## Abstract

Time-reversal (TR) phase prints are first used in far-field (FF) detection of sub-wavelength (SW) deformable scatterers without any extra metal structure positioned in the vicinity of the target. The 2D prints derive from discrete short-time Fourier transform of 1D TR electromagnetic (EM) signals. Because the time-invariant intensive background interference is effectively centralized by TR technique, the time-variant weak indication from FF SW scatterers can be highlighted. This method shows a different use of TR technique in which the focus peak of TR EM waves is unusually removed and the most useful information is conveyed by the other part.

© 2014 Optical Society of America

## 1. Introduction

To detect or control the scattering of sub-wavelength (SW) structure beyond Rayleigh resolution limit, a massive amount of additional metal parts must be brought to the vicinity of the structure in many known approaches based on plasmon [1], such as hyperlenses [2], perfect lenses [3], or adiabatic concentrators [4] including those based on metal-wire arrays [5,6]. Heisenberg uncertainty principle always limits the minimum width of the far-field (FF) interference fringes of electromagnetic (EM) waves. When we only use the FF scattering signals to backtrack and reconstruct the near-field (NF) details of the scatterer, the uncertainty limitation will conceal the NF SW details, which can be just called Rayleigh resolution limit. So it is an inspiring challenge to detect FF SW deformable scatterers without any extra metal structure positioned in the vicinity of the target. Fortunately, Rayleigh limit only restricts the spatial distribution of the EM amplitude, in other words, it is still possible to accurately obtain the phase and polarization information from NF SW structures. That is the core physical idea of this paper about the FF phase prints (PP) detection of the SW deformable Tetris. In fact, the phase information from the SW structure has been perfectly used in some time-reversal (TR) EM systems [1,7,8], and the polarization information has also been ideally used in SW full-vectorial profiling of optical focus [9] and spatiotemporal SW NF light localization [10], although the physical idea is distinctly proposed in this paper. We give a further explanation of the idea. After an SW structure is deformed by changing the geometry of the structure or the physical parameter in the structure, due to the uncertainty principle we cannot expect the emergence of the FF interference fringes of the scattered EM waves beyond the Rayleigh limit but we can expect the accurate measurement of the fringes’ movements between before and after deformation of the SW structure and also can expect the accurate measurement of the polarization rotation of the scattering EM waves in the fringes.

To meet this inspiring challenge needs both the physical idea and the technical support. The weak scattering indication from FF SW blocks will be submerged in the intensive background interference, especially in dense multipath environments. Based on the previous physical analysis, what really get us concerned will be the phases of the scattered EM waves from FF SW blocks rather than the amplitudes, but the dense multipath of the background will not only affect the amplitude measurement but also affect the extraction of the scattering phases. TR operation or phase conjugation [11] can compensate the dispersion of the EM waves from the multipath background in frequency domain, which means all phases of the EM waves from the multipath background will be merged into one and the same phase at the focus point. TR operation can also effectively centralize the EM energy from the dense multipath background at the focus moment, which means the phases of the scattered EM waves from FF SW blocks will be easier to emerge at the other moment. As if a bush is deprived of shrubs, then the weeds in which there is our real concern will be fully exposed. Hence TR technique is the core technical support of this paper. About the spatiotemporal focusing of TR EM waves there have been many beautiful experimental reports [12–16] in dense multipath environments, which we call direct use (Because the most useful information is always conveyed by the temporal focus peak or the spatial focus point, and aside from the peak or point, the other part of TR EM waves is grossly neglected.). This paper will first show an indirect use of TR EM waves in which the TR focus peak will be removed and the most useful information is highlighted in the other part. With the help of time-dependence Fourier analysis [17, 18], the PP database of the TR scattering waves can be established to detect different FF SW blocks of Tetris. Discrete short-time Fourier transform (DSTFT) [19] is the other technical support.

## 2. Model and theory

In Fig. 1 there is a cubic vacuum resonant cavity ($L\times L\times L$, where $L=60cm$) to cause multiple interactions of the TR EM waves with SW blocks needed to transfer the scattering information to the far field. Tetris contains 5 blocks. Every block contains 4 cubic conductor cells. To be detected block will be located at the origin which is the center of the cavity. The deformation between different blocks can be realized by moving some cubic cells. The dimension of every cell is $a\times a\times a$, where $a=3mm$. The transmitter-receiver polarizations are both $\left(1,1,1\right)/\sqrt{3}$ and the positions are ${r}_{T}:\left(12.9,11.3,4.7\right)cm$ and ${r}_{R}:\left(-9.9,-12.1,-13\right)cm$ respectively, which have deliberately averted the polarization selectivity and the polycentric focus effect [20] from spatial centrosymmetry and mirror symmetry of the cavity.

First, one pulse signal $s\left(t\right)$ with a frequency spectrum range of 2~3$\text{GHz}$ is transmitted at the point ${r}_{T}$ in the empty cavity without any block, then the receiver at the point ${r}_{R}$ will receive the multipath signal $s\left(t\right)\otimes {h}_{0}\left(t\right)$ where ${h}_{0}\left(t\right)$ is the channel impulse response (CIR) of the static empty cavity with intensive scattering and $\otimes $ represents convolution. Second, the signal $s\left(-t\right)\otimes {h}_{0}\left(-t\right)$ from the TR signal sequence of $s\left(t\right)\otimes {h}_{0}\left(t\right)$ will be transmitted at the point ${r}_{T}$ if the $i\text{th}$ SW block to be detected is located at the origin, and then the receiver at the point ${r}_{R}$ will receive the signal $s\left(-t\right)\otimes {h}_{0}\left(-t\right)\otimes {h}_{i}\left(t\right)$ where ${h}_{i}\left(t\right)$ represents the CIR of the cavity including the $i\text{th}$ SW block.

In Eq. (1), ${T}_{i}\left(t\right)$ represents the perturbation from the $i\text{th}$ SW block and generally satisfies## 3. Numerical demonstration and discussion

Microwave CST studio [21] is used to numerically calculate $s\left(-\tau \right)\otimes {h}_{0}\left(-\tau \right)\otimes {h}_{i}\left(\tau \right)$ and then Eq. (6) can be calculated by Digital Visual Fortran programs. Figure 2 demonstrates the FF TR PPs of the 5 blocks of SW Tetris in turn, where the differences between any two PPs are evident.

Figure 2(a) is the TR PP of Block 1, (b) is the one of Block 2, (c) is the one of Block 3, (d) is the one of Block 4, and (e) is the one of Block 5. Each block can be uniquely characterized by its TR PP. The specific belts around the frequency of $2.5\text{GHz}$ related to the resonance of the cavity show evidently different quasi-periods. The belt in Fig. 2(a) has about 3 complete cycles, the one in Fig. 2(b) has about 2.5 cycles, the one in Fig. 2(c) has about 7.5 cycles, the one in Fig. 2(d) has about 2 complete cycles, and the one in Fig. 2(e) has about 8.5 cycles. These differences have verified the method of the FF SW detection based on TR PP, although there may be other different strategies about the comparison between any two PPs.

The stability of the detection method becomes an urgent issue because the TR PP appears reasonably effective. Supposing that the CIR ${h}_{0}\left(t\right)$ has a small synchronization error $\Delta t$, the phase deviation of ${h}_{0}\left(\omega \right)$ can be written as $\varphi =\omega \cdot \Delta t$ in frequency domain. If $\text{Arg}\left\{{T}_{i}\left(\omega \right)\right\}$ is directly extracted, the deviation can be estimated as

## 4. Conclusion

Both theory and simulation demonstrate a new detection method of FF SW deformable Tetris without extra NF metal parts, which is based on the 2D TR PP from DSTFT of the 1D TR scattering fields. The successful detection verifies the physical idea that the uncertainty principle does not restrict the accuracy of the phase measurement although it restricts the spatial distribution of the amplitude, which will probably help engineers find new approaches to super resolution. TR PP can effectively capture weak time-variant signals in intensive time-invariant backgrounds by removing TR focus peak and inherently have the linear stability, which will partially relieve the urgency of the amplitude in the analysis process of the weak signal.

This paper does not yet explore and analyze the polarization advantages of TR EM waves concretely. In geophysics, Goldstein [22] and Foster [23] have both presented the new concept of Subauroral Polarization Stream (SAPS) which is a disturbance time effect in the dusk-to-midnight magnetic local time sector. Being inspired by SAPS, our next work will focus on an imaging method based on TR Polarization Stream.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61331007 and No. 61361166008), the Research Fund for the Doctoral Program of Higher Education of China (No. 20120185130001), the Fundamental Research Funds for the Central Universities (No. ZYGX2012YB020), the HBNU Research Funds for Young (No. 700702), and the Project ITR1113.

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