Plane-wave expansion

In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves:
where

is the imaginary unit,
is a wave vector of length,
is a position vector of length,
are spherical Bessel functions,
are Legendre polynomials, and
the hat denotes the unit vector.

In the special case where is aligned with the z axis,
where is the spherical polar angle of.

Expansion in spherical harmonics

With the spherical-harmonic addition theorem the equation can be rewritten as
where

are the spherical harmonics and
the superscript denotes complex conjugation.

Note that the complex conjugation can be interchanged between the two spherical harmonics due to symmetry.

Applications

The plane wave expansion is applied in