1. Introduction

This Letter is motivated by the observations of toroidal dipole interactions in metamaterials [¹,²] and nanomaterials [³], and attempts to establish the difference between electric and toroidal moment electromagnetic radiations using the familiar spherical multipole expansions in the far-away zone [⁴]. The works in [¹-³] and [⁵,⁶] have cited our work [⁷] on the complete electromagnetic multipole expansion including toroidal moments, which is complete on two accounts: 1) it describes the fields inside and outside a spherical toroidal surface where the sources are confined, and 2) it includes the toroidal moments. The use of the multipole expansion in the far-away zone ignores the roles of the sources and of Maxwell’s equations in the boundary condition form, which are indispensable to connect the inner and outer solutions satisfying the homogeneous Helmholtz equation. The citations of our work motivated us to revisit it [⁸], and we have just realized that when going from sources confined on a spherical surface [⁹] to sources confined on a spherical toroidal surface, the spherical symmetry of the first one is broken in the second one. Consequently, this contribution is focused on the manifestations of the symmetry breaking and its general consequences, which hopefully will be particularly useful for the metamaterials and nanomaterials communities. We may add that we have just completed the identification of the complete and exact solutions of a family of electromagnetic multipole moments in cylindrical toroids as reported in [¹⁰], which has been an important point of reference and guidance in our writing of this Letter. It is also important to mention that Refs. [¹-³] were reviewed in [¹¹], recognizing that the ideas of Superintegrability and Symmetry Breaking apply not only for atoms and molecules, but also to the electromagnetic multipole expansion, when its sources are confined on toroidal surfaces.

The confinement of the sources and Dirichlet and Neumann boundary conditions for resonant cavities and optimum-efficiency antennas, respectively, lead to quantization of wave numbers and frequencies, and to complete and orthonormal sets of eigenfunctions. Symmetry breaking involves the same constants of motion but different eigenvalues: l becomes λ with non integer values. Correspondingly, the associated Legendre Polynomials become Legendre functions with (λ,m) parameters in their hypergeometric function representation, with a well defined parity for symmetric toroids. The radial spherical functions become R _λ , wellbehaved superpositions of spherical Bessel functions j _λ and n _λ inside the toroid, and outside j _λ in the hole and hλ(1) outgoing spherical wave.

The longitudinal, toroidal and poloidal current densities of a given multipolarity, on each section of the spherical toroidal surface, are constructed from a common scalar Debye potential involving one eigenfunction of the sets mentioned above, via the application of the gradient ∇, angular momentum L = −i r×∇ and curl ∇× operators, in the succession already identified for cylindrical toroids [¹⁰]. They are the sources of the electric intensity fields in the inhomogeneous Helmholtz equation. The complementary sources of the magnetic induction field in the same equation are also identified.

The Green functions for the spherical toroids are also constructed using the bases of the above-mentioned radial and polar angle eigenstates, as the counterpart of those in [¹⁰]. In [⁷] the point unit charge Green function was used, with the sources of Eqs. (23) and (24) involving Heaviside step functions, representing uniform distributions on each section of the spherical toroidal surface. It is appropriate at this point to recognize that the orthonormality and completeness of the radial and polar angle eigenfunctions at the boundary defined by the other variable provide the representations of the difference of the respective Dirac-delta functions, and that their integrations become the difference of the Heaviside step functions. These relationships are important to understand the connections between the Green functions and sources in [⁷,⁸] and this contribution, as well as the equivalence of their integrated results.

Consequently, the successive sections of this Letter in 2. Spherical Symmetry Breaking, explain the reasons behind it and identify its explicit manifestations, 3. Sources Distributed on the Surfaces of Spherical Toroids, identify them from a common Debye scalar potential for the successive electric, magnetic and toroidal moments on the basis of their multipolarities and directionalities, including their successive connections, 4. Scalar Green Functions, to be constructed using the solutions of the homogeneous Helmholtz equations inside and outside the toroidal surface, satisfying Dirichlet boundary conditions for resonant cavities and Neumann boundary conditions for optimum-efficiency antennas, respectively, and 5. Discussion of the anticipated results for the electromagnetic fields, to be evaluated from the integrations of the sources in Sec. 3 and the Green functions in Sec. 4. The Appendix serves as a guide to the readers interested in understanding the details of the text.

2. Spherical symmetry breaking

The restriction of distributing the sources on the surface of a spherical toroid formed by spherical concentric rings of radius r = α and r = b, and symmetric conical rings at the polar angles θ = θ ₁ y θ ₂ = π − θ ₁, and 0 < ( < 2π, in spherical coordinates, allows the construction of solutions of Maxwell equations using the basis of solutions for the scalar homogeneous Helmholtz equation, inside and outside the toroid, as Debye potentials subject to Dirichlet and Neumann boundary conditions for the electromagnetic radiation fields in resonant cavities and optimum-efficiency antennas, respectively. In general, for arbitrary values of the polar angle θ ₁, the above mentioned boundary conditions can not be satisfied by the familiar associated Legendre polynomials of degree (ℓ,m) . The quantum label ℓ takes non integer values λ, and the polynomials are replaced by Legendre functions of order λ, associativity m, and definite parity, represented as infinite series superpositions of hypergeometric functions with arguments (1−cosθ)/2 and (1+cosθ)/2, with common labels in λ and m = 1:

The order λ is determined by the nodes and the extreme points of the series in θ₁. The companion radial functions are superpositions of the corresponding ordinary and Neumann type spherical Bessel functions inside the toroid r ∈ [α,b]:

ordinary spherical Bessel function in the hole of the toroid r ∈ [0,α], and outgoing wave spherical Bessel function outside r ∈ [b,∞]. The respective boundary conditions on the radial eigenfunctions at the inner and outer spherical rings can be satisfied only for specific choices of coefficients A _λ (k), B _λ (k), leading to resonant wave numbers and their associated frequencies, which turn out to be interlaced for the successive modes of the cavities and antennas. Additionally, their nodal lines combined with those of their companion polar eigenfunctions determine the positions of foursided perimeters in each meridian plane; as well as the corresponding positions of extreme amplitudes inside the perimeter. The product of the radial and polar eigenfunctions apply to the magnetic moments. For the electric and toroidal moments there is an extra radial factor, argument of the spherical Bessel functions, as justified at the end of Sec. 3. Reference [¹¹] illustrates the symmetry breaking in the context of confined atoms and molecules, and also included reviews of [¹-³].

3. Sources distributed on the surfaces of spherical toroids

The sources of the electromagnetic fields distributed on the surface of the spherical toroids are identified from a common Debye scalar potential, for each spherical and conical ring sections at the respective positions r = a, b and θ = θ₁, θ₂, respectively; with multipole distributions and directionalities to be described next for the successive multipole moments. For the electric moments, the starting point is the longitudinal current density with a gradient of the Debye scalar potential of the chosen multipolarity:

Its divergence leads to the surface charge density via the continuity equation,

In turn, the cross product of (r sinθ)φ^ and r + rθ^) with the longitudinal current density leads to the poloidal current density K ^Ep of the electric intensity field, and the source density S ^Et of the magnetic induction, in the respective Helmholtz equations:

In turn, the curl of K ^Ep becomes the toroidal current density for the magnetic moments K ^Mt as the source density of the electric intensity field:

Additionally, the curl of K leads to the poloidal source density S ^Mp of the magnetic induction field:

Similarly, the curl of S ^Et is identified as the poloidal current density of the toroidal moments K ^Tp , source of E ^Tp :

Additionally, the curl of K ^Tp yields the toroidal source density S ^Tt for B ^Tt :

Notice that the longitudinal, poloidal and toroidal source densities involve first derivatives of the radial and polar angle eigenfunctions, for the electric moments. The curls of the last two transverse current densities yield the toroidal and poloidal current densities, for the magnetic and toroidal moments, respectively, involving second derivatives associated with the Laplace operator, and yielding the radial and polar eigenvalues and eigenfunctions of the Helmholtz equation. In turn, the additional curls of the current densities lead to the source densities for the respective magnetic induction fields, with third order derivatives in the radial and polar angle eigenfunctions. These dependences are behind the difference by a factor of k² between the electric and toroidal moment electromagnetic fields, sharing otherwise the same space dependences in their transverse fields. as well as their resonant frequencies for cavities and antennas. On the other hand, the electromagnetic fields for magnetic moments differ from the other two by exchange of the directionalities of their electric intensity and magnetic induction fields, and even versus odd orders of derivatives, as well as different resonance frequencies. In the case of electric moments, it is important to point out the presence of the common radial factor in the vectors r, rθ^ and r sinθφ^, which via their successive vector products with the longitudinal current density yield the respective components of the toroidal source of B ^Et and the poloidal current source of E ^Ep . The extra factor of the spherical Bessel functions mentioned in Sec. 2 for electric and toroidal moments is thus justified.

4. Green functions

This section contains the scalar Green functions of the Helmholtz equation with a unit charge at the point r’, on the toroidal surface, using the inner solutions of the homogeneous equation, Eqs. (1) and (2), satisfying the Dirichlet boundary condition for the resonant cavities, and the Neumann boundary conditions for the optimum-efficiency antennas, as well as for the near fields produced by the longitudinal current and charge densities in the electric moments:

The outgoing Green function in (11) uses the same angular bases and the linear superposition of the products of the inner and outer Bessel functions with coefficients gnrnθ(k), taking into account the quantization and order of excitation in the respective degrees of freedom. The coefficients are evaluated by integrating the radial part of the Helmholtz equation involving the Wronskian of the inner and outer spherical Bessel functions:

5. Discussion

This section describes the anticipated results for the electromagnetic fields to be obtained from the integration of the source densities identified in Sec. 3 and the Green functions of Sec. 4, for the successive magnetic, toroidal and electric moments. This order is followed because the first two share transverse sources and only radiation fields, while the last one also involves the longitudinal current and charge densities and their near fields in the vicinity of the spherical toroid. The presence of the latter illustrates clearly the difference between the toroidal and electric moments.

For the magnetic moments, the toroidal surface current densities K ^Mt yield the toroidal electric intensity field E(r,θ,φ), while their poloidal curls, S ^Mp , yield the poloidal magnetic induction fields B(r,θ,φ), for the interior of the resonant cavities and for the interior and exterior of the toroidal optimum-efficiency antennas, respectively. The vanishing of the radial functions inside the toroidal surface at the inner and outer spherical rings of radii a and b, required by the Dirichlet boundary condition, guarantees that the outer fields vanish in the first case. On the other hand, in the second case, the extensions of the radial functions with Neumann boundary conditions to the outer intervals, [0,α] in the hole and [b,∞] to the far-away zone, guarantee their proper behaviors in their respective domains. In both cases, the electric intensity field turns out to be consistent with Gauss’ and Faraday’s laws, while the magnetic induction field is also consistent with Gauss’, Ampere’s and Maxwell’s laws in their boundary condition forms. Additionally, their field lines may be written in their differential forms and turn out to be integrable for the poloidal fields, yielding their vortex structure in each meridian plane within four-sided radial and circular separatrices.

For the toroidal moments, the poloidal current densities K ^Tp yield the poloidal electric intensity fields E ^Mp , while the toroidal source S ^Tt yields the toroidal magnetic induction field B ^Mt . Notice the changes in directionalities of the respective sources and fields when going from the magnetic multipoles to the toroidal ones, directly connected with the connections of the fields themselves via Faraday and Maxwell laws in both cases. This observation is pertinent here, in order to avoid the repetition of the discussion of the last part in the previous paragraph.

For the electric moments, the difference with respect to the toroidal moments, besides the extra k² in B ^Mt compared to B ^Et , is obviously the presence of the longitudinal current density and its associated charge density. The evaluation of the longitudinal electric field from the integration of the longitudinal current density and the respective Green functions for the resonant cavities and antennas, E ^El , yields the surface charge density distribution at the toroidal surface via Gauss’s law in its boundary condition form; on the other hand, its normal components at the first separatrices of the radiation fields, inside and outside the toroid, vanish at the separatrices becoming tangential near them, and consistent with Faraday’s law. The reader may see their counterparts for the sphere in [⁹], and for circular cylindrical toroids in [¹⁰].

The analytical elements described in the successive sections are the basis for the numerical computations of the respective electromagnetic fields, for specific values of the geometrical parameters of the spherical toroids. The corresponding results will be reported in the near future.

The comparison of the contents of this Letter and our previous works may help the readers to appreciate the differences in them. In fact, the Letter is the natural extension of [⁷,⁸], recognizing explicitly the spherical symmetry breaking and its implementation. The comparison with [⁹] shows their common structure allowing for the difference in their respective inner and outer solutions. In the comparison with [¹⁰], its separation into independent axial and circular radial components is replaced by the separation into spherical radial and common polar angle components; in the first one, both degrees of freedom determine the resonant frequencies, and in the second one, the spherical radial component determines by itself the resonant frequencies.

We conclude this Letter by pointing out that its contents provide the new elements of spherical symmetry breaking eigenfunctions of the scalar homogeneous Helmholtz equations, the connections between the sources of the successive electromagnetic multipole fields for electric, magnetic and toroidal moments, and the respective Green functions for spherical toroidal resonant cavities and optimum-efficiency antennas, needed for the evaluation of the respective electric intensity and magnetic induction fields. One of its important results is the identification of the difference between the toroidal and electric moments in the presence of the localized near field in the latter produced by the longitudinal current and its associated charge density. Hopefully, this will be useful and of interest for the Metamaterials and Nanomaterials communities, by providing exact results to compare with their different interpretations of the toroidal moments, as illustrated in [⁴].