Spherical wave functions

The probability density of an electron with amplitude (wave function) / is /2. The s-type (spherical) wave functions, / for the first few principal quantum numbers (n = 1,2,3. ..) are ... 

Within the PWE approach, using the expansion of the bound state a) in terms of the free-electron spherical wave functions, we write the counterterms in the form ... 

For describing the photons with the angular set of quantum numbers jlM the vector spherical functions are introduced. The three-component photon spin wave function can be considered as the three-dimensional vector x. Then the spherical wave function YjiM(f) in momentum space is defined as ... 

S. Stein, Addition theorems for spherical wave functions. Q. Appl. Math. 19(1), 15-24 (1961) R.T. Wang, B.A.S. Gustafson, Angular scattering and polarization by randomly oriented dumbbells and chains of spheres, in Proceedings of the 1983 Scientific Conference on Obscuration and Aerosol Research, eds. by J. Farmer, R. Kohl (U.S. Army Aberdeen, Md.,... 

The derivation of the transition matrix in the framework of the null-field method requires the expansion of the incident field in terms of (localized) vector spherical wave functions. This expansion must be provided in the particle coordinate system, where in general, the particle coordinate system Oxyz is obtained by rotating the global coordinate system OXYZ through the Euler angles ap, j3p and 7p (Fig. 1.5). In our analysis, vector plane waves and Gaussian beams are considered as external excitations. 

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... 

As a result, we obtain the familiar expansions of the electromagnetic fields in terms of vector spherical wave functions of the interior wave equation ... 

Although the derivation of X and Y differs from that of Kiselev et al. [119], the resulting systems of vector functions are identical except for a multiplicative constant. Accordingly to Kiselev et al. [119], this system of vector functions will be referred to as the system of vector quasi-spherical wave functions. In (1.43)-(1.46) the integration over a can be analytically performed by using the relations... 

In the above analysis, and Y are expressed in the principal coordinate system, but in general, it is necessary to transform these vector functions from the principal coordinate system to the particle coordinate system through a rotation. The vector quasi-spherical wave functions can also be defined for biaxial media (e 7 y z) by considering the expansion of the tangential vector function T>c (3,a)Va + T>is j3, a)vjj in terms of vector spherical harmonics. 

If the transition matrix is known, the scattering characteristics (introduced in Sect. 1.4) can be readily computed. Taking into account the asymptotic behavior of the vector spherical wave functions we see that the far-field pattern can be expressed in terms of the elements of the transition matrix by the relation... 

Taking into account the orthogonality relations of the vector spherical wave functions on a spherical surface (cf. (B.18) and (B.19)) we obtain... 

Assuming the T-matrix equations s = Te and s = Te our task is to express the transition matrix in the global coordinate system T in terms of the transition matrix in the particle coordinate system T. Defining the augmented vectors of spherical wave functions in each coordinate sjretem... 

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 conventional derivation of the T matrix relies on the approximation of the surface fields by the system of localized vector spherical wave functions. Although these wave functions appear to provide a good approximation to the solution when the surface is not extremely aspherical, they are disadvantageous when this is not the case. The numerical instability of the T-matrix calculation arises because the elements of the matrix differ by many orders of magnitude and the inversion process is ill-conditioned. As a result, slow convergence or divergence occur. If instead of localized vector spherical... 

In the following analysis we summarize the basic concepts of the null-field method with distributed sources. The distributed vector spherical wave functions are defined as... 

TV) (hr ), respectively, while Q (kg, kg) contains as rows and columns the vectors M hr ), M (ksr ) and Mj (ksr ), Nj (ksr ), respectively. To compute the scattered field we proceed as in the case of locahzed sources. Application of the Huygens principle yields the expansion of the scattered field in terms of localized vector spherical wave functions as in (2.15) and (2.16). Inserting (2.22) into (2.16) gives... 

The use of distributed vector spherical wave functions is most effective for axisymmetric particles because, in this case, the T matrix is diagonal with respect to the azimuthal indices. For elongated particles, the sources are distributed on the axis of rotation, while for flattened particles, the sources are distributed in the complex plane (which is the dual of the symmetry plane). 

The expressions of the distributed vector spherical wave functions with the origins located in the complex plane are given by (B.31) and (B.32). 

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... 

The expressions of the elements of the Qaiis matrix are similar but with M and 7V in place of M and iV, respectively. Using the properties of the vector quasi-spherical wave functions (cf. (1.47)) it is simple to show that for Siz = Si, the present approach leads to the T-matrix solution of an isotropic particle. 

In the present analysis we will derive the expression of the transition matrix by using the translation properties of the vector spherical wave functions. The completeness property of the vector spherical wave functions on two enclosing surfaces, which is essential in our analysis, is established in Appendix D. 

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