Abstract

Current experimental investigations about time reversal (TR) electromagnetic (EM) fields always depended on TR mirror (TRM). However, EM fields can perform reversal operation invariance in four domains connected by Fourier Transform. A multiplication table and an appropriate operating figure about EM fields’ invariance are derived to describe a series of dual combined operations in the four domains, in which there are at least 10 dual-combination operations different from current TRM operations which can equivalently actualize TR symmetry. Theoretically, centrosymmetric operations of spatial position vector and spatial spectrum vector may have the potential to promote different reversal mirrors.

© 2013 Optical Society of America

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) [17]. 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 [813]. 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 [1316]. 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., (r,t), (r,ω), (k,t), and (k,ω) 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 [E(r,t),H(r,t)] in the linear, lossless and static medium satisfies Maxwell equations and the constitutive relations

{×E(r,t)=B(r,t)t×H(r,t)=D(r,t)t·D(r,t)=0·B(r,t)=0
{E(r,t)=+E(r,ω)ejωtdωH(r,t)=+H(r,ω)ejωtdωε¯¯(r,ω)=ε¯¯(r,ω)μ¯¯(r,ω)=μ¯¯(r,ω)}Lossless&SaticD(r,t)=+ε¯¯(r,ω)·E(r,ω)ejωtdωB(r,t)=+μ¯¯(r,ω)·H(r,ω)ejωtdω}Linear
then a[E(r,t),H(r,t)] must also satisfy Eq. (1) and Eq. (2), here a, that is TRS in time domain. We call a[E(r,t),H(r,t)] TR EM fields of [E(r,t),H(r,t)]. Especially, a[E(r,t),H(r,t)] are not TR EM fields, but that is easily overlooked when TRS is analyzed by Helmholtz equations of EM fields. a[E(r,T0t),H(r,T0t)], T0 can also satisfy Eq. (1) and Eq. (2), and we call that time translation symmetry (TTS) of TR EM fields. Almost all of actual TR EM experiments have to use TTS in the light of the causality between the signal to be reversed and the reversed signal.

If the formal solution [E(r,ω),H(r,ω)] in the linear, lossless and static medium satisfies Maxwell equations and the constitutive relations

{×E(r,ω)=jωB(r,ω)×H(r,ω)=jωD(r,ω)·D(r,ω)=0·B(r,ω)=0
{ε¯¯(r,ω)=ε¯¯(r,ω)μ¯¯(r,ω)=μ¯¯(r,ω)}Lossless&SaticD(r,ω)=ε¯¯(r,ω)·E(r,ω)B(r,ω)=μ¯¯(r,ω)·H(r,ω)}Linear
then a[E(r,ω),H(r,ω)] must also satisfy Eq. (3) and Eq. (4), which is TRS in frequency domain. And the formal solution has phase translation symmetry (PTS) which can be derived from the arbitrariness of Arg(a).

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 δ2/ωμσ, and the average power loss per unit area will be PLossα02/2σδ, here α0σδE0/2 is the peak of the surface current density, E0 is the peak of the surface electric field. Thus, when σ+, we will get δ0+ and PLoss0+. That means EM fields will be compressed in a thin surface layer of the good conductor (i.e. skin effect).

{en×(E2E1)=0en×(H2H1)=α1en·(D2D1)=β1en·(B2B1)=0
where en points medium 2 from conductor 1. If [β1(r,t),α1(r,t),E1,2(r,t),H1,2(r,t)] can satisfy the boundary Eq. (5), TR EM fields a[β1(r,t),α1(r,t),E1,2(r,t),H1,2(r,t)] will also do. Seemingly, TR EM fields should be forbidden by the principle of entropy increase because the directions of α1 and E1 became opposite. But the surface entropy production rate per unit area will be PLoss/T0+ because the good conductor approaches almost lossless. Accordingly we can say that good conductors have the surface TRS, which is important for the realization of TR EM fields experimentally (EM fields are constrained, radiated and received by conductor surfaces in general.).

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,

{LE(r,t)·dl=tSB(r,t)·dSLH(r,t)·dl=tSD(r,t)·dSD(r,t)·dS˜=0B(r,t)·dS˜=0
{LE(r,ω)·dl=jωSB(r,ω)·dSLH(r,ω)·dl=jωSD(r,ω)·dSD(r,ω)·dS˜=0B(r,ω)·dS˜=0
where the constitutive relations are the same as Eq. (2). Based on Eq. (6) and Eq. (7), we can verify that a[E(r,t),H(r,t)] and a[E(r,ω),H(r,ω)] are TR EM fields. Therefore, in the linear, lossless and static media, EM fields have TRS.

3. Four-domain invariance of EM fields

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

{[E(k,t),H(k,t)]=[E(r,t),H(r,t)]ejk·rdr[E(k,ω),H(k,ω)]=[E(k,t),H(k,t)]ejωtdt

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

{Lejk·rE(k,t)·dl=tSejk·rB(k,t)·dSLejk·rH(k,t)·dl=tSejk·rD(k,t)·dSejk·rD(k,t)·dS˜=0ejk·rB(k,t)·dS˜=0
{Lejk·rE(k,ω)·dl=jωSejk·rB(k,ω)·dSLejk·rH(k,ω)·dl=jωSejk·rD(k,ω)·dSejk·rD(k,ω)·dS˜=0ejk·rB(k,ω)·dS˜=0

[E(k,t),H(k,t)] and [E(k,ω),H(k,ω)] will respectively have TR EM fields a[E(k,t),H(k,t)] in the (k,t) domain and a[E(k,ω),H(k,ω)] in the (k,ω) 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

{LE(r,t)·dl=tSB(r,t)·dSLH(r,t)·dl=tSD(r,t)·dSD(r,t)·dS˜=0B(r,t)·dS˜=0
{LE(r,ω)·dl=jωSB(r,ω)·dSLH(r,ω)·dl=jωSD(r,ω)·dSD(r,ω)·dS˜=0B(r,ω)·dS˜=0

A chiral operator C¯¯ can be defined to express the transition of EM fields between a right-handed frame and a left-handed frame

C¯¯[E,H]=[E,H],C¯¯2=I¯¯

Similarly, we can introduce conjugation operator P¯¯, time reverse operator T¯¯, negative frequency operator N¯¯, space vector centrosymmetry operator Χ¯¯ and space spectrum vector centrosymmetry operator K¯¯, as follows

{P¯¯[E,H]=[E,H]T¯¯(t)=(t),T¯¯2=I¯¯N¯¯(ω)=(ω),N¯¯2=I¯¯Χ¯¯(r)=(r),Χ¯¯2=I¯¯K¯¯(k)=(k),K¯¯2=I¯¯
where I¯¯ is an identical operator. Four-domain invariance of EM fields can be expressed as the dual operating invariance,
{[E(r,t),H(r,t)]=T¯¯(r,t)C¯¯(r,t)[E(r,t),H(r,t)][E(r,ω),H(r,ω)]=P¯¯(r,ω)C¯¯(r,ω)[E(r,ω),H(r,ω)][E(k,t),H(k,t)]=T¯¯(k,t)C¯¯(k,t)[E(k,t),H(k,t)][E(k,ω),H(k,ω)]=P¯¯(k,ω)C¯¯(k,ω)[E(k,ω),H(k,ω)]
where the subscripts identify the operating domains. And by using the principle of linear superposition, the four formal solutions are equivalent.

In fact, if the symmetries of EM fields are discussed on the operator set Ο={I¯¯,C¯¯,T¯¯,P¯¯,N¯¯,Χ¯¯,K¯¯}, the symmetry operations will not be only four, i.e., T¯¯(r,t)C¯¯(r,t), P¯¯(r,ω)C¯¯(r,ω), T¯¯(k,t)C¯¯(k,t), and P¯¯(k,ω)C¯¯(k,ω). Table 1 depicts the dual multiplications induced by the operator set Ο, which has main diagonal symmetry due to the reciprocity of any two operators. By the definition, eleven operations (C¯¯Κ¯¯, C¯¯T¯¯, C¯¯N¯¯, C¯¯Χ¯¯, Κ¯¯T¯¯, Κ¯¯N¯¯, Κ¯¯Χ¯¯, T¯¯N¯¯, T¯¯Χ¯¯, N¯¯Χ¯¯, I¯¯) have the invariance of EM fields in linear and lossless media, and the seven undetermined operations (P¯¯, C¯¯P¯¯, Κ¯¯P¯¯, T¯¯P¯¯, N¯¯P¯¯, Χ¯¯P¯¯,P¯¯P¯¯) may have the invariance determined by the operating domains, and the five remaining operations (C¯¯, Κ¯¯, T¯¯, N¯¯, Χ¯¯) have no invariance. In the whole set Ο except for 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 P¯¯ must be in the same domain, then twice operations will perform P¯¯2=I¯¯.

Tables Icon

Table 1. Dual Multiplication Table Induced on the Operator Set Ο

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 T¯¯(r,t) or P¯¯(r,ω) will lead to SS, so we need the operator C¯¯ to perform [E,H][E,H] for the correction of the reversed chirality. The dual combined operators T¯¯(r,t)C¯¯(r,t) and P¯¯(r,ω)C¯¯(r,ω) 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 P¯¯I¯¯ to depict 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 rr 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

{Lejk·rE(k,ω)·dl=jωSejk·rB(k,ω)·dSLejk·rH(k,ω)·dl=jωSejk·rD(k,ω)·dSejk·rD(k,ω)·dS˜=0ejk·rB(k,ω)·dS˜=0

Accordingly, if the formal solution [E(k,ω),H(k,ω)] satisfies Eq. (10), then P¯¯(k,ω)[E(k,ω),H(k,ω)]=[E(k,ω),H(k,ω)] will also do, which directly verifies the invariance of P¯¯(k,ω)I¯¯ in Fig. 1. By means of the invariance of P¯¯(k,ω), the invariance of the dual combined operator P¯¯(r,ω)X¯¯(r,t) can be indirectly verified as follow

P¯¯(r,ω)X¯¯(r,t)++++[E(k,ω),H(k,ω)]ejk·rejωtdkdω=+{P¯¯(r,ω)+++[E(k,ω),H(k,ω)]ejk·rdk}ejωtdω=++++{P¯¯(k,ω)[E(k,ω),H(k,ω)]}ejk·rejωtdkdω

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 X¯¯(r,ω)P¯¯(r,ω), which can reradiate incident EM signal in the arrival direction regardless of the array orientation relative to the direction [2224]. According to the definition, EM fields after X¯¯(r,ω)P¯¯(r,ω) operation can still propagate in the same space (r,ω). The operator P¯¯(r,ω) will lead to SS, which has enough provided the retrodirective mechanism. But Fig. 1 shows that only P¯¯(r,ω) cannot perform invariance! Hence, P¯¯(r,ω) needs X¯¯(r,ω) 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 P¯¯(r,ω) operation and X¯¯(r,ω) 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 (r,ω).

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

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 [1113, 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.

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.

References and links

1. G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004). [CrossRef]   [PubMed]  

2. R. Carminati, R. Pierrat, J. de Rosny, and M. Fink, “Theory of the time reversal cavity for electromagnetic fields,” Opt. Lett. 32(21), 3107–3109 (2007). [CrossRef]   [PubMed]  

3. D. H. Chambers and J. G. Berryman, “Time-reversal analysis for scatterer characterization,” Phys. Rev. Lett. 92(2), 023902 (2004). [CrossRef]   [PubMed]  

4. 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). [CrossRef]  

5. P. Sundaralingam and V. Fusco, “Effect of quantisation and under sampling on time reversal spatial and temporal focusing in highly reverberant environment,” Proceedings of the 41st European Microwave Conference (2011).

6. Y. M. Chen and B.-Z. Wang, “Seeing time-reversal transmission characteristics through kinetic anti-ferromagnetic Ising chain,” Chinese Phys. B 21(2), 026401 (2012). [CrossRef]  

7. V. Dmitriev, “Space-time reversal symmetry properties of electromagnetic green’s tensors for complex and bianisotropic media,” PIER 48, 145–184 (2004). [CrossRef]  

8. T. Strohmer, M. Emami, J. Hansen, G. Papanicolaou, and A. J. Paulraj, “Application of time-reversal with MMSE equalizer to UWB communications,” IEEE Global Telecommunications Conferenc 5, 3123–3127 (2004).

9. 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]  

10. G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-Field time reversal,” Science 315(5815), 1120–1122 (2007). [CrossRef]   [PubMed]  

11. I. H. Naqvi, G. E. Zein, G. Lerosey, J. de Rosny, P. Besnier, A. Tourin, and M. Fink, “Experimental validation of time reversal ultra wide-band communication system for high data rates,” IET Microw. J. Antenna Propag. 4(5), 643–650 (2010). [CrossRef]  

12. C. Zhou, N. Guo, and R. C. Qiu, “Time reversed ultra-wideband (UWB) multiple-input multiple-output (MIMO) based on measured spatial channels,” IEEE Trans. Vehicular Technol. 58(6), 2884–2898 (2009). [CrossRef]  

13. R. C. Qiu, C. Zhou, N. Guo, and J. Q. Zhang, “Time reversal with MISO for ultra-wideband communications: experimental results,” IEEE Antennas Wirel. Propag. Lett. 5(1), 269–273 (2006). [CrossRef]  

14. P. Sundaralingam, V. Fusco, and N. B. Buchanan, “Spatial field localisation and data transfer in a reverberant environment using analog real time phase conjugation,” Proceedings of the 40th European Microwave Conference (Paris, 2010), pp. 565–568.

15. Y. Zhang, X. Z. Zhu, and Z. Q. Zhao, “Application of TRM in detection of metal buried in wall,” International Conference on Computational Problem-Solving (2012), pp. 287–290.

16. J. Feng, C. Liao, L. L. Chen, and H. J. Zhou, “Amplification of electromagnetic waves by time reversal mirror in a leaky reverberation chamber,” International Conference on Microwave and Millimeter Wave Technology (5, 2012), pp. 1–4.

17. D. A. B. Miller, “Time reversal of optical pulses by four-wave mixing,” Opt. Lett. 5(7), 300–302 (1980). [CrossRef]   [PubMed]  

18. D. Marom, D. Panasenko, R. Rokitski, P.-C. Sun, and Y. Fainman, “Time reversal of ultrafast waveforms by wave mixing of spectrally decomposed waves,” Opt. Lett. 25(2), 132–134 (2000). [CrossRef]   [PubMed]  

19. O. Kuzucu, Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Spectral phase conjugation via temporal imaging,” Opt. Express 17(22), 20605–20614 (2009). [CrossRef]   [PubMed]  

20. Y. Sivan and J. B. Pendry, “Broadband time-reversal of optical pulses using a switchable photonic-crystal mirror,” Opt. Express 19(15), 14502–14507 (2011). [CrossRef]   [PubMed]  

21. E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic Press, 1959), Chap. 26.

22. A. Yu. Butrym, O. V. Kazanskiy, and N. N. Kolchigin, “Van Atta’s array consist of tapered slot antennas for wideband pulse signals,” 5th International Conference on Antenna Theory and Techniques (Ukraine, 2005), pp. 221–223.

23. K. W. Wong, L. Chiu, and Q. Xue, “A 2-D Van Atta Array Using Star-Shaped Antenna Elements,” IEEE Trans. Antenn. Propag. 55(4), 1204–1206 (2007). [CrossRef]  

24. D. D. Ivanchenko, O. V. Kazansky, and N. N. Kolchigin, “About influence of mutual coupling on pattern and wave form of pulse reradiated by Van Atta's array of tapered slot antennas,” 6th International Conference on Ultrawideband and Ultrashort Impulse Signals (Sevastopol, Ukraine, 2012), pp. 81–82. [CrossRef]  

25. A. Khaleghi, “Measurement and analysis of ultra-wideband time reversal for indoor propagation channels,” Wirel. Pers. Commun. 54(2), 307–320 (2010). [CrossRef]  

References

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  1. G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
    [CrossRef] [PubMed]
  2. R. Carminati, R. Pierrat, J. de Rosny, and M. Fink, “Theory of the time reversal cavity for electromagnetic fields,” Opt. Lett. 32(21), 3107–3109 (2007).
    [CrossRef] [PubMed]
  3. D. H. Chambers and J. G. Berryman, “Time-reversal analysis for scatterer characterization,” Phys. Rev. Lett. 92(2), 023902 (2004).
    [CrossRef] [PubMed]
  4. 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).
    [CrossRef]
  5. P. Sundaralingam and V. Fusco, “Effect of quantisation and under sampling on time reversal spatial and temporal focusing in highly reverberant environment,” Proceedings of the 41st European Microwave Conference (2011).
  6. Y. M. Chen and B.-Z. Wang, “Seeing time-reversal transmission characteristics through kinetic anti-ferromagnetic Ising chain,” Chinese Phys. B 21(2), 026401 (2012).
    [CrossRef]
  7. V. Dmitriev, “Space-time reversal symmetry properties of electromagnetic green’s tensors for complex and bianisotropic media,” PIER 48, 145–184 (2004).
    [CrossRef]
  8. T. Strohmer, M. Emami, J. Hansen, G. Papanicolaou, and A. J. Paulraj, “Application of time-reversal with MMSE equalizer to UWB communications,” IEEE Global Telecommunications Conferenc 5, 3123–3127 (2004).
  9. 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]
  10. G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-Field time reversal,” Science 315(5815), 1120–1122 (2007).
    [CrossRef] [PubMed]
  11. I. H. Naqvi, G. E. Zein, G. Lerosey, J. de Rosny, P. Besnier, A. Tourin, and M. Fink, “Experimental validation of time reversal ultra wide-band communication system for high data rates,” IET Microw. J. Antenna Propag. 4(5), 643–650 (2010).
    [CrossRef]
  12. C. Zhou, N. Guo, and R. C. Qiu, “Time reversed ultra-wideband (UWB) multiple-input multiple-output (MIMO) based on measured spatial channels,” IEEE Trans. Vehicular Technol. 58(6), 2884–2898 (2009).
    [CrossRef]
  13. R. C. Qiu, C. Zhou, N. Guo, and J. Q. Zhang, “Time reversal with MISO for ultra-wideband communications: experimental results,” IEEE Antennas Wirel. Propag. Lett. 5(1), 269–273 (2006).
    [CrossRef]
  14. P. Sundaralingam, V. Fusco, and N. B. Buchanan, “Spatial field localisation and data transfer in a reverberant environment using analog real time phase conjugation,” Proceedings of the 40th European Microwave Conference (Paris, 2010), pp. 565–568.
  15. Y. Zhang, X. Z. Zhu, and Z. Q. Zhao, “Application of TRM in detection of metal buried in wall,” International Conference on Computational Problem-Solving (2012), pp. 287–290.
  16. J. Feng, C. Liao, L. L. Chen, and H. J. Zhou, “Amplification of electromagnetic waves by time reversal mirror in a leaky reverberation chamber,” International Conference on Microwave and Millimeter Wave Technology (5, 2012), pp. 1–4.
  17. D. A. B. Miller, “Time reversal of optical pulses by four-wave mixing,” Opt. Lett. 5(7), 300–302 (1980).
    [CrossRef] [PubMed]
  18. D. Marom, D. Panasenko, R. Rokitski, P.-C. Sun, and Y. Fainman, “Time reversal of ultrafast waveforms by wave mixing of spectrally decomposed waves,” Opt. Lett. 25(2), 132–134 (2000).
    [CrossRef] [PubMed]
  19. O. Kuzucu, Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Spectral phase conjugation via temporal imaging,” Opt. Express 17(22), 20605–20614 (2009).
    [CrossRef] [PubMed]
  20. Y. Sivan and J. B. Pendry, “Broadband time-reversal of optical pulses using a switchable photonic-crystal mirror,” Opt. Express 19(15), 14502–14507 (2011).
    [CrossRef] [PubMed]
  21. E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic Press, 1959), Chap. 26.
  22. A. Yu. Butrym, O. V. Kazanskiy, and N. N. Kolchigin, “Van Atta’s array consist of tapered slot antennas for wideband pulse signals,” 5th International Conference on Antenna Theory and Techniques (Ukraine, 2005), pp. 221–223.
  23. K. W. Wong, L. Chiu, and Q. Xue, “A 2-D Van Atta Array Using Star-Shaped Antenna Elements,” IEEE Trans. Antenn. Propag. 55(4), 1204–1206 (2007).
    [CrossRef]
  24. D. D. Ivanchenko, O. V. Kazansky, and N. N. Kolchigin, “About influence of mutual coupling on pattern and wave form of pulse reradiated by Van Atta's array of tapered slot antennas,” 6th International Conference on Ultrawideband and Ultrashort Impulse Signals (Sevastopol, Ukraine, 2012), pp. 81–82.
    [CrossRef]
  25. A. Khaleghi, “Measurement and analysis of ultra-wideband time reversal for indoor propagation channels,” Wirel. Pers. Commun. 54(2), 307–320 (2010).
    [CrossRef]

2012 (1)

Y. M. Chen and B.-Z. Wang, “Seeing time-reversal transmission characteristics through kinetic anti-ferromagnetic Ising chain,” Chinese Phys. B 21(2), 026401 (2012).
[CrossRef]

2011 (2)

2010 (2)

A. Khaleghi, “Measurement and analysis of ultra-wideband time reversal for indoor propagation channels,” Wirel. Pers. Commun. 54(2), 307–320 (2010).
[CrossRef]

I. H. Naqvi, G. E. Zein, G. Lerosey, J. de Rosny, P. Besnier, A. Tourin, and M. Fink, “Experimental validation of time reversal ultra wide-band communication system for high data rates,” IET Microw. J. Antenna Propag. 4(5), 643–650 (2010).
[CrossRef]

2009 (2)

C. Zhou, N. Guo, and R. C. Qiu, “Time reversed ultra-wideband (UWB) multiple-input multiple-output (MIMO) based on measured spatial channels,” IEEE Trans. Vehicular Technol. 58(6), 2884–2898 (2009).
[CrossRef]

O. Kuzucu, Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Spectral phase conjugation via temporal imaging,” Opt. Express 17(22), 20605–20614 (2009).
[CrossRef] [PubMed]

2007 (3)

K. W. Wong, L. Chiu, and Q. Xue, “A 2-D Van Atta Array Using Star-Shaped Antenna Elements,” IEEE Trans. Antenn. Propag. 55(4), 1204–1206 (2007).
[CrossRef]

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-Field time reversal,” Science 315(5815), 1120–1122 (2007).
[CrossRef] [PubMed]

R. Carminati, R. Pierrat, J. de Rosny, and M. Fink, “Theory of the time reversal cavity for electromagnetic fields,” Opt. Lett. 32(21), 3107–3109 (2007).
[CrossRef] [PubMed]

2006 (1)

R. C. Qiu, C. Zhou, N. Guo, and J. Q. Zhang, “Time reversal with MISO for ultra-wideband communications: experimental results,” IEEE Antennas Wirel. Propag. Lett. 5(1), 269–273 (2006).
[CrossRef]

2004 (4)

D. H. Chambers and J. G. Berryman, “Time-reversal analysis for scatterer characterization,” Phys. Rev. Lett. 92(2), 023902 (2004).
[CrossRef] [PubMed]

V. Dmitriev, “Space-time reversal symmetry properties of electromagnetic green’s tensors for complex and bianisotropic media,” PIER 48, 145–184 (2004).
[CrossRef]

T. Strohmer, M. Emami, J. Hansen, G. Papanicolaou, and A. J. Paulraj, “Application of time-reversal with MMSE equalizer to UWB communications,” IEEE Global Telecommunications Conferenc 5, 3123–3127 (2004).

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[CrossRef] [PubMed]

2000 (1)

1980 (1)

Alrubaiee, M.

Appleby, R.

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).
[CrossRef]

Berryman, J. G.

D. H. Chambers and J. G. Berryman, “Time-reversal analysis for scatterer characterization,” Phys. Rev. Lett. 92(2), 023902 (2004).
[CrossRef] [PubMed]

Besnier, P.

I. H. Naqvi, G. E. Zein, G. Lerosey, J. de Rosny, P. Besnier, A. Tourin, and M. Fink, “Experimental validation of time reversal ultra wide-band communication system for high data rates,” IET Microw. J. Antenna Propag. 4(5), 643–650 (2010).
[CrossRef]

Buchanan, N. B.

P. Sundaralingam, V. Fusco, and N. B. Buchanan, “Spatial field localisation and data transfer in a reverberant environment using analog real time phase conjugation,” Proceedings of the 40th European Microwave Conference (Paris, 2010), pp. 565–568.

Cai, W.

Carminati, R.

Chambers, D. H.

D. H. Chambers and J. G. Berryman, “Time-reversal analysis for scatterer characterization,” Phys. Rev. Lett. 92(2), 023902 (2004).
[CrossRef] [PubMed]

Chen, L. L.

J. Feng, C. Liao, L. L. Chen, and H. J. Zhou, “Amplification of electromagnetic waves by time reversal mirror in a leaky reverberation chamber,” International Conference on Microwave and Millimeter Wave Technology (5, 2012), pp. 1–4.

Chen, Y. M.

Y. M. Chen and B.-Z. Wang, “Seeing time-reversal transmission characteristics through kinetic anti-ferromagnetic Ising chain,” Chinese Phys. B 21(2), 026401 (2012).
[CrossRef]

Chiu, L.

K. W. Wong, L. Chiu, and Q. Xue, “A 2-D Van Atta Array Using Star-Shaped Antenna Elements,” IEEE Trans. Antenn. Propag. 55(4), 1204–1206 (2007).
[CrossRef]

de Rosny, J.

I. H. Naqvi, G. E. Zein, G. Lerosey, J. de Rosny, P. Besnier, A. Tourin, and M. Fink, “Experimental validation of time reversal ultra wide-band communication system for high data rates,” IET Microw. J. Antenna Propag. 4(5), 643–650 (2010).
[CrossRef]

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-Field time reversal,” Science 315(5815), 1120–1122 (2007).
[CrossRef] [PubMed]

R. Carminati, R. Pierrat, J. de Rosny, and M. Fink, “Theory of the time reversal cavity for electromagnetic fields,” Opt. Lett. 32(21), 3107–3109 (2007).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[CrossRef] [PubMed]

Derode, A.

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[CrossRef] [PubMed]

Dmitriev, V.

V. Dmitriev, “Space-time reversal symmetry properties of electromagnetic green’s tensors for complex and bianisotropic media,” PIER 48, 145–184 (2004).
[CrossRef]

Emami, M.

T. Strohmer, M. Emami, J. Hansen, G. Papanicolaou, and A. J. Paulraj, “Application of time-reversal with MMSE equalizer to UWB communications,” IEEE Global Telecommunications Conferenc 5, 3123–3127 (2004).

Fainman, Y.

Feng, J.

J. Feng, C. Liao, L. L. Chen, and H. J. Zhou, “Amplification of electromagnetic waves by time reversal mirror in a leaky reverberation chamber,” International Conference on Microwave and Millimeter Wave Technology (5, 2012), pp. 1–4.

Fink, M.

I. H. Naqvi, G. E. Zein, G. Lerosey, J. de Rosny, P. Besnier, A. Tourin, and M. Fink, “Experimental validation of time reversal ultra wide-band communication system for high data rates,” IET Microw. J. Antenna Propag. 4(5), 643–650 (2010).
[CrossRef]

R. Carminati, R. Pierrat, J. de Rosny, and M. Fink, “Theory of the time reversal cavity for electromagnetic fields,” Opt. Lett. 32(21), 3107–3109 (2007).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-Field time reversal,” Science 315(5815), 1120–1122 (2007).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[CrossRef] [PubMed]

Foster, M. A.

Fusco, V.

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).
[CrossRef]

P. Sundaralingam and V. Fusco, “Effect of quantisation and under sampling on time reversal spatial and temporal focusing in highly reverberant environment,” Proceedings of the 41st European Microwave Conference (2011).

P. Sundaralingam, V. Fusco, and N. B. Buchanan, “Spatial field localisation and data transfer in a reverberant environment using analog real time phase conjugation,” Proceedings of the 40th European Microwave Conference (Paris, 2010), pp. 565–568.

Gaeta, A. L.

Gayen, S. K.

Guo, N.

C. Zhou, N. Guo, and R. C. Qiu, “Time reversed ultra-wideband (UWB) multiple-input multiple-output (MIMO) based on measured spatial channels,” IEEE Trans. Vehicular Technol. 58(6), 2884–2898 (2009).
[CrossRef]

R. C. Qiu, C. Zhou, N. Guo, and J. Q. Zhang, “Time reversal with MISO for ultra-wideband communications: experimental results,” IEEE Antennas Wirel. Propag. Lett. 5(1), 269–273 (2006).
[CrossRef]

Hansen, J.

T. Strohmer, M. Emami, J. Hansen, G. Papanicolaou, and A. J. Paulraj, “Application of time-reversal with MMSE equalizer to UWB communications,” IEEE Global Telecommunications Conferenc 5, 3123–3127 (2004).

Khaleghi, A.

A. Khaleghi, “Measurement and analysis of ultra-wideband time reversal for indoor propagation channels,” Wirel. Pers. Commun. 54(2), 307–320 (2010).
[CrossRef]

Kuzucu, O.

Lerosey, G.

I. H. Naqvi, G. E. Zein, G. Lerosey, J. de Rosny, P. Besnier, A. Tourin, and M. Fink, “Experimental validation of time reversal ultra wide-band communication system for high data rates,” IET Microw. J. Antenna Propag. 4(5), 643–650 (2010).
[CrossRef]

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-Field time reversal,” Science 315(5815), 1120–1122 (2007).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[CrossRef] [PubMed]

Liao, C.

J. Feng, C. Liao, L. L. Chen, and H. J. Zhou, “Amplification of electromagnetic waves by time reversal mirror in a leaky reverberation chamber,” International Conference on Microwave and Millimeter Wave Technology (5, 2012), pp. 1–4.

Lipson, M.

Marom, D.

Miller, D. A. B.

Montaldo, G.

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[CrossRef] [PubMed]

Naqvi, I. H.

I. H. Naqvi, G. E. Zein, G. Lerosey, J. de Rosny, P. Besnier, A. Tourin, and M. Fink, “Experimental validation of time reversal ultra wide-band communication system for high data rates,” IET Microw. J. Antenna Propag. 4(5), 643–650 (2010).
[CrossRef]

Okawachi, Y.

Panasenko, D.

Papanicolaou, G.

T. Strohmer, M. Emami, J. Hansen, G. Papanicolaou, and A. J. Paulraj, “Application of time-reversal with MMSE equalizer to UWB communications,” IEEE Global Telecommunications Conferenc 5, 3123–3127 (2004).

Paulraj, A. J.

T. Strohmer, M. Emami, J. Hansen, G. Papanicolaou, and A. J. Paulraj, “Application of time-reversal with MMSE equalizer to UWB communications,” IEEE Global Telecommunications Conferenc 5, 3123–3127 (2004).

Pendry, J. B.

Pierrat, R.

Qiu, R. C.

C. Zhou, N. Guo, and R. C. Qiu, “Time reversed ultra-wideband (UWB) multiple-input multiple-output (MIMO) based on measured spatial channels,” IEEE Trans. Vehicular Technol. 58(6), 2884–2898 (2009).
[CrossRef]

R. C. Qiu, C. Zhou, N. Guo, and J. Q. Zhang, “Time reversal with MISO for ultra-wideband communications: experimental results,” IEEE Antennas Wirel. Propag. Lett. 5(1), 269–273 (2006).
[CrossRef]

Rokitski, R.

Salem, R.

Sivan, Y.

Strohmer, T.

T. Strohmer, M. Emami, J. Hansen, G. Papanicolaou, and A. J. Paulraj, “Application of time-reversal with MMSE equalizer to UWB communications,” IEEE Global Telecommunications Conferenc 5, 3123–3127 (2004).

Sun, P.-C.

Sundaralingam, P.

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).
[CrossRef]

P. Sundaralingam and V. Fusco, “Effect of quantisation and under sampling on time reversal spatial and temporal focusing in highly reverberant environment,” Proceedings of the 41st European Microwave Conference (2011).

P. Sundaralingam, V. Fusco, and N. B. Buchanan, “Spatial field localisation and data transfer in a reverberant environment using analog real time phase conjugation,” Proceedings of the 40th European Microwave Conference (Paris, 2010), pp. 565–568.

Tourin, A.

I. H. Naqvi, G. E. Zein, G. Lerosey, J. de Rosny, P. Besnier, A. Tourin, and M. Fink, “Experimental validation of time reversal ultra wide-band communication system for high data rates,” IET Microw. J. Antenna Propag. 4(5), 643–650 (2010).
[CrossRef]

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-Field time reversal,” Science 315(5815), 1120–1122 (2007).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[CrossRef] [PubMed]

Turner-Foster, A. C.

Wang, B.-Z.

Y. M. Chen and B.-Z. Wang, “Seeing time-reversal transmission characteristics through kinetic anti-ferromagnetic Ising chain,” Chinese Phys. B 21(2), 026401 (2012).
[CrossRef]

Wong, K. W.

K. W. Wong, L. Chiu, and Q. Xue, “A 2-D Van Atta Array Using Star-Shaped Antenna Elements,” IEEE Trans. Antenn. Propag. 55(4), 1204–1206 (2007).
[CrossRef]

Wu, B.

Xu, M.

Xue, Q.

K. W. Wong, L. Chiu, and Q. Xue, “A 2-D Van Atta Array Using Star-Shaped Antenna Elements,” IEEE Trans. Antenn. Propag. 55(4), 1204–1206 (2007).
[CrossRef]

Zein, G. E.

I. H. Naqvi, G. E. Zein, G. Lerosey, J. de Rosny, P. Besnier, A. Tourin, and M. Fink, “Experimental validation of time reversal ultra wide-band communication system for high data rates,” IET Microw. J. Antenna Propag. 4(5), 643–650 (2010).
[CrossRef]

Zelenchuk, D.

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).
[CrossRef]

Zhang, J. Q.

R. C. Qiu, C. Zhou, N. Guo, and J. Q. Zhang, “Time reversal with MISO for ultra-wideband communications: experimental results,” IEEE Antennas Wirel. Propag. Lett. 5(1), 269–273 (2006).
[CrossRef]

Zhang, Y.

Y. Zhang, X. Z. Zhu, and Z. Q. Zhao, “Application of TRM in detection of metal buried in wall,” International Conference on Computational Problem-Solving (2012), pp. 287–290.

Zhao, Z. Q.

Y. Zhang, X. Z. Zhu, and Z. Q. Zhao, “Application of TRM in detection of metal buried in wall,” International Conference on Computational Problem-Solving (2012), pp. 287–290.

Zhou, C.

C. Zhou, N. Guo, and R. C. Qiu, “Time reversed ultra-wideband (UWB) multiple-input multiple-output (MIMO) based on measured spatial channels,” IEEE Trans. Vehicular Technol. 58(6), 2884–2898 (2009).
[CrossRef]

R. C. Qiu, C. Zhou, N. Guo, and J. Q. Zhang, “Time reversal with MISO for ultra-wideband communications: experimental results,” IEEE Antennas Wirel. Propag. Lett. 5(1), 269–273 (2006).
[CrossRef]

Zhou, H. J.

J. Feng, C. Liao, L. L. Chen, and H. J. Zhou, “Amplification of electromagnetic waves by time reversal mirror in a leaky reverberation chamber,” International Conference on Microwave and Millimeter Wave Technology (5, 2012), pp. 1–4.

Zhu, X. Z.

Y. Zhang, X. Z. Zhu, and Z. Q. Zhao, “Application of TRM in detection of metal buried in wall,” International Conference on Computational Problem-Solving (2012), pp. 287–290.

Chinese Phys. B (1)

Y. M. Chen and B.-Z. Wang, “Seeing time-reversal transmission characteristics through kinetic anti-ferromagnetic Ising chain,” Chinese Phys. B 21(2), 026401 (2012).
[CrossRef]

IEEE Antennas Wirel. Propag. Lett. (1)

R. C. Qiu, C. Zhou, N. Guo, and J. Q. Zhang, “Time reversal with MISO for ultra-wideband communications: experimental results,” IEEE Antennas Wirel. Propag. Lett. 5(1), 269–273 (2006).
[CrossRef]

IEEE Global Telecommunications Conferenc (1)

T. Strohmer, M. Emami, J. Hansen, G. Papanicolaou, and A. J. Paulraj, “Application of time-reversal with MMSE equalizer to UWB communications,” IEEE Global Telecommunications Conferenc 5, 3123–3127 (2004).

IEEE Trans. Antenn. Propag. (1)

K. W. Wong, L. Chiu, and Q. Xue, “A 2-D Van Atta Array Using Star-Shaped Antenna Elements,” IEEE Trans. Antenn. Propag. 55(4), 1204–1206 (2007).
[CrossRef]

IEEE Trans. Vehicular Technol. (1)

C. Zhou, N. Guo, and R. C. Qiu, “Time reversed ultra-wideband (UWB) multiple-input multiple-output (MIMO) based on measured spatial channels,” IEEE Trans. Vehicular Technol. 58(6), 2884–2898 (2009).
[CrossRef]

IET Microw. J. Antenna Propag. (1)

I. H. Naqvi, G. E. Zein, G. Lerosey, J. de Rosny, P. Besnier, A. Tourin, and M. Fink, “Experimental validation of time reversal ultra wide-band communication system for high data rates,” IET Microw. J. Antenna Propag. 4(5), 643–650 (2010).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. Lett. (2)

D. H. Chambers and J. G. Berryman, “Time-reversal analysis for scatterer characterization,” Phys. Rev. Lett. 92(2), 023902 (2004).
[CrossRef] [PubMed]

G. Lerosey, J. de Rosny, A. Tourin, A. Derode, G. Montaldo, and M. Fink, “Time reversal of electromagnetic waves,” Phys. Rev. Lett. 92(19), 193904 (2004).
[CrossRef] [PubMed]

PIER (1)

V. Dmitriev, “Space-time reversal symmetry properties of electromagnetic green’s tensors for complex and bianisotropic media,” PIER 48, 145–184 (2004).
[CrossRef]

Science (1)

G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “Focusing beyond the diffraction limit with far-Field time reversal,” Science 315(5815), 1120–1122 (2007).
[CrossRef] [PubMed]

Wirel. Pers. Commun. (1)

A. Khaleghi, “Measurement and analysis of ultra-wideband time reversal for indoor propagation channels,” Wirel. Pers. Commun. 54(2), 307–320 (2010).
[CrossRef]

Other (8)

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).
[CrossRef]

P. Sundaralingam and V. Fusco, “Effect of quantisation and under sampling on time reversal spatial and temporal focusing in highly reverberant environment,” Proceedings of the 41st European Microwave Conference (2011).

P. Sundaralingam, V. Fusco, and N. B. Buchanan, “Spatial field localisation and data transfer in a reverberant environment using analog real time phase conjugation,” Proceedings of the 40th European Microwave Conference (Paris, 2010), pp. 565–568.

Y. Zhang, X. Z. Zhu, and Z. Q. Zhao, “Application of TRM in detection of metal buried in wall,” International Conference on Computational Problem-Solving (2012), pp. 287–290.

J. Feng, C. Liao, L. L. Chen, and H. J. Zhou, “Amplification of electromagnetic waves by time reversal mirror in a leaky reverberation chamber,” International Conference on Microwave and Millimeter Wave Technology (5, 2012), pp. 1–4.

D. D. Ivanchenko, O. V. Kazansky, and N. N. Kolchigin, “About influence of mutual coupling on pattern and wave form of pulse reradiated by Van Atta's array of tapered slot antennas,” 6th International Conference on Ultrawideband and Ultrashort Impulse Signals (Sevastopol, Ukraine, 2012), pp. 81–82.
[CrossRef]

E. P. Wigner, Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic Press, 1959), Chap. 26.

A. Yu. Butrym, O. V. Kazanskiy, and N. N. Kolchigin, “Van Atta’s array consist of tapered slot antennas for wideband pulse signals,” 5th International Conference on Antenna Theory and Techniques (Ukraine, 2005), pp. 221–223.

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Figures (1)

Fig. 1
Fig. 1

The invariance of seven undetermined operations in appropriate operating domains.

Tables (1)

Tables Icon

Table 1 Dual Multiplication Table Induced on the Operator Set Ο

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

{ ×E( r,t )= B( r,t ) t ×H( r,t )= D( r,t ) t ·D( r,t )=0 ·B( r,t )=0
{ E( r,t ) = + E( r,ω ) e jωt dω H( r,t ) = + H( r,ω ) e jωt dω ε ¯ ¯ ( r,ω )= ε ¯ ¯ ( r,ω ) μ ¯ ¯ ( r,ω )= μ ¯ ¯ ( r,ω ) }Lossless&Satic D( r,t )= + ε ¯ ¯ ( r,ω )·E( r,ω ) e jωt dω B( r,t )= + μ ¯ ¯ ( r,ω )·H( r,ω ) e jωt dω }Linear
{ ×E( r,ω )=jωB( r,ω ) ×H( r,ω )=jωD( r,ω ) ·D( r,ω )=0 ·B( r,ω )=0
{ ε ¯ ¯ ( r,ω )= ε ¯ ¯ ( r,ω ) μ ¯ ¯ ( r,ω )= μ ¯ ¯ ( r,ω ) }Lossless&Satic D( r,ω )= ε ¯ ¯ ( r,ω )·E( r,ω ) B( r,ω )= μ ¯ ¯ ( r,ω )·H( r,ω ) }Linear
{ e n ×( E 2 E 1 )=0 e n ×( H 2 H 1 )= α 1 e n ·( D 2 D 1 )= β 1 e n ·( B 2 B 1 )=0
{ L E( r,t )·dl = t S B( r,t )·dS L H( r,t )·dl = t S D( r,t )·dS D( r,t )·d S ˜ =0 B( r,t )·d S ˜ =0
{ L E( r,ω )·dl =jω S B( r,ω )·dS L H( r,ω )·dl =jω S D( r,ω )·dS D( r,ω )·d S ˜ =0 B( r,ω )·d S ˜ =0
{ [ E( k,t ),H( k,t ) ]= [ E( r,t ),H( r,t ) ] e jk·r dr [ E( k,ω ),H( k,ω ) ]= [ E( k,t ),H( k,t ) ] e jωt dt
{ L e jk·r E( k,t )·dl = t S e jk·r B( k,t )·dS L e jk·r H( k,t )·dl = t S e jk·r D( k,t )·dS e jk·r D( k,t )·d S ˜ =0 e jk·r B( k,t )·d S ˜ =0
{ L e jk·r E( k,ω )·dl =jω S e jk·r B( k,ω )·dS L e jk·r H( k,ω )·dl =jω S e jk·r D( k,ω )·dS e jk·r D( k,ω )·d S ˜ =0 e jk·r B( k,ω )·d S ˜ =0
{ L E( r,t )·dl = t S B( r,t )·dS L H( r,t )·dl = t S D( r,t )·dS D( r,t )·d S ˜ =0 B( r,t )·d S ˜ =0
{ L E( r,ω )·dl =jω S B( r,ω )·dS L H( r,ω )·dl =jω S D( r,ω )·dS D( r,ω )·d S ˜ =0 B( r,ω )·d S ˜ =0
C ¯ ¯ [ E,H ]=[ E,H ], C ¯ ¯ 2 = I ¯ ¯
{ P ¯ ¯ [ E,H ]=[ E , H ] T ¯ ¯ ( t )=( t ), T ¯ ¯ 2 = I ¯ ¯ N ¯ ¯ ( ω )=( ω ), N ¯ ¯ 2 = I ¯ ¯ Χ ¯ ¯ ( r )=( r ), Χ ¯ ¯ 2 = I ¯ ¯ K ¯ ¯ ( k )=( k ), K ¯ ¯ 2 = I ¯ ¯
{ [ E( r,t ),H( r,t ) ]= T ¯ ¯ ( r,t ) C ¯ ¯ ( r,t ) [ E( r,t ),H( r,t ) ] [ E ( r,ω ), H ( r,ω ) ]= P ¯ ¯ ( r,ω ) C ¯ ¯ ( r,ω ) [ E( r,ω ),H( r,ω ) ] [ E( k,t ),H( k,t ) ]= T ¯ ¯ ( k,t ) C ¯ ¯ ( k,t ) [ E( k,t ),H( k,t ) ] [ E ( k,ω ), H ( k,ω ) ]= P ¯ ¯ ( k,ω ) C ¯ ¯ ( k,ω ) [ E( k,ω ),H( k,ω ) ]
{ L e jk·r E ( k,ω )·dl =jω S e jk·r B ( k,ω )·dS L e jk·r H ( k,ω )·dl =jω S e jk·r D ( k,ω )·dS e jk·r D ( k,ω )·d S ˜ =0 e jk·r B ( k,ω )·d S ˜ =0
P ¯ ¯ ( r,ω ) X ¯ ¯ ( r,t ) + + + + [ E( k,ω ),H( k,ω ) ] e jk·r e jωt dk dω = + { P ¯ ¯ ( r,ω ) + + + [ E( k,ω ),H( k,ω ) ] e jk·r dk } e jωt dω = + + + + { P ¯ ¯ ( k,ω ) [ E( k,ω ),H( k,ω ) ] } e jk·r e jωt dk dω

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