Fresnel formulae

Here E and E are electric field amplitudes on the surface and in vacuo interrelated by the Fresnel formulae c0 is the velocity of light in vacuo x°jf (K,susceptibility tensor defined by the equation ... 

Since adsorbed molecules are exposed both to the incident and reflected light waves whose electric field amplitudes are interrelated by the Fresnel formulae, the vectors in s- and -polarizations appear as ... 

A simple and practical way to achieve the field enhancement is to use backside illumination of a dielectric plate, for instance a cover glass, in a standard DLW geometry with an oil-immersion focusing lens. According to the Fresnel formulas for the right angle incidence (0, = 0°), the coefficients of the in-plane ( ) polarized amphtudes of transmitted and reflected electric fields are, respectively ... 

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

Measurement of reflectances for incident light of various polarization states and two oblique angles of incidence the results are analyzed with the Fresnel formulas. Large angles are required for high accuracy, and this requires large sample surfaces. 

Up to this point we have only assumed that k n subject to this restriction and, of course, the assumption that geometrical optics combined with the Fresnel formulas is a good approximation, (7.1) is completely general. Let us further assume that the sphere is sufficiently weakly absorbing that 2 aa 1 with this assumption... 

Equations (1.74) to (1.77) are the Fresnel formulae. Examination of equation (1.77) reveals that light with p polarization will not be reflected from a plane interface (/ = 0) at a specific angle of incidence, Brewster angle and, for the case... 

To get the dielectric functions e(A), since the geometrical shape of the crystal is not perfectly known, we ought to select experimental conditions so as to have the simplest possible relation between e(k) and r(A), namely, the Fresnel formula for the normal reflection amplitude of a semi-infinite dielectric ... 

Ellipsometry measures the orientation of polarized light undergoing oblique reflection from a sample surface. Linearly polarized light, when reflected from a surface, will become elliptically polarized, because of presence of the thin layer of the boundary surface between two media. Dependence between optical constants of a layer and parameters of elliptically polarized light can be found on basis of the Fresnel formulas described above. 

Using these values in FresnelS formula, the amplitude of the reflected beam perpendicular to the plane of incidence (Ari.) is given by Equation (4),... 

Qo is the solar radiation at the top of the atmosphere and Tu = 0.95 is the transmissivity of the upper atmosphere. The albedo a is calculated from the Fresnel formula (Kondratyev, 1969). The quantities Tr and Ab are transmissivity and absorption in the atmospheric boundary layer. 

The well known solutions for every single boundary between two optical media, the Fresnel equations, are applied. The reflected portions of the waves are added with respect to amplitudes and phases. Only one way of calculating the reflectance of multilayers applying the Fresnel formula will be demonstrated here. 

Powder samples were deposited on glass slides and carefully flattened to obtain as smooth as possible surfaces. Die incident angle was usually ranged between 0.5 and 2°. The penetration depth calculated owing to Fresnel formula [6] was equal to 1.5 mm in pure alumina for an incident angle of 1°. X-ray spectra were recorded for 2q varying from 20 to 80° at a constant speed of 0.027s. 

For the calculations of the optical properties of polymer films with embedded nanoparticles, two routes can be selected. In the exact route, the extinction cross sections Cact(v) of single particles are calculated. The calculated extinction spectra for single particles—or, better, a summation of various excitation spectra for a particle assembly—can be compared with the experimental spectra of the embedded nanoparticles. In the statistic route, an effective dielectric function e(v) is calculated from the dielectric function of the metal e(T) and of the polymer material po(v) by using a mixing formula, the so-called effective medium theory. The optical extinction spectra calculated from the effective dielectric functions by using the Fresnel formulas can be compared with the experimental spectra. 

For the following calculation, experimentally determined dielectric functions for silver [30] and for a plasma polymer [31] were taken. The effective dielectric functions e(v) were calculated with the Maxwell Garnett theory for parallel-oriented particles, equation (13). From the effective dielectric function, transmission or extinction spectra can be calculated by using the Fresnel formulas [10] for the optical system air-composite media-quartz substrate. As a further parameter, the thickness of the film with embedded particles and the thickness of other present layers that do not contain metal nanoparticles have to be included. The calculated extinction spectra can be compared with the experimental spectra. 

Fig. 9.12 (a) Total reflection from the empty cell (b) reflection from the filled cell (c) reflectivity spectrum of the setup (dashed line) experimental values (solid line) calculated using Fresnel formulae for Bap2/D20/Au interface and the angle of incidence of 60°. Thin-cavity thickness was determined to be 2.4 pm. 

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°) ... 

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 radiation reflected from a two-layer system can be represented as a superposition of the components h, I2, , In, which arise at the interface of each different layer (Fig. 2.16). These components are functions of the optical constants and thicknesses of the upper and lower layers ( 2, fe, < 2 and 3, k, d ), the optical constants of the metal ( 4, 4), and the angle of incidence of the light, Fresnel formulas as described in Section 1.7. In the computations [62], values of the optical constants of the upper... 

Numerous experimental IR spectra of ultrathin films confirm the above theoretical interpretation (see Ref. [26] and literature therein). The jo-polarized emission spectra of a ZnSe film on the Si-Al BML substrate (Fig. 3.9) are given as an example [26]. As the Si underlayer thickness increases, the intensity of the ZnSe band near vtq (206 cm ) increases. In fact, according to the quasi-static interpretation, the dipole moment of mode 3 (mainly responsible for the band near vto) is directed along the x-axis, parallel to the interface. At small thicknesses, mode 3 will be quenched by the image in the metal (Section 1.8.2), but as the film thickness increases, this effect diminishes. Mode 2, whose band is near vlo (252 cm ) with polarization parallel to the surface, is unaffected by the presence of the underlayer. It is noteworthy that the experimental spectra agree so well with the spectral simulations from the Fresnel formulas. Another result, which agrees with the theoretical treatment, is the frequency dependence of the band near vto in spectra of the ZnSe layers on metals with different permittivities Sm... 

The most rigorous approach is to generate spectral simulations using the exact Fresnel formulas (Sections 1.5-1.7) however, this requires a knowledge of the optical constants of the film over the spectral range of interest. Special methods may be used to measure the optical parameters of thin films in their final form (Sections 3.10 and 3.11). To simplify the problem, differences between the optical... 

As shown in Chapter 2, the electromagnetic enhancement can be generated within a film that has a smooth metal under- or overlayer when there is a specific combination of the refractive indices of the various components of the system. This effect is described by the Fresnel formulas (Sections 1.5-1.7) when the surfaces of the metal film are assumed to be smooth [400, 401]. 

Figure 3.72. Dependence of reflectivity in IRRAS spectrum of film on metal on film thickness, calculated with (1) linear approximation and (2) exact Fresnel formulas = 75°, 02 = 0.5 - 0.12/, / 3 = 15 - 60/, V = 1000 cm" . Reprinted, by permission, from V. P. Tolstoy, Methods of UV-Vis and IR Spectroscx)py of Nanolayors, St. Petersburg University Press, St. Petersburg, 1998, p. 189, Fig. 5.20. Copyright St. Petersburg University Press.

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