Multipole Expansion of Vector Wave Equation

Maxwell's equations in free space reduce to

$$\nabla\cdot {\bf E} =0,$$ (1449)

$$\nabla \cdot c\,{\bf B} =0,$$ (1450)

$$\nabla\times {\bf E} = {\rm i}\,k\,c\,{\bf B},$$ (1451)

$$\nabla\times c\,{\bf B} =-{\rm i}\,k \,{\bf E},$$ (1452)

assuming an ${\rm e}^{-{\rm i}\,\omega\, t}$ time dependence of all field quantities. Here, $k=\omega/c$ . Eliminating ${\bf E}$ between Equations (1453) and (1454), we obtain

$$(\nabla^{\,2} + k^{\,2}) \,{\bf B} =0,$$ (1453)

$$\nabla\cdot{\bf B} =0,$$ (1454)

with ${\bf E}$ given by

$${\bf E} = \frac{\rm i}{k}\,\nabla\times c\,{\bf B}.$$ (1455)

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

$$(\nabla^{\,2} + k^{\,2})\, {\bf E} =0,$$ (1456)

$$\nabla\cdot {\bf E} =0,$$ (1457)

with ${\bf B}$ given by

$$c\,{\bf B} =- \frac{\rm i}{k}\,\nabla\times{\bf E}.$$ (1458)

It is clear that each Cartesian component of ${\bf B}$ and ${\bf E}$ satisfies the homogeneous Helmholtz wave equation, (1413). Hence, according to the analysis of Section 11.2, these components can be written as a general expansion of the form

$$\psi({\bf r}) = \sum_{l,m}\left[A_{lm}^{(1)} \,h_l^{(1)}(k\,r) +A_{lm}^{(2)} \,h_l^{(2)}(k\,r)\right] Y_{lm}(\theta,\varphi),$$ (1459)

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

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

$$\nabla^{\,2}({\bf r}\cdot{\bf A}) = {\bf r}\cdot(\nabla^{\,2}{\bf A}) + 2 \,\nabla\cdot{\bf A}.$$ (1460)

It follows from Equations (1455)-(1456) and (1458)-(1459) that the scalars ${\bf r}\cdot {\bf E}$ and ${\bf r}\cdot {\bf B}$ both satisfy the homogeneous Helmholtz wave equation: that is,

$$(\nabla^{\,2} + k^{\,2})\,({\bf r}\cdot{\bf E}) =0,$$ (1461)

$$(\nabla^{\,2}+k^{\,2})\,({\bf r}\cdot{\bf B}) =0.$$ (1462)

Thus, the general solutions for ${\bf r}\cdot {\bf E}$ and ${\bf r}\cdot {\bf B}$ can be written in the form (1461).

Let us define a magnetic multipole field of order $l, m$ as the solution of

$${\bf r}\cdot c\,{\bf B}_{lm}^{(M)} = \frac{l\,(l+1)}{k} \,g_l(k\,r) \,Y_{lm}(\theta, \varphi),$$ (1463)

$${\bf r}\cdot{\bf E}_{lm}^{(M)} =0,$$ (1464)

where

$$g_l(k\,r) = A_l^{(1)} \,h_l^{(1)} (k\,r) + A_l^{(2)} \,h_l^{(2)}(k\,r).$$ (1465)

The presence of the factor $l\,(l+1)/k$ in Equation (1465) is for later convenience. Equation (1460) yields

$$k\,{\bf r}\cdot c\,{\bf B} = -{\rm i}\,{\bf r}\cdot (\nabla\times{\bf E}) = -{\rm i} \,({\bf r}\times\nabla)\cdot {\bf E} = {\bf L}\cdot{\bf E},$$ (1466)

where ${\bf L}$ is given by Equation (1438). Thus, with ${\bf r}\cdot c\,{\bf B}$ taking the form (1465), the electric field associated with a magnetic multipole must satisfy

$${\bf L}\cdot{\bf E}_{lm}^{(M)} (r,\theta,\varphi) =l\,(l+1)\, g_l(k\,r) \,Y_{lm}(\theta,\varphi),$$ (1467)

as well as ${\bf r}\cdot{\bf E}_{lm}^{(M)}=0$ . Recall that the operator ${\bf L}$ acts on the angular variables $\theta, \varphi$ only. This implies that the radial dependence of ${\bf E}_{lm}^{(M)}$ is given by $g_l(k\,r)$ . It is easily seen from Equations (1436) and (1442) that the solution to Equations (1466) and (1469) can be written in the form

$${\bf E}_{lm}^{(M)} = g_l(k\,r)\,{\bf L}\, Y_{lm}(\theta,\varphi).$$ (1468)

It follows from the analysis Section 11.3 that the angular dependence of ${\bf E}_{lm}^{(M)}$ consists of a linear combination of $Y_{l,m-1}(\theta,\varphi)$ , $Y_{lm}(\theta,\varphi)$ , and $Y_{l,m+1}(\theta,\varphi)$ functions. Equation (1470), together with

$$c\,{\bf B}_{lm}^{(M)} = -\frac{\rm i}{k}\, \nabla\times{\bf E}_{lm}^{(M)},$$ (1469)

specifies the electromagnetic fields of a magnetic multipole of order $l, m$ . According to Equation (1442), the electric field (1470) 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 $l, m$ satisfy

$${\bf r}\cdot{\bf E}_{lm}^{(E)} = -\frac{l\,(l+1)}{k}\,f_l(k\,r)\, Y_{lm}(\theta,\varphi),$$ (1470)

$${\bf r}\cdot{\bf B}_{lm}^{(E)} =0.$$ (1471)

It follows that the fields of an electric multipole are

$$c\,{\bf B}_{lm}^{(E)} = f_l(k\,r)\,{\bf L}\, Y_{lm}(\theta,\varphi),$$ (1472)

$${\bf E}_{lm}^{(E)} = \frac{\rm i}{k} \nabla\times c\,{\bf B}_{lm}^{(E)}.$$ (1473)

Here, the radial function $f_l(k\,r)$ is an expression of the form (1467).

The two sets of multipole fields, (1470)-(1471), and (1474)-(1475), form a complete set of vector solutions to Maxwell's equations in free space. Because the vector spherical harmonic ${\bf L}\, Y_{lm}$ plays an important role in the theory of multipole fields, it is convenient to introduce the normalized form

$${\bf X}_{lm}(\theta,\varphi) = \frac{1}{\sqrt{l\,(l+1)}}\,{\bf L} \, Y_{lm}(\theta,\varphi).$$ (1474)

It can be demonstrated that these forms possess the orthogonality properties

$$\oint {\bf X}_{l'm'}^{\,\ast}\cdot{\bf X}_{lm}\,d{\mit\Omega} =\delta_{ll'}\, \delta_{m m'},$$ (1475)

$$\oint {\bf X}_{l'm'}^{\,\ast} \cdot({\bf r}\times {\bf X}_{lm})\,d {\mit\Omega} =0,$$ (1476)

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

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

$$c\,{\bf B} = \sum_{l,m}\left[a_E(l,m)\,f_l(k\,r)\,{\bf X}_{lm} -\frac{\rm i}{k}\, a_M(l,m) \,\nabla\times g_l(k\,r) \,{\bf X}_{lm}\right],$$ (1477)

$${\bf E} = \sum_{l,m}\left[ \frac{\rm i}{k}\,a_E(l,m) \,\nabla \times f_l(k\,r)\, {\bf X}_{lm} + a_M(l,m)\, g_l(k\,r)\, {\bf X}_{lm}\right],$$ (1478)

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 $f_l(k\,r)$ and $g_l(k\,r)$ are both of the form (1467). The coefficients $a_E(l,m)$ and $a_M(l,m)$ , as well as the relative proportions of the two types of Hankel functions in the radial functions $f_l(k\,r)$ and $g_l(k\,r)$ , are determined by the sources and the boundary conditions.

Equations (1479) and (1480) yield

$${\bf r}\cdot c\,{\bf B} = \frac{1}{k}\sum_{l,m} a_M(l,m) \,g_l(k\... ... X}_{lm}=\frac{1}{k} \sum_{l,m} a_M(l,m) \,g_l(k\,r) \sqrt{l\,(l+1)} \, Y_{lm},$$ (1479)

and

$${\bf r}\cdot{\bf E} = -\frac{1}{k} \sum_{l,m}a_E(l,m)\, f_l(k\,r)... ..._{lm} = - \frac{1}{k} \sum_{l,m} a_E(l,m)\, f_l(k\,r)\sqrt{l\,(l+1)} \, Y_{lm},$$ (1480)

where use has been made of Equations (1436), (1438), (1442), and (1476). It follows from the well-known orthogonality property of the spherical harmonics that

$$a_M(l,m) \,g_l(k\,r) = \frac{k}{\sqrt{l\,(l+1)}}\oint Y_{lm}^{\,\ast} \,{\bf r} \cdot c\,{\bf B}\,d{\mit\Omega},$$ (1481)

$$a_E(l,m)\,f_l(k\,r) = - \frac{k}{\sqrt{l\,(l+1)}} \oint Y_{lm}^{\,\ast} \,{\bf r} \cdot{\bf E}\,d{\mit\Omega}.$$ (1482)

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