1. Introduction

When we discuss time-reversal (TR) symmetry of Electromagnetic (EM) fields, the forward process and the reverse process will usually be compared in order to emphasize the TR focusing performance and TR symmetry (TRS) [1–7]. Many TR EM experiments have confirmed that TRS can bring temporal and spatial focusing of EM energy in multipath circumstances, which has triggered widespread concerns of engineers [8–13]. In such TR experiments, the reverse fields could perform the temporal focusing, because the optical path differences were compensated by TR mirror (TRM) which can perform timing reversal in time domain or phase conjugation in frequency domain [1,10,11,13,14]. Physically, the spatial focusing will come directly from the interference of EM waves, but the focal point position and the high focusing gain are still due to TRM considerably [13–16]. TRM played a key role in a TR EM system, but it also became a bottleneck because of its complexity or high cost or non-real time.

An earlier research about TRM is four-wave mixing with optical pulses considered analytically by Miller in 1980 [17], but this theory is nonlinear. Marom etc. illustrated the difference between the spectral inversion (negative frequency operation, actually) and phase conjugation in 2000 [18], however, lack of the frame analysis of multi-domain symmetry is a pity. Kuzucu etc. experimentally demonstrated wavelength-preserving spectral phase conjugation for compensating chromatic dispersion and self-phase modulation in optical fibers in 2009 [19], which is an implementation of broadband four-wave mixing. Sivan and Pendry analyzed TR efficiency based on spatially-periodic modulation in 2011 [20], physically, where they used the symmetry of EM fields in the spatial spectrum domain. This paper will verify that TRS of EM fields can always be assumed to be a dual operating invariance under TR operation and chiral operation in the relevant experimental reports. So that theoretically means we probably find new dual or multiple combined operation on a much broader horizon to replace the existing dual-combination operation based on TRM.

The first part of this paper will discuss TRS of EM fields. Based on the first part, the following part will show the operating invariance of EM fields in four domains, i.e., $\left(r,t\right)$, $\left(r,\omega \right)$, $\left(k,t\right)$, and $\left(k,\omega \right)$ domains, and provide with intuitive charts of the operator multiplications. The third part will apply the four-domain invariance to the explanation of the retrodirective mechanism on Van Atta’s array. The last part will be some discussions and conclusions.

2. TRS of EM fields

Definition: If a formal solution after a certain operation still satisfies the original equation, then the formal solution has the symmetry or invariance under the operation.

The discussion about TRS has crossed mechanical determinism. Since the principle of entropy increase was established, the perfect TRS has been driven to the microcosm [21]. In the macrocosm, the thermodynamic arrow of time would direct the evolution of an isolated system, and lead to TRS breaking [6].

Engineers may be more concerned about the sufficient condition of TRS of time-varying EM fields, or at least, the discussion of the possibility of realization in engineering terms. If the formal solution $\left[E\left(r,t\right),H\left(r,t\right)\right]$ in the linear, lossless and static medium satisfies Maxwell equations and the constitutive relations

If the formal solution $\left[E\left(r,\omega \right),H\left(r,\omega \right)\right]$ in the linear, lossless and static medium satisfies Maxwell equations and the constitutive relations

Good conductors are usually used to carry most of currents in many transceivers of EM energy. The shallow penetration depth of microwave in good conductor will be $\delta \approx \sqrt{2/\omega \mu \sigma }$, and the average power loss per unit area will be $P_{Loss}\approx {\alpha }_0^2/2\sigma \delta$, here ${\alpha }_0^{}\approx \sigma \delta E_0/\sqrt{2}$ is the peak of the surface current density, $E_0$ is the peak of the surface electric field. Thus, when $\sigma \to +\infty$, we will get $\delta \to 0^{+}$ and $P_{Loss}\to 0^{+}$. That means EM fields will be compressed in a thin surface layer of the good conductor (i.e. skin effect).

The complexities of an EM system derive mainly from the complex constitutive relations and the complex geometrical boundaries. For more general cases, integral equations will be more suitable,

3. Four-domain invariance of EM fields

In the domains $\left(r,t\right)$ and $\left(r,\omega \right)$, TRS of EM fields has be discussed. Furthermore, we will get two expanding domains $\left(k,t\right)$ and $\left(k,\omega \right)$ by Fourier Transform,

Accordingly, in the linear and lossless media the integral equations can be written as

$\left[E\left(k,t\right),H\left(k,t\right)\right]$ and $\left[E\left(k,\omega \right),H\left(k,\omega \right)\right]$ will respectively have TR EM fields $a\left[E\left(k,-t\right),-H\left(k,-t\right)\right]$ in the $\left(k,t\right)$ domain and $a\left[E^{\ast }\left(k,\omega \right),-H^{\ast }\left(k,\omega \right)\right]$ in the $\left(k,\omega \right)$ domain.

Maxwell equations in the previous sections are always written in a right-handed frame. Whereas, in a left-handed frame, Eq. (6) and Eq. (7) can be rewritten as

A chiral operator $\overline{\overline{C}}$ can be defined to express the transition of EM fields between a right-handed frame and a left-handed frame

Similarly, we can introduce conjugation operator $\overline{\overline{P}}$, time reverse operator $\overline{\overline{T}}$, negative frequency operator $\overline{\overline{N}}$, space vector centrosymmetry operator $\overline{\overline{{\rm X}}}$ and space spectrum vector centrosymmetry operator $\overline{\overline{K}}$, as follows

In fact, if the symmetries of EM fields are discussed on the operator set ${\rm O}=\left\{\overline{\overline{I}},\overline{\overline{C}},\overline{\overline{T}},\overline{\overline{P}},\overline{\overline{N}},\overline{\overline{{\rm X}}},\overline{\overline{K}}\right\}$, the symmetry operations will not be only four, i.e., ${\overline{\overline{T}}}_{\left(r,t\right)}{\overline{\overline{C}}}_{\left(r,t\right)}$, ${\overline{\overline{P}}}_{\left(r,\omega \right)}{\overline{\overline{C}}}_{\left(r,\omega \right)}$, ${\overline{\overline{T}}}_{\left(k,t\right)}{\overline{\overline{C}}}_{\left(k,t\right)}$, and ${\overline{\overline{P}}}_{\left(k,\omega \right)}{\overline{\overline{C}}}_{\left(k,\omega \right)}$. Table 1 depicts the dual multiplications induced by the operator set ${\rm O}$, which has main diagonal symmetry due to the reciprocity of any two operators. By the definition, eleven operations ($\overline{\overline{C}}\overline{\overline{{\rm K}}}$, $\overline{\overline{C}}\overline{\overline{T}}$, $\overline{\overline{C}}\overline{\overline{N}}$, $\overline{\overline{C}}\overline{\overline{{\rm X}}}$, $\overline{\overline{{\rm K}}}\overline{\overline{T}}$, $\overline{\overline{{\rm K}}}\overline{\overline{N}}$, $\overline{\overline{{\rm K}}}\overline{\overline{{\rm X}}}$, $\overline{\overline{T}}\overline{\overline{N}}$, $\overline{\overline{T}}\overline{\overline{{\rm X}}}$, $\overline{\overline{N}}\overline{\overline{{\rm X}}}$, $\overline{\overline{I}}$) have the invariance of EM fields in linear and lossless media, and the seven undetermined operations ($\overline{\overline{P}}$, $\overline{\overline{C}}\overline{\overline{P}}$, $\overline{\overline{{\rm K}}}\overline{\overline{P}}$, $\overline{\overline{T}}\overline{\overline{P}}$, $\overline{\overline{N}}\overline{\overline{P}}$, $\overline{\overline{{\rm X}}}\overline{\overline{P}}$,$\overline{\overline{P}}\overline{\overline{P}}$) may have the invariance determined by the operating domains, and the five remaining operations ($\overline{\overline{C}}$, $\overline{\overline{{\rm K}}}$, $\overline{\overline{T}}$, $\overline{\overline{N}}$, $\overline{\overline{{\rm X}}}$) have no invariance. In the whole set ${\rm O}$ except for $\overline{\overline{P}}$, that any operator is operated twice will be always equivalent to an identical operator, even the twice operations are done in different domains. Only $\overline{\overline{P}}$ must be in the same domain, then twice operations will perform ${\overline{\overline{P}}}^2=\overline{\overline{I}}$.

Table 1. Dual Multiplication Table Induced on the Operator Set ${\rm O}$

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If the medium chirality is given, the chiral relation among $E$, $H$ and the energy flux vector $S$ will also be given. Physically the operator ${\overline{\overline{T}}}_{\left(r,t\right)}$ or ${\overline{\overline{P}}}_{\left(r,\omega \right)}$ will lead to $S\to -S$, so we need the operator $\overline{\overline{C}}$ to perform $\left[E,H\right]\to \left[E,-H\right]$ for the correction of the reversed chirality. The dual combined operators ${\overline{\overline{T}}}_{\left(r,t\right)}{\overline{\overline{C}}}_{\left(r,t\right)}$ and ${\overline{\overline{P}}}_{\left(r,\omega \right)}{\overline{\overline{C}}}_{\left(r,\omega \right)}$ are just the reported TR symmetries based on TRM operations.

Figure 1 gives a detailed indication of the invariance of seven undetermined operations in appropriate operating domains, in which we use $\overline{\overline{P}}\overline{\overline{I}}$ to depict $\overline{\overline{P}}$ equivalently. Two ends of each line indicate the appropriate operating domains. All dual combined operators linked by lines have the invariance, and the operators not linked by lines do not show the invariance.

 

Fig. 1 The invariance of seven undetermined operations in appropriate operating domains.

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Taking conjugate and $r\to -r$ transformations on both sides of Eq. (10) (these are mathematical identical transformations aimed at the equations, however, in Table 1 those operators are applied to the formal solutions.), we will get

Accordingly, if the formal solution $\left[E\left(k,\omega \right),H\left(k,\omega \right)\right]$ satisfies Eq. (10), then ${\overline{\overline{P}}}_{\left(k,\omega \right)}\left[E\left(k,\omega \right),H\left(k,\omega \right)\right]=\left[E^{\ast }\left(k,\omega \right),H^{\ast }\left(k,\omega \right)\right]$ will also do, which directly verifies the invariance of ${\overline{\overline{P}}}_{\left(k,\omega \right)}\overline{\overline{I}}$ in Fig. 1. By means of the invariance of ${\overline{\overline{P}}}_{\left(k,\omega \right)}$, the invariance of the dual combined operator ${\overline{\overline{P}}}_{\left(r,\omega \right)}{\overline{\overline{X}}}_{\left(r,t\right)}$ can be indirectly verified as follow

Similarly we can prove the invariance of all dual combined operators linked by lines in Fig. 1. It is conceivable that triple or more multiple operations will show more complex and richer multiplication tables and links of operating domains.

4. An application of four-domain invariance

Physically, Van Atta’s array has used the invariance of the dual combined operator ${\overline{\overline{X}}}_{\left(r,\omega \right)}{\overline{\overline{P}}}_{\left(r,\omega \right)}$, which can reradiate incident EM signal in the arrival direction regardless of the array orientation relative to the direction [22–24]. According to the definition, EM fields after ${\overline{\overline{X}}}_{\left(r,\omega \right)}{\overline{\overline{P}}}_{\left(r,\omega \right)}$ operation can still propagate in the same space $\left(r,\omega \right)$. The operator ${\overline{\overline{P}}}_{\left(r,\omega \right)}$ will lead to $S\to -S$, which has enough provided the retrodirective mechanism. But Fig. 1 shows that only ${\overline{\overline{P}}}_{\left(r,\omega \right)}$ cannot perform invariance! Hence, ${\overline{\overline{P}}}_{\left(r,\omega \right)}$ needs ${\overline{\overline{X}}}_{\left(r,\omega \right)}$ to help EM fields backtracking in the premise of no damage to the retrodirective mechanism. Any pair of antennas with coelongate transmission line will be equidistant from the center of Van Atta’s array, which can provide ${\overline{\overline{P}}}_{\left(r,\omega \right)}$ operation and ${\overline{\overline{X}}}_{\left(r,\omega \right)}$ operation simultaneously (If the directly reflected fields are ignored, by centrosymmetric array, the fields will enter into one end and radiate out from the centrosymmetric other end.). That is the physical performance of Van Atta’s array in the domain $\left(r,\omega \right)$.

Unfortunately, Fig. 1 also tells that ${\overline{\overline{X}}}_{\left(r,t\right)}{\overline{\overline{P}}}_{\left(r,t\right)}$ cannot perform invariance! That is to say, Van Atta’s array without any reform in the domain $\left(r,t\right)$ is not TRM, although it just likes TRM in the domain $\left(r,\omega \right)$. Fortunately, ${\overline{\overline{X}}}_{\left(r,t\right)}{\overline{\overline{P}}}_{\left(k,t\right)}$ can perform invariance, in other words, if Van Atta’s array with some reform can perform ${\overline{\overline{P}}}_{\left(k,t\right)}$ operation, it will have the potential to promote new TRM in the domain $\left(r,t\right)$.

5. Discussion and conclusion

Lossless medium space and linear response will be the sufficient conditions of TRS of EM fields. For engineers, there are three constant problems in EM systems, i.e., loss, nonlinearity and time varying. It is obviously ridiculous to discuss TR focusing performance if EM energy has been depleted before the energy effectively reaches the focal point, so we need further quantitative analysis on TRS breaking caused by loss. Some related references have reported some improved TR communications in a fading channel [11–13, 25]. Near equilibrium usually implies linearity, but non-equilibrium in many EM systems, e.g. containing active devices, widely exists. So it is also a challenge to better understand the relations between nonlinearity and four-domain invariance of EM fields. In time varying environments, how to amend TTS of TR EM fields will directly affect the applications of TRS in high-speed communications and high resolution imaging of moving objects.

TRS of EM fields can be performed in four domains connected by Fourier Transform. And more symmetries can be performed by introducing centrosymmetric operations of spatial position vector and spatial spectrum vector. Many combined operations different from TRM operations can equivalently actualize TRS. In this paper, all symmetries are generalized forms. Accordingly, we will have further works to do in many concrete EM problems, such as the relation between TR spatial focusing and space symmetry, the relation between the robustness of TR temporal focusing and spectrum space symmetry in a time varying environment, the relation between TR focusing gain and spatial distribution symmetry of TRM array.