Objective The semiconductor laser with external optical feedback (EOF-SL) provides a good platform for the study of nonlinear effects and complex photon dynamics. The high-dimensional chaotic signal output of an EOF-SL has been widely used in chaotic secure communications and physical random number generation. However, the external optical feedback gives rise to a time-delay signature, which reduces the entropy and randomness of a chaotic entropy source. In addition, quantifying the randomness of a physical process from the original analog signals remains to be explored. The randomness of a chaotic laser originates from the inherent quantum noise. Quantum noise can provide a stochastic initial value for the chaotic system, and the randomness of the quantum noise can be rapidly and nonlinearly amplified by the chaotic system. In this study, we propose a technique to effectively enhance the entropy of a chaotic laser via quantum noise. Here a method is proposed to accurately quantify entropy evolution of a chaotic laser and evaluate noise characteristics of the chaotic system. We hope that our basic strategy and findings are potentially beneficial to improve the quality of random number generation and the security of chaotic secure communications.
Methods In this work, a chaotic laser prepared by a semiconductor laser with external optical feedback is used as an entropy source. The effects of quantum noise intensity and bandwidth on the entropy enhancement of the chaotic laser are studied numerically and experimentally. The mean value of permutation entropies at the feedback delay time τext, τext/2, and τext/3 is used for quantifying the complexity hdcomp of the chaotic signals. The permutation entropy growth Gd between the adjacent embedding dimensions is used to evaluate whether the chaotic dynamics is noise-dominated or not. In addition, to investigate the influence of quantum noise on the dynamics of the chaotic laser, the theoretical model can be described by Lang-Kobayashi rate equations with Gaussian white noise (GWN-LK). Quantum noise is experimentally prepared through balanced homodyne detection, which is injected into the chaotic entropy source. Then the time sequences of chaotic lasers are recorded by an oscilloscope for the analysis of complexity and entropy growth.
Results and Discussion The theoretical results show that the entropy growth Gd is greater than the noise-dominated threshold GdNthr after injecting quantum noise, and Gd increases monotonically and rapidly as the embedding dimension d increases, that is, the chaotic dynamics process is located in the noise-dominated region (Fig. 2). The effects of quantum noise intensity and bandwidth on the hdcomp and Gd show that the injection of quantum noise can effectively enhance the entropy of the chaotic laser, and the effect of entropy enhancement becomes more significant as the quantum noise intensity and bandwidth increase (Fig. 3). In the experiment, the mean entropy hdcomp of the 3.8 GHz bandwidth chaotic laser is increased to 0.999 in the case of quantum noise injection with an intensity of 15 dBm and a bandwidth of 500 MHz, and it demonstrates that narrow-bandwidth quantum noise substantially enhances the entropy of the wide-bandwidth chaotic laser (Fig. 5). The experimental results for hdcomp and Gd of the chaotic laser versus quantum noise intensity agree well with the theoretical results in Fig. 3. As the quantum noise intensity increases, the entropy and complexity of the chaotic laser enhance significantly. The maximum entropy output can be achieved by using quantum noise with a narrower bandwidth, and the chaotic laser with noise domination and entropy enhancement is obtained (Fig. 6).
Conclusions In our study, a technique for evaluating and enhancing the entropy of a chaotic laser is proposed. And the complexity of the chaotic laser is quantified by the mean value of the permutation entropies at the feedback delay time τext, τext/2, and τext/3. We observe the entropy enhancement of the 3.8 GHz bandwidth chaotic laser versus quantum noise intensity for different quantum noise bandwidths: 100 MHz, 300 MHz, and 500 MHz. The results demonstrate that narrow-bandwidth quantum noise substantially enhances the entropy of the wide-bandwidth chaotic laser, and the mean entropy or whole complexity of the 3.8 GHz bandwidth chaotic laser is increased to 0.998 in the case of quantum noise injection with an intensity of 15 dBm and a bandwidth of 100 MHz. Moreover, the quantum noise bandwidth, which is needed to enhance the entropy of the chaotic laser to the maximum, decreases as the quantum noise intensity increases. In addition, the entropy growth Gd is used to determine whether the chaotic dynamics process is located in the noise-dominated region. As the quantum noise intensity increases, Gd is greater than the noise-dominated threshold GdNthr. And Gd increases monotonically as the embedding dimension d increases. The results show that the chaotic dynamic process is dominated by random noise, that is, the chaotic laser output dominated by random noise is obtained. The improved chaotic laser has potential applications in random number generation and secure communications.
