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Energy conservation of coherently oscillating charged particles in classical electrodynamics Pardis Niknejadi, John M. J. Madey, and Jeremy M. D. Kowalczyk University of Hawaii at M a noa, Department of Physics and Astronomy, Honolulu,

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R E V I E W C O P Y N O T F O R D I S T R I B U T I O N

Energy conservation of coherently oscillating charged particles in classical electrodynamics

Pardis Niknejadi,∗ John M. J. Madey, and Jeremy M. D. Kowalczyk

University of Hawai‘i at M¯ anoa, Department of Physics and Astronomy, Honolulu, HI 96822 (Dated: August 25, 2014) As the technology of particle accelerators and light sources has advanced, the significance of the process of coherent radiation, in which a number of charges within a fraction of a wavelength are induced by their motion to emit electromagnetic radiation, has increased. Examples of such light sources are THz coherent synchrotron radiation sources as well as self amplified spontaneous emission (SASE) free electron lasers (FELs). Established theory for these sources is classical, but

ur understanding of the process of coherent radiation in the classical limit is far from complete.

For instance, classical field theory has, for more than 100 years, failed to provide a non-diverging solution and an origin for the radiation reaction field: the electric field responsible for energy conservation in the process of radiation into free-space. In this paper we will look at an elementary model of two coherently oscillating charged particles that provides a non-diverging solution and

rigin of the radiation reaction field. We show that an alternative approach to classical field theory,

analysis provided by Wheeler and Feynman in their 1945 paper “Interaction with the Absorber as the Mechanism of Radiation”, yields, for the first time, an exact match between the radiated power and the rate of change of the particle’s kinetic energy. These developments seem likely to clarify our understanding of the role of advanced forces in electrodynamics and may contribute to the advancement of synchrotron radiation and FEL light sources that rely on the coherent radiation mechanism. PACS numbers: 04.40.Nr, 52.59.Ye, 52.40.Fd I. INTRODUCTION

Unlike the case of coherent radiation into conducting

r reflecting cavities, which has been studied by Slater,

Lamb and Siegman [1] in their reduction of the problem to the analysis of the evolution of the normal modes of these cavities, a number of issues for radiation into free- space (here free-space refers to space containing only an absorber of arbitrary density throughout) including the nature of energy conservation in the case of coherent ra- diation remain unresolved. However, a previously un- remarked aspect of Wheeler and Feynman’s analysis in their 1945 paper [2] (page 164, Equations 15-17, see Ap- pendix A) appears to hold the key to the solution of this

problem. Here we present an application of Wheeler and

Feynman’s analysis. Our results add weight to the via- bility of their, long discussed and controversial in some aspects, model as a more realistic description of the in- teractions of radiating and absorbing systems of charges and currents. Since the introduction of the covariant Lienard- Weichert [3, 4] general solution to the inhomogeneous wave equation, the nature of energy conservation in the case of radiation into free-space has remained unresolved. In contrast, the results demonstrated in Section III.C not only present a perfect solution to the problem of conservation of energy, they are elegantly simple. This

∗ pardis@hawaii.edu

must surely raise the question of why neither Wheeler nor Feynman drew attention to these implications of the results in their paper. Contemporary accounts indicate that even though both authors were inspired by Dirac, who had suggested a non-diverging finite solution for ra- diation emitted by individual charges [5], their personal research interest was not focused on the topic. Wheeler seem to have been interested in the space-time implica- tion of their analysis [6] and Feynman was in search of a non-divergent quantizable field model [7] (a description of the means by which the action-at-a-distance model exam- ined by Wheeler and Feynman can be quantized was not published until 1995 [8]). Furthermore, not only does the model converge to the established single point radiation reaction force, as we will show it provides for conserva- tion of energy for coherently radiation pairs of particles, a problem which was not at the time raised or understood. Serious consideration of the problems of energy conser- vation for the case of radiation into free space had to await the development of free electron laser light sources decades later. The underlying theory for these sources was quantum based [9] which automatically assured com- pliance with energy conservation, but some years later, when the classical electrodynamics based theory emerged [10–12], the idea of SASE FELs was introduced [13–16] and its theory was developed [17–22]. Initial experimen- tal results for SASE FELs were obtained starting in 1985 [23] and operational results in the EUV and x-ray regimes were obtained starting in the late 90s [24]. The energy conservation problem emerged with the introduction of these classical models. Isidore Kimel and Luis Elias have

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2 previously considered the issue of coherent radiation re- action for FELs in the context of the fully covariant the-

ry [25, 26] and proposed a possible analytical solution to

resolve the problem. In this paper we extend that anal- ysis to a proper laboratory frame in which the average velocities of the oscillating charges are zero. The result- ing picture is advantageous with respect to the analysis

f the requirements for energy conservation in the case
f non-relativistic coherent radiation and also the explo-

ration of the possibilities for an experimental test of the first-principles theories which may have relevance to the resolution of the energy conservation problem.

II. HISTORY

By the early decades of twentieth century, attempts to describe radiation had favored classical field theory over the idea of action-at-a-distance. In the early 1900s the derivations by Lienard and Wiechert offered the first de- tailed covariant picture of radiation by accelerating point

particles. The boundary conditions applicable to this so-

lution have traditionally been taken to follow from Som- merfelds analysis of the uniqueness of solutions to the wave equation. In 1912, Sommerfeld established the “ra- diation condition”, a boundary condition outside the vol- ume occupied by radiating charges, that he applied to analyze the implications of the advanced solutions to the wave equation. Sommerfeld’s approach requires that ad- vanced solutions outside this far off boundary must be excluded to achieve uniqueness [27], and that result has continued to be relied upon through present date. Although the Sommerfeld radiation condition holds a very strong position in the classical theory of radiation, there have been cases where it has been enforced unnec-

essarily. For instance, the advanced solutions for cases of

radiation in conducting cavities were ignored until Ein- stein argued that the two solutions to Maxwell equations were critical to conservation of energy and momentum for a radiating particle in a “box” (conducting cavity) [28] which served as the intellectual basis for the sub- sequent treatment of radiation into microwave and laser cavities by Slater, Lamb and Siegman [29–31]. It is of great interest to us that Einstein’s original remarks and the cavities analysis were not intended to deal with the case of radiation into free-space [32], therefore most clas- sical analysis of radiating sources completely ignored the effect of an absorbing media in the system until the anal- ysis by Wheeler and Feynman. Similarly, it has also been brought to our attention that the Wheeler and Feynman analysis has been criticized for not agreeing with Sommerfeld’s radiation condition [33]. However it must be noted that a main requirement

f Wheeler and Feynman’s analysis is that the radiation

fields of all the accelerated charges in the universe are to be specified as one half of the advanced plus one half of the retarded solutions to the inhomogeneous wave equa- tion. Since the relative contributions of the advanced and retarded solutions are specified in this model, there can be no ambiguity regarding the fields that are actually present in the system, and as we show here, superposition

f these fields yields the retarded field of experience.

Sommerfeld’s condition was intended to restrict solu- tions to the wave equation to yield only the retarded fields of experience but Wheeler and Feynman’s condi- tion of half advanced plus half retarded fields, also lead- ing to the same retarded field of experience, provides an alternative model with the added benefit of energy con-

servation. We must therefore conclude that Sommerfeld’s

“radiation condition” does not constitute an appropriate basis for the assessment of the roles of advanced and re- tarded fields in electrodynamics. During the development of the classical theory of radi- ation, many efforts were dedicated to describe a force needed for conservation of energy of an accelerating charge in free-space. Abraham and Lorenz [34, 35] mod- ified the equation of motion for an accelerating particle and assumed the particle to be a spherical cloud (i.e. not a point particle) to account for conservation of energy via a radiation reaction force. However, their formula resulted in an equation of motion that is unacceptable for two reasons. First, the acceleration increased expo- nentially even if there was no external force. Second, if boundary conditions were provided to prevent this expo- nential increase, the particle accelerated before any ex- ternal force was applied [36]. Later, an alternative for- mula by Landau and Lifshitz was presented that does not include either of these two defects [37] and is a good ap- proximation for the radiation reaction force for a single particle but still fails to describe an acceptable physical source responsible for conservation of energy. In other words their solution was only phenomenological and it has not been applied to the case of coherent radiation. Nonetheless, the Lorenz and Abraham model can be considered the first attempt to analyze the coherent ra- diation mechanism given their model of the radiation emitted by the coherently accelerated charge distributed around the spherical shell of their model of an elec- tron. Based on these efforts an important conclusion was reached: given that the electron is a point particle to the limit of experimental resolution, retarded classical electrodynamics is fundamentally incapable of generat- ing the ‘first principles’ internal forces that can account for the conservation of energy [38]. Parallel to developments in classical field theory dur- ing the second decade of 20th century, Schwarzschild and Fokker showed that Gauss’s concept of electromagnetic action-at-a-distance is mathematically self consistent and in complete harmony with the Maxwell’s equations. This led to the Wheeler and Feynman analysis of the action- at-a-distance model of Fokker [39–41], Tetrode [42] and Schwarzschild [43]. Although initially further work on the Wheeler-Feynman 1945 analysis was abandoned by Feynman in favor of quantizable electrodynamics (which later came to be known as quantum electrodynamics (QED)), it was later shown by Hoyle and Narlikar that

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3 the 1945 analysis is quantizable and for perfect absorp- tion, “in the action-at-a-distance picture, the role of the vacuum is taken over by the response of the Universe” [8]. The Wheeler-Feynman analysis can thus also serve as an alternative divergence-free version of QED. Although Wheeler and Feynman did not put any emphasis in the derivation presented in their paper on the coherent radia- tion emitted by multiple accelerated charges, their model can be considered an extension of classical electrodynam- ics which can be used to solve for the response of systems with large numbers of radiating particles. Since QED cannot be applied to such systems at high radiated pow- ers, the Wheeler-Feynman analysis can thus serve as an alternative divergence-free approach to development of solutions for these systems in the classical limit.

III. REVISITING THE PROBLEM OF CONSERVATION OF ENERGY FOR RADIATING CHARGES

When in equilibrium, the electromagnetic energy stored within a spherical shell surrounding a periodically

scillating charge is constant, requiring that surface inte-

gral of the Poynting vector must be equal to the volume integral of E·j within the sphere. The surface integral of the Poynting vector can thus be used to infer the ampli- tude of the electric field acting on the electrons in phase with their velocities. The equilibrium between the radi- ated power as calculated using Poynting’s vector and the work done on the radiating charges by the electric field they generate is a requirement that follows directly from Maxwell’s equations, and requires the rigorous detailed balance of these two quantities. Consider the geometry of the case presented in Figure (1), where two charged particles are oscillating in phase and are separated by distance r in the direction parallel to the direction of their oscillation. Power radiated by this system, P =

E × H · da [44], integrated over the

surface of a large diameter sphere centered on midpoint between the two charges is: P ∝ 2 π sin3(θ) cos2[kr cos(θ) 2 ]dθ, (1) where k = ω/c and k = 2π/λ. When the separation r = 0, the total power radiated by the two electrons is proportional to 2 π

0 sin3(θ)dθ = 8 3 which is used to nor-

malize Equation (1) in the subsequent plots. The normal- ized power radiated by these two coherently oscillating charged particles is shown in Figure (2). Since, as shown in Figure (2), the power radiated by the particles varies with their separation, there must be a force acting on the two electrons that oscillates in phase with their velocities and has an amplitude that varies with spacing to match the variation with spacing of their radiated power [45]. We can then ask the question of whether any of the existing theories for the radiation

FIG. 1. Two coherently oscillating charged particles with dis-

tance r between the center of their oscillation and amplitude

f oscillation of x(t) where r is parallel to the direction of
scillation. (x1(t) = x2(t) = x(t))
FIG. 2. Plot of normalized total radiated power from two co-

herently oscillating charged particles showing dependence on the separation between the center of oscillation of two charges. When r << λ the radiated power has twice the magnitude compare to when r >> λ

reaction force provide an estimate of that force, which agree with the electrons’ radiated power and has the same variation of power with the vector separation of the two charges.

A. Role of the induction field

Since there is no component of the particles’ radiation fields along the axis on which they are positioned, the particles’ radiation fields cannot contribute to the radia- tion reaction force needed to conserve energy for the case

f radiation into free-space. The retarded electric induc-

tion field, however, does include an oscillating component when evaluated at the extrapolated position given by: xret = x|t − x|t− r

c + r

c dx dt |t− r

c ,

(2) yielding the dependence on particle spacing r [46]. In Equation (2), xret is the apparent position of the sec-

nd particle from the perspective of the first particle

at present time. The retarded induction field of a non- relativistic particle due to the other is Eret = e 4πǫ0r3 xret (3) and the power, Pind = < e E2ret · u1(t) >, (4) is plotted in Figure (3).

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4 However, the force due to the induction field also fails to comprise a component of the radiation reaction force, not only because it is much weaker, as evident by compar- ing Figure (2) and (3), but also because it diverges as the particles’ separation approaches zero. The only way to cancel the divergent term is to add a component to the field equal to the advanced component of the Lienard- Wiechert induction fields which has a divergence of the

pposite sign. Nonetheless, if that advanced component

is also added to the two particles’ radiation fields, the two particle system no longer emits any induced radia-

tion. (See Appendix B)

It is important to note that the Sommerfeld condition, when applied to isolated pairs of oscillating charges, does not allow the waves converging on the particles from a distance that are required to satisfy energy conservation as we are about to show. So any search for a source of the radiation reaction fields responsible for energy con- servation in such a simple two-particle system under the Sommerfeld condition must inevitably fail to find a solu- tion on fundamental grounds.

FIG. 3. Amplitude of the component of the retarded electric

induction field generated by the lower charge in Figure 1 and evaluated at the position of the upper charge. The Figure shows the amplitude of that component of the induction field

scillating in phase with the velocity of the upper charge and

the field diverges as 1/r2 at small separations. This plot is set to have the right ratio compared to the power normalized to 1.0 by matching the amplitude and frequency of oscillation to those in Figure 2. B. Dirac’s Solution

In his 1938 “Classical Theory of Radiating Electrons”, in favor of conservation of energy, Dirac offered an ex- pression for the radiation reaction field (rrf) of a single particle: FDirac−rrf ∝ [exp(iu) − exp(−iu)] u . (5) where u = kr for our physical interpretation. While this seems to have been a big step in the right direction for ex- plaining the phenomena of radiation in free-space, Dirac did not provide a physical theory as a basis for his radi- ation reaction field. He noted that the real part of the radiated Lienard- Weichert field at the position of the particle is 90 degree

ut of phase with the particle’s velocity. The imaginary

part, on the other hand, is in-phase with the particle’s velocity and is therefore capable of reducing the particle’s kinetic energy during the process of radiation. By eval- uating half the difference of the advanced and retarded solutions for the field equations he was able to demon- strate that the electric field at r = 0 had precisely the value required to insure conservation of energy. Although Dirac’s radiation reaction field at finite values of r in the near field is also defined by (5), these values are only close to what is required for energy balance for coherent emission in vicinity of the particle. Dirac’s result stem from the need to find a non- divergent expression for the radiation reaction field that leads to energy conservation and oscillates in phase with the radiating particle’s velocity. His motivation for ex- pressing (5) as half the difference of the advanced and retarded components of the field seems purely mathe- matical. Dirac’s field fails in two ways to account for the power actually radiated by a pair of coherently oscillat- ing charged particles. Since such charges emit no electric fields along the direction of their acceleration, the Dirac model is not capable of preserving energy conservation for vector displacements parallel to the direction of the charges’ oscillation. So in the case of two particle os- cillating coherently with r being parallel to direction of

scillation Dirac formula does not provide a suitable so-

lution as demonstrated in Figure (4) (See Appendix C). Also as stated, the Dirac field for two coherently oscillat- ing charges, PDirac−rrf,2e ∝ 2FDirac−rrf(0) + 2FDirac−rrf(ωr c ) ∝ 1 + sin(kr)/kr 2 , (6) does not match the variation of coherently radiated power by particles with vector separations r normal to the direction of acceleration Figure (5).

FIG. 4. Comparison of the power extracted by the Dirac co-

herent radiation force with the power radiated by the two os- cillating charged particles’ for displacements parallel to their vector accelerations. The non-local component of the Dirac coherent radiation reaction force falls to zero for any finite displacement of the two charges along this direction, leaving

nly each particle’s single point radiation reaction force to
ppose their oscillating velocities.
FIG. 5.

Comparison of the power extracted by the Dirac coherent radiation force with the power radiated by the two

scillating charged particles’ for displacements perpendicular

to their vector accelerations.

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5

C. Wheeler and Feynman Approach

Fortunately, Dirac’s radiation reaction field inspired Wheeler and Feynman to more directly explore the phys- ical basis of the radiation reaction field for point parti-

cles. In their 1945 paper, Wheeler and Feynman were

able to demonstrate that, when formulated in the lan- guage of covariant action-at-a-distance, the solution of the boundary value problem corresponding to an oscil- lating particle within a spherical absorbing shell (repre- senting free-space) of arbitrary composition was domi- nated by the interference of the retarded and advanced forces originating in the accelerated and absorbing parti-

cles. By including both advanced and retarded solution

for the source and absorber they were able to define a radiation reaction force exactly equal to that needed to match the power carried by radiation to the particles in the absorbing shell (see Appendix A for details). Wheeler and Feynman introduce Equation (A1a) as the force field for the radiation reaction field (rrf) for a single particle which is also finite when r = 0. They explicitly note that the adjunct field they had derived converged to Dirac’s radiation reaction field at r = 0 and also at large distances from the accelerated parti- cle. However they do not mention that their adjunct radiation reaction field diverged from Dirac’s radiation reaction field at distances between a few tenths of a wave- length and 4-5 wavelengths. Nevertheless, since Equation (A1a) is accounting for a conservative force, total power

f FWF −rrf for the system of two coherently oscillating

charged particles is PWF −rrf,2e ∝ 2FWF −rrf(0) + 2FWF −rrf( ωr

c )

(7) with ω/c = 2π/λ, and is shown in Figures (6) and (7).

FIG. 6.

Comparison of the amplitude of the Wheeler- Feynman coherent radiation force with the dependence of the two oscillating charged particles radiated power for displace- ments parallel to their vector accelerations.

FIG. 7.

Comparison of the amplitude of the Wheeler- Feynman coherent radiation force with the dependence of the two oscillating charged particles’ radiated power for displace- ments at right angles to their vector accelerations.

Figure (6) and Figure (7) suggest that the derivation is highly significant with respect to the process of co- herent radiation in free-space, for in contrast to Dirac’s field, the Wheeler-Feynman adjunct radiation reaction field accounts exactly for the force needed to preserve energy conservation when compared with the strongly enhanced radiated power emitted by two coherently oscil- lating charged particles at varying spacings and angular displacements in their mutual near fields. It’s important to note that their approach is not of- fensive to causality. To satisfy conservation of energy the the advanced field of the absorber must be added to half retarded and half advanced field of the source. The advanced field of absorber is equal to the difference be- tween half retarded and half advanced field of the source. Therefore including the advanced field of the absorber produces the fully retarded field known from experience and all the components of the radiation field (advanced and retarded) fall to zero outside of the “absorbing shell” in their model.

IV. IMPLICATIONS

The results demonstrated in Section III.C suggest that the limitations of established radiation theory with re- spect to the case of coherent radiation in free-space are due to omitting the radiating particles’ interactions via the advanced terms in the general solution of the inho- mogeneous wave equation with the distant charges and currents that absorb that radiation. The general solution of an inhomogeneous problem is made up of any particular solution of the problem plus the general solution of the corresponding homogeneous problem for which both the equation and boundary con- ditions are homogeneous [47]. In field theory, the solu- tion to the radiation problem requires the inclusion of the boundary conditions that fix the values of the so- lution at the boundaries of the region surrounding the

source. In particular, such solutions typically require the

addition of a combination of the solutions to the homo- geneous equation (with no sources) to the particular so- lution (satisfying the wave equation with the inclusion

f all source terms) to satisfy the conditions applicable

at the boundaries of the region surrounding the source. Those conditions will differ depending on the nature of the boundary. This is similar to the case of conducting cavity where by defining the spectrum of solutions to the homogeneous wave equation which need to be added to the solution of the inhomogeneous wave equation to sat- isfy the boundary conditions at the conducting walls of the cavity the nature of the radiation that can be emit- ted by an oscillating charge distribution is profoundly altered. The fields which have traditionally been ascribed to the radiation emitted by a single oscillating charge constitute a “special solution” to the inhomogeneous wave equation for that problem, but do not include any of the solu- tions to the homogeneous wave equation needed to fulfill the boundary conditions applicable to the problem, pre- sumably those appropriate to the distant absorbing shell assumed in the Wheeler Feynman model for free-space. It is thus not a surprise that the fields traditionally at- tributed to a single oscillating charge without consider-

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6 ation of the relevant boundary conditions, either those

f the conducting microwave or reflecting optical cav-

ities long familiar from cavity electrodynamics, or the absorbing walls the anechoic chambers used to simulate radiation into free-space should fail to satisfy the test

f energy conservation in the case of coherent radiation

into free-space. Except for the inclusion of the specific homogeneous terms needed to satisfy the boundary con- ditions applicable to radiation into free-space, there ob- viously can be no confidence that any special solution to the wave equation for this case can accurately define the fields acting on other oscillating charges in the vicinity

f the assumed source charge [48].

The results presented in this paper, if verified by an experiment, will require an extension of our understand- ing of the nature of the radiative interactions that occur in the limit of classical electrodynamics. In particular, since the influence of the absorber at the time the source has just started to radiate is what is responsible for con- servation of energy, we need to imagine that at the very moment the source begins to oscillate, it is also subject to a connection between itself and its surrounding medium through which energy will be transmitted or shared, as unlikely as this new model for radiative interactions may

seem. Fortunately we are in an era of technology that

allows us to test these concepts.

A. Measurement of the advanced radiation reaction field

The radiation emitted by macroscopic antennas bears many similarities to the radiation emitted by elementary

scillating charges [49]. In particular, it has long been

known that the driving point impedance of an antenna is affected by presence of other antenna systems (reso- nant or non-resonant) surrounding them. In particular, it has recently been shown in general that if both ad- vanced and retarded Green’s functions are included for the case of a single antenna, the single antenna in free- space will not radiate [50] [51]. In Section III we dis- cussed radiation from two oscillating charged particles and showed the variation of the radiation reaction fields with distance between the oscillating charged particles. Now if we consider a small dipole antenna; it will generate a field attributable to the superposition of many oscillat- ing elementary charges set in motion along the surfaces

f its conducting elements by its signal source [52]. We

assume that the current along the elements of the dipole decreases linearly from a maximum feed point to zero at the ends of the dipole as is conventional in analysis of short antennas [53]. For the purpose of the proposed ex- periment, it is important to note that if the elementary

scillating charged particles comprising the superposition

are not closely spaced or not limited to a small volume in space, the interference of their individual coherent ra- diation reaction forces will generally result in a net force field that bore little resemblance to the force fields of the antenna’s individual oscillating charged particles. To minimize this distortion, we have determined that the functional form of the Wheeler-Feynman radiation reac- tion field for an antenna (modeled by the superposition

f single oscillating particles) converges to the form of

the single-particle radiation reaction field as the dimen- sions of the antenna are reduced to λ/10 or less. The current distribution of the antenna can be constructed by adjusting the amplitudes and positions of the single

scillating particles to match the known current distribu-

tion for such a short dipole antenna as shown in Figure

8. The predicted advanced radiation reaction field for

the antenna from the radiated power and that of a single

scillating particle as a function of distance r perpendic-

ular to the direction of the current is shown in Figure (9); they are nearly identical. By measurement of the power attributed to these fields present in this antenna’s vicinity it is possible to test the existence and functional form of the non-local coherent radiation reaction field of the Wheeler-Feynman model. Still, this is not a simple experiment. Although we live an era of technology where minimally perturbing multi- axis electric field probes are now commercially available for use in the characterization of new antenna systems, the additional means needed to measure only that com- ponent of the field that oscillates in phase with the ve- locity of the oscillating charged particles in the nearby radiation source appears to represent a new requirement for these field probes. We nonetheless believe that the design of such a phase sensitive field probe is within the current state of the art, though requiring a signifi- cant commitment with respect to engineering and com- missioning. Also it will be required to implement the boundary conditions surrounding the transmitting an- tenna and field probe which should match as closely as possible the absorbing boundary conditions assumed in the Wheeler-Feynman analysis. This challenge can be

vercome through the use of an anechoic chamber with

absorbing walls of the kind used in the field of antenna development.

FIG. 8. Drawing showing the distribution of elementary os-

cillating charged particles used to derive the form of the Wheeler-Feynman coherent radiation reaction field for the short dipole antenna with constant current I distributed along the length l of the antenna which is located at distance r from the field probe or test charge. Here r is the distance between the probe and the antenna and α = 0. B. Conservation of Energy in FELs

If verified by experiment, the implications of the Wheeler-Feynman model for our understanding of the na- ture of the universe in which we live would be profound.

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7

FIG. 9.

Comparison of the amplitude of the Wheeler- Feynman Coherent radiation reaction force for a single os- cillating particle with the superposed and scaled Wheeler- Feynman fields of an array of individual oscillating particles distributed along the length of the elements of a short dipole antenna shown in Figure (8) according to the known distri- bution of current in these elements.

For instance, although Kimel and Elias have previously described the derivation of a covariant force field capable

f preserving energy conservation when added to the lo-

cally sourced electric and magnetic fields in free electron lasers emitting coherent radiation into free-space [25, 26], the Wheeler-Feynman model is the only known analysis to date to explain that force field on the basis of clearly defined and physically valid first principles. There is also implicit in the Wheeler-Feynman model the possibility

f enhancing the capabilities of the new SASE FELs by

altering the structure of the target to alter the nature

f the forces acting on the radiating electrons and op-

timize the spectrum of the emitted coherent radiation, relying on the cavity’s normal modes or the dispersion of the cavity’s mirrors to improve the lasers temporal co- herence instead of enclosing the system in a conducting

r reflecting cavity.

V. CONCLUSION

We have shown that the Wheeler-Feynman derivation for a moving charge interacting with the absorber pro- vides a perfect match to the radiated energy for the case

f two coherently oscillating charged particles.

There- fore, even though some aspects of the model are con- troversial and it has been considered a conceptually de- manding theory [54], in addition to being consistent with the existing theory with respect to conservation of en- ergy for a single charge uniquely amongst the possible physical models provides conservation of energy for co- herently radiating pairs of particles. This suggests that the action-at-a-distance concept presented by Tetrode, Wheeler, Feynman and others for radiation is capable of providing a valid physical first principle picture for the problem of radiation in classical electrodynamics. We have also shown that it is possible to experimentally test the ideas presented in this paper with existing technology. Since the classical electrodynamic treatment of radiation is the foundation for theoretical advancements in the cur- rent and future generation light sources where systems as large as 1010 radiating electrons are considered, we hope that by experimentally testing the case presented here we will take a step toward comprehensive analysis of these sources.

ACKNOWLEDGMENTS

The authors would like to thank Professors Isidore Kimel and Luis Elias for their pioneering theoretical ex- plorations of the coherent radiation reaction force, Ming Xie for his contribution to theory of high gain FELs, Ian Howe for critical discussions regarding the power radiated by a pair of oscillating charges, Julius Madey for his advise regarding the engineering issues for the proposed experiment and Eric Reckwerdt for elucidating remarks concerning the Sommerfeld radiation condition.

Appendix A: Summary of the Wheeler and Feynman approach and the resultant formulae

The Wheeler and Feynman 1945 paper [2] illustrates how advanced potentials from a far off absorber can be employed to build an action-at-a-distant model of radi- ation that intrinsically includes the radiation reaction force in a covariant and causal formulation consistent with observation. Wheeler and Feynman present four increasing complex derivations using an absorber of ar- bitrary density. First, they derive an expression for the radiation reaction force on a non-relativistic accelerated source charge. Second, they derive the fields responsible for the radiation reaction force on that source charge for all space and show how the advanced fields cancel ev- erywhere except at the source charge leading to only re- tarded radiation, consistent with experience. Third, they consider the source charge to be moving with arbitrary velocity, and forth they take a general approach. The re- sult established in the second derivation, Equation A1a, is the backbone and reason for this paper. Here we sum- marize how Wheeler and Feynman arrive at this formula. To calculate the effect of absorber in vicinity of an accelerating source charge first the retarded field of the source charge traveling outbound is used to calculate the motions of absorber particles then the sum of advanced fields from the absorber near the source is calculated. It is shown that addition of this field to the half advanced plus half retarded field of the source gives the expected fully retarded field of the source while producing the cor- rect radiation reaction force. In other words, the paper defines the origin of casualty for the case of radiation from the accelerating particle. Using their formulation, the force on a particle of charge e at distance d from the source charge is:

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8 FWF −rrf(ωd c ) = 2e2 3c3 (−iωa) exp(−iωt) [F0(ωd c ) − P2(cos(a, d))F2(ωd c )] (A1a) F0(u) = 1 2 1

−1

exp(iu cosθ)d cos θ (A1b) F2(u) = 1 2 1

−1

exp(iu cosθ)P2(cos θ)d cos θ. (A1c) For a large d both the F0 and F2 terms reduce to [exp(iu) − exp(−iu)] 2iu (A2) and FWF −rrf becomes FDirac−rrf which suggest that the advanced field of the absorber in the vicinity of a source charge is equal to the half advanced minus half retarded field of the source itself.

Appendix B: Advanced component of the radiation reaction force from the induction field

To calculate and plot Equation 4, we considered the general oscillating motion where uo = ωxo and ao = ω2xo: x = xo cos(ωt) (B1a) u = −uo sin(ωt) (B1b) a = ao cos(ωt). (B1c) Plugging these into Equation 2 xret = xo[cos(ωt)−cos(ω(t−r c))−ωr c sin(ω(t−r c))] (B2) and the power dissipated by charge 1 due to the induction field of charge 2 (Pind,ret) becomes: < e E2ret.u1(t) > =< e2ωx2

sin(ωt)

4πǫ0r3

cos(ωt) − cos(ω(t − r

c)) + ωr c sin(ω(t − r c))

= e2ωx2

8πǫ0r3
sin(ωr

c ) − ωr c cos(ωr c )

(B3)

and Equation B3 shows the power where we have used < sin(ωt) cos(ωt) >= 0 and expanded the trigonometric terms to simplify the result. Now consider the advanced term: Eadv = e 4πǫ0r3 xadv, (B4) where xadv = x|−t − x|−t− r