Plane-wave expansion

Template:Short description In physics, the plane-wave expansion or Rayleigh expansion expresses a plane wave as a linear combination of spherical waves: $$e^{i𝐤\cdot 𝐫}=\sum _{\ell =0}^{\mathrm{\infty }}(2\ell +1)i^{\ell }{j}_{\ell }(kr)P_{\ell }(\widehat{𝐤}\cdot \widehat{𝐫}),$$ where

• Template:Math is the imaginary unit,
• Template:Math is a real or complex wave vector of length Template:Math,
• Template:Math is a position vector of length Template:Math,
• Template:Math are spherical Bessel functions,
• Template:Math are Legendre polynomials, and
• the hat Template:Math denotes the unit vector.

In the special case where Template:Math 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 Template:Math is the spherical polar angle of Template:Math.

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 $$e^{i𝐤\cdot 𝐫}=4\pi \sum _{\ell =0}^{\mathrm{\infty }}\sum _{m=-\ell }^{\ell }{i}^{\ell }{j}_{\ell }(kr)Y_{\ell }^m(\widehat{𝐤})Y_{\ell }^{m\ast }(\widehat{𝐫}),$$ where

• Template:Math are the spherical harmonics and
• the superscript Template:Math 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

• Template:Citation
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