A Theory for Electromagnetic Radiation and Coupling

—A theory for analyzing the radiative and reactive electromagnetic energies of a radiator in vacuum is presented. In vacuum, the radiative electromagnetic energies will depart from their sources and travel to infinity, generating a power flux in the space. However, the reactive electromagnetic energies are bounded to their sources. They appear and disappear almost in the same time with their sources, and their fluctuation also causes a power flux in the space. In the proposed theory, the reactive electromagnetic energies of a radiator are defined by postulating that they have properties similar to the self-energies in the charged


I. INTRODUCTION
The electromagnetic radiation problems have been intensively investigated for more than a hundred years. It is a little bit strange that there is still no widely accepted formulation for evaluating the stored reactive energies and Q factors of radiators [1]- [14]. The main difficulty may come from the fact that there is no clear definition in macroscopic electromagnetic theory for the reactive electromagnetic energy. It is commonly known in classical charged particle theory that the fields associated with charged particles can be divided into self-fields and radiative fields [15] [16]. The self-fields include the Coulomb fields and the velocity fields, carrying self-energies, part of it also referred to as Schott energy in some literatures [17]- [19]. The radiative fields are generated by the acceleration of charged particles, emitting radiative energies to the surrounding space. The self-fields/self-energies are considered to be attached to the charged particles, or simply speaking, they appear with the charged particles and disappear with the charged particles. On the contrary, after being radiated by the charged particles, the radiative fields/energies will depart from the sources and propagate to the remote infinity. They exist after their generation sources disappeared and can couple with other sources they encountered in their journey. Although it is natural to consider that the reactive energies in macroscopic electromagnetics is similar to the self-energies or the Schott energy, no successful attempt has been found or well accepted to handle the reactive energies in this manner. No expressions for reactive energies are established in macroscopic electromagnetics that can be derived rigorously from the self-fields of charged particles.
On the other hand, Poynting vector is widely considered as the electromagnetic power flux density [20]. Poynting Theorem describes the relationship between the Poynting vector, the varying rate of the total electromagnetic energy densities, and the work rate done by the exciting source. It provides an intuitive description of the propagation of the electromagnetic energy. However, interpreting the Poynting vector as the electromagnetic power flux density has always been controversial [21]- [31], and some researchers have pointed out that Poynting Theorem may have not been used in the correct way in some situations [32] [33]. This difficulty is largely due to the fact that it is not easy to separate from the Poynting vector the true radiative power flux. It is known that the Poynting Theorem is not convenient to use for evaluating the reactive energies stored by radiators in an open space [5] [13], which has been investigated for decades. For harmonic fields, the total electromagnetic energy obtained by integrating the conventional energy densities of   0.5  D E and   0.5  B H over the infinite three-dimensional volume is infinite because they account for the total energy consisting of the radiative energy and the reactive energy. For harmonic fields over the time interval ( t     ), the radiative energy occupies the whole space and is infinitely large [14]. Some researchers suggested that those fields associated with the propagating waves should not contribute to the stored reactive energies. The reactive energies can be made finite by subtracting from the total energy density an additional term associated with the radiative power. However, it is not easy to give a general expression for that term because the propagation patterns are quite different for different radiators [1] [5].
Based on these observations, the macroscopic electromagnetic radiation issue is revisited and a new energy/power balance equation at a certain instant time is proposed, which may give an intuitive and reasonable demonstration that the Poynting vector does not only contain the radiative power flux density but also a pseudo power flux caused by the fluctuation of the reactive energies.
It is not the aim of this paper to argue that the reactive energies in the macroscopic electromagnetics are exactly the selfenergy or the Schott energy in the classical charged particles. Instead, a definition for the reactive electromagnetic energies is proposed based on the hypothesis that the reactive energies in the macroscopic electromagnetics bear the same characteristics as the self-energies or Schott energy: (1) they are attached to the sources. They will disappear after their sources disappeared; (2) the definition is in consistent with the stored energies associated with static charges and steady state currents; (3) the reactive energies do not propagate like the radiative energies, but their fluctuation may propagate at the light velocity in vacuum just like the radiative fields. A theory is proposed based on these considerations, in which the radiative energies and the reactive energies can be separated. A special energy term is included in the reactive energy, which performs like the Schott energy in charged particle theory [18]. As a consequence, the Poynting vector is divided into two vectors. One vector mainly accounts for the power flux density associated with the radiative energies and the other vector accounts for the fluctuation of the reactive energies. The theory also provides a simple way to define the mutual electromagnetic couplings between two radiators. One radiator may exert electromagnetic couplings to other sources through its potentials instead of fields.

II. FORMULATIONS FOR REACTIVE AND RADIATIVE ENERGIES
For the sake of convenience, we define the energy term associated with a charge density and a current density in free space as follows, respectively, where the scalar potential  and the vector potential A evaluated at the observation point r and the time t are defined in their usual way, In the above equations,   where a S is the surface enclosing a V with outward normal unit n . (7) shows that   W t  can be separated into two parts, one part is stored in the domain a V , the other part will pass through a S and be stored in the region outside a V .
Especially, recalling that , where r is the unit radial vector, the surface integral at the RHS of (7) approaches zero at S  with r   . Therefore, the electric energy defined by   W t  really has the sense of being stored in the space with no energy leaking to the infinity. Furthermore, it can be checked that   W t  satisfies the three terms listed in the previously specified hypothesis. Therefore, it is reasonable to define the reactive electric energy of the radiator as . Therefore,   J W t is not an energy purely stored in the whole space V  because it contains a part of energy that always flows towards the infinity, relating to the electromagnetic radiation. Secondly, in vacuum, the total radiative electric energy of a radiator should equal its total radiative magnetic energy. Denote the total magnetic energy as W t is defined as the reactive magnetic energy, it can be checked from (9) that the radiative magnetic energy does not equal the corresponding radiative electric energy   e rad W t . Thirdly, as has been verified in our previous works [14][34] [35], in the case of the Hertzian dipole, the reactive electric energy defined by   e react W t is exactly in agreement with the electric energy stored in the capacitor in its equivalent circuit model proposed by Chu [36]. However, the reactive magnetic energy calculated with   J W t does not exactly equal to the magnetic energy stored in the equivalent inductor. Only their time average values are equal. Taking into account of these factors, the definition of the reactive magnetic energy of a radiator is modified by making the total radiative magnetic energy equal the total radiative electric energy. Explicitly, we define . Making use of (6), the reactive magnetic energy can be expressed by Apparently, an additional term, the second volume integral on the RHS, is used to balance the radiative magnetic energy and the radiative electric energy. The leakage energy to infinity contained in   J W t is now accounted by the surface integral on the RHS. For pulse radiators, the surface integral vanishes because their waves never reach the infinity. It is clear that   J W t appears/disappears with the current source   ,t J r . In the following, we will show that the second volume integral on the RHS will also disappear not simultaneously but soon after its sources disappeared.
The electric flux density can be expressed by where the superscript "." means derivative with respect to time. The time domain Green's function can be expressed with the Dirac delta function, Denote the second volume integral in the RHS of (10) as   AD W t . Substituting (4) and (12) into it yields integrations over source region With the derivations detailed in the Appendix, the integral can be explicitly expressed by an integration of the source distributions, is not zero everywhere in the space, its volume integral over the whole space, i.e.,   AD W t , will soon become zero. In the espresions, 21,max r is the largest distance between two source positions.
The reactive electromagnetic energy is the sum of the reactive electric energy and the reactive magnetic energy, (16) or numerically, it is equal to The total radiative energy is the sum of the radiative electric energy and the radiative magnetic energy, which is For static electromagnetic fields, the radiative energy is zero, and the reactive electric (magnetic) energy is exactly the stored electric(magnetic) energy associated with the static charge sources (the steady state current sources). Denote The radiative energy can then be divided into With these definitions, the total electromagnetic energy can be expressed with It can be checked that for a pulse source over   for t T  . However, as seen from (20), the total radiative energy will continue to vary in a small time period 21,max Now we may give an interpretation to the radiation process of a pulse radiator with sources existing in   0,T . The reactive electromagnetic energy generated by the pulse radiator includes two parts, as shown in (17) [19]. As is discussed in [19], the radiation rate is always nonnegative and it describes an irreversible loss of energy, while the Schott energy changes reversibly. Judging from (17) and (20) where the Poynting vector   S E H is conventionally regarded as the power flux density, like in the antenna society. With the definition of (16) and (18), it can be rewritten as 1 1 1 which implies that the Poynting vector contains the contribution from the propagation of the radiative energy and the fluctuation of the reactive energy. Now we will show that the radiative energy part   0 rad W t associated with a bounded volume is a convenient quantity for engineering application. Substituting (5) and (6) into (23) and reorganizing it gives For the sake of convenience, a new vector is introduced for the integrand of the surface integral in (25), It has to be noted that 0 rad S is not the radiative power density. Denote its surface integration as The total work done by the source is Integrating both side of (25) gives J J r r r r r J J r r r r (32) and for that from source-1 to source-2, In this theory, the mutual coupling energies also include a Schott energy like term, which may be denoted by   A D W t . It is proposed in this theory that the mutual electromagnetic coupling occurs when the potentials from one source propagate to other sources and interact with them or their nearby fields. The mutual coupling is exerted through potentials instead of fields, so the electric and magnetic Aharonov-Bohm effect [38][39] may be interpreted with the classical electromagnetic theory in a simple way, as the change of the mutual coupling energies will inevitably introduce a force on the sources influenced by the mutual coupling.

IV. RADIATION OF HARMONIC SOURCES
For harmonic fields with time convention of j t e  , the radiation is assumed to last temporally from  to  , so the radiative energy is infinitely large. The Poynting theorem can be applied to describe the balance between the time average powers and the varying rate of the energies, 1 1 The same symbols are used for the corresponding phasors for the sake of convenience. From which the time average radiative power at infinity can be evaluated with source distributions, However, the evaluation of the reactive energies in conventional formulation requires to subtract the radiative energy from the total energy. Since both the energies are unbounded, all those formulations based on energy subtraction are not quite satisfactory so far.
With the theory proposed here, the power balance can be evaluated within any domain enclosed by an observation surface a S enclosing the source region s V , The time average radiative power crossing the observation surface can be obtained using the radiative power flux vector Note that the observation surface is not required to approach infinity for evaluating the radiative power. It can be checked that the result is in consistent with that obtained using the Poynting vector, as has been shown in [14] The average reactive energy can be calculated with the fields and the vector potential, which is exactly in agreement with the result shown in [41].
The well-established equivalent circuit model proposed by Chu [36] for Hertzian dipole is shown in Fig.1 With the wave travels to infinity, the total radiative energy    Fig. 3(a). In this case, the reactive energy includes the contribution from the current alone since the corresponding charge is zero, so it is denoted as J W in the figures. J W oscillates with the source and admits negative values periodically. In the proposed theory, it is acceptable because the reactive energy is dependent on the potentials, which are values relative to their reference zero points. When the current varies and changes its direction periodically, the retarded vector potential in the resource region lags behind and may point in direction opposite to that of the current, causing negative values. The energy   AD W t is also plotted in Fig.3(a), and is zoomed in in Fig.3(b) together with J W . It is shown that   AD W t oscillates like J W , but continue to exist for about 0.33ns after the source disappeared at 1ns. Note that the Schott energy in the charged particle theory may also be negative [19] [42].
The energies passing through sphere-1 are shown in Fig. 3(c). The smallest and the largest distance between the source and sphere-1 are respectively 0.1m and 0.3m. The total radiative energy passed at t=2ns is equal to that evaluated at the source region.
The excitation power, radiative power Srad P and the time varying rate of the reactive energy are shown in Fig. 4(a).
The powers crossing-1 and sphere-2 are shown in Fig.4(b) and ( VII. CONCLUSIONS Some issues concerning with the electromagnetic radiation and mutual couplings remain confusing or even controversial for decades long, especially the definitions for the reactive energy and Q factors of radiators. This theory proposes clear definitions and explicit expressions for the reactive energy and the radiative energy of a radiator. The introduction of a Schott energy like term in the reactive energy makes it possible to separate the radiative energy and the reactive energy in a reasonable manner. Consequently, a new power balance equation is given by modifying the Poynting relation so that the Poynting vector is divided into two parts, accounting for the contribution from the radiative energy propagation and the fluctuation of the reactive energy. The newly defined reactive energy term,   0 rad W t , and its flux,   Srad P t , can characterize the radiative energy almost satisfactorily. Furthermore, they can be numerically evaluated more efficiently, so are the mutual electromagnetic coupling energies defined with potentials.
Although in the theory, expressions for the reactive electric energy and the reactive magnetic energy are also separately provided, it is strongly recommended to combine the two reactive energies together and treating them as a whole.
The theory is different from Carpenter formulation [23], in which it was proposed to use the source-potential combination terms in (16) as the total electromagnetic energy, and to replace the Poynting Theorem with a new equation. The formulation, as well as the power flow vector J   by Slepian [43], was pointed to be mathematically flawed by Dr.
Endean [44] . In the theory proposed here, the source-potential terms are considered to form the reactive energy together with a Schott energy like term. They compose only part of the total electromagnetic energy. So the theory does not suffer from the mathematical flaws checked by Dr. Endean since there is no modification to the total electromagnetic energy and the correspondent Poynting Theorem.