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
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 , such as hyperlenses , perfect lenses , or adiabatic concentrators  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  and spatiotemporal SW NF light localization , 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  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)  is the other technical support.
2. Model and theory
In Fig. 1 there is a cubic vacuum resonant cavity (, where ) 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 , where . The transmitter-receiver polarizations are both and the positions are and respectively, which have deliberately averted the polarization selectivity and the polycentric focus effect  from spatial centrosymmetry and mirror symmetry of the cavity.
First, one pulse signal with a frequency spectrum range of 2~3 is transmitted at the point in the empty cavity without any block, then the receiver at the point will receive the multipath signal where is the channel impulse response (CIR) of the static empty cavity with intensive scattering and represents convolution. Second, the signal from the TR signal sequence of will be transmitted at the point if the SW block to be detected is located at the origin, and then the receiver at the point will receive the signal where represents the CIR of the cavity including the SW block.Eq. (1), represents the perturbation from the SW block and generally satisfiesEquation (2) sets up a challenging condition for SW detection, because it is a mathematic difficulty to accurately extract the phase information of the perturbation from the difference between and directlyEquation (3) can be easily proved under condition of Eq. (2). In fact, the purpose of the extra metal parts positioned in the vicinity of the target in many known approaches [1–4, 7, 8] is to amplify until Eq. (3) is completely invalid. The motivation of TR operation in this paper, however, is to eliminate the background effect of and then can be highlighted relatively.Eq. (4), the phase from the pulse excitation and the one from the intensive background scattering and the one from the weak perturbation of the SW block are linearly dismantled, which will be free from the difficulty of the Eq. (2) condition, and where the first two phases are time-invariant and the third phase is time-variant with the deformation of SW block. Third, with the help of DSTFT, the time-frequency analysis of the 5 SW blocks can be calculated as followsEq. (5) can be simplified as follows
3. Numerical demonstration and discussion
Microwave CST studio  is used to numerically calculate 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 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 has a small synchronization error , the phase deviation of can be written as in frequency domain. If is directly extracted, the deviation can be estimated asEquation (7) actually estimates the deviation of Eq. (3), where the condition of Eq. (2) will probably amplify the deviation of . For instance, given , , and , we can find the deviation of will be over , in other words, the direct extraction of becomes simply incredible. In contrast, if is extracted by the method of the TR PP, the deviation can be calculated asEquation (8) shows that the TR PP will linearly respond to the error in the CIR, which is independent of the condition of the Eq. (2). Therefore, the stability of the TR PP method is inherently better than the one of the direct extraction method. In theory, the stability has nothing to do with the strength of the background interference. Of course, the transceiver’s locations, the size of the resonant cavity and the signal bandwidth etc. are all important in a real application system, because they are all the freedoms of the active detection; in other words, the sensitivity of the detection is a complex function about those active freedoms, but the influence of the background is weak due to TR operation.
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  and Foster  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.
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.
References and links
1. X. Li and M. I. Stockman, “Highly efficient spatiotemporal coherent control in nanoplasmonics on a nanometer-femtosecond scale by time reversal,” Phys. Rev. B 77(19), 195109 (2008). [CrossRef]
6. G. Shvets, S. Trendafilov, J. B. Pendry, and A. Sarychev, “guiding, focusing, and sensing on the subwavelength scale using metallic wire arrays,” Phys. Rev. Lett. 99(5), 053903 (2007). [CrossRef] [PubMed]
8. G.-D. Ge, B.-Z. Wang, D. Wang, D. Zhao, and S. Ding, “Subwavelength array of planar monopoles with complementary split rings based on far-field time reversal,” IEEE Trans. Antenn. Propag. 59(11), 4345–4350 (2011). [CrossRef]
9. H. Yi, J. Long, H. Li, X. He, and T. Yang, “Sub-wavelength full-vectorial profiling of optical focus,” Frontiers in Optics/Laser Science (2013), FW4F.3.
12. P. Sundaralingam, V. Fusco, D. Zelenchuk, and R. Appleby, “Detection of an object in a reverberant environment using direct and differential time reversal,” 6th European Conference on Antennas and Propagation (2012), pp. 1115–1117. [CrossRef]
13. N. Guo, B. M. Sadler, and R. C. Qiu, “Reduced-complexity UWB time-reversal techniques and experimental results,” IEEE Trans. Wirel. Comm. 6(12), 4221–4226 (2007). [CrossRef]
14. B. Wu, W. Cai, M. Alrubaiee, M. Xu, and S. K. Gayen, “Time reversal optical tomography: locating targets in a highly scattering turbid medium,” Opt. Express 19(22), 21956–21976 (2011). [CrossRef] [PubMed]
15. A. Khaleghi, G. El Zein, and I. H. Naqvi, “Demonstration of time-reversal in indoor ultra-wideband communication: time domain measurement,” IEEE ISWCS (2007), pp. 465–468.
16. Y. W. Jin and J. M. F. Moura, “Time-reversal detection using antenna arrays,” IEEE Trans. Signal Process. 57(4), 1396–1414 (2009). [CrossRef]
17. S. Yu, J. Zhang, M. S. Moran, J. Q. Lu, Y. Feng, and X.-H. Hu, “A novel method of diffraction imaging flow cytometry for sizing microspheres,” Opt. Express 20(20), 22245–22251 (2012). [CrossRef] [PubMed]
19. T.-H. Sang, “The self-duality of discrete short-time Fourier transform and its applications,” IEEE Trans. Signal Process. 58(2), 604–612 (2010). [CrossRef]
22. J. Goldstein and J. L. Burch, “Magnetospheric model of subauroral polarization stream,” J. Geophys. Res. 110(A9), A09222 (2005). [CrossRef]
23. J. C. Foster, P. J. Erickson, A. J. Coster, J. Goldstein, and F. J. Rich, “Ionospheric signatures of plasmaspleric tails,” Geophys. Res. Lett. 29(13), 1623 (2002). [CrossRef]