Plane wave expansion


In physics, the plane wave expansion expresses a plane wave as a linear combination of spherical waves,
where
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
Note that the complex conjugation can be interchanged between the two spherical harmonics due to symmetry.

Applications

The plane wave expansion is applied in