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. For proof, expand e i k r cos θ {\displaystyle e^{ikr\cos \theta }} in Legendre polynomials P l ( cos θ ) {\displaystyle P_{l}(\cos \theta )} , and evaluate the coeffient integrals.
Expansion in spherical harmonics With the spherical-harmonic addition theorem the equation can be rewritten as
e i k ⋅ r = 4 π ∑ ℓ = 0 ∞ ∑ m = − ℓ ℓ i ℓ j ℓ ( k r ) Y ℓ m
( k ^ ) Y ℓ m ∗ ( r ^ ) , {\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
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.
Applications The plane wave expansion is applied in
Acoustics Optics S-matrix Quantum mechanics
See also Helmholtz equation Plane wave expansion method in computational electromagnetism Weyl expansion
References Digital Library of Mathematical Functions, Equation 10.60.7, National Institute of Standards and Technology Rami Mehrem (2009), The Plane Wave Expansion, Infinite Integrals and Identities Involving Spherical Bessel Functions, arXiv:0909.0494, Bibcode:2009arXiv0909.0494M