Axially symmetric solutions of the Maxwell equations are quite important, at least as an often relevant approximation. For instance, axisymmetric magnetic fields occur naturally as produced by systems possessing an axis of revolution, such as disks or coils [1,2], or astrophysical systems like accretion disks  or disk galaxies . Axisymmetric solutions are also used to model EM beams and their propagation (e.g., refs. [5,6]). In particular, nondiffracting beams are usually endowed with axial symmetry, see, e.g., refs. [7,8,9]. Naturally, one often considers time-harmonic solutions, since a general time dependence is obtained by summing such solutions. Two classes of time-harmonic axisymmetric solutions of the free Maxwell equations, mutually associated by EM duality, have been introduced recently . The main aim is to “describe the propagation of nonparaxial scalar and electromagnetic beams in exact and analytic form.” However, as noted by the authors of ref. , the analytical expression for a totally propagating time-harmonic axisymmetric solution Ψ of the scalar wave equation, from which they start [equation (1) below], covers all such solutions  – thus not merely ones corresponding to nonparaxial scalar beams. The first class of EM fields defined in ref.  is obtained by associating with any such scalar solution Ψ a vector potential A by equation (10) below . The second class is deduced from the first one by EM duality .
The aim of the present work is to show that, by combining these two classes, one is able to describe all totally propagating time-harmonic axisymmetric EM fields – and thus, by summing on frequencies, all totally propagating axisymmetric EM fields. To this aim, we shall prove the following theorem: Any time-harmonic axisymmetric EM field (whether totally propagating or not) is the sum of two EM fields, say (E1, B1) and
This article is organized as follows. Section 2 presents and comments on the results of ref. , and somewhat extends them in particular by noting that equations (11)–(13) apply to any time-harmonic axisymmetric solution of the scalar wave equation. Also, equations (15)–(20) are new. Section 3 presents the main results of this work – it gives the proof of the theorem. That proof is not immediate but uses standard mathematics with which one is familiar from classical field theory. Section 4 summarizes main results and presents a method to obtain an explicit representation for all totally propagating axisymmetric solutions of the free Maxwell equations.
2 From scalar waves to Maxwell fields
2.1 Axially symmetric scalar waves
We adopt cylindrical coordinates ρ, ϕ and z about the symmetry axis, that is, the z axis. Any totally propagating, time-harmonic, axisymmetric solution of the scalar wave equation (of d’Alembert) can be written [9,10] as a sum of scalar Bessel beams:
Thus, the (axial) “wave vector spectrum” S is a (generally complex) function of the real variable k = kz (−K ≤ k ≤ K), that is, the projection of the wave vector on the z-axis. This function S determines the spatial dependency of the time-harmonic solution (1) in the two-dimensional space left by the axial symmetry, i.e., the half-plane (ρ ≥ 0, z ∈] −∞, +∞[). Thus, any totally propagating, time-harmonic, axisymmetric scalar wave Ψ can be put in the explicit form (1), in which no restriction has to be put on the “wave vector spectrum” S (except for a minimal regularity ensuring that the function Ψ is at least twice continuously differentiable: the integrability of S, S ∈ L1([−K, + K]), would be enough). The general totally propagating axisymmetric solution of the scalar wave equation can be obtained from (1) by an appropriate summation over a frequency spectrum: an integral (inverse Fourier transform) in the general case or a discrete sum if a discrete frequency spectrum (ωj)(j = 1,…,Nω) is considered for simplicity:
2.2 Reminder on time-harmonic free Maxwell fields
In this section, we recall equations more briefly recalled in ref. . The electric and magnetic fields in SI units are given in terms of the scalar and vector potentials V and A by
2.3 Time-harmonic axisymmetric fields: from a scalar wave to a Maxwell field
An “azimuthally polarized” solution (E′, B′) (in the sense that
Now we observe this: the fact that the function Ψ have the form (1) plays no role in the derivation of the exact solution (11)–(13) to the free Maxwell equations. The only relevant fact is that Ψ = Ψ(t, ρ, z) is a time-harmonic axisymmetric solution of the scalar wave equation. Thus, with any axisymmetric time-harmonic solution Ψ of the scalar wave equation, we can associate two axisymmetric time-harmonic solutions of the free Maxwell equations: the solution (11)–(13) and the one deduced from it by the duality (14). These two solutions will be called here the “GAZR1 solution” and the “GAZR2 solution,” respectively, because both were derived in ref.  – although this was then for a totally propagating solution having the form (1), and we have just noted that this is not necessary.
It is implicit that, in equations (11)–(13), Bϕ, Eρ and Ez are the real parts of the respective right hand side [as are also E and B in equations (3), (4) and (9)]. Thus, if Az is totally propagating and hence may be written in the form (1), then by dJ0/dx = −J1(x), we obtain the following equations:
3 From Maxwell fields to scalar waves
Now an important question arises: Do the GAZR solutions generate all axisymmetric time-harmonic solutions of the Maxwell equations (in which case, by summation on frequencies, they would generate all axisymmetric solutions of the Maxwell equations)? Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. Can one find a GAZR1 solution and a GAZR2 solution, whose sum gives just that starting solution?
Note from equations (11)–(14) that the GAZR1 solution and the GAZR2 solution are complementary: in cylindrical coordinates, the GAZR1 solution provides non-zero components Bϕ, Eρ and Ez, the other components Eϕ, Bρ and Bz being zero – and the exact opposite is true for the GAZR2 solution. In view of this complementarity, we can consider separately the two sets of components: Bϕ, Eρ and Ez on one side and Eϕ, Bρ and Bz on the other side.
3.1 Sufficient conditions for the existence of the decomposition
For the “GAZR1” solution, which gives nonzero values to the first among the two sets of components just mentioned, we have the following result:
Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. In order that a time-harmonic axisymmetric solution (A1z, B1ϕ, E1ρ, E1z; E1ϕ = B1ρ = B1z = 0), of the form (11)–(13), and having the same frequency ω as the starting solution (A, E, B), be such that B1ϕ = Bϕ, E1ρ = Eρ, E1z = Ez, it is sufficient that we have just
Let A1z(t, ρ, z) be a time-harmonic axisymmetric solution of the wave equation, with frequency ω, and assume that B1ϕ as defined by equation (11) [with A1z in the place of Az] is equal to Bϕ, where B is defined by equation (4). That is, assume that
It follows for the dual (“GAZR2”) solution:
Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. In order that a time-harmonic solution (A2z,
The GAZR2 solution
3.2 Generality of the decomposition
Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. There exists a time-harmonic axisymmetric solution A2zof the wave equation, with the same frequency, such that the associated GAZR2 solution
In view of Corollary 1, we merely have to prove that there exists a time-harmonic axisymmetric solution A2z of the wave equation, such that equation (37) is satisfied. From equations (9) and (25), we have
Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. There exists a time-harmonic axisymmetric solution A1zof the wave equation, with the same frequency, such that the associated GAZR1 solution (E1, B1), deduced from A1zby equations (11)–(13), satisfy
Let A1z be a time-harmonic axisymmetric solution of the wave equation, and consider:
- (i)the GAZR1 solution defined from A1z by equations (11)–(13) (thus with A1z, B1ϕ, E1ρ, E1z,… instead of Az, Bϕ, Eρ, Ez,…, respectively).
- (ii)the GAZR2 solution defined from the same A1z by applying the duality (14) to the said GAZR1 solution:
Accounting for Proposition 2 and Corollary 2, and remembering the “complementarity” of the GAZR1 and GAZR2 solutions, we thus can answer positively to the question asked at the beginning of this section.
Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. There exist a unique GAZR1 solution (E1, B1) and a unique GAZR2 solution
Thus, the uniqueness of the representation concerns the electric and magnetic fields. It does not concern the potentials A1 = A1zez and A2 = A2zez that generate (E1, B1) and
4 Discussion and conclusion
The authors of ref.  introduced two classes of axisymmetric solutions of the free Maxwell equations, and they showed that these two classes allow one to obtain nonparaxial EM beams in explicit form. It has been proved that, by combining these two classes, one can define a method that allows one to get all totally propagating, time-harmonic, axisymmetric free Maxwell fields – and thus, by the appropriate summation on frequencies, all totally propagating axisymmetric free Maxwell fields. This method results immediately from the aforementioned theorem and from the general form (1) of a totally propagating, time-harmonic, axisymmetric solution of the scalar wave equation. However, that theorem is not an obviously expected result, and its proof is not immediate. We thus have now a constructive method to obtain all totally propagating axisymmetric free Maxwell fields. Namely, considering a discrete frequency spectrum
In the forthcoming work, we shall apply this to model the interstellar radiation field in a disc galaxy as an (axisymmetric) exact solution of the free Maxwell equations. In this application, it is very important that, due to the present work, one knows that any (totally propagating) axisymmetric free Maxwell field can be obtained in this way.
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Beware that instead K := 2ω/c in ref. . Our notation seems as natural and gives more condensed formulas.
The explicit expression of the corresponding vector potential