Multipole expansions and plane wave representations of the electromagnetic field

A new and conceptually simple derivation is presented of the multipole expansion of an electromagnetic field that is generated by a localized, monochromatic charge‐current distribution. The derivation is obtained with the help of a generalized plane wave representation (known also as the angular spectrum representation) of the field. This representation contains both ordinary plane waves, and plane waves that decay exponentially in amplitude as the wave is propagated. The analysis reveals an intimate relationship between the generalized plane wave representation and the multipole expansion of the field and leads to a number of new results. In particular, new expressions are obtained for the electric and magnetic multipole moments in terms of certain components of the spatial Fourier transform of the transverse part of the current distribution. It is shown further that the electromagnetic field at all points outside a sphere that contains the charge‐current distribution is completely specified by the radiation pattern (i.e., by the field in the far zone). Explicit formulas are obtained for all the multipole moments in terms of the radiation pattern.

[1]  Fritz Rohrlich.  Classical charged particles , 1965 .

[2]  J. Goodman.  Introduction to Fourier optics , 1969 .

[3]  H. Casimir,et al.  On multipole expansions in the theory of electromagnetic radiation , 1954 .

[4]  R. B. Zipin,et al.  Principles of Atomic Spectra , 1968 .

[5]  A. Devaney,et al.  Plane-Wave Representations for Scalar Wave Fields , 1973 .

[6]  M. Saporoschenko.  Drift Velocities of O2+, CO2+, and CO4+ Ions in CO 2 Gas , 1973 .

[7]  W. Heitler,et al.  On the Radiation Emitted by a Multipole and its Angular Momentum , 1936, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  T. A. Bromwich.  The Scattering of Plane Electric Waves by Spheres , 1920 .

[9]  Emil Wolf,et al.  Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave—Part I† , 1962 .

[10]  William H. Carter,et al.  Bandlimited angular spectrum approximation to a scalar dipole field , 1970 .

[11]  M. Golay.  Note on the Possibility of a Parabolic Fabry-Perot Interferometer , 1962 .

[12]  A. Messiah.  Quantum Mechanics , 1961 .

[13]  A. Erdélyi.  Zur theorie der kugelwellen , 1937 .

[14]  A. Nisbet.  Source representations for Debye's electromagnetic potentials , 1955 .

[15]  P. Debye,et al.  Der Lichtdruck auf Kugeln von beliebigem Material , 2014 .

[16]  Edmund Taylor Whittaker,et al.  On the partial differential equations of mathematical physics , 1903 .

[17]  W. W. Hansen.  A New Type of Expansion in Radiation Problems , 1935 .

[18]  E. L. Hill.  The Theory of Vector Spherical Harmonics , 1954 .

[19]  P. Wallace.  THEORY OF MULTIPOLE RADIATIONS , 1951 .

[20]  A. Baños.  Dipole radiation in the presence of a conducting half-space , 1966 .

[21]  V. Weisskopf,et al.  Theoretical Nuclear Physics , 1953 .

[22]  W. Heitler,et al.  The quantum theory of radiation , 1936 .

[23]  G. Mie.  Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen , 1908 .

[24]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[25]  Georges de Rham,et al.  Les formes différentielles harmoniques , 1946 .

[26]  W. Franz.  Multipolstrahlung als Eigenwertproblem , 1950 .

[27]  E. A. Power.  Introductory Quantum Electrodynamics , 1964 .