Multipole expansion of the scalar wave equation

Consider the emission and scattering of electromagnetic radiation. This type of problem involves solving the vector wave equation. The solutions of this equation in free space are conveniently written as an expansion in orthogonal spherical waves. This expansion is known as the multipole expansion. Let us examine this expansion in more detail.

Before considering the vector wave equation, let us consider the somewhat simpler scalar wave equation. A scalar field $\psi({\bfm r}, t)$ satisfying the homogeneous wave equation

$$\nabla^2\psi - \frac{1}{c^2} \frac{\partial^2\psi}{\partial t^2} = 0$$ (1159)

can be Fourier analyzed in time

$$\psi({\bfm r}, t) = \int_{-\infty}^{\infty} \psi({\bfm r}, \omega)\, {\rm e}^{-{\rm i}\,\omega t} \,d\omega$$ (1160)

with each Fourier harmonic satisfying the Helmholtz wave equation

$$(\nabla^2 + k^2)\,\psi({\bfm r}, \omega) = 0,$$ (1161)

where $k^2=\omega^2/c^2$. We can write the Helmholtz equation in terms of spherical polar coordinates ($r$, $\theta$, $\varphi$):

$$\left[\frac{1}{r^2}\frac{\partial}{\partial r}\,r^2 \frac{\p... ...a}\frac{\partial^2 }{\partial \varphi^2}+k^2\right]\!\psi = 0.$$ (1162)

As is well known, it is possible to solve this equation via the separation of variables:

$$\psi({\bfm r}, \omega) = \sum_{l,m} f_{lm}(r)\, Y_{lm}(\theta,\varphi).$$ (1163)

Here, we restrict our attention to physical solutions which are well behaved in the angular variables $\theta$ and $\varphi$. The spherical harmonics $Y_{lm}(\theta,\varphi)$ satisfy the following equations:

$$-\frac{\partial^2 Y_{lm}}{\partial\varphi^2} = m^2\, Y_{lm},$$ (1164)

$$-\left[\frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \,\sin... ...eta}+ \frac{1}{\sin^2\theta}\frac{\partial^2 }{\partial \varphi^2}\right]Y_{lm} = l(l+1)\,Y_{lm},$$ (1165)

where $l$ is a non-negative integer, and $m$ is an integer which satisfies the inequality $\vert m\vert\leq l$. The radial functions $f_{lm}(r)$ satisfy

$$\left[\frac{d^2}{dr^2} + \frac{2}{r}\frac{d}{dr} + k^2 - \frac{l(l+1)}{r^2} \right] \!f_l(r) =0,$$ (1166)

where there is no dependence on $m$. With the substitution

$$f_l(r) = \frac{u_l(r)}{r^{1/2}},$$ (1167)

Eq. (7.7) is transformed into

$$\left[\frac{d^2}{dr^2} + \frac{1}{r}\frac{d}{dr} +k^2 - \frac{(l+1/2)^2}{r^2}\right] u_l(r) =0.$$ (1168)

It can be seen, by comparison with Eq. (5.39), that this is a type of Bessel's equation of half-integer order $l+1/2$. Thus, we can write the solution for $f_{lm}(r)$ as

$$f_{lm}(r) = \frac{A_{lm}}{r^{1/2}}\,J_{l+1/2}(kr) + \frac{B_{lm}}{r^{1/2}} \,Y_{l+1/2}(kr),$$ (1169)

where $A_{lm}$ and $B_{lm}$ are arbitrary constants. The half-integer order Bessel functions $J_{l+1/2}(z)$ and $Y_{l+1/2}(z)$ have analogous properties to the integer order Bessel functions $J_m(z)$ and $Y_m(z)$. In particular, the $J_{l+1/2}(z)$ are well behaved in the limit $\vert z\vert\rightarrow 0$, whereas the $Y_{l+1/2}(z)$ are badly behaved. The asymptotic expansions (5.43) remain valid when $m\rightarrow l+1/2$.

It is convenient to define the spherical Bessel functions $j_l(r)$ and $y_l(r)$, where

$$j_l(z) = \left(\frac{\pi}{2 z}\right)^{1/2} J_{l+1/2}(z),$$ (1170)

$$y_l(z) = \left(\frac{\pi}{2 z}\right)^{1/2} Y_{l+1/2}(z).$$ (1171)

It is also convenient to define the spherical Hankel functions

$$h_l^{(1,2)} (z) = j_l(z)\pm {\rm i}\,y_l(z).$$ (1172)

For real $z$, $h^{(2)}_l(z)$ is the complex conjugate of $h^{(1)}_l(z)$. It turns out that the spherical Bessel functions can be expressed in the closed form

$$j_l(z) = (-z)^l \left(\frac{1}{z}\frac{d}{dz}\right)^l \left(\frac{\sin z}{z}\right),$$ (1173)

$$y_l(z) = -(-z)^l \left(\frac{1}{z}\frac{d}{dz}\right)^l \left(\frac{\cos z} {z}\right).$$ (1174)

In the limit of small argument

$$j_l(z) \rightarrow \frac{z^l}{(2l+1)!!}\left[1+ O(z^2)\right],$$ (1175)

$$y_l(z) \rightarrow -\frac{(2l-1)!!}{z^{l+1}}\left[1+ O(z^2)\right],$$ (1176)

where $(2l+1)!! = (2l+1)(2l-1) (2l-3)\cdots 5\cdot 3\cdot 1$. In the limit of large argument

$$j_l(z) \rightarrow \frac{\sin(z-l\pi/2)}{z},$$ (1177)

$$y_l(z) \rightarrow -\frac{\cos(z-l\pi/2)}{z},$$ (1178)

and

$$h_l^{(1)} \rightarrow (-{\rm i})^{l+1}\, \frac{ {\rm e}^{\,{\rm i}\,z} }{z}.$$ (1179)

The inhomogeneous Helmholtz equation is conveniently solved using the Green's function $G_\omega({\bfm r}, {\bfm r}')$, which satisfies (see Eq. (2.109))

$$(\nabla^2 + k^2)\,G_\omega({\bfm r}, {\bfm r}') =-\delta({\bfm r}-{\bfm r}').$$ (1180)

The solution of this equation, subject to the Sommerfeld radiation condition, which ensures that sources radiate waves instead of absorbing them, is written (see Section 2.13)

$$G_\omega ({\bfm r}, {\bfm r}') =\frac{{\rm e}^{\,{\rm i}\,k\... ...bfm r} -\bfm{r}'\vert}} {4\pi \,\vert{\bfm r}-{\bfm r}'\vert}.$$ (1181)

The spherical harmonics satisfy the completeness relation

$$\sum_{l=0}^\infty \sum_{m=-l}^{l} Y^\ast_{lm}(\theta',\varph... ...) = \delta(\varphi-\varphi')\,\delta(\cos\theta -\cos\theta').$$ (1182)

Now the three dimensional delta function can be written

$$\delta({\bfm r}-{\bfm r}') = \frac{\delta(r-r')}{r^2}\, \delta(\varphi-\varphi')\, \delta(\cos\theta-\cos\theta').$$ (1183)

It follows that

$$\delta({\bfm r}-{\bfm r}') = \frac{\delta(r-r')}{r^2}\sum_{l... ...l}^{l} Y^\ast_{lm}(\theta',\varphi')\, Y_{lm}(\theta,\varphi).$$ (1184)

Let us expand the Green's function in the form

$$G_\omega({\bfm r}, {\bfm r}') =\sum_{l,m} g_l(r, r') \,Y_{lm}^\ast(\theta',\varphi') \,Y_{lm}(\theta,\varphi).$$ (1185)

Substitution of this expression into Eq. (7.17) yields

$$\left[\frac{d^2}{dr^2} +\frac{2}{r}\frac{d}{dr}+k^2 -\frac{l(l+1)}{r^2}\right] \!g_l = - \frac{\delta(r-r')}{r^2}.$$ (1186)

The appropriate boundary conditions are that $g_l$ is finite at the origin and corresponds to an outgoing wave at infinity (i.e., $g\propto {\rm e}^{\,{\rm i}\,kr}$ in the limit $r\rightarrow\infty$). The solution of the above equation which satisfies these boundary conditions is

$$g_l(r, r') = A \,j_l(kr_<)\, h_l^{(1)}(k r_>),$$ (1187)

where $r_<$ and $r_>$ are the greater and the lesser of $r$ and $r'$, respectively. The correct discontinuity in slope at $r=r'$ is assured if $A = {\rm i}\,k$, since

$$\frac{d h_l^{(1)}(z)}{dz}\, j_l (z) - h_l^{(1)}(z)\, \frac{d j_l(z)}{dz} = \frac{{\rm i}} {z^2}.$$ (1188)

Thus, the expansion of the Green's function is

$$\frac{{\rm e}^{\,{\rm i}\,k\vert{\bfm r} -\bfm{r}'\vert}} {4\... ...=-l}^l Y_{lm}^\ast(\theta',\varphi') \,Y_{lm}(\theta,\varphi).$$ (1189)

This is a particularly useful result, as we shall discover, since it easily allows us to express the general solution of the inhomogeneous wave equation as a multipole expansion.

It is well known in quantum mechanics that Eq. (7.6b) can be written in the form

$$L^2\, Y_{lm} = l(l+1)\,Y_{lm}.$$ (1190)

The differential operator $L^2$ is given by

$$L^2 = L_x^{~2} + L_y^{~2} + L_z^{~2},$$ (1191)

where

$${\bfm L} = -{\rm i}\,{\bfm r}\wedge \nabla$$ (1192)

is $1/\hbar$ times the orbital angular momentum operator of wave mechanics.

The components of ${\bfm L}$ can be conveniently written in the combinations

$$L_+ = L_x + {\rm i}\,L_y = {\rm e}^{\,{\rm i}\,\varphi} \left(\frac{\pa... ...\partial\theta} +{\rm i}\,\cot\theta\, \frac{\partial}{\partial\varphi}\right),$$ (1193)

$$L_- = L_x - {\rm i}\,L_y={\rm e}^{-{\rm i}\,\varphi} \left(-\frac{\part... ...\partial\theta} +{\rm i}\,\cot\theta\, \frac{\partial}{\partial\varphi}\right),$$ (1194)

$$L_z = -{\rm i}\, \frac{\partial}{\partial\varphi}.$$ (1195)

We note that ${\bfm L}$ operates only on angular variables and is independent of $r$. From the definition (7.29) it is evident that

$${\bfm r}\!\cdot\!{\bfm L} = 0$$ (1196)

holds as an operator equation. It is easily demonstrated from Eqs. (7.30) that

$$L^2 = - \frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\... ...}-\frac{1}{\sin^2\theta} \frac{\partial^2}{\partial\varphi^2}.$$ (1197)

The following results are well known in quantum mechanics:

$$L_+ \,Y_{lm} = \sqrt{(l-m)(l+m+1)}\,Y_{l,m+1},$$ (1198)

$$L_-\,Y_{lm} = \sqrt{(l+m)(l-m+1)}\,Y_{l,m-1},$$ (1199)

$$L_z\,Y_{lm} = m\,Y_{lm}.$$ (1200)

In addition,

$$L^2\,{\bfm L} = {\bfm L} \,L^2,$$ (1201)

$${\bfm L}\wedge{\bfm L} = {\rm i}\,{\bfm L},$$ (1202)

$$L_j \nabla^2 = \nabla^2 L_j,$$ (1203)

where

$$\nabla^2 = \frac{1}{r^2}\frac{\partial}{\partial r} \,r^2\frac{\partial}{\partial r} - \frac{L^2}{r^2}.$$ (1204)