Plane-wave expansion
In physics, the plane-wave expansion or Rayleigh expansion expresses a plane wave as a linear combination of spherical waves: e i k ⋅ r = ∑ ℓ = 0 ∞ ( 2 ℓ + 1 ) i ℓ j ℓ ( k r ) P ℓ ( k ^ ⋅ r ^ ) , {\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, 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, e i k r cos θ = ∑ ℓ = 0 ∞ ( 2 ℓ + 1 ) i ℓ j ℓ ( k r ) P ℓ ( cos θ ) , {\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.