Multipole expansions and plane wave representations of the electromagnetic field

Journal of Mathematical Physics

Volumn 15, Issue 2, 1973, Pages 234-244

  Devaney, A J ^a   Wolf, E ^a  

^a UNIVERSITY OF ROCHESTER   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 36849104619     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1666629     Document Type: Article

References (42)

1. For a very readable account of the early studies of scattering of plane waves by spheres see
2. See also, a note by the same author published
3. For a detailed list of references to papers on multipole expansions up to 1953 see
4. For a discussion of this representation see, for example
5. The relationship between homogeneous plane wave expansions and angular spectrum representations is discussed, for certain classes of scalar wave fields, in
6. The customary term “multipole field” for the field [formula omitted] defined by Eq. (2.4) is rather inappropriate since this field is everywhere well-behaved. This terminology is undoubtedly responsible for incorrect statements in the literature about the behavior of such a field at the origin. [See, for example, W. Heitler. The Quantum Theory of Radiation (Clarendon, Oxford, 1954), 3rd ed., statement following Eq. (9) on p. 402.]
7. We use the same definitions of the spherical harmonics and of the spherical Bessel and Hankel functions as those given in A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, third printing 1965), Vol. I, App. BII and BIV.
8. If the representation (3.4a), rather than (3.4), is employed one obtains, instead of (3.5), the following form of the angular spectrum representation, which is commonly employed in the literature: [formula omitted]where [formula omitted] is defined by (3.4b), and the positive signs are used when [formula omitted] and the negative signs when [formula omitted]
9. The effect of discarding all evanescent plane waves in the angular spectrum representation of the scalar dipole field [formula omitted] [see Eq. (3.14)] has been very clearly analyzed by
10. This result is, essentially, a three-dimensional analogue of a well-known theorem that the Fourier transform of a continuous function which vanishes outside a finite interval is a boundary value of an entire analytic function.
11. See, for example, the Appendix in
12. It is planned to present the corresponding analysis for the source-free, classical electromagnetic field in another publication. An analogous treatment of the quantized, free electromagnetic field has been given by
13. If, in addition to oscillating charges and current densities, there is a contribution from density of magnetization [formula omitted] one must replace [formula omitted] by [formula omitted] The introduction of magnetization makes it possible to take into account the effect of spin in the corresponding quantum mechanical formulation.
14. Conversely, it is not difficult to show that the E and H fields that are the solutions to Eqs. (4.1) and which behave at infinity as outgoing spherical waves satisfy the full set of Maxwell equations everywhere.
15. The angular spectrum representation (4.2) may, of course, be expressed in the vectorial generalization of the alternative form discussed in Footnote 21.
16. See, for example
17. This fact may readily be deduced from the customary form of the completeness theorem (discussed, for example, in Ref. 11, pp. 798–799), which may be stated as follows: An arbitrary, well-behaved vector field [formula omitted] may be expanded in terms of three types of vector spherical harmonics [formula omitted] [formula omitted] and [formula omitted] in the form [formula omitted] where [formula omitted] [formula omitted] and [formula omitted] are functions of the radial coordinate [formula omitted] only. (The vector spherical harmonics [formula omitted] [formula omitted] and [formula omitted] correspond to [formula omitted] [formula omitted] and [formula omitted] respectively, of Ref. 11).
18. By making use of elementary vector identities and the fact that [formula omitted] one can rewrite (4.12a) in the form [formula omitted] Now, [formula omitted] is the Fourier transform [formula omitted] of the transverse part [formula omitted] of the current distribution [cf.
19. 14.
20. The decomposition (5.3) of the spectral amplitudes [formula omitted] and [formula omitted] which we obtained as a consequence of the completeness of the vector spherical harmonics [formula omitted] and [formula omitted] with respect to all well behaved fields [formula omitted] that are orthogonal to s [i.e., such that [formula omitted]], may also be obtained as a direct consequence of the so-called Hodge’s decomposition theorem [See, for example
21. This identity follows from a well known general expression for vector spherical harmonics in terms of ordinary spherical harmonics. [See, for example, Eq. (1.5), p. 797 in Ref. 11].
22. See, for example