Plane-wave reflection

The double-crystal rocking curve is symmetric, though the plane wave reflectivity curve is not. This is a consequence of the autocorrelation, since the autocorrelation of any function is an even fimction. 

If the second crystal is the specimen rather than a beam conditioner element, we shall have got close to the aim of measiuing the plane wave reflectivity of a material. The narrow rocking curve peaks permit us to separate closely matched layer and substrate reflections and complex interference details, as already seen in Figure 1.6. The sensitivity limit depends on the thickness of the layer but for a 1 micrometre layer it is about 50 ppm in the 004 symmetric geometry with GaAs and CuK radiation. This method has been used extensively to study narrow crystal reflections since the invention of the technique. 

Thus we calculate the reflectivity of a whole layered material from the bottom up, using the amplitude ratio of the thick crystal as the input to the first lamella, the output of the first as the input to the second, and so on. At the top of the material the amplitude ratio is converted into intensity ratio. This calculation is repeated for each point on the rocking curve, corresponding to different deviations from the Bragg condition. This results in the plane wave reflectivity, appropriate for synchrotron radiation experiments and others with a highly collimated beam from the beam conditioner. 

Dipole scattering does not require an atomistic theory. A phenomenological theory suffices, which includes a response function dependent on dielectric constants. The cross-section for dipole scattering based on these assumptions is given in Eqs, 3.7 and 3.9 of Ibach and Mills./61/ These formulae include plane-wave reflection coefficients from the surface, which are solutions of the standard LEED problem. Since dipole scattering involves essentially only forward scattering, it is not necessary in practice to adopt the spherical-wave picture of our step 2 (cf. section 3.4.3), the plane-wave approach is adequate in this situation. 

Note that expression (14.64) resembles formula (14.53) for quasi-linear approximation. However, the reflection coefficient A now has a much clearer physical meaning. It has been demonstrated by Bleistein et al. (2001), that to leading order (high frequency asymptotics), this reflection coefficient is the same as that derived for plane waves reflected by planar interfaces in the media with a piecewise-constant distribution of the parameters. Note also that to leading order in the frequency w, the... 

Figure 3 Graphic illustration of Snell s law of refraction. (A) Refracted and reflected light beam at the refractive index boundary for n, > ri2. (B) The refraction of the plane wave. Reflected wave is not shown.

The fractional energy loss per transit due to diffraction of a plane wave reflected back and forth between the two plane mirrors is approximately given by... 

The reflection loss factor is computed by considering the electric, magnetic and plane wave reflection losses separately according to the following set of equations ... 

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]... 

If we substitute for W and from inside the back cover, the eigenvalue equations for TM modes in Table 12-2 show that U depends on both Fand A. The dependence on A is a manifestation of the polarization properties of the waveguide contained in the V, In terms in Eq. (12-2), or, equivalently, in the polarization-dependent property of plane-wave reflection from a planar interface, discussed in Section 11-16. The influence of waveguide polarization increases with A. 

Conversely, in the above example, the variation of the phase change with polarization of the plane wave electric field is small when the difference between and ng is small, although the incident wave is still totally reflected. Thus, if the refractive indices are nearly equal, the slight nonuniformity maintains total internal reflection, but the medium is virtually homogeneous as far as polarization effects are concerned. Further, the fields associated with plane-wave reflection satisfy the scalar wave equation, as discussed in Section 35-6. With this perspective, we anticipate that waveguides of arbitrary refractive-index profile have some analogous simplification in the description of their modal fields, provided only that the profile height parameter is small, i.e. A < 1, or n,. ... 

The relationships between E, H, and the incident beam fields Ej and H are provided by the Fresnel reflection coefficients for plane-wave reflection at a dielectric interface, discussed in Section 35-6. In general, these coefficients depend on the polarization of the beam, but, if 0j is small, they are independent of polarization and are given by Eq. (35-18) with d replaced by Ujand... 

Consider two semi-infinite media of refractive indices and < n , separated by the planar interface x = 0 in Fig. 35-3(a). A ray, or plane wave, is incident on the interface from the denser medium at angle 6 to the z-direction. Plane wave reflection in this situation is well-known and the power transmission coefficient of Eq. (35-11) is identical to the classical Fresnel coefficient [2]. 

In this section, two illustrative numerical results, obtained by means of the described reconstruction algorithm, are presented. Input data are calculated in the frequency range of 26 to 38 GHz using matrix formulas [8], describing the reflection of a normally incident plane wave from the multilayered half-space. 

Flere we model the pump beams associated with fields E(a> ) and (102) as plane waves with wavevectors Jti = and Jt, = feiwiv/fii (wil/r - The directions of tlie reflected and transmitted beams can... 

W L Bragg [7] observed that if a crystal was composed of copies of identical unit cells, it could then be divided in many ways into slabs with parallel, plane faces whose distributions of scattering matter were identical and that if the pathlengths travelled by waves reflected from successive, parallel planes differed by integral multiples of the... 

Figure 4 Interference pettern created when regularly spaced atoms scatter an incident plane wave. A spherical wave emanates from each atom diffracted beams form at the directions of constructive interference between these waves. The mirror reflection—the (00) beam—and the first- and second-order diffracted beams are shown.

Kijute 2.71 IIImttalion erf the X-ray standing wave held formed by the interference between the incident and reflected plane waves above a mirror surface t ee teat for detail ) After Bedivk ei cl (1990) Copyright 1990 by the AAAS... 

Figure 2.79 For 0 = 0. (a) The angular dependence of the reflectivity R and relative phase V of the reflected plane wave, (b) The angular dependence of the electric field intensity at =0 and 2 = 2DC for Ej = I. After Bedzyk et at. (1990) and M. J. Bedzyk, Synchrotron Radiation News. 3 (1990) 25, Copyright 1990 Gordon and Breach Science Publishers, S.A.

The more common type of spalling failure of concrete occurs when (and where) the transmitted compressive wave reflects from the free surface back face of the slab as a tensile wave, and the head of the reflected tensile wave and tail of the transmitted compressive wave combine to produce net tensile stress exceeding the dynamic tensile strength of the concrete. This process is shown schematically in Figure 21 for the simplified case of a plane, triangular compressive... 

Figure 1. Schematic view of the one-dimensional layered structure. The vertical lines denote discontinuities of refractive index distribution. The monochromatic plane wave is incident in the y direction with the electric field amplitude The amplitudes of reflected and transmitted waves are ref and, respectively.

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