Plane-wave expansion

Expressing a plane wave as a combination of spherical waves

In physics, the plane-wave 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 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.

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

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


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