Expansion of a Plane Wave in Vector Spherical Harmonics

2 EXPANSION OF A PLANE WAVE IN VECTOR SPHERICAL HARMONICS 

Expansion of a plane wave in vector spherical harmonics is a lengthy, although straightforward, procedure. In this section we outline how one goes about determining the coefficients in such an expansion. 

The problem with which we are concerned is scattering of a plane x-polarized wave, written in spherical polar coordinates as 

Because sin m f is orthogonal to cos m j for all m and m it follows that Memn and MOWJ/J are orthogonal in the sense that 

Similarly, (Now ,NewM), (Mom ,Now ) and (M N J are mutually orthogonal sets of functions. The orthogonality properties of cos m and sin m f imply that all vector harmonics of different order m are mutually orthogonal. 

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