Plane-wave expansion
In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves: [math]\displaystyle{ 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}), }[/math] 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, [math]\displaystyle{ e^{i k r \cos \theta} = \sum_{\ell = 0}^\infty (2 \ell + 1) i^\ell j_\ell(k r) P_\ell(\cos \theta), }[/math] 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 [math]\displaystyle{ 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}), }[/math] 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
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
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