Local plane waves polarization

The transmission coefficient T is found by using the local plane-wave description of a ray. We regard the local plane wave as part of an infinite plane-wave incident on a planar interface between unbounded media, whose refractive indices coincide with the core and cladding indices and of the waveguide, as shown in Fig. l-3(b). For the step interface, Tis identical to the Fresnel transmission coefficient for plane-wave reflection at a planar dielectric interface [6]. In the weak-guidance approximation, when s n, the transmission coefficient is independent of polarization, and is derived in Section 35-6. From Eq. (35-20) we have [7]... 

Thus the modal and ray transit times are equal only when tj - 1. This condition is satisfied only by those rays belonging to modes well above cutoff, i.e. when Vp U, or, equivalently, when 0 < 0c- Hence is inaccurate for arbitrary values of 9. This inaccuracy arises because the ray transit time ignores diffraction effects, which were discussed in Chapter 10. The step-profile planar waveguide is a special case, however, because all diffraction effects can be accounted for exactly by including the lateral shift at each reflection, together with recognizing the preferred ray directions. TWs was carried out in Section 10-6, and for rays, or local plane waves, whose electric field is polarized in the y-direction in Fig. 10-2, leads to the modified ray transit time of Eq. (10-13). If we use Table 36-1 to express 0, and 0(.in terms of U, Vand Wand substitute rj for TE modes from Table 12-2, we find that Eqs. (10-13) and (12-8) are identical since 0 = 0. It is readily verified that the same conclusion holds for TM modes and local plane waves whose magnetic field is polarized in the y-direction of Fig. 10-2. 

We now consider illumination by light beams that are parallel to, or subtend only small angles 0i to the axis of the fiber. The fields of the beam at the endface of the fiber have the form of a local plane wave whose wave vector is parallel to the direction of the beam. If the electric field, Ej, is uniformly polarized parallel to the x-axis in Fig. 20-2 (a) and the beam is parallel to the x-z plane, with a symmetric amplitude distribution/(r), then... 

The higher-order bulk contribution to the nonlmear response arises, as just mentioned, from a spatially nonlocal response in which the induced nonlinear polarization does not depend solely on the value of the fiindamental electric field at the same point. To leading order, we may represent these non-local tenns as bemg proportional to a nonlinear response incorporating a first spatial derivative of the fiindamental electric field. Such tenns conespond in the microscopic theory to the inclusion of electric-quadnipole and magnetic-dipole contributions. The fonn of these bulk contributions may be derived on the basis of synnnetry considerations. As an example of a frequently encountered situation, we indicate here the non-local polarization for SFIG in a cubic material excited by a plane wave (co) ... 

SIESTA code, the interactions of valence electrons with the atomic ionic cores are described by the norm-conserving pseudopotentials with the partial core correction of 0.6 au. on the oxygen atom. We used the optimized-zeta plus polarization (DZP) basis sets with medium localization in the SIESTA code. A mesh cutloff energy of 350 Ry, which defines the equivalent plane wave cut-off for the grid, was used. The forces on atomic ions are obtained by the Heilman IFeynman theorem and were used to relax atomic ionic positions to the minimum energy. The atomic forces within the supercell were minimized to within 0.035 eV/A and 0.05 eV/A in the SIESTA and CASTEP codes respectively. 

Now we turn to the situation when the QW width fluctuations, alloy disorder or impurities localize the 2D exciton (such a situation is more frequent for II-VI semiconductor quantum wells than for III—V ones). Then, the wavefunction of the center-of-mass exciton motion (ry) is no longer just a plane wave, and the corresponding polarization is given by... 

We examined the effect of coupling between surface plasmon polariton (SPP) modes on the optical activity of metal nanostructures. By measuring the in-plane wave vector dependence on transmission and polarization azimuth rotation, we show that coupling of the SPP modes with orthogonal polarization localized at different interfaces is responsible for the optical activity in metal nanostructures. 

As we have seen, on the whole the agreement with theory for the localized form factor associated with the 4f electrons in lanthanide metals and compounds is satisfactory provided one is careful to use relativistic calculations. The situation for the conduction electron polarization distribution is less clear. Conduction electron form factors were obtained for Gd by Moon et al. (1972) and for Er by Stassis et al. (1976). In both cases, these were obtained by separating from the measured form factor the localized 4f contribution, and in both cases appear to be different from either a 5d or 6s atomic form factor. A spin-polarized augmented-plane-wave (APW) calculation of the conduction electron polarization in ferromagnetic Gd was performed by Harmon and Freeman (1974). Their results are, however, only in qualitative agreement with the results of Moon et al. The theoretical form factor of Harmon and Freeman is in somewhat better agreement with the experimental results of Stassis et al. on Er. 

We consider an elliptically polarized plane wave of frequency normal incidence onto an inhomogeneous, nonabsorbing, locally uniaxial film of thickness d. The dielectric tensor of the medium has the form... 

The permittivity discussed so far depends only on frequencies and, through the relaxation times, on the particle size. It is well-known that the dielectric response of materials, in particular metal, is non-local, i.e. the polarization vector induced at a certain point depends on the values of the electric field in all other points. In the reciprocal space language, we can say that e(plane-waves in which the probing electric field can be decomposed. The permittivity of metals such as Ag, Au and Cu at optical frequencies mainly depends on the behavior of both the valence electrons, which is close to that of a free-electron gas, and the core of the metal. As we did before, the total dielectric constant of the metal ([Pg.239]

Figure 7. Local field intensity for a gold triangle antenna with a length of 200 nm, radius of curvature at the apex of 20 nm, apex angle of 45°, and thickness of 80 nm. The incident plane wave of unit amplitude is polarized along the x axis. The FDTD cell size is (2.5 nm).2...

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