Open Access
Add to BibSonomy
Abstract
Aiming at the difficulty of traditional chaotic-shift-keying (CSK) systems in resisting return map attacks, we propose an optical chaotic communication system based on time-delayed shift keying and common-signal-induced synchronization. This scheme combines amplified spontaneous emission (ASE) noise, phase modulator (PM), and fiber Bragg grating (FBG) to achieve dual masking in both intensity and phase fields, achieving 10Gb/s information transmission. A common-signal-induced method is used to achieve the synchronization of the system. Moreover, by shifting the time delay as the message-feeding method, the return map attack is effectively resisted, to prevent the amplitude and frequency information of the chaotic attractor from being exposed. In terms of confidentiality and communication performance, this scheme demonstrates good performance of time delay signatures (TDSs) concealment and long-distance transmission capability. In addition, this scheme maintains high sensitivity to key parameters and achieves better confidentiality while increasing the key space.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Nowadays, with the rapid growth of optical fiber communication data, people's social demand for information confidentiality is rising year by year. Therefore, it is crucial to seek effective encryption methods. As a kind of physical layer encryption technology, chaotic optical communication has been widely studied because of its sensitivity to initial values, noise-like temporal features, and unpredictability. In terms of chaotic signal generation, three commonly used methods were reported such as optical feedback [1–4], optical injection [5–8], and electro-optical feedback [9–13]. Among them, optical feedback uses a mirror as a feedback medium to generate chaotic signals by disturbing the internal optical field of the laser. The optical injection destroys the stable state of the response laser through the output from the driving laser, resulting in a complex nonlinear dynamic state. The electro-optical feedback modulation technique based on Ikeda’s equation has the advantages of higher bandwidth, easier synchronization, and lower cost, so it has wide prospects and is favored by many researchers.
In terms of chaotic signal loading and demodulation methods, there are three main methods: chaos masking (CM), chaotic parameters modulation (CPM), and chaotic-shift-keying (CSK). The principle of CM is to mix the chaotic signal with the message at the transmitter and directly subtract the synchronous chaotic signal from the encrypted signal during decryption. For example, in 2023, Y. Zhang et al. proposed a secure optical communication scheme using optical chaos to mask M-ary signals, and the secure transmission of 1024-ary signals was verified using CM at a rate of 1.25 Gbps [14]. However, in CM systems, the power of the information must be much smaller than that of the chaotic carrier, which makes CM relatively sensitive to channel noise. The CPM directly uses the message to modulate the system parameters at the transmitter, which directly affects the chaotic dynamic state. For example, in 2023, Y. Xie et al. designed a chaotic optical communication system based on electro-optical feedback using CPM and investigated the signal-to-noise ratio (SNR) degradation of 40 Gbit/s phase chaos and intensity chaos models with a 15 GHz wideband chaotic carrier under different channel SNRs. The experimental results guide long-distance transmission [15]. However, in a CPM system, the message is directly involved in the chaotic generation process, and it is necessary to design a specific adaptive controller with a certain mathematical form for a given chaotic system. In addition, compared with CM, CPM has a more complex structure and requires higher hardware costs. Compared with CPM and CK, CSK does not directly transmit messages to the channel for transmission, but only transmits chaotic signals with alternative effects, making it more secure. In 2020, L. Wang et al. proposed a high-speed physical key distribution method without external feedback for semiconductor lasers using CSK, and numerically demonstrated a secure key distribution at a rate of 1.2 Gb/s [16]. In 2023, Z. Deng et al. proposed a new scheme of high-speed key distribution based on interference spectrum-shift keying with signal mutual modulation in commonly driven chaos synchronization and experimentally proved a 6.7 Gbit/s key distribution rate with a bit error rate (BER) below 3.8 × 10−3 at a distance of 40 km [17]. However, already in 1995, G. Pérez and H. Cerdeira proposed the return map attack specifically targeting CSK, which was able to extract masked messages, exposing the shortcomings of CSK [18]. The inadequacy of the above CSK systems is that none of them discusses the impact of the return map attack.
Aiming to address the vulnerability of CSK systems to return map attacks, an optical chaotic communication system based on time-delayed shift keying and common-signal-induced synchronization is proposed and demonstrated in this paper. In this scheme, the common-signal induction method is used to realize the synchronization of the system, and the amplified spontaneous emission (ASE) noise is introduced as the entropy source to effectively enhance the security of the system. Based on achieving good synchronization, the encryption and decryption processes are successfully achieved by using the time-delayed shift keying encryption method and synchronous power error method, respectively. Simulation results show that the system effectively suppresses time delay signatures (TDSs) and has high security against the return map attack. In addition, some key parameters in the system have high sensitivity to mismatch, which can effectively enhance the key space.
2. Principle and system architecture
Figure 1 shows the basic principle and system architecture of the proposed secure communication system, which consists of three parts: drive source, transmitter and receiver. In the drive source module, ASE is used as the driving signal of the phase modulator (PM1), and the phase chaotic signal is obtained after phase modulation of the laser output from the semiconductor laser (SL). It should be noted that ASE refers to the random incoherent spontaneous emission of excited particles in an optical amplifier when the particles return from the excited state to the ground state and amplify the optical signal. For example, it can be generated by an erbium-doped fiber amplifier (EDFA) to achieve optical steganography [19]. And ASE noise is a true random signal, with much higher randomness than chaotic signals with class noise, and is often used as an ultra-wideband entropy source to provide a wider bandwidth, so it can effectively enhance the security of the system. After that, the photodetector (PD1) is used to convert the optical signal into an electrical signal, and the radio frequency amplifier (RF1) amplifies the electrical signal to a suitable range for phase modulation. Subsequently, the fiber Bragg grating (FBG1) converts the phase signal into an intensity signal, so that the drive source module obtains a signal with both phase and intensity in a chaotic state, which serves as a common driving signal for the transmitter and receiver, inducing synchronization between them. The transmitter consists of FBG2 and an electro-optical self-feedback loop based on PM2. In this loop, the switch is controlled to connect the time delay line (TDL1) and TDL2 with different delay parameters when m(t) is “0” and “1”, respectively, thereby mapping the binary message into different chaotic attractors. The output of the optical coupler (OC3) is divided into two parts. One part is converted into an electrical signal by PD2 by entering the loop as described earlier, and then further amplified by RF2 as the driving signal for PM2. The other part enters into FBG2 to realize the redistribution of chaotic signals in the time domain, which can greatly suppress the TDS. The time delay, the depth of phase modulation, and the dispersion coefficient of FBG2 in the self-feedback loop jointly affect the encryption parameters. For the eavesdroppers who do not exactly match the hardware parameters, only the noisy signal with both intensity and phase disturbed can be obtained, which fully guarantees security. The transmission channel consists of a single-mode fiber (SMF), an EDFA, and a dispersion compensated fiber (DCF). At the receiver, the structure is symmetric with that of the transmitter, and the parameters of TDL3 and FBG3 are chosen to be the same as those of TDL1 and FBG3, respectively, to generate signals that correspond to the signals generated at the transmitter when m(t) is “0”. Finally, the balanced photodetector (BPD) outputs the difference signals between the transmitter and receiver, and then gets the recovered message through the low pass filter (LPF). The decryption principle will be explained in detail in Section 3.3.
Fig. 1. Schematic of the proposed scheme. SL, semiconductor laser; PM, phase modulator; FBG, fiber Bragg grating; PD, photodetector; RF, radio frequency amplifier; OC, optical coupler; TDL, time delay line; SMF, single-mode fiber; DCF, dispersion compensated fiber; EDFA, erbium-doped fiber amplifier; BPD, balanced photodetector; LPF, low pass filter.
In general, referring to the Lang-Kobayashi differential equations [1], the output ${E_{\textrm{SL}}}(t)$ of laser SL can be solved numerically as follows:
Table 1. Simulation parameters.
3. Numerical results and discussions
3.1 Chaotic states and synchronization
In order to prove the chaotic state of the proposed system, we generate a chaotic bifurcation diagram by plotting the amplitude of local extremes of chaotic time series, which can directly reflect the nonlinear dynamic behavior of the system. Figure 2(a) shows the chaotic bifurcation diagram affected by the modulation depth ${K_{pm1}}$ of the phase modulator PM1. As can be seen from Fig. 2(a), when ${K_{pm1}}$ is small, the extreme value distribution range is narrow, and it is in the early stage of chaos development. From the general trend, it can be seen from Fig. 2(a) that the distribution of extreme values increases rapidly with the increase of ${K_{pm1}}$. When ${K_{pm1}}$ is large, the distribution in the chaotic bifurcation diagram is much larger than that in the previous stage, and it is in the full chaotic region. Similarly, as the modulation depth ${K_{pm2}}$ of the phase modulator PM2 increases, the same trend can be observed in Fig. 2(b), which also means that the signal enters a more complex state. In order to accurately analyze the complexity of the time series and to further demonstrate the chaotic state, we use the permutation entropy (PE) algorithm proposed by C. Bandt et al. in 2002 [20]. It calculates the entropy of the relevant information based on the probability of different alignment patterns in the time series and is one of the most widely used methods to estimate the complexity of chaotic signals. As the normalized PE value gets closer to 1, the stochastic and unpredictable nature of the time series becomes stronger [21]. In Fig. 2(c), when ${K_{pm1}}$ is low, PE is at a lower level relative to the PE value at stability, corresponding to the earlier stage of chaos development. Subsequently, PE increases rapidly with the increase of the modulation coefficient ${K_{pm1}}$, and then stabilizes above 0.9995, which is very close to 1, indicating that the signal complexity is extremely high. This corresponds to a state of higher dynamic complexity in the full chaotic development region at high ${K_{pm1}}$ in Fig. 2(a). In addition, chaotic attractor can also help us better understand the complexity and aperiodicity of chaotic systems. In the chaotic attractor shown in Fig. 2(d), there are no repeated periodic patterns, but rather intricate, irregular shapes. Moreover, its trajectory converges globally to the attractor region in Fig. 2(d), which reflects the characteristics of chaos.
Fig. 2. Chaotic bifurcation diagrams (a) influenced by ${K_{pm1}}$, (b) influenced by ${K_{pm2}}$, (c)PE influenced by ${K_{pm1}}$ (d)Chaotic attractor of ${E_t}(t)$ versus ${E_t}(t - \tau )$.
After ensuring that the system has entered a chaotic state, it is necessary to test the synchronization of the system, because good synchronization is the basis for correct decryption. To test the synchronization of the system, fix the value of m(t) to “0”, so that the switch at the transmitter is fixedly connected to TDL1 with the same parameter as TDL3. This allows the parameters of the transmitter and receiver to be matched. Then, the synchronization of the system is judged by calculating the correlation between the signal ${E_t}$ generated at the transmitter and the signal ${E_r}$ generated at the receiver. In Fig. 3(a), the chaotic waveforms of ${E_t}$ and ${E_r}$ exhibit similar fluctuations, with a cross-correlation coefficient (CC) of 0.989, which is very close to 1. In order to show the details more clearly, we enlarged the part of Fig. 3(a) to get Fig. 3(b). Moreover, the scatter points fit in Fig. 3(c) is near the diagonal, indicating that the common signal-induced structure has achieved high-quality chaotic synchronization.
Fig. 3. (a) Signals ${E_t}$ and ${E_r}$ generated at the transmitter and receiver, respectively. (b) The partial enlarged view of (a). (c) The corresponding correlation diagrams.
3.2 TDSs concealment
Usually, an eavesdropper cracks the intercepted signal through statistical analysis and then reconstructs the system with relevant parameters to obtain the original message. Due to the exposure of signal features by TDSs, it is necessary to hide them [22]. The effect of TDSs concealment is measured by calculating the autocorrelation function (ACF) of the encrypted signal, and the statistical method used is shown in Eq. (8):
3.3 Encryption and decryption process and system performance
Next, we will turn to the encryption and decryption process of the proposed system, as shown in Figs. 5(a)-5(e). The message m(t) to be encrypted in Fig. 5(a) is “0110010110101010”. After time-delayed shift keying encryption, m(t) is mapped to different chaotic attractors, then FBG2 distorts the signal in the time domain, and the resulting output signal at the transmitter is shown in Fig. 5(b). At the receiver, since TDL3 and FBG3 have the same parameters as TDL1 and FBG2, respectively, they can generate signals as shown in Fig. 5(c) corresponding to the signals generated when m(t) is “0” at the transmitter. Finally, the BPD in the receiver subtracts the two signals in Figs. 5(b) and 5(c), and the resulting difference signal is shown in Fig. 5(d). LPF filters the difference signal to obtain the decrypted signal as shown by the red line in Fig. 5(e), and then the recovered $m^{\prime}(t )$ obtained by the threshold judgment is shown by the blue dashed line. When m(t) is “0”, the parameter-matched transmitter and receiver produce synchronized signals, so the amplitude of the difference signal obtained is very close to 0. But when m(t) is “1”, the switch at the transmitter is connected to TDL2, and the parameter mismatch between the transmitter and receiver causes the signal to be out of synchronization, resulting in a significant synchronization error. When decrypting, it is judged to be “1”. This is the principle of synchronous power error decryption.
Fig. 5. Process diagrams of encryption and decryption: (a) Initial message m(t). (b) Encrypted signal generated at the transmitter. (c) The signal is generated at the receiver. (d) Differential signal output from BPD. (e) Decrypted signal filtered by LPF (red line) and recovered signal (blue dashed line).
An eye diagram is a tool used to evaluate and analyze the performance of communication systems, which can reflect the stability, noise, and distortion in signals. The output signal of FBG2 in the transmitter is the encrypted signal ${E_t}(t) = {F^{ - 1}}\{ F[{E_{p2}}(t)]{H_2}(\omega )\}$. When the signal ${E_t}(t)$ is directly intercepted in the transmission channel, the eye diagram of the encrypted signal obtained is shown in Fig. 6(a). Its complete closure indicates that the signal is distorted by interference, which comes from the time-delayed shift keying encryption method. It is proved that after the encryption process is completed, even if the signal is intercepted by the eavesdropper, the effective information can not be obtained from it, which indicates that the security of our proposed system has been effectively guaranteed. Figure 6(b) shows the eye diagram of the decrypted signal, which is fully open indicating good signal quality. It is easy to distinguish different signal levels, which shows that the proposed scheme exhibits excellent chaotic synchronization and decryption performance.
Figure 7 shows the spectrum of the encrypted signal, which is flatter compared to the electrical chaotic signal, reflecting the excellent broadband characteristics of the optical chaos. Figure 8 depicts the waveforms of the decrypted signal obtained after transmission over different distances. The peak-valley value of the decrypted signal gradually decreases with the increase of the transmission distance. When the transmission distance is 80km, it can still meet the requirement of the BER required for communication (the hard-decision forward error correction (HD-FEC) threshold is 3.8 × 10−3), which reflects the capability of long-distance transmission of our proposed scheme.
3.4 Return map attack
The return map attack proposed by G. Pérez and H. Cerdeira is a kind of attack method specifically targeting CSK, which is also a major challenge faced by traditional CSK communication systems [18]. The principle of CSK is to map different symbols into different chaotic attractors. Since the local maximum and local minimum values of encrypted signals expose the amplitude information of attractors, changes in the size of attractors can cause shifts or structural deformation of the return map, and even change the intrinsic frequency of attractors. Therefore, different return maps of chaotic signals can reveal the amplitude and frequency information of signals [23–25].
To verify the confidentiality of the proposed scheme, a return map attack is used to test it. Firstly, define the following variables:
Define ${V_{\max }}(n)$ and ${V_{\min }}(n)$ be the nth local maximum and the nth local minimum of the encrypted signal V(t), and their corresponding times are $t_n^{\max }$ and $t_n^{\min }$, respectively. The time-domain diagram of the encrypted signal V(t) of the proposed scheme is shown in Fig. 5(b) of Section 3.3. Equations (9) and (10) can respectively reveal the amplitude and frequency information of the attractor. Equation (9) is used to calculate the encrypted signal V(t) generated when m(t) is “0” and “1”, respectively, to obtain Fig. 9(a) which represents the amplitude information. Similarly, Fig. 9(b) characterizing the frequency information is obtained using Eq. (10). In the return map of traditional CSK, the switching of m(t) results in two different strip tracks, and the return map of traditional CSK can be referred to in Refs. [24–26]. The information of the signal can be easily uncovered by checking which strip the point falls on. However, in Figs. 9(a) and 9(b), the points characterizing the amplitude information and frequency information show a diffuse state, and the trace of switching m(t) cannot be found, which demonstrates that the proposed scheme can resist return map attack and therefore has a high degree of confidentiality.
Fig. 9. Return maps of (a)${A_n}$ and ${B_n}$ in the proposed scheme, and (b)${T_{\max }}(n)$ and ${V_{\max }}(n)$, ${T_{\min }}(n)$ and ${V_{\min }}(n)$ in the proposed scheme.
3.5 Parameter mismatch
In theory, perfect synchronization can be achieved when the parameters in the transmitter and receiver are precisely matched. However, in practical applications, minor parameter mismatches caused by device production and other reasons are inevitable. Since parameter mismatch may affect the synchronization quality and decryption performance, after discussing the confidentiality performance of the system, we turn to discuss the sensitivity of the parameters in the system to mismatch. The sensitivity of parameter mismatch can be evaluated by calculating the BER of the decrypted signal, and the BER is calculated by comparing the recovered signal with the original message m(t). We discuss three key parameters at the transmitter, including ${K_{pm2}}$, ${\tau _1}$ and ${B_2}$. The BER of the decrypted signal in Figs. 10(a)-10(c) increases with the degree of parameter mismatch. When the BER exceeds the threshold of HD-FEC, it means that the received message is unreliable. In Fig. 10(a), even if the mismatch degree of ${K_{pm2}}$ reaches 26%, BER can still be lower than the threshold of HD-FEC. This result indicates that this parameter has strong robustness. When the mismatch degree is at a high level, the system can still achieve good synchronization, which enhances the feasibility of this scheme in reality. However, in Figs. 10(b) and 10(c), when the mismatch between ${\tau _1}$ and ${B_2}$ is less than 0.5%, BER is already higher than the threshold of HD-FEC, which indicates that these parameters have strong sensitivity and can effectively prevent the third party from independently reconstructing the illegal receiver. The parameters sensitive to mismatch introduce a larger key space into the system, which ensures the security of the system.
4. Conclusions
In this paper, from the perspective of overcoming the susceptibility to return map attacks that traditional CSK systems have, an optical chaotic communication scheme based on time-delayed shift keying and common-signal-induced synchronization was proposed to successfully simulate high-quality encryption and decryption at a rate of 10Gb/s over 80km transmission. This scheme achieved high-quality synchronization using the common-signal-induced method. Moreover, the optimum parameter group selected by using the bifurcation diagram and TDS evolution diagram as reference effectively covered the TDSs. When attacked by the return map attack, no trace of keying switching can be found, and no amplitude and frequency information will be leaked. In addition, the effects of parameter mismatch on system security was also investigated, and the simulation results demonstrated the feasibility of the proposed confidential communication scheme. In future experiments, we can choose different devices to combine with CSK, or improve the shift keying method, to obtain better communication performance.
Funding
Key Research and Development Program of Zhejiang Province (2022C03037); Primary Research and Development Plan of Zhejiang Province (2023C03014).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]
2. O. Spitz, A. Herdt, J. Wu, et al., “Private communication with quantum cascade laser photonic chaos,” Nat. Commun. 12(1), 3327 (2021). [CrossRef]
3. L. Wang, J. Wang, Y. Wu, et al., “Chaos synchronization of semiconductor lasers over 1040-km fiber relay transmission with hybrid amplification,” Photonics Res. 11(6), 953–960 (2023). [CrossRef]
4. B. M. Al Bayati, A. K. Ahmad, and K. A. M. Al Naimee, “Effect of control parameters on chaos synchronization by means of optical feedback,” Opt. Commun. 472, 126032 (2020). [CrossRef]
5. V. Kovanis, A. Gavrielides, T. B. Simpson, et al., “Instabilities and chaos in optically injected semiconductor lasers,” Appl. Phys. Lett. 67(19), 2780–2782 (1995). [CrossRef]
6. S. Xiang, M. Yang, and J. Wang, “Chaotic optical communications of 12.5-Gbaud OOK and 10-Gbaud QPSK signals based on mutual injection of semiconductor lasers,” Opt. Lett. 47(11), 2818–2821 (2022). [CrossRef]
7. S. Ebisawa and S. Komatsu, “Orbital instability of chaotic laser diode with optical injection and electronically applied chaotic signal,” Photonics 7(2), 25 (2020). [CrossRef]
8. X. Mao, Y. Sun, L. Wang, et al., “Instability of optical phase synchronization between chaotic semiconductor lasers,” Opt. Lett. 46(12), 2824–2827 (2021). [CrossRef]
9. 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]
10. G. Zou, H. Wang, and Y. Ji, “Electro-optic chaos system with time delay signature concealment based on XOR operation and multi-bit PRBS,” Opt. Express 29(5), 7327–7341 (2021). [CrossRef]
11. M. Yu, H. Wang, and Y. Ji, “Secure WDM communication based on second order differential feedback electro-optical chaotic mutual injection synchronization,” Opt. Laser Technol. 169, 110147 (2024). [CrossRef]
12. Z. Yang, L. Yi, J. Ke, et al., “Chaotic optical communication over 1000 km transmission by coherent detection,” J. Lightwave Technol. 38(17), 4648–4655 (2020). [CrossRef]
13. J. Bai, H. Wang, and Y. Ji, “Time-delay signature concealing electro-optic chaotic system with multiply feedback nonlinear loops,” Opt. Express 29(2), 706–718 (2021). [CrossRef]
14. Y. Zhang, Z. Li, Z. Tao, et al., “Secure optical communication based on M-ary signals masked by optical chaos,” Opt. Commun. 528, 129017 (2023). [CrossRef]
15. Y. Xie, Z. Yang, M. Shi, et al., “Signal-to-noise ratio degradation analysis for optoelectronic feedback-based chaotic optical communication systems,” Opt. Lett. 48(19), 5005 (2023). [CrossRef]
16. L. Wang, M. Chao, A. Wang, et al., “High-speed physical key distribution based on dispersion-shift-keying chaos synchronization in commonly driven semiconductor lasers without external feedback,” Opt. Express 28(25), 37919 (2020). [CrossRef]
17. Z. Deng, X. Gao, Y. An, et al., “High-speed secure key distribution based on interference spectrum-shift keying with signal mutual modulation in commonly driven chaos synchronization,” Opt. Express 31(25), 42449 (2023). [CrossRef]
18. G. Pérez and H. Cerdeira, “Extracting messages masked by chaos,” Phys. Rev. Lett. 74(11), 1970–1973 (1995). [CrossRef]
19. B. Wu, Z. Wang, Y. Tian, et al., “Optical steganography based on amplified spontaneous emission noise,” Opt. Express 21(2), 2065–2071 (2013). [CrossRef]
20. C. Bandt and B. Pompe, “Permutation Entropy: A Natural Complexity Measure for Time Series,” Phys. Rev. Lett. 88(17), 174102 (2002). [CrossRef]
21. M. Chao, D. Wang, L. Wang, et al., “Permutation entropy analysis of chaotic semiconductor laser with chirped FBG feedback,” Opt. Commun. 456, 124702 (2020). [CrossRef]
22. M. D. Prokhorov, V. I. Ponomarenko, A. S. Karavaev, et al., “Reconstruction of time-delayed feedback systems from time series,” Phys. D 203(3-4), 209–223 (2005). [CrossRef]
23. S. Li, G. Chen, and G. Álvarez, “RETURN-MAP CRYPTANALYSIS REVISITED,” Int. J. Bifurcation Chaos 16(05), 1557–1568 (2006). [CrossRef]
24. T. Yang, L.-B. Yang, and C.-M. Yang, “Cryptanalyzing chaotic secure communications using return maps,” Phys. Lett. A 245(6), 495–510 (1998). [CrossRef]
25. J. Liu, X. Zhou, Q. Li, et al., “Secondary-encryption optical chaotic communication system based on one driving laser and two responding lasers,” Chaos, Solitons Fractals 163(11), 112554 (2022). [CrossRef]
26. X. Gao, M. Cheng, L. Deng, et al., “Robust chaotic-shift-keying scheme based on electro-optical hybrid feedback system,” Opt. Express 28(8), 10847 (2020). [CrossRef]
