Plane-wave expansion
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In physics, the plane-wave expansion or Rayleigh 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 real or complex 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 zaxis, 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.
For proof, expand e i k r cos θ {\displaystyle e^{ikr\cos \theta }} in Legendre polynomials P l ( cos θ ) {\displaystyle 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 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
See also
- Helmholtz equation
- Plane wave expansion method in computational electromagnetism
- Weyl expansion