Plane-wave expansion

In physics, the plane-wave expansion or Rayleigh expansion expresses a plane wave as a linear combination of spherical waves:

where

• is the imaginary unit,
• is a real or complex 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,

where is the spherical polar angle of .

For proof, expand in Legendre polynomials , and evaluate the coeffient integrals.

Expansion in spherical harmonics

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

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

References