... 

Two important cases of negative differential conductivity (NDC) are described by an iV-shaped or an -shaped j (F) characteristic, and denoted by NNDC and SNDC, respectively. However, more complicated forms like Z-shaped, loop-shaped, or disconnected characteristics are also possible [15]. NNDC and SNDC are associated with voltage- or current-controlled instabilities, respectively. In the NNDC case the current density is a singlevalued function of the field, but the field is multivalued the F j) relation has three branches in a certain range of j. The SNDC case is complementary in the sense that F and j are interchanged. In case of NNDC, the NDC branch is often but not always - depending upon external circuit and boundary conditions - unstable against the formation of nonuniform field... 

Za and Zi are the charges of the ion in the adsorbed state and in the bulk solution. A somewhat more general model of the compact layer as a nonuniform dielectric slab whose local dielectric permittivity depends on the normal coordinate, (z), results in the formula (88b) containing the ratio of the effective thickness for the overall compact layer, Lh = (43tCh) , and that of its inner part between the bulk metal and the adsorption plane, L (again, the charge of the adsorbed ion is considered here as the point charge localized at the distance a from the surface) [262] ... 

X = 0, there is specified a uniform distribution of isotropic sources of strength go neutrons per unit area per unit time. Because of the finite extent of the x dimension, and the nonsymmetry of the problem, there will be a nonuniform distribution of the neutron population along the x direction the dependence on the y and z directions for given x, however, will be uniform. Thus, the flux is given by the function (x), and the appropriate differ-... 

The vectors a, b and c are defined with respect to a SmC layer, as shown in Fig. 10(a). The coefficients A describe curvature of the planes, B nonuniform rotations of the optic axis about z in a flat structure, and the C coefficients are coupling terms between A and B, and are shown schematically in Fig. 10(b). All of the temperature dependences of these coefficients may be included in the tilt angle. 

We next consider nonuniform fibers with clad power-law profiles. These profiles have the form given by Eq. (5-15), with A and p dependent on z, but we assume the exponent q remains constant. Substituting into Eq. (5-41) we have... 

We define a transit time for the beam to transverse the shift z. The concept of transit time implies that we are considering a beam which exists for only a finite duration in time. This is dual to the concept of the shift, involving beams of finite (or nonuniform) extent in space. Hence the beam cannot be monochromatic, and has a small spread in frequency 8co about a central frequency coq. The phase function (0z) depends on frequency through the relationship j8 = 2nn cos 0 /2. If we express 2 in terms of co using 2o) = 2nc, where c is the free-space speed of light, then / = con cos OJc. We expand (co) about u>o and, correct to first order, obtain... 

The general conclusions on the thermodynamic restrictions for empty routes are also vahd in the case of surface nonuniformity or appearance of lateral interactions in the adsorbed layer. For the non-ideal surfaces when the nonlinearity of macroscopic rate laws manifests itself, the rate constants of adsorption and desorption depend on surface coverage because of lateral interactions. Imagine that in the reaction mechanism there are steps of types t AZ + Z products and AZ + BZproducts. The reaction rates for steps of type i and j could be given by the following expressions ... 

The basis for the decision of this problan is formula (5.1.32) a force working on a magnetic dipole depends on both the nonuniformity of the magnetic field and the orientation of the atomic magnetic momoits relative to the quantization axis z. If the atomic magnetic moments submit to laws of classical physics, there could be any values of angles and the experiment would result in a fuzzy maximum. However, in quantum mechanics this is not a case ... 

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