Consequences of the Potential Gauging Process for Modeling Electromagnetic Wave Propagation

This predominantly theoretical article focuses on a qualitative discussion of peculiarities, which are introduced in practical electromagnetic (EM) wave propagation scenarios when the gauge for the electrodynamic potentials is not chosen in accordance to the appropriate space-time metric of the underlying physical framework. Based on ordinary vector calculus, this is done for the viewpoint of radio frequency (RF) engineers by using two examples of guided EM waves: one large-scale case of a terrestrial scenario and one small-scale case involving a device level setup. Readers may benefit especially from this practical orientation, since gauging is often analyzed primarily mathematical by solely arguing on terms of equations instead of discussing concrete applications. The provided context aims to enhance the usual perspective and is applicable for a wide class of situations involving various wave types at any frequency.


I. INTRODUCTION
B ESIDES direct formulations which describe the electro- magnetic (EM) field (see, for instance, [1], [2], or [3]) often used in numerical calculation, it is possible to utilize potentials for its representation.The reason for choosing a specific gauge in this context for EM modeling is typically a desired simplification of the underlying partial differential equations (PDEs) involving the ordinary electrodynamic potentials ϕ and A. Many textbooks, as, for instance, [3], [4], [5], [6], [7], [8] or [9] introduce them among other motivations to model EM wave propagation.This article addresses such a customary approach in which potentials are applied to obtain scalar wave equations (see especially [5]) instead of vectorial ones for the electric and magnetic field.
The general opinion (see, e.g., [9], [10], [11] or [12], a discussion from a historical point of view is presented in [13]) assumes -based on gauge freedom 1 -the possibility to impose additional conditions on the potentials, which does not lead to consequences for the physical EM field when it is derived from these potentials.This assumption is confirmed throughout this article, at least for the EM field itself.With the aid of guided wave propagation examples from radio frequency (RF) engineering, it will be illustrated that the choice of gauge has consequences in this case with respect to the behavior of certain EM quantities.This concerns the electric scalar potential and one part of the magnetic vector potential.The reason is, that the choice of gauge determines the space-time [14] of the application.That is specifically due to the combined use of the ordinary electrodynamic potentials, as will be shown throughout this article.To investigate this, it is useful to distinguish EM quantities between the two types "irrotational" and "solenoidal", indicated by subscript "i" or "s", respectively, for everything below.In accordance to, e.g., [2], [9] or [15], an irrotational vector field features a vanishing curl and a solenoidal one has the property of zero divergence.If one argues in temporally varying EM scenarios with electrodynamic potentials of irrotational nature2 (see, for example, [16], [17] or [18] for applications in several disciplines), these investigations may be influenced by gauging in an inappropriate manner.According to, e.g., [2], [19] or [20], the usual approach to calculate surface currents -placed along the propagation path -is to determine first the traveling EM field described by the Poynting vector's real part (possibly with the aid of the solenoidal magnetic vector potential) and then derive the former quantities from this solution.Subsequently, the electric scalar potential and the irrotational magnetic vector potential are commonly calculated, which together gives the irrotational part of the electric field (see, e.g., [21] or [22] for this distinction 3 ).That approach is used throughout this work.
The high relevance of potential formulations as an established topic results from the broad scope of application in, among others, teaching, research and industry.A practiceoriented analysis based on guided EM wave propagation examples involving changed phase velocities caused by the potential gauging process with reference to the used space-time does not yet exist.In this context, the behavior of individual potentials in such scenarios is analyzed in detail whether the usual gauge is used or not.Throughout this article, electrodynamic potentials are understood as mathematical auxiliary quantities, their physical nature is not explicitly addressed.In order to distinguish the influence of the medium's permittivity and permeability in which the waves propagate from the impact of gauging the potentials, vacuum is assumed as the carrying medium throughout all presented investigations.That does not form a restriction of generality, rather it simplifies the discussion's content.Everything shown justifies the choice of the Lorenz gauge when modeling practical wave propagation scenarios.
This article is organized as follows: As a first major topic, a theoretical analysis is broadly provided in Section II.Starting with a detailed distinction of two topological kinds of field lines, a Helmholtz decomposition of the magnetic vector potential follows.The concept of retardation outlined next helps to explain a possible delay between cause and effect.Phase velocities of considered EM field quantities are recapitulated afterwards.The relation of both the Coulomb and the Lorenz gauge to the underlying space-time is discussed subsequently.Due to the special behavior of the irrotational electric field's shares, they are examined in more detail.Section III contains as a second major topic an investigation regarding the effects of gauging to several EM field quantities involved in two simplified practical wave propagation applications.After some preliminary considerations, the mentioned examples are comprehensively addressed.These are the first transatlantic radio communication and a typical RF scenario involving a rectangular waveguide.On this basis, important findings are finally summarized.

II. THEORETICAL ANALYSIS
Throughout this section, the different topics which are used to justify the reasoning on possible consequences induced by the potential gauging process are discussed.Alternatively to analyzing the single terms described in [21], the adjustment of the gauge to the appropriate space-time metric is motivated.This makes the irrotational potentials' propagation behavior more intuitive instead of accepting the potential gauging process in its original form, which means especially the peculiarity of changed phase velocities.

A. TWO TOPOLOGICAL KINDS OF FIELD LINES
The distinction between two specific kinds of field lines is highly important for the following discussion in terms of which quantities are affected by gauging.An exemplary irrotational electric field pattern E i as well as a solenoidal one E s are illustrated in Fig. 1 (a) and (b), respectively.
The Coulomb field between two monopole charges with different signs +Q and −Q is selected in (a).All field lines point to the right, start at +Q and end at −Q.Therefore, they represent open paths.If the magnitudes of the charges placed in the setup are time-harmonic, both intensity and direction of the field lines change over time, which means they exhibit a wave-like character.The example just explained can be described by the electric scalar potential ϕ, which in turn has the volume charge density as a source.Alternatively, it is also possible to replace the charges +Q and −Q in (a) by two irrotational electric current densities J i located at different conducting areas and to consider the corresponding distribution of the irrotational magnetic vector potential A i .Then, the source would be described by a vectorial J i instead of a scalar .Nevertheless, the field pattern would be also purely irrotational (here: the whole field is longitudinal).The second example in (b) contains two dipole charges Q 1 and Q 2 , where one of them acts as a transmitter and the other one represents a receiver of EM waves.Between them, the field lines of the radiated E s are sketched, which can be described by the solenoidal magnetic vector potential A s .Obviously, the field pattern consists of closed lines.
They are oriented in a transversal manner (orthogonal to the propagation direction, which points to the right) at almost all locations.In [23], a more detailed investigation of the electric field E around a Hertzian dipole is reported.
In many practical cases, the description with transversal and longitudinal field components is sufficient.Because of, for instance, boundary conditions or obstacles, this reasoning does not hold in every situation.Therefore, this article refers to the terms "solenoidal" for field lines described by closed paths and "irrotational" for those which exhibit open paths.By the use of a Helmholtz decomposition, every vector field can be separated into (at least) these two shares, which will be demonstrated in the subsequent section.Often in literature, the harmonic component is also considered (see, e.g., [24] for a discussion regarding vector calculus or [25] concerning computational applications).It acts like a uniform directional offset.Additionally, other decompositions exist, which can possibly further subdivide vector fields.Some authors -such as [26] or [27] -argue that the lines of a solenoidal field are not closed in every case.This view will not be rated here.By aiming to retain the possibility to discuss some practical applications at a sufficiently deep degree of precision, the field lines of a solenoidal field are assumed to be closed.

B. HELMHOLTZ DECOMPOSITION OF THE MAGNETIC VECTOR POTENTIAL
With the aim of providing an insight into the propagation behavior of several EM quantities, a Helmholtz decomposition of the magnetic vector potential A is discussed in this section.Note that the gradient of the electric scalar potential ϕ does not have to be decomposed, since it is purely irrotational.In [28], it is stated that the Helmholtz decomposition is physically meaningful in terms of determining radiating wave components.The latter are conceived in the language of this article as the solenoidal ones.On the other side, there are non-radiating components, which are related to irrotational quantities.This motivates in particular the consideration of practical examples relating to guided EM waves in this work.The separation into (at least) these two kinds of fields is known as the Helmholtz theorem (see, e.g., [9] or [29]).According to [14], the Coulomb gauge corresponds to an elliptic Helmholtz decomposition, whereas the Lorenz gauge corresponds to a hyperbolic Helmholtz decomposition.For what follows, no distinction is made between them.Details regarding these two as well as to further kinds of Helmholtz decompositions are discussed, for example, in [14].
In the following derivation, the International System of Units is applied.It is oriented on [29], where both A and E are decomposed.See also, for example, [21] for such a decomposition concerning E. The velocity gauge is used in this article to describe the interval of possible phase velocities v p obtained between the Coulomb and the Lorenz gauge.It is utilized to disclose possibilities apart from the usual choice, see the Appendix for details.At first, (32) is rewritten using both the speed of light c defined in (35) and v p introduced in (37).Usage of the velocity gauge (36)which is utilized here in a converse sense -produces Application of ( 31) on (1) and some rearrangement leads to ( The Helmholtz decomposition separates A into the sum of A s and A i .Whenever the colors violet or red are used in this article, they refer to traveling solenoidal or irrotational vector quantities, respectively.With regard to the current density J, the associated sources of A s are J s , whereas J i are understood as the sources of A i .This is exemplarily illustrated on a Hertzian dipole in [23].Note that J s do not have to be necessarily solenoidal themselves, rather they are the sources of the vector potential A s , which is observed far enough from the location of J s .The same reasoning applies for J i .Therefore, current densities are not highlighted in color in the following figures if their assignment is not clear.Similarly to [29], a decomposition of both A = A s + A i and J = J s + J i turns (1) into and (2) appears as Because of the uniqueness of the Helmholtz theorem (see especially [9]), the solenoidal part of (3) is In (5), the solenoidal part of (3) instead of its irrotational part is extracted, since a relatively simple PDE is obtained.The third term on the left-hand side of (3) vanishes, since the divergence "∇•" of a solenoidal field quantity yields zero.Analogously deduced, the irrotational part of (4) reads Here, the third term on the left-hand side of (4) vanishes, since the curl "∇×" of an irrotational vector quantity yields zero.Obviously, ( 5) is a wave equation with the phase velocity v p = c.By assuming a propagation path placed in vacuum and further a non-moving J s , the solenoidal component A s propagates with the speed of light c away from its source J s .In contrast, the connection between the propagating A i and the non-moving J i described in (6) shows a different behavior.It is determined by the phase velocity v p , which in turn depends on the chosen gauge.
To clarify the spatial orientations of the two kinds of the magnetic vector potential A s and A i , they are illustrated in Fig. 2, where a purely transversal wavefront 4 is depicted.For waveguide modes -such as the transverse electric (TE) 10mode of the second practical example, discussed in Section III -both A s and A i may have components in transversal and longitudinal directions due to boundary conditions.Nevertheless, their solenoidal / irrotational nature is preserved.As evident from ( 5) and ( 6) as well as from (8) presented below, current densities are oriented in the same spatial direction as their corresponding magnetic vector potential.
For the purpose of expressing the dependency of ϕ on v p , the velocity gauge (36) is inserted in (24), which yields Just like (6), the wave equation ( 7) features a phase velocity which depends on the gauge.This behavior is in direct contrast to that of the commonly used wave equation (5).

C. THE CONCEPT OF RETARDATION
For the progress of investigating phase velocities, the concept of retardation is useful.Especially in [8] or [9], it is extensively covered.In the Lorenz gauge, the explicit solutions for the retarded potentials by assuming three spatial dimensions (see, e.g., [2], [9] or [30] for similar formulations) are and 4 Imagine the transverse electromagnetic (TEM)-mode of a lossless coaxial wire or see the first practical example presented in Section III. in vacuum, where d = x 2 −x 1 is the distance vector between that one which starts at the origin and points to the source point x 1 and the one which describes the observation point x 2 with respect to the origin.The retarded time τ is the difference between the start t 1 and end time event t 2 .It is expressed in the velocity gauge by using d for the magnitude of d as This means in (8), that τ can be calculated using c in order to describe A s (cf.( 5)) or, alternatively, based on v p , which would yield A i (cf.( 6)).Both ( 8) and ( 9) describe the timedependent (non-static) case.The travel of an EM quantity defined in the Minkowski space-time5 over a distance d naturally takes time.This one is the retarded time τ defined in (10) for causal fields (the object starts to travel at a source located at one point and arrives later at another point), which is illustrated in Fig. 3.It is obvious from the occurrence of v p in (10), that τ is also affected by gauging.In the Coulomb gauge, τ = 0 applies and in the Lorenz gauge, τ = d / c is valid.Even a moving constellation of ϕ and would be instantaneously coupled together in the Coulomb gauge, since no time derivative occurs in (24) after this gauge is applied, see the first equation of (39).A completely different behavior is observed for J s .It can be understood as that part of a current distribution located at an antenna, which produces based on (5) a solenoidal field observed far from the antenna with τ = d / c in every gauge.
In this work, both the electric and the magnetic field are derived from the electrodynamic potentials.In what follows, the former is further subdivided.The first share of E defined in (23) can be identified as the irrotational part E i by Since the EM field itself is -due to gauge freedomunaffected by gauging the potentials, the physical nature of E i has to be preserved.In [21], contributions of ϕ and A i to the electric field are outlined for the Coulomb and the velocity gauge.There it is shown, that all components of the electric field remaining after cancellation of terms are retarded.This is a clear contradiction to [22] or [29], where it is assumed that E i travels instantaneously in every gauge.In the Coulomb gauge, A i vanishes (apply v p → ∞ for (6)) and E i is completely determined by ϕ, where the latter is a solution of the Poisson equation ( 24) resulting for ∇ • A = 0.If another gauge is applied (such as the Lorenz gauge), the longitudinal part of "−∂ / ∂t A i " compensates the longitudinal "−∇ϕ" each induced by this gauge, which yields together in (11) the same E i as in every case.In this context it is mentioned again that this process is described in detail for the Coulomb and the velocity gauge in [21].
The second share of E described in (23) is the solenoidal electric field which is in [28] understood as a radiated field without a connection to charges.To complete the quantities which contribute to the EM field, the whole magnetic field given in ( 29) is now identified as a solenoidal one and suitably rewritten as Irrotational components of the magnetic field would be based on magnetic monopoles, which have not yet been experimentally observed (the right-hand side of ( 21) is zero).The sources of A i and ϕ arising in ( 6) and ( 7) are connected by the continuity equation (see, e.g., [3] or [30]) In short, this means that a current source is formed by the movement of charges.Here, it is directly assumed that only J i fits into (14).Since the divergence of curl is zero, J s does not contribute to (14).For further reading, J is decomposed into its solenoidal J s and irrotational J i components, for instance, in [29].
In the next section, EM quantities which are affected by the velocity gauge in terms of v p will be addressed in detail.

D. PHASE VELOCITIES OF EM QUANTITIES
It is convenient to recapitulate the phase velocities of all quantities discussed so far in Table 1.Both J s and J i as well as are non-moving, which are those used in ( 5), ( 6) and (7).Usually, only the three solenoidal components A s , E s and H s are in relation to the real part of the Poynting vector, since they inherently describe active power transfer.In this case, the phase velocity equals the group velocity (cf., e.g., [2]).As well as these solenoidal components, E i propagates in every gauge with v p = c in vacuum, as discussed by means of (11).The induced current density J e is to be introduced here as well, which travels also with v p = c.It is located alongside the propagation path of a guided wave and produced by the traveling mode pattern of the solenoidal EM field.In the course of Section III, it is identified as an irrotational eddy current density (therefore: subscript "e").In contrast, the potentials ϕ and A i propagate with v p between c and → ∞, depending on the chosen gauge.If one assumes v p → ∞ in (6), no non-trivial solution exists for A i .However, an insertion of v p → ∞ in (7) shows a non-trivial solution for ϕ, which explains the different behavior of ϕ and A i .Note that the interval c ≤ v p < ∞ used in the highlighted column of Table 1 is valid for gauges, whose phase velocity lies between them of the discussed limit cases Coulomb and Lorenz gauge.As addressed in the Appendix, the Kirchhoff gauge shows an imaginary phase velocity for ϕ, which is outside this interval.Therefore, for an investigation of a wider range of the velocity gauge, that interval has to be adapted.Nevertheless, the quantities affected by every choice of "∇ •A" are the same, which allows for general statements.

E. METRICS AND RELATED SYMMETRIES
Based on the velocities of the quantities just recapitulated, the relation of the Coulomb and Lorenz gauge to their associated space-times is addressed below.In this article, symmetries are understood as a class of transformations regarding the position or orientation of objects (more specific: EM quantities), which preserve the related distance between two points located inside a space-time.According to [34], a generic general relativistic space-time admits no nontrivial symmetries.This does not apply to the Euclidean space or the Minkowski space-time.Thus, it follows that symmetries are closely related to the metric, since no metric is defined, no symmetries are present.Following [14], the used metric to describe a physical setting determines the kind of Helmholtz decomposition as well as the correct gauge for the electrodynamic potentials.In other words, an appropriate space-time embeds the proper phase velocities of all quantities considered in the physical setting.By using the common way of choosing the Lorenz gauge when dealing with RF applications, the Minkowski space-time fits well with the desired propagation behavior.
The metric of the 3-dimensional Euclidean space R 3 is defined by the Euclidean distance of the coordinate differentials dx, dy and dz as the distance differential dl, which reads as (see, for instance, [32] or [35]) The Euclidean group (see, e.g., [36] for further reading) includes the symmetries of the Euclidean space, those who are of interest in this work are spatial translations.Since no time is involved in (15), only spatial symmetries exist there.
For the four-dimensional Minkowski space-time R 4 , the concept of ( 15) is enhanced by the time differential dt under consideration of the speed of light c to the Minkowski distance (see, for example, [32] or [35]) The definition of ( 16) can be equivalently expressed with the metric signature (−, +, +, +), which elucidates the opposite handling of time and space.Symmetries induced by ( 16) are spatial rotations and boosts.They are collected in the Lorentz group [31].Following [33], boosts are transformations which allow to transfer the description of one observer to the description of a second observer moving with constant speed relative to the first observer.This describes well the propagation of a wave field with regard to a non-moving reference system.The boosts of Minkowski space-time -which are the symmetries of interest covered in this article -are similar to the spatial translations of the Euclidean space with the exception that they consider in addition the time dimension.In contrast to R 3 , which is equipped with an absolute time and space, instantaneous changes of the EM field quantities' locations are not described by the symmetries of R 4 .For waves propagating in the direction of ascending spatial coordinates, ( 16) leads with the time-harmonic convention "exp(−j ω t)" to the undamped exponential function where ω is the angular frequency and k is the wave number.
It is generally known that the latter is defined for vacuum by which shows together with the definition of the retarded time in (10) the relation between the two terms in the exponent of (17).Since ( 17) is derived from ( 16), the Minkowski space-time is implicitly assumed in general when dealing with EM wave propagation.That behavior in R 4 is depicted in Fig. 4, where a mode pattern of the electric wave field's real part Re with magnitude E = |E| is indicated for an arbitrarily chosen point in time.There, the mode pattern propagates in vacuum with the velocity c.For R 3 , wave propagation cannot be described well, since the time dimension is not involved in (15).By limiting the solution space of the potential PDEs (38) to outgoing waves, only retarded solutions remain.Incoming waves can be eliminated from a wave equation's solutions by consideration of the Sommerfeld radiation condition, see, for instance, [37].As discussed throughout this section, if the chosen gauge differs from the Lorenz gauge when the Minkowski spacetime is assumed, a symmetry break (a violation of at least one symmetry) occurs for ϕ and A i .In this case, the Minkowski distance ( 16) is not preserved, what justifies the Lorenz gauge with reference to the metric.

F. SHARES OF THE IRROTATIONAL ELECTRIC FIELD
The behavior of potentials affected by gauging -which form together the irrotational electric field -are exemplified in the following.Note that for this analysis, all propagation processes are stopped and the contributions of ϕ and A i are evaluated at the same location.
In order to visualize the two shares of E i defined in (11) for the two limit cases Lorenz and Coulomb gauge, a qualitative representation is given in Fig. 5. Everywhere in this section, the magnitude over distance is used for the argumentation.At the top, both irrotational potentials are considered, whereas at the bottom, the two terms forming E i are examined.In (a), the electric scalar potential of a point charge is depicted over the distance d, which is proportional to d −1 .Its spatial derivative in the form of a gradient is depicted in (c), which decays in contrast with d −2 .For both diagrams, the magnitude in the case of the Coulomb gauge is doubled in comparison to that of the Lorenz gauge.The two charts (b) and (d) show no contribution in the case of the Coulomb gauge, since A i vanishes as already outlined.The progression for both the solid and the dashed curves are assumed to be equally besides their different magnitudes.If the dashed curve of the Lorenz gauge depicted in (c) is added to that one pictured out in (d), exactly that of the Coulomb gauge shown in (c) is generated.This means that E i analyzed over d is the same in both gauges, as gauge freedom implies.

III. PRACTICAL EM WAVE PROPAGATION SCENARIOS
Before the examination of practical examples, it is convenient to specify certain things more precisely and to give an outline which additional assumptions are made as follows.
1) The examples presented in this section exhibit a supporting structure.These can be traveling induced current densities J e as well as the potentials ϕ and A i which in turn form E i based on (11).In particular, the presence of a supporting structure means the waves are bound to a medium with non-zero conductivity.If freespace waves instead of these guided ones are assumed, no contradictions would arise since the irrotational potentials are not considered in this case.Then, the solenoidal components E s and H s could be derived from the likewise solenoidal A s by using ( 12) and ( 13).2) Note the distinction between traveling current densities of the supporting structure J e -which are caused by H s of the traveling EM wave field -and nonmoving current densities J s and J i , which excite the propagating potentials.The magnetic field generated by J e (directed opposite to H s ) is neglected for the sake of simplicity.Only J s , J i and are considered as sources of the potentials, see ( 5), ( 6) and (7).3) For further simplification, the surrounding region close to the feeding structure is not discussed in the following.See, e.g., [23] for such an investigation.In terms of 2), the relation between H s and J e is of interest, which are perpendicular to each other on the guiding surface of the conducting material as (see, e.g., [4] or [20]) where n is the surface normal vector of the conductor.The relation (19) was mentioned in Section I "the usual approach to calculate surface currents".To explain that by means of a concrete application, the cross-sectional view of a coaxial wire is depicted in Fig. 6 for the lossless case.This can be understood as a guided version of Fig. 2. Therein, E s builds together with H s the well-known field pattern of the TEM-mode.In accordance to (12), A s is spatially oriented in parallel to E s .As illustrated in Fig. 2, A i points in Fig. 6 again completely towards the longitudinal direction.Due to the consideration of the supporting structure (here the inner and outer conductor's surfaces of the coaxial wire),  traveling J e which propagate as fast as H s are also present.That induced current density exhibits solely longitudinal components for this scenario.In addition to the application of RF engineering presented here, the direction of J e in Fig. 6 describes well the case of direct current.The stationary mode pattern of a start t 1 and an end time event t 2 in a general space-time (the number as well as the behavior of the space-time dimensions are not specified, only a distinction between them is noted) is indicated in Fig. 7. Based on (17), a periodic variation of a propagating wave's amplitude appears if the time or the location varies.Furthermore, the propagation of a wavefront takes the retarded time τ = t 2 − t 1 to travel from location x 1 to x 2 , expressed by d = x 2 − x 1 as discussed earlier.Ordinary technical studies regarding the mode pattern at a certain location x 1 or x 2 assume a particular time t 1 or t 2 for all involved quantities, a distinction between them is usually not made.
To establish a practical reference of those quantities, they are distinguished -as a general allocation -with respect to the material section that primarily support them in Table 2.This assignment is not always like that, since, for instance, H s can enter the supporting structure because of the skin depth.Another exception would be if E i is partially placed in vacuum caused by the mode pattern under consideration.Of course, A i or ϕ can also be analyzed in vacuum, but this is not the intention here.The first line of Table 2 contains the source region, where J s , J i and are located.Solenoidal quantities are usually considered for active power transfer in vacuum, which are inserted in the second line: A s , E s and H s .The mode pattern's share located at the supporting structure is built by A i , ϕ and E i , noted in the third line of Table 2. There, also J e is located.By applying a gauge, the phase velocity v p of A i and ϕ is set between c and → ∞, as shown earlier in Table 1.

A. FIRST TRANSATLANTIC RADIO COMMUNICATION
In 1901, Guglielmo Marconi has built the first transatlantic radio communication in history between Cornwall in England and Newfoundland in Canada (see, for instance, [38] for details).
A vast simplification of this terrestrial large-scale EM wave field scenario is shown in Fig. 8, where the transmitting antenna is drawn on the left and the receiving antenna is depicted on the right.Earth curvature, finite conductivity of soil / water, charge separation effects, atmospheric variation of the permittivity as well as the consideration of obstacles -among other things -are neglected in this simplified model.All wave quantities should travel with the same phase velocity, since the solenoidal fields are bound to the irrotational current density distribution J e (drawn in red) placed at the supporting structure.According to (19), the pattern of J e travels as the solenoidal quantities with v p = c.In the limit case of the Coulomb gauge, ϕ considered in soil / water arrives instantaneously at the receiving antenna, whereas A i vanishes.
The power transfer in air needs according to (10) the retarded time τ of about 11.67 • 10 −3 s to travel, whereby d ≈ 3500 • 10 3 m and v p = c ≈ 3 • 10 8 m/s.This leads to a significant difference if ϕ or A i in soil / water is analyzed together with the EM field described by the real part of the Poynting vector.As stated earlier, that contradiction occurs because EM wave propagation is implicitly based on the Minkowski space-time.Here, the Coulomb gauge is chosen for demonstration purposes, which is premised on the Euclidean space.By assuming v p → ∞ for the irrotational potentials, (7) yields an instantaneously moving ϕ, whereas A i in (6) vanishes.For the solenoidal components, (5) was utilized, which is independent of the gauge.

B. RECTANGULAR WAVEGUIDE
Waveguides are often used in RF engineering to transport energy or information between components.The commonly used rectangular waveguide suits well as an example of a guided small-scale EM wave field scenario at the device level.Concerning a waveguide whose interior is limited by metallic walls, the natural propagation behavior is clearly a combined travel of the mode pattern build by E s and H s .It travels inside the hollow space together with quantities associated to the supporting structure located at the inner surfaces.In this view, that structure mainly comprises propagating induced current densities described by J e .Charge separation effects are neglected in this simplified investigation.Again, the retarded time τ represents the delay of interaction between the potentials and their sources.
In Fig. 9, the quantity J e located at the supporting structure that belongs to the TE 10 -mode is shown.It is the fundamental mode of the rectangular waveguide, see, e.g., [2], [8] or [20] for details.The red pattern of J e has to be in phase with the solenoidal field components, which in turn travel inside the waveguide in vacuum with the velocity of light.This is given again by (19).Concerning Fig. 9, the solenoidal electric flux density D s is not drawn, which forms together with J e closed rings [20].Due to this closed flow enabled by boundary conditions, which is built by the aforementioned two fundamentally different quantities, J e is identified as a construct of eddy currents.That pattern travels in accordance to the discussion by means of Fig. 6 alongside the propagating wave and is assumed to produce no magnetic field itself to simplify the investigation.Even for the Coulomb gauge, (5) fully describes active power transfer inside the waveguide.Exactly as in the first example, the retarded time is shrunk to zero (see Fig. 3), ϕ travels instantaneously and A i vanishes for this gauge.
In this application, the power transfer in vacuum inside the waveguide needs -by using (10) -about 0.33 • 10 −9 s when assuming d = 100 • 10 −3 m (see Fig. 9) and again v p = c ≈ 3 • 10 8 m/s.It is striking that this time is much smaller than that calculated for the first example in the previous section, which is explained by the different scales of both setups.
The following passages aim to reinforce the chain of reasoning by applying the aforementioned identification of two types of field lines based on this practical RF scenario.It was stated just now, that the distribution of J e located at the waveguide's inner surface induced by the traveling mode pattern of the solenoidal EM field was identified as a construct of irrotational eddy currents.Following the viewpoint in [23], open field lines are understood as irrotational ones.This description holds for the lines of J e in a cross-sectional view placed at the middle of the rectangular waveguide, which is exemplarily depicted in Fig. 10.Without the consideration of D s (solenoidal nature, looks like irrotational due to boundary conditions), which crosses the hollow space inside the waveguide from the top to the bottom, the resulting field lines are open.Therefore, J e is identified as irrotational.On the other hand, if D s is included, closed field lines are build (cf.[20] or [39]), which would lead to an assignment to the solenoidal type.
Obviously, an ambiguity occurs if one argues in a local sense by using the specific argumentation just described on the transversal field pattern.A unique determination can be achieved by imagining globally, that the longitudinal part of J e travels along the propagation direction of the wave field (cf.Fig. 6).For the transversal part of the field pattern, another explanation fits better.Since the carrier of field lines changes between the two fundamentally different EM quantities J and D, this construct is not identified as closed, rather both of them feature an irrotational field pattern (cf.[23]).In this way, a clear identifiability as a construct of irrotational eddy currents J e follows.

C. FINDINGS CONCERNING GAUGE FREEDOM
The two illustrative examples have revealed the peculiarity induced by applying the Coulomb gauge to physical settings, which are naturally based on the Minkowski space-time.As shown in the Appendix, the Lorenz gauge leads to two independent scalar wave equations for both the electric scalar potential ϕ and the magnetic vector potential A in (38).That describes well the correct physical behavior of EM wave propagation.In contrast, the Coulomb gauge would lead with v p → ∞ to an instantaneous movement of ϕ based on (7), whereas A i vanishes (see (6) in this article or, e.g., [29, eq. ( 16)]).This means for the first example a premature arrival of ϕ by about 11.67 • 10 −3 s and for the second example that it arrives around 0.33 • 10 −9 s too early its destination.Both propagation processes are depicted in Fig. 11, where the Lorenz gauge yields the solid lines.These linearly progressing lines might be steeper depending on the chosen gauge, which is indicated by the gray shaded area, respectively.Thereby, the Coulomb gauge would yield vertical lines.If, for instance, evanescent waveguide modes with superluminal v p are to be modeled, the line in (b) would be steeper. 6As previously mentioned, the large difference between the times given in Fig. 11 is caused by the different spatial scales of the scenarios.For both examples, frequencies of operation are intentionally not explicitly stated.The number of wavelengths distributed over the propagation path can be derived -suitably for the reader's application (such as the microwave regime) -from the solution of the solenoidal wave components.As already discussed, the solenoidal wave part is usually sufficient in RF applications, since this part is directly related to active power transfer.It is also quite common to handle scenarios without a supporting structure or neglect it, which explains why there the outlined propagation behavior does not arise.With this in mind, it is concluded that it only occurs for specific types of EM wave applications which means those where the chosen gauge of the ordinary potentials is not in accordance to the underlying physical description and, moreover, irrotational potentials are analyzed.Therefore, gauge freedom is limited to a certain extent when these potentials are individually considered in this way.In commercial EM field modeling softwares like ANSYS Electronics Desktop TM [40], CST Studio Suite [41] or COMSOL Multiphysics [42], a behavior of irrotational potentials similar to that shown by analytical examples is expected since numerical models are also based on gauge-dependent PDEs such as (38) or (39).

IV. CONCLUSION
In this article, a side effect of gauging the magnetic vector potential's divergence has been extensively presented with reference to the metric and applied by means of two practical guided EM wave propagation examples.It is not commonly known to this extent, since often the part of the magnetic vector potential A which is correctly considered serves as the basic quantity for calculating the EM field.The study has revealed in detail affected objects, which provides the reader with a comprehensive overview and enables him to a well-founded handling of his own applications.That includes the possibility to adjust mathematical phase velocities v p by gauging to a desired physical behavior, for instance, in order to model quantities which propagate faster than with the speed of light in a sophisticated way.
A decomposition of A has made it possible to determine its affected part and to examine each behavior individually.It has been shown that if a specific space-time is suitable to describe an EM wave propagation scenario, a changed v p occurs for the electric scalar potential ϕ as well as for the irrotational magnetic vector potential A i if the chosen gauge is not appropriate.This concerns models, in which physical meanings are assigned to these potentials or those for which they are used separately as a foundation for calculating fields by aiming to design devices such as RF components.The outlined side effect is based exclusively on the mathematical construct of precisely ϕ and A. Often, its consequences are not noticeable, since it is usually not intuitive nor obviously useful to apply a gauge which does not fit into the space-time metric of the underlying physical setting.

APPENDIX
To facilitate derivations, this Appendix contains the potential gauging process deduced from Maxwell's equations by assuming vacuum as the medium of wave propagation.Special emphasis is placed on the Coulomb and the Lorenz gauge, which is enhanced by the more general velocity gauge.It is convenient to derive the non-gauged potential PDEs with Gauss' law of electricity as a starting point, which reads where ∇ is the Nabla operator, D the electric flux density and is the volume charge density.Gauss' law of magnetism is which describes the zero divergence of the magnetic flux density B. Insertion of the constitutive equation (where ε 0 is the permittivity of vacuum) in (20), consideration of the electric field E derived from the electric scalar potential ϕ and the magnetic vector potential A through (see, for example, [3] or [43]) yields after some rearrangement the non-gauged PDE for ϕ as To derive the non-gauged PDE for the magnetic vector potential A, it is usual to start with Ampère's law, which reads There, H is the magnetic field and J is the electric current density.Faraday's law is given here as The constitutive equation for B is where μ 0 is the permeability of vacuum.The constitutive equation for J reads with the conductivity σ Both H and D of (25) are replaced by ϕ and A using ( 22), ( 23) and (see, for instance, [5] or [6]) which yields for the assumption of homogeneous vacuum The solution of a scalar wave equation can be easily expressed using a scalar Green's function, whereas the solution of a vectorial wave equation involves a more sophisticated dyadic Green's function.This illustrates that it is a general aim in EM wave propagation modeling to use the magnetic vector potential A to calculate a scalar instead of a vectorial wave equation, which means "∇ × (∇ × A)" must be converted into "∇ 2 A".To do this, the ordinary vector identity (see, e.g., [2] or [4]) Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
is suitable, where the first term on the right-hand side is affected by gauging.After the application of ( 31) to (30) and some rearrangement follows finally Note that ( 24) and ( 32) describe the same physics as the Maxwell system (20), ( 21), ( 25) and ( 26) does.In both (24) and (32), the divergence of the magnetic vector potential appears.A specific choice of "∇ • A" -which can also be called a gauge transformation, since a specific set of potentials satisfies (24) and ( 32) -can simplify one or both equations.Ordinarily used selections are the Lorenz gauge (see, for instance, [3] or [11]), which is given here for vacuum by and the Coulomb gauge (see, e.g., [2] or [3]), which reads In (33), the speed of light c is defined as usual by c = 1 √ μ 0 ε 0 (35) and serves as the basic phase velocity for vacuum.If the Lorenz gauge is not chosen, the basic phase velocity of the irrotational potentials differs from c.Both ( 33) and ( 34) are special cases of the velocity gauge, which reads See, e.g., [11] for expressions in the International System of Units used here.The Gaussian System of Units is utilized, for instance, in [14] or [43].The phase velocity v p is usually defined in accordance to, for example, [4] or [5] as Note that in this work μ and ε in (37) are not physically meaningful in terms of a material distribution, rather they refer to the phase velocity of the irrotational potentials determined by the chosen gauge.The ordinary definition of v p is based on the material properties permeability and permittivity representing the physical behavior.No further explanation of (37) is provided here, since in this article, v p is used directly to express consequences of gauging.It is obvious from (36), that v p equals c in (33) and that v p goes towards infinity in (34).A modern discussion of several values for v p is presented in [44].Various other gauges -which might be further special cases of the velocity gauge -are possible, see, for instance, [44].One of them is the Kirchhoff gauge (see, for example, [43] or [45]), where an imaginary phase velocity follows for the electric scalar potential ϕ.A discussion of the velocity gauge from a historical perspective is presented, e.g., in [43].The free choice of "∇ •A" in the velocity gauge is illustrated by means of Fig. 12, where the cases Lorenz and Coulomb gauge are highlighted.They limit the interval considered in this article.
If the Lorenz gauge ( 33) is applied to ( 24) and ( 32), both equations become separate scalar wave equations and describe wave propagation for ϕ and A, respectively, as (cf.[9]) In (38), both potentials travel with v p = c in vacuum, which suits well for RF scenarios.The other limit case is achieved by applying the Coulomb gauge (34), which yields a Poisson equation for (24) with infinite phase velocity v p for ϕ and a mixed formulation for (32) containing ϕ and A as (cf.[9]) Obviously, this gauge yields only a simplified first equation in (39), which leads to an easier calculability of ϕ.There arise different consequences for A, which are discussed throughout this article with special emphasis on other field quantities than purely on the EM field and the potential itself.This context is not widely discussed in the standard literature.In contrast, the different propagation speeds of the electrodynamic potentials ϕ and A induced by gauging them are outlined in several publications, such as [21], [22], 7  or [43].

FIGURE 1 .
FIGURE 1. Principle meaning of (a) irrotational and (b) solenoidal field lines.In (a), an example of an irrotational electric field pattern Ei generated by monopole charges is depicted, which exhibits open paths.Contrary to that, the solenoidal electric field pattern Es caused by dipole charges in (b) consists only of closed paths.

FIGURE 2 .
FIGURE 2. Spatial orientation of the two kinds of A: As (indicated in violet) in the transversal plane and Ai (drawn in red) in the longitudinal direction valid for point-shaped source and sink.Note that for waveguide modes, both As and Ai may have components in transversal and longitudinal directions.

FIGURE 3 .
FIGURE 3. Retarded time τ as the delay between sources and associated irrotational potentials in vacuum.The interval of τ is limited in this article by properties caused by the Coulomb and the Lorenz gauge.

FIGURE 4 .FIGURE 5 .
FIGURE 4. Wave propagation behavior with suitable finite velocity in Minkowski space-time for an exemplary point in time.

FIGURE 6 .FIGURE 7 .
FIGURE 6. Cross-sectional view of a lossless coaxial wire aiming to explain (19) by assuming for simplicity a TEM-mode.The surface normal vector n of the inner conductor points parallel to As and Es in the radial direction, that of the outer conductor is directed to the center axis.

FIGURE 8 .
FIGURE 8. Ground wave propagation along the earth's surface.Only the traveling current density Je is illustrated (drawn in red, because uniquely identifiable as irrotational), since it builds up a physical distribution.Furthermore, the non-moving Js and Ji are assigned to the transmitting antenna, which are the sources of the EM wave.

FIGURE 9 .
FIGURE 9. Wave propagation in positive z-direction inside a rectangular waveguide by assuming a TE10-mode.As done in Fig. 8, the traveling pattern of Je is indicated alongside the propagation path.

FIGURE 10 .
FIGURE 10.Cross-sectional view (xy-plane) of a TE10-mode's field pattern inside the metallic walls of the rectangular waveguide presented in Fig. 9.If the dashed Ds is considered, closed paths are present together with the solid drawn Je located at the inner surface of the waveguide, if not, open paths follow.Indicated by a frame, the latter interpretation is utilized throughout this work.

FIGURE 11 .
FIGURE 11.Linear propagation behavior of irrotational potentials in the Lorenz gauge considering (a) the large-scale case depicted in Fig. 8 and (b) the small-scale case, see Fig. 9.The gray shaded area indicates possibilities for curves representing other gauges, respectively.

FIGURE 12 .
FIGURE 12. Relation of ordinary gauges to the velocity gauge, which provides the option to specify "∇ • A" by an arbitrary choice of the phase velocity vp.