Complex Fresnel coefficients

Since p is a complex number, it may be expressed in terms of the amplitude factor tan P, and the phase factor exp jA or, more commonly, in terms of just P and A. Thus measurements of P and A are related to the properties of matter via Fresnel coefficients derived from the boundary conditions of electromagnetic theory. ... 

Information about the properties of the sample are contained in the complex ratio, p, of the Fresnel coefficients of reflection of the parallel (rp) and perpendicular (rg) incident plane polarized electrical field vectors. 

To handle the three-layer problem quantitatively, we write the z = 0 electric field components in terms of complex three-layer Fresnel coefficients ... 

Parameters e, ( = h2) are the dielectric constants of the respective media (which may be complex for light-absorbing materials). Parameters Tp s are the Fresnel coefficients for transmission through a stratified three-medium system with the beam incident from the medium 3 side and an intermediate medium 2 of thickness (5(l0) ... 

The Fresnel coefficients are complex numbers, describing the amplitude and phase changes of the p and s components. 

After reflection at the interface, the reflected wave is in general elliptically polarized. The shape and orientation of the ellipse depend on the incident polarization, incident angle and reflection properties of the interface, and can be described with the ratio p of the complex Fresnel s reflection coefficients for the p and s polarizations ... 

The ray tubes formed by the incident and reflected rays in Fig. 35-3 (a) have the same z-directed cross-section, and, since 4 is complex, the power density in each varies as 14 p, as is clear from Table 13-2, page 292. We deduce from Eqs. (35-11) and (35-18) that the transmission coefficient is given by 1 — B/A for 0. < 0 < n/2, leading to the Fresnel coefficient... 

To describe transmission and refraction we have Introduced the transmission and reflection coefficients called t and r, respectively. Generally these are complex quantities, i.e. they eure written as ( and r, but for non-adsorbing media and Fresnel surfaces they become real, and we recall from (1.7.10.6 and 7) that... 

In view of the experimental difficulties a theory for radiation properties is desirable. The classical theory of electromagnetic waves from J.C. Maxwell (1864), links the emissivity e x with the so-called optical constants of the material, the refractive index n and the extinction coefficient k, that can be combined into a complex refractive index n = n — ik. The optical constants depend on the temperature, the wavelength and electrical properties, in particular the electrical resistivity re of the material. In addition, the theory delivers, in the form of Fresnel s equations, an explicit dependence of the emissivity on the polar angle / , whilst no dependence on the circumferential angle ip appears, as isotropy has been assumed. 

Using a similar approach as in the case of the top antireflection coating above, and assuming normal incidence at the boundary between two media, for instance medium 1 (photoresist) and medium 2 (BARC), the reflection (rj2) and transmission (ti2) coefficients are given by the Fresnel equations above [Eqs. (9.5) and (9.6)]. Multiplication of the reflection amplitude with its complex conjugate yields... 

The reflectivity is calculated using the Fresnel reflection coefficient r, which is defined as 7 rp/ ip, where rp and Eip are the complex amplitudes of the electric vectors of the reflected and incident p-polarized light respectively. Values may also be calculated for = Ers/Eis, the reflection... 

In Figure 10.1, the electric vectors of the incident and reflected rays on the metal surface are shown as illustrated, the electric vector of the reflected ray for the s-wave always has a phase shift of about -180 from that of the incident ray (effectively creating a node at the metal surface), whereas the phase shift for the reflected p-wave depends on the angle of incidence as described below. The definition of the phase shift follows established practice [1,4-6]. In Figure 10.2, the amplitude of the reflection coefficients (p, p ) and phase shifts (Tj, Tp) calculated by the Fresnel equations for the radiation reflected from a metal surface are shown. The definition of the phase shifts in this figure is the same as those shown in Figure 10.1. For this calculation, the complex refractive-index values of gold (n = 15.9... 

The reflectivity of an interface is calculated from Fresnel s equations, which are derived from the boundary conditions that the tangential (t) component of the electric field vector, and the normal (n) component of the electric displacement vector, D = e - E , are continuous across the inter-face. We define the so-called Fresnel reflection coefficient r as the ratio of the complex amplitudes of the electric field vectors of the reflected and the incident waves ... 

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