Plane-wave expansion

Short description: Expressing a plane wave as a combination of spherical waves

In physics, the plane-wave expansion or Rayleigh expansion expresses a plane wave as a linear combination of spherical waves: $$e^{i\mathbf{𝐤}\cdot \mathbf{𝐫}}=\sum _{\ell =0}^{\mathrm{\infty }}(2\ell +1)i^{\ell }{j}_{\ell }(kr)P_{\ell }(\widehat{\mathbf{𝐤}}\cdot \widehat{\mathbf{𝐫}}),$$ 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, $$e^{ikr\cos \theta }=\sum _{\ell =0}^{\mathrm{\infty }}(2\ell +1)i^{\ell }{j}_{\ell }(kr)P_{\ell }(\cos \theta ),$$ where θ is the spherical polar angle of r.

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

Expansion in spherical harmonics

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

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, http://dlmf.nist.gov/10.60.E7 
• Rami Mehrem (2009), The Plane Wave Expansion, Infinite Integrals and Identities Involving Spherical Bessel Functions, Bibcode: 2009arXiv0909.0494M