Reflected vector plane wave

Fig. 2.15. Reflection and refraction of a vector plane wave (propagating in the ambient medium) at the interface S...

Equations (1) are the von Laue conditions, which apply to the reflection of a plane wave in a crystal. Because of eqs. (1), the momentum normal to the surface changes abruptly from hk to the negative of this value when k terminates on a face of the BZ (Bragg reflection). At a general point in the BZ the wave vector k + bm cannot be distinguished from the equivalent wave vector k, and consequently... 

As mentioned above, the XSW field arises from the interference between the coherently related incident and Bragg-diffracted beams from the surface of a perfect crystal. In the vicinity of a Bragg reflection (Fig. 24A-B), an incident plane wave (with wave vector fco) and a reflected wave (with wave vector kfi) interfere to generate a standing wave with a periodicity equivalent to that of the (h, k, 1) diffracting planes. The ratio of the electric field amplitudes of the reflected and incident waves is given by... 

Fig. 4 A Schematic cross section of metal film growth and corresponding scanning electron micrographs (below) of the gold nanocavities fabricated with a = 350 nm latex spheres of thickness t for (a) ajl, (b) a, and (c) 2.1a [91]. B Measured energy dispersion of the reflectivity for TM polarized light as a function of the in-plane wave vector for increasing relative void depth, i=t/(2a) (a-c). Log color scale white dotted lines show a zone-folded plasmon dispersion, sample orientations of 4> = 30° in all cases, (i-iv) k space cuts through dispersion relation at (i) (i,E) = (0.25,2.2 eV) (ii) (i,E) = (0.4,2.2 eV) (in) (i,E) = (0.4,1.7 eV) (iv) (f, ) = (0.6,2.2 eV), symmetry shown above (i). Light shade corresponds to absorption features [93]...

Figure 6.9a-f illustrates a variety of the accepted band structure representations for nearly-free electron model. The Figure introduces the repeated-zone, extended-zone and reduced-zone images. The original free-electron parabola E = fi k Klme) is shown in Figure 6.9a. To leading order in the weak one-dimension periodic potential this curve remains correct except the value of k near the reciprocal lattice vector g. One can imagine that in this point the Bragg plane reflects the electron wave since the Bragg condition holds. Another free-electron parabola is centered at fe = g, and two parabolas are crossed each other at the...

Fig. la. Schematic showing the optical field (magnetic component) at an interface which supports surface plasmons. The dielectric function in the dielectric medium is the diectric function in the metal can be approximated hy the Drude-Lorentz expression given in the upper right hand corner. Notice that the field extends much farther into the dielectric than the metal, b. The reflectivity in an ATR configuration. The 0 is the critical angle and 0gp is the angle at which the surface plasmon is excited. Reflectivity extends from zero to one. Notice that the reflectivity from s waves, i.e., those waves with their electric vector perpendicular to the plane of incidence do not excite a surface mode.. 

A plane wave incident on a surface at an oblique angle can in general be decomposed into two polarization components, which are typically referred to as the p and s waves, whose electric field vectors lie in and normal to, respectively, the plane of incidence. Upon reflection, these two waves may each undergo changes of amplitude and phase. In ellipsometric measurement it is the change in the relative amplitude and phase of the two components that is measured. That is to say, if we designate phases and anplitudes by p and A, respechvely, and denote p and s waves by the corresponding subscripts, then the two quantities to be determined are (Archer, 1962) ... 

For normal incidence, we have that 0, = 0 = 0 = 0. In this case the parallel or perpendicular orientations coincide and the plane-wave transmitted in the second medium is uniform. Considering the power (i.e. from the Po3mting vector) instead of the amplitude coefficients, we obtain the reflectance (R) and the transmittance (T)... 

The vectorial nature of the electromagnetic fields representing light implies polarization, a fundamental property of light. The particular orientation of the e-vector of a plane wave incident on an interface has a profound effect on the detailed interaction that takes place and influences, for example, the amount of light that is reflected or transmitted. If the electric field vector is confined to oscillate in a plane (e.g. the X- z plane in Figure 1), the wave is said to be linearly polarized. If the tip of the electric field vector traverses an elliptical path around the direction of... 

Figure 9.7 Definition of the plane of incidence (p plane) and the incidence angle 4>o through the wave vectors of the incident and emerging (reflection set up) plane waves. Ap, A, Bp, and 6, denote the complex amplitudes of the p and s modes...

If E,o is perpendicular to the plane of incidence containing the k,- vector and the normal to the boundary, the polarization state is called s-polarized. The symmetry of the problem with respect to this plane indicates that the same polarization is obtained for the amplitudes of the reflected and refracted waves. With the y-axis along the E,-o vector, we can write the boundary conditions at z = 0 as... 

Fig. 11-2 The electric field vector is orthogonal to the ray, or local plane-wave direction. On the step-profile fiber, the direction of e for (a) a meridional ray is parallel to a fixed direction, and for (b) a skew ray it changes direction at each reflection. On the parabolic-profile fiber the direction of e changes continuously along the skew-ray path (c).

In terms of the x-component of the local wave vector of Eq. (35-27) and the WKB solutions of Eq. (35-29), the solutions of the scalar wave equation for the incident, reflected and transmitted local plane waves, respectively, are expressible as... 

Strictly monochromatic radiation propagating in a unique direction (e.g., from a point source) is never realized. A monochromatic wave implies a periodic process of infinite duration. Such waves do not exist, although the signal from a stable, singlemode laser provides a fair approximation. Ordinary incoherent radiation emitted and reflected from real atmospheres and surfaces consists of individual wave packets of finite length and duration a few meters and 10 seconds are typical values. Similarly, point sources are replaced by extended sources in practice. Radiation from such sources tends to be incoherent and covers a range of frequencies and directions. Thus, it is more convenient to work with a distribution of plane waves and their associated Poynting vectors. 

The direction of propagation, parallel to the wave vector k, and the unit vector n perpendicular to the plane of reflection P define the plane of incidence P (Figure 1). Both the electric field vectors, E, for the incident plane wave and E . for the reflected plane wave, can be decomposed into two independent perpendicular components, linearly polarized, respectively, in the plane of incidence, Ep (p-polarization), and perpendicular to the plane of incidence, E (5-polarization). For these two components, there is no change in the polarization state upon reflection, as seen by symmetry considerations, but a change in amplitude and phase results. 

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