On the dynamic toroidal multipoles from localized electric current distributions

We analyze the dynamic toroidal multipoles and prove that they do not have an independent physical meaning with respect to their interaction with electromagnetic waves. We analytically show how the split into electric and toroidal parts causes the appearance of non-radiative components in each of the two parts. These non-radiative components, which cancel each other when both parts are summed, preclude the separate determination of each part by means of measurements of the radiation from the source or of its coupling to external electromagnetic waves. In other words, there is no toroidal radiation or independent toroidal electromagnetic coupling. The formal meaning of the toroidal multipoles is clear in our derivations. They are the higher order terms of an expansion of the multipolar coefficients of electric parity with respect to the electromagnetic size of the source.

I. CONTRIBUTIONS TO b ω jm , a ω jm AND c ω jm Equation (14) in [1] is an exact expression for the {a ω jm , b ω jm , c ω jm } coefficients in terms of integrals in both momentum and coordinate space. With Q jm (p) standing for any of the {X jm (p), Z jm (p), W jm (p)} and q ω jm for any of the corresponding {a ω jm , b ω jm , c ω jm }, Eq. (14) in [1] reads where j l (·) are the spherical Bessel functions of the first kind.

II. CANCELLATION OF LONGITUDINAL FIELDS OUTSIDE THE SOURCE
We show that the longitudinal field with |p| = ω/c is zero outside the source region.
We consider spatially confined monochromatic electric charge and current density distributions ρ ω (r) and J ω (r) embedded in an isotropic and homogeneous medium with constant permittivity and permeability µ. We assume them to be confined in space. In the Lorenz gauge, the scalar and vector potentials meet the following inhomogeneous wave equations: where c = 1/ √ µ. Outside the source region, it can be shown that the spatial Fourier transforms of φ ω (r) and A ω (r) are non-zero only for |p| = ω/c (see [ In coordinate space, the electric field as a function of the potentials is which, in momentum space (∇ → ip) reads The longitudinal electric field is hence Using Eq. (4) and that p is restricted to p = ω cp we can write Eq. (7) as The term inside the brackets in Eq. (8) is equal to zero because of the continuity equation in momentum space We conclude that, outside the source region, the longitudinal field with |p| = ω/c produced by the current density exactly cancels the one produced by the charge density. Note that the result is gauge independent since it is a statement about the electric field.

III. THE SPLIT OF ELECTRIC AND TOROIDAL PARTS INTRODUCES OUT OF SHELL COMPONENTS IN BOTH OF THEM
In this appendix we show that the independent measurement of electric and toroidal parts is impossible.
Let us consider the expression of the exact frequencydependent multipoles of electric parity a ω jm in Eq. (2). The monochromatic current J ω (r) appears in two different spatial integrals, and where k = ω/c, j l (·) are spherical Bessel functions, r = |r|,r = r/|r|, and Y ln (·) are scalar spherical harmonics.
Let us now split Eq. (11) into two parts by means of the small argument expansion of j j−1 (kr). We isolate the first term of the expansion, which is of order (kr) j−1 and obtain: where n!! = n(n − 2)(n − 4) . . . is the double factorial. As we will now show, this is the split that gives rise to the electric and toroidal parts in the original literature [3,4]. The first term in Eq. (13) corresponds to the electric part, and the second term is contained in the toroidal part. The toroidal part also contains the whole contribution of the integrals involving j j+1 (kr) in Eq. (12).
Let us now see this splitting in the original literature [3,4]. We start from the definition of the time-dependent exact multipoles of electric parity, which can be written from Eqs. 20 and 24 in Ref. 3 where Y jlm (·) are vector spherical harmonics.
where the electric part is (Eq and T jm (k, t) is the toroidal part.
It can be seen from Eqs.
(14) to (16) that the integrand that defines T jm (k, t) must contain the j j+1 (kr)Y j,j+1,m (r) contribution plus the j j−1 (kr)Y j,j−1,m (r) contribution except for the first term in the small argument expansion of j j−1 (kr), which is of order r j−1 and has been split up. This splitting corresponds to the one we have performed in Eq. (13). It causes the appearance of out of shell components in both electric and toroidal parts.
As mentioned in the main text and explained at the end of Sec. III in Ref. 1, the spherical Bessel functions inside the spatial integrals act as a filter that completely reject the out of shell (|p| = ω/c) components of the current. After the split, the term proportional to (kr) j−1 by itself does not provide such rejection. This can be appreciated in the first line of Eq. (13): r j−1 Y * j−1m (r) is a frequency independent function which cannot remove the |p| = ω/c components present in J ω (r). Multiplication by a factor of k j−1 does not change this. Since a ω jm are physically measurable quantities without out of shell components, the presence of |p| = ω/c components in the electric part implies their presence in the toroidal part with opposite sign, as it is obvious from Eq. (13).
The out of shell components do not couple to the electromagnetic field, and therefore preclude the independent physical measurement of the electric and toroidal parts.

IV. AN ALTERNATIVE DEFINITION OF THE TOROIDAL MULTIPOLES
In this appendix we first show that the definition of toroidal multipoles recently given in [6, Box 2] is different from the original definition in [3,4]. Both definitions involve the split of the multipoles of electric parity into two parts, but the splits are different in [6, Box 2] and [3,4]. We demonstrate this by showing that the well known expression of the toroidal dipole in the limit of small source [4, Eq. 2.11] cannot be recovered from the definition of [6, Box 2]. Some terms are missing. We then also show that the missing terms are contained in a coefficient which is explicitly excluded from the definition of toroidal multipoles in [6,Box 2]. The derivations in this appendix recover one of the results from the main text: The toroidal dipole is just the next to leading order term in the small source expansion of the exact electric dipole. The difference is that here the result is obtained directly in coordinate (r) space, while Eq. (4) from the main text was obtained in [1] by first going to momentum (p) space, and then going back to r space. Finally, we show that, in this alternative definition, both parts contain longitudinal terms, which render them non-separable.
We start by examining the definitions in [6, Box 2] where E sca (r) is the field produced by the sources, and Ψ jm (r) and Φ jm (r) are the multipolar fields of electric and magnetic parity, respectively. The Q jm are said to be charge excitations yielding electric multipoles, the M jm transverse (w.r.t r) current excitations yielding magnetic multipoles, and the T jm radial current excitations yielding toroidal multipoles. We first note that Q jm and T jm are both multiplying the same multipolar field of electric parity Ψ jm (r). This implies that, together, Q jm and T jm must completely determine the multipolar coefficients of electric parity. This can be readily checked by setting the magnetic currents to zero in Jackson's [7, Eq. 9.167] expression for the exact multipoles of electric parity a E (j, m), namely: It is clear from Eq. (17) that Q jm corresponds to the first term of the sum in Eq. (18), and iT jm to the second term. The sum Q jm +iT jm determines a E (j, m). The definition of T jm in Eq. (17) involves a split of the a E (j, m) into two parts. We now show that it is a different split from the one in the original definition of toroidal multipoles [3,4].
Let us use iT 1m to attempt to recover the toroidal dipole in the small source approximation [4, Eq. 2.11], and which follows from the original definition ([3, Eq. 39], [4, Eq. 2.11]). We start by arranging the three components corresponding to m = {1, 0, −1} into a vector, and consider the correspondence between Y * 1m (r) andr in the spherical basiŝ (22) Equation (21) is a vector in the spherical vector basis except that, when both r and J ω (r) are expressed in the spherical vector basis, the term [r · J ω (r)] should be written r † J ω (r) , where r † denotes the hermitian conjugate of the position vector r. This is due to the fact that, in the spherical vector basis, r has complex components. In the Cartesian basis used in [6,Box 2] and Eq. (20), r is real valued and the dot product r † J ω (r) can be written as [r · J ω (r)]. In this appendix we will use the Cartesian basis from now on. We can change T sph 1 to the Cartesian basis by multiplying T sph 1 itself, and the other vectors involved in the expression with the change of basis matrix After using Eq. (22) and changing the basis, the Cartesian expression reads: We now use j 1 (kr) ≈ kr/3 in the limit of small kr, to approximate Eq. (24) by  [3,4]. We now show that the missing terms are contained in Q 1m : They are the terms of second lowest order in the small source approximation of Q 1m . This reproduces our results from the main text.
We start with take the same steps as before regarding vectors and basis, and use the continuity equation ρ ω (r) = ∇ · J ω (r)/(ikc), to obtain: We now consider the term in the shaded box, which, using the derivative of spherical Bessel functions 1 can be written as We now plug Eq. (31) into the right hand side of Eq. (28) 1 d dx j l (x) = 1 2l+1 (lj l−1 (x) − (l + 1)j l+1 (x))