Scattered field

An exhaustive treatment of electromagnetic wave propagation in isotropic, chiral media has been given by Lakhtakia et al. [136]. This analysis deals with the conservation of energy and momentum, properties of the infinite-medium Green s function and the mathematical expression of Huygens s principle. 

In this section we consider the basic properties of the scattered field as they are determined by energy conservation and by the propagation properties of the fields in source-free regions. The results are presented for electromagnetic scattering by dielectric particles, which is modeled by the transmission boundary-value problem. To formulate the transmission boundary-value problem we consider a bounded domain Di (of class with boimdary S and exterior D, and denote by n the unit normal vector to S directed into Ds (Fig. 1.8). The relative permittivity and relative permeability of the domain Dt are et and /it, where t = s,i, and the wave number in the domain Dt is kt = where ko is the wave number in free space. The imbounded 

Given E, as an entire solution to the Maxwell equations representing the external excitation, find the vector fields Es,Hs C Ds) n C Ds) and Ei, Hi e C Di) n C Di) satisfying the Maxwell equations 

It should be emphasized that for the assumed smoothness conditions, the transmission boundary-value problem possesses an unique solution [177]. 

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. 

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To express the fact that the scattered field is induced, we let E, = Tfi, where T is in general complex, T = T0e. Now we allow the dipole to oscillate as... 

Figure 8.18. The phase of the scattered field, evaluated at the position of the source dipole, as a function of free-space wavenumber (cm-1).

Here subscripts s and 1 refer to the scattered field and internal field, respectively. 

Let us write the expansions of the internal electric vector E and the scattered field electric vector E, in the forms... 

The scattered energy is obtained from the scattered fields, Eq. (63), and the associated magnetic vector. 

Figure 1.4 The total scattered field at P is the resultant of all the wavelets scattered by the regions into which the particle is subdivided. 

The Inverse Problem. By a suitable analysis of the scattered field, describe the particle or particles that are responsible for the scattering. This is the hard problem it consists of describing a dragon from an examination of its tracks (Fig. 1.56). 

Figure 3.1 The incident field (E,H,) gives rise to a field (Ei,Hi) inside the particle and a scattered field (EJ,HJ) in the medium surrounding the particle.

The basis vector is parallel and is perpendicular to the scattering plane. Note, however, that Es and E, are specified relative to different sets of basis vectors. Because of the linearity of the boundary conditions (3.7) the amplitude of the field scattered by an arbitrary particle is a linear function of the amplitude of the incident field. The relation between incident and scattered fields is conveniently written in matrix form... 

S, the Poynting vector associated with the incident wave, is independent of position if the medium is nonabsorbing Ss is the Poynting vector of the scattered field and we may interpret Sext as the term that arises because of interaction between the incident and scattered waves. 

The boundary conditions (4.39), the orthogonality of the vector harmonics, and the form of the expansion of the incident field dictate the form of the expansions for the scattered field and the field inside the sphere the coefficients in these expansions vanish for all m = = 1. Finiteness at the origin requires that we take y (kjr), where kj is the wave number in the sphere, as the appropriate spherical Bessel functions in the generating functions for the vector harmonics inside the sphere. Thus, the expansion of the field (Ej,H,) is... 

In the region outside the sphere jn and yn are well behaved therefore, the expansion of the scattered field involves both of these functions. However, it is convenient if we now switch our allegiance to the spherical Hankel functions h[]) and h% We can show that only one of these functions is required by considering the asymptotic expansions of the Hankel functions of order v for large values of p (Watson, 1958, p. 198) ... 

The first of these asymptotic expressions corresponds to an outgoing spherical wave the second corresponds to an incoming spherical wave. If, on physical grounds, the scattered field is to be an outgoing wave at large distances from the particle, then only should be used in the generating functions. When we consider the scattered field at large distances we shall also need the asymptotic expression for the derivative of h it follows from the identity... 

Note that the denominators of cn and bn are identical as are those of an and dn. If for a particular n the frequency (or radius) is such that one of these denominators is very small, the corresponding normal mode will dominate the scattered field. The a mode is dominant if the condition... 

Note that an and bn vanish as m approaches unity this is as it should be when the particle disappears, so does the scattered field. 

Although we considered only scattering of x-polarized light in the preceding section, the scattered field for arbitrary linearly polarized incident light, and... 

We assume that the series expansion (4.45) of the scattered field is uniformly convergent. Therefore, we can terminate the series after nc terms and the resulting error will be arbitrarily small for all kr if nc is sufficiently large. If, in addition, kr n, we may substitute the asymptotic expressions (4.42) and (4.44) in the truncated series the resulting transverse components of the scattered electric field are... 

We showed in Section 2.3 that the real and imaginary parts of the electric susceptibility are connected by the dispersion relations (2.36) and (2.37). This followed as a consequence of the linear causal relation between the electric field and polarization together with the vanishing of x(<°) in the limit of infinite frequency to. We also stated that, in general, similar relations are expected to hold for any frequency-dependent function that connects an output with an input in a linear causal way. An example is the amplitude scattering matrix (4.75) the scattered field is linearly related to the incident field. Moreover, this relation must be causal the scattered field cannot precede in time the incident field that excited it. Therefore, the matrix elements should satisfy dispersion relations. In particular, this is true for the forward direction 6 = 0°. But 5(0°, to) does not have the required asymptotic behavior it is clear from the diffraction theory approximation (4.73) that for sufficiently large frequencies, 5(0°, to) is proportional to to2. Nevertheless, only minor fiddling with S makes it behave properly the function... 

The electromagnetic field (E H,) inside the sphere is obtained from (8.13) and the transformation (8.9). We saw in Chapter 4 that, for given , there are four unknown coefficients in the expansions for the fields when the sphere is nonactive optical activity doubles the number of coefficients, which are determined by applying the conditions (4.39) at the boundary between sphere and surrounding medium and solving the resulting system of eight linear equations. We are interested primarily in the coefficients of the scattered field ... 

If the fields in (3.12) are transformed according to (8.14) and (8.15), the relation between incident and scattered fields becomes... 

Therefore, if the scattered field (ES,HS) is to be an outgoing wave at large distances from the cylinder, the generating functions in the expansions... 

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