Plane-wave expansion explained

In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves:$$e^\; =\; \backslash sum\_^\backslash infty\; (2\; \backslash ell\; +\; 1)\; i^\backslash ell\; j\_\backslash ell(k\; r)\; P\_\backslash ell(\backslash hat\; \backslash cdot\; \backslash hat),$$where

• is the imaginary unit,
• is a wave vector of length,
• is a position vector of length,
• are spherical Bessel functions,
• are Legendre polynomials, and
• the hat denotes the unit vector.

In the special case where is aligned with the z axis,$$e^\; =\; \backslash sum\_^\backslash infty\; (2\; \backslash ell\; +\; 1)\; i^\backslash ell\; j\_\backslash ell(k\; r)\; P\_\backslash ell(\backslash cos\; \backslash theta),$$where is the spherical polar angle of .

Expansion in spherical harmonics

With the spherical-harmonic addition theorem the equation can be rewritten as$$e^\; =\; 4\; \backslash pi\; \backslash sum\_^\backslash infty\; \backslash sum\_^\backslash ell\; i^\backslash ell\; j\_\backslash ell(k\; r)\; Y\_\backslash ell^m(\backslash hat)\; Y\_\backslash ell^(\backslash hat),$$where

• 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