Vector wave solutions in electrodynamics: the Heaviside-Larmor symmetry and tiered potential invariance

The tiered symmetric structure of the electromagnetic equations, in particular the Heaviside Larmor (HL) symmetry, is used to obtain vector wave solutions of the wave equation obtained from Maxwell's equations. In this HL vector solutions procedure (HL-VS), the seed vector need not be constant but must satisfy an inhomogeneous Helmholtz equation. This seed can be used either to obtain the polarization basis in a given coordinate system or a particular polarization state. The spherical vector waves (SVW) basis is economically derived retaining the physical meaning throughout the procedure. Examples of low order modes are expounded. The cylindrical vector wave basis is also derived and particular polarization states are compared with previous results. Plane waves are used to elucidate the HL symmetry terms and their controlled manipulation in order to tailor specific structured light states. The angular momentum of plane waves with elliptical polarization is obtained. The controversy as to whether circularly polarized plane waves carry angular momentum is answered in the affirmative.


Introduction
The electromagnetic equations require vector solutions even in scenarios where the wave equation is satisfied by scalar fields. For this reason, various proposals have been put forward in order to obtain full vector solutions from either exact or paraxial solutions to the scalar wave equation. The procedure introduced by Stratton [1] and other venerable works [2], as we shall see, alters the state of polarization rather than just introducing the necessary vector components in order to satisfy Maxwellʼs equations. Furthermore, their approach identified the seed vector with the Hertz potentials. However, it has recently been shown that this identification is not adequate if the symmetric tiered structure of the potentials is maintained [3]. Moreover, methods in theoretical physics often emphasize the mathematical aspect and lose sight of the physical context. If the system exhibits spherical symmetry, the Debye potentials provide a scalar solution that can be used to generate the vector solutions [4]. These and other potentials are discussed in ( [5], M Phillips, Classical Electrodynamics). The paraxial approximation has also been used to generate vector solutions [6]. Corrections to the paraxial vector solutions have been made via iterative solutions [7,8], Lax series expansions [9], associated Legendre functions [10] and others [11].
The tiered structure of the electromagnetic formalism developed in [3] has a fourfold advantage that will be exploited here, (i) the seed polarization is preserved, components are adjusted so that the vector electromagnetic equations are satisfied. (ii) The vector solutions do not involve the paraxial approximation, they are only limited to monochromatic fields in the absence of sources. (iii) Seed vectors in coordinate systems other than Cartesian are allowed, in particular, spherical and cylindrical coordinates. (iv) The physical meaning is clear throughout the derivations.
In section 2, the general HL vector solutions formalism is developed. In section 3, spherical vector waves are obtained in a remarkably straightforward fashion. There is no need to invoke the multipole expansion nor the quantum angular momentum operator nor a particular gauge. The l = 1, m = 0 term is explicitly evaluated in

Vector solutions
The set of electromagnetic equations () for isotropic and homogeneous non dispersive linear media in the absence of sources are where the ∇ 2 operator represents the vector Laplacian when operating over a vector field V, and · ( ) y y  º   2 when operating over a scalar field ψ. Only in Cartesian coordinates if V = ψv, v a constant vector, the vector Laplacian is equal to the scalar Laplacian times the vector ( ) ( ) y y  =  v v 2 2 [12]. Moon and Spencer have proposed the ✡ symbol for the vector Laplacian [13]; however, its use has not become widespread. Since the fields are divergence free, they can be written as the curl of another vector, say U, = É U i k . The imaginary unit insures that the polarization of the U seed vector field is maintained and the k −1 inverse wave number constant preserves the same units for E and U. The seed vector U can be, but does not have to be solenoidal. That is, U need not be in the  set. Substitution in (3) gives, , where the precedence of the vector Laplacian and curl operators has been interchanged. Therefore U must satisfy the inhomogeneous vector Helmholtz equation, field solutions are given by where the ( ) g superindex has been dropped for economy. The ℘ 1 and ℘ 2 terms are the HL symmetric solutions, that is, solutions that arise from the interchange of the electric and magnetic fields. This physical meaning has been obscure or non existent in previous derivations. A pictorial representation of this HL vector solutions procedure (HL-VSP) is depicted in figure 1. If ℘ 1 ≠ 0 and ℘ 2 = 0, the electric field maintains, if possible, the polarization of the seed vector U. The two gradient terms in (6a) modify the polarization only to the extent necessary to make E satisfy the  equations. If ℘ 1 = 0, the electric field polarization is orthogonal to the seed vector polarization. However, it preserves the state of polarization provided that Ã Î ; 2 that is, if the polarization is linear it remains linear or if it is circular it remains circular. If the ℘ 1 terms are considered to be the reference polarization, (in contrast with ℘ 2 taken as reference) the ℘ 2 term represents a p 2 clockwise rotation with respect to the ℘ 1 polarization. The explicit i factor in (5a) to (6b), not present in most derivations, is necessary to preserve the state of polarization. The k factors insure the correct units in all terms, for example, if U is adimensional, all terms are adimensional and the fields units are attached to ℘ 1 , ℘ 2 . These assertions will become clearer in the following sections.
The vector U has often been used as the starting point with H = ∇ × U and then the electric field is evaluated from ∇ × H. For this reason, it has been identified with the vector potential A. The above results exhibit the full symmetry of the problem and evince that it is better not to identify U with the vector potential A (For The vector potential A and the Bateman [14,15] potential Ç can also be written in the same way as (6a) and (6b) in terms of U. This tiered symmetry is addressed in section 6.

Vector solutions from scalar solutions
Given a general solution ψ to the scalar homogeneous Helmholtz equation ∇ 2 ψ + k 2 ψ = 0, the directional property of U = ψu can be obtained from multiplication of ψ times a vector u, sometimes called a pilot vector [16], such that (4) is satisfied. The vector field solutions are Figure 1. Vector solutions procedure in order to obtain the electromagnetic fields or the potentials in any tier. A source free homogeneous region (pale yellow) is considered so that the fields are divergence free. A seed vector U (light blue) that satisfies the inhomogeneous Helmholtz equation (4) is proposed. The vector field solutions (dark green) are obtained from the seed field using (6a) and (6b). In this example, a spherical coordinates basis is depicted, sources are isolated (red), and the seed vector field U = ψ r is radial.
If u is a constant Cartesian vector, the derivatives of u are zero and thus ∇ϒ = 0. However, a constant vector u in other coordinate systems can give rise to ∇ϒ ≠ 0. Furthermore, the inhomogeneous Helmholtz equation (4) is also satisfied for certain non constant vectors, for example if u = r, where r is the position vector. A particular wave polarization can also be favoured by choosing an adequate u vector. For these reasons, U = ψu is labeled as the seed vector rather than the more restrictive u constant pilot conception. These various possibilities are expounded in the following sections.

Spherical vector waves
The usual procedure of introducing vector spherical harmonics from the scalar potential multipole expansion and then constructing the spherical vector waves is unnecessary in this exposition. There is no need to invoke a quantum angular momentum operator [17] borrowed from quantum theory nor is it necessary to separate the longitudinal and transverse contributions [2].
Recall that Helmholtz equation ∇ 2 ψ + k 2 ψ = 0 in spherical coordinates can be separated in radial and angular variables ⎛ ⎝ ⎞ ⎠ and the angular differential equation is The radial solutions R l in terms of the spherical Hankel functions are where P l m are the associated Legendre polynomials. Consider spherical coordinates withˆˆq f e e e , , r unit vectors corresponding to the radial, polar and azimuthal variables. Let the seed vector U = ψ k r be the product of ψ, solution to the scalar Helmholtz equation, with the position vector r scaled by k, that is equal to the dimensionless radial vector in the spherical coordinate system = = k k r u r e r . The seed proposal must satisfy the Helmholtz equation possibly with an inhomogeneous gradient term. The vector Laplacian ofÂ e r r is ( ) . An equivalent Cartesian component wise derivation of this result is given in [16]. Equation (4) is then but r∇ 2 ψ + k 2 ψr = 0, since ψ satisfies the scalar Helmholtz equation. Thus y ¡ =  k 2 .
Substitution of u = k r in the electric field equation (7a) gives ⎛ ⎝ ⎞ ⎠ If the radial and angular functions separation is written explicitly ψ = R l Y lm , the ( · ) y   r term becomes r l lm r l lm r r l lm 2 ⎛ ⎝ ⎞ ⎠ where ¶ R r l 2 has been rewritten from the radial differential equation (8a). The electric field is then 1 0 lm l lm r r l lm l lm 1 2 ⎛ ⎝ ⎞ ⎠ Notice that the seed ℘ 1 ψkr term in (9) is canceled out with the-rk R Y e l lm r 2 coming from ( · ) y   r . The seed, in this and other cases, is always canceled out if it is not part of the solution to the vector wave equation. Summation over l, m gives the spherical vector wave (SVW) mode expansion of the electric field where A lm , B lm are constant coefficients. The SVW expansion for the magnetic field, from (7b) is A remarkably simple derivation of the field in terms of the vector spherical harmonic basis is thus obtained. The symmetry of the electric and magnetic field solutions is manifest from these results. The angular dependent functions within each term areŶ e lm r , r∇Y lm and r × ∇Y lm , a minus sign in the last term is required to reverse the cross product factors. They are often dubbed as the vector spherical harmonic (VSH) functions. It often takes a considerable preamble [18] or educated guessing [19] to obtain the VSH functions.
The first term is the far field radiation term E rad , whereas the remaining ℘ 1 terms represent the E r and E θ components in the near and intermediate zones. The E f term with ℘ 2 coefficient involves a radiative and a non radiative part. The electric field with This field is readily identified with the field produced by an oscillating electric dipole [17], In this case, the ik factor included in ℘ 1 establishes the correct slope between the solution at the source and the free field ( [17], p. 428, equation (9.97)).
The terms with ℘ 1 and ℘ 2 coefficients arise, as we have seen, from the exchange of the electric and magnetic fields, namely the HL symmetry. No wonder then, that the electric field azimuthal component in (12) corresponds to the field produced by a magnetic dipole arising, for example, from a circular current loop with e − i ω t modulation. If ℘ 1 = 0 and , this expression reproduces the electric field produced by an oscillating magnetic dipole ( [17], p. 413, equation (9.36)).

Divergence terms
Let us make a digression regarding the transversality of the fields. Recall that the present derivation requested that ∇ · E = 0 and ∇ 2 E + μεω 2 E = 0. Therefore, the solutions (11a), (11b) are divergence free. This fact is not usually recognized because the vector wave solutions are obtained from a multipole expansion that acts as a source for the fields. Consider the divergence of the l = 1, m = 0 electric field. The divergence of the far field radiation term is However, integration over the volume of a sphere centered at the origin for a sphere centered at the origin the E rad · ds dot product is zero since E rad lies in theˆq e direction and ds is radial. The wave propagates inê r , thus the wave vector is= k k e r . Therefore, E rad · k is zero but ∇ · E rad ≠ 0. The identification of transverse fields with divergence free fields is not true in this case. The divergence of the near and intermediate field terms component wise are The sum of these two terms cancel their inverse quartic and cubic r terms The remaining inverse quadratic term from the radial gradient cancels out with the divergence of the far field radiation term. Thus the divergence of the electric field is strictly zero when none of the terms are neglected. However, notice that E · k is not zero in the near/intermediate zone, although ∇ · E = 0. The divergence free field has a longitudinal component. We must conclude that identifying the Helmholtz decomposition with longitudinal and transverse components of the field is not always appropriate. In this example ∇ · U ≠ 0, that is, the seed is not in the  set. The electric field from (7a) iŝ The first two gradient terms are

Spherically symmetric seed polarized in the radial direction
. The ∇ϒ term is obtained from the inhomogeneous Helmholtz equation (4), the U vector Laplacian is  The remaining terms involving the ℘ 2 coefficient are clearly zero. The field for a spherically symmetric wave with radial vector is then zeroˆˆˆŷ , where Ei is the error function. Usually, all that is needed is ∇ϒ.
Notice that if U is the product of a scalar function ψ, solution to the Helmholtz equation and u a unit vector, ∇ 2 U + k 2 U is not necessarily solenoidal, that is, it may not be possible to write it as a gradient. For example, q e e r ikr 1 cannot be written as a gradient. Such a U proposal is then not admissible.

Cylindrical vector waves
The scalar Helmholtz equation , gives the longitudinal Z, azimuthal Φ and radial P scalar solutions, where the argument of the Bessel functions is omitted for economy. For m = 0, the ℘ 1 term is mostly radial if k z ? κ and the ℘ 2 term is azimuthal, as has been shown before [21]. The relationship between spherical and cylindrical vector waves has been addressed by Han et al [22].

Linearly polarized Bessel beams
Linearly polarized Bessel beams have been expressed in two different ways, both will be obtained here in a unified formulation. In the present approach, a linearly polarized seed vector is proposed, say in the x direction y = U e x . The electric field, from (7a) iŝ These are the components that were obtained in [23], the procedure there was to consider a vector potential = J e A e ik z x 0 z . Clearly ∇ · A ≠ 0 and the Lorenz gauge had to be invoked. In order to have circular symmetry, two solutions had to be added, one with ( · ) and another with = ´¢ E A ,¢ = J e A e ik z y 0 z . This latter expression is reminiscent of the remark made in the introduction regarding the inadequate association of the seed vector with A.
The ℘ 2 term in (14) [21], provided that i k is factored. In [21], this result was attained by a judicious combination of A m=±1 , B m=±1 terms in (13a).
Other polarization states can be obtained either by choosing the appropriate values of ℘ 1 and ℘ 2 or evaluating the cylindrical vector waves with a specific initial seed vector.

Plane waves
In amplitude and phase variables, the complex scalar solution is ψ where the wave vector is defined by the gradient of the phase k ≡ ∇j. The electric field solution in amplitude and phase variables is  Ã Ã Î  , 1 2 . The linear polarization is maintained but is rotated clockwise in theˆê e , Notice that the ℘ 1 seed has become the reference, thus the rotation is opposite to the HL rotation where ℘ 2 was taken as reference. Thus, the polarization of the ℘ 2 term is rotated clockwise by p 2 . If Ã + Ã = 1 , and from (17), x y x y x y 1 0 2 0 1 0 2 0 In contrast, consider Strattonʼs proposal where ( ) . The state of polarization of the seed vector is altered, a linearly polarized seed becomes circularly polarized. The reason can be traced back to the imaginary unit included in (5a) and (5b), that is missing in the Stratton expression. This i factor is due to the linear dependence between the curl electromagnetic equations and the time derivatives of the monochromatic fields [3]. . The expected result is obtained: there is no electric field in the direction of propagation for a plane wave in free space. Notice that the term ( ·ˆ) y  ik u is essential to obtain this result. This term adds whatever is necessary to the seed vectorŷ e z so that their sum satisfies the  equations. Then, if the seed cannot be modified in such a way that the field satisfies the  set, it will give a null outcome for the vector field. The null spherically symmetric radial field in section 3.2 is another such case. This type of null results can be used as a test, in order to verify that the formalism is giving the correct vector expressions.

Elliptical polarization
Let us now turn to plane waves with elliptical polarization. These results will be particularly relevant for the evaluation of the helicity and chirality of EM waves. . For u 0x , u 0y > 0, the upper negative sign corresponds to right hand elliptical polarization with clockwise rotation as a function of time looking at the incoming wave. Since U is already a solution to the electromagnetic equations it can be identified with the fields. The elliptically polarized electric field in terms of the real function is , and from (17), Therefore, the circular polarization seed is maintained but a  p 2 out of phase circular polarization of the same handedness is added with a ℘ 2 weighing factor. Generalized to an arbitrary elliptical polarization seed,

Modification of the state of polarization
Propagation of a wave in free space cannot alter its state of polarization. This statement is consistent with the superposition of Ã Ã Î  , 1 2 terms. Recall that the difference between these two terms come from the electricmagnetic HL symmetry.
However, it is possible to allow for complex coefficients Ã ¢ Ã ¢ Î  , 1 2 (primes denote complex values). In the polar representation, 2 . The relevant quantity regarding retardation being the phase difference ( ) a a e i 2 1 . In order to obtain a field with Ã ¹ Î  0 2 from a field with ℘ 1 ≠ 0, requires an anisotropic medium, so that the phase retardation in one direction differs from the retardation in an orthogonal direction. Quarter and half wave plates are the usual way to alter the state or direction of polarization of a given field. Nonetheless, any complex phase difference between ℘ 1 ', ℘ 2 ' is also a free space vector solution. Therefore, an arbitrary polarization state with complex ℘ 1 ', ℘ 2 ' is possible. For example, a linearly polarized plane wavê via a quarter wave plate. From the linear polarization expression (18), but allowing for x y 1 2 that is equal to the elliptical polarization seed function (19). Furthermore, each mode in the different coordinate systems could have a different retardation for the ℘ 1 ' and ℘ 2 ' solutions. This degree of freedom can be incorporated by allowing the mode coefficients to be complex, for example, in the spherical vector basis (11a)- . Structured light could then be shaped such that different modes are in different states of polarization. Recall that since the mode expressions are true for an arbitrary position and the modes are orthogonal, they do not 'mix' as they propagate.

Potentials and tier symmetry
In the tiered structure of the electromagnetic equations, the free fields are given by the time derivative of the vector potentials A and Ç, t t for monochromatic frequency ω components, . Notice that the starting point is not the usual H = ∇ × A curl relationship. This curl result is obtained as a consequence because the fields and potentials satisfy the  equations. In the absence of an external magnetic polarization, Ç = m C 1 , where C is the Bateman second potential [14,15]. The potentials in turn can be written in terms of the potpotentials, The recursive relationship between potentials as the process is repeated is The electric field E is identified with A 0 and the magnetic field H with Ç 0 . The vector potentials A, Ç are equal to A 1 , Ç 1 and the potpotentials A 2 , Ç 2 are the Hertz potentials [24]. The curl equalities are a consequence of the  equations. The symmetric tiered structure of the  set in the absence of sources, requires divergenceless potentials for all tiers In Fourier space, the homogeneous vector Helmholtz equation is satisfied by the fields or the potentials in any tier, for each monochromatic wave of frequency ω. Since the divergence of A j (or Ç j ) is zero, this vector can be written as the curl of another vector, say U, ( ) = Á U j p i k . The argument in section 2 then follows for the potentials for all Î  j . The vector fields or the vector potentials in any tier are given by provided that U satisfies the inhomogeneous Helmholtz equation, ∇ 2 U + k 2 U = ∇ϒ. Let us insist that the seed U is not necessarily in the  set. It is clear from (23) or (25a)-(25b) that the fields and their corresponding potentials have the same number of non vanishing components. In the tiered formalism, for each Fourier component, E and A have the same non vanishing vector components scaled by iω, since E = − ∂ t A = iωA. The same is true for the magnetic field H and the vector potential Ç, In other approaches, U has been identified with the vector potential A. The electric field E and the potential A can then have a different number of nonvanishing components [25]. In that case, cross products of these vectors have to be very carefully handled because they can lead to peculiar results.
If a gauge transformation is performed A → A + ∇f A . From (24), ∇ · E = 0 and ∇ · A = 0, thus , ∇f A = 0. If the potential falls off to zero at infinity f A = 0, this gauge is equivalent to the temporal gauge [26]. There is therefore no gauge freedom or at most the highly restrictive vector gauge, ∂ t ∇f A = 0. The scalar potential f A is then electrostatic (or an additive space independent linear time term) and plays no role in electrodynamic phenomena. The corrections to the helicity and its flow in order to make the conservation equations gauge invariant [27], are not required in this scheme. Furthermore, since the Ç j vector potential is also present, the six vector components of the fields are replaced by the six vector components of the potentials. The same is true for any tier. The  equations set then retains the same number of variables in any tier [27]. The remaining of this subsection elaborates on this fourth point. The helicity and flux for an elliptically polarized plane wave propagating in the z direction represented by the real field (20c) are [29], where the ( ) e 1 2 2 factor comes from the constant intensity condition instead of constant E 0x = E 0 imposed in [29]; the upper sign corresponds to right hand elliptical polarization. These quantities are not zero provided that the polarization ellipse does not collapse onto a line, e ≠ 1.
However, if a complex representation of the fields is used The corresponding vector potentials are and has the opposite sign of theˆê e y x term. All other inner and outer products are also zero. At first sight, this is a somewhat perplexing result, since the helicity and its flow evaluated from the real fields give non zero results but its evaluation with complex fields vanish. The underlying reason is that linear independence is a necessary condition to obtain non zero assessed quantities in the continuity equations obtained via the complementary fields procedure [46]. The real potential fields that play the role of complementary fields in order to obtain the helicity continuity equation are A and Ç. These fields, in the particular case of plane waves with elliptical polarization are  (27). This problem is often encountered in nonlinear optics where field products are at the fore, and has led to a variety of awkward definitions [48].

Conclusions
A systematic way to obtain vector wave solutions using the Heaviside Larmor symmetry vector solutions (HL-VS) procedure has been presented. A compact HL-VS derivation is possible in different coordinate systems for arbitrary polarization states. The most common cases of Cartesian, cylindrical and spherical coordinates have been treated in a unified way. In Euclidean three dimensional space, there are eleven orthogonal coordinate systems obtained from orthogonal intersections of first and second degree surfaces, that allow the separation of variables for the Laplace and Helmholtz equations [13]. In principle, these systems are amenable to the present formalism. However, not for all of them a transverse solution to the vector Helmholtz equation tangential to one of the coordinates is possible. Some of these coordinate systems will be treated in a forthcoming communication. The seed vector can be used to obtain the function basis expansion in a given coordinate system or to obtain a particular polarization state. For the latter, it is advantageous to use the desired polarization state as seed vector. However, care should be taken to insure that the seed proposal is admissible, namely, that it satisfies the Helmholtz vector equation with an inhomogeneous gradient term. The physical meaning is retained throughout the procedure. The two terms with coefficients ℘ 1 , ℘ 2 in the vector solutions correspond to the HL symmetry, that is, the exchange of the electric and magnetic fields. Depending on the ℘ 1 , ℘ 2 values, a continuous rotation symmetry is achieved. In the absence of sources, for each monochromatic component, the tiered symmetry establishes electromagnetic equations with the same form for the vector potentials in any tier. The potentials and the fields thus have the same number of non vanishing components. The possibility of a gauge transformation is then highly restricted to gradient free electrostatic fields. Since these fields play no role in dynamic phenomena, there is no gauge freedom in the electrodynamic tiered symmetric formalism. Therefore, an asset of the present approach is that it avoids the problem of observables that depend on gauge dependent potentials, such as rotational content related quantities. The angular momentum of elliptically polarized plane waves, evaluated via the helicity and helicity flow in this scheme, has been shown to be finite if the polarization ellipse minor semi-axis is different from zero. According to this view, circularly polarized plane waves do carry an angular momentum density. The helicity and its flow have to be defined in terms of complex conjugate quantities if the fields are represented by complex functions, otherwise, spurious vanishing results can be obtained.