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Spherical wave expansion of a vector plane wave

In discussing the scattering or absorption of electromagnetic radiation by localized systems, it is useful to be able to express a plane electromagnetic wave as a superposition of spherical waves.

Consider, first of all, the expansion of a scalar plane wave as a set of scalar spherical waves. This expansion is conveniently obtained from the expansion (7.26) for the Green's function of the scalar Helmholtz equation. Let us take the limit of this equation. We can make the substitution on the left-hand-side, where is a unit vector pointing in the direction of . On the right-hand side, and . Furthermore, we can use the asymptotic form (7.16) for . Thus, we obtain

 (1312)

Canceling the factor on either side, and taking the complex conjugate, we get the following expansion for a scalar plane wave,
 (1313)

where is the wave vector with the spherical coordinates , , . The well known addition theorem for the spherical harmonics states that
 (1314)

where is the angle subtended between the vectors and . Consequently,
 (1315)

It follows from Eqs. (7.113) and (7.114) that
 (1316)

or
 (1317)

since
 (1318)

Let us now make an equivalent expansion for a circularly polarized plane wave incident along the -axis:

 (1319) (1320)

Since the plane wave is finite everywhere (including the origin), its multipole expansion (7.54) can only involve the well behaved radial eigenfunctions . Thus,
 (1321) (1322)

To determine the coefficients and we make use of a slight generalization of the standard orthogonality properties (7.53) of the vector spherical harmonics:
 (1323) (1324)

The first of these follows directly from Eq. (7.53a). The second follows from Eqs. (7.31), (7.53b), (7.59), and the identity
 (1325)

The coefficients and are obtained by taking the scalar product of Eqs. (7.120) with and integrating over all solid angle, making use of the orthogonality relations (7.121). This yields

 (1326) (1327)

Substitution of Eqs. (7.52) and (7.120a) into Eq. (7.123a) gives
 (1328)

where the operators are defined in Eqs. (7.30). Making use of Eqs. (7.33), the above expression reduces to
 (1329)

If the expansion (7.117) is substituted for , and use is made of the orthogonality properties of the spherical harmonics, then we obtain the result
 (1330)

It is clear from Eqs. (7.119b) and (7.123b) that
 (1331)

Thus, the general expansion of a circularly polarized plane wave takes the form
 (1332)

The expansion for a linearly polarized plane wave is easily obtained by taking the appropriate linear combination of the above two expansions.

Next: Mie scattering Up: The multipole expansion Previous: Radiation from a linear
Richard Fitzpatrick 2002-05-18