Vector spherical wave functions regular

To solve the scattering problem in the framework of the null-field method it is necessary to approximate the internal field by a suitable system of vector functions. For isotropic particles, regular vector spherical wave functions of the interior wave equation are used for internal field approximations. In this section we derive new systems of vector functions for anisotropic and chiral particles by representing the electromagnetic fields (propagating in anisotropic... 

In (1.38)-(1,39), the electromagnetic fields are expressed in terms of the unknown scalar functions T>a and V/3, while in (1.41) and (1.42), the electromagnetic fields are expressed in terms of the unknown expansion coefficients Cmn and dmn These unknowns will be determined from the boundary conditions for each specific scattering problem. The vector functions and can be regarded as a generalization of the regular vector spherical wave functions and For isotropic media, we have eXfSfs = 1, = 0 and... 

Considering the general null-field equation (2.4), we restrict r to lie on a spherical surface enclosed in D expand the incident field and the dyad gl in terms of regular vector spherical wave functions (cf. (1.25), (B.21) and (B.22)), and use the orthogonality of the vector spherical wave functions on spherical surfaces to obtain... 

An approximate solution to the null-field equations can be obtained by approximating the surface fields e-, and h, by the complete set of regular vector spherical wave functions for the interior domain (or the interior wave... 

The surface fields ei i and hi i are the tangential components of the electric and magnetic fields in the domain bounded by the closed surfaces Sx and 52. Taking into account the completeness property of the system of regular and radiating vector spherical wave functions on two enclosing surfaces... 

The surface fields e i, h[ i and e 2, are the tangential components of the electric and magnetic fields in the domains D[ i and -Di,2, respectively, and the surface fields approximations can be expressed as linear combinations of regular vector spherical wave functions,... 

Finally, using the addition theorem for the regular vector spherical wave functions... 

Thus, the field exciting the /th particle can be expressed in terms of regular vector spherical wave functions centered at the origin O ... 

The surface fields ei i, /ti,i and e, 2, /ti,2 are approximated by finite expansions in terms of regular vector spherical wave functions as in (2.132) and (2.133) respectively. Inserting these expansions into the null-field equations (2.159) and (2.160), jdelds the system of matrix equations... 

The incident wave strikes the particle either directly or after interacting with the surface. The direct and the reflected incident fields are expanded in terms of regular vector spherical wave functions... 

The superscript T stands for the regular vector spherical wave functions while the superscript 3 stands for the radiating vector spherical wave functions. It is useful to note that for n = m = 0, we have mI q = = 0. Mi, ,... 

As in the scalar case, integral and series representations for the translation coefficients can be obtained by using the integral representations for the vector spherical wave functions. First we consider the case of regular vector spherical wave functions. Using the integral representation (B.26), the relation r = 0 + T" ) and the vector spherical wave expansion... 

The regular and radiating spherical vector wave functions can be expressed as integrals over vector spherical harmonics [26]... 

The following theorems state the completeness and linear independence of the system of regular and radiating spherical vector wave functions on two enclosing surfaces. 

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