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On-chip dynamic time reversal of light in a coupled-cavity system: APL Photonics: Vol 4, No 3
ABSTRACT . We theoretically and experimentally demonstrate dynamic, all-linear time-reversal of infrared light in planar optical circuits for the first time. We propose that the oscillatory motion of the light stored in cavities can be time-reversed by fast nonadiabatic tuning of the frequency of eigenmodes of a coupled cavity system and experimentally demonstrate it using a system consisting of distant high- Q -factor two-dimensional photonic crystal cavities between which effective direct couplings are formed via line-defect waveguides. We also analyze the loss and methods to reduce the loss, as well as a theory that expands our system to realize general time-reversal operation for any input light. I. INTRODUCTION Section: The time reversal of light, which reverses the time order of incident light, can reverse not only the propagation direction of the light but also additional phase distortions in the transmitted optical media, resulting in the elimination of dispersions, fluctuations, or scattering in the media. 1–5 1. D. M. Pepper, Laser Handbook ( North-Holland Physics , 1988). 2. T. Honda and H. Matsumoto, “ Aberration correction of acousto-optically modulated laser beams by phase conjugation ,” Opt. Lett. 15 , 308 (1990). https://doi.org/10.1364/ol.15.000308 3. Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “ Optical phase conjugation for turbidity suppression in biological samples ,” Nat. Photonics 2 , 110 (2008). https://doi.org/10.1038/nphoton.2007.297 4. G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “ Focusing beyond the diffraction limit with far-field time reversal ,” Science 315 , 1120 (2007). https://doi.org/10.1126/science.1134824 5. J. B. Pendry, “ Time reversal and negative refraction ,” Science 322 , 71 (2008). https://doi.org/10.1126/science.1162087 All previous experimental studies on the time-reversal of light have been based on nonlinear optical systems. One method is optical phase conjugation (OPC) with four-wave mixing, 6,7 6. B. Y. Zel’Dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation ( Springer , 2013). 7. G. He, “ Optical phase conjugation: Principles, techniques, and applications ,” Prog. Quantum Electron. 26 , 131 (2002). https://doi.org/10.1016/s0079-6727(02)00004-6 which operates only with a monochromatic wave due to the requirement for phase-matching. Other techniques include spectral hole burning (SHB) 8,9 8. P. Saari, R. Kaarli, and A. Rebane, “ Picosecond time- and space-domain holography by photochemical hole burning ,” J. Opt. Soc. Am. B 3 , 527 (1986). https://doi.org/10.1364/josab.3.000527 9. A. Rebane, J. Aaviksoo, and J. Kuhl, “ Storage and time reversal of femtosecond light signals via persistent spectral hole burning holography ,” Appl. Phys. Lett. 54 , 93 (1989). https://doi.org/10.1063/1.101226 and photon echo, 10 10. N. W. Carlson, W. R. Babbitt, T. W. Mossberg, L. J. Rothberg, and A. G. Yodh, “ Storage and time reversal of light pulses using photon echoes ,” Opt. Lett. 8 , 483 (1983). https://doi.org/10.1364/ol.8.000483 which are difficult to integrate in planar optical circuits such as silicon (Si) photonics platforms, which limits their ability to be used with other optical functionalities. Recently, all-linear schemes that utilize dynamic modulation of artificial crystals have been proposed. 11–15 11. M. F. Yanik and S. Fan, “ Time reversal of light with linear optics and modulators ,” Phys. Rev. Lett. 93 , 173903 (2004). https://doi.org/10.1103/physrevlett.93.173903 12. S. Sandhu, M. L. Povinelli, and S. Fan, “ Stopping and time reversing a light pulse using dynamic loss tuning of coupled-resonator delay lines ,” Opt. Lett. 32 , 3333 (2007). https://doi.org/10.1364/ol.32.003333 13. S. Longhi, “ Stopping and time reversal of light in dynamic photonic structures via Bloch oscillations ,” Phys. Rev. E 75 , 026606 (2007). https://doi.org/10.1103/physreve.75.026606 14. Y. Sivan and J. B. Pendry, “ Time reversal in dynamically tuned zero-gap periodic systems ,” Phys. Rev. Lett. 106 , 193902 (2011). https://doi.org/10.1103/physrevlett.106.193902 15. M. Minkov and S. Fan, “ Localization and time-reversal of light through dynamic modulation ,” Phys. Rev. B 97 , 060301 (2018). https://doi.org/10.1103/physrevb.97.060301 Such all-linear schemes can provide an on-chip platform for time-reversal of light with wide-bandwidth operation and a high dynamic range of input power. These can then be used in large-scale multi-functional photon manipulation platforms, which can be applied to dispersion compensation, adaptive optics, focusing beyond the diffraction limit, imaging through random media, and other applications. 1–5 1. D. M. Pepper, Laser Handbook ( North-Holland Physics , 1988). 2. T. Honda and H. Matsumoto, “ Aberration correction of acousto-optically modulated laser beams by phase conjugation ,” Opt. Lett. 15 , 308 (1990). https://doi.org/10.1364/ol.15.000308 3. Z. Yaqoob, D. Psaltis, M. S. Feld, and C. Yang, “ Optical phase conjugation for turbidity suppression in biological samples ,” Nat. Photonics 2 , 110 (2008). https://doi.org/10.1038/nphoton.2007.297 4. G. Lerosey, J. de Rosny, A. Tourin, and M. Fink, “ Focusing beyond the diffraction limit with far-field time reversal ,” Science 315 , 1120 (2007). https://doi.org/10.1126/science.1134824 5. J. B. Pendry, “ Time reversal and negative refraction ,” Science 322 , 71 (2008). https://doi.org/10.1126/science.1162087 Yanik and Fan proposed the use of adiabatic tuning of a coupled resonator optical waveguide (CROW), 11 11. M. F. Yanik and S. Fan, “ Time reversal of light with linear optics and modulators ,” Phys. Rev. Lett. 93 , 173903 (2004). https://doi.org/10.1103/physrevlett.93.173903 Sandhu et al. proposed the use of dynamical tuning of distributed loss, 12 12. S. Sandhu, M. L. Povinelli, and S. Fan, “ Stopping and time reversing a light pulse using dynamic loss tuning of coupled-resonator delay lines ,” Opt. Lett. 32 , 3333 (2007). https://doi.org/10.1364/ol.32.003333 Longhi proposed the use of Bloch oscillation, 13 13. S. Longhi, “ Stopping and time reversal of light in dynamic photonic structures via Bloch oscillations ,” Phys. Rev. E 75 , 026606 (2007). https://doi.org/10.1103/physreve.75.026606 Sivan and Pendry proposed the use of dynamic transition of light between zero-gap bands, 14 14. Y. Sivan and J. B. Pendry, “ Time reversal in dynamically tuned zero-gap periodic systems ,” Phys. Rev. Lett. 106 , 193902 (2011). https://doi.org/10.1103/physrevlett.106.193902 and Minkov and Fan proposed the use of dynamic localization of light by a fast resonant frequency modulation. 15 15. M. Minkov and S. Fan, “ Localization and time-reversal of light through dynamic modulation ,” Phys. Rev. B 97 , 060301 (2018). https://doi.org/10.1103/physrevb.97.060301 However, none of these have been realized yet in the optical region. One reason for this is that it is difficult to realize dynamic modulation of the refractive index spatially distributed with the order of the optical wavelength. Also, in order to apply a fast, deep modulation of the refractive index in a linear material such as silicon, we need to use the carrier–plasma dispersion effect, 16,17 16. G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “ Silicon optical modulators ,” Nat. Photonics 4 , 518 (2010). https://doi.org/10.1038/nphoton.2010.179 17. Y. Tanaka, J. Upham, T. Nagashima, T. Sugiya, T. Asano, and S. Noda, “ Dynamic control of the Q factor in a photonic crystal nanocavity ,” Nat. Mater. 6 , 862 (2007). https://doi.org/10.1038/nmat1994 which absorbs light in the structure. To solve these issues, we propose a different scheme based on the indirect and non-adiabatic control of the eigenmodes of a coupled cavity system. We experimentally demonstrate this scheme on a Si chip using distant photonic crystal (PC) nanocavities that are directly coupled to each other effectively via the virtual excitation of intermediate line-defect waveguides. By connecting our proposed system in series, time-reversal of any input light is possible (see Sec. V ). II. THEORY Section: We consider a coupled system of three optical cavities, as schematically shown in Fig. 1(a) , denoted by A, C, and B, coupled in series. The resonant frequencies of A and B match each other ( ω A = ω B = ω 0 ) and that of C is slightly detuned ( ω C = ω 0 ± Δ). Figures 1(b) and 1(c) show the resonant frequencies for ω C = ω 0 ? Δ and ω C = ω 0 + Δ, respectively. It is well known that if two resonators couple with each other, the eigenmodes of the system split into two. 18,19 18. H. Haus, Waves and Fields in Optoelectronics ( Prentice-Hall , 1984). 19. Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “ Strong coupling between distant photonic nanocavities and its dynamic control ,” Nat. Photonics 6 , 56 (2012). https://doi.org/10.1038/nphoton.2011.286 For the cases of Figs. 1(b) and 1(c) , a theoretical model (see Sec. I of the supplementary material ) shows that the eigenmodes split into three, as shown in Figs. 1(d) and 1(e) , respectively. Here, the S and AS modes have symmetric and anti-symmetric electric field distributions between A and B, respectively, and in the C′ mode, the light is concentrated in C. We assume that the detuning Δ is larger than the coupling strength between each cavity μ and much smaller than ω 0 . Here, the S and AS modes can also be written as S ≈ A + B / 2 and A S ≈ A ? B / 2 , respectively, where A and B are the original modes of the cavities A and B normalized appropriately. Half of the frequency separation between the S and AS modes, denoted by Ω, can be written in the following form: Ω ≈ ? μ 2 / Δ . . (1) . In this paper, we excite both the S and AS modes simultaneously. A slowly varying envelope of an equal superposition of the two modes ψ t results in ψ t = cos Ω t + ? 0 A + i ?sin Ω t + ? 0 B , . (2) . where the cos and sin terms correspond to a “beat” signal between the two different frequencies of S and AS. As a result, light moves back and forth between A and B [see the left half of Fig. 1(f ) ]. This is called an optical Rabi oscillation. 19 19. Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “ Strong coupling between distant photonic nanocavities and its dynamic control ,” Nat. Photonics 6 , 56 (2012). https://doi.org/10.1038/nphoton.2011.286 FIG. 1. Schematic illustration of the time-reversal of optical Rabi oscillation. (a) Schematic representation of a coupled system of three optical cavities. [(b) and (c)] Resonant frequencies of cavities A, C, and B where the detuning of cavity C is negative and positive, respectively. [(d) and (e)] Schematic representation of the frequencies and the electric field distributions of three eigenmodes of the coupled cavity system where the detuning of cavity C is negative and positive, respectively. The horizontal axis shows the spectral intensity of light I ( ω ) trapped in the S and AS modes. (f) Schematic illustration of the time-reversal of the optical Rabi oscillation. At t = 0, ω C is switched from ω 0 ? Δ to ω 0 + Δ , resulting in swapping of the S and AS modes and substitution of ? Ωt for Ωt , bringing about the time-reversal. PPT . High-resolution . The most important point of our proposed system is that if the detuning of cavity C or Δ is negative or positive, the beat frequency Ω is, respectively, positive or negative according to Eq. (1) . This is because the S mode and the cavity C mode have the same symmetry and their eigenfrequencies repel each other. This is illustrated in Figs. 1(d) and 1(e) , where the S and AS modes “swap” their relations in the frequency axis by switching ω C from ω 0 ? Δ to ω 0 + Δ. This jump of states in the frequency domain can be non-adiabatic 20 20. A. Messiah, Quantum Mechanics ( North-Holland , 1961), Vol. I. when the speed of change of ω C is much faster than the square of the coupling strength ( dω C / dt ? μ 2 ). Physically, ω C can be dynamically controlled by changing the refractive index around cavity C by, for example, the carrier–plasma dispersion effect. 16,17 16. G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “ Silicon optical modulators ,” Nat. Photonics 4 , 518 (2010). https://doi.org/10.1038/nphoton.2010.179 17. Y. Tanaka, J. Upham, T. Nagashima, T. Sugiya, T. Asano, and S. Noda, “ Dynamic control of the Q factor in a photonic crystal nanocavity ,” Nat. Mater. 6 , 862 (2007). https://doi.org/10.1038/nmat1994 Here, we consider switching ω C instantaneously at t = 0. After the switching ( t > 0), the modulated envelope ψ ′ t results in ψ ′ t = cos ? Ω t + ? 0 A + i ?sin ? Ω t + ? 0 B = ψ ? t . . (3) . From Eqs. (2) and (3) , we see that the exchange oscillation of light between cavities A and B is time-reversed [ Fig. 1(f ) ]. The second important point is that dynamic modulation occurs in the cavity, in which the light does not concentrate [ Figs. 1(d) and 1(e) ]. Therefore, any additional loss in C due to the dynamic modulation has a small effect on the loss of light trapped in the structure, leading to high transparency in spite of the introduction of free carriers. Here, we discuss the requirements of the parameters. As mentioned above, we need to excite both the S and AS modes simultaneously in order to observe the optical Rabi oscillation. Therefore, the spectral width of the excitation pulse should be larger than the frequency difference between the S and AS modes (2Ω). There are two requirements related to the frequency difference between the S and AS modes. (a) The photon lifetime (= Q / ω ) in the cavities A and B should be longer than the period of the optical Rabi oscillation 2 π /2Ω in order to observe the oscillation of light in the system. (b) To obtain low-loss operation, the population of light in cavity C should be small, i.e., the dynamic amount of change (=2 Δ ) of the frequency of cavity C should be sufficiently larger than 2Ω. The possible Q value of the cavities determines the lower limit of Ω, and the possible dynamic amount of change of cavity C determines the upper limit of Ω. Here, the Q value is typically hundreds of thousands for the coupled cavity system. The dynamic amount of change of cavity C is approximately 0.1% of the frequency of cavity C. Therefore, the available range of Ω is approximately 10 10 –10 11 rad/s for optical communication wavelengths under current nanophotonic technologies. III. DEVICE AND EXPERIMENTAL SETUP Section: For the physical implementation, we use Si photonic crystal (PC) nanocavities formed on a chip. PC nanocavities 21–23 21. S. Noda, A. Chutinan, and M. Imada, “ Trapping and emission of photons by a single defect in a photonic bandgap structure ,” Nature 407 , 608 (2000). https://doi.org/10.1038/35036532 22. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “ Ultra-high- Q photonic double-heterostructure nanocavity ,” Nat. Mater. 4 , 207 (2005). https://doi.org/10.1038/nmat1320 23. H. Sekoguchi, Y. Takahashi, T. Asano, and S. Noda, “ Photonic crystal nanocavity with a Q -factor of ?9 million ,” Opt. Express 22 , 916 (2014). https://doi.org/10.1364/oe.22.000916 are of great interest because they can trap light for a long time in a tiny region in the material on the order of cubic wavelengths. They are a key component for the on-chip manipulation of light as they can be integrated, coupled with each other, and dynamically modulated on a chip. 17,19,24 17. Y. Tanaka, J. Upham, T. Nagashima, T. Sugiya, T. Asano, and S. Noda, “ Dynamic control of the Q factor in a photonic crystal nanocavity ,” Nat. Mater. 6 , 862 (2007). https://doi.org/10.1038/nmat1994 19. Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “ Strong coupling between distant photonic nanocavities and its dynamic control ,” Nat. Photonics 6 , 56 (2012). https://doi.org/10.1038/nphoton.2011.286 24. R. Konoike, H. Nakagawa, M. Nakadai, T. Asano, Y. Tanaka, and S. Noda, “ On-demand transfer of trapped photons on a chip ,” Sci. Adv. 2 , e1501690 (2016). https://doi.org/10.1126/sciadv.1501690 Figure 2 shows a schematic illustration of the sample and the measurement setup. We used a triangular-lattice two-dimensional (2D) slab PC structure with a lattice constant of 410 nm and slab thickness of ?220 nm. On this structure, we formed a coupled system of three multistep heterostructure (MH) high- Q nanocavities 25,25 22. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “ Ultra-high- Q photonic double-heterostructure nanocavity ,” Nat. Mater. 4 , 207 (2005). https://doi.org/10.1038/nmat1320 25. Y. Tanaka, T. Asano, and S. Noda, “ Design of photonic crystal nanocavity with Q -factor of ?10 9 ,” J. Lightwave Technol. 26 , 1532 (2008). https://doi.org/10.1109/jlt.2008.923648 and three heaters made of Au for the initialization of each resonance. The three nanocavities were separated by 41 ? m, but effective direct couplings were formed between them via the virtual excitation of intermediate waveguides. 19 19. Y. Sato, Y. Tanaka, J. Upham, Y. Takahashi, T. Asano, and S. Noda, “ Strong coupling between distant photonic nanocavities and its dynamic control ,” Nat. Photonics 6 , 56 (2012). https://doi.org/10.1038/nphoton.2011.286 Thanks to the large separation between the nanocavities, the resonant frequency of each nanocavity can be individually controlled either by the heaters (slow, static control) or by irradiation with control light pulses (fast, dynamic control). FIG. 2. Schematic illustration of the experimental setup. The sample consists of three coupled nanocavities formed on a silicon photonic crystal (Si-PC). Au heaters are used to initialize the resonances. A Ti:S laser and an OPO generate an excitation pulse (1.5 ? m) and a control pulse (820 nm). After the excitation (i), a control pulse is irradiated on cavity C at a certain timing (ii), inducing an instantaneous change of ω C . The time-resolved amplitudes of the light dropped from A and B are measured using a cross correlation technique. Ref. = reference light pulse, BS = beam splitter, and Det. = balanced detectors. PPT . High-resolution . We first measured the spectrum of the sample by inputting continuous wave (CW) light of various wavelengths through the input waveguide placed to the left of cavity A. The Q -factors of the three resultant eigenmodes were approximately 400 000 around a wavelength of 1547 nm, which agrees with our design. We also found that the coupling strength between adjacent cavities was 25 GHz between A and C and 16 GHz between B and C (see Sec. II of the supplementary material ). The heaters at the three cavities can tune the resonant frequency of each nanocavity using the thermo-optic effect. 26 26. T. Asano, W. Kunishi, M. Nakamura, B. S. Song, and S. Noda, “ Dynamic wavelength tuning of channel-drop device in two-dimensional photonic crystal slab ,” Electron. Lett. 41 , 37 (2005). https://doi.org/10.1049/el:20057116 If we supply constant electric powers of 16.5 mW and 15 mW to heaters B and C, respectively, the resonant wavelength of cavities A and B becomes matched (1546.8 nm) and that of cavity C is detuned by approximately +0.3 nm (?37.5 GHz). Thus, we aligned the resonances as shown in Fig. 1(b) , where ω C = ω 0 ? 37.5 GHz. For the time-reversal experiments, a pulsed laser and an optical parametric oscillator (OPO) were used to generate synchronized excitation pulses (?1547 nm) and control pulses (?820 nm). The excitation pulses were shaped to have a time duration of ?20 ps and were input to excite cavity A. The optical Rabi oscillation started at this timing. At a certain time after the excitation, we irradiated cavity C by a control pulse to locally generate free carriers in the Si slab, which lowered the refractive index and increased ω C by ?125 GHz. The control pulse was shaped to have a time duration of approximately 3 ps, which satisfied the non-adiabatic condition. This fast control resulted in the dynamic time-reversal of the optical Rabi oscillation. We measured absolute values of the time-resolved amplitudes of light that were vertically dropped from cavities A and B based on a cross correlation technique. 27 27. J. Upham, Y. Tanaka, Y. Kawamoto, Y. Sato, T. Nakamura, B. S. Song, T. Asano, and S. Noda, “ Time-resolved catch and release of an optical pulse from a dynamic photonic crystal nanocavity ,” Opt. Express 19 , 23377 (2011). https://doi.org/10.1364/oe.19.023377 IV. RESULTS AND DISCUSSION Section: We first measured the exchange oscillation of light between cavities A and B, or an optical Rabi oscillation, without applying a dynamic control. The red and blue lines in Fig. 3(a) show the measured amplitudes of light dropped from cavities A and B. The dashed curve in Fig. 3(a) shows a fitted decay curve with a time constant of τ ? 185 ps. Figure 3(b) shows the energy ratio of cavities A and B, obtained by dividing the measured energy of cavities A and B by their sum, where “energy” is the square of the measured amplitude. From the figure, we can clearly observe the exchange oscillation of light between the cavities originating from the superposition of the S and AS modes. We then irradiated a control pulse around cavity C with a spot size of several μ m and pulse energy of ?48 pJ at 135 ps after the excitation, which switched ω C from ω 0 ? 37.5 GHz to ω 0 + 87.5 GHz nearly instantaneously. Figures 3(c) and 3(d) show the amplitudes and energy ratio of the light in the cavities, respectively. From the energy ratio shown in Fig. 3(d) , it is clear that the oscillatory motion of light was time-reversed at the timing of the dynamic control ( t = t 0 ). This was due to the dynamic change of the detuning between the S and AS modes from positive to negative values induced by the dynamic change of the detuning of cavity C, as discussed before. We consider that the results of Figs. 3(c) and 3(d) show the first demonstration of the linear time-reversal of light on a chip. The period of the oscillation before and after the dynamic control was T 1 ? 31 ps and T 2 ? 66 ps, respectively, which corresponded to the absolute detuning of cavity C before and after the dynamic control (37.5 GHz and 87.5 GHz) [see Eq. (1) ]. If the absolute detuning of cavity C after the dynamic control is matched to that before the control, T 1 = T 2 is achieved. FIG. 3. Measured results (solid lines) of the optical Rabi oscillation between cavities A and B (a) without and (c) with irradiation by the control pulse. (b) and (d) show the results in which the loss of the system is eliminated from (a) and (c), respectively. The horizontal axis is shifted so that the excitation timing becomes 0 ps, and t 0 is the timing of the dynamic control ( t 0 = 135 ps). PPT . High-resolution . Next, we effectively changed the timing of the time-reversal operation by varying the timing of the excitation while fixing the timing of the control pulse irradiation. The energy of the control light pulses was ?36 pJ. The measured time-resolved amplitudes from cavity A are shown in the color plot of Fig. 4 , where the horizontal axis shows the time and the vertical axis shows the timing of the excitation, which was varied from 50 ps to 65 ps. The intense peaks around the tilted dashed line in Fig. 4 correspond to scattered light of the excitation pulses. From 50 ps to 190 ps, Fig. 4 shows a series of tilted stripes, which corresponds to the exchange oscillation of light between cavities A and B. After the irradiation by control light pulses at 190 ps, we can clearly see that the tilted lines are “mirrored” in the time axis. These results show that the on-chip time-reversal operation can be performed at any time. FIG. 4. Measured results of time-reversal operation with varying excitation timings. The time-resolved amplitudes (color plot) of cavity A for various excitation timings (y-axis) are plotted. Control light pulses were irradiated at a fixed timing (190 ps). Time-reversed (mirrored) signals were observed after the irradiation. PPT . High-resolution . Here, we discuss the measured losses in our experiments and consider ways to reduce the losses. From Fig. 3(d) , we see that (i) the exchange of light between cavities A and B does not reach 100% after the dynamic control. We also see from Fig. 3(c) that (ii) the time constant of the decay of the total amplitude (dashed line) reduces at the dynamic control and that (iii) there is an instantaneous drop of the amplitude at the dynamic control (denoted by L C ). These effects arise from the same origin: light absorption by the carriers generated in cavity C. Although cavity C was detuned from cavities A and B after the time-reversal, slight non-resonant coupling with cavity C reduced their decay time constant, causing (ii). In addition, loss in the S mode is larger than that in the AS mode due to the preferential mode overlap with cavity C, and the unbalanced amplitudes between S and AS prevent the exchange of light from reaching 100%, causing (i). This loss after the time-reversal operation can be eliminated by further detuning cavity C from cavities A and B by an amount larger than the coupling strengths between cavity C and cavities A and B. There is also an instantaneous loss due to the crossing of cavity C with cavities A and B, which induces both direct absorption of light by cavity C at resonance and an adiabatic transition of a part of the light to the lossy cavity C mode, causing both (i) and (iii). The amount of L c was 3–4 dB per control pulse, and it can also be reduced by reducing the coupling strength and/or increasing the speed of the dynamic change. Detailed theoretical analyses of the reduction in the losses and experimental data showing the loss and time-constant values are given in Secs. III and IV of the supplementary material . V. THEORY FOR EXPANDING THE SYSTEM Section: Here, we theoretically show that our scheme can be expanded to perform time-reversal of any input light pulse. Figure 5(a) shows a schematic illustration of the system. In Fig. 5(a) , cavities a i (1 ≤ i < n + 1) and b i (1 ≤ i ≤ n ) are alternately coupled with each other. The cavities a i and b i have resonant frequencies of ω 0 and ω 0 + Δ, respectively, and the detuning between b i and a i is Δ. The coupling strength between adjacent cavities is represented as μ . As is well known, such a chain of coupled cavities forms optical band structures called coupled resonator optical waveguide (CROW) bands. 28 28. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “ Coupled-resonator optical waveguide: A proposal and analysis ,” Opt. Lett. 24 , 711 (1999). https://doi.org/10.1364/ol.24.000711 In this case, two bands appear due to the coupling between the two dissimilar cavities. We calculated the band structures of an infinitely long system with two parameters Δ = ?6 μ and Δ = +6 μ and plotted them in Figs. 5(b) and 5(c) , respectively. The lower (“L”) and higher (“H”) frequency bands in Fig. 5(b) (Δ = ?6 μ ) consist almost entirely of cavities b i or a i . We inject light pulses into the H band, as shown by the red lines in the right and lower boxes of Fig. 5(b) . The space-time distribution of light propagating in the system can be expressed in the following form: ψ x , t 2 = ∑ i c i e j ω i t ? k i x 2 , . (4) . where c i is a complex spectral component and j is an imaginary unit. Here, we consider irradiation by identical control pulses into all the cavities b i simultaneously at a time when the light is propagating in the system. To realize such control pulses, we can use, for example, a single, spatially wide optical control pulse that covers all cavities b i when cavities a i and b i are arranged in a zigzag shape. We assume that the control pulse changes the detuning Δ from ?6 μ to 6 μ (by the carrier–plasma effect for example). As a result, the L band in Fig. 5(b) , which mainly consists of cavities b i , moves to the H band in Fig. 5(c) . Associated with this change, the H band in Fig. 5(b) , which mainly consists of cavities a i , changes its dispersion to that of the L band in Fig. 5(c) due to the repulsion from the H band in Fig. 5(c) . When the speed of change of Δ is much faster than the square of the coupling strength, this change of modes in the frequency domain can be non-adiabatic, and the light injected in the H band of Fig. 5(b) remains in the L band in Fig. 5(c) . As we can see from these figures, this operation flips the sign of the group velocity v g ≡ dω / dk owing to the symmetry of the system. In addition, because we induce a spatially uniform change in the entire system, the distribution of light in k space [lower part of Fig. 5(b) ] is preserved. Therefore, the space-time distribution of the light ψ ′ x , t 2 after the dynamic control at t = 0 can be expressed in the following form: ψ ′ x , t 2 = ∑ i c i e j ? ω i t ? k i x 2 = ψ x , ? t 2 . . (5) . From Eq. (5) , we can see that any light pulse can be time-reversed using the method described here. Moreover, we expect that our method is robust against losses caused by free carriers because the light is concentrated in cavities a i in which free carriers do not exist. FIG. 5. (a) Schematic illustration of the expanded system. Control pulses are identically irradiated on cavities b i . [(b) and (c)] Calculated band structures (black lines) and spectral components of light (red lines) (b) before and (c) after irradiation by the control pulses. PPT . High-resolution . We performed a numerical simulation based on a coupled mode theory (CMT). 29 29. C. Manolatou, M. J. Khan, S. Fan, P. R. Villeneuve, H. A. Haus, L. Fellow, and J. D. Joannopoulos, “ Coupling of modes analysis of resonant channel add-drop filters ,” IEEE J. Quantum Electron. 35 , 1322 (1999). https://doi.org/10.1109/3.784592 We assumed that the cavities b i initially had a detuning of Δ = ?6 μ , where μ = 77 GHz. The calculated size n of the system was 20. In the calculation, we ignored any intrinsic loss of the cavities and absorption losses caused by the carrier–plasma effect. A pulse train is input from the left port at 0 ps, and the dynamic control is applied at 120 ps, which instantaneously changes the detuning of cavities b i from ?6 μ to +6 μ . The band structures and the spectral components of the input light are shown in Fig. 5(b) as black lines and red lines, respectively. Figure 6(a) shows the calculated energy of the pulse train propagating in the system. From this figure, we see that the direction of the propagation of the light pulses changes at the time of the dynamic control. Figures 6(b) and 6(c) show the envelope of the input and output pulse trains propagating through the input/output port. These figures show that the envelope of the pulse train is time-reversed by the dynamic control. From the above discussions, we have theoretically shown that our scheme can be expanded to perform time-reversal of any input light pulse. We note that our scheme cannot be operated for monochromatic waves because it is essentially based on the time-reversal of the envelope of the light. There are many applications for this type of time-reversal, such as dispersion compensation. 11–15 11. M. F. Yanik and S. Fan, “ Time reversal of light with linear optics and modulators ,” Phys. Rev. Lett. 93 , 173903 (2004). https://doi.org/10.1103/physrevlett.93.173903 12. S. Sandhu, M. L. Povinelli, and S. Fan, “ Stopping and time reversing a light pulse using dynamic loss tuning of coupled-resonator delay lines ,” Opt. Lett. 32 , 3333 (2007). https://doi.org/10.1364/ol.32.003333 13. S. Longhi, “ Stopping and time reversal of light in dynamic photonic structures via Bloch oscillations ,” Phys. Rev. E 75 , 026606 (2007). https://doi.org/10.1103/physreve.75.026606 14. Y. Sivan and J. B. Pendry, “ Time reversal in dynamically tuned zero-gap periodic systems ,” Phys. Rev. Lett. 106 , 193902 (2011). https://doi.org/10.1103/physrevlett.106.193902 15. M. Minkov and S. Fan, “ Localization and time-reversal of light through dynamic modulation ,” Phys. Rev. B 97 , 060301 (2018). https://doi.org/10.1103/physrevb.97.060301 FIG. 6. (a) Calculated energy of the propagating pulses in the system where the dynamic control is applied at 120 ps. [(b) and (c)] Calculated input and output envelopes of the light through the input/output ports. PPT . High-resolution . VI. CONCLUSION Section: In this paper, we theoretically and experimentally demonstrated on-chip, dynamic, all-linear time-reversal of light using non-adiabatic transition of light. We emphasize that our low-loss scheme has enabled the first experimental observation of the time-reversal of light in a linear system. The results also demonstrate the usefulness of the indirect and non-adiabatic control of the eigenmodes of an optical system. By repeating the proposed structure in series, we can extend our system to perform time-reversal of light with an arbitrary envelope, and parallel integration of such units where the input/output ports are aligned on a one- or two-dimensional grid would allow for control of a wavefront based on the dynamic time-reversal. SUPPLEMENTARY MATERIAL Section: See supplementary material for details regarding the derivation of the eigenmodes of the system, the preliminary experiments, analysis of loss in the time-reversal operation, and experimental verification of the time-reversal of light at a different timing. ACKNOWLEDGMENTS . This paper is partially based on the results obtained from a project commissioned by the New Energy and Industrial Technology Development Organization (NEDO). 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