Plane wave expansion

In physics, the plane wave expansion expresses a plane wave as a sum of spherical waves,

$e^{i \mathbf k \cdot \mathbf r} = \sum_{\ell = 0}^\infty (2 \ell + 1) i^\ell j_\ell(k r) P_\ell(\hat{\mathbf k} \cdot \hat{\mathbf r})$,

where

• i is the imaginary unit,
• k is a 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,

$e^{i k r \cos \theta} = \sum_{\ell = 0}^\infty (2 \ell + 1) i^\ell j_\ell(k r) P_\ell(\cos \theta)$,

where θ is the spherical polar angle of r.

Expansion in spherical harmonics

With the spherical harmonic addition theorem the equation can be rewritten as

$e^{i \mathbf{k} \cdot \mathbf{r}} = 4 \pi \sum_{\ell = 0}^\infty \sum_{m = -\ell}^\ell i^\ell j_\ell(k r) Y_\ell^m{}^*(\hat{\mathbf k}) Y_\ell^m(\hat{\mathbf r})$,

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.

Applications

The plane wave expansion is applied in

• Acoustics
• Optics
• Quantum scattering theory

See also

• Helmholtz equation
• Plane wave expansion method in computational electromagnetism

References

• Digital Library of Mathematical Functions, Equation 10.60.7, National Institute of Standards and Technology
• Rami Mehrem, The Plane Wave Expansion, Infinite Integrals and Identities Involving Spherical Bessel Functions