Programmable synthetic temporal lattices for high-fidelity chaotic synchronization

Abstract

In this paper, the authors demonstrate high-fidelity chaotic synchronization in a programmable synthetic temporal lattice and achieve accurate signal transmission for arbitrary-intensity messages using an alternate encoding strategy. By constructing a synthetic temporal lattice with truncated boundaries via coupled fiber loops and incorporating optoelectronic feedback to introduce controllable artificial nonlinearity, they generate discrete chaotic pulse sequences with tunable entropy. A chaotic synchronization scheme is then implemented within the same synthetic temporal lattice for signal transmission. In this scheme, each message bit is alternately loaded onto the chaotic carrier at single-pulse intervals, allowing the unmodulated carrier pulses at the transmitter to precisely drive synchronized chaotic pulses at the receiver. Experimental results show that nearly perfect synchronization and near-zero decoding error are achieved regardless of the injected message intensity. The study establishes a reconfigurable synthetic-dimensional platform for controllable optical chaos and its high-fidelity application in secure signal transmission.

Graphical abstract

Keywords

Synthetic dimension / Synthetic temporal lattices / Chaos / Chaotic synchronization

1 Introduction

Chaos refers to the complex dynamic behavior in deterministic systems that appears random and long-term unpredictable due to extreme sensitivity to initial conditions [1]. Over the past decades, chaotic dynamics has been observed in numerous optical systems, including semiconductor lasers [2–4], nonlinear optical fibers [5], and optical cavities [6]. Although governed entirely by deterministic physical laws, these systems can produce broadband, noise-like signals, making them attractive not only for studying nonlinear dynamics but also for applications such as secure communication [7–11]. So far, diverse types of feedback systems based on chaotic lasers [12], all-optical fiber loops [13], and opto-electronic hybrid systems [14] have been employed to achieve optical chaos synchronization and communication. However, adjusting the delay time and feedback strength requires hardware modifications, which impedes the implementation of rapid reconfiguration. In addition, when message injection is introduced, maintaining reliable synchronization requires the message intensity to remain within an appropriate dynamic range relative to the chaotic carrier [15,16]. If the intensity is too small, the embedded information becomes difficult to extract, whereas excessively strong intensity may overload the transmitter or introduce detection distortion at the receiver, thereby imposing additional constraints on signal transmission.

Recently, synthetic temporal lattices constructed from coupled fiber loops enable the mapping of pulse sequences onto discrete lattice sites, forming a controllable dynamical space in the time domain [17–27]. Leveraging this highly controllable platform, various complex physical phenomena have been demonstrated, including parity-time symmetry [19], Anderson localization [22] and topological light funneling [26]. Moreover, by incorporating optoelectronic feedback, programmable artificial nonlinearities can be introduced into temporal lattices [27]. Since temporal lattices possesses structural reconfigurability and a tunable nonlinear response, they are expected to perform controllable chaotic synchronization without physical reconfiguration of the hardware. Meanwhile, temporal lattices enable selective modulation of pulses [28,29], providing a new route for message injection and potentially alleviating the constraints on injection intensity in chaos-based signal transmission.

In this work, we construct a synthetic temporal lattice with truncated boundaries based on coupled fiber loops. An effective feedback opto-electronic circuit is incorporated in the fiber loops to introduce artificial nonlinearity. By adjusting the architecture and nonlinearity of the lattice, we can generate a chaotic pulse sequence with high chaos entropy at the output. When the output pulse sequence from the transmitter acts as a modulation signal for a replicated receiver system, the two systems may reach synchronization. To achieve accurate demodulation, each bit of the message is encoded onto the chaotic carrier at intervals of one pulse. In this manner, the message signal with any intensity can be accurately decoded. This study establishes a programmable platform for controllable chaotic synchronization in synthetic dimensions, offering enhanced robustness of synchronization against message disturbances.

2 Methods

The architecture for generating optical chaos is depicted in Fig. 1a, which consists of two coupled fiber loops with different lengths and an opto-electronic feedback circuit. The long and short loops are connected via a variable optical coupler (VOC) at the center. The detailed experimental setup is shown in the supplementary material. The arriving time of the pulses can be mapped to a two-dimensional temporal lattice, as shown in Fig. 1b. The pulse dynamics of the synthetic temporal lattice is described by the discretized iteration equations

(1)\begin{array}{rl}{u}_n^{m+1}& =\cos \beta \cdot u_{n+1}^m+\text{i}\sin \beta \cdot v_{n+1}^m,\\ v_n^{m+1}& =\cos \beta \cdot v_{n-1}^m+\text{i}\sin \beta \cdot u_{n-1}^m,\end{array}

where u_n^m and v_n^m denote the pulse amplitudes in the short and long loops at the lattice site n and time step m. As the fibers have an average length of 5 km and a length difference of 300 m, the intervals of adjacent lattice sites and time steps are 1.5 μs and 25 μs, respectively. The coupling ratio of the VOC is given by cos²β/sin²β with β being the time-variable coupling angle and 0 ≤ β ≤ π/2. By setting β = π/2 at even steps, the optical pulses are restrained within three lateral sites (n = 0, ± 1), as shown in Fig. 1b. It is worth noting that although the two fiber loops are approximately 5 km in length, the evolution of a single pulse over hundreds of steps takes only a few milliseconds, which is much shorter than the characteristic time scales of temperature drift and mechanical vibrations.

We first inject a single optical pulse in the short ring via an optical coupler (OC). After the pulse passes through the VOC, another OC in the loop feeds its partial power in the modulation circuit, where a single mode fiber (SMF) with sufficient length will delay the optical signal. Then the electronic output of the photodetector (PD) is amplified and fed to the phase modulator (PM). An electric switch is placed in the circuit that allows the modulating signal to only take effect after the optical pulse evolves for N lattice periods at the VOC, as shown in Fig. 1c. Note that each lattice period is equivalent to two time steps for the pulse evolution. Once the power of the pulse is fed in the feedback circuit at this time step, the AOM works to prevent it from continuing its evolution. Meanwhile, another optical pulse, which is phase relevant with the former, is generated from the laser and injected into the optical loop. After evolving for N_q lattice periods (N_q ≤ N), the new pulse will encounter the electric modulating signal originated from the former pulse of incidence. Therefore, we establish a feedback strategy when a periodic pulse sequence is incident and the modulation process repeats. The period number of the incident pulse is denoted by p with each pulse period containing N lattice periods labeled by q = 1, 2,...,N. The phase of each pulse in the sequence is determined by the intensity of its former, as depicted in Fig. 1c. According to Eq. (1), the transmittance of the p-th incident pulse after evolving N lattice periods under a feedback delay of N_q lattice periods is given by

(2)T_p={\cos }^2(N\alpha )+1/2\sin [2(N-N_q)\alpha ]\sin (2N_q\alpha )\left(1-\cos {\varphi }_p\right),

where α = β + π/2 and ϕ_p represents the modulation phase of the p-th incident pulse. The phase obeys the feedback relation ϕ_p = χI_p−1 with I_p = T_p(ϕ_p)I₀, where I₀ and I_p are intensities of the pulse at incidence and after evolving N lattice periods. The normalized nonlinear feedback coefficient is given by χ = πG_eRA_o/V_π, where A_o represents the attenuation ratio after the pulse passes through OC, R is the voltage responsivity of PD, G_e is the amplification of the electric signal by AMP, and V_π is the half-wave voltage of PM. The detailed derivation process of transmittance is presented in the supplementary material.

3 Results and discussion

The iteration of the pulse intensity is plotted in Figs. 2a and 2b. Here we set β = π/3 at odd time steps, N = 2, and N_q = 1. For small feedback coefficient as χ = π, the final intensity reaches to a single point, indicating the system remains at steady-state. As the coefficient increases to χ = 1.5π, the intensities finally hover at two fixed values. For even higher feedback coefficient χ = 1.7π, as shown in Fig. 2b, the system becomes chaotic and the intensity cannot be restricted to a finite number of fixed values. Figure 2c shows the bifurcation diagram as the feedback coefficient varies, which illustrates the transition of the system from steady to chaotic states through a sequence of period-doubling bifurcations. The circles represent the experimental data for different values of χ. They coincide fairly with the simulation results. Figure 2d plots the maximal Lyapunov exponent λ_max versus the nonlinear coefficient χ as the number of lattice periods N varies from 1 to 6 for β = π/3 and N_q = 1. The system enters chaos as long as λ_max > 0. Note that the output signal is not affected by the modulation phase as N = 1 and 4. Moreover, the system is fixed at a steady-state as N = 3, according to Eq. (2). In contrast, for N = 2, 5, and 6, the chaos is achievable. As χ increases, λ_max becomes greater than zero at first for N = 2, implying that the system is more likely to enter chaos. The experimental results are also shown in Fig. 2e. The largest entropy appears at β = 0.29π and χ = 1.7π, which coincides well with the result of maximal Lyapunov exponent, as depicted in Fig. 2f. The experimental results are also shown in Fig. 2g. As χ increases from π to 1.5π and 1.7π, ones clearly see the process of the system changing from steady to period bifurcation and chaos. The last plot with largest entropy manifests a more random distribution of pulse intensity.

Based on the chaotic signals in the synthetic temporal lattice, we propose a discrete-time-based chaotic synchronization scheme for signal transmission, as shown in Fig. 3a. The transmitter employs two coupled optical fiber rings with an additional feedback circuit. The message signal is loaded onto the chaotic carrier (I_p) through an arbitrary waveform generator (AWG) and a VOC. At the receiver, the encoded message signal (I'_p) is divided into two paths. One is directly converted into an electrical signal by a PD and recorded by an oscilloscope (OSC), while the other serves as a driving signal to make the receiver generate a chaotic signal (R_p) synchronized with the transmitter. To ensure chaotic synchronization, the receiver also adopts the same coupled optical fiber rings structure as the transmitter. However, the modulation is fed by the encoded signal from the transmitter rather than self-feedback. Specifically, the encoded optical signal is converted into an electrical signal and directly applied to the phase modulator of the receiver, with the modulation phase given by ϕ'_p = χI'_p−1, where I'_p−1 is the intensity of the received encoded signal. This synchronization method requires that all parameters of the two systems be exactly identical. By comparing the encoded signal with the chaotic synchronization signal using the OSC, the decoded message is ultimately obtained. Figure 3b illustrates the chaotic carrier generated at the transmitter. The black arrows indicate that the phase of each current carrier pulse is determined by the intensity of its former. The original message signal is depicted in Fig. 3c. Differing from conventional sequential loading, we alternately load the message signal onto the carrier pulses at a single-pulse interval, as shown in Fig. 3d. For each chaotic pulse without being loaded with message, it will modulate the receiver to generate a synchronized chaotic pulse in the subsequent time slot, which is exactly the same with the original chaotic pulse used to load message, as shown in Fig. 3e. By comparing the encoded pulses with its corresponding synchronized pulses, the encoded message can be accurately recovered without errors.

The experimental results are presented in Fig. 4. The message composed of a binary sequence is depicted in Fig. 4a. The data 0 and 1 are generated by the voltages V₀ and V₁ in the experiment with V₀ = 823 mV and V₁ = 522 mV. Then the normalized intensity of the encoded signal is expressed as I'_p = I_pexp[h(V_j-\overline{V})/(V₁−V₀)] with j = 0 or 1 and \overline{V} = (V₁ + V₀)/2. The variable h denotes a scaling factor that controls the strength of the message signal relative to the carrier. Utilizing a VOC, the signal is loaded onto the chaotic pulses generated by the transmitter. Figure 4c displays the encoded message for h = 0.2, with red arrows indicating the embedding pulse slots. The modulation phase generated by the PM at the receiver is plotted in Fig. 4d, which is governed by ϕ_p= χI_p−1. The synchronized chaotic signal is shown in Fig. 4e, where the black dotted curve represents the original chaotic signal from the transmitter. It is evident that the signals maintain high-fidelity synchronization at the data loading positions, with negligible deviation between the transmitter and receiver. By reversing the encoding process, the normalized voltage of the decoded signal is illustrated in Fig. 4f. Given a judgment voltage of 0.5, normalized voltages greater and less than this value are designated as 1 and 0, respectively. Consequently, the retrieved binary message aligns precisely with the original, as indicated by the red bars in the figure.

We also conducted a comparative analysis with the sequential pulse encoding method, where the discrete message signal with a random intensity profile is sequentially loaded on each pulse of the chaotic carrier, as shown in Fig. 5a. The synchronized chaotic signal exhibits obvious errors at each data bit compared to the transmitter chaotic signal, as illustrated in Fig. 5b. The alternate encoding method, however, loads data at every other pulse position, as shown in Fig. 5c. At the data-loaded positions, the synchronized chaotic signal perfectly matches the chaotic carrier generated from the transmitter, as depicted in Fig. 5d. The performance of chaotic synchronization is depicted in Fig. 5e. It is apparent that the alternate encoding approach achieves high-fidelity synchronization, as the data points of original and synchronized chaotic signals are distributed nearly on a single line. In contrast, the sequential mapping leads to a dispersed distribution, reflecting degradation of synchronization. We also consider a regular message signal of simple harmonic wave such that the encoded signal reads I'_p = I_pexp[hsin(Ωp_m)] with Ω = 0.8 being the normalized frequency of the signal and p_m the sampling index of the message signal, as shown in Fig. 5f. Both encoding methods are employed. The results verify again the high accuracy with alternate encoding. The slight discrepancies primarily originate from the finite noise of the experimental system. It should be noted that we selected the moderate intensity parameters h = 0.2 and Ω = 0.8 as an example. The sequential encoding works only under small intensity, with error increasing significantly as intensity rises. Although our alternate encoding achieves zero-error decoding under larger intensity, excessive intensity compromises message concealment. Thus, this parameter set balances performance and security. Finally, we numerically investigate the influence of message signal on the synchronization accuracy and decoding error. As the scaling factor h increases, one can observe the enlargement of decoding error [Fig. 5g] for sequential pulse encoding, which is calculated by the root-mean-square error (RMSE) of the original and decoded message signals. This behavior originates from disruption of the synchronization condition under strong perturbations, as evidenced by the decreasing synchronization accuracy, which is characterized by the normalized cross-correlation (NCC) of the chaotic carrier and synchronized chaotic signals. The calculation formula for NCC is

(3)\text{NCC}=\frac{{\displaystyle \sum _{p=1}^N\left(I_p-\overline{I}\right)\left(R_p-\overline{R}\right)}}{\sqrt{{\displaystyle \sum _{p=1}^N{\left(I_p-\overline{I}\right)}^2}}\cdot \sqrt{{\displaystyle \sum _{p=1}^N{\left(R_p-\overline{R}\right)}^2}}},

where I_p and R_p are the intensities of the p-th pulse in the output chaotic pulse sequences of the transmitter and the receiver, respectively. \overline{I} and \overline{R} are the mean values of these two chaotic sequences, respectively. N is the length of the pulse sequence. Here, we regard NCC > 0.99 as indicating nearly perfect synchronization. While perfect synchronization would theoretically yield NCC = 1, the presence of computational errors leads us to set 0.99 as an acceptable practical criterion. Note that for much larger intensity, the message cannot be hidden well by the chaotic carrier and thus is decoded easily. Similarly, increasing the modulation frequency reduces synchronization quality and enlarges reconstruction error by using traditional sequential encoding method, as shown in Fig. 5h. In our scheme of alternate encoding, the synchronization accuracy always remains at high level with nearly zero decoding error. This is because the strategy loads message bits only onto every other pulse, while the intervening pulses remain unmodulated and are used to drive the receiver. Therefore, regardless of the message intensity, the driving signal remains unaffected and the synchronization quality does not degrade. This capability is enabled by the independent manipulation of individual pulses in the synthetic temporal lattice platform.

It should be noted that although a low transmission rate of ~5 kbit/s in the present experiment is demonstrated, the system offers substantial potential for rate enhancement without altering its underlying chaos synchronization principle. With shorter fiber loops or integrated photonic waveguides, the round-trip time can be reduced to tens of nanoseconds, and using high-speed modulators could increase the pulse repetition rate to tens of megahertz. Under the alternate encoding scheme, the effective data rate could reach the order of 100 Mbit/s or higher, highlighting the platform’s potential for scalable, high-speed secure communication. Furthermore, the long optical fibers used in this work result in a bulky system that is susceptible to temperature drift and vibrations during long-term operation. For practical applications, future implementations could replace them with chip-scale waveguides to reduce the loop length to the centimeter scale and introduce active stabilization, thereby improving compactness and environmental robustness.

4 Conclusion

In conclusion, we have experimentally demonstrated high-fidelity chaotic synchronization in a programmable synthetic temporal lattice formed by coupled fiber loops with tunable optoelectronic feedback. The introduction of artificial nonlinearity enables controllable nonlinear evolution in the temporal lattices and provides a reconfigurable platform for exploring chaotic dynamics within synthetic dimensions. Building upon the generated chaotic pulse sequences, we implement a chaotic synchronization scheme in synthetic temporal lattices to enable signal transmission. To ensure secure and reliable message transmission, we propose an alternate encoding strategy that embeds message into high-entropy chaotic carriers with minimal perturbation to system synchronization. The results confirm robust synchronization with negligible decoding errors. Moreover, the system offers high configurability through electric control of nonlinear parameters, making it a flexible and scalable platform for investigating nonlinear dynamics and chaos-based signal processing.

References

[1] Lorenz , E.N. : Deterministic nonperiodic flow. J. Atmos. Sci 20(2), 130–141(1963)

[2] Sciamanna , M. , Shore , K.A. : Physics and applications of laser diode chaos. Nat. Photonics 9(3), 151–162(2015)

[3] Yang , L. : Fighting chaos with chaos in lasers. Science 361(6408), 1201(2018)

[4] Bachelard , N. , Gigan , S. , Noblin , X. , Sebbah , P. : Adaptive pumping for spectral control of random lasers. Nat. Phys 10(6), 426–431(2014)

[5] Le Boudec, P. , Jaouen , C. , François , P.L. , Bayon , J.F. , Sanchez , F. , Besnard , P. , Stéphan , G. : Antiphase dynamics and chaos in self-pulsing erbium-doped fiber lasers. Opt. Lett 18(22), 1890–1892(1993)

[6] Liu , C. , Di Falco, A. , Molinari , D. , Khan , Y. , Ooi , B.S. , Krauss , T.F. , Fratalocchi , A. : Enhanced energy storage in chaotic optical resonators. Nat. Photonics 7(6), 473–478(2013)

[7] Pecora , L.M. , Carroll , T.L. : Synchronization in chaotic systems. Phys. Rev. Lett 64(8), 821–824(1990)

[8] Argyris , A. , Syvridis , D. , Larger , L. , Annovazzi-Lodi , V. , Colet , P. , Fischer , I. , García-Ojalvo , J. , Mirasso , C.R. , Pesquera , L. , Shore , K.A. : Chaos-based communications at high bit rates using commercial fibre-optic links. Nature 438(7066), 343–346(2005)

[9] Mao , X. , Wang , A. , Wang , L. , Fu , S. , Wang , Y. , Qin , Y. : 100-Gbit/s 100-km physical-layer secure fiber-optic communication using wideband chaotic semiconductor lasers. J. Lightwave Technol 43(5), 2176–2183(2025)

[10] Hayes , S. , Grebogi , C. , Ott , E. : Communicating with chaos. Phys. Rev. Lett 70(20), 3031–3034(1993)

[11] Spitz , O. , Herdt , A. , Wu , J. , Maisons , G. , Carras , M. , Wong , C.W. , Elsäßer , W. , Grillot , F. : Private communication with quantum cascade laser photonic chaos. Nat. Commun. 12(1), 3327(2021)

[12] Kusumoto , K. , Ohtsubo , J. : 1.5-GHz message transmission based on synchronization of chaos in semiconductor lasers. Opt. Lett 27(12), 989–991(2002)

[13] Sivaprakasam , S. , Shahverdiev , E.M. , Spencer , P.S. , Shore , K.A. : Experimental demonstration of anticipating synchronization in chaotic semiconductor lasers with optical feedback. Phys. Rev. Lett 87(15), 154101(2001)

[14] Goedgebuer , J.P. , Larger , L. , Porte , H. : Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode. Phys. Rev. Lett 80(10), 2249–2252(1998)

[15] Cuomo , K.M. , Oppenheim , A.V. : Circuit implementation of synchronized chaos with applications to communications. Phys. Rev. Lett 71(1), 65–68(1993)

[16] Kocarev , L. , Parlitz , U. : General approach for chaotic synchronization with applications to communication. Phys. Rev. Lett 74(25), 5028–5031(1995)

[17] Bersch , C. , Onishchukov , G. , Peschel , U. : Spectral and temporal Bloch oscillations in optical fibres. Appl. Phys. B 104(3), 495–501(2011)

[18] Regensburger , A. , Bersch , C. , Hinrichs , B. , Onishchukov , G. , Schreiber , A. , Silberhorn , C. , Peschel , U. : Photon propagation in a discrete fiber network: an interplay of coherence and losses. Phys. Rev. Lett 107(23), 233902(2011)

[19] Regensburger , A. , Bersch , C. , Miri , M.A. , Onishchukov , G. , Christodoulides , D.N. , Peschel , U. : Parity-time synthetic photonic lattices. Nature 488(7410), 167–171(2012)

[20] Wimmer , M. , Regensburger , A. , Bersch , C. , Miri , M.A. , Batz , S. , Onishchukov , G. , Christodoulides , D.N. , Peschel , U. : Optical diametric drive acceleration through action–reaction symmetry breaking. Nat. Phys 9(12), 780–784(2013)

[21] Wimmer , M. , Miri , M.A. , Christodoulides , D. , Peschel , U. : Observation of Bloch oscillations in complex PT-symmetric photonic lattices. Sci. Rep 5(1), 17760(2015)

[22] Vatnik , I.D. , Tikan , A. , Onishchukov , G. , Churkin , D.V. , Sukhorukov , A.A. : Anderson localization in synthetic photonic lattices. Sci. Rep 7(1), 4301(2017)

[23] Wimmer , M. , Price , H.M. , Carusotto , I. , Peschel , U. : Experimental measurement of the Berry curvature from anomalous transport. Nat. Phys 13(6), 545–550(2017)

[24] Wang , S. , Qin , C. , Wang , B. , Lu , P. : Discrete temporal Talbot effect in synthetic mesh lattices. Opt. Express 26(15), 19235–19246(2018)

[25] Wang , S. , Wang , B. , Lu , P. : PT-symmetric Talbot effect in a temporal mesh lattice. Phys. Rev. A (Coll. Park) 98(4), 043832(2018)

[26] Weidemann , S. , Kremer , M. , Helbig , T. , Hofmann , T. , Stegmaier , A. , Greiter , M. , Thomale , R. , Szameit , A. : Topological funneling of light. Science 368(6488), 311–314(2020)

[27] Wang , S. , Wang , B. , Liu , C. , Qin , C. , Zhao , L. , Liu , W. , Longhi , S. , Lu , P. : Nonlinear non-Hermitian skin effect and skin solitons in temporal photonic feedforward lattices. Phys. Rev. Lett 134(24), 243805(2025)

[28] Wang , S. , Qin , C. , Zhao , L. , Ye , H. , Longhi , S. , Lu , P. , Wang , B. : Photonic Floquet Landau-Zener tunneling and temporal beam splitters. Sci. Adv 9(18), eadh0415(2023)

[29] Ye , H. , Wang , S. , Qin , C. , Zhao , L. , Hu , X. , Liu , C. , Wang , B. , Lu , P. : Observation of generalized dynamic localizations in arbitrary‐wave driven synthetic temporal lattices. Laser Photonics Rev 17(9), 2200505(2023)

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