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} }}),}
- 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 z axis,
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 ),}
For proof, expand
e
i
k
r
cos
θ
{\displaystyle e^{ikr\cos \theta }}
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} }}),}
- 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
- Rami Mehrem (2009), The Plane Wave Expansion, Infinite Integrals and Identities Involving Spherical Bessel Functions, arXiv:0909.0494, Bibcode:2009arXiv0909.0494M