Efficient Recursive Generation of the Scalar Spherical Multipole Translation Matrix

Abstract

New efficient recursive procedures for generating the translation matrix of the scalar spherical multipole field are describd. They are based on a new set of recurrence relations that result when the angular-momentum operator is applied to the spherical multipole field. Their efficiency and accuracy are compared analytically and through a computer experiment with those of the brute-force method and an existing recursive procedure.

Theory of Near-Field Scanning with a Probe Array

“…Note that these are the only paths from excitation voltage to measured voltage that traverse a branch with large load impedance only once. 24 Approved for public release; distribution is unlimited. It is clear from Equations (28), (29) and (30), that if the Thévenin load impedances create open circuit conditions at the terminals of each probe array element, then the measured voltage, E m , will be proportional to only the plane wave induced voltage at the terminal with the modulated impedance.…”

“…Also, E pw i , i = 1, 2, 3, m is the open circuit voltage induced at each terminal due to the plane wave excitation of the array and measurement dipole. Each terminal of the probe array is loaded with a Thévenin equivalent source as discussed in connection with Equations (23) through (24) giving, from Equation (24)…”

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

“…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.…”

Translational transformations of tensor solutions of the Helmholtz equation and their application to describe interactions in force fields of various physical nature

“…Chew [13] presented an alternative derivation of the vector addition theorem using the completeness of vector wave functions and integration by parts. Mackowsky [14], Chew and Wang [15], and Kim [16], [17] rederived vector addition theorems and presented simple recurrence relations for the addition coefficients. Kim also discussed the symmetry relationships for the translation coefficients [18].…”