Fresnel reflection-transmission

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). 

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. 

Complex Index of Refraction of Soot. Soot refractive index has been measured by several researchers. The experimental techniques used can be broadly categorized as in situ and ex situ techniques. In the former, the measurements are performed nonintrusively in a flame environment. The necessary information is retrieved either from spectral transmission data or both the transmission and scattering information, as in Refs. 215-224. The ex situ measurements involve the reflection/transmission of incident spectral radiation on planar pellets of soot, and the optical properties are determined using the Fresnel relations [225]. An alternative ex situ technique was used by Janzen [226], who dispersed the soot particles in a KBr matrix and used transmission measurements to extract the required optical properties. 

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 ... 

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 ... 

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). 

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. 

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. 

Light striking a dielectric interface can undergo a number of processes, the most important of which are reflection, absorption, and scattering. In a spectroscopic experiment, the intensity of light incident on a surface, Iq, is compared to that which has been transmitted through the medium, T = Iq/I, scattered from the medium S = IJIq or reflected from the medium, where R = IJI. A material s optical parameters (i.e., reflectivity, transmissivity, and absorbtivity) are described by the Fresnel equations which define these properties in terms of the materials refractive index, n, where n = n, + ik or complex dielectric constant e = + ik. These two parameters are interrelated since n = Ve. Fresnel also... 

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). 

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. 

Ie is the true scattered intensity I e is the measured scattered intensity at angle 6, and Z i8o-e is at the supplementary angle. fa and fi are the Fresnel s coefficients for the fractions of light reflected at perpendicular incidence at the glass-air and glass-liquid interfaces, respectively ta and tx are the corresponding transmission coefficients. They are defined by the following equations ... 

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 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]. 

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... 

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