# Electromagnetic sources beyond common multipoles

Nikita A. Nemkov<EMAIL_ADDRESS>Russian Quantum Center, Skolkovo, Moscow
143025, Russia  Vassili A. Fedotov<EMAIL_ADDRESS>Optoelectronics
Research Centre, University of Southampton, Southampton SO17 1BJ, UK

# On the Role of Longitudinal Currents in Radiating Systems of Charges

Nikita A. Nemkov<EMAIL_ADDRESS>Russian Quantum Center, Skolkovo, Moscow
143025, Russia  Vassili A. Fedotov<EMAIL_ADDRESS>Optoelectronics
Research Centre, University of Southampton, Southampton SO17 1BJ, UK

###### Abstract

The time derivative of the charge density is linked to the current density by
the continuity equation. However, it features only the longitudinal part of a
current density, which is known to produce no radiation. This fact usually
remains unnoticed though it poses a seemingly serious paradox, suggesting that
the temporal variation of a charge density should be also irrelevant for
radiation. We resolve this paradox by showing that the effective longitudinal
currents are not spatially confined even when the time-dependent charge
density that generates them is. This enforces the co-existence of the
complementary (i.e. transverse) part of the current density through the
Helmholtz decomposition. We illustrate the mechanics of the underlying non-
locality of the Helmholtz decomposition in the case of a dynamic electric
dipole, discussing its practical implication for underwater antenna
communications. More generally, we explain the role of the Helmholtz
decomposition in shaping the structure of the conventional multipole
expansion.

## Introduction

In many problems of classical electrodynamics involving electromagnetic
radiation it is methodologically useful to substitute the time-dependent
charge density $\rho(t)$, with its current density equivalent
$\boldsymbol{j}(t)$, related by the continuity equation (see, for example,
[1])

$\displaystyle\partial_{t}\rho+\operatorname{div}\boldsymbol{j}=0\ .$ (1)

However, this generally adopted approach leads to an apparent paradox which,
to the best of our knowledge, has not been discussed (let alone alleviated) in
textbooks on electrodynamics. The tension is caused by the following
observation. On the one hand, the time-dependent charge density gives rise to
radiation – accelerating particles and oscillating electric dipoles are the
well-known examples here. On the other hand, the continuity equation is
missing (though implicitly) the transverse part of the current density
$\boldsymbol{j}_{\perp}$, which is held responsible for generating radiation
fields, i.e. electric and magnetic fields outside the volume taken by the
charge density, see e.g. classic paper [2] or textbook [3]. Colloquially
speaking, longitudinal currents are usually considered to produce no
radiation.

Indeed, assuming that the vector field $\boldsymbol{j}(r,t)$ is sufficiently
smooth and decays sufficiently fast at infinity, it can be split via the
Helmholtz decomposition into the longitudinal
($\operatorname{rot}\boldsymbol{j}_{\parallel}=0$) and transverse
($\operatorname{div}\boldsymbol{j}_{\perp}=0$) components, and so in the
continuity equation
$\operatorname{div}\boldsymbol{j}=\operatorname{div}(\boldsymbol{j}_{\parallel}+\boldsymbol{j}_{\perp})=\operatorname{div}\boldsymbol{j}_{\parallel}$.
Since longitudinal currents do not radiate, the time-dependent charge density
must also be irrelevant for the radiation.

To resolve this apparent paradox one needs to accept that the longitudinal and
transverse parts of a spatially localized current density are not completely
independent, as was briefly noted in [4]. Moreover, even when a system of non-
stationary charges is confined to some finite region of space, the transverse
and longitudinal components of a current density not only can exist outside
that region but extend to infinity, provided that
$\boldsymbol{j}_{\parallel}=-\boldsymbol{j}_{\perp}$ there [5]. This simple
yet, perhaps, counterintuitive observation is one of the main points that we
elaborate on in this paper.

##  I Longitudinal and transverse currents of a localized source

We begin by clarifying how (or rather when) a vector field can be
simultaneously transverse and longitudinal. One usually identifies a vector
field with such a property while deriving the Helmholtz decomposition, but
here we are going to avoid the lengthy derivation and offer a brief and
perhaps simpler alternative in the style of [6].

If a vector field $\boldsymbol{j}$ (which at the moment is not necessarily a
current density) is longitudinal, i.e. $\operatorname{rot}\boldsymbol{j}=0$,
it can always be represented as the gradient of some scalar potential $\psi$

$\displaystyle\boldsymbol{j}=\boldsymbol{\nabla}\psi\ .$ (2)

The requirement for this field to be simultaneously transverse, i.e.
$\operatorname{div}\boldsymbol{j}=0$, leads to $\Delta\psi=0$. If this
condition holds everywhere in space the corresponding current is
vanishing.111Assuming that $\boldsymbol{j}$ decays sufficiently fast at
spatial infinity it follows $\int
d^{3}\boldsymbol{r}\,\,\boldsymbol{j}(\boldsymbol{r})\cdot\boldsymbol{j}(\boldsymbol{r})=-\int
d^{3}\boldsymbol{r}\,\,\psi\Delta\psi=0$ and hence
$\boldsymbol{j}(\boldsymbol{r})=0$ everywhere. However, if $\Delta\psi=0$
holds only outside some region $R$, then $\psi$ can be perfectly non-trivial,
and so $\boldsymbol{j}$ will be both transverse and longitudinal outside $R$.

In fact, this type of fields is not at all exotic. Electric field of (almost)
any static distribution of charges is of this kind. Indeed, consider a static
charge density $\rho_{0}$ confined to $R$. Its scalar potential $\phi_{0}$ is
governed by the Poisson equation222We introduce subscript 0 to indicate
quantities related to this auxiliary static problem.

$\displaystyle\Delta\phi_{0}=-4\pi\rho_{0}\ .$ (3)

Outside $R$ such a potential naturally satisfies $\Delta\phi_{0}=0$ and,
therefore, the electric field generated there
$\boldsymbol{E}_{0}=-\boldsymbol{\nabla}\phi_{0}$ is simultaneously
longitudinal and transverse. Importantly, this general example also shows that
the leakage of the longitudinal field component outside the confined source
and up to infinity is almost inevitable.333There are exceptions though, as
discussed in appendix B.

Now let us consider a general current $\boldsymbol{j}$ with its Helmholtz
decomposition into transverse and longitudinal parts
$\boldsymbol{j}=\boldsymbol{j}_{\parallel}+\boldsymbol{j}_{\perp}$. We will
show that both $\boldsymbol{j}_{\parallel}$ and $\boldsymbol{j}_{\perp}$
generically extend to infinity even when $\boldsymbol{j}$ is confined. Scalar
potential $\psi$ that defines the longitudinal part
$\boldsymbol{j}_{\parallel}=\boldsymbol{\nabla}\psi$ can be found from
condition $\Delta\psi=\operatorname{div}\boldsymbol{j}$ or

$\displaystyle\Delta\psi=-\partial_{t}\rho\ ,$ (4)

where we have used the continuity equation (1). This is mathematically the
same problem as (3) with $\psi$ in place of $\phi_{0}$ and
$-\frac{1}{4\pi}\partial_{t}\rho$ in place of $\rho_{0}$. This means that
$\boldsymbol{j}_{\parallel}$ is proportional to the time-derivative of the
longitudinal part of the electric field produced by the actual charge density

$\displaystyle\boldsymbol{j}_{\parallel}=-\frac{1}{4\pi}\partial_{t}\boldsymbol{E}_{\parallel}\
.$ (5)

One could have derived this result directly from the Maxwell equation for the
current density

$\displaystyle
4\pi\boldsymbol{j}=-\partial_{t}\boldsymbol{E}+c\operatorname{rot}\boldsymbol{H}$
(6)

by considering its longitudinal component. However, it would then be unclear
that $\boldsymbol{j}_{\parallel}$ was not simply vanishing. Indeed, one is
naturally tempted to conclude that $\boldsymbol{E}_{\parallel}=0$ because the
electric field is transverse in the radiation zone (i.e.
$\operatorname{div}\boldsymbol{E}=0$). This conclusion is obviously premature
since $\boldsymbol{E}_{\parallel}$ and $\boldsymbol{E}_{\perp}$ must be
defined everywhere in space (and not only in the radiation zone), and hence
$\boldsymbol{E}_{\parallel}$ is (typically) non-zero.

As we have demonstrated above, the longitudinal electric field usually extends
beyond the confines of a static distribution of charges and, thus, the
longitudinal component of the current density $\boldsymbol{j}_{\parallel}$
does the same in the dynamic case. The most important consequence of the
spatially delocalized nature of $\boldsymbol{j}_{\parallel}$ is that the
transverse component of the current density $\boldsymbol{j}_{\perp}$ must also
be non-vanishing, even though this cannot be inferred directly from the
continuity equation. This simply follows from the fact that the total current
density $\boldsymbol{j}=\boldsymbol{j}_{\parallel}+\boldsymbol{j}_{\perp}$
vanishes outside the source and hence

$\displaystyle\boldsymbol{j}_{\perp}=-\boldsymbol{j}_{\parallel}\qquad(\text{outside
$R$})\ .$ (7)

Now we are ready to revisit the apparent paradox posed in the introduction.
Given that $\boldsymbol{j}_{\parallel}$ and $\boldsymbol{j}_{\perp}$ of a
localised source are linked, a more precise statement is not that the
longitudinal currents are irrelevant to the radiation, but rather that the
radiation fields can be expressed solely via the transverse currents. Indeed,
a vector potential $\boldsymbol{A}$ can be written in the following form (see
app. A)

$\displaystyle\boldsymbol{A}(\boldsymbol{r})=-\frac{4\pi}{k^{2}c}\boldsymbol{j}_{\parallel}(\boldsymbol{r})+\frac{1}{c}\int_{R}d\boldsymbol{r}^{\prime}\,G(\boldsymbol{r}-\boldsymbol{r}^{\prime})\boldsymbol{j}_{\perp}(\boldsymbol{r^{\prime}})\
.$ (8)

Here $\boldsymbol{A}$ is non-locally related to $\boldsymbol{j}_{\perp}$, i.e.
the distribution of $\boldsymbol{j}_{\perp}$ inside $R$ crucially affects the
radiation field outside $R$. However, the longitudinal part of the current
density enters $\boldsymbol{A}$ only locally. Since Eq. (8) holds in any
region of space, one can replace $\boldsymbol{j}_{\parallel}$ with
$-\boldsymbol{j}_{\perp}$ outside $R$ and, as a result, the radiation fields
will be expressed solely via $\boldsymbol{j}_{\perp}$.

At the same time, one usually cannot alter $\boldsymbol{j}_{\perp}$ without
altering $\boldsymbol{j}_{\parallel}$ (or, equivalently, the charge density of
the source) and so it would be misleading to state that
$\boldsymbol{j}_{\parallel}$ is completely irrelevant to the radiation. This
is especially true if the radiation occurs in a partially conducting media
(such as sea water), where the total current density outside the source does
not have to be zero. We consider this case in more detail in the next section.

##  II Electric dipole

The preceding discussion was rather abstract and we would now like to
illustrate it using the simplest non-trivial example of a radiating localized
source – a non-stationary electric dipole of negligible size. For the dipole
moment $\boldsymbol{d}$ placed at the origin $r=0$ the charge and current
densities are

$\displaystyle\rho=-(\boldsymbol{d},\nabla)\,\,\delta(\boldsymbol{r}),\qquad\boldsymbol{j}=\dot{\boldsymbol{d}}\,\delta(\boldsymbol{r})\
.$ (9)

Neither $\operatorname{rot}\boldsymbol{j}$ nor
$\operatorname{div}\boldsymbol{j}$ vanish everywhere, so $\boldsymbol{j}$
contains both longitudinal and transverse parts. Let us explicitly construct
$\boldsymbol{j}_{\parallel}$ corresponding to the point dipole. Since
$\operatorname{div}\boldsymbol{j}_{\perp}=0$ by definition and
$\boldsymbol{j}_{\parallel}=\nabla\phi$, Eq. (1) transforms into

$\displaystyle\Delta\phi=(\dot{\boldsymbol{d}},\boldsymbol{\nabla})\delta(\boldsymbol{r})\
,$ (10)

which has the following solution

$\displaystyle\phi=-(\dot{\boldsymbol{d}},\boldsymbol{\nabla})\frac{1}{4\pi
r}\ .$ (11)

The longitudinal part of the corresponding current density is given by the
gradient of $\phi$

$\displaystyle\boldsymbol{j}_{\parallel}=\boldsymbol{\nabla}\phi=-\frac{\dot{\boldsymbol{d}}-3(\dot{\boldsymbol{d}},\hat{\boldsymbol{r}})\hat{\boldsymbol{r}}}{4\pi
r^{3}}\ ,$ (12)

where $\hat{\boldsymbol{r}}=\boldsymbol{r}/r$. At any given moment this is
nothing but the electric field produced by the static electric dipole with the
moment $d_{0}=-\partial_{t}\boldsymbol{d}$ 444Of course for time-dependent
$\boldsymbol{d}$ the ”electric field” in Eq.(12) is time-dependent as well,
but at each moment in time it has a shape of the electric field from a static
dipole.. We emphasize that Eq.(12) makes it evident that
$\boldsymbol{j}_{\parallel}$ extends beyond $r=0$, and, in fact, up to
infinity.

If the electric dipole resides in vacuum, $\boldsymbol{j}_{\parallel}$ must be
canceled outside $r=0$ so that the total current density produced externally
will remain vanishing. This is possible only by admitting the co-existence of
the spatially non-localized complementary, transverse part of the current
density of the form

$\displaystyle\boldsymbol{j}_{\perp}=\operatorname{rot}^{2}\frac{\dot{\boldsymbol{d}}}{4\pi
r}\ .$ (13)

Explicit computation gives

$\displaystyle\boldsymbol{j}_{\perp}=(\operatorname{grad}\operatorname{div}-\Delta)\frac{\dot{\boldsymbol{d}}}{4\pi
r}=\nabla(\dot{\boldsymbol{d}},\nabla)\frac{1}{4\pi
r}+\dot{\boldsymbol{d}}\,\delta(\boldsymbol{r})\ ,$ (14)

where we used identity $\delta(\boldsymbol{r})=-\Delta\frac{1}{4\pi r}$. It is
easy to see that the first term in Eq.(14) is equal to
$-\boldsymbol{j}_{\parallel}$, while the second term gives the spatially
localized current density of the point electric dipole (9).

Figure 1: (a) An oscillating electric dipole shown schematically as a pair of
non-stationary charge distributions of opposite signs at a moment in time when
the charge densities increase. Curved black arrows represent longitudinal
currents, which ensure the change in accordance with the continuity equation.
(b) The distribution of transverse currents, which can counteract longitudinal
currents beyond the confines of the electric dipole. (c) The electric dipole
radiates in non-conducting medium, where the cancellation of longitudinal
currents by transverse currents must be complete. (d) An electric dipole
radiates less efficiently in partially conducting medium since the complete
cancellation of longitudinal currents is not required, and so transverse
currents (responsible for radiation) are weaker.

Fig.1 offers a graphical illustration of the above analysis. Without loss of
generality we assume that the radiating electric dipole is formed by a pair of
spatially localized positive and negative charge distributions (see Fig.1a).
We further assume that the change of the electric dipole moment occurs due to
oscillations of the charge densities (rather than the distance between the
two).

We now consider a moment in time when the charge densities increase. According
to the continuity equation, an increase in charge density is ensured by
currents flowing in (or out) of the volume containing the charges, and those
currents are longitudinal. In our case the currents must flow towards the
positive charge distribution, where positive charges are accumulated, and away
from the negative charge distribution, where positive charges are lost.
Considering each charge distribution separately one can already appreciate the
fact that the longitudinal currents will formally extend to infinity.

In a physical system longitudinal currents must find their source and drain
(due to charge conservation). In our example this means that the currents
leaving the negative charge distribution and entering positive charge
distribution must be connected everywhere (including infinity), as illustrated
in Fig.1a. However, in a typical situation – when an electric dipole is
located in a dielectric or vacuum – charges cannot be physically transported
outside the confines of the dipole and, hence, the longitudinal currents must
appear somehow negated.

At this point one has to admit the presence of complementary oscillating
transverse currents (see Fig.1b). Such currents require neither a source nor a
drain, and ”can be arranged” in space in a way that they will cancel
longitudinal currents everywhere apart from the central region, where they
have the same density and flow in the same direction as the longitudinal
currents. The resulting picture (Fig.1c) is now intuitively more acceptable,
since it shows the electric dipole as a spatially localised charge-current
distribution, the radiation of which is allowed through the presence of
confined transverse currents.

One may still wonder why going to the trouble of introducing into the picture
spatially non-localized currents for an a priory localized electromagnetic
source, especially that those currents cannot be sustained outside the source,
and thus appear purely virtual. As we have pointed out earlier, such an
approach enables one to show explicitly why a system of non-stationary charges
acquires a transverse component of its real current density and, therefore,
can radiate.

Furthermore, and perhaps more importantly, admitting spatially non-localized
currents helps one to understand the peculiarities of antenna radiation in
partially conducting media, such as sea water or soil [7, 8]. Indeed, in such
media the longitudinal currents outside the electric dipole in Fig.1a can be
partially sustained (via real conductivity) and, therefore, are not required
to be completely negated by transverse currents. Correspondingly, the current
balance outside the electric dipole (7) is modified as follows

$\displaystyle\boldsymbol{j}_{\perp}+\boldsymbol{j}_{\parallel}=\boldsymbol{j}_{\sigma}\
,$ (15)

where $\boldsymbol{j}_{\sigma}$ is the density of the actual longitudinal
currents supported by the medium due to its non-vanishing electric
conductivity. Since $\boldsymbol{j}_{\parallel}$ remains fixed by the
continuity equation, $|\boldsymbol{j}_{\perp}| < |\boldsymbol{j}_{\parallel}|$
and this inequality becomes only stronger with increasing conductivity of the
medium. As a result, not only the emitted electromagnetic waves will decay
faster than in vacuum/air (due to absorption) but also the overall radiation
efficiency of an electric dipole antenna will become lower even in a well
matched case (see Fig.1d).

In plain words, the fraction of energy that would have been radiated by the
antenna is now circulated in space (and dissipated) via real longitudinal
currents. This, in particular, might explain anomalously high attenuation of
radiation from electric dipole antennas in sea water and shows why loop (i.e.,
magnetic dipole) antennas, which do not feature longitudinal currents, are
generally better emitters in partially conducting media [7, 8].

##  III Implications for the multipole expansion

Another way to formulate the puzzle that we have started with is to note that
in a typical electrodynamic system the charge multipoles, such as an electric
dipole or quadrupole, are central to radiation. Their moments, however, are
defined solely by the charge density. For instance

$\displaystyle\boldsymbol{D}=\int
d\boldsymbol{r}\,\boldsymbol{r}\rho(\boldsymbol{r})\ ,$ (16) $\displaystyle
Q_{\alpha\beta}=3\int
d\boldsymbol{r}\,\left(3r_{\alpha}r_{\beta}-r^{2}\delta_{\alpha\beta}\right)\rho(\boldsymbol{r})\
,$ (17)

where $ve{D}$ and $Q$ are the electric dipole and quadrupole moments,
respectively. The charge density is formally related only to the longitudinal
currents $\boldsymbol{j}_{\parallel}$ (1) and, therefore, it is seemingly
irrelevant for the radiation fields, which can be defined only through
transverse currents $\boldsymbol{j}_{\perp}$. Now, of course, we appreciate
that the paradox is resolved because $\boldsymbol{j}_{\perp}$ and
$\boldsymbol{j}_{\parallel}$ are not independent. In particular, the time-
derivatives of all electric multipole moments can be expressed via
$\boldsymbol{j}_{\perp}$. For instance

$\displaystyle\partial_{t}\boldsymbol{D}=\int
d\boldsymbol{r}\,\,\boldsymbol{r}\partial_{t}\rho(\boldsymbol{r})=-\int
d\boldsymbol{r}\,\,\boldsymbol{j}=-\int
d\boldsymbol{r}\,\,\boldsymbol{j}_{\perp}\ .$ (18)

Here we used the Helmholtz decomposition and the fact that $\int
d\boldsymbol{r}\boldsymbol{j}_{\parallel}=0$, which is not entirely trivial
but true, as explained in 34.

It seems however that the non-locality of the Helmholtz decomposition brings
about an additional confusion to the structure of the multipole expansion. The
full multipole expansion of an arbitrary current density consists of the
charge, magnetic and toroidal multipoles. For instance, magnetic and toroidal
dipoles are defined by

$\displaystyle\boldsymbol{M}=\frac{1}{2}\int
d\boldsymbol{r}\,\,\boldsymbol{r}\times\boldsymbol{j}\ ,$ (19) $\displaystyle
T_{\alpha}=\frac{1}{10}\int
d\boldsymbol{r}\,\,\left(r_{\alpha}r_{\beta}-2r^{2}\delta_{\alpha\beta}\right)j_{\beta}\
.$ (20)

It is sometimes claimed that the toroidal multipole family represents merely
higher-order corrections to the charge multipoles. This is, of course, not
correct – the two families are independent of each other, just as they are
independent of the magnetic multipoles, with one notable exception, the
leading order.

To explain this we need to recall additional terms of the multipole expansion
of a non-stationary charge-current distribution, the mean-square radii [9].
For any given multipole moment (which can be viewed as a measure of certain
angular properties of the charge-current distribution) there is a series of
the mean square radii, which characterizes the radial profile the
corresponding multipolar mode. For example, the mean-square radii of an
electric dipole moment are defined by555Here and further in this section
$\propto$ means equality up to non-zero numerical factors which we do not keep
track of.

$\displaystyle\boldsymbol{D}^{(n)}\propto\int
d\boldsymbol{r}\,\,r^{2n}\boldsymbol{r}\rho(\boldsymbol{r})$ (21)

and one simply inserts an additional factor of $r^{2}$ to obtain
$\boldsymbol{D}^{(n+1)}$ from $\boldsymbol{D}^{(n)}$. One can also move in the
reverse direction, i.e. from $\boldsymbol{D}^{(n+1)}$ to
$\boldsymbol{D}^{(n)}$, by inserting $\Delta\rho$ in place of $\rho$:

$\boldsymbol{D}^{(n)}\propto\int
d\boldsymbol{r}\,\,r^{2n+2}\boldsymbol{r}\Delta\rho(\boldsymbol{r})\propto\int
d\boldsymbol{r}\,\,r^{2n}\boldsymbol{r}\rho(\boldsymbol{r})\ .$ (22)

If one continuous the reverse transformation further, one will see that the
lowest-order term in the series, i.e. the ”parent” term (given by $n=0$) is,
of course, the electric dipole moment itself,
$\boldsymbol{D}^{(0)}=\boldsymbol{D}$.

The point we wish to make is that the same procedure applied to the mean-
square radii of a toroidal dipole will also yield an electric dipole moment as
the lowest-order term of the series (but now corresponding to $n=-1$)

$\displaystyle T^{(-1)}_{\alpha}\propto\int
d\boldsymbol{r}\left(r_{\alpha}r_{\beta}-2r^{2}\delta_{\alpha\beta}\right)\Delta
j_{\beta}\propto D_{\alpha}\ ,$ (23)

since
$\Delta\left(r_{\alpha}r_{\beta}-2r^{2}\delta_{\alpha\beta}\right)=-10\delta_{\alpha\beta}$.
This invites one to conclude that the electric dipole moment not only is the
actual ”parent” term of the series of mean-square radii here, but also is the
lowest order term of the toroidal multipole family. Note also that for the
mean-square radii of any familiar electric or magnetic multipole the above
procedure cannot extended beyond $n=0$, yielding vanishing result for $n=-1$.
Indeed, consider, for example, an electric quadrupole – the $r$-dependent
weight in the integrand satisfies
$\Delta(3r_{\alpha}r_{\beta}-r^{2}\delta_{\alpha\beta})=0$. So despite the
fact that the charge and toroidal multipoles are independent families their
lowest orders seem to agree.

To show that this coincidence is a direct consequence of the non-locality in
the Helmholtz decomposition, as well as to place the above heuristic
discussion on a firm ground, we need to define multipole moments precisely. A
succinct way to do so is to introduce the following orthonormal basis of
vector fields [4, 10]

$\displaystyle\boldsymbol{\mathcal{F}}_{lmk}^{(-)}(\boldsymbol{r})=\frac{\nabla\mathcal{F}_{lmk}(\boldsymbol{r})}{ik}\
,$
$\displaystyle\boldsymbol{\mathcal{F}}_{lmk}^{(0)}(\boldsymbol{r})=\frac{\operatorname{rot}\left(\boldsymbol{r}\mathcal{F}_{lmk}(\boldsymbol{r})\right)}{-i\sqrt{l(l+1)}}\
,$
$\displaystyle\boldsymbol{\mathcal{F}}_{lmk}^{(+)}(\boldsymbol{r})=\frac{\operatorname{rot}^{2}\left(\boldsymbol{r}\mathcal{F}_{lmk}(\boldsymbol{r})\right)}{-ik\sqrt{l(l+1)}}\
.$ (24)

where $\mathcal{F}_{lmk}=j_{l}(kr)Y_{lm}(\boldsymbol{n})$, $j_{l}$ is the
Bessel function and $Y_{lm}$ are the spherical harmonics. Their origin and
properties are explained in appendix D. For our purposes it is most important
to note that the so-called form-factors [4, 11, 12, 13]

$\displaystyle m^{(\lambda)}_{lm}(k)=\int
d\boldsymbol{r}\,\boldsymbol{j}(\boldsymbol{r})\left(\boldsymbol{\mathcal{F}}^{(\lambda)}_{lmk}(\boldsymbol{r})\right)^{*}$
(25)

directly encode both the multiple moments and mean-square radii a current
distribution. More specifically, form-factor $m^{(-)}_{lm}(k)$ corresponds to
the charge multipoles666This is clear for example from the fact that
$\boldsymbol{\mathcal{F}}^{(0)}$ and $\boldsymbol{\mathcal{F}}^{(+)}$ are
orthogonal to $\boldsymbol{j}_{\parallel}$., $m^{(0)}_{lm}(k)$ – to magnetic
multipoles and $m^{(+)}_{lm}(k)$ – to toroidal multipoles. Indices $lm$ in
each form-factor describe the multipole being probed (dipole, quadrupole
etc.), while $k$ defines the radial profile of each multipolar mode and
corresponds to the mean-square radii.

The multipoles themselves are proportional to form-factors at $k=0$. For
instance (up to the usual transformation between the Cartesian and spherical
basis)

$\displaystyle\boldsymbol{D}\propto m^{(-)}_{1,i}(k=0)\ .$ (26)

The mean square radii appear as coefficients in the Taylor expansion of
$m^{(\lambda)}_{lm}(k)$. For example, the mean-square radii of the dipole
moment (21) are proportional to

$\displaystyle
D^{(n)}\propto\frac{d^{2n}}{dk^{2n}}m^{(-)}_{1,i}(k)\Big{|}_{k=0}\ .$ (27)

Now, form-factors $m^{(\lambda)}_{lm}(k)$ are in general independent functions
and describe three independent multipole families. However it turns out that
functions $\boldsymbol{\mathcal{F}}_{lmk}^{(-)}$ and
$\boldsymbol{\mathcal{F}}_{lmk}^{(+)}$ have the same behavior at small $k$

$\displaystyle\boldsymbol{\mathcal{F}}_{lmk}^{(+)}\approx\sqrt{l(l+1)}\boldsymbol{\mathcal{F}}_{lmk}^{(-)},\qquad
k\to 0\ .$ (28)

As a consequence, the multipole moments defined by $m^{(-)}_{lm}(0)$ and
$m^{(+)}_{lm}(0)$ coincide (up to numerical factors). Crucially though,
$m^{(-)}_{lm}(k)$ and $m^{(+)}_{lm}(k)$ cease to agree beyond $k=0$ and,
hence, the charge and toroidal mean-square radii are in general all different.

The above analysis formalizes our heuristic derivation at the beginning of
this section, which showed that the toroidal dipole moment has the usual
electric dipole moment as its ”parent multipole” and that the same applies to
all toroidal multiples. Because of this coincidence, the toroidal family is
usually defined to start one order higher, i.e. the 1st mean square radii of
form-factors $m^{(+)}_{lm}(k)$ are considered to be the primary toroidal
multipoles [4]. In retrospect, it would be, perhaps, more consistent to refer
to the quantities $m^{(+)}_{lm}(k=0)$ (and thus $\propto m^{(-)}_{lm}(k=0)$)
as toroidal rather than charge multipole moments, because the mean-square
radii of the toroidal multipoles all contribute to radiation, while the charge
mean-square radii do not [9].

Finally, let us emphasize that the noted relation between the charge and
toroidal multipoles is conditioned by two factors. The first is the apparent
numerical coincidence777Form-factor functions (24) are in fact uniquely fixed
by Helmholtz decomposition together with requirements of irreducibility and
parity (42), so this apparent numerical coincidence has deeper roots.  stated
in Eq.(28). The second is that although charge multipole moments are
originally defined only via the longitudinal currents, while toroidal moments
only via transverse currents, the non-locality of the Helmholtz decomposition
imposes a connection between them. In particular, note that $k\to 0$ limit
(when the charge and toroidal form-factors agree) corresponds to the infinite-
wavelength limit. In this regime the distinction between
$\boldsymbol{j}_{\parallel}$ and $\boldsymbol{j}_{\perp}$ becomes negligible,
as they only differ in a finite region, which can not be resolved by a wave
with an infinitely large wavelength.

## Acknowledgments

We thank Alexey Basharin for useful discussions. The work of NN is partly
supported by Leading Research Center on Quantum Computing (Agreement no.
014/20).

##  Appendix A Maxwell’s equations

Here we collect our conventions regarding Maxwell’s equations. We work with
harmonic time-dependence, frequency is denoted by $\omega$, $k=\omega/c$ is
the wavenumber. Maxwell’s equations can be written as

$\displaystyle\boldsymbol{j}(\boldsymbol{r})=\frac{c}{4\pi}\left(-(k^{2}+\Delta)\boldsymbol{A}(\boldsymbol{r})+\boldsymbol{\nabla}\operatorname{div}\boldsymbol{A}(\boldsymbol{r})\right)\
,$ (29)

where $\boldsymbol{A}(\boldsymbol{r})$ is a vector potential in the Weyl gauge
(vanishing scalar potential). In this gauge the electric and magnetic fields
are related to the vector potential as follows

$\displaystyle\boldsymbol{E}(\boldsymbol{r})=ik\boldsymbol{A}(\boldsymbol{r}),\qquad\boldsymbol{H}(\boldsymbol{r})=\operatorname{rot}\boldsymbol{A}(\boldsymbol{r})\
.$ (30)

Conversely, assuming that the vector potential decays at infinity it can be
found by solving (29)

$\displaystyle\boldsymbol{A}(\boldsymbol{r})=\frac{1}{k^{2}c}\int
d\boldsymbol{r}^{\prime}\,G(\boldsymbol{r}-\boldsymbol{r}^{\prime})\left(k^{2}\boldsymbol{j}(\boldsymbol{r}^{\prime})+\boldsymbol{\nabla}\operatorname{div}\boldsymbol{j}(\boldsymbol{r^{\prime}})\right)\
,$ (31)

where
$G(\boldsymbol{r}-\boldsymbol{r}^{\prime})=\frac{e^{ik|\boldsymbol{r}-\boldsymbol{r}^{\prime}|}}{|\boldsymbol{r}-\boldsymbol{r}^{\prime}|}$
is the Green function for Helmholtz equation
$(\Delta+k^{2})G(\boldsymbol{r})=-4\pi\delta^{(3)}(\boldsymbol{r})$.

The transverse part of the current satisfies
$\operatorname{div}\boldsymbol{j}_{\perp}=0$ and gives rise to the transverse
part of the vector potential

$\displaystyle\boldsymbol{A}_{\perp}(\boldsymbol{r})=\frac{1}{c}\int
d\boldsymbol{r}^{\prime}\,G(\boldsymbol{r}-\boldsymbol{r}^{\prime})\boldsymbol{j}_{\perp}(\boldsymbol{r^{\prime}})\
.$ (32)

The longitudinal part of the current satisfies
$\boldsymbol{\nabla}\operatorname{div}\boldsymbol{j}_{\parallel}=\Delta\boldsymbol{j}_{\parallel}$
and, hence

$\displaystyle\boldsymbol{A}_{\parallel}(\boldsymbol{r})=-\frac{4\pi}{k^{2}c}\boldsymbol{j}_{\parallel}(\boldsymbol{r})\
.$ (33)

Outside of a localized source
$\boldsymbol{j}_{\perp}=-\boldsymbol{j}_{\parallel}=0$ and, therefore, both
$\boldsymbol{A}_{\perp}$ and $\boldsymbol{A}_{\parallel}$ can be expressed via
$\boldsymbol{j}_{\perp}$ alone.

##  Appendix B Non-radiating charge densities

In the main text we have been careful to make reservations while claiming that
any charge density will be consequential for the radiation fields. There is a
notable exception. Let us reconsider the Poisson equation (3) and ask whether
there exist a charge density $\rho_{0}$ which actually does not produce a
potential $\phi_{0}$ (and hence the electric field) outside its domain of
definition. The answer is yes, and it is simple to describe all such
densities. Assume that $\rho_{0}$ is a Laplacian of some function $\phi$, i.e.
$\rho_{0}=\Delta\phi$ and $\phi$ is zero outside $R$. Then from (3) it follows
that the potential $\phi_{0}$ is just equal to $\phi$ up to constants and
hence is itself confined (the argument is almost tautological).

This implies the existence of charge densities that do not produce any
electric fields outside. Translated into the language of the Helmholtz
decomposition this means that some modifications of the longitudinal current
do not affect the transverse part. Namely, one can add to $\boldsymbol{j}$ a
term of the form $\boldsymbol{\nabla}\phi$ with any confined function $\phi$.

##  Appendix C Volume integral of longitudinal currents

Here we show that for a spatially confined current $\boldsymbol{j}$ its
longitudinal part $\boldsymbol{j}_{\parallel}$ satisfies

$\displaystyle\int d\boldsymbol{r}\,\boldsymbol{j}_{\parallel}=0\ ,$ (34)

which makes it possible to express all multipole terms as integrals of
$\boldsymbol{j}_{\perp}$ alone. As explained in Sec. I relation between
$\boldsymbol{j}$ and $\boldsymbol{j}_{\parallel}$ arises through a Poisson
equation

$\displaystyle\Delta\psi(\boldsymbol{r})=\operatorname{div}\boldsymbol{j}(\boldsymbol{r}),\qquad\boldsymbol{j}_{\parallel}(\boldsymbol{r})=\boldsymbol{\nabla}\psi\
,$ (35)

which is solved by

$\displaystyle\psi(\boldsymbol{r})=-4\pi\int
d\boldsymbol{r}^{\prime}\,\frac{\operatorname{div}\boldsymbol{j}(\boldsymbol{r}^{\prime})}{|\boldsymbol{r}-\boldsymbol{r}^{\prime}|}\
.$ (36)

Because the source is spatially confined the range of
$\boldsymbol{r}^{\prime}$ is bounded and for large enough $\boldsymbol{r}$ we
can write

$\psi(\boldsymbol{r})=-\frac{4\pi}{r}\int
d\boldsymbol{r}^{\prime}\operatorname{div}\boldsymbol{j}(\boldsymbol{r}^{\prime})+\\\
\frac{4\pi}{r^{2}}\int
d\boldsymbol{r}^{\prime}(\boldsymbol{n},\boldsymbol{r^{\prime}})\operatorname{div}\boldsymbol{j}(\boldsymbol{r}^{\prime})+O\left(\frac{1}{r^{3}}\right),$
(37)

where $\boldsymbol{n}=\boldsymbol{r}/r$. The first term here is zero by the
Gauss theorem, while the second can be rewritten

$\displaystyle\psi(\boldsymbol{r})=\frac{4\pi}{r^{2}}(\boldsymbol{n},\boldsymbol{V})+O\left(\frac{1}{r^{3}}\right),\quad\boldsymbol{V}=\int
d\boldsymbol{r}^{\prime}\boldsymbol{r^{\prime}}\operatorname{div}\boldsymbol{j}(\boldsymbol{r}^{\prime})\
.$ (38)

Here $\boldsymbol{V}$ is an analog of the dipole moment but for our purposes
this interpretation is not important. $\boldsymbol{V}$ is just a constant
vector characterizing current distribution $\boldsymbol{j}$.

Now we are ready to prove (34). Consider

$\int d\boldsymbol{r}\,\boldsymbol{j}_{\parallel}=\int
d\boldsymbol{r}\,\boldsymbol{\nabla}\psi=\\\ \lim_{R\to\infty}\int_{r=R}4\pi
r^{2}d\boldsymbol{n}\left(\frac{(\boldsymbol{n},\boldsymbol{V})}{r^{2}}+O\left(\frac{1}{r^{3}}\right)\right)\
.$ (39)

The leading term vanishes due to identity $\int
d\boldsymbol{n}\,\boldsymbol{n}=0$ (the average of the normal vector over a
unit sphere is zero), while the subleading terms vanish in the $R\to\infty$
limit. This establishes (34).

##  Appendix D Multipole form-factors

Functions $\boldsymbol{\mathcal{F}}_{lmk}^{(\lambda)}$ introduced in (24) are
regular solutions to the Helmholtz equation

$\displaystyle(\Delta+k^{2})\boldsymbol{\mathcal{F}}_{lmk}^{(\lambda)}=0$ (40)

and satisfy orthogonality and completeness relations

$\displaystyle\int
d\boldsymbol{r}\,\boldsymbol{\mathcal{F}}^{(\lambda)}_{lmk}(\boldsymbol{r})\cdot\boldsymbol{\mathcal{F}}^{*(\lambda^{\prime})}_{l^{\prime}m^{\prime}k^{\prime}}(\boldsymbol{r})=\frac{(2\pi)^{3}}{k^{2}}\delta^{\lambda\lambda^{\prime}}\delta_{ll^{\prime}}\delta_{mm^{\prime}}\delta(k-k^{\prime})\
,$
$\displaystyle\sum_{lmk\lambda}\left[\boldsymbol{\mathcal{F}}^{(\lambda)}_{lmk}(\boldsymbol{r})\right]_{i}\left[\boldsymbol{\mathcal{F}}^{*(\lambda)}_{lmk}(\boldsymbol{r}^{\prime})\right]_{j}=(2\pi)^{3}\delta_{ij}\delta(\boldsymbol{r}-\boldsymbol{r}^{\prime})\
.$ (41)

Under parity transformations $\boldsymbol{x}\to-\boldsymbol{x}$ they behave as

$\displaystyle\mathcal{F}_{lmk}^{\lambda}(-\boldsymbol{x})=(-)^{l+\lambda}\mathcal{F}_{lmk}^{\lambda}(\boldsymbol{x})\
.$ (42)

The scalar functions $\mathcal{F}_{lmk}=j_{l}(kr)Y_{lm}(\boldsymbol{n})$ that
are used to define $\boldsymbol{\mathcal{F}}^{(\lambda)}_{lmk}$ also solve the
Helmholtz equation and have the following $k\to 0$ asymptotic

$\displaystyle\mathcal{F}_{lmk}(\boldsymbol{r})=\frac{4\pi
i^{l}}{(2l+1)!!}(kr)^{l}Y_{lm}(\boldsymbol{n})(1+O(k))\ ,$ (43)

from which (28) can be derived.

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