Fresnel reflection and transmission

The reflectance (also called reflectivity) (R) and transmittance (T) are, respectively, given by the intensities of the reflected and transmitted radiation normalized by the intensity of the incident radiation. They are related to the Fresnel reflection and transmission coefficients in the following way ... 

The physics of RAM is well described by the Fresnel reflection and transmission equations. The material composite is driven by appHcation, and includes concerns such as weight, weather durability, flexibility, stability, life cycle, heat resistance, and cost. 

The exact solution for the unperturbed system can be obtained by the Parratt formalism.For a thin film sample consisting of three layers (layers j= 1 vacuum, 2 thin film, and 3 substrate), as shown in Figure 3(a), the refractive index rij of layer j isnj=l-Sj + ifSj with a dispersion Sj and an absorption pj. The Fresnel reflection and transmission coefficients for each sharp interface arer j+j = + and... 

Equations (2.67)-(2.70) are the Fresnel formulas for reflection and transmission of light obliquely incident on a plane boundary. 

In addition to the tensor element dependence of the sum-frequency intensity, there is also a dependence on the geometry of the experiment that manifests itself in the linear and non-linear Fresnel factors that describe the behaviour of the three light beams at the interface. Fresnel factors are the reflection and transmission coefficients for electromagnetic radiation at a boundary and depend on the frequency, polarization and incident angle of the electromagnetic waves and the indices of refraction for the media at the boundary [16,21]. 

The reflection and transmission (refraction) of light obliquely incident on the interface between two isotropic media is entirely controlled by the angle of incidence and the complex refractive indices of the media, being described by the Fresnel reflection and refraction equations (see Appendix). Originally worked out for transparent materials, these equations apply with complete generality when the refractive indices are complex rather than simple numbers. If the refractive indices are complex numbers, the angles of refraction must also be complex. For a description of the meaning of such quantities, see ref. 3. 

The Fresnel equations describe the reflection and transmission coefficients at the interface of two optical media. The polarisation of the incident Hght affects the magnitude of these coefficients. It is possible to derive expressions for the intensities of the reflected and refracted rays. These differ for the TE and TM polarisations as follows ... 

For nonabsorbing materials, the boundary conditions (1.4.7°) lead to the Fresnel formulas for the amplitude of reflection and transmission coefficients (1.4.5°) ... 

By using the elements of the M matrix, the Fresnel amplitude reflection and transmission coefficients (1.58) of the A -isotropic-phase medium are found from... 

Reflected and transmitted radiation from a powder layer can be either specular or diffuse (Fig. 1.22). The specular (Fresnel) component Isr reflected from the external boundary, which is comprised of all parts of the interface that have faces oriented in the direction of the averaged common interface. The magnitude of this component and its angular dependence can be determined by the Fresnel formulas (1.62). The specular (regular) transmission 7rt is the fraction of radiation that travels through the sample without any inclination. The other fractions of the radiation, the so-called diffuse reflection and transmission, /dr and /dt. respectively, are generated by the incoherent (independent) scattering and absorption by particles and do not satisfy the Fresnel formulas. 

The refractive index depends on the propagation of light as well as the reflected and transmitted fractions of incident waves on an interface. The Fresnel coefficients for reflection and transmission are given by the refractive indices of the two adjacent materials. For many applications of porous silicon in optics or optoelectronics, it is necessary to know the exact refractive index (see, e.g., chapters Porous Silicon Photonic Crystals, Porous Silicon Optical Waveguides, Porous... 

Reflection and transmission coefficients for multiple interference antireflection layers may be calculated in the following manner. For each y-th layer, we write the expression for Fresnel amplitude reflection coefiicient r for the next m j layers located behind it. This expression features the reflection at the surface of this layer rj, its phase qr, and the amplitude reflection coefiicient of all the remaining m-j-l layers. By writing successively in the same manner the expressions for the remaining layers, we obtain a recursive system with an infinite number of solutions... 

The theory of reflection and transmission of an electromagnetic wave by a plane boundary was first derived by Fresnel. The geometry of specular reflection and transmission is depicted in Figure 1. The incident (i) plane wave consists of the parallel ( ) and perpendicular polarized ( ) electric field components E, and E j., respectively. The corresponding components of the reflected (r) and refracted (transmitted t) field components are denoted by Ej. E j, E, and Fresnel s equations relate the reflected and transmitted components to the corresponding incident components. 

The reflection and transmission properties of multiple layers of materials with different refractive indices can be treated either as a ray tracing or as a boundary value problem (e.g., Wolter, 1956 Bom Wolf, 1975). The ray tracing method leads to summations where it is sometimes difficult to follow the phase relations, especially if several layers are to be treated. We follow closely the boundary value method reviewed by Wolter (1956). In effect this method is a generalization of the one-interface boundary problem that led to the formulation of the Fresnel equations in Section 1.6. 

The natural extension of this model is to consider a free-standing film, i.e., a thin transmitting sample not deposited on a substrate. In this case we have two interfaces (assumed to be flat) and transmission and reflection Fresnel coefficients at both interfaces (air/material and material/air). Even though it is not easy to produce such films, some examples are reported in the CP literature [13,14,26,27,32], Assuming that the medium is in vacuum (no = 2 = 1) with thickness d, it is easy to calculate the total reflectance R, and transmittance T of the sample as [21-23]... 

In spite of these problems, the Fresnel reflection-transmission technique is relatively easy to use, and, because of that, it has been employed by various researchers to determine the coal refractive index (see Refs. 227 and 228 for reviews). 

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... 

Fresnel equations relate the electric field strength amphtudes of the incident, reflected, and transmitted waves. They are solutions of Maxwells equations by applying the above-mentioned boundary conditions. It can be shown that for a plane boundary between two non-magnetic isotropic phases of infinite thickness, schematically depicted in Fig. 9.1, the Fresnel reflection (r) and transmission (t) coefficients for s- and p-polarized light are given by the following equations ... 

One can distinguish the surface and volume components in the diffuse transmission /dt and the diffuse reflection /dr (Fig. 1.22) [224-227]. The surface component, which is referred to as Fresnel diffiise reflectance, is the radiation undergoing mirrorlike reflection and still obeying the Fresnel reflection law but arising from randomly oriented faces. This phenomenon was first described by Lambert in 1760 [228] to account for the colors of opaque materials. The volume, or Kubelka-Munk (KM), component is the radiation transmitted through at least one particle or a bump on the surface (Fig. 1.22). 

Purely diffuse reflection occurs at microscopically irregular surfaces, while purely specular reflection occurs when the surface is perfectly smooth, like a minor. If the reflected radiance from a surface is completely uniform with angle of observation, it is called a Lambert surface. The BRDF for a Lambert surface is independent of both the direction of incidence and the direction of observation. Then, the reflectance simplifies to p(v, —h, = where pi is the Lambert reflectance. Specular reflection from and transmission throngh a smooth dielectric surface can be calculated from Snell s law and Fresnel s equations, given the optical constants of air and the dielectric material. 

When considering external attenuation affecting the transmission of light through optical fibres, one finds that, for short lengths of fibres, external attenuation normally is substantially larger than internal attenuation. At the entrance and exit face, both Fresnel reflection Lr and Fraunhofer diffraction Ld losses occur. 

Measurement of the Fresnel reflectance spectrum is a very useful way of obtaining the spectrum of solids with flat surfaces when sample preparation is not possible. For example, to measure the spectrum of an oriented polymer, the sample cannot be melted, dissolved, or finely ground. A microscopic sample of a hard polymer may be available that is too thick for transmission spectrometry. If the sample is so hard, rough, and/or thick that a good transmission or attenuated total reflection (see Chapter 15) spectrum cannot be measured, Fresnel reflection spectrometry presents a very useful means of obtaining the spectrum. 

Diffuse reflection (DR) spectra result from the radiation incident on a powdered sample that is absorbed as it refracts through each particle and is scattered by the combined process of reflection, refraction, and diffraction. That fraction of the incident radiation that reemerges from the upper surface of the sample is said to be diffusely reflected. Because DR spectra result from an absorption process, they have the appearance of transmission spectra (i.e., bands appear in absorption), unlike the case for Fresnel reflection spectra of bulk samples (see Chapter 13). When DR spectra are acquired on Fourier transform spectrometers, the singlebeam spectra of the sample and a nonabsorbing reference are measured separately and ratioed to produce the reflectance spectrum, Rfy). 

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