Optically thick media

Here, oR is the Raman cross section, a constant whose magnitude depends on the excitation and collection geometry. In optically thick media, as in a geometrically thin but optically dense tissue, a deviation from the linear Raman response of Is versus concentration N is to be expected. This can... 

J. T. Farmer and J. R. Howell, Hybrid Monte Carlo/Diffusion Method for Enhanced Solution of Radiative Transfer in Optically Thick Media, AIAA/ASME Heat Transfer Conf., Colorado Springs, June, 1994. 

From the computational point of view, the (o-transformation is most advantageous for multiple scattering in optically thick media with small tme absorption. In particular, for semi-infinite media, the simple transformation formulae, Eqs. (44) and (48), offer the possibihties for highly efficient computational schemes. The ( -transformation also may be useful for avoiding problems of convergence, which may occur for adding and doubling computational schemes. 

In the case of optically thick media, e.m. field propagation effects and beam absorption should be taken into account. In particular. for inhomogeneous1y-broadened gas media. pump absorption gets verv large-even complete-inside... 

P-n Radiation Model The P-n model (method of spherical harmonics) transforms the equations of radiative transfer into a set of partial differential equations that are simultaneously solved. Usually, low-order approximations, i.e. P-1, P-3 are used in which the angular dependence of the real radiation field is reduced to spherical functions. The low-order approximations are only accurate in optically thick media. 

Based on the energy conservation law, the total heat flux through a foam board equals the sum of the radiative heat flux and the conductive heat flux. Thermal radiation in the foam board can be described using an integro-differential equation which reduces to the diffusive equation for the optically thick media [8]. Thermal conduction of the foam board can be determined by the Fourier law. Thus, the total thermal conductivity of the foam board is composed of the radiative conductivity plus the thermal conductivity of the gaseous phase and polymer matrix. 

Fig. 3. Various situations encountered in ATR spectroscopy. Medium 1 represents the IRE. (a) Bulk rare (optically thin) medium 2. (b) Thin film with thickness d much less than the penetration depth dp. (c) General case with N layers of different optical properties and thickness. The electric field depicted schematically on the left decays exponentially into the rare medium. This situation applies for case (a). In the more general case (c), the electric field does not decay smoothly.

In the case of high liquid water clouds, the D(X) value can be as high as 10 as noted by Madronich [109,110] and co-workers. For example, they note that measurements of spectral ultraviolet-B irradiance under optically thick clouds show strongly enhanced attenuation by molecular and particulate absorbers and that the photon path is enhanced due to the presence of the highly scattering medium, leading to an amplification of absorption by chromophores. Using discrete ordinate and Monte Carlo model caculations, they [110] showed that photon paths (i.e., D(X)) in realistic water clouds could be enhanced by factors of 10 and more compared to cloudless sky. ... 

Pollack and Ackerman (1983) have reported the results of calculation with a one-dimensional radiative-convective model which predict the El Chichon cloud to have caused an increase of planetary albedo of 10%, a decrease in total radiation at the ground of 2-3%, and an increase in temperature of 3.5 degrees at the 30 mbar level. The GCM of the European Center for Medium Range Forecast was utilised to model the perturbation introduced by a fixed layer with an optical thickness of 0.15 added to the background a stratospheric warming of 3.5 C in the stratosphere and a cooling of about 0.1 C near the surface was obtained (Tanre and Geleyn, 1984). 

Figure 16.19 shows the spatial-frequency distributions of bit data recorded with focused laser beam and coherent optical transfer function (CTF) of reflection type confocal microscopeFigure 16.19a shows a spatial-frequency distribution of bit datum recorded in very thick medium. This distribution coincides with the spatial-frequency distribution of the focused light to record the bit datum, because the bit is recorded with the focused beam. It is assumed that the NA of the objective lens is given by n sin a and k =l ulk, where A denotes the wavelength. 

The emissivity of the gas media is a function of many parameters including gas pressure, temperature, partial pressures of radiatively participating species, and optical path length or characteristic dimension. Thus, if the concentration of the absorb-ing/emitting species is increased, the emissivity of the media increases as well. If the optical thickness of a medium tends to infinity, then the emissivity of such a medium tends to 1, which corresponds to the blackbody limit. At this limit, radiation becomes a totally diffusive process. 

At high temperature, radiation conduction can be the predominant mode of heat transfer, eg. in glassmaking more than 90% of the thermal transfer occurs by radiation conduction. The radiation conductivity can be calculated for an optically-thick sample if steady-state conditions apply and if it is assumed that the absorption coefficient of the medium, a, is independent of... 

Note that these coefficients are obtained over the entire wavelength range therefore, the spectral details are lost. For practical applications where the spectral structure of radiation field is important, the use of these mean coefficients is not recommended. The Planck coefficient yields acceptable results if the medium optical thickness is small (less than unity), whereas the Rosseland coefficient is more useful at large optical thicknesses (i.e., larger than five over the entire spectrum). 

Simplified but useful solutions can be obtained for some cases. For example, if the medium is optically thick at all important wavelengths, then the local radiative flux can be related to the gradient in the local medium emissive power, or... 

Part of the incident light will be reflected at the top and bottom surfaces of the antireflection film, and in both cases the reflection will take place in a medium of lower refractive index than the adjoining medium. Thus, to ensure that the relative phase shift is 180°, the optical thickness of the film should be made one quarter wavelength so that the total difference in phase between the two beams will correspond to twice one quarter wavelength, that is 180°. This is the phase condition ... 

Let us now return to the general transport equation (1) for polarized radiation in a plane medium. As a typical transfer problem, we consider the surface Greens function matrix G(t,m 0,), fiQ e[0,l], for a slab with an optical thickness b defined as the solution to the homogeneous transfer equation... 

For a medium of finite optical thickness b, the -transformation formulae become a bit more complex. On starting with the transfer equation Eq. (37) for the surface Green s function, we find a partial solution... 

In this section, we introduce a set of dimensionless quantities commonly used to characterize radiative transfer in specific configurations. These quantities are necessary for a detailed understanding of radiative transfer in photobioreactors. Depending on the value of the albedo a, the asymmetry parameter of the phase function, and the optical thickness of the medium, it is possible to conduct first analysis of the scattering regime and to identify the appropriate approximations. 

Let us consider a sample of an optically anisotropic medium. The sample is in the shape of an infinite slab of thickness d, where the dielectric tensor continuously rotates within the sample thickness, forming the angles 0(z), q>( z) with respect to a fixed reference system (x,y,z). The slab is parallel to the (x,y) plane (see Figure 1). 

---