Time-reversal Symmetry in Antenna Theory

Here, I discuss some implications of the time-reversal invariance of lossless radiating systems. I highlight that time-reversal symmetry provides a rather intuitive explanation for the conditions of polarization and impedance matching of a receiving antenna. Furthermore, I describe a solution to generate the time-reversed electromagnetic field through the illumination of a matched receiving antenna with a Herglotz wave.


Introduction
Time-reversal is the operation that flips the arrow of time such that t t   [1,2]. Remarkably, the laws that rule the microscopic dynamics of most physical systems are invariant under a time-reversal transformation (the exceptions occur in some nuclear interactions and are evidently irrelevant in context of this study). This property implies that under suitable initial conditions, the time reversed dynamics may be generated and observed in a real physical setting, similar to a movie played backwards. Ultimately, the invariance under time-reversal implies that at a microscopic level the physical phenomena are intrinsically reversible: if a particular time evolution is compatible with the physical laws, then the time-reversed dynamics also is.
A consequence of "time-reversal invariance" is that the propagation of light in standard waveguides is inherently bi-directional, even if the system does not have any particular spatial symmetry. For example, if some electromagnetic wave can go through some metallic pipe with no back-reflections, then the time-reversed wave also can, but propagating in the opposite direction. This rather remarkable property is usually explained with the help of the Lorentz reciprocity theorem [3][4], but it is ultimately a consequence of microscopic reversibility and time reversal invariance [2,5].
In this article, I reexamine the consequences of time-reversal invariance in antenna theory. I show that time-reversal invariance provides a rather intuitive explanation for the conditions of impedance and polarization matching in the theory of the receiving antenna. In addition, I prove that in time-harmonic regime the time-reversed wave can be generated through the illumination of the receiving antenna with a superposition of plane waves generated in the far-field.

Time-reversal symmetry
It is well known that the equations of macroscopic electrodynamics are not time reversal invariant when the system has dissipative elements. This is so because the description provided by macroscopic electrodynamics is incomplete, as it only models the time evolution of the electromagnetic degrees of freedom. In contrast, in a microscopic formalism -with all the light and matter degrees of freedom included in the analysis-the system dynamics is time-reversal symmetric. Thus, in some sense, macroscopic dissipative systems (e.g., lossy dielectrics) have a hidden time-reversal symmetry. To circumvent this complication, here I will focus on systems with negligible material absorption, so that the dynamics determined by the macroscopic Maxwell equations is time-reversal symmetric.

General case
Consider the propagation of light in some lossless, dispersion-free, dielectric system described by the Maxwell equations: E r E r . In particular, suppose that some wave incident in port 1 is fully transmitted to port 2. Then, if port 2 is illuminated with the time-reversed transmitted signal it will reproduce the original signal in port 1, but reversed in time. Thereby, time-reversal invariant systems are intrinsically bi-directional, independent of any spatial asymmetry.
The enunciated results can be generalized in a straightforward way to dispersive lossless dielectrics, e.g., to material structures characterized by some real-valued scalar permittivity   ,     r (e.g., a Lorentz dispersive model with no dissipation). The reason is that the electrodynamics of lossless dispersive systems can be formulated as a Schrödinger-type time evolution problem [6][7][8], which in the case of reciprocal media (e.g., standard dielectrics) is time-reversal invariant.
Furthermore, as discussed in [9], most lossless nonlinear systems are time-reversal symmetric and hence are also bi-directional. For example, for an instantaneous Kerr-type nonlinear response the Maxwell equations (1) remain time-reversal invariant. Interestingly, the time-harmonic response of a two-port microwave network with nonlinear components is generically asymmetric [10,11,12]. Indeed, if the ports are individually excited by the same time-harmonic signal the level of the transmitted signal depends on which port is excited; thus nonlinear systems are nonreciprocal [10,11,12]. In summary, lossless nonlinear systems are usually both time-reversal invariant and nonreciprocal, the two conditions are not incompatible [9]. In the rest of the article, I focus on linear systems.

Time-harmonic variation
Consider a time-harmonic solution of the Maxwell equations, such that the electromagnetic fields and current density are of the form: , where the symbol "*" stands for complex conjugation. Hence, the complex amplitudes of the fields and current density are transformed as: Thus, in the frequency domain the time-reversal operation is closely linked to phase conjugation [13][14]. Similarly, voltages and currents are transformed as: For example, consider a N-port microwave network such that the voltages and currents at a generic port i are of the form: represents an incoming (incident) wave and is the scattering matrix. The time reversal operation exchanges the roles of the incident and scattered waves, such that TR, Thus, the scattering matrix must satisfy On the other hand, for a lossless system the incident power must equal to the scattered power: To satisfy this additional constraint the scattering matrix must be unitary †   S S 1 . Combining the two results, one finds that the scattering matrix must be transpose symmetric: Thus, any time-reversal invariant linear lossless system is necessarily reciprocal ( ij ji S S  ) [15].
Here, I note in passing that in electromagnetic theory the time-reversal operator  is idempotent, such that 2  1  . In contrast, in condensed matter theory the time-reversal operator satisfies 2  1  , and because of this property the scattering matrix of fermionic systems is anti-symmetric, T   S S [15]. It was recently shown that photonic systems protected by a special parity-time-duality (  ) symmetry are constrained by

Application to antenna theory
The time-reversal property may be used to explain several well known properties of radiating systems. Similar to the previous section, I assume that the antennas are formed by lossless materials, e.g., lossless dielectrics or perfect conductors. In particular, the radiation efficiency of the antennas is 100%.

Polarization and impedance matching
Consider now the time-reversed problem represented in Fig. 2b, where all the radiated energy is returned back to the antenna. The time reversed voltage and current at the antenna terminals are For example, an antenna that radiates a right-circularly polarized (RCP) wave in some direction of space is polarization matched to an incoming plane wave with RCP polarization. Even though the wave and antenna polarizations are identical, the geometrical senses of rotation of the relevant electric fields are opposite. This otherwise intriguing property can be understood as a simple consequence of time-reversal invariance.

Time-reversed field generated with a far-field illumination
The problem of generating a time-reversed field distribution is of practical interest, as it enables concentrating and focusing energy from the far-field into some desired region of space. The theory and application of time-reversed fields were developed and extensively explored by Fink and co-authors [17][18][19][20][21]. Here, I revisit the problem and highlight some features that were not discussed in Ref. [19].
In the time-reversed problem of Fig. 2b  Notably, I prove in the Appendix that the solution of the scattering problem formulated in the previous paragraph can be constructed from the fully-time reversed field TR  E . Specifically, when an impedance-matched antenna is illuminated by the incident field the field scattered by the antenna is precisely given by scat Comparing Eqs. (6) and (8)

Conclusions
I revisited the topic of time-reversal symmetry in macroscopic electromagnetism. I showed that under a time-reversal transformation a transmitting antenna becomes the impedance matched receiving antenna. Heuristically, the excitation with the time-reversed wave must be the most effective way of delivering power to an antenna. Thus, the time-reversal invariance provides a simple and intuitive understanding of the conditions of impedance and polarization matching in antenna theory. In particular, it elucidates why a polarization matched incident wave has an electric field that rotates geometrically in a direction opposite to that of the field radiated by the antenna in the same direction. In addition, I generalized the ideas of [19] and showed that the time-reversal of the field emitted by a lossless transmitting antenna can be created by illuminating an impedance matched receiving antenna with the far-field excitation associated with the Herglotz wave (7). In such a scenario, the power captured by the matched load is precisely the same as the power scattered by the antenna.  is field associated with an incident plane wave (arriving from direction r ) evaluated at the origin [16]. Thus, from the superposition principle, the voltage induced by the Herglotz wave (7)  . This direct analysis confirms that when the antenna is illuminated by the Herglotz wave the power absorbed by a matched load is exactly the power radiated by the antenna in the scenario of Fig. 2a.