An explicit representation for the axisymmetric solutions of the free Maxwell equations
aa r X i v : . [ phy s i c s . c l a ss - ph ] J u l An explicit representation for theaxisymmetric solutions of the free Maxwellequations
Mayeul Arminjon
Univ. Grenoble Alpes, CNRS, Grenoble INP, 3SR, F-38000 Grenoble, FranceE-mail: [email protected]
Short title:
Explicit representation for axisymmetric free Maxwell fields
Abstract
Garay-Avenda˜no & Zamboni-Rached (2014) defined two classes of ax-isymmetric solutions of the free Maxwell equations. We prove thatthe linear combinations of these two classes of solutions cover all to-tally propagating time-harmonic axisymmetric free Maxwell fields —and hence, by summation on frequencies, all totally propagating ax-isymmetric free Maxwell fields. It provides an explicit representationfor these fields. This will be important, e.g., to have the interstellarradiation field in a disc galaxy modelled as an exact solution of thefree Maxwell equations.
Keywords:
Maxwell equations; axial symmetry; exact solutions;electromagnetic duality.
Axially symmetric solutions of the Maxwell equations are quite important,at least as an often relevant approximation. For instance, axisymmetricmagnetic fields occur naturally as produced by systems possessing an axis ofrevolution, such as disks or coils [1, 2], or astrophysical systems like accretion1isks [3] or disk galaxies [4]. Axisymmetric solutions are also used to modelEM beams and their propagation (e.g. [5, 6]). In particular, non-diffractingbeams are usually endowed with axial symmetry, see e.g. Refs. [7, 8, 9].Naturally, one often considers time-harmonic solutions, since a general timedependence is got by summing such solutions. Two classes of time-harmonicaxisymmetric solutions of the free Maxwell equations, mutually associatedby EM duality, have been introduced recently [10]. The main aim was to“describe in exact and analytic form the propagation of nonparaxial scalarand electromagnetic beams.” However, as noted by the authors of Ref. [10],the analytical expression for a totally propagating time-harmonic axisym-metric solution Ψ of the scalar wave equation, from which they start [Eq. (1)below], covers all such solutions [9] — thus not merely ones correspondingto 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 Eq. (10) below, and the second class is deduced from the first one byEM duality [10].The aim of the present work is to show that, by combining these twoclasses, one is able to describe actually all totally propagating time-harmonicaxisymmetric EM fields — and thus, by summing on frequencies, all totallypropagating axisymmetric EM fields. To do that, we shall prove the followingTheorem: Any time-harmonic axisymmetric EM field (whether totally prop-agating or not) is the sum of two EM fields, say ( E , B ) and ( E ′ , B ′ ) ,deduced from two time-harmonic axisymmetric solutions of the scalar waveequation, say Ψ and Ψ . The first EM field derives from the vector po-tential A = Ψ e z , and the second one is deduced by EM duality from theEM field that derives from the vector potential A = Ψ e z . Because thisresult is based on determining two vector potentials from merely two scalarfields, it was not necessarily expected.The paper is organized as follows. Section 2 presents and comments theresults of Ref. [10], and somewhat extends them, in particular by notingthat Eqs. 11)–(13) apply to any time-harmonic axisymmetric solution of thescalar wave equation. Also, Eqs. (15)–(20) are new. Section 3 contains themain results of this work — it gives the proof of the Theorem. That proofis not immediate but uses standard mathematics with which one is familiarfrom classical field theory. In Section 4, we summarize our main resultsand we note that they lead straightforwardly to a method to get an explicit2epresentation for all totally propagating axisymmetric solutions of the freeMaxwell equations. We adopt cylindrical coordinates ρ, φ, z about the symmetry axis, that is the z axis. Any totally propagating, time-harmonic, axisymmetric solution ofthe scalar wave equation (of d’Alembert) can be written [9, 10] as a sum ofscalar Bessel beams:Ψ ω S ( t, ρ, z ) = e − i ωt Z + K − K J (cid:16) ρ √ K − k (cid:17) e i k z S ( k ) d k, (1)with ω the angular frequency, K := ω/c , and J the first-kind Bessel func-tion of order 0. (Here c is the velocity of light.) The Bessel beams were firstintroduced by Durnin [7]. A physical discussion of these beams and their“non-diffracting” property can be found in Ref. [11]. On the other hand,a “totally propagating” solution of the wave equation is one that does nothave any evanescent mode. Loosely speaking, an evanescent mode can bedescribed as a wave that behaves as a plane wave in some spatial direction,though with an imaginary wavenumber so that its amplitude decreases expo-nentially. See e.g. Refs. [12, 13]. In the present case, the totally propagatingcharacter 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 − k isreal as is also k z .Thus the (axial) “wave vector spectrum” S is a (generally complex) func-tion of the real variable k = k z ( − K ≤ k ≤ K ), that is the projectionof the wave vector on the z axis. This function S determines the spatialdependency of the time-harmonic solution (1) in the two-dimensional spaceleft by the axial symmetry, i.e. the half-plane ( ρ ≥ z ∈ ] − ∞ , + ∞ [). Thus,any totally propagating, time-harmonic, axisymmetric scalar wave Ψ can beput 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 Beware that instead K := 2 ω/c in Ref. [10]. Our notation seems as natural and givesmore condensed formulas. S , S ∈ L ([ − K, + K ]), would be enough for this). Of course, the general totally propagating axisymmetric solution of the scalar wave equation canbe got from (1) by an appropriate summation over a frequency spectrum: anintegral (inverse Fourier transform) in the general case, or a discrete sum if adiscrete frequency spectrum ( ω j ) ( j = 1 , ..., N ω ) is considered for simplicity:Ψ( t, ρ, z ) = N ω X j =1 Ψ ω j S j ( t, ρ, z ) , (2)where, for j = 1 , ..., N ω , Ψ ω j S j is the time-harmonic solution (1), correspond-ing with frequency ω j and wave vector spectrum S j . The different weights w j which may be affected to the different frequencies can be included in thefunctions S j , replacing S j by w j S j . In this subsection, we recall equations more briefly recalled in Ref. [10]. Theelectric and magnetic fields in SI units are given in terms of the scalar and(3-)vector potentials V and A by E = −∇ V − ∂ A ∂t , (3) B = rot A . (4)These equations imply that E and B obey the first group of Maxwell equa-tions. If one imposes the Lorenz gauge condition1 c ∂V∂t + div A = 0 , (5)then the validity of the second group of the Maxwell equations in free spacefor E and B is equivalent to ask that V and A verify d’Alembert’s waveequation [14, 15]. Moreover, if one assumes a harmonic time-dependence for V and A : V ( t, x ) = e − i ωt ˆ V ( x ) , A ( t, x ) = e − i ωt ˆ A ( x ) , (6)4hen the wave equation for A becomes the Helmholtz equation: ∆ A + ω c A = , (7)and the Lorenz gauge condition (5) rewrites as V = − i c ω div A . (8)If A is time-harmonic [Eq. (6) ] and obeys (7), then V given by (8) is time-harmonic and automatically satisfies the wave equation. The electric field(3) is then easily rewritten as E = i ω A + i c ω ∇ (div A ) . (9)Thus, the data of a time-harmonic vector potential A obeying the waveequation, or equivalently obeying Eq. (7), determines a unique solution ofthe free Maxwell equations, by Eqs. (4) and (9), and that solution is time-harmonic with the same frequency ω as for A . To any solution Ψ( t, ρ, z ) = e − i ωt ˆΨ( ρ, z ) of the scalar wave equation havingthe form (1), the authors of Ref. [10] associate a vector potential A by A := Ψ e z , or A z := Ψ , A ρ = A φ = 0 . (10)(We shall denote by ( e ρ , e φ , e z ) the standard point-dependent orthonormalbasis associated with the cylindrical coordinates.) The form (1) applies, as wementioned, to totally propagating, axisymmetric, time-harmonic solutions ofthe scalar wave equation. Thus, in the way recalled in the foregoing subsec-tion, they define a unique solution of the free Maxwell equations, which istime-harmonic. Their equations for the different components of this solution Of course, just the same equation (7) applies to the “amplitude field” ˆ A . E , B ) are as follows: B φ = − ∂A z ∂ρ , E φ = 0 , (11) E ρ = i c ω ∂ A z ∂ρ ∂z , B ρ = 0 , (12) E z = i c ω ∂ A z ∂z + i ωA z , B z = 0 . (13)These equations follow easily from Eqs. (4), (9) and (10), and from theaxisymmetry of A z = Ψ( t, ρ, z ), by using the standard formulas for thecurl and divergence in cylindrical coordinates. Equations (11)–(13) providean axisymmetric EM field whose electric field is radially polarized ( E = E ρ e ρ + E z e z ), in short a “radially polarized” EM field.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 thedata Ψ, by transforming the solution (11)–(13) through the EM duality, thatis: E ′ = c B , B ′ = − E /c. (14)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 freeMaxwell equations. The only relevant fact is that Ψ = Ψ( t, ρ, z ) is a time-harmonic axisymmetric solution of the scalar wave equation. Thus, withany axisymmetric time-harmonic solution Ψ of the scalar wave equation, wecan associate two axisymmetric time-harmonic solutions of the free Maxwellequations: 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.As usual, it is implicit that, in Eqs. (11)–(13), B φ , E ρ and E z are actuallythe real parts of the respective r.h.s. [as are also E and B in Eqs. (3), (4),and (9)]. Thus, if A z is totally propagating and hence may be written in the6orm (1), we obtain using the fact that d J / d x = − J ( x ): B φ ω S = R e (cid:20) e − i ωt Z + K − K √ K − k J (cid:16) ρ √ K − k (cid:17) S ( k ) e ikz d k (cid:21) , (15) E ρ ω S = R e (cid:20) − i c ω e − i ωt Z + K − K √ K − k J (cid:16) ρ √ K − k (cid:17) i k S ( k ) e ikz d k (cid:21) , (16) E z ω S = R e (cid:20) i e − i ωt Z + K − K J (cid:16) ρ √ K − k (cid:17) (cid:18) ω − c ω k (cid:19) S ( k ) e ikz d k (cid:21) , (17)where K := ω/c . In the case with a (discrete) frequency spectrum, onejust has to sum each component: (15), (16), or (17), over the differentfrequencies 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): B φ = N ω X j =1 B φ ω j S j , (18) E ρ = N ω X j =1 E ρ ω j S j , (19) E z = N ω X j =1 E z ω j S j . (20) Now an important question arises: Do the GAZR solutions generate all ax-isymmetric time-harmonic solutions of the Maxwell equations (in which case,by summation on frequencies, they would generate all axisymmetric solutionsof the Maxwell equations)? That is: let ( A , E , B ) be any time-harmonicaxisymmetric solution of the free Maxwell equations. Can one find a GAZR1solution and a GAZR2 solution, whose sum give just that starting solution? Note from Eqs. (11)–(13) and (14) that the GAZR1 solution and theGAZR2 solution are complementary: in cylindrical coordinates, the GAZR17olution provides non-zero components B φ , E ρ , E z , the other components E φ , B ρ , 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 ofcomponents: B φ , E ρ , E z on one side, and E φ , B ρ , B z on the other side. For the “GAZR1” solution, which gives non-zero values to the first amongthe two sets of components just mentioned, we have the following result:
Proposition 1.
Let ( A , E , B ) be any time-harmonic axisymmetric solutionof the free Maxwell equations. In order that a time-harmonic axisymmetricsolution ( A z , B φ , E ρ , E z ; E φ = B ρ = B z = 0) , of the form (11)–(13),and having the same frequency ω as the starting solution ( A , E , B ) , be suchthat B φ = B φ , E ρ = E ρ , E z = E z , it is sufficient that we have just B φ = B φ . (21) Proof.
Let A z ( t, ρ, z ) be a time-harmonic axisymmetric solution of the waveequation, with frequency ω , and assume that B φ as defined by Eq. (11)[with A z in the place of A z ] is equal to B φ , where B is defined by Eq. (4).I.e., assume that − ∂A z ∂ρ = ∂A ρ ∂z − ∂A z ∂ρ . (22)Denoting by A := A z e z the vector potential that provides the GAZR1solution ( B φ , E ρ , E z ; E φ = B ρ = B z = 0), let us compute E ρ − E ρ and E z − E z . We have by Eq. (9): ω i c ( E − E ) = ∇ (div A ′ ) + ω c A ′ , (23)where A ′ := A − A := A − A z e z . (24)In order that the vector potential A of the a priori given solution ( A , E , B )be axisymmetric, its components A ρ , A φ , A z must depend only on t, ρ, z , i.e.,be independent of φ . Therefore:div A = 1 ρ ∂ ( ρA ρ ) ∂ρ + ∂A z ∂z , (25)8nd, using this and (24): div A ′ = 1 ρ ∂ ( ρA ρ ) ∂ρ + ∂A ′ z ∂z . (26)Hence, in (23), we have ∇ (div A ′ ) = ∇ (cid:18) ∂A ρ ∂ρ + A ρ ρ + ∂A ′ z ∂z (cid:19) (27)= (cid:18) ∂ A ρ ∂ρ + 1 ρ ∂A ρ ∂ρ − ρ A ρ + ∂ A ′ z ∂ρ∂z (cid:19) e ρ + (cid:18) ∂ A ρ ∂z∂ρ + 1 ρ ∂A ρ ∂z + ∂ A ′ z ∂z (cid:19) e z . The radial component of the vector (23) is thus: ω i c ( E − E ) ρ = ∂ A ρ ∂ρ + 1 ρ ∂A ρ ∂ρ − A ρ ρ + ∂ A z ∂ρ∂z − ∂ A z ∂ρ∂z + ω c A ρ . (28)However, the vector potential A obeys the Helmholtz equation (7), that isfor the radial component (using the fact that ∂A ρ ∂φ = ∂A φ ∂φ ≡ A ) ρ + ω c A ρ ≡ ∆ A ρ − A ρ ρ − ρ ∂A φ ∂φ + ω c A ρ ≡ ∂ A ρ ∂ρ + 1 ρ ∂A ρ ∂ρ + ∂ A ρ ∂z − A ρ ρ + ω c A ρ = 0 . (29)Inserting (29) into (28) gives us: ω i c ( E − E ) ρ = − ∂ A ρ ∂z + ∂ A z ∂ρ∂z − ∂ A z ∂ρ∂z = − ∂∂z (cid:18) ∂A ρ ∂z − ∂A z ∂ρ + ∂A z ∂ρ (cid:19) . (30)Therefore, if Eq. (22) is satisfied, then we have E ρ = E ρ .Similarly, from (27), the axial component of the vector (23) is ω i c ( E − E ) z = ∂ A ρ ∂z∂ρ + 1 ρ ∂A ρ ∂z + ∂ A z ∂z − ∂ A z ∂z + ω c ( A z − A z ) . (31)On the other hand, the axial component of the Helmholtz equation (7) is(∆ A ) z + ω c A z ≡ ∆ A z + ω c A z ≡ ∂ A z ∂ρ + 1 ρ ∂A z ∂ρ + ∂ A z ∂z + ω c A z = 0 . (32)9f Eq. (22) is satisfied, we have ∂ A z ∂ρ = ∂ A ρ ∂ρ∂z + ∂ A z ∂ρ . (33)In Eq. (32), we replace ∂ A z ∂ρ by its value given on the r.h.s. above, and wereplace ∂A z ∂ρ by its value given by Eq. (22). This gives: ∂ A ρ ∂ρ∂z + ∂ A z ∂ρ + 1 ρ ∂A ρ ∂z + 1 ρ ∂A z ∂ρ + ∂ A z ∂z + ω c A z = 0 . (34)Using this equation in Eq. (31), we rewrite the latter as ω i c ( E − E ) z = − ∂ A z ∂ρ − ρ ∂A z ∂ρ − ∂ A z ∂z − ω c A z . (35)We recognize the r.h.s as that of Eq. (32), though with the minus sign, andwith A z in the place of A z . I.e., Eq. (35) is just ω i c ( E − E ) z = − ∆ A z − ω c A z . (36)But this is zero, since A z is by assumption a time-harmonic solution of thewave equation, with frequency ω . Therefore, if Eq. (22) is satisfied, then wehave E z = E z , too. This completes the proof of Proposition 1. (cid:3) It follows for the dual (“GAZR2”) solution:
Corollary 1.
Let ( A , E , B ) be any time-harmonic axisymmetric solution ofthe free Maxwell equations. In order that a time-harmonic solution ( A z , E ′ φ , B ′ ρ , B ′ z ; B ′ φ = E ′ ρ = E ′ z = 0) with the same frequency, deduced from Eqs.(11)–(13) by the duality (14), be such that E ′ φ = E φ , B ′ ρ = B ρ , B ′ z = B z ,it is sufficient that we have just E ′ φ = E φ . (37) Proof.
The GAZR2 solution ( A z , E ′ φ , B ′ ρ , B ′ z ; B ′ φ = E ′ ρ = E ′ z = 0) isdeduced from the GAZR1 solution ( A z , B φ , E ρ , E z ; E φ = B ρ = B z = 0),10ssociated with the same potential A z , by the duality relation (14). Supposethat Eq. (37) is satisfied. With the starting solution ( A , E , B ) of the freeMaxwell equations, we may associate another solution, by the inverse duality: e B = 1 c E , e E = − c B . (38)The assumed relation (37) means that B φ = e B φ . (39)Indeed, by applying successively (14) , (37), and (38) , we obtain: B φ = 1 c E ′ φ = 1 c E φ = e B φ . (40)In turn, the relation (39) means that we may apply Proposition 1 to theGAZR1 solution ( A z , B φ , ... ) and the solution ( e A , e E , e B ). Thus, Proposition 1tells us here that, since B φ = e B φ , we have also E ρ = e E ρ and E z = e E z , (41)i.e., in view of (14) and (38) : − cB ′ ρ = − cB ρ and − cB ′ z = − cB z . (42)This proves Corollary 1. (cid:3) Proposition 2.
Let ( A , E , B ) be any time-harmonic axisymmetric solutionof the free Maxwell equations. There exists a time-harmonic axisymmetric The explicit expression of the corresponding vector potential e A as function of( A , E , B ) is not needed: only the existence of an axisymmetric e A , such that e B = rot e A , isneeded. Precisely, e A is got as a solution of the PDE rot e A = e B . (Such a solution alwaysexists in a topologically trivial domain, thus in particular if the domain is the whole space.)Hence e A can indeed be chosen axisymmetric, i.e. with its components in cylindrical coor-dinates being independent of φ , because with this choice the independent variables of thePDE rot e A = e B simply do not include φ — since e B = c E is axisymmetric, as is E byassumption. olution A z of the wave equation, with the same frequency, such that theassociated GAZR2 solution ( E ′ , B ′ ) , deduced from A z by Eqs. (11)–(13)followed by the duality transformation (14), satisfy E ′ φ = E φ , B ′ ρ = B ρ , B ′ z = B z . (43) Proof.
In view of Corollary 1, we merely have to prove that there exists atime-harmonic axisymmetric solution A z of the wave equation, such thatEq. (37) is satisfied. From Eqs. (9) and (25), we have E φ = i ωA φ . (44)Using this with Eqs. (11) and (14) , we may rewrite the sought-for relation(37) as: − ∂A z ∂ρ = i ωc A φ . (45)This equation can be solved by a quadrature: A z ( t, ρ, z ) = h ( t, z ) + Z ρρ − i ωc A φ ( t, ρ ′ , z ) d ρ ′ . (46)We thus have to find out if it is possible to determine the function h so that A z given by (46) obey the wave equation. Moreover, the unknown function A z must have a harmonic time dependence with frequency ω as has A φ , i.e., A z ( t, ρ, z ) = ψ ( ρ, z ) e − i ωt , A φ ( t, ρ, z ) = ˆ A φ ( ρ, z ) e − i ωt , (47)so we must have h ( t, z ) = e − i ωt g ( z ), too. Hence we may rewrite (46) as ψ ( ρ, z ) = g ( z ) + Z ρρ − i ωc ˆ A φ ( ρ ′ , z ) d ρ ′ , (48)and now the question is to know if g can be determined so that ψ obey thescalar Helmholtz equation, i.e. [cf. Eq. (32)]: H ψ := ∆ ψ + ω c ψ ≡ ∂ ψ∂ρ + 1 ρ ∂ψ∂ρ + ∂ ψ∂z + ω c ψ = 0 (49)— knowing that A φ or ˆ A φ does obey the φ component of the vector Helmholtzequation (7), i.e.: (cid:16) ∆ ˆA (cid:17) φ + ω c ˆ A φ ≡ ∆ ˆ A φ − ˆ A φ ρ + 2 ρ ∂ ˆ A ρ ∂φ + ω c ˆ A φ ≡ ∂ ˆ A φ ∂ρ + 1 ρ ∂ ˆ A φ ∂ρ + ∂ ˆ A φ ∂z − ˆ A φ ρ + ω c ˆ A φ = 0 . (50)12e have from Eqs. (45) and (47): ∂ψ∂ρ = − i ωc ˆ A φ , (51)hence ∂ ψ∂ρ = − i ωc ∂ ˆ A φ ∂ρ . (52)And we get from (48): ∂ψ∂z = d g d z − Z ρρ i ωc ∂ ˆ A φ ∂z ( ρ ′ , z ) d ρ ′ , (53)whence ∂ ψ∂z = d g d z − Z ρρ i ωc ∂ ˆ A φ ∂z ( ρ ′ , z ) d ρ ′ . (54)Entering Eqs. (51), (52) and (54) into (49) , we obtain: H ψ = − i ωc ∂ ˆ A φ ∂ρ − i ωc ˆ A φ ρ + d g d z − Z ρρ i ωc ∂ ˆ A φ ∂z ( ρ ′ , z ) d ρ ′ (55)+ ω c (cid:18) g − i ωc Z ρρ ˆ A φ ( ρ ′ , z ) d ρ ′ (cid:19) . Therefore, the scalar Helmholtz equation (49) rewrites asd g d z + ω c g = i ωc " ∂ ˆ A φ ∂ρ + ˆ A φ ρ + Z ρρ ∂ ˆ A φ ∂z + ω c ˆ A φ ! d ρ ′ , (56)or, using (50):d g d z + ω c g = i ωc " ∂ ˆ A φ ∂ρ + ˆ A φ ρ + Z ρρ − ∂ ˆ A φ ∂ρ ′ − ρ ′ ∂ ˆ A φ ∂ρ ′ + ˆ A φ ρ ′ ! d ρ ′ . (57)An integration by parts gives us: Z ρρ − ∂ ˆ A φ ∂ρ ′ + ˆ A φ ρ ′ ! d ρ ′ = − " ∂ ˆ A φ ∂ρ ′ + ˆ A φ ρ ′ ρρ + Z ρρ ρ ′ ∂ ˆ A φ ∂ρ ′ d ρ ′ , (58)13o Eq. (57) rewrites asd g d z + ω c g = i ωc " ∂ ˆ A φ ∂ρ ( ρ , z ) + ˆ A φ ( ρ , z ) ρ . (59)As is well known and easy to check, this very ordinary differential equationcan be solved explicitly by the method of variation of constants. (The gen-eral solution g of (59) depends linearly on two arbitrary constants.) And byconstruction, any among the solutions g of (59) is such that, with that g ,the function ψ in Eq. (48) obeys the scalar Helmholtz equation (49). Thisproves Proposition 2. (cid:3) Corollary 2.
Let ( A , E , B ) be any time-harmonic axisymmetric solutionof the free Maxwell equations. There exists a time-harmonic axisymmetricsolution A z of the wave equation, with the same frequency, such that theassociated GAZR1 solution ( E , B ) , deduced from A z by Eqs. (11)–(13),satisfy B φ = B φ , E ρ = E ρ , E z = E z . (60) Proof.
Let A z be a time-harmonic axisymmetric solution of the wave equa-tion, and consider:(i) the GAZR1 solution defined from A z by Eqs. (11)–(13) (thus with A z , B φ , E ρ , E z , ... instead of A z , B φ , E ρ , E z , ... respectively).(ii) the GAZR2 solution defined from the same A z by applying the duality(14) to the said GAZR1 solution: E ′ φ := cB φ , B ′ ρ := − c E ρ , B ′ z := − c E z , (61) E ′ ρ := cB ρ = 0 , E ′ z := cB z = 0 , B ′ φ := − c E φ = 0 . (62)On the other hand, consider the free Maxwell field ( E ′ , B ′ ) deduced fromthe given time-harmonic axisymmetric solution ( E , B ) of the free Maxwellequations by the same duality relation: E ′ := c B , B ′ := − E /c. (63)14ust in the same way as it was shown in Note 3, we know that a vectorpotential A ′ such that B ′ = rot A ′ does exist and can be chosen to be ax-isymmetric (and is indeed chosen so) — as are E and B , and hence E ′ and B ′ . Clearly, the sought-for relation (60) is equivalent to E ′ φ = E ′ φ , B ′ ρ = B ′ ρ , B ′ z = B ′ z . (64)Therefore, the existence of A z as in the statement of Corollary 2 is ensuredby Proposition 2. (cid:3) Accounting for Proposition 2 and for Corollary 2, and remembering the“complementarity” of the GAZR1 and GAZR2 solutions, we thus can answerpositively to the question asked at the beginning of this section:
Theorem.
Let ( A , E , B ) be any time-harmonic axisymmetric solution ofthe free Maxwell equations. There exist a unique GAZR1 solution ( E , B ) and a unique GAZR2 solution ( E ′ , B ′ ) , both with the same frequency as has ( A , E , B ) , and whose sum gives just that solution: E = E + E ′ , B = B + B ′ . (65) Remark.
Thus, the uniqueness of the representation concerns the electricand magnetic fields. It of course does not concern the potentials A = A z e z and A = A z e z that generate respectively ( E , B ) and ( E ′ , B ′ ). The authors of Ref. [10] introduced two classes of axisymmetric solutionsof the free Maxwell equations, and they showed that these two classes allowone to obtain in explicit form nonparaxial EM beams. It has been provedhere that, by combining these two classes, one can define a method thatallows one to get all totally propagating, time-harmonic, axisymmetric freeMaxwell fields — and thus, by the appropriate summation on frequencies, alltotally propagating axisymmetric free Maxwell fields. This method resultsimmediately from the Theorem just above, and from the general form (1)of a totally propagating, time-harmonic, axisymmetric solution of the scalarwave equation. However, that theorem is not an obviously expected result,15nd its proof is not immediate. We thus have now a constructive methodto obtain all totally propagating axisymmetric free Maxwell fields. Namely,considering a discrete frequency spectrum ( ω j ) j =1 ,...,N ω for simplicity: thereare 2 N ω functions, k S j ( k ), and k S ′ j ( k ) ( j = 1 , ..., N ω ), such that thecomponents B φ , E ρ , E z of the field are given by Eqs. (18), (19), (20) respec-tively — while the components E φ , B ρ , B z are given by these same equationsapplied with the primed spectra S ′ j , followed by the duality transformation(14).In a forthcoming work, we shall apply this to model the interstellar ra-diation field in a disc galaxy as an (axisymmetric) exact solution of the freeMaxwell equations. In this application, it is very important that, due to thepresent work, one knows that any (totally propagating) axisymmetric freeMaxwell field can be got in this way. References [1] Garrett M.W., Axially symmetric systems for generating and measuring mag-netic fields. Part I, J. Appl. Phys., 1951, 22, 1091–1107.[2] Boridy E., Magnetic fields generated by axially symmetric systems, J. Appl.Phys., 1989, 66, 5691–5700.[3] Wang J.C.L., Sulkanen M.E., Lovelace R.V.E., Self-collimated electromagneticjets from magnetized accretion disks: the even-symmetry case, Astrophys. J.,1990, 355, 38–43.[4] Beck R., Wielebinski, R., Magnetic fields in the Milky Way and in galaxies,In: Oswalt T.D., Gilmore G. (Eds.), Planets, Stars and Stellar Systems Vol.5, Springer, Dordrecht, 2013, 641–723[5] Nesterov A.V., Niziev V.G., Propagation features of beams with axially sym-metric polarization, J. Opt. B: Quantum and Semiclassical Optics, 2001, 31,215–219.[6] Borghi R., Ciattoni A., Santarsiero M., Exact axial electromagnetic field forvectorial Gaussian and flattened Gaussian boundary distributions, J. Opt. Soc.Am. A, 2002, 19, 1207–1211.[7] Durnin J., Exact solutions for nondiffracting beams. I. The scalar theory, J.Opt. Soc. Am. A, 1987, 4, 651–654.
8] Durnin J., Miceli J.J., Jr, Eberly J.H., Diffraction-free beams, Phys. Rev. Lett.,1987, 58, 1499–1501.[9] Zamboni-Rached M., Recami E., Hern´andez-Figueroa H.E., Structure of non-diffracting waves and some interesting applications, In: Hern´andez-FigueroaH.E., Zamboni-Rached M., Recami E. (Eds.), Localized Waves, John Wiley &Sons, Hoboken, 2008, 43–77[10] Garay-Avenda˜no R.L., Zamboni-Rached M., Exact analytic solutions ofMaxwell’s equations describing propagating nonparaxial electromagneticbeams, Appl. Opt., 2014, 53, 4524–4531.[11] McGloin D., Dholakia K., Bessel beams: diffraction in a new light, Contem-porary Physics, 2005, 46, 15–28.[12] Jackson J.D., Classical electrodynamics, 3rd edition, John Wiley & Sons,Hoboken, 1998, 360[13] Mikki S.M., Antar Y.M.M., Physical and Computational Aspects of AntennaNear Fields: The Scalar Theory, Progr. Electromag. Res. B, 2015, 63, 67–78.[14] Landau L.D., Lifshitz E.M., The classical theory of fields, 3rd English edition,Pergamon, Oxford, 1971, 108–109[15] Jackson J.D., Classical electrodynamics, 3rd edition, John Wiley & Sons,Hoboken, 1998, 239–2408] Durnin J., Miceli J.J., Jr, Eberly J.H., Diffraction-free beams, Phys. Rev. Lett.,1987, 58, 1499–1501.[9] Zamboni-Rached M., Recami E., Hern´andez-Figueroa H.E., Structure of non-diffracting waves and some interesting applications, In: Hern´andez-FigueroaH.E., Zamboni-Rached M., Recami E. (Eds.), Localized Waves, John Wiley &Sons, Hoboken, 2008, 43–77[10] Garay-Avenda˜no R.L., Zamboni-Rached M., Exact analytic solutions ofMaxwell’s equations describing propagating nonparaxial electromagneticbeams, Appl. Opt., 2014, 53, 4524–4531.[11] McGloin D., Dholakia K., Bessel beams: diffraction in a new light, Contem-porary Physics, 2005, 46, 15–28.[12] Jackson J.D., Classical electrodynamics, 3rd edition, John Wiley & Sons,Hoboken, 1998, 360[13] Mikki S.M., Antar Y.M.M., Physical and Computational Aspects of AntennaNear Fields: The Scalar Theory, Progr. Electromag. Res. B, 2015, 63, 67–78.[14] Landau L.D., Lifshitz E.M., The classical theory of fields, 3rd English edition,Pergamon, Oxford, 1971, 108–109[15] Jackson J.D., Classical electrodynamics, 3rd edition, John Wiley & Sons,Hoboken, 1998, 239–240