Abstract
Optical chaos is vital for various applications such as private communication, encryption, antiinterference sensing, and reinforcement learning. Chaotic microcombs have emerged as promising sources for generating massive optical chaos. However, their interchannel correlation behavior remains elusive, limiting their potential for onchip parallel chaotic systems with high throughput. In this study, we present massively parallel chaos based on chaotic microcombs and highnonlinearity AlGaAsOI platforms. We demonstrate the feasibility of generating parallel chaotic signals with interchannel correlation <0.04 and a high random number generation rate of 3.84 Tbps. We further show the application of our approach by demonstrating a 15channel integrated random bit generator with a 20 Gbps channel rate using silicon photonic chips. Additionally, we achieved a scalable decisionmaking accelerator for up to 256armed bandit problems. Our work opens new possibilities for chaosbased information processing systems using integrated photonics, and potentially can revolutionize the current architecture of communication, sensing and computations.
Introduction
Chaos is a fundamental phenomenon in physics that exhibits random behaviors due to its great sensitivity to small changes of conditions^{1}. It has been playing key roles behind a wide range of applications for modern society: in communications, the generation of chaos guarantees the integrity of cryptographic protocols for secure networks^{2,3,4,5}; in computations, the simulation of Monte Carlo problems and reinforcement learning relies on random numbers^{6} generated from chaos, which is particularly essential for Artificial Intelligence (AI). Recently, chaos has also been used to enable advanced sensing technologies, such as Multiple Input Multiple Output (MIMO) radar^{7,8} or Random Modulation Continuous Wave (RMCW) lidar^{9,10,11}, which are immune to interference and thus can overcome the time/frequency congestion in ranging. Conventionally, the chaotic source used in information systems is generated from the electronic chaos (Fig. 1a1) induced by nonlinear circuits^{12,13} such as application specific integrated circuit (ASIC) and field programmable gate array (FPGA)^{14,15}. Despite their integration compatibility with CMOS electronics, these chaotic sources suffer from low bandwidth (on the order of hundreds of MHz^{12}) and the combination of several sources is necessary for high throughput rate, which is well behind the electronic processing speed and therefore lagging the system performance. This problem gets further highlighted in nowadays parallel information architectures, where multiple data channels need to be proceeded simultaneously for high system throughput.
Optical chaos^{16,17,18,19,20} has been studied for decades to overcome the limitations of electronic approaches, by utilizing the ultrafast nature of optical processes (Figs. 1a, 2–3). Chaotic lasers^{21,22,23} have been widely used for this purpose, by perturbing the laser cavities with external injection or optical feedback^{24}, which leads to highly nonlinear dynamics and chaotic behavior. Chaotic lasers have achieved up to 50 GHz^{25} chaotic bandwidth, surpassing electronic approaches. However, the integration and parallelization of chaotic lasers have encountered remarkable challenges. Multiwavelength lasers often exhibit high correlation between different wavelengths due to the competition between longitudinal modes^{26}, making chaotic lasers singlechannel^{27} devices. Previously, parallelization relied on incompatible spatial multiplexing^{16,28} or laser arrays^{29,30}. Moreover, the requirement for strict injection or feedback configurations complicates miniaturization and widespread usage. These limitations hinder the practical deployment of optical chaos^{30}.
The use of a chaotic comb (Figs. 1a, 4 as a parallel chaotic source offers a promising alternative, as spatiotemporal chaos can be achieved by pumping an optical nonlinear microcavity with sufficient power. In frequency domain, the comb is consisted of wellseparated comb lines, each exhibiting poor coherence^{31,32,33}. This promises tremendous parallelism provided by optics through wavelengthdivisionmultiplexing technology. Recently, chaotic combs have been employed as parallel optical sources for optical coherence tomography^{34,35} and parallel ranging^{11}. Especially for parallel ranging, the chaotic properties of chaotic combs were harnessed for unambiguous and interferencefree Lidar with simplified systems^{11,36}. However, the parallelization in previous works was based on frequency domain rather than on the chaotic signal carried by each channel. For parallel chaotic sources, channels should be uncorrelated with each other, which is vital to applications ranging from private communication^{4,37} and key distribution^{5} to chaotic lidar/radar systems^{7,36}. However, it is still unclear whether chaotic combs can function as parallel chaotic sources.
In this work, we fill this gap by providing a full investigation of chaotic comb lines, and introduce the optical chaotic comb as a massively parallel chaotic source with low interchannel correlation as shown in Fig. 1b. For the first time, to the best of our knowledge, we verified interchannel orthogonality relation between each tooth of a chaotic microcomb in both theory and experiment, which demonstrated its parallelization capability. By using an advanced heterogeneous photonic platform, aluminium gallium arsenide on insulator (AlGaAsOI) on silicon substrate^{38,39}, we demonstrate a massively parallelized chaotic source with hundreds of wavelength channels on chip, that is compatible with current photonic foundry production. Based on this capability, we explore two advanced applications. A 15channel random number generation system with 20 Gbps bit channel rate is achieved using silicon photonic WDM and receivers, whose aggregate rate is one order of magnitude higher than previous onchip random number generators. Detecting two chaotic microcombs with commercial photodetectors, the aggregation rate can reach 3.84 Tbps, with 32 channels and 120 Gbps per channel rate. This is the highest generation rate for opticalchaosbased systems. We also show the unprecedented advantages of this strategy in optical computations by accelerating the decision making of multiarmed bandit problem. Our work paves the way for information processing on chip using integrated, high performance chaotic sources, which will lead to many new opportunities in private communications, computation and ranging for integrated photonics (Fig. 1c).
Results
Properties of massively parallel chaotic combs
Firstly, we verify the ability of microcombs in chaotic states to act as massively parallel chaotic sources. Chaotic bandwidth is one of the key metrics of chaos and Fig. 2c shows the radio frequency spectra of chaotic combs under different pump powers. Due to the high nonlinear effect, a chaotic comb with ∼GHz bandwidth is reachable with tens of mW pump power. The bandwidth of the chaotic signal broadens with a stronger pump. As the onchip pump power reaches 130 mW, the chaotic comb shows a broad radio frequency noise with a 10 dBbandwidth of up to 5.6 GHz. The bandwidth is currently limited by the freecarrierabsorption (FCA) and the threephotonabsorption (3PA) in the AlGaAsOI waveguide, as discussed in the Supplementary Note 8. Nonetheless, the bandwidth is already compatible with that of the opticalfeedback chaotic laser. In addition, the broad chaos, starting from the very lowfrequency baseband, holds consecutively strong components within its noise band, which is powerefficient for radio applications considering a lowpass response in the optical receiver for RF signal generation^{7}.
To characterize the intra and interchannel chaotic properties (see Supplementary Note 1), each comb line is filtered out from a chaotic microcomb with 130 mW pump power (Fig. 2a). The comb is rendered as the trapezoidal shape in log scale, with an approximate power variation of 7 dB within the C band. Before each filtered comb line is sent into the photodetector and afterward recorded, the pump mode is suppressed and the remaining signals are amplified together, as shown in Fig. 2b. The time domain signal of a comb line is depicted in Fig. 2d. Due to the intracavity field undergoing the spatiotemporal chaos, the amplitude of the comb line changes rapidly and intensely, totally different from the case of the localized comb state. The extreme events are also captured in the amplitude distribution (Fig. 2e), where a long tail exists at high intensity. The quality of the chaotic signal is valued by the autocorrelation function (ACF) in Fig. 2f. The ACFs of the 51 comb lines possess a Diraclike shape. Without the need for a feedback loop to estimate a chaotic state as the chaotic laser, no time delayed signature is observed in the ACF of chaotic combs. It is also worth noting that the strong Kerr effect in an AlGaAsOI microresonator, which holds the highest thirdorder nonlinear coefficient among all integrated nonlinear platforms^{36,38}, helps to achieve the full width at half maximum (FWHM) of all comb lines smaller than 0.15 ns. Figure 2g shows the FWHM of the ACF changes with the detuning. Due to the selflocking induced by the intracavity thermal effect^{39,40}, the FWHM for all comb lines can be maintained within 0.2 ns for a detuning range larger than 25 GHz, indicating remarkable stability and interchannel consistency.
Despite the simultaneous generation of multi highquality chaotic signals demonstrated above, the chaotic comb cannot be employed as a parallel chaotic source unless channels are orthogonal to each other. The orthogonality is quantified by the correlation between channels. In our experiment, we examine the interchannel correlation by filtering and detecting every two comb lines simultaneously. The correlation of each pair of comb lines in the C band is shown in Fig. 2h. Because the pump mode is suppressed, the correlation between the pump mode and others is not presented here. Two significant lines with relatively strong correlations can be observed. One is the main diagonal, which is autocorrelation. The other is symmetric with respect to the pump mode, indicating an obvious correlation between symmetrical comb lines. The correlation between symmetrical comb lines is also captured in our simulation based on the LugiatoLefever equation (LLE), shown in Fig. 2i and Supplementary Note 4. As the energy to stimulate comb lines is originated from the intracavity pump laser, the symmetrical comb lines should be naturally correlated with each other due to the fourwavemixing process. Except for the symmetrical modes, the correlation between comb lines is lower than 0.04, which is low enough for applications requiring parallel chaotic sources^{7,36,37,41}.
Hundreds of channels, the uniform power distribution, the negligible crosscorrelation between different channels, and the comparable chaotic bandwidth with chaotic lasers, all these properties suggest the chaotic comb is an attractive massively parallel chaotic source. In addition, different from its localized partner, the soliton comb, the chaotic comb can be generated in a simple and distinct manner, by slowly sweeping into a highquality optical nonlinear microcavity without the blocking of the intracavity thermal effect. More attractively, the intracavity thermal effect can facilitate the stabilization of the chaotic state^{39,40}.
Integrated parallel random bit generation
As proof of the chaotic comb functioning as a parallel chaotic source, we combine the microcomb and a silicon photonic (SiPh) integrated circuit to realize massively parallel random bit generation (RBG). Figure 3b shows the detection scheme using the SiPh receiver. Before injection into the SiPh chip, the chaotic comb output from the microring chip passes a FBG to suppress the pump mode and is amplified by a commercial DWDM EDFA. To compensate for the insertion loss of the SiPh chip, one selected comb line is filtered and amplified before injection into the SiPh chip. The SiPh chip is constituted of a 16channel arrayed wavelength grating (AWG) and arrayed GeSi photodetectors. The average channel spacing of the AWG is about 180 GHz, fitted with the 2 FSR of the chaotic comb. The selected comb line is selectively coupled into one PD passing the AWG. The signal detected by the GeSi PD is sent to the oscilloscope, recorded at the sampling rate of 20 GSa/s.
Ensuring the relatively low correlation between adjacent sampled data, the recorded data is downsampled to 6.67 GSa/s. As shown in Fig. 2e, the intensity distribution of the output I shows an asymmetric shape. To symmetrize the distribution, a delay difference is employed to the raw data, obtaining ∆I with a symmetric distribution as shown in Fig. 3d. The symmetric distribution is favorable for unbiased bit extraction^{42}. The differential data ∆I is digitalized to 8 bits and the 3 least significant bits (LSB) are used for random bit generation. A normal distribution of the extracted 3 LSBs (see Fig. 3e) indicates that the generated bit sequence contains different bit patterns with equal frequencies. Then a selfdelayed exclusiveOR (XOR) process is employed to remove the residual bias as shown in Fig. 3b. Figure 3f shows the ACF of the generated bit sequence and the correlation is around the lower limit 1/\(\sqrt{n}\), where n = 10^{6} is the length of bit sequence. A strict test of the generated random bits is carried out by the NIST SP 80022, a standard statistical test suite. As shown in Fig. 3g, all of the 15 channels obtain pass rates >96.5%, within the acceptance range for the NIST SP 80022 test. The generated random bits also pass the Diehard test successfully, shown in Supplementary Note 11.
The generation rate of single chaotic comb is mainly limited by the chaotic bandwidth, which is degenerated by the nonlinear absorption. To compensate this, here we provide a method by a dualcomb scheme. The schematic is shown is Fig. 3c, where two combs are pumped by two lasers with a frequency difference around 4 GHz. In experiment, two microcavities are packaged with temperature controllers and the chaotic combs are generated by sweeping the resonant frequency by tuning the chip temperature, while the pump lasers are kept fixed. The comb teeth of the two chaotic combs are filtered, amplified and combined, before being sent to the photodiode. The time domain waveforms of each tooth pairs are recorded at a sampling rate of 80 GSa/s. The raw data are downsampling to 40 GSa/s and processed as described above. In this case, the single channel generation rate can be increased to 120 Gbps, which is comparable to the random bit generator based on chaotic lasers. 32 channels on the same side of the pump mode are recorded, processed, and successfully pass the NIST SP 80022 test (see in Fig. 3h), corresponding to an aggregation rate of 3.84 Tbps.
In Table 1, we present a comparation of different random bit generation schemes. Due to the massively parallelism provided by chaotic combs, the total bit rate of chaoticcombbased RBG shows the highest generation rate among opticalchaosbased methods. As the lack of data in published articles, the power consumption is not listed in Table 1. Considering simple setup with one pump laser and one microcavity, the generation of optical chaotic sources based on chaotic combs is powerefficient and lowcost compared with chaotic lasers^{43,44}, which require at least one laser diode per channel. In addition, our work gives the demonstration of parallel chaotic signal detection employing the integrated SiPh chip. Although the generation rate of chaoticcombbased RBG is lower than ASE (amplified spontaneous emission)based RBG with spectral or spatial parallelism, the use of chaotic systems offers the ability to synchronize the output of two chaos generators^{4,37}, which has been well studied for chaotic lasers^{2,5} but not demonstrated yet for chaotic combs. The synchronized systems can be arranged at the transmitter and receiver sides respectively. This synchronization enables key distribution and private communication, which is not possible with stochastic sources such as ASE and random lasers.
Computation acceleration based on chaotic combs
Optical chaos, with its fast and complex internal time evolution, is a powerful entropy source that can be used for exploration purposes^{45}. Chaotic lasers have been successfully employed to solve the multiarmed bandit problem (MAB)^{29,46,47}, a fundamental problem of reinforcement learning. To scale up the problem exponentially, a parallel scheme is required that employs parallel chaotic signals or is based on timedivisionmultiplexing^{48}. In this section, we propose the use of the chaotic comb as a massively parallel chaotic source to solve the MAB. Figure 4a shows the schematic of the optical decision making. There are N slot machines with a reward possibility p_{i}(i = 0 ∼ N − 1) respectively. The decision maker needs to find out the slot machine with the highest reward possibility by consecutively playing the slot machines. For each play, the decision maker chooses and plays one slot machine. The decision maker will change its selection strategy based on whether a reward is obtained. In the decision maker, N chaotic signals ∆I_{i}(t)(i = 0, 1, 2, 3) are detected and add with bias values B_{i}(t) respectively to produce A_{i}(t). Each channel is correlated with one slot machine. For each play, the slot machine with the highest biased signal A_{i}(t) is selected and played. As continuously playing the game, bias values will be tuned based on tugofwar method^{47} (see Methods).
In experiment, we detect 32 channels at the shortwavelength side of the pump mode one by one, to solve the 32armed bandit problem. The hit probabilities of the 32 slots are set as follows: P_{1} = 0.7, P_{2} = 0.5, P_{3} = 0.9, P_{4} = 0.1,…, P_{2j−1} = 0.7 and P_{2j} = 0.5. As shown in Fig. 4b, the chaosbased decision maker would initially explore a wide range to identify the slot machine with the highest hit probability. After sufficient exploration, the slot machine 3 is identified as the best one and frequently selected, indicating successful decision making. To evaluate the performance of the decision making based on chaotic combs, the slot machines are played C = 1000 cycles consecutively, and this process is repeated T = 1000 times. For tcycle, the correct decision ratio (CDR) is defined as T_{hit}/T, where T_{hit} represents the number of times a reward is obtained at the tth cycle among the T processes. A CDR above 95% is viewed as a successful decision and the first cycle getting a CDR > 95% is defined as the convergence cycle (CC). Figure 4c shows the CDR evolution along with the increase of cycles. Under the same system condition, the CDR evolution remains consistent across different N_{target}, which represent the number of the slot with the highest hit probability.
The scalability is a critical characteristic of the decision maker, where the convergence cycle will exponentially increase as the rise of problem scales. To assess the ability of solving large scale problems, we use the chaotic comb to solve the MBA problems with the number of slot machines N = 4, 8, 16, 32, 64, 128, 256. Limited by the gain bandwidth and the spectrum width of the chaotic comb, the number of available channels is 32. For N > 32, we record the 32 channels repeatedly to obtain enough channels of chaotic signals. As shown in Fig. 4d, the chaoticcombbased decision marker can solve the MBA problems even with a large scale. Figure 4e gives the performance comparation between chaoticcombbased decision marker and other methods^{47}. Fitted with the power function, the relationship between the convergence cycle CC and the number of slot machines N can be described as CC = 26.48∙N^{0.89}. Compared with widelyused algorithm such as UCB1tuned (upper confidence bound 1tuned) algorithm and Thompson sampling algorithm, methods based on opticalchaos show smaller scaling exponents, suitable for largescale problems. Among the optical chaos, the decision maker shows a superb convergence speed, close to the spatialchaos based method. While compared with the spatial chaos system, the chaotic microcomb benefits from integrated optoelectronics, providing a compact, lowcost, massproducible decision marker. Despite having fewer channels compared to the spatial chaos system^{47}, the chaotic comb feathers high generation rate (40 GSa/s employed here for decision making) and the adoption of timemultiplexing^{48} can be a promising approach to scale up the system, even though it may result in a tradeoff between speed and scalability.
Discussion
The bandwidth and the parallelization capability demonstrated here can be further improved by optimizing the design as well as fabrication for the microresonator. Currently the microresonator in this work is critically coupled, and by using an overcoupled structure instead, higher conversion efficiency, wider optical spectrum and higher power per comb line could be obtained without degenerating the chaotic properties. Another factor which limits the chaotic bandwidth currently is the intracavity nonlinear loss^{49,50} (discussed in Supplementary Note 8), which mainly comes from the three photon absorption and consequently free carrier absorption, leading to relatively high side lobes. This can be solved by employing an integrated PIN structure for the waveguide or working at a longerwavelength band^{51}. Combining these strategies, combs with chaotic bandwidth above 10 GHz can be expected with 100 mW pump powers. Furthermore, the spectral coverage of the comb can be further extended through dispersion engineering, accessing much more chaotic comb channels than those at C band in this work. By combining all these strategies, a parallel random number generator with beyond 3 Tbps total rate can be achieved, by using only one comb source (detailed discussions can be found in Supplementary Note 6 and 10), which is even better than the best benchtop chaotic system.
One key advantage of our approach is the scalability. The AlGaAsOI platform is fabricated by heterogeneous integration on Si substrate^{52}, and thus is compatible with the most widely used platform of silicon photonics^{53,54,55}. Such integration has recently been demonstrated^{56}, suggesting that our chaotic microcomb can be seamlessly implemented in fully integrated photonic systems, together with diverse silicon photonic engines such as optical I/O, optical computation^{57,58} unit or optical phase array for sensing^{59}. Other materials, such as silicon nitride^{60}, hydex^{61,62} or lithium niobite^{63}, can also be used for realizing chaotic microcombs, and potentially can also be integrated with silicon photonics.
Besides the applications we showed in this work, the parallel chaotic sources can also benefit many other applications. It can be used to generate chaotic signals for MIMO radar^{7,8}/LiDAR^{9,10,11}, which will lead to higher resolution due to the large bandwidth and higher energy efficiency. One great advantage of the chaotic microcomb, compared with previous widely used electronic chaotic source, is the capability of directly generating optical signals. Its output thus can be transmitted over long distance for communications, or emitted directly to freespace for sensing. As a result, this integrated, massively parallel chaotic source holds the promise to rewrite the paradigm of information technologies in the future.
Methods
Design and fabrication of the devices
The ring waveguides of the AlGaAsOI resonator were designed to work in anomalous dispersion with a crosssection of 400 nm × 650 nm. The width of the bus waveguide at the facet was designed to be 200 nm for efficient chiptofiber coupling. The fabrication of AlGaAs microresonators was based on heterogeneous wafer bonding technology. The epitaxial wafer growth was accomplished using molecularbeam epitaxy (MBE). A 248 nm deepultraviolet (DUV) stepper was used for the lithography. A photoresist reflow process and an optimized dry etch process were applied in waveguide patterning to minimize waveguide scattering loss. More fabrication details can be found in^{38} and^{52}. The silicon photonics PIC was fabricated on a 200 mm SOI wafer with a siliconlayer thickness of 220 μm and a BOX layer thickness of 2 μm using CMOScompatible processes at CompoundTek Pte. The PD exhibits 3 dB bandwidths more than 20 GHz. In our experiment, lensed fibers with different mode field diameter (MFD) were selected for the AlGaAsOI and SOI chips; the coupling loss is 3–5 dB per facet for AlGaAsOI waveguides and 2–3 dB per facet for Si waveguides.
Characterization of chaotic combs
The pump laser is provided by an externalcavitydiode laser (Toptica CTL 1550). For the dispersion estimation, the pump laser is scanning from 1515 nm to 1630 nm. The pump power is attenuated by a tunable optical attenuator to avoid the resonance distortion caused by the thermal drift. The resonant frequencies are recorded for dispersion calculation by \({D}_{{{{{\mathrm{int}}}}}}\left(\mu \right)={\omega }_{\mu }{\omega }_{0}\mu {D}_{1}\approx \frac{{D}_{2}}{2}{\mu }^{2}\), where ω_{μ} is frequency of the μ−th resonance and μ = 0 is the central mode. D_{1}/2π is the free spectral range (FSR) and D_{2} is the secondorder dispersion. A positive D_{2} indicates the anomalous dispersion shown in the Supplementary Note 6.
For chaotic comb generation, the pump laser is boosted to 26 dBm by a highpower EDFA and the ASE is partially filtered by a tunable bandpass filter. Lensed fibers are employed to couple the pump laser into the microresonator chip and collect the output chaotic comb into the test link. The remaining pump light is suppressed by a fiber bragg grating and amplified by a DWDM EDFA. Each comb line in the C band is filtered and the chaotic signal is recorded by an electrical spectrum analyzer in the frequency domain and the oscilloscope in the time domain. For the dataIrecorded by the oscilloscope, the autocorrelation function ACF and the crosscorrelation function XCF
are calculated. δI_{i}(t) is the fluctuation of the recorded data I_{i}(t) from channel i and is equal to I_{i}(t)−〈I_{i}(t)〉_{t}. ACF_{i}(τ) indicates the autocorrelation function of channel i and XCF_{ij}(τ) represents the crosscorrelation function between channel i and j. The maximum of XCF_{ij}(τ) is used to valuate the correlation between channel i and j, as shown in Fig. 3h.
Random bit generation
In the experiment of random bit generation based on the SiPh chip^{55}, the microring with FSR ∼90 GHz is used to fit the average channel spacing of the integrated AWG. With 26 dBm pump laser, a chaotic comb is generated and individual comb lines are coupled into the SiPh chip and selectively injected into certain GeSi PDs, after being filtered by a notch filter and amplified. The input power to the SiPh chip is maintained around 14 dBm to compensate the insertion loss of the edge coupler (∼2 dB per facet) and the AWG (5–8 dB). The bandwidth of the GeSi PDs is about 20 GHz with a bias voltage of −3 V, enough for the detection of the chaotic comb. For each channel, 2 × 108 points are recorded at the sampling rate of 20 GSa/s. For the random bit extraction, the raw data detected by the GeSi PD are downsampled to 6.67 GSa/s for a low correlation between continuous sampled data. Then, a delay difference (I(i + 6) − I(i)) is employed to the raw data I, obtaining ∆I with a symmetric distribution. The differential data is then digitalized into an 8 bit binary number. After that, we discard the most significant bits (MSBs) and keep 3 bits of LSBs (MLSBs) as the final output of RBG. The generation rate is thus equal to 3 times the sampling rate for each channel, where the random bit generation of 20 Gbps per channel is demonstrated for the SiPh chip. For the NIST SP 80022 test, 200 and 1000 bit sequences are employed as the input of the test respectively for Fig. 3g and Fig. 3h, where each bit sequence contains 10^{6} bits.
Optical decision making
The optical decision making process presented here is based on^{47}. In our experiment, 32 channels are filtered and recorded one by one. The recorded data are employed for an offline decisionmaking process. At cycle t, the slot machine with the highest biased value A_{i} is selected and played. The biased value A_{i} is calculated based on
B_{i} is the bias, which is tuned based on the estimated hit probabilities \({\hat{P}}_{i}\), given by:
Where T_{i} is the time that the slot machine i is selected, W_{i} (L_{i}) is the time of win (loss) as selecting slot machine i. \({\hat{P}}_{{top}1}\) and \({\hat{P}}_{{top}2}\) are the highest and second highest estimated hit probabilities. All parameters above are determined by the game process, except for k. The value of k is tuned to obtain the balance between exploration and convergence. The decision process will converge fast under a large k, where a large bias is obtained after a few cycles, and vice versa. For N = 4, 8, 16, 32, 64, 128, 256, k is set to 0.11, 0.17, 0.23, 0.33, 0.48, 0.58, 0.67 respectively, determined by sweep the k value
under different N as illustrated in Supplement note 13.
Numerical simulation
To get a deep insight into the process of the chaotic comb evolution, numerical simulations based on the LugiatoLefever equation
are carried out under different conditions. E stands for the intracavity temporal fields and α is the roundtrip cavity loss factor. δ_{0} is the detuning between coldcavity resonant frequency and pump laser. t_{R} is the roundtrip time of the primary mode and L is the roundtrip length. The pump filed is coupled into the cavity by \(\sqrt{\theta }\)E_{in}, where θ is the waveguide coupling coefficient and E_{in} is the pump field. β_{2} represents the secondorder dispersion coefficients. More simulation result about the chaotic comb can be found in Supplementary Note.
Data availability
The data that supports the plots within this paper and other findings of this study are available on Zenodo (https://doi.org/10.5281/zenodo.8105301). All other data used in this study are available from the corresponding authors upon reasonable request.
Code availability
The codes that support the findings of this study are available from the corresponding authors upon reasonable request.
References
Strogatz, S. H. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering (CRC press, 2018).
Argyris, A. et al. Chaosbased communications at high bit rates using commercial fibreoptic links. Nature 438, 343–346 (2005).
Wang, L. et al. Scheme of coherent optical chaos communication. Opt. Lett. 45, 4762–4765 (2020).
GarciaOjalvo, J. & Roy, R. Spatiotemporal communication with synchronized optical chaos. Phys. Rev. Lett. 86, 5204 (2001).
Gao, H. et al. 0.75 gbit/s highspeed classical key distribution with modeshift keying chaos synchronization of fabry–perot lasers. Light.: Sci. Appl. 10, 1–9 (2021).
Metropolis, N. & Ulam, S. The monte carlo method. J. Am. Stat. Assoc. 44, 335–341 (1949).
Feng, W. et al. Pulsedchaos mimo radar based on a single flatspectrum and deltalike autocorrelation optical chaos source. Opt. Express 30, 4782–4792 (2022).
Cheng, C.H., Chen, Y.C. & Lin, F.Y. Generation of uncorrelated multichannel chaos by electrical heterodyning for multipleinput–multipleoutput chaos radar application. IEEE Photon. J. 8, 1–14 (2015).
Lin, F.Y. & Liu, J.M. Chaotic lidar. IEEE J. Sel. Top. Quant. Electron. 10, 991–997 (2004).
Chen, J.D., Wu, K.W., Ho, H.L., Lee, C.T. & Lin, F.Y. 3d multiinput multioutput (mimo) pulsed chaos lidar based on timedivision multiplexing. IEEE J. Sel. Top. Quant. Electron. 28, 1–9 (2022).
Lukashchuk, A., Riemensberger, J., Tusnin, A., Liu, J. & Kippenberg, T. Chaotic microcomb based parallel ranging. arXiv Prepr. arXiv 2112, 10241 (2021).
Gong, L., Zhang, J., Liu, H., Sang, L. & Wang, Y. True random number generators using electrical noise. IEEE Access 7, 125796–125805 (2019).
Hamburg, M., Kocher, P. & Marson, M. E. Analysis of intel’s ivy bridge digital random number generator. Online: https://cdn.atraining.ru/docs/Intel_TRNG_Report_20120312.pdf (2012).
Wang, Y., Hui, C., Liu, C. & Xu, C. Theory and implementation of a very high throughput true random number generator in field programmable gate array. Rev. Sci. Instrum. 87, 044704 (2016).
Yang, S.S., Lu, Z.G. & Li, Y.M. Highspeed postprocessing in continuousvariable quantum key distribution based on fpga implementation. J. Lightwave Technol. 38, 3935–3941 (2020).
Stipčević, M. & Bowers, J. Spatiotemporal optical random number generator. Opt. Exp. 23, 11619–11631 (2015).
Monifi, F. et al. Optomechanically induced stochastic resonance and chaos transfer between optical fields. Nat. Photon. 10, 399–405 (2016).
Wu, J. et al. Mesoscopic chaos mediated by drude electronhole plasma in silicon optomechanical oscillators. Nat. Commun. 8, 1–7 (2017).
Virte, M., Panajotov, K., Thienpont, H. & Sciamanna, M. Deterministic polarization chaos from a laser diode. Nat. Photon. 7, 60–65 (2013).
Sciamanna, M. & Shore, K. A. Physics and applications of laser diode chaos. Nat. Photon. 9, 151–162 (2015).
Kanter, I., Aviad, Y., Reidler, I., Cohen, E. & Rosenbluh, M. An optical ultrafast random bit generator. Nat. Photon. 4, 58–61 (2010).
Deng, Y. et al. Midinfrared hyperchaos of interband cascade lasers. Light.: Sci. Appl. 11, 1–10 (2022).
Albert, F. et al. Observing chaos for quantumdot microlasers with external feedback. Nat. Commun. 2, 1–5 (2011).
Mørk, J., Mark, J. & Tromborg, B. Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback. Phys. Rev. Lett. 65, 1999 (1990).
Qiao, L. et al. Generation of flat wideband chaos based on mutual injection of semiconductor lasers. Opt. Lett. 44, 5394–5397 (2019).
Li, P. et al. Observation of flat chaos generation using an optical feedback multimode laser with a bandpass filter. Opt. Express 27, 17859–17867 (2019).
Li, S.S., Li, X.Z. & Chan, S.C. Chaotic timedelay signature suppression with bandwidth broadening by fiber propagation. Opt. Lett. 43, 4751–4754 (2018).
GarcíaOjalvo, J. & Roy, R. Parallel communication with optical spatiotemporal chaos. IEEE Trans. Circuits Syst. I: Fundamental Theory Appl. 48, 1491–1497 (2001).
Han, Y. et al. Generation of multichannel chaotic signals with time delay signature concealment and ultrafast photonic decision making based on a globallycoupled semiconductor laser network. Photon. Res. 8, 1792–1799 (2020).
Zhao, A. et al. Parallel generation of lowcorrelation wideband complex chaotic signals using cw laser and externalcavity laser with selfphasemodulated injection. OptoElectron. Adv. 5, 200026–1 (2022).
Coulibaly, S. et al. Turbulenceinduced rogue waves in kerr resonators. Phys. Rev. X 9, 011054 (2019).
Anderson, M., Leo, F., Coen, S., Erkintalo, M. & Murdoch, S. G. Observations of spatiotemporal instabilities of temporal cavity solitons. Optica 3, 1071–1074 (2016).
Liu, Z. et al. Characterization of spatiotemporal chaos in a kerr optical frequency comb and in all fiber cavities. Opt. Lett. 42, 1063–1066 (2017).
Ji, X. et al. Chipbased frequency comb sources for optical coherence tomography. Opt. Express 27, 19896–19905 (2019).
Marchand, P. J. et al. Soliton microcomb based spectral domain optical coherence tomography. Nat. Commun. 12, 427 (2021).
Chen, R. et al. Breaking the temporal and frequency congestion of lidar by parallel chaos. Nat. Photon. 17, 306–314 (2023).
Xiao, J., Hu, G. & Qu, Z. Synchronization of spatiotemporal chaos and its application to multichannel spreadspectrum communication. Phys. Rev. Lett. 77, 4162 (1996).
Chang, L. et al. Ultraefficient frequency comb generation in algaasoninsulator microresonators. Nat. Commun. 11, 1–8 (2020).
Shu, H. et al. Microcombdriven silicon photonic systems. Nature 605, 457–463 (2022).
Carmon, T., Yang, L. & Vahala, K. J. Dynamical thermal behavior and thermal selfstability of microcavities. Opt. Express 12, 4742–4750 (2004).
Kim, K. et al. Massively parallel ultrafast random bit generation with a chipscale laser. Science 371, 948–952 (2021).
Reidler, I., Aviad, Y., Rosenbluh, M. & Kanter, I. Ultrahighspeed random number generation based on a chaotic semiconductor laser. Phys. Rev. Lett. 103, 024102 (2009).
Jørgensen, A. et al. Petabitpersecond data transmision using a chipscale microcomb ring resonator source. Nat. Photon. 16, 798–802 (2022).
MarinPalomo, P. et al. Microresonatorbased solitons for massively parallel coherent optical communications. Nature 546, 274–279 (2017).
Kitayama, K.i et al. Novel frontier of photonics for data processing—photonic accelerator. Apl. Photon. 4, 090901 (2019).
Naruse, M., Terashima, Y., Uchida, A. & Kim, S.J. Ultrafast photonic reinforcement learning based on laser chaos. Sci. Rep. 7, 1–10 (2017).
Morijiri, K. et al. Parallel photonic accelerator for decision making using optical spatiotemporal chaos. Optica 10, 339–348 (2023).
Naruse, M. et al. Scalable photonic reinforcement learning by timedivision multiplexing of laser chaos. Sci. Rep. 8, 1–16 (2018).
Benis, S. et al. Threephoton absorption spectra and bandgap scaling in directgap semiconductors. Optica 7, 888–899 (2020).
Espinosa, D. H., Harrigan, S. R., Awan, K. M., Rasekh, P. & Dolgaleva, K. Geometrydependent twophoton absorption followed by freecarrier absorption in algaas waveguides. JOSA B 38, 3765–3774 (2021).
Yu, M., Okawachi, Y., Griffith, A. G., Lipson, M. & Gaeta, A. L. Modelocked midinfrared frequency combs in a silicon microresonator. Optica 3, 854–860 (2016).
Xie, W. et al. Ultrahighq algaasoninsulator microresonators for integrated nonlinear photonics. Opt. Express 28, 32894–32906 (2020).
Thomson, D. et al. Roadmap on silicon photonics. J. Opt. 18, 073003 (2016).
Soref, R. The past, present, and future of silicon photonics. IEEE J. Sel. Top. Quant. Electron. 12, 1678–1687 (2006).
Liu, Z. et al. 25× 50 gbps wavelength division multiplexing silicon photonics receiver chip based on a silicon nanowirearrayed waveguide grating. Photon. Res. 7, 659–663 (2019).
Xie, W. et al. Siliconintegrated nonlinear iiiv photonics. Photon. Res. 10, 535–541 (2022).
Feldmann, J. et al. Parallel convolutional processing using an integrated photonic tensor core. Nature 589, 52–58 (2021).
Xu, X. et al. 11 tops photonic convolutional accelerator for optical neural networks. Nature 589, 44–51 (2021).
Poulton, C. V. et al. 8192element optical phased array with 100° steering range and flipchip cmos. In 2020 Conference on Lasers and ElectroOptics (CLEO), JTh4A.3 (Optical Society of America, 2020).
Liu, J. et al. Highyield, waferscale fabrication of ultralowloss, dispersionengineered silicon nitride photonic circuits. Nat. Commun. 12, 1–9 (2021).
Razzari, L. et al. Cmoscompatible integrated optical hyperparametric oscillator. Nat. Photon. 4, 41–45 (2010).
Xu, X. et al. 11 tops photonic convolutional accelerator for optical neural networks. Nature 589, 44–51 (2021).
ShamsAnsari, A. et al. Reduced material loss in thinfilm lithium niobate waveguides. Apl. Photon. 7, 081301 (2022).
Monet, F., Boisvert, J.S. & Kashyap, R. A simple highspeed random number generator with minimal postprocessing using a random raman fiber laser. Sci. Rep. 11, 1–8 (2021).
Liu, J. et al. 117 gbits/s quantum random number generation with simple structure. IEEE Photon. Technol. Lett. 29, 283–286 (2016).
Okawachi, Y. et al. Quantum random number generator using a microresonatorbased kerr oscillator. Opt. Lett. 41, 4194–4197 (2016).
Bai, B. et al. 18.8 gbps realtime quantum random number generator with a photonic integrated chip. Appl. Phys. Lett. 118, 264001 (2021).
Cao, G., Zhang, L., Huang, X., Hu, W. & Yang, X. 16.8 tb/s true random number generator based on amplified spontaneous emission. IEEE Photon. Technol. Lett. 33, 699–702 (2021).
Wang, A., Wang, L., Li, P. & Wang, Y. Minimalpostprocessing 320gbps true random bit generation using physical white chaos. Opt. Express 25, 3153–3164 (2017).
Sakuraba, R., Iwakawa, K., Kanno, K. & Uchida, A. Tb/s physical random bit generation with bandwidthenhanced chaos in threecascaded semiconductor lasers. Opt. Express 23, 1470–1490 (2015).
Xiang, S. et al. 2.24tb/s physical random bit generation with minimal postprocessing based on chaotic semiconductor lasers network. J. Lightwave Technol. 37, 3987–3993 (2019).
Ma, C.G., Xiao, J.L., Xiao, Z.X., Yang, Y.D. & Huang, Y.Z. Chaotic microlasers caused by internal mode interaction for random number generation. Light.: Sci. Appl. 11, 187 (2022).
Ugajin, K. et al. Realtime fast physical random number generator with a photonic integrated circuit. Opt. Express 25, 6511–6523 (2017).
Acknowledgements
The authors thank Fenghe Yang, Yan Zhou in Peking University Yangtze Delta Institute of Optoelectronics for microcomb packaging support. This work was supported by the National Key Research and Development Program of China (2021YFB2800400, X.W.), National Natural Science Foundation of China (62235002, 12204021, X.W.), Beijing Municipal Science and Technology Commission (Z221100006722003, X.W.), Beijing Municipal Natural Science Foundation (Z210004, X.W.), Nantong Science and Technology Bureau (JC2021002, Z.G.).
Author information
Authors and Affiliations
Contributions
The experiments were conceived by B.S. and H.S. The devices were designed by H.S., L.C., Z.L., and B.C. The AlGaAsOI microresonators were fabricated by W.X. and L.C. The chip was packaged by Z.G. The microcomb simulation and modelling were conducted by B.S. The experiments were performed by B.S. and H.S., with the assistance by R.C. X.Z., Y.W., and Y.Z. The results were analyzed by B.S. All authors participated in writing the paper. The project was under the supervision of H.S., S.Y., L.C., and X.W.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Communications thanks Anton Lukashchuk and the other, anonymous, reviewers for their contribution to the peer review of this work. A peer review file is available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Shen, B., Shu, H., Xie, W. et al. Harnessing microcombbased parallel chaos for random number generation and optical decision making. Nat Commun 14, 4590 (2023). https://doi.org/10.1038/s4146702340152w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s4146702340152w
This article is cited by

Scalable parallel ultrafast optical random bit generation based on a single chaotic microcomb
Light: Science & Applications (2024)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.