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Next: Radiation from a linear Up: The multipole expansion Previous: Properties of multipole fields

Sources of multipole radiation

Let us now examine the connection between multipole fields and their sources. Suppose that there exist localized distributions of electric change $\rho({\bfm r}, t)$, true current ${\bfm j}({\bfm r}, t)$, and magnetization ${\bfm M}({\bfm r}, t)$. We assume that the time dependence can be analyzed into its Fourier components, and we therefore only consider harmonically varying sources, $\rho({\bfm r}) \,{\rm e}^{-{\rm i}\,\omega t}$, ${\bfm j}({\bfm r}) \,{\rm e}^{-{\rm i}\,\omega t}$, and ${\bfm M}({\bfm r}) \,{\rm e}^{-{\rm i}\,\omega t}$, where it is understood that we take the real parts of complex quantities.

Maxwell's equations can be written

$\displaystyle \nabla\!\cdot\!{\bfm E}$ $\textstyle =$ $\displaystyle \frac{\rho}{\epsilon_0},$ (1262)
$\displaystyle \nabla\!\cdot\!{\bfm B}$ $\textstyle =$ $\displaystyle 0,$ (1263)
$\displaystyle \nabla\wedge{\bfm E} -{\rm i}\,k \,c{\bfm B}$ $\textstyle =$ $\displaystyle 0,$ (1264)
$\displaystyle \nabla\wedge c{\bfm B} + {\rm i}\,k \,{\bfm E}$ $\textstyle =$ $\displaystyle \mu_0 c \,({\bfm j} + \nabla\wedge {\bfm M}),$ (1265)

with the continuity equation
{\rm i}\,\omega \rho = \nabla\!\cdot\!{\bfm j}.
\end{displaymath} (1266)

It is convenient to deal with divergenceless fields. Thus, we use as the field variables, ${\bfm B}$ and
{\bfm E}' = {\bfm E} + \frac{\rm i}{\epsilon_0 \omega} \,{\bfm j}.
\end{displaymath} (1267)

In the region outside the sources ${\bfm E}'$ reduces to ${\bfm E}$. When expressed in terms of these fields, Maxwell's equations become
$\displaystyle \nabla\!\cdot\!{\bfm E}'$ $\textstyle =$ $\displaystyle 0,$ (1268)
$\displaystyle \nabla\!\cdot\!{\bfm B}$ $\textstyle =$ $\displaystyle 0,$ (1269)
$\displaystyle \nabla\wedge{\bfm E}' -{\rm i}\,k \,c{\bfm B}$ $\textstyle =$ $\displaystyle \frac{\rm i}{\epsilon_0
\omega} \nabla\wedge{\bfm j},$ (1270)
$\displaystyle \nabla\wedge c{\bfm B} + {\rm i}\,k \,{\bfm E}'$ $\textstyle =$ $\displaystyle \mu_0 c \,\nabla\wedge {\bfm M}.$ (1271)

The curl equations can be combined to give two inhomogeneous Helmholtz wave equations:
(\nabla^2 + k^2)c{\bfm B} = -\mu_0 c\, \nabla\wedge({\bfm j}+\nabla\wedge{\bfm M}),
\end{displaymath} (1272)

(\nabla^2+k^2){\bfm E}' = -{\rm i}\,k\,\mu_0 c\,\nabla\wedge\left(
{\bfm M} + \frac{\nabla\wedge{\bfm j}}{k^2}\right).
\end{displaymath} (1273)

These equations, together with $\nabla\!\cdot\!{\bfm B} = 0$, and $\nabla\!\cdot\!{\bfm E}' = 0$, and the curl equations giving ${\bfm E}'$ in terms of ${\bfm B}$ and vice versa, are the analogues to Eqs. (7.37)-(7.40) when sources are present.

Since the multipole coefficients in Eqs. (7.54) are determined according to Eqs. (7.57) from the scalars ${\bfm r}\!\cdot\!{\bfm B}$ and ${\bfm r}\!\cdot\!{\bfm E}'$, it is sufficient to consider wave equations for these quantities, rather than the vector fields ${\bfm B}$ and ${\bfm E}'$. From Eqs. (7.42), (7.81), (7.82), and the identity

{\bfm r}\!\cdot\!(\nabla\wedge{\bfm A}) = ({\bfm r}\wedge\nabla)\!\cdot\!
{\bfm A} = {\rm i}\,{\bfm L}\!\cdot\!{\bfm A}
\end{displaymath} (1274)

for any vector field ${\bfm A}$, we obtain the inhomogeneous wave equations
$\displaystyle (\nabla^2 + k^2)\,{\bfm r}\!\cdot\!c{\bfm B}$ $\textstyle =$ $\displaystyle -{\rm i}\,\mu_0 c\,{\bfm L}\!\cdot\!({\bfm j} + \nabla\wedge{\bfm M}),$ (1275)
$\displaystyle (\nabla^2+k^2)\,{\bfm r}\!\cdot\!{\bfm E}'$ $\textstyle =$ $\displaystyle k\,\mu_0 c\,
{\bfm L}\!\cdot\!\left({\bfm M} + \frac{\nabla\wedge{\bfm j}}{k^2}\right).$ (1276)

Now the Green's function for the inhomogeneous Helmholtz equation (defined by Eq. (7.17)), subject to the boundary condition of outgoing waves at infinity, is given by Eq. (7.18). It follows that Eqs. (7.84) can be inverted to give

$\displaystyle {\bfm r}\!\cdot \!c{\bfm B}({\bfm r})$ $\textstyle =$ $\displaystyle \frac{{\rm i}\,\mu_0 c}{4\pi}
\int\frac{{\rm e}^{\,{\rm i}\,k\,\v...
...ft[{\bfm j}({\bfm r}') +\nabla'\wedge
{\bfm M}({\bfm r}')\right]\,d^3{\bfm r}',$ (1277)
$\displaystyle {\bfm r}\!\cdot \!{\bfm E}'({\bfm r})$ $\textstyle =$ $\displaystyle -\frac{k\,\mu_0 c}{4\pi}\int \frac{{\rm e}^{\,{\rm i}\,k\,\vert{\...
...({\bfm r}')+\frac{\nabla'\wedge
{\bfm j}({\bfm r}')}{k^2}\right]\,d^3{\bfm r}'.$  

In order to evaluate the multipole coefficients by means of Eqs. (7.57), we first observe that the requirement of outgoing waves at infinity makes $A_l^{(2)} = 0$ in Eq. (7.45). Thus, we choose $f_l(kr) =g_l(kr) = h_l^{(1)}(kr)$ in Eqs. (7.54) as the radial eigenfunctions of ${\bfm E}$ and ${\bfm B}$ in the source free region. Next, let us consider the expansion (7.26) of the Green's function for the Helmholtz equation in terms of spherical harmonics. We assume that the point ${\bfm r}$ lies outside some spherical shell which completely encloses the sources. It follows that $r_< = r'$ and $r_>=r$ in all of the integrations. Making use of the orthogonality property of the spherical harmonics, it follows from Eq. (7.26) that
\int Y^\ast_{lm}(\theta,\varphi)\,\frac{{\rm e}^{\,{\rm i}\,...
...,k\, h_l^{(1)}(kr) \,j_l(kr')\, Y_{lm}^\ast(\theta',\varphi').
\end{displaymath} (1279)

Finally, Eqs. (7.57), (7.85), and (7.86) yield
$\displaystyle a_E(l,m)$ $\textstyle =$ $\displaystyle \frac{\mu_0 c\,\,{\rm i}\,k^3}{\sqrt{l(l+1)}}\int
...\!\cdot\!\left({\bfm M} +\frac{\nabla\wedge
{\bfm j}}{k^2}\right)\,d^3{\bfm r},$ (1280)
$\displaystyle a_M(l,m)$ $\textstyle =$ $\displaystyle -\frac{\mu_0 c\,k^2}{\sqrt{l(l+1)}}\int
j_l(kr)\,Y_{lm}^\ast\, {\bfm L}\!\cdot\!({\bfm j} +\nabla\wedge{\bfm M})\,
d^3{\bfm r}.$ (1281)

The expressions (7.87) give the strengths of the various multipole fields outside the source in terms of integrals over the source densities ${\bfm j}$ and ${\bfm M}$. They can be transformed into more useful forms by means of the following arguments. The results

$\displaystyle {\bfm L}\!\cdot\!{\bfm A}$ $\textstyle =$ $\displaystyle {\rm i}\,\nabla\!\cdot\!({\bfm r}\wedge
{\bfm A}),$ (1282)
$\displaystyle {\bfm L}\!\cdot\!(\nabla\wedge{\bfm A})$ $\textstyle =$ $\displaystyle {\rm i}\,\nabla^2({\bfm r}
\!\cdot\!{\bfm A}) -{\rm i}\,\frac{1}{r}\frac{\partial (r^2 \,\nabla\!\cdot\!
{\bfm A})}{\partial r}$ (1283)

follow from the definition (7.29) of ${\bfm L}$, and simple vector identities. Substituting into Eq. (7.87a), we obtain
$\displaystyle a_E(l,m)$ $\textstyle =$ $\displaystyle - \frac{\mu_0 c\,\,k^3}{\sqrt{l(l+1)}}
\int j_l(kr)\,Y_{lm}^\ast\left[\nabla\!\cdot\!({\bfm r}\wedge{\bfm M})\right.$  
    $\displaystyle \left.\mbox{\hspace{1cm}}+\frac{\nabla^2({\bfm r}\!\cdot\!{\bfm j...
...m i}\,\frac{c}{k\,r}
\frac{\partial(r^2 \rho)}{\partial r}\right]\,d^3{\bfm r},$ (1284)

where use has been made of Eq. (7.78). Use of Green's theorem on the second term replaces $\nabla^2$ by $-k^2$ (since we can neglect the surface terms, and $j_l(kr)\,Y_{lm}^\ast$ is a solution of the Helmholtz equation). A radial integration by part on the third term (again neglecting surface terms) casts the radial derivative over onto the spherical Bessel function. The result for the electric multipole coefficient is
$\displaystyle a_E(l,m)$ $\textstyle =$ $\displaystyle \frac{\mu_0 c\,k^2}{{\rm i}\,\sqrt{l(l+1)}}
\int Y_{lm}^\ast \lef...
... [r\, j_l(kr)]}{d r}
+ {\rm i}\,k\, ({\bfm r}\!\cdot\!{\bfm j})\,j_l(kr)\right.$  
    $\displaystyle \left.\mbox{\hspace{2cm}}-{\rm i}\,k\,\nabla\!\cdot\!({\bfm r}\wedge{\bfm M})\,j_l(kr)\right]\,d^3{\bfm r}.$ (1285)

The analogous set of manipulations using Eq. (7.87b) leads to an expression for the magnetic multipole coefficient:
$\displaystyle a_M(l,m)$ $\textstyle =$ $\displaystyle \frac{\mu_0 c\,k^2}{{\rm i}\,\sqrt{l(l+1)}}
\int Y_{lm}^\ast \lef...
...\bfm j})
\,j_l(kr) +\nabla\!\cdot\!{\bfm M} \,\frac{d
[r\,j_l(kr)]}{d r}\right.$  
    $\displaystyle \left.\mbox{\hspace{2cm}}-k^2\,({\bfm r}\!\cdot\!{\bfm M})\,j_l(kr)
\right]\,d^3{\bfm r}.$ (1286)

Both the above results are exact, and are valid for arbitrary wavelength and source size.

In the limit in which the source dimensions are very small compared to a wavelength (i.e., $kr\ll 1$) the expressions for the multipole coefficients can be considerably simplified. Using the asymptotic form (7.14a), and keeping only lowest powers in $kr$ for terms involving $\rho$, ${\bfm j}$, and ${\bfm M}$, we obtain the approximate electric multipole coefficient

a_E(l,m) \simeq \frac{\mu_0 c\, k^{l+2}}{{\rm i}\,
\end{displaymath} (1287)

where the multipole moments are
$\displaystyle Q_{lm}$ $\textstyle =$ $\displaystyle \int r^l \,Y_{lm}^\ast\,c\rho\,d^3{\bfm r},$ (1288)
$\displaystyle Q_{lm}'$ $\textstyle =$ $\displaystyle \frac{-{\rm i}\,k}{l+1} \int r^l \,Y_{lm}^\ast\,
\nabla\!\cdot\!({\bfm r}\wedge{\bfm M})\,d^3{\bfm r}.$ (1289)

The moment $Q_{lm}$ has the same form as a conventional electrostatic multipole moment. The moment $Q_{lm}'$ is an induced electric multipole moment due to the magnetization. It is generally a factor $kr$ smaller than the normal moment $Q_{lm}$. For the magnetic multipole coefficient $a_M(l,m)$ the corresponding long wavelength approximation is
a_M(l,m) \simeq \frac{\mu_0 c\, \,{\rm i}\, k^{l+2}}{
({\cal M}_{lm}+{\cal M}_{lm}'),
\end{displaymath} (1290)

where the magnetic multipole moments are
$\displaystyle {\cal M}_{lm}$ $\textstyle =$ $\displaystyle -\frac{1}{l+1} \int r^l\,Y_{lm}^\ast\,
\nabla\!\cdot\!({\bfm r}\wedge{\bfm j})\,d^3{\bfm r},$ (1291)
$\displaystyle {\cal M}_{lm}'$ $\textstyle =$ $\displaystyle -\int r^l\,Y_{lm}^\ast\, \nabla\!\cdot\!{\bfm M} \,d^3{\bfm r}$ (1292)

Note that for a system with intrinsic magnetization the magnetic moments ${\cal M}_{lm}$ and ${\cal M}_{lm}'$ are generally of the same order of magnitude.

Thus, in the long wavelength limit the electric multipole fields are determined by the charge density $\rho$, whereas the magnetic multipole fields are determined by the magnetic moment densities ${\bfm r}\wedge{\bfm j}/2$ and ${\bfm M}$.

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Next: Radiation from a linear Up: The multipole expansion Previous: Properties of multipole fields
Richard Fitzpatrick 2002-05-18