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Concept
Plane-wave expansion
Summary
In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves:
where
i is the imaginary unit,
k is a wave vector of length k,
r is a position vector of length r,
jl are spherical Bessel functions,
Pl are Legendre polynomials, and
the hat ^ denotes the unit vector.
In the special case where k is aligned with the z axis,
where θ is the spherical polar angle of r.
With the spherical-harmonic addition theorem the equation can be rewritten as
where
Ylm 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.
Official source
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