Complex reflectance coefficients

A new one-dimensional mierowave imaging approaeh based on suecessive reeonstruetion of dielectrie interfaees is described. The reconstruction is obtained using the complex reflection coefficient data collected over some standard waveguide band. The problem is considered in terms of the optical path length to ensure better convergence of the iterative procedure. Then, the reverse coordinate transformation to the final profile is applied. The method is valid for highly contrasted discontinuous profiles and shows low sensitivity to the practical measurement error. Some numerical examples are presented. 

Consider the reflection of a normally incident time-harmonic electromagnetic wave from an inhomogeneous layered medium of unknown refractive index n(x). The complex reflection coefficient r(k,x) satisfies the Riccati nonlinear differential equation [2] ... 

Krishnaswamy. They emphasize the importance of a phase factor in the complex reflection coefficient of electrons from the surface, and propose that field ion energy distributions may be used to measure the phase of electron reflection at the crystal surface. However, neither is the potential barrier known accurately enough nor are the available experimental data good enough, and to this date no such information has been obtained. 

An acoustic wave which is traveling in a medium characterized by an impedance encounters a change of impedance Z at the face of the sample. This change causes the wave to be partially reflected with a complex reflection coefficient r, given by... 

Equation (2) can be combined with equation (1) to yield an expression for r, the complex reflection coefficient. 

In practice, we can use the impedance tube to find the complex coefficient of reflection and then vary the two propagation constants in the theory to produce the same complex reflection coefficient. These variations are not easy to perform as the equation is transcendental however, there are computer programs available to do this (7). 

Polystyrene Solutions 23—300 MHz. Moore et al. (85,86) obtained complex moduli data in the range of 23-300 MHz by measuring the complex reflection coefficient of shear waves reflected from a solid-liquid interface. The measurements were performed for solutions of polystyrene in dibutyl phthalate over the concentration range of 3-20 Vol.-% (86). The result is shown in Fig. 4.4 where (rf — v1 rjs)/ (t] - vt rjs) is plotted against f(t] — v1 f]s) M/cR T, vx being the volume fraction of the solvent and / the frequency in Hz. It is seen that the... 

In Fig. 1, we display the frequency dependence of the complex reflection coefficients, R(cd), for two different bend geometries. In particular, the reflection amplitude p(a>) of the roundish bend (b) vanishes at several resonance frequencies and we want to emphasize that at exactly these resonance frequencies, the phase of the reflection coefficient experiences a non-trivial discontinuity. The complex transmission coefficients T w) display an analogous behavior and, together with the reflection coefficients R(o)), completely determine the bends 5-matrix if we neglect the evanescent modes as discussed above. 

Infrared Imaging and Mapping for Biosensors, Fig. 3 (a) Principle of IR synchrotron mapping ellipsometry. Due to reflection at the sample, the polarization of the radiation is changed. For example, the incident linearly polarized radiation becomes elliptically polarized. The sample Is moved by a two-dimensional mapping table, and a spectrum Is taken for every probed spot. The elllpsometiic parameters defined by the quantity p, which is the ratio of the complex reflection coefficients Tp and r, are measured for every spot. A is the phase shift... 

The first and most direct is shown in Fig. D.12. Here all elements are fed via a harness as shown. The impedance Za of a typical element is detennined by insertion of an appropriate reflectometer or impedance measuring device. Neither of these devices should have any attenuation, and their electrical length must be incorporated in the feed cable. From the complex reflection coefficient F at the element terminal we find... 

The two-microphone impedance tube determines the ratio of absorption coefficient and complex reflection coefficient based on the transfer function between the two microphones. The integrated PULSE software that calculates the various data ensures that the sum of the two coefficients shall always conform to unity. The plots in Fig. 7.5b show that AG2 and AGS have the best absorption coefficients of 0.52 and 0.48, respectively, compared to the other sizes when measured at 15 mm thickness. The absorption coefficients lowered to 0.32 and 0.29, respectively, when the thickness is reduced to 10 mm as shown in Fig. 7.5a. Naturally it can be seen that mixed granules, AGMX, for which 70 % of the composition come from the two sizes mentioned, has an absorption coefficient that is close to both the 1.2 and 1.7 mm sizes. The porous ply shows negligible absorption similar to air. Table 7.2 lists the absorption coefficients abstracted from Fig. 7.5 at the center and maximum frequency. 

Eqn. (11) relates the quantities I and A with the reflectivity properties of the sample. The following two section discuss various ways to measure A and and the theory and algorithm used for a calculation of the complex reflectivity coefficient. Some ellipsometers are designed in a way that they measure directly the real and imaginary part of the complex quantity p instead of 4 and A. Ekjn. (11) allows a conversion between both notations. 

If the layer thickness h is much smaller than the wavelength A of light it is justified to expand the complex reflectivity coefficients in a power series in terms of h/X. The first... 

In ellipsometry, the quantity measured is the ratio (p) between the complex reflection coefficients with polarization parallel (Rp) and perpendicular (Rs) to the plane formed by the incident and reflected light beams. 

If the sample surface is isotropic and has no film or other overlayer (d = 0 in Figure 3), then one can use the Maxwell equations to calculate the complex reflection coefficients ... 

There are many different kinds of ellipsometers, some of which are shown schematically in Figure 4. All ellipsometers measure one or more quantities that can be related to the complex reflection coefficient ratio p shown in Equations [8]. 

Tp, Tj, etc. = complex reflection coefficients 5 = sin(2 /) sin(A) 5 = Stokes vector V = voltage, element of Stokes vector z = vector of fitted parameters (Eqn [14]) a = absorption coefficient 8c = constant phase shift A = ellipsometry angle 0 = azimuthal angle X = wavelength p = complex reflection ratio ( ) = angle of incidence ( )f, = complex... 

The orthogonal vibrational amplitudes tp and (complex reflection coefficients) for p- and s-polarized incident light are defined according to the expressions given by Equation 2.15 ... 

Here and rj denote the p- and s-polarized complex reflection coefficients. 

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