An explicit representation for the axisymmetric solutions of the free Maxwell equations

Mayeul Arminjon

doi:10.1515/phys-2020-0117

Article Open Access

An explicit representation for the axisymmetric solutions of the free Maxwell equations

Mayeul Arminjon

Published/Copyright: June 29, 2020

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From the journal Open Physics Volume 18 Issue 1

Abstract

Garay-Avendaño and Zamboni-Rached defined two classes of axisymmetric solutions of the free Maxwell equations. We prove that the linear combinations of these two classes of solutions cover all totally propagating time-harmonic axisymmetric free Maxwell fields – and hence, by summation on frequencies, all propagating axisymmetric free Maxwell fields. It provides an explicit representation for these fields. This will be important, e.g., to have the interstellar radiation field in a disc galaxy modeled as an exact solution of the free Maxwell equations.

Keywords: Maxwell equations; axial symmetry; exact solutions; electromagnetic duality

1 Introduction

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 [3] or disk galaxies [4]. 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 [10]. 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. [10], 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 [9] – thus not merely ones corresponding to nonparaxial scalar beams. The first class of EM fields defined in ref. [10] is obtained by associating with any such scalar solution Ψ a vector potential A by equation (10) below [10]. The second class is deduced from the first one by EM duality [10].

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 (E ₁, B ₁) and ( E 2 ′ , B 2 ′ ) , deduced from two time-harmonic axisymmetric solutions of the scalar wave equation, say Ψ ₁ and Ψ ₂. The first EM field is derived from the vector potential A ₁ = Ψ ₁ e _z, and the second one is deduced by EM duality from the EM field that is derived from the vector potential A ₂ = Ψ ₂ e _z. As this result is based on determining two vector potentials from merely two scalar fields, it was not necessarily expected.

This article is organized as follows. Section 2 presents and comments on the results of ref. [10], 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:

(1) Ψ ω S ( t , ρ , z ) = e − i ω t ∫ − K + K J 0 ( ρ K 2 − k 2 ) e i k z S ( k ) d k ,

where ω is the angular frequency, K := ω/c ^[1] and J ₀ is the first-kind Bessel function of order 0. (Here, c is the velocity of light.) The Bessel beams were first introduced by Durnin [7]. A physical discussion of these beams and their “nondiffracting” property can be found in ref. [11]. On the other hand, a “totally propagating” solution of the wave equation is one that does not have any evanescent mode. In simple words, an evanescent mode can be described as a wave that behaves as a plane wave in some spatial direction, however with an imaginary wavenumber so that its amplitude decreases exponentially. See, e.g., refs. [12,13]. In the present case, the totally propagating character of the wave (1) means precisely that the axial wavenumber k = k _z verifies −K ≤ k ≤ K, [10], so that the radial wavenumber k ρ = K 2 − k 2 is real, as is also k _z.

Thus, the (axial) “wave vector spectrum” S is a (generally complex) function of the real variable k = k _z (−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 (ω _j)(j = 1,…,N _ω) is considered for simplicity:

(2) Ψ ( t , ρ , z ) = ∑ j = 1 N ω Ψ ω j S j ( t , ρ , z ) ,

where, for j = 1,…,N _ω, Ψ ω j S j is the time-harmonic solution (1), corresponding with frequency ω _j and wave vector spectrum S _j. The different weights w _j, which may be applied at the different frequencies, can be included in the functions S _j, replacing S _j by w _j S _j.

2.2 Reminder on time-harmonic free Maxwell fields

In this section, we recall equations more briefly recalled in ref. [10]. The electric and magnetic fields in SI units are given in terms of the scalar and vector potentials V and A by

(3) E = − ∇ V − ∂ A ∂ t ,

(4) B = rot A .

These equations imply that E and B obey the first group of Maxwell equations. If one imposes the Lorenz gauge condition

(5) 1 c 2 ∂ V ∂ t + div A = 0 ,

then the validity of the second group of the Maxwell equations in free space for E and B is equivalent to asking that V and A verify d’Alembert’s wave equation [14,15]. Moreover, if one assumes a harmonic time-dependence for V and A:

(6) V ( t , x ) = e − i ω t V ˆ ( x ) , A ( t , x ) = e − i ω t A ˆ ( x ) ,

then the wave equation for A becomes the Helmholtz equation: ^[2]

(7) Δ A + ω 2 c 2 A = 0 ,

and the Lorenz gauge condition (5) can be rewritten as follows:

(8) V = − i c 2 ω div A .

If A is time-harmonic [equation (6)₂] and obeys (7), then V given by (8) is time-harmonic and satisfies the wave equation. Then, the electric field (3) can be rewritten as follows:

(9) E = i ω A + i c 2 ω ∇ ( div A ) .

Thus, the data of a time-harmonic vector potential A obeying the wave equation, or equivalently obeying equation (7), determine a unique solution of the free Maxwell equations, by equations (4) and (9), and that solution is time-harmonic with the same frequency ω as for A.

2.3 Time-harmonic axisymmetric fields: from a scalar wave to a Maxwell field

To any solution Ψ ( t , ρ , z ) = e − i ω t Ψ ˆ ( ρ , z ) of the scalar wave equation having the form (1), the authors of ref. [10] associate a vector potential A as follows:

(10) A := Ψ e z , or A z := Ψ , A ρ = A ϕ = 0 .

(We shall denote the standard point-dependent orthonormal basis associated with the cylindrical coordinates by (e _ρ, e _ϕ, e _z).) Equation (1) is valid, as we mentioned earlier, for totally propagating, axisymmetric, time-harmonic solutions of the scalar wave equation. Thus, in the way recalled in the previous section, a unique solution of the free Maxwell equations is defined, which is time-harmonic. The equations for the different components of this solution (E, B) are as follows:

(11) B ϕ = − ∂ A z ∂ ρ , E ϕ = 0 ,

(12) E ρ = i c 2 ω ∂ 2 A z ∂ ρ ∂ z , B ρ = 0 ,

(13) E z = i c 2 ω ∂ 2 A z ∂ z 2 + i ω A z , B z = 0 .

These equations follow easily from equations (4), (9) and (10), and from the axisymmetry of A _z = Ψ(t, ρ, z), by using the standard formulas for the curl and divergence in cylindrical coordinates. Equations (11)–(13) provide an axisymmetric EM field whose electric field is radially polarized (E = E _ρ e _ρ + E _z e _z).

An “azimuthally polarized” solution (E′, B′) (in the sense that E ′ = E ′ ϕ e ϕ ) of the free Maxwell equations can alternatively be deduced from the data Ψ by transforming the solution (11)–(13) through the EM duality, that is:

(14) E ′ = c B , B ′ = − E / c .

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. [10] – 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), B _ϕ, E _ρ and E _z are the real parts of the respective right hand side [as are also E and B in equations (3), (4) and (9)]. Thus, if A _z is totally propagating and hence may be written in the form (1), then by dJ ₀/dx = −J ₁(x), we obtain the following equations:

(15) B ϕ ω S = ℛ e [ e − i ω t ∫ − K + K K 2 − k 2 J 1 ( ρ K 2 − k 2 ) S ( k ) e i k z d k ] ,

(16) E ρ ω S = ℛ e [ − i c 2 ω e − i ω t ∫ − K + K K 2 − k 2 J 1 ( ρ K 2 − k 2 ) i k S ( k ) e i k z d k ] ,

(17) E z ω S = ℛ e [ i e − i ω t ∫ − K + K J 0 ( ρ K 2 − k 2 ) ( ω − c 2 ω k 2 ) S ( k ) e i k z d k ] ,

where K := ω/c. In the case with a (discrete) frequency spectrum, one just has to sum each component: (15), (16) or (17), over the different frequencies ω _j, with the corresponding values K _j = ω _j/c and spectra S _j = S _j(k)(−K _j ≤ k ≤ + K _j) – as with a scalar wave (2):

(18) B ϕ = ∑ j = 1 N ω B ϕ ω j S j ,

(19) E ρ = ∑ j = 1 N ω E ρ ω j S j ,

(20) E z = ∑ j = 1 N ω E z ω j S j .

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 B _ϕ, E _ρ and E _z, the other components E _ϕ, B _ρ and B _z 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: B _ϕ, E _ρ and E _z on one side and E _ϕ, B _ρ and B _z 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:

Proposition 1

Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. In order that a time-harmonic axisymmetric solution (A _1z, B _1ϕ, E _1ρ, E _1z; E _1ϕ = B _1ρ = B _1z = 0), of the form (11)–(13), and having the same frequency ω as the starting solution (A, E, B), be such that B _1ϕ = B _ϕ, E _1ρ = Eρ, E _1z = E _z, it is sufficient that we have just

(21) B 1 ϕ = B ϕ .

Proof

Let A _1z(t, ρ, z) be a time-harmonic axisymmetric solution of the wave equation, with frequency ω, and assume that B _1ϕ as defined by equation (11) [with A _1z in the place of A _z] is equal to B _ϕ, where B is defined by equation (4). That is, assume that

(22) − ∂ A 1 z ∂ ρ = ∂ A ρ ∂ z − ∂ A z ∂ ρ .

Denoting by A ₁ := A _1z e _z the vector potential that provides the GAZR1 solution (B _1ϕ, E _1ρ, E _1z; E _1ϕ = B _1ρ = B _1z = 0), let us compute E _ρ − E _1ρ and E _z − E _1z. We obtain the following equation by equation (9):

(23) ω i c 2 ( E − E 1 ) = ∇ ( div A ′ ) + ω 2 c 2 A ′ ,

where

(24) A ′ := A − A 1 := A − A 1 z e z .

In order that the vector potential A of the a priori given solution (A, E, B) be axisymmetric, its components A _ρ, A _ϕ and A _z must depend only on t, ρ and z, i.e., be independent of ϕ. Therefore:

(25) div A = 1 ρ ∂ ( ρ A ρ ) ∂ ρ + ∂ A z ∂ z .

By using this and (24), we obtain

(26) div A ′ = 1 ρ ∂ ( ρ A ρ ) ∂ ρ + ∂ A ′ z ∂ z .

Hence, in (23), we have

(27) ∇ ( div A ′ ) = ∇ ( ∂ A ρ ∂ ρ + A ρ ρ + ∂ A z ′ ∂ z ) = ( ∂ 2 A ρ ∂ ρ 2 + 1 ρ ∂ A ρ ∂ ρ − 1 ρ 2 A ρ + ∂ 2 A z ′ ∂ ρ ∂ z ) e ρ + ( ∂ 2 A ρ ∂ z ∂ ρ + 1 ρ ∂ A ρ ∂ z + ∂ 2 A z ′ ∂ z 2 ) e z .

The radial component of the vector (23) is thus:

(28) ω i c 2 ( E − E 1 ) ρ = ∂ 2 A ρ ∂ ρ 2 + 1 ρ ∂ A ρ ∂ ρ − A ρ ρ 2 + ∂ 2 A z ∂ ρ ∂ z − ∂ 2 A 1 z ∂ ρ ∂ z + ω 2 c 2 A ρ .

However, the vector potential A obeys the Helmholtz equation (7), that is, for the radial component (using the fact that ∂ A ρ ∂ ϕ = ∂ A ϕ ∂ ϕ ≡ 0 ):

(29) ( Δ A ) ρ + ω 2 c 2 A ρ ≡ Δ A ρ − A ρ ρ 2 − 2 ρ 2 ∂ A ϕ ∂ ϕ + ω 2 c 2 A ρ ≡ ∂ 2 A ρ ∂ ρ 2 + 1 ρ ∂ A ρ ∂ ρ + ∂ 2 A ρ ∂ z 2 − A ρ ρ 2 + ω 2 c 2 A ρ = 0 .

Inserting (29) into (28) gives:

(30) ω i c 2 ( E − E 1 ) ρ = − ∂ 2 A ρ ∂ z 2 + ∂ 2 A z ∂ ρ ∂ z − ∂ 2 A 1 z ∂ ρ ∂ z = − ∂ ∂ z ( ∂ A ρ ∂ z − ∂ A z ∂ ρ + ∂ A 1 z ∂ ρ ) .

Therefore, if equation (22) is satisfied, then we have E _1ρ = E _ρ.

Similarly, from (27), the axial component of the vector (23) is expressed as follows:

(31) ω i c 2 ( E − E 1 ) z = ∂ 2 A ρ ∂ z ∂ ρ + 1 ρ ∂ A ρ ∂ z + ∂ 2 A z ∂ z 2 − ∂ 2 A 1 z ∂ z 2 + ω 2 c 2 ( A z − A 1 z ) .

The axial component of the Helmholtz equation (7) is expressed as follows:

(32) ( Δ A ) z + ω 2 c 2 A z ≡ Δ A z + ω 2 c 2 A z ≡ ∂ 2 A z ∂ ρ 2 + 1 ρ ∂ A z ∂ ρ + ∂ 2 A z ∂ z 2 + ω 2 c 2 A z = 0 .

If equation (22) is satisfied, we have

(33) ∂ 2 A z ∂ ρ 2 = ∂ 2 A ρ ∂ ρ ∂ z + ∂ 2 A 1 z ∂ ρ 2 .

In equation (32), we replace ∂ 2 A z ∂ ρ 2 by its value given on the right hand side in equation (33), and we replace ∂ A z ∂ ρ by its value given in equation (22). Thus, we obtain

(34) ∂ 2 A ρ ∂ ρ ∂ z + ∂ 2 A 1 z ∂ ρ 2 + 1 ρ ∂ A ρ ∂ z + 1 ρ ∂ A 1 z ∂ ρ + ∂ 2 A z ∂ z 2 + ω 2 c 2 A z = 0 .

By substituting this equation in equation (31), we rewrite the latter as follows:

(35) ω i c 2 ( E − E 1 ) z = − ∂ 2 A 1 z ∂ ρ 2 − 1 ρ ∂ A 1 z ∂ ρ − ∂ 2 A 1 z ∂ z 2 − ω 2 c 2 A 1 z .

We recognize the right-hand side as that of equation (32), although with the minus sign, and with A _1z in the place of A _z. That is, equation (35) is just

(36) ω i c 2 ( E − E 1 ) z = − Δ A 1 z − ω 2 c 2 A 1 z .

But this is zero, since A _1z is by assumption a time-harmonic solution of the wave equation, with frequency ω. Therefore, if equation (22) is satisfied, then we have E _1z = E _z too. This completes the proof of Proposition 1.□

It follows for the dual (“GAZR2”) solution:

Corollary 1

Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. In order that a time-harmonic solution (A _2z, E 2 ϕ ′ , B 2 ρ ′ , B 2 z ′ ; B 2 ϕ ′ = E 2 ρ ′ = E 2 z ′ = 0 ) with the same frequency, deduced from equations (11)–(13) by the duality (14), be such that E 2 ϕ ′ = E ϕ , B 2 ρ ′ = B ρ , B 2 z ′ = B z , it is sufficient that we have just

(37) E ′ 2 ϕ = E ϕ .

Proof

The GAZR2 solution ( A 2 z , E 2 ϕ ′ , B 2 ρ ′ , B 2 z ′ ; B 2 ϕ ′ = E 2 ρ ′ = E 2 z ′ = 0 ) is deduced from the GAZR1 solution ( A 2 z , B 2 ϕ , E 2 ρ , E 2 z ; E 2 ϕ = B 2 ρ = B 2 z = 0 ) , associated with the same potential A _2z, by the duality relation (14). Suppose that equation (37) is satisfied. With the starting solution (A, E, B) of the free Maxwell equations, we may associate another solution by the inverse duality:

(38) B ˜ = 1 c E , E ˜ = − c B .

The assumed relation (37) means that

(39) B 2 ϕ = B ˜ ϕ .

Indeed, by applying successively (14)₁, (37) and (38)₁, we obtain:

(40) B 2 ϕ = 1 c E 2 ϕ ′ = 1 c E ϕ = B ˜ ϕ .

In turn, the relation (39) means that we may apply Proposition 1 to the GAZR1 solution (A _2z, B _2ϕ,…) and the solution ( A ˜ , E ˜ , B ˜ ) . ^[3] Thus, Proposition 1 reveals that, since B 2 ϕ = B ˜ ϕ , we have also

(41) E 2 ρ = E ˜ ρ and E 2 z = E ˜ z ,

i.e., in view of (14)₂ and (38)₂:

(42) − c B 2 ρ ′ = − c B ρ and − c B 2 z ′ = − c B z .

This proves Corollary 1.□

3.2 Generality of the decomposition

Proposition 2

Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. There exists a time-harmonic axisymmetric solution A _2z of the wave equation, with the same frequency, such that the associated GAZR2 solution ( E 2 ′ , B 2 ′ ) , deduced from A _2z by equations (11)–(13) followed by the duality transformation (14), satisfy

(43) E 2 ϕ ′ = E ϕ , B 2 ρ ′ = B ρ , B 2 z ′ = B z .

Proof

In view of Corollary 1, we merely have to prove that there exists a time-harmonic axisymmetric solution A _2z of the wave equation, such that equation (37) is satisfied. From equations (9) and (25), we have

(44) E ϕ = i ω A ϕ .

By using this with equations (11) and (14)₁, we may rewrite the sought-for relation (37) as follows:

(45) − ∂ A 2 z ∂ ρ = i ω c A ϕ .

This equation can be solved by a quadrature:

(46) A 2 z ( t , ρ , z ) = h ( t , z ) + ∫ ρ 0 ρ − i ω c A ϕ ( t , ρ ′ , z ) d ρ ′ .

We thus have to find out if it is possible to determine the function h, so that A _2z given by (46) obeys the wave equation. Moreover, the unknown function A _2z must have a harmonic time dependence with frequency ω as has A _ϕ, i.e.,

(47) A 2 z ( t , ρ , z ) = ψ ( ρ , z ) e − i ω t , A ϕ ( t , ρ , z ) = A ˆ ϕ ( ρ , z ) e − i ω t ,

so we must have h(t, z) = e ^−iωt g(z), too. Hence, we may rewrite (46) as follows:

(48) ψ ( ρ , z ) = g ( z ) + ∫ ρ 0 ρ − i ω c A ˆ ϕ ( ρ ′ , z ) d ρ ′ ,

and now the question is to know if g can be determined so that ψ obeys the scalar Helmholtz equation, i.e., [cf. equation (32)]:

(49) ℋ ψ ≔ Δ ψ + ω 2 c 2 ψ ≡ ∂ 2 ψ ∂ ρ 2 + 1 ρ ∂ ψ ∂ ρ + ∂ 2 ψ ∂ z 2 + ω 2 c 2 ψ = 0

knowing that A _ϕ or A ˆ ϕ does obey the ϕ component of the vector Helmholtz equation (7), i.e.,

(50) ( Δ A ˆ ) ϕ + ω 2 c 2 A ˆ ϕ ≡ Δ A ˆ ϕ − A ˆ ϕ ρ 2 + 2 ρ 2 ∂ A ˆ ρ ∂ ϕ + ω 2 c 2 A ˆ ϕ ≡ ∂ 2 A ˆ ϕ ∂ ρ 2 + 1 ρ ∂ A ˆ ϕ ∂ ρ + ∂ 2 A ˆ ϕ ∂ z 2 − A ˆ ϕ ρ 2 + ω 2 c 2 A ˆ ϕ = 0 .

We have from equations (45) and (47):

(51) ∂ ψ ∂ ρ = − i ω c A ˆ ϕ ,

hence,

(52) ∂ 2 ψ ∂ ρ 2 = − i ω c ∂ A ˆ ϕ ∂ ρ .

And we obtain from (48):

(53) ∂ ψ ∂ z = d g d z − ∫ ρ 0 ρ i ω c ∂ A ˆ ϕ ∂ z ( ρ ′ , z ) d ρ ′ ,

whence,

(54) ∂ 2 ψ ∂ z 2 = d 2 g d z 2 − ∫ ρ 0 ρ i ω c ∂ 2 A ˆ ϕ ∂ z 2 ( ρ ′ , z ) d ρ ′ .

Entering equations (51), (52) and (54) into (49)₁, we obtain:

(55) ℋ ψ = − i ω c ∂ A ˆ ϕ ∂ ρ − i ω c A ˆ ϕ ρ + d 2 g d z 2 − ∫ ρ 0 ρ i ω c ∂ 2 A ˆ ϕ ∂ z 2 ( ρ ′ , z ) d ρ ′ + ω 2 c 2 ( g − i ω c ∫ ρ 0 ρ A ˆ ϕ ( ρ ′ , z ) d ρ ′ ) .

Therefore, the scalar Helmholtz equation (49)₂ can be rewritten as follows:

(56) d 2 g d z 2 + ω 2 c 2 g = i ω c [ ∂ A ˆ ϕ ∂ ρ + A ˆ ϕ ρ + ∫ ρ 0 ρ ( ∂ 2 A ˆ ϕ ∂ z 2 + ω 2 c 2 A ˆ ϕ ) d ρ ′ ] ,

or, using (50):

(57) d 2 g d z 2 + ω 2 c 2 g = i ω c [ ∂ A ˆ ϕ ∂ ρ + A ˆ ϕ ρ + ∫ ρ 0 ρ ( − ∂ 2 A ˆ ϕ ∂ ρ ′ 2 − 1 ρ ′ ∂ A ˆ ϕ ∂ ρ ′ + A ˆ ϕ ρ ′ 2 ) d ρ ′ ] .

An integration by parts gives us:

(58) ∫ ρ 0 ρ ( − ∂ 2 A ˆ ϕ ∂ ρ ′ 2 + A ˆ ϕ ρ ′ 2 ) d ρ ′ = − [ ∂ A ˆ ϕ ∂ ρ ′ + A ˆ ϕ ρ ′ ] ρ 0 ρ + ∫ ρ 0 ρ 1 ρ ′ ∂ A ˆ ϕ ∂ ρ ′ d ρ ′ ,

so equation (57) can be rewritten as follows:

(59) d 2 g d z 2 + ω 2 c 2 g = i ω c [ ∂ A ˆ ϕ ∂ ρ ( ρ 0 , z ) + A ˆ ϕ ( ρ 0 , z ) ρ 0 ] .

As is well known and easy to check, this very ordinary differential equation can be solved explicitly by the method of variation of constants. (The general solution g of (59) depends linearly on two arbitrary constants.) And by construction, any among the solutions g of (59) is such that, with that g, the function ψ in equation (48) obeys the scalar Helmholtz equation (49). This proves Proposition 2.□

Corollary 2

Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. There exists a time-harmonic axisymmetric solution A _1z of the wave equation, with the same frequency, such that the associated GAZR1 solution (E ₁, B ₁), deduced from A _1z by equations (11)–(13), satisfy

(60) B 1 ϕ = B ϕ , E 1 ρ = E ρ , E 1 z = E z .

Proof

Let A _1z be a time-harmonic axisymmetric solution of the wave equation, and consider:

the GAZR1 solution defined from A _1z by equations (11)–(13) (thus with A _1z, B _1ϕ, E _1ρ, E _1z,… instead of A _z, B _ϕ, E _ρ, E _z,…, respectively).
the GAZR2 solution defined from the same A _1z by applying the duality (14) to the said GAZR1 solution:

(61) E 1 ϕ ′ := c B 1 ϕ , B 1 ρ ′ := − 1 c E 1 ρ , B 1 z ′ := − 1 c E 1 z ,

(62) E 1 ρ ′ := c B 1 ρ = 0 , E 1 z ′ := c B 1 z = 0 , B 1 ϕ ′ := − 1 c E 1 ϕ = 0 .

On the other hand, consider the free Maxwell field (E′, B′) deduced from the given time-harmonic axisymmetric solution (E, B) of the free Maxwell equations by the same duality relation:

(63) E ′ := c B , B ′ := − E / c .

Just in the same way as it was shown in Note 1, we know that a vector potential A′ such that B′ = rot A′ does exist and can be chosen to be axisymmetric (and is indeed chosen so) – as are E and B, and hence E′ and B′. The sought-for relation (60) is equivalent to

(64) E 1 ϕ ′ = E ϕ ′ , B 1 ρ ′ = B ρ ′ , B 1 z ′ = B z ′ .

Therefore, the existence of A _1z as in the statement of Corollary 2 is ensured by Proposition 2.□

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.

Theorem

Let (A, E, B) be any time-harmonic axisymmetric solution of the free Maxwell equations. There exist a unique GAZR1 solution (E ₁, B ₁) and a unique GAZR2 solution ( E 2 ′ , B 2 ′ ) , both with the same frequency as has (A, E, B), and whose sum gives just that solution:

(65) E = E 1 + E 2 ′ , B = B 1 + B 2 ′ .

Remark

Thus, the uniqueness of the representation concerns the electric and magnetic fields. It does not concern the potentials A ₁ = A _1z e _z and A ₂ = A _2z e _z that generate (E ₁, B ₁) and ( E 2 ′ , B 2 ′ ) , respectively.

4 Discussion and conclusion

The authors of ref. [10] 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 ( ω j ) j = 1 , .. . , N ω for simplicity: there are 2N _ω functions, k ↦ S j ( k ) and k ↦ S j ′ ( k ) ( j = 1 , .. . , N ω ) , such that the components B _ϕ, E _ρ, E _z of the field are given by equations (18), (19) and (20), respectively – while the components E _ϕ, B _ρ and B _z are given by these same equations applied with the primed spectra S j ′ , followed by the duality transformation (14).

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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Received: 2020-01-28

Revised: 2020-03-30

Accepted: 2020-04-02

Published Online: 2020-06-29

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2020-0117

Keywords for this article

Maxwell equations; axial symmetry; exact solutions; electromagnetic duality

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