The Weyl expansion for the scalar and vector spherical wave functions

The Weyl expansion technique, also known as the angular spectrum expansion, expresses an outgoing spherical wave as a linear combination of plane waves. The scalar spherical waves are the solutions of the homogeneous Helmholtz equation and therefore have direct relation to the scalar multipole fields. This paper gives the Weyl expansion of multipole fields, scalar and vector, of any degree and order for spherical wave functions. The expressions are given in closed form for the scalar, psi & ell;m(tau)$$ {\psi}_{\mathit{\ell m}}^{\left(\tau \right)} $$, and vector, M & ell;m(tau),N & ell;m(tau)$$ {\mathbf{M}}_{\mathit{\ell m}}^{\left(\tau \right)},{\mathbf{N}}_{\mathit{\ell m}}^{\left(\tau \right)} $$, L & ell;m(tau)$$ {\mathbf{L}}_{\mathit{\ell m}}^{\left(\tau \right)} $$, multipole fields, evaluated across a plane orthogonal to any given direction. In the case of scalar spherical multipoles, the spherical gradient operator has been used, while for the vector spherical multipoles, the vector spherical wave operator has been constructed.