Multipole expansion of the vector wave equation

Maxwell's equations in free space reduce to

$$\nabla\!\cdot\!{\bfm E} = 0,$$ (1205)

$$\nabla \!\cdot\! c{\bfm B} = 0,$$ (1206)

$$\nabla\wedge{\bfm E} = {\rm i}\,k\,c{\bfm B},$$ (1207)

$$\nabla\wedge c{\bfm B} = -{\rm i}\,k \,{\bfm E},$$ (1208)

assuming an ${\rm e}^{-{\rm i}\,\omega t}$ time dependence of all field quantities. Here, $k=\omega/c$. Eliminating ${\bfm E}$ between Eqs. (7.36c) and (7.36d), we obtain the following equations for ${\bfm B}$:

$$(\nabla^2 + k^2) {\bfm B} = 0,$$ (1209)

$$\nabla\!\cdot\!{\bfm B} = 0,$$ (1210)

with ${\bfm E}$ given by

$${\bfm E} = \frac{\rm i}{k}\nabla\wedge c{\bfm B}.$$ (1211)

Alternatively, ${\bfm B}$ can be eliminated to give

$$(\nabla^2 + k^2) {\bfm E} = 0,$$ (1212)

$$\nabla\!\cdot\!{\bfm E} = 0,$$ (1213)

with ${\bfm B}$ given by

$$c{\bfm B} =- \frac{\rm i}{k}\nabla\wedge{\bfm E}.$$ (1214)

It is clear that each Cartesian component of ${\bfm B}$ and ${\bfm E}$ satisfies the Helmholtz wave equation (7.3). Hence, these components can be written in a general expansion of the form

$$\psi({\bfm r}) = \sum_{l,m}\left[A_{lm}^{(1)} \,h_l^{(1)}(kr) +A_{lm}^{(2)} \,h_l^{(2)}(kr)\right] Y_{lm}(\theta,\varphi),$$ (1215)

where $\psi$ stands for any Cartesian component of ${\bfm E}$ or $c{\bfm B}$. Note, however, that the three Cartesian components of ${\bfm E}$ or ${\bfm B}$ are not entirely independent, since they must also satisfy the constraints $\nabla\!\cdot\!{\bfm E} = 0$ and $\nabla\!\cdot\!{\bfm B} = 0$. Let us examine how these constraints can be satisfied with the minimum labour.

Consider the scalar ${\bfm r}\!\cdot\!{\bfm A}$, where ${\bfm A}$ is a well behaved vector field. It is easily verified that

$$\nabla^2({\bfm r}\!\cdot\!{\bfm A}) = {\bfm r}\!\cdot\!(\nabla^2{\bfm A}) + 2 \nabla\!\cdot\!{\bfm A}.$$ (1216)

It follows from Eqs. (7.37) and (7.39) that the scalars ${\bfm r}\!\cdot {\bfm E}$ and ${\bfm r}\!\cdot {\bfm B}$ both satisfy the Helmholtz wave equation:

$$(\nabla^2 + k^2)({\bfm r}\!\cdot\!{\bfm E}) = 0,$$ (1217)

$$(\nabla^2+k^2)({\bfm r}\!\cdot\!{\bfm B}) = 0.$$ (1218)

Thus, the general solutions for ${\bfm r}\!\cdot {\bfm E}$ and ${\bfm r}\!\cdot c{\bfm B}$ can be written in the form (7.41).

Let us define a magnetic multipole field of order $(l, m)$ by the conditions

$${\bfm r}\!\cdot\!c{\bfm B}_{lm}^{(M)} = \frac{l(l+1)}{k} \,g_l(kr) \,Y_{lm}(\theta, \varphi),$$ (1219)

$${\bfm r}\!\cdot\!{\bfm E}_{lm}^{(M)} = 0,$$ (1220)

where

$$g_l(kr) = A_l^{(1)} \,h_l^{(1)} (kr) + A_l^{(2)} \,h_l^{(2)}(kr).$$ (1221)

The presence of the factor $l(l+1)/k$ is for later convenience. Equation (7.40) yields

$$k\,{\bfm r}\!\cdot\!c{\bfm B} = -{\rm i}\,{\bfm r}\!\cdot\! ... ...r}\wedge\nabla)\!\cdot\! {\bfm E} = {\bfm L}\!\cdot\!{\bfm E},$$ (1222)

where ${\bfm L}$ is given by Eq. (7.29). With ${\bfm r}\!\cdot\!{\bfm B}$ given by Eq. (7.44a), the electric field associated with a magnetic multipole must satisfy

$${\bfm L}\!\cdot\!{\bfm E}_{lm}^{(M)} (r,\theta,\varphi) =l(l+1)\, g_l(kr) \,Y_{lm}(\theta,\varphi)$$ (1223)

and ${\bfm r}\!\cdot\!{\bfm E}_{lm}^{(M)}=0$. Note that the operator ${\bfm L}$ acts only on the angular variables $(\theta, \varphi)$. This means that the radial dependence of ${\bfm E}_{lm}^{(M)}$ must be given by $g_l(kr)$. Note also, from Eqs. (7.33), that the operator ${\bfm L}$ acting on $Y_{lm}$ transforms the $m$ value but does not change the $l$ value. It is easily seen from Eqs. (7.27) and (7.31) that the solution to Eqs. (7.44b) and (7.47) can be written in the form

$${\bfm E}_{lm}^{(M)} = g_l(kr)\,{\bfm L}\, Y_{lm}(\theta,\varphi).$$ (1224)

Thus, the angular dependence of ${\bfm E}_{lm}^{(M)}$ consists of some linear combination of $Y_{l,m-1}$, $Y_{lm}$, and $Y_{l,m+1}$. Equation (7.48), together with

$$c{\bfm B}_{lm}^{(M)} = -\frac{\rm i}{k}\, \nabla\wedge{\bfm E}_{lm}^{(M)},$$ (1225)

specifies the electromagnetic fields of a magnetic multipole of order $(l, m)$. Note from Eq. (7.31) that the electric field given by Eq. (7.48) is transverse to the radius vector. Thus, magnetic multipole fields are sometimes termed transverse electric (TE) multipole fields.

The fields of an electric or transverse magnetic (TM) multipole of order $(l, m)$ are specified by the conditions

$${\bfm r}\!\cdot\!{\bfm E}_{lm}^{(E)} = -\frac{l(l+1)}{k}\,f_l(kr)\, Y_{lm}(\theta,\varphi),$$ (1226)

$${\bfm r}\!\cdot\!{\bfm B}_{lm}^{(E)} = 0.$$ (1227)

It follows that the fields of an electric multipole are given by

$$c{\bfm B}_{lm}^{(E)} = f_l(kr)\,{\bfm L}\, Y_{lm}(\theta,\varphi),$$ (1228)

$${\bfm E}_{lm}^{(E)} = \frac{\rm i}{k} \nabla\wedge c{\bfm B}_{lm}^{(E)}.$$ (1229)

The radial function $f_l(kr)$ is given by an expression like (7.45).

The two sets of multipole fields (7.48), (7.49), and (7.51), form a complete set of vector solutions to Maxwell's equations in free space. Since the vector spherical harmonic ${\bfm L}\, Y_{lm}$ plays an important role in multipole fields, it is convenient to introduce the normalized form

$${\bfm X}_{lm}(\theta,\varphi) = \frac{1}{\sqrt{l(l+1)}}\,{\bfm L} \, Y_{lm}(\theta,\varphi).$$ (1230)

It can be demonstrated that the vector spherical harmonics possess the orthogonality properties

$$\int {\bfm X}_{l'm'}^\ast\!\cdot\!{\bfm X}_{lm}\,d\Omega = \delta_{ll'}\, \delta_{m m'},$$ (1231)

$$\int {\bfm X}_{l'm'}^\ast \!\cdot\! ({\bfm r}\wedge {\bfm X}_{lm})\,d \Omega = 0,$$ (1232)

for all $l$, $l'$, $m$, and $m'$.

By combining the two types of fields we can write the general solution to Maxwell's equations in free space in the form

$$c{\bfm B} = \sum_{l,m}\left[a_E(l,m)\,f_l(kr)\,{\bfm X}_{lm} -\frac{\rm i}{k}\, a_M(l,m) \,\nabla\wedge g_l(kr) {\bfm X}_{lm}\right],$$ (1233)

$${\bfm E} = \sum_{l,m}\left[ \frac{\rm i}{k}\,a_E(l,m) \,\nabla \wedge f_l(kr) {\bfm X}_{lm} + a_M(l,m)\, g_l(kr)\, {\bfm X}_{lm}\right],$$ (1234)

where the coefficients $a_E(l,m)$ and $a_M(l,m)$ specify the amounts of electric $(l, m)$ and magnetic $(l, m)$ multipole fields. The radial functions $f_l(kr)$ and $g_l(kr)$ are of the form (7.45). The coefficients $a_E(l,m)$ and $a_M(l,m)$, as well as the relative proportions in (7.45), are determined by the sources and the boundary conditions.

Equations (7.54) yield

$${\bfm r}\!\cdot\!c{\bfm B} = \frac{1}{k}\sum_{l,m} a_M(l,m) \,g_l(kr)\,{\bfm L} \,{\bfm X}_{lm}$$

$$= \frac{1}{k} \sum_{l,m} a_M(l,m) \,g_l(kr) \sqrt{l(l+1)} \, Y_{lm},$$ (1235)

and

$${\bfm r}\!\cdot\!{\bfm E} = -\frac{1}{k} \sum_{l,m}a_E(l,m)\, f_l(kr)\, {\bfm L} \,{\bfm X}_{lm}$$

$$= - \frac{1}{k} \sum_{l,m} a_E(l,m)\, f_l(kr)\sqrt{l(l+1)} \, Y_{lm},$$ (1236)

where use has been made of Eqs. (7.27), (7.29), and (7.31). It follows from the well known orthogonality property of the spherical harmonics that

$$a_M(l,m) \,g_l(kr) = \frac{k}{\sqrt{l(l+1)}}\int Y_{lm}^\ast \,{\bfm r} \!\cdot\!c{\bfm B}\,d\Omega,$$ (1237)

$$a_E(l,m)\,f_l(kr) = - \frac{k}{\sqrt{l(l+1)}} \int Y_{lm}^\ast \,{\bfm r} \!\cdot\!{\bfm E}\,d\Omega.$$ (1238)

Thus, knowledge of ${\bfm r}\!\cdot\!{\bfm B}$ and ${\bfm r}\!\cdot\!{\bfm E}$ at two different radii in a source free region permits a complete specification of the fields, including the relative proportions of $h_l^{(1)}$ and $h_l^{(2)}$ in $f_l$ and $g_l$.