Extinction theorem

There are also more limited treatments of scattering. McCartney (1976, Chaps. 4-6) confines his attention to scattering by atmospheric particles. This is also discussed by Twomey (1977, Chaps. 9-10) in his treatise on atmospheric aerosols. In Goody (1964, Chap. 7) there are discussions of absorption by gases and, in less detail, extinction by molecules and by droplets. Parts of books on electromagnetic theory or optics include the theory of scattering by a sphere, most notably Stratton (1941, pp. 563-573) and Born and Wolf (1965, pp. 633-664). The latter also derive the Ewald-Oseen extinction theorem and apply it to reflection and refraction at a plane interface (pp. 98-104). 

The above relation is a representation of the optical theorem, and since the extinction cross-section is in terms of the scattering amplitude in the forward direction, the optical theorem is also known as the extinction theorem or the forward scattering theorem. This fundamental relation can be used to compute the extinction cross-section when the imaginary part of the scattering amplitude in the forward direction is known accurately. In view of (1.88) and (1.74), and taking into account the explicit expressions of the elements of the extinction matrix we see that... 

Applications of the extinction theorem and Huygens principle yield the null-field equations (2.6) and the integral representations for the scattered field coefficients (2.16). Taking into account that the electromagnetic fields propagating in an isotropic, chiral medium can be expressed as a superposition of vector spherical wave functions of left- and right-handed type (cf. Sect. 1.3), we represent the approximate surface fields as... 

Substituting (2.177) and (2.182) into (2.176) gives two types of terms. One type of terms has a exp(jA s 2oi) dependence and corresponds to waves traveling with the wave number of the incident wave, while the other type of terms has a exp(jifg o/) dependence and corresponds to waves traveling with the wave number of the effective medium. The terms with wave number should balance each other giving the generalized Ewald-Oseen extinction theorem,... 

After computing the singular solution K, the elements of the s vector can be expressed in terms of an arbitrary constant, and this constant will be determined from the Ewald-Oseen extinction theorem. Certainly, this strategy tacitly assumes that the system of equations (2.185) reduces to a single scalar equation and to prove this assertion we introduce the notations... 

In previous chapters we have always taken particles to be in a nonabsorbing medium. We now briefly remove this restriction. The notion of extinction by particles in an absorbing medium is not devoid of controversy more than one interpretation is possible. But Bohren and Gilra (1979) showed that if the extinction cross section is interpreted as the reduction in area of a detector because of the presence of a particle [see Section 3.4, particularly the development leading up to (3.34)], then the optical theorem for a spherical particle in an absorbing medium is formally similar to that for a nonabsorbing medium ... 

Let us point out, at this stage, that real stars of course stop on their evolutionary tracks in the log P — log p-diagram for long time when they burn a nuclear fuel in their core. However, every fuel extincts at one point, and then the contraction of the core goes on It can only be stopped when the core becomes unstable, i.e. collapses or explodes — note that the virial theorem is not valid any more in this cases —, or when it hits Pauli s exclusion principle, i.e. when the core is extremely degenerate. 

This is based on the evaluation of the joint probability /7(Egr P), where Egj-denotes the extinction symbol, and P = (A, G... /n) are the correlated reflection intensities obtained from the linear least-squares Pawley refinement, when the most general extinction group of the crystal system under consideration is adopted. From Bayes s theorem ... 

Using Poynting s theorem, finally, after some mathematics, the extinction cross section Cext for the dipole is obtained as... 

The basic definition of an effective medium is that the ESU, when embedded in the effective medium, should not be detectable in an experiment using electromagnetic measurement. In other words, the extinction of the ESU should be the same as if it were replaced with a material characterized by Ceff. This criterion makes it fruitful to use a recently derived [12] optical theorem for absorbing media it relates the extinction of the spherical cell compared to that of the surrounding medium with the scattering amplitude in the direction of the impinging beam S(0) (forward scattering amplitude) by... 

The ecological interpretation of part (2) of the theorem lies in the fact that it may give an answer to questions like this can a reaction modelling an ecological system evolve towards an equilibrium population in which all species coexist, or, to put it another way, none of them become extinct Statements on coexistence have been formulated by several authors recently, as, for example, by Epstein (1979), by Goh (1975), or by Freedman So (1985). 

Equation (1.290) represents the optical theorem. This theorem is also know as the extinction or the forward scattering theorem, because it allows to compute the extinction cross section only in terms of the scattering dyad in the forward (i.e. the incident) direction. 

According to Poynting s theorem, the extinction spectra, Pexi (o), the scattering spectra, Pscaipi) and the absorption spectra, Pabs oi), are given respectively by ... 

Our presentation is focused on the analysis of the scattered field in the far-held region. We begin with a basic representation theorem for electromagnetic scattering and then introduce the primary quantities which dehne the single-scattering law the far-held patterns and the amplitude matrix. Because the measurement of the ampUtude matrix is a comphcated experimental problem, we characterize the scattering process by other measurable quantities as for instance the optical cross-sections and the phase and extinction matrices. 

The elements of the extinction matrix have the dimension of area and only seven components are independent. Equation (1-78) is an interpretation of the so-called optical theorem which will be discussed in the next section. This relation shows that the particle changes not only the total electromagnetic power received by a detector in the forward scattering direction, but also its state of polarization. 

The expression of extinction has been derived by integrating the Po3mting vector over an auxiliary surface around the particle. This derivation emphasized the conservation of energy aspect of extinction extinction is the combined effect of absorption and scattering. A second derivation emphasizes the interference aspect of extinction extinction is a result of the interference between the incident and forward scattered light [17]. Applying Green s second vector theorem to the vector fields Eg and El in the domain D bounded by S and Sc, we obtain... 

In view of the optical theorem, the orientation-averaged extinction cross-section is (cf. (1.88) with jEgoj = 1)... 

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