Complex Polar Coordinates in Electromagnetics

Abstract

The definition of a complex polar coordinate system that unifies complex distances and complex angles together with their relations with the real observation space is presented in this paper. Its utility is shown through some applications to electromagnetic problems: (i) new solutions are obtained from well-known solutions in real polar coordinates, (ii) the parameterization of the real space in terms of the complex polar coordinates helps to understand the physical behaviour of those solutions, and (iii) the results provide a better physical insight of the complex distances and angles.

ELECTRIC AND MAGNETIC FIELD PROBLEMS WITH PERIODIC CIRCULAR CYLINDRICAL SYMMETRY AND THEIR CONNECTION WITH A NOVEL GEOMETRICAL INTERPRETATION OF THE ALGEBRAIC OPERATION aN±bN

“…The conclusions in (12) and (15) prove the assertion in (9). The proof presented here considered the case when the focus point is internal to the source points.…”

“…The results in (6) are absolutely general and valid for any values of a, b, and N . The conclusions in (6) were obtained resorting to simple trigonometric operations, however, we believe that, the same results could as well be derived by using complex polar coordinates, as in [15].…”

Angular spectral framework to test full corrections of paraxial solutions

“…This correction scheme was used in scattering problems with Gaussian beams, for instance [15]. Couture and Bélanger [8] pursued a correction scheme that could invert the paraxial approximation of the complex polar coordinates [16,17] in complex source point solutions of the Helmholtz equation [18][19][20][21][22]. Such solutions have been used to find solutions of scattering problems under beam incidence [23][24][25][26][27] and further studied in [12,13,[28][29][30][31][32][33][34][35][36][37]].…”

Complex Point Source for the 3d Laplace Operator

“…The analysis of the potential given in (5) is based on the complex distance properties, which may be found in [12][13][14][15][16]. We point out two useful results concerning complex distance.…”

“…Radiation and scattering problems have been constructed within the framework of these solutions, with special attention to paraxial approximations as Gaussian beams [6][7][8][9][10][11]. A more general description of the exact complex beams and the complex polar and spherical coordinates resulting from the complex extension of the coordinates has been studied in [12][13][14][15][16]. The aforementioned papers implied that there are some sources which may be formally expressed as a Dirac delta function of a complex argument, but omitting its analysis and meaning.…”