Plane-wave expansion

In physics, the plane-wave expansion or Rayleigh expansion expresses a plane wave as a linear combination of spherical waves: $${\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }=\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }j_{\ell }(kr)P_{\ell }({\hat {\mathbf {k} }}\cdot {\hat {\mathbf {r} }}),}$$ where

• i is the imaginary unit,
• k is a real or complex wave vector of length k,
• r is a position vector of length r,
• j_ℓ are spherical Bessel functions,
• P_ℓ are Legendre polynomials, and
• the hat ^ denotes the unit vector.

In the special case where k is aligned with the z axis, $${\displaystyle e^{ikr\cos \theta }=\sum _{\ell =0}^{\infty }(2\ell +1)i^{\ell }j_{\ell }(kr)P_{\ell }(\cos \theta ),}$$ where θ is the spherical polar angle of r.

For proof, expand ${\displaystyle e^{ikr\cos \theta }}$ in Legendre polynomials ${\displaystyle P_{l}(\cos \theta )}$, and evaluate the coeffient integrals.

With the spherical-harmonic addition theorem the equation can be rewritten as $${\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }=4\pi \sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }i^{\ell }j_{\ell }(kr)Y_{\ell }^{m}{}({\hat {\mathbf {k} }})Y_{\ell }^{m*}({\hat {\mathbf {r} }}),}$$ where

• Y_ℓ^m 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.

The plane wave expansion is applied in

• Acoustics
• Optics
• S-matrix
• Quantum mechanics