Plane wave expansion

In physics, the plane wave expansion expresses a plane wave as a sum of spherical waves,

$e^{i\mathbf{k}\cdot\mathbf{r}} = e^{ikr\cos\theta}=\sum_{l=0}^\infty i^l (2l+1) j_l(kr)P_l(\cos\theta),$

where $i=\sqrt{-1}$. The wave vector $\mathbf{k}=(k_x,k_y,k_z)$ has length $k=|\mathbf{k}|$ and the vector $\mathbf{r}=(x,y,z)$ has length $r=|\mathbf{r}|$. The angle between the vectors $\mathbf{k}$ and $\mathbf{r}$ is $\theta$. The functions $j_l$ are Spherical Bessel functions and $P_l$ are Legendre polynomials.

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

$e^{i\mathbf{k}\cdot\mathbf{r}} = 4\pi\sum_{l=0}^\infty\sum_{m=-l}^l i^l j_l(kr) Y_{lm}(\theta_r,\phi_r)Y^\ast_{lm}(\theta_k,\phi_k),$

where $(r,\theta_r,\phi_r)$ and $(k,\theta_k,\phi_k)$ are the spherical coordinates of the vectors $\mathbf{r}$ and $\mathbf{k}$, respectively, and the functions $Y_{lm}$ are spherical harmonics.

Applications

The plane wave expansion is applied in

• Acoustics
• Optics
• Quantum scattering theory

See also

• Helmholtz equation

References

• Digital Library of Mathematical Functions, Equation 10.60.7, National Institute of Standards and Technology 
• Rami Mehrem, The Plane Wave Expansion, Infinite Integrals and Identities Involving Spherical Bessel Functions