## Abstract

Polycentric focus effect of time-reversal (TR) electromagnetic field is found in a rectangular resonant cavity. Theoretical deduction shows that the effect is due to the mirror symmetry of the cavity and the maximum number of focus points is 27 including 1 main focus point and 26 secondary focus points. A case of 6 focus points is calculated, in which the numerical results are consistent with the theoretical predictions, and particularly the 5 secondary focus points have directly resulted in inaccurate imaging and pulse signal interception.

© 2013 Optical Society of America

## 1. Introduction

In nearly closed metal cavity structures such as container, armored car, submarine, spacecraft, etc., the multipath effects of pulse propagation are evident. However, many experiments have confirmed that time-reversal (TR) electromagnetic (EM) field can perform temporal and spatial focusing, especially in dense multipath circumstances [1–3]. And such high TR focus gains have attracted extensive attentions from engineers [1–10]. Recently, TR EM field has even been studied to be applied in ultra-wideband communication [4,6,8,9] and high resolution imaging [5,10,11]. It seems that TR EM field should ONLY focus back on its originator. Is it true?

In the phenomenological analysis, TR EM focus arises from the temporal and spatial coherent superposition of EM waves [12–14]. In dense multipath environments, TR operation to reverse impulse response will compensate multipath delays, which results in the temporal coherent superposition (TR temporal focus). If the compensation effect of TR operation will be lost quickly after the transceiver moves a short distance, it will be called the spatial de-correlation [6] (TR spatial focus). In case the distance is far less than Rayleigh limit, super resolution would be reported [1,10]. The theory of monochromatic TR mirrors or equivalently phase conjugate mirrors has been developed for electromagnetic waves [11,12,15], but the uniqueness of TR focus position is still uncertain.

## 2. Theoretical derivation

The final analysis of the uniqueness can only come from Maxwell equations and boundary conditions. It is possible to get the analytical expression of TR EM fields in some cases of symmetrical boundaries. In a rectangular vacuum resonant cavity (${L}_{a}\times {L}_{b}\times {L}_{c}$), one pulse is transmitted by the current element $i\left(t\right)l$ at the point ${R}_{T}$, then the electric field probes at the point ${R}_{R}$ will receive $E\left({R}_{R},t\right)$. The method of images can be deliberately used to calculate $E\left({R}_{R},t\right)$ in the presence of conducting boundaries with mirror symmetry [12,15]

After TR operation and retransmission, the probes at the point $R$ will receive TR electric field as follow

The reason of 27 is $3\times 3\times 3$, where each dimension contributes three possibilities, namely, twice mirror and self. However, observable focus points must be positioned in the cavity, so 27 is the most. And the 27 possible points depend on the size of the cavity and the positions of the transceivers.

## 3. Numerical verification

The vacuum cavity sizes in Fig. 1 are ${L}_{a}=40cm$, ${L}_{b}=50cm$, ${L}_{c}=60cm$, and the transmit-receive positions are ${R}_{T}:\left(20,38,49\right)cm$, ${R}_{R}:\left(10,19,20\right)cm$, and the band width of the pulse excitation is $\left(3.1,10.7\right)\text{GHz}$, then Eq. (11) predicts 6 possible focus points exhibited in Table 1. To verify the theoretical predictions, we will use Microwave CST studio to numerically simulate TR focus process in accordance with 5 steps tagged in Fig. 1.

In Table 1, the numerical simulation results derive from TR imaging in the plane $x=20cm$ at the focus moment, which are consistent with the theoretical predictions. Furthermore, if the theoretical focus positions are brought back into Eq. (5), we can also calculate TR gains of all focus points theoretically.

In Fig. 2, the polycentric TR focus effect has directly affected TR imaging quality because the accurate image should be only one focus point. To improve the quality or to inhibit the polycentric effect, we need to further investigate the relationship between the spatial coherent superposition of EM waves and the spatial symmetry of EM boundaries.

Figure 3 shows another consequence of the polycentric TR focus effect, i.e. signal interception. Although TR focus gain at MFP is highest, TR focus gains at SFP are high enough to intercept the pulse signal. Thus, TR post-filter scheme [9] seems to be a better scheme for anti interception than TR pre-filter scheme (The complexity at the receiving end will be higher in post-filter scheme than in pre-filter scheme.).

The multipath delay set can be grouped into 8 regular delay-compensation subsets due to the mirror symmetry of the cavity. In the case of MFP, the bijective relationship of $D$ is

## 4. Conclusion

Both theory and simulation negate the uniqueness of TR focus between two points in rectangular resonant cavity, which suggests that more structures with spatial symmetries need to be further investigated about TR focus patterns. For TR users, actively, TR polycentric focus effect has the potential to help design new ultra-wide band analog signal distributors and multiplex wireless chargers. Passively, it may be also interesting to regulate the polycentric effect by EM polarization, because in the presence of spatial symmetries the effect may also appear in other imaging processes not only TR imaging process.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61071031 and No. 61107018), the Research Fund for the Doctoral Program of Higher Education of China (No. 20100185110021 and No. 20120185130001), the Fundamental Research Funds for the Central Universities (No. ZYGX2012YB020), and the Project ITR1113.

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