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Abstract
Recent demonstrations of chaos-based secure communication have proven the feasibility of secured transmission of high-speed (tens of Gbit/s) signals over certain distances (∼100-km), which bring hope for secure communication from theoretical analysis to practical applications. So far, the chaos-based secure communication system with chaos-masking (CMS) encryption is considered as one of the most important and feasible schemes. In this paper, an optical chaotic carrier generated by an opto-electronic oscillator is used to encrypt 112-Gbit/s message by CMS encryption for data transmission over a 1040-km single-mode-fiber. The message is successfully decrypted by combining coherent detection and our proposed blind decryption algorithms, which can successfully separate the chaotic carrier and the message with the bit-error-rate (BER) below the forward error correction (FEC) threshold. Experimental results show that the coherent detection combined digital signal processing algorithms may be a possible way to promote the practical applications of chaotic optical communication in the future. In addition, this paper reveals that the security of the CMS encryption may be not high enough for those systems requiring rigorous confidentiality. Subsequently, we further discuss the bottlenecks encountered in current high-speed chaotic optical communication systems and analyze how to improve and weight the security and practicability.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Driven by the ever-growing demands for higher bandwidth and faster speed connections of various multimedia and data services (e.g., big data, cloud computing, streaming video, Internet of Things, machine-to-machine communication, remote surgery and so on), global network traffic has presented explosive growth over the past decade [1]. Optical fiber communication with the advantages of high bandwidth and anti-electromagnetic interference is increasingly being used in place of metal wires, and is widely used for data transmission in modern society. Due to the diversity of application scenarios of optical fiber transmission networks, the risk of eavesdropping or interception may be increased for message in transmission networks. Therefore, the development of new strategies or schemes (e.g. quantum key distribution (QKD) [2–4] and chaos-based secure communication [5–8]) to protect transmission message from eavesdropping or interception has gained significant attention from researchers. Among these strategies, QKD generates shared secret keys to encrypt communication channels with unconditional security [9]. However, it has been just demonstrated at some advanced systems [10–13] which yield secret key rate of bit/s with few hundreds of kilometers ultralow-loss fiber, so that it would be difficult to be compatible with future high-speed optical communication systems. By contrast, another secure communication scheme, chaos-based secure communication that is via robust hardware encryption to achieve message masking and has been verified in various high-speed (Gbit/s) transmission systems [5,14,15] can be regarded as the most potential complement to secure communication.
In chaos-based secure communication systems, optical chaotic carrier can be generated by all optical feedback [16–21] and opto-electronic feedback [5,14,15,22–24] based on the nonlinear chaotic behavior of laser or modulator. Among these, the opto-electronic feedback based on modulator which dates from the Ikeda ring cavity [25] has been widely used to generate broadband chaotic carrier whose bandwidth is just limited by electric devices. In particular, the time-delay, feedback strength and filter parameters of that can be tuned for the generation of robust, high-dimensional chaos. At present, various verification experiments of high-speed chaotic secure transmission have been reported by using opto-electronic feedback oscillator based on modulator. In 2004, N. Gastaud et al. had reported a bit-rate encryption up to 3-Gbit/s with acceptable masking efficiency and excellent decoding quality [20]. In 2005, A. Argyris et al. had successfully demonstrated the first field trial of 2.4-Gbit/s chaos-based secure communication under 120-km standard single mode fiber (SSMF) and dispersion compensation fiber (DCF) transmission in the metropolitan area network of Athens, Greece [5]. In 2010, R. Lavrov et al. had reported 10-Gbit/s field experiment under 120-km dispersion managed transmission link in Besancon, France [14]. In 2017, J. Ai et al. performed a series of 5-Gbit/s carrierless amplitude/phase modulation and 10-Gbit/s on-off key secure transmission experiments over 2.6-km multimode fiber [26]. In 2018, J. Ke et al. had successfully verified a 30-Gbit/s signal transmission of a duo-binary message hidden in a chaotic optical carrier over 100-km dispersion managed transmission link [15]. Subsequently, the group had further demonstrated a 32-Gbit/s message transmission over 20-km SSMF transmission assisted by deep learning to achieve chaos synchronization [27]. As described, these works based on opto-electronic feedback had brought hope to practical applications of secure communication.
In this paper, we focus on using opto-electronic feedback oscillator with the nonlinear dynamics of modulator to generate optical chaotic carrier, then using chaos masking (CMS) encryption to mask high-speed transmission message. Without either hardware synchronization or machine-learning based decoding, we have successfully decrypted the polarization and phase domain chaotic-embedded message by combining coherent detection and our proposed blind decryption algorithms. For polarization domain chaotic encryption, a blind polarization tracking chaotic decryption (PTCD) scheme is proposed and experimentally verified in chaotic-embedded 28-Gbit/s on-off-keying (OOK) and 56-Gbit/s quadrature phase-shift-keying (QPSK) systems over 2000-km dispersion unmanaged transmission link, and 112-Gbit/s 16-level quadrature amplitude modulation (16QAM) system over 1040-km dispersion unmanaged transmission link. Meanwhile, for the phase domain chaotic encryption, a chaotic-embedded 28-Gbit/s OOK signal over 1440-km dispersion unmanaged transmission link is successfully recovered by our proposed carrier frequency and phase chaotic decryption (CFPCD) scheme. The bit-error-rate (BER) performance of the proposed two decryption schemes are all below 20% FEC threshold (BER=2.4×10−2), which means the proposed decryption schemes can separate the chaotic carrier and the transmission message in high-speed chaotic optical communication systems. Next, we further discuss the security of encryption mechanism and analyze how to improve the security of high-speed chaotic optical communication systems.
2. Principle
As one of the most important components of secure communication, the chaotic optical communication systems with CMS encryption have obtained extensive attention over the past years. The structure of the chaotic optical communication systems is mainly composed of three parts: transmitter with CMS encryption way, fiber link without inline dispersion compensation, and receiver with advanced chaotic decryption technique, as depicted in Fig. 1. In this paper, the advanced decryption technique combining coherent detection and digital signal processing module is introduced to compensate transmission impairments and to separate the chaotic carrier and the transmission message. Here, for polarization and phase domain chaotic encryption, we present two effective chaotic decryption algorithms embedded in digital signal processing module to achieve chaotic decryption process.
On the transmitter side of the chaotic optical communication systems, the optical chaotic carrier is generated by applying opto-electronic feedback loop based on intensity modulator. As shown in Fig. 2, the feedback loop is composed of a continuous-wave (CW) laser with output power P0, a Mach-Zehnder modulator (MZM) with half-wave voltage Vπ, an erbium doped fiber amplifier (EDFA) with amplification gain g1, two variable optical attenuators (VOAs) with attenuation η1 and η2, a variable optical delay line (VODL) with delay time Δτ, an electrical amplifier (EA) with amplification gain g2, and a photodetector (PD) with responsivity R, which all are connected together to generate chaotic carrier c(t). Then, the generated chaotic carrier c(t) and arbitrary modulation format message m(t) are coupled together to achieve chaotic encryption and obtain the chaotic-embedded signal CM(t). In here, the process of chaotic encryption [15,28] (CMS encryption) can be expressed as
On the receiver side, the traditional decryption way is to construct a feedback loop composing of various optical and electrical devices to realize chaos synchronization, which can be called as hardware synchronization as shown in Fig. 1. Note that these optical and electrical devices in the feedback loop must be well matched with those in transmitter for achieving chaos synchronization. It is difficult to find two identical devices in the real world so that a well-matched hardware cannot always be guaranteed. Besides, the transmission link impairments need to be compensated by DCF or tunable dispersion compensation (TDC) module before hardware synchronization, so the hardware synchronization may be still a long way to practical applications. Furthermore, in order to simplify chaos synchronization operation and improve system practicability, a deep learning-based software synchronization receiver which can maintain the performance consistency with the chaotic-embedded transmitter has been proposed and verified in chaotic optical transmission systems [27]. As mentioned in [27], the deep learning-based chaotic receiver and all digital signal processing functions can be integrated into a chip [29] so that the case can achieve consistent synchronization performance for point to multipoint networking, which is almost impossible to be achieved by traditional hardware synchronization. Although the deep learning-based receiver presents good performance, it is still powerless for long-distance transmission systems with serious dispersion and nonlinear effects. Different from the traditional decryption technique, this paper presents an advanced decryption technique combining coherent detection and digital signal processing module, which would be able to achieve chaotic decryption of high-speed and long-distance chaotic optical transmission systems. For polarization domain chaotic encryption, we propose a blind and effective PTCD scheme to decode the chaotic-embedded signal. Subsequently, a blind CFPCD scheme is proposed for phase domain chaotic encryption.
2.1 Principle of chaotic decryption for polarization domain chaotic encryption
Before performing chaotic decryption, digital backward propagation (DBP) algorithm and timing recovery algorithm would be utilized to compensate almost all linear and nonlinear impairments suffered in long-distance transmission link. After that, the chaotic-embedded signal Einx, Einy enter into the proposed blind PTCD scheme to realize chaotic decryption as depicted in Fig. 3. Polarization rotation in transmission link can be described by azimuth angle and ellipticity angle [30]. To recover polarization state by using a pure feedforward approach, we can firstly set test range of azimuth angle θ and ellipticity angle φ as
The polarization rotation process can be expressed as
Next, we utilize Godard’s error module to analyze signal quality of the rotated signals Eoutx,k,m and Eouty,k,m. Here, the Godard’s error [31] can be defined as
In here, in order to make the blind PTCD scheme more simplified and faster, we introduce two iterations of azimuth angle and ellipticity angle estimation in the scheme. The first iteration of the angle estimation process is increased in steps of δ1= π/10 to obtain a set of sub-optimum anglesθsubopt, φsubopt as depicted in Fig. 4(a). Hereafter, we apply the set of sub-optimum angles and the step size δ1 to construct two new test ranges (from θsubopt-δ1 to θsubopt+δ1, and from φsubopt-δ1 to φsubopt+δ1) for the second iteration of the angle estimation process. The process is performed with a higher resolution of δ2=π/50 to obtain the exact set of optimum angles θopt, φopt as depicted in Fig. 4(b). It is no doubt that the step sizes δ1 and δ2 are smaller, the accuracy and complexity are all higher. Therefore, it is necessary to comprehensively weigh between complexity and accuracy when choosing step size.
2.2 Principle of chaotic decryption for phase domain chaotic encryption
As shown in Fig. 5, we present a blind CFPCD scheme for phase domain chaotic encryption. Before performing decryption process, various linear and nonlinear impairments suffered in long-distance transmission link require to be compensated by DBP algorithm and timing recovery algorithm in digital signal processing module. After compensation, the chaotic-embedded signal Ein firstly carries out fast Fourier transform (FFT) operation to obtain frequency spectrum, and then frequency offset preliminary value fc can be determined by searching maximum pilot of spectrum. Next, we can use the preliminary value fc to recovery frequency offset of the signal Ein, which can be described as
After that, the signal Einc is converted from a serial sequence to multiple parallel sequences Einc,k. Due to the frequency shift of two lasers under long-term operation, this paper further utilizes phase slope lf of parallel sequences Einc,k to estimate exact frequency offset for achieving accurate frequency offset compensation. After that, we apply the mean value of phase information whose modulus must be greater than a threshold Rth to estimate the laser phase noise, because these phase information would not be effected by transmission message. Finally, these parallel sequences after carrier frequency and phase recovery are converted back into a serial sequence. In here, we can obtain the complex constellation diagram and the message eye-diagram after chaotic decryption as depicted in Fig. 5.
3. Experimental setup and results
3.1 Experimental setup and results for polarization domain chaotic encryption
The experimental setup of 28-Gbit/s chaotic-embedded OOK, 56-Gbit/s chaotic-embedded QPSK, and 112-Gbit/s chaotic-embedded 16QAM optical transmission systems for polarization domain chaotic encryption are shown in Fig. 6. In message generation part, a test sequence with 215−1 pseudo-random bit sequence (PRBS) is generated and maps into OOK or mQAM with 2 samples per symbol. The up-sampled signals are shaped by using a square root raised cosine (SRRC) with a roll-off factor of 1.0. Next, a pre-distortion operation is utilized to overcome the frequency roll-off of digital to analog converter (DAC). The light from CW1 laser at ∼1549.32-nm with ∼100-kHz linewidth is modulated by a MZM or IQ modulator driven by the DAC operation 64-GSa/s and 25-GHz analog bandwidth to generate 28-Gbit/s OOK, 56-Gbit/s QPSK, 112-Gbit/s 16QAM message. The other part is to generate chaotic carrier by the electro-optic feedback loop based on intensity modulator. The light from CW2 laser at ∼1549.32-nm with ∼100-kHz linewidth and 10-dBm output power is injected into a MZM intensity modulator with a 3-dB bandwidth of 40-GHz driven by an EA with 75-kHz to 10-GHz frequency response and 14-dB to 26-dB variable gain. The insertion loss of MZM cannot be ignored, so the optical power after MZM modulator needs to be adjusted by an EDFA and a VOA. After that, chaotic carrier and message under polarization orthogonality are combined into polarization beam combiner (PBC) to obtain the chaotic-embedded signal, and then the signal is divided into two parts by an optical coupler, where one part is used for secure transmission and the other part is fed into the electro-optic feedback loop to participate the chaotic generation of the next period. Note that the chaotic mask coefficient between the message and the chaotic carrier can be adjusted by VOA1 and VOA2 before PBC module. Here, it is defined as the ratios of the average power of the message to the chaotic carrier. The chaotic-embedded signal is delayed by a delay line (DL) to produce about a 75-ns time delay. The VOA3 before PD is applied to control feedback strength of the feedback loop, then PD with a 3-dB bandwidth of 17-GHz and a trans-impedance amplifier converts the optical signal to electric signal. Next, the electric signal amplified by EA is modulated by the MZM modulator in the loop to generate chaotic carrier of the next period. The transmission link is composed of multi-spans SSMFs and multiple EDFAs. The dispersion parameter, attenuation, and nonlinear coefficient of SSMF are 16.9-ps/nm/km, 0.2-dB/km, 1.27-km−1·W−1, respectively. Fiber loss of each span is compensated using an EDFA with a noise figure of ∼5-dB. An optical band-pass filter (OBPF) with 0.4-nm bandwidth is applied to filter out-of-band amplified spontaneous emission (ASE) noise. In receiver side, the chaotic-embedded signal and local oscillator (LO) light are combined into an integrated coherent receiver to obtain the electrical signals with phase, intensity, and polarization information, after that the signals are sampled at 80-GSamples/s by real-time digital oscilloscope with 33-GHz electrical bandwidth. Finally, the digital signals are processed in off-line digital signal processing (DSP) module. A ∼ MHz digital polarization rotation [32] is applied in the module to improve the security of the chaotic optical transmission systems, while the proposed chaotic decryption scheme is used to decrypt the embedded signals in the module. It is worth noting that the polarization rotation should be applied to the transmitter to improve the security. Because there is no such high-speed polarization rotator in our laboratory, the digital polarization rotation is used to simulate the function of high-speed polarization rotator.
The performance of the blind PTCD scheme for polarization domain chaotic encryption has been verified in a series of chaotic-embedded 28-Gbit/s OOK, 56-Gbit/s QPSK, and 112-Gbit/s 16QAM systems as depicted in Fig. 7. We firstly calculate the BER performance of the chaotic-embedded and decrypted 28-Gbit/s OOK signal after 2000-km transmission as shown in Fig. 7(a). Here, we find that BER of the chaotic-embedded signal before decryption is higher than 3×10−1 under all mask coefficients (i.e. from 0.6 to 1.8), which is defined as the optical power of a message compared to the chaos power. After decryption by the proposed scheme, the BER performance of message under 2000-km fiber transmission is below 20%FEC threshold (BER=2.4×10−2). Furthermore, we also calculate the BER performance of chaotic-embedded and decrypted signals (e.g. 56-Gbit/s QPSK, and 112-Gbit/s 16QAM) under different transmission distances. The BER of two chaotic-embedded signals before decryption are all higher than 3×10−1 for all mask coefficients. Thus, it can be considered that the two chaotic-embedded transmission systems are also hard to directly be decrypted. Assisted by the proposed PTCD decryption scheme, the chaotic-embedded 56-Gbit/s QPSK signal after 2000-km fiber transmission can be effectively decrypted below 7%FEC threshold (BER=3.8×10−3), and the decrypted performance for the chaotic-embedded 112-Gbit/s 16QAM signal with 1040-km can be achieved below 20%FEC threshold. In addition, we also find the decryption performance would become worse when the mask coefficient decreases. This is because that the residual error of chaos cancellation is equivalent to large noise power, which would directly lead to the performance of the proposed PTCD decryption scheme to decrease.
Fig. 7. BER performance of (a) the chaotic-embedded 28-Gbit/s OOK signal and decrypted 28-Gbit/s OOK signal over 2000-km transmission, (b) the chaotic-embedded 56-Gbit/s QPSK signal and decrypted 56-Gbit/s QPSK signal over 2000-km transmission, and (c) the chaotic-embedded 112-Gbit/s 16QAM signal and decrypted 112-Gbit/s 16QAM signal over 1040-km transmission. The inserted constellation diagrams are before and after chaotic decryption, respectively.
3.2 Experimental setup and results for phase domain chaotic encryption
We further study the other secure optical transmission system based on phase domain chaotic encryption, and the experimental setup is shown in Fig. 8. The light from CW1 at ∼1549.32-nm with ∼100-kHz linewidth is modulated using an integrated LiNbO3 I/Q modulator. The light firstly is split into two parts (e.g. the one for message generation, the other one for generation of chaotic carrier) in IQ modulator which is composed of an upper arm modulator MZM1, a 90° phase shifter, and a lower arm modulator MZM2. The MZM1 is driven by the DAC operation 64-GSa/s and 25-GHz analog bandwidth to generate 10-Gbit/s or 28-Gbit/s OOK signal m(t), and the MZM2 is applied to modulate chaotic carrier c(t) which is generated by an electro-optic feedback loop whose device parameters are the same as Fig. 6. Then, two lights with a phase difference of 90 degrees are coupled through an optical coupler to obtain a chaotic-embedded signal CM(t). Next, the CM(t) is transmitted via fiber transmission link whose structure and device parameters are also the same as the setup of polarization domain chaotic encryption system in Fig. 6. After long-haul transmission, the chaotic-embedded signal is detected by the coherent receiver and decrypted by the proposed CFPCD decryption scheme.
The BER performance of the proposed CFPCD decryption scheme for phase domain chaotic encryption has been measured in the chaotic-embedded 10-Gbit/s and 28-Gbit/s OOK systems under different fiber transmission distances as depicted in Fig. 9. For these two bit rates, the BER of the chaotic-embedded signal are all higher than 1×10−1, so that we can consider an eavesdropper would be difficult to directly decrypt the chaotic-embedded signal without the aid of the proposed scheme. As shown in Fig. 9(a), the proposed scheme can successfully decrypt the chaotic-embedded 10-Gbit/s OOK signal, and the BER performance after about 2000-km fiber transmission is below 20%FEC threshold. The decrypted performance of the chaotic-embedded 28-Gbit/s OOK signal after 1440-km fiber transmission is also below 20%FEC threshold in Fig. 9(b). Certainly, the decryption performance can be maintained below 7%FEC threshold when fiber transmission distance is less than 1800-km for 10-Gbit/s OOK system, and less than 1100-km for 28-Gbit/s OOK system. It is well known that the signal with higher bit-rate would be more sensitive to transmission link impairments, so the decryption performance of chaotic-embedded 28-Gbit/s OOK signal is worse than that of lower bit-rate transmission system.
Fig. 9. BER performance of the chaotic optical communication based on phase domain chaotic encryption with different fiber lengths. (a) BER performance of the chaotic-embedded 10-Gbit/s OOK signal and decrypted 10-Gbit/s OOK signal; (b) BER performance of 28-Gbit/s OOK signal and decrypted 28-Gbit/s OOK signal;
4. Discussion and conclusions
At present, researchers have been paying attention to improve transmission rate and distance for making chaos-based secure communication compatible with current commercial optical fiber communication systems. Another important factor, security which would determine success or failure of chaotic-embedded transmission, cannot be ignored in the next high-speed, large-capacity, long-distance optical communication systems. In our previous works [33–35], we mainly focus on how to increase the complexity (the security) of chaotic carrier. However, we find that the security of encryption mechanism between the chaotic carrier and the message may be more important, because it may determine whether the chaotic-embedded signal would be directly decrypted by eavesdropper. The CMS encryption way with simple implementation process is one of the most commonly used encryption ways in chaos-based secure communication. In traditional low-speed and short-distance systems, the transmission message are masked though CMS encryption under the same wavelength and polarization state. The case with the same polarization state would show high security, but the beat frequency effect between chaotic carrier and message would inevitably affect the quality of chaotic decryption. Once the transmission distance becomes longer (i.e. ∼100-km) and the transmission rate becomes higher (i.e. tens of Gbit/s), the beat frequency effect could directly lead to chaotic decryption failure. In this paper, we explore to introduce orthogonal bases (with polarization in one case, with I + Q in the other) between chaotic carrier and message to alleviate the serious beat frequency effect, while the security of chaotic-embedded transmission may be weakened to some extent. Then, for trading off security and practicability, this paper further applies fast polarization perturbation to further supply and improve the security. While the preliminary exploration scheme in this paper may not be the optimal or only high-speed chaos-based secure communication solution, researchers may need to further explore how to improve the security in the future. However, we believe that the case with better decryption performance and relatively high security may be able to promote the chaotic transmission system into the practical application.
In order to realize large-capacity, long-distance, high-security optical fiber transmission systems, we may explore combine the chaotic encryption method with other effective encryption methods (e.g. digital chaotic ciphers [36], optical steganography [37,38], optical CDMA [39], quantum stream cipher [40]) to further improve or enhance the security and practicability. In addition, the chaos-based secure communication systems need to be compatible with high-speed and long-haul transmission simultaneously in the future, so that the accurate and effective impairments compensation would be very necessary no matter online or offline. Here, we predict that coherent detection and digital signal processing technology would play an important role for link impairments compensation in chaos-based transmission system in the future. Furthermore, it will be very important to introduce or propose new security mechanism in the future high-speed optical transmission systems.
Funding
National Key Research and Development Program of China (2019YFB1803500); National Natural Science Foundation of China (61860206006, 62005228).
Disclosures
The authors declare no conflicts of interest.
References
1. F. Al-Turjman, E. Ever, and H. Zahmatkesh, “Small Cells in the Forthcoming 5G/IoT: Traffic Modelling and Deployment Overview,” IEEE Commun. Surv. Tutorials 21(1), 28–65 (2019). [CrossRef]
2. M. Lucamarini, Z. Yuan, J. Dynes, and A. Shields, “Overcoming the rate–distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018). [CrossRef]
3. S. Liao, W. Cai, W. Liu, L. Zhang, Y. Li, J. Ren, J. Yin, Q. Shen, Y. Cao, Z. Li, F. Li, X. Chen, L. Sun, J. Jia, J. Wu, X. Jiang, J. Wang, Y. Huang, Q. Wang, Y. Zhou, L. Deng, T. Xi, L. Ma, T. Hu, Q. Zhang, Y. Chen, N. Liu, X. Wang, Z. Zhu, C. Lu, R. Shu, C. Peng, J. Wang, and J. Pan, “Satellite-to-ground quantum key distribution,” Nature 549(7670), 43–47 (2017). [CrossRef]
4. V. Scarani, H. Bechmann-Pasquinucci, N. Cerf, D. Miloslav, L. Norbert, and P. Momtchil, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009). [CrossRef]
5. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. Mirasso, L. Pesquera, and K. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005). [CrossRef]
6. U. Atsushi, A. Kazuya, I. Masaki, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2(12), 728–732 (2008). [CrossRef]
7. M. Sciamanna and K. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015). [CrossRef]
8. D. Navarro-Urrios, N. Capuj, M. Colombano, P. García, M. Sledzinska, F. Alzina, A. Griol, A. Martínez, and C. Sotomayor-Torres, “Nonlinear dynamics and chaos in an optomechanical beam,” Nat. Commun. 8(1), 14965–10 (2017). [CrossRef]
9. I. Osborne, “Securing quantum key distribution,” Science 368, 382–383 (2020). [CrossRef]
10. H. Semenenko, P. Sibson, A. Hart, M. Thompson, J. Rarity, and C. Erven, “Chip-based measurement-device-independent quantum key distribution,” Optica 7(3), 238–242 (2020). [CrossRef]
11. X. Fang, P. Zeng, H. Liu, M. Zou, and J. Pan, “Implementation of quantum key distribution surpassing the linear rate-transmittance bound,” Nat. Photonics 14(7), 422–425 (2020). [CrossRef]
12. H. Lo, M. Curty, and K. Tamaki, “Secure quantum key distribution,” Nat. Photonics 8(8), 595–604 (2014). [CrossRef]
13. A. Boaron, G. Boso, D. Rusca, C. Vulliez, C. Autebert, M. Caloz, M. Perrenoud, G. Gras, F. Bussières, M. Li, D. Nolan, A. Martin, and H. Zbinden, “Secure quantum key distribution over 421 km of optical fiber,” Phys. Rev. Lett. 121(19), 190502 (2018). [CrossRef]
14. R. Lavrov, M. Jacquot, and L. Larger, “Nonlocal nonlinear electro-optic phase dynamics demonstrating 10 Gb/s chaos communications,” IEEE J. Quantum Electron. 46(10), 1430–1435 (2010). [CrossRef]
15. J. Ke, L. Yi, G. Xia, and W. Hu, “Chaotic optical communications over 100-km fiber transmission at 30-Gb/s bit rate,” Opt. Lett. 43(6), 1323–1326 (2018). [CrossRef]
16. K. Kusumoto and J. Ohtsubo, “1.5-GHz message transmission based on synchronization of chaos in semiconductor lasers,” Opt. Lett. 27(12), 989–991 (2002). [CrossRef]
17. J. Ohtsubo, “Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback,” IEEE J. Quantum Electron. 38(9), 1141–1154 (2002). [CrossRef]
18. N. Li, W. Pan, L. Yan, B. Luo, X. Zou, and S. Xiang, “Enhanced two-channel optical chaotic communication using isochronous synchronization,” IEEE J. Sel. Top Quant. 19(4), 0600109 (2013). [CrossRef]
19. S. Xiang, W. Pan, B. Luo, L. Yan, X. Zou, N. Li, and H. Zhu, “Wideband unpredictability-enhanced chaotic semiconductor lasers with dual-chaotic optical injections,” IEEE J. Quantum Electron. 48(8), 1069–1076 (2012). [CrossRef]
20. A. Wang, Y. Wang, and H. He, “Enhancing the bandwidth of the optical chaotic signal generated by a semiconductor laser with optical feedback,” IEEE Photonic. Tech. L. 20(19), 1633–1635 (2008). [CrossRef]
21. M. Zhang, T. Liu, P. Li, A. Wang, J. Zhang, and Y. Wang, “Generation of broadband chaotic laser using dual-wavelength optically injected Fabry–Pérot laser diode with optical feedback,” IEEE Photonic. Tech. L. 23(24), 1872–1874 (2011). [CrossRef]
22. S. Tang and J. Liu, “Chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed opto-electronic feedback,” IEEE J. Quantum Electron. 37(3), 329–336 (2001). [CrossRef]
23. J. Goedgebuer, L. Laurent, and P. Henri, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80(10), 2249–2252 (1998). [CrossRef]
24. N. Gastaud, S. Poinsot, L. Larger, J. Merolla, M. Hanna, J. Goedgebuer, and F. Malassenet, “Electro-optical chaos for multi-10 Gbit/s optical transmissions,” Electron Lett. 40(14), 898–899 (2004). [CrossRef]
25. K. Ikeda, “Multiple-valued stationary state and its instability of the transmitted light by a ring cavity system,” Opt. commun. 30(2), 257–261 (1979). [CrossRef]
26. J. Ai, L. Wang, and J. Wang, “Secure communications of CAP-4 and OOK signals over MMF based on electro-optic chaos,” Opt. Lett. 42(18), 3662–3665 (2017). [CrossRef]
27. J. Ke, L. Yi, Z. Yang, Y. Yang, Q. Zhuge, Y. Chen, and W. Hu, “32 Gb/s chaotic optical communications by deep-learning-based chaos synchronization,” Opt. Lett. 44(23), 5776–5779 (2019). [CrossRef]
28. T. Murphy, A. Cohen, B. Ravoori, K. Schmitt, A. Setty, F. Sorrentino, C. Williams, E. Ott, and R. Roy, “Complex dynamics and synchronization of delayed-feedback nonlinear oscillators,” Philos. Trans. R. Soc., A 368(1911), 343–366 (2010). [CrossRef]
29. D. Liang and J. Bowers, “Recent Progress in Heterogeneous III-V-on-Silicon Photonic Integration,” Light Adv. Manuf. 2, doi:10.37188/lam.2021.005 (2021).
30. Z. Chen, L. Yan, Y. Pan, L. Jiang, A. Yi, W. Pan, and B. Luo, “Use of polarization freedom beyond polarization-division multiplexing to support high-speed and spectral-efficient data transmission,” Light: Sci. Appl. 6(1), e16190 (2017). [CrossRef]
31. L. Jiang, L. Yan, A. Yi, Y. Pan, M. Hao, W. Pan, B. Luo, and Y. Jaouën, “Chromatic dispersion, nonlinear parameter, and modulation Format monitoring based on Godard's error for coherent optical transmission systems,” IEEE Photonics J. 10(1), 1–12 (2018). [CrossRef]
32. Y. Yang, G. Cao, K. Zhong, X. Zhou, Y. Yao, A. Lau, and C. Lu, “Fast polarization-state tracking scheme based on radius-directed linear Kalman filter,” Opt. Express 23(15), 19673–19680 (2015). [CrossRef]
33. N. Jiang, W. Pan, B. Luo, L. Yan, S. Xiang, L. Yang, D. Zheng, and N. Li, “Properties of leader-laggard chaos synchronization in mutually coupled external-cavity semiconductor lasers,” Phys. Rev. E 81(6), 066217 (2010). [CrossRef]
34. P. Mu, W. Pan, L. Yan, B. Luo, N. Li, and M. Xu, “Experimental evidence of time-delay concealment in a DFB laser with dual-chaotic optical injections,” IEEE Photon. Technol. Lett. 28(2), 131–134 (2016). [CrossRef]
35. N. Li, W. Pan, A. Locquet, and D. Citrin, “Time-delay concealment and complexity enhancement of an external-cavity laser through optical injection,” Opt. Lett. 40(19), 4416–4419 (2015). [CrossRef]
36. A. Gonzalo and S. Li, “Some basic cryptographic requirements for chaos-based cryptosystems,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 16(08), 2129–2151 (2006). [CrossRef]
37. B. Wu, Z. Wang, Y. Tian, M. P. Fok, B. J. Shastri, D. R. Kanoff, and P. R. Prucnal, “Optical steganography based on amplified spontaneous emission noise,” Opt. Express 21(2), 2065–2071 (2013). [CrossRef]
38. M. Fok and P. Prucnal, “Compact and low-latency scheme for optical steganography using chirped fibre Bragg gratings,” Electron. Lett. 45(3), 179–180 (2009). [CrossRef]
39. S. Andrew and E. Sargent, “The role of optical CDMA in access networks,” IEEE Commun. Mag. 40(9), 83–87 (2002). [CrossRef]
40. O. Hirota, M. Sohma, M. Fuse, and K. Kato, “Quantum stream cipher by the Yuen 2000 protocol: Design and experiment by an intensity-modulation scheme,” Phys. Rev. E 72(2), 022335 (2005). [CrossRef]
