Optical theorem

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 

This result together with (1.82) and the identity Re. EeXff = Re ElxHs give 

In the integral representation for the electric far-held pattern (cf. (1.69)) we set r = Cfc, take the dot product between Esooi k) and EIq, and obtain 

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 

Using these dehnitions into Eq. (1.284) it is possible to show that [15]  

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. 

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The optical theorem relates the integral cross section to the unaginary part of the forward scattering amplitude by... 

The one-to-one correspondence of alloy and host sites is seen explicitly in Eq. (4). For the moment we now concentrate on the transition probability This quantity is proportional to the density of impurities and, according to the optical theorem, is given by... 

Since ft and /" represent an absorption of the propagating beam, they are related to the linear absorption coefficient p oS). This relation is called the optical theorem... 

From (3.39) and the optical theorem (3.24) it follows that the irradiance is attenuated according to /, = / exp( —aext/j) as the incident beam traverses the slab of particles, where the attenuation coefficient aext is... 

The first term in (4.71) is just the incident wave the second term is the wave scattered (diffracted) by the obstacle. The optical theorem for scalar waves is... 

From these relations, the optical theorem (4.76), and (2.51) it follows that... 

The optical theorem yields the absorption cross section... 

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 ... 

As a check on the amplitude scattering matrix elements, we compute Qcxt in BHMIE from the optical theorem (4.76), whereas Qsca is computed from the series (4.61). POL, the degree of polarization, must vanish for scattering angles of 0 and 180°, as must 34- Also, the 4x4 scattering matrix elements must satisfy... 

It is useful to consider rare gas scattering, for which a = a and ft = / . Using the optical theorem we can relate the imaginary part of the forward scattering amplitude to the total cross section.18 Explicitly, we can apply the optical theorem to the scattering amplitudes on both the left hand and right hand sides of Eq. (11.29). This procedure yields... 

The elastic scattering cross section for a perturber in state (i incident on a Rydberg atom in state a with momentum K is obtained from the optical theorem. Explicitly3... 

A more common means of calculating aT at intermediate and high energies is to use the optical theorem, which expresses the conservation of... 

This is the optical theorem, and it expresses the conservation of the number of particles in the scattering process. As already mentioned in section 2.2, it is valid even when inelastic processes can occur, although dei is then replaced by the total scattering cross section [Pg.96]

For 5 = 0 (limit point at infinity), one obtains the corresponding Coulomb modifications, see Refs. [36,42] for the complications at origin. Note also the strong dependence on the incident directions for "ellipsoidal" potentials yet the optical theorem holds, see Ref. [39]. 

The Optical Theorem Form Dichroism and Birefringence From Dilute Suspensions 71... 

Although the physical processes responsible for X-ray and neutron resonance scattering are vastly different a unified approach of resonance (or anomalous) scattering can be given on the basis of the famous optical theorem ... 

The optical theorem relates the total cross section, which includes both elastic and inelastic contributions, to the imaginary component of the scattering amplitude in the exact forward direction. It is given by... 

Some important implications of the optical theorem may be summarized as follows ... 

The optical theorem provides a method of determining the imaginary component of the scattering amplitude in forward direction from the experimentally obtained total scattering scattering cross Section. 

This important relationship is the optical theorem. The imaginary part of the forward scattering amplitude is proportional to the total cross section. 

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