Vector spherical harmonics

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

Several conventions have been used to define the VSH. We follow that of Barrera et al.. Given a scalar spherical harmonic Y_ℓm(θ, φ), we define three VSH:

with ${\displaystyle {\hat {\mathbf {r} }}}$ being the unit vector along the radial direction in spherical coordinates and r the vector along the radial direction with the same norm as the radius: (r, 0, 0). The radial factors are included to guarantee that the dimensions of the VSH are the same as those of the ordinary spherical harmonics and that the VSH do not depend on the radial spherical coordinate.

The interest of these new vector fields is to separate the radial dependence from the angular one when using spherical coordinates, so that a vector field admits a multipole expansion

The labels on the components reflect that ${\displaystyle E_{lm}^{r}}$ is the radial component of the vector field, while ${\displaystyle E_{lm}^{(1)}}$ and ${\displaystyle E_{lm}^{(2)}}$ are transverse components.

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