Transmission boundary-value problem

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... 

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

The transmission boundary-value problem for homogeneous and isotropic particles has been formulated in Sect. 1.4 but we mention it in order for our analysis to be complete. We consider an homogeneous, isotropic particle occupying a domain D with boundary S and exterior (Fig. 2.1). The imit normal vector to S directed into is denoted by n. The exterior domain Ds is assumed to be homogeneous, isotropic, and nonabsorbing, and if t and jM. are the relative permittivity and permeability of the domain Ht, where t = s, i, we have s > 0 and ps > 0. The wave number in the domain Dt is kt = ko, /etPt, where ko is the wave number in the free space. The transmission boundary-value problem for a homogeneous and isotropic particle has the following formulation. 

The null-field method leads to a nonsingular integral equation of the first kind. However, in the framework of the surface integral equation method, the transmission boundary-value problem can be reduced to a pair of singular integral equations of the second kind [97]. These equations are formulated in terms of two surface fields which are treated as independent unknowns. In order to elucidate the difference between the null-field method and the surface integral equation method we follow the analysis of Martin and Ola [155] and review the basic boundary integral equations for the transmission boundary-value problem. We consider the vector potential Aa with density a... 

The problem of scattering by isotropic, chiral spheres has been treated by Bohren [16], and Bohren and Huffman [17] using rigorous electromagnetic field-theoretical calculations, while the analysis of nonspherical, isotropic, chiral particles has been rendered by Lakhtakia et al. [135]. To accoimt for chirality, the surface fields have been approximated by left- and right-circularly polarized fields and the same technique is employed in our analysis. The transmission boundary-value problem for a homogeneous and isotropic, chiral particle has the following formulation. 

The solution of the transmission boundary-value problem for each particle. 

The considerations in the preceding section make it worthwhile to discuss reflection and transmission at plane boundaries first, one plane boundary separating infinite media, then in the next section two successive plane boundaries forming a slab. In addition to providing useful results for bulk materials, these relatively simple boundary-value problems illustrate methods used in more complicated small-particle problems. Also, the optical properties of slabs often will be compared to those of small particles—both similarities and differences—to develop intuitive thinking about particles by way of the more familiar properties of bulk matter. 

The reflection and transmission properties of multiple layers of materials with different refractive indices can be treated either as a ray tracing or as a boundary value problem (e.g., Wolter, 1956 Bom Wolf, 1975). The ray tracing method leads to summations where it is sometimes difficult to follow the phase relations, especially if several layers are to be treated. We follow closely the boundary value method reviewed by Wolter (1956). In effect this method is a generalization of the one-interface boundary problem that led to the formulation of the Fresnel equations in Section 1.6. 

The uitrasonic properties of emuisions are usuaiiy frequency dependent, as are the effects of diffraction and transmission-refiection at muitiiayer boundaries. Thus, the frequency content of a US pulse used in an experiment is important. If a pulse contains a wide range of frequencies, then each frequency component will travel through a material at a different velocity and will be attenuated to a different extent, so only average values will be measured. This problem can be overcome by conducting experiments at a particular frequency or by using Fourier analysis to examine the frequency content of broad-band pulses. 

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