Symmetry Relations of the Translation Coefficients of the Spherical Scalar and Vector Multipole Fields

Abstract

Abstract-We offer symmetry relations of the translation coefficients of the spherical scalar and vector multi-pole fields. These relations reduce the computational cost of evaluating and storing the translation coefficients and can be used to check the accuracy of their computed values. The symmetry relations investigated herein include not only those considered earlier for real wavenumbers by Peterson and Ström [9], but also the respective symmetries that arise when the translation vector is reflected about the xy-, yz-, and zx-planes. In addition, the symmetry relations presented in this paper are valid for complex wavenumbers and are given in a form suitable for exploitation in numerical applications.

Efficient Recursive Generation of the Scalar Spherical Multipole Translation Matrix

“…Therefore, it is necessary to develop a technique that is applicable to the near-field SSMTCs. [9] discusses the respective symmetry relations of SSMTCs and VSMTCs and they are shown to reduce the cost of generating and storing the translation coefficients of the scalar [5] and vector [8] multipole fields. While they produce a relatively modest reduction in comparison with the aforementioned diagonalization of the far-field SSMTCs, these symmetry relations can be applied to the near-field SSMTCs and their reduction factor is independent of the particle distribution.…”

Unified Derivation of the Translational Addition Theorems for the Spherical Scalar and Vector Wave Functions

“…The above reasoning, however, does not work in the vector case treated in the next section. So, for the sake of completeness and as an introduction to the vector case, let us derive (49) anew starting from (41) when the wave function on the left-hand-side is outgoing and the ones on the right-hand-side incoming, and t > r. We assume that a out-in l,m;n,p has a far field limit according to (18) and a radiation pattern denoted by (a out-in l,m;n,p ) ∞ . Then, by taking the limits on both sides, now with respect to t instead of r, we obtain…”

“…Consider first the out-to-out translation, in which case the wave functions on both sides of (41) are of an outgoing type and t < r. Then, by taking far field limits on both sides according to (18) and using (19), as explained in Sec. 3, the equation (41) reduces to…”

“…Kim [17] derived the theorems as well as recurrence formulas by applying similar spherical tensor technique as in [12] and [13]. He also discussed the symmetry relations of the coefficients of both the scalar and vector theorems in [18] and presented more efficient recurrence procedure for the coefficients of the scalar theorem in [19]. Quite recently, Chew presented a new derivation of the vector theorems [20], which is similar to his derivation of the scalar theorems [15], but still relies on some peculiar integral results.…”

Rapid Generation of the Scalar Spherical Multipole Translation Matrix using a New Set of Recurrence Relations

“…When one attempts to solve these equations on a computer, one computational bottleneck is to compute and store the multipole translation matrix. To reduce this computational burden, [3] discusses the respective symmetry relations of SSMTCs and VSMTCs. These symmetry relations are shown to reduce both the CPU and memory requirements of the scalar [4] and vector [5] T-matrix multiple scattering equation solution methods.…”