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Permutation entropy analysis of chaotic semiconductor laser with chirped FBG feedback
Optics Communications 456 (2020) 124702
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Permutation entropy analysis of chaotic semiconductor laser with chirped FBG feedback Meng Chao a,b , Daming Wang a,b , Longsheng Wang a,b , Yuchuan Sun a,b , Hong Han a,b , Yuanyuan Guo a,b , Zhiwei Jia a,b , Yuncai Wang a,b,c , Anbang Wang a,b ,∗ a
Key Laboratory of Advanced Transducers & Intelligent Control System, Ministry of Education, Taiyuan University of Technology, Taiyuan 030024, China Institute of Optoelectronic Engineering, College of Physics & Optoelectronics, Taiyuan University of Technology, Taiyuan 030024, China c Institute of Advanced Photonics Technology, Guangdong University of Technology, Guangzhou 510006, China b
ARTICLE
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Keywords: Chaos Chirped fiber Bragg grating Complexity Permutation entropy Semiconductor laser Time delay signature
ABSTRACT We numerically analyze the chaos complexity of a semiconductor laser with chirped fiber Bragg grating (CFBG) feedback by utilizing the permutation entropy (PE). Due to dispersion-induced nonlinearity of chirped grating, the time delay signature (TDS) can be suppressed or erased and the complexity enhancement is expected. The effects of laser parameters on PE as well as the TDS are studied in detail. The results show that for CFBG feedback, the TDS can be suppressed in the range of parameters where linewidth enhancement factor exceeds 3 and gain saturation coefficient is lower than 4×10−4 μm3 , and the complexity of chaotic signals is enhanced. As the feedback strength and bias current increase, the PE value under CFBG feedback increases linearly to a maximum of 0.994 and remains stable. Compared with mirror feedback, CFBG feedback can achieve increased complexity over a wider range of parameters. In addition, when the dispersion value of the chirped grating exceeds 2000 ps/nm, the frequency detuning range of -15 GHz to 10 GHz is beneficial to obtain chaotic signals with higher PE values.
1. Introduction Semiconductor lasers have been widely used in many fields due to its enormous potential, such as chaotic secure communication [1–4], physical random bit generation [5–9], secure key distribution [10,11], photonic spiking neural network [12], delay-coupled semiconductor lasers networks [13]. Obtaining broadband and high complexity signals is a key issue in laser chaos. An external cavity feedback semiconductor laser is the most attractive choice in view of its simple and integrated setup. Unfortunately, due to the linear feedback caused by the external cavity, semiconductor laser with conventional mirror optical feedback has time delay signature (TDS), which makes the chaotic signals have a high correlation with its previous states, thus reducing its complexity [14,15]. Some methods have been proposed to eliminate the TDS, including dual-cavity optical feedback [16–18], additional optical injection [19– 21], and uniform grating feedback [22,23]. Guo et al [18] have numerically studied the complexity of a chaotic laser system with dual-cavity mirror feedback using permutation entropy (PE), and found the regions of feedback rate and bias current for high complexity and low TDS. Li et al numerically shown that for optically injected semiconductor lasers the TDS can be concealed and the dynamic complexity can be
enhanced under large linewidth enhancement factor [21]. It was also experimental demonstrated that optical feedback from uniform fiber Bragg grating (FBG) can depress TDS [23] and enhance the PE [24], which is beneficial to random number generation [9]. Note that uniform FBG feedback should have a detuned frequency relative to laser frequency in order to suppress TDS greatly due to that remarkable dispersion effect appears in sideband of uniform grating spectrum [23]. Our previous work experimentally and theoretically proved that optical feedback from a chirped fiber Bragg grating (CFBG) can eliminate the TDS without necessary frequency detuning [25]. Compared with uniform grating feedback, CFBG feedback is complete dispersive feedback without filtering effect, because of CFBG reflection spectrum is much wider than that of chaotic laser. In this paper, we quantitatively analyze the complexity of a semiconductor laser with CFBG feedback by using PE as well as TDS. Theoretically, dispersion feedback from CFBG causes irregular external cavity modes [25], and resultantly laser chaos with high complexity can be expected. By analyzing the laser parameters and grating parameters, the parameter regions that can generate high complexity chaotic signals are successfully identified.
∗ Corresponding author. E-mail address: [email protected] (A. Wang).
https://doi.org/10.1016/j.optcom.2019.124702 Received 24 June 2019; Received in revised form 18 September 2019; Accepted 2 October 2019 Available online 4 October 2019 0030-4018/© 2019 Published by Elsevier B.V.
M. Chao, D. Wang, L. Wang et al.
Optics Communications 456 (2020) 124702
2.2. Permutation entropy calculation In 2002, Bandt and Pompe proposed a permutation entropy (PE) algorithm for quantitative analysis of the time series complexity [28]. Embed an arbitrary time series {𝑥𝑡 , 𝑡 = 1,. . . , N} into the m-dimensional space [28]:
Fig. 1. The simulation model of a CFBG feedback semiconductor laser system. DFB: distributed feedback; SL: semiconductor laser; CFBG: chirped fiber Bragg grating.
𝑋𝑡 = {𝑥[𝑡], 𝑥[𝑡 + 𝜏𝑒 ], … , 𝑥[𝑡 + (𝑚 − 1)𝜏𝑒 ]}
where m and 𝜏𝑒 are two basic physical quantities in the permutation entropy algorithm, which are the embedding dimension and the embedding time delay, respectively. The 𝑋𝑡 will be rearranged in ascending order of {𝑥[𝑡 + (𝑗1 − 1)𝜏𝑒 ] ≤ 𝑥[𝑡 + (𝑗2 − 1)𝜏𝑒 ] ≤ ⋯ ≤ 𝑥[𝑡 + (𝑗𝑚 − 1)𝜏𝑒 ]}. If there are two identical numbers, they are distinguished by the 𝑗𝑚 size. Therefore, for any 𝑋𝑡 , it can be uniquely mapped to an ‘‘ordered pattern’’ s = (𝑗1 , 𝑗2 , … , 𝑗𝑚 ), where s is one of m! permutations consisting of m symbols (1, 2, . . . , m) and 𝑗𝑚 is the order in which 𝑥[𝑡 + (𝑗𝑚 − 1)𝜏𝑒 ] is placed in the original 𝑥[𝑡 + (𝑚 − 1)𝜏𝑒 ]. Bandt and Pompe suggested that the embedding dimension should be 3 ≤ m ≤ 7 and pointed out that the length of the sequence N should be large enough to satisfy N ≥ m! in order to obtain more reliable probability distribution [28]. In this paper, for the convenience of calculation, we use 𝑚 = 4 and N = 10 000 to simulate.
2. Theory and model 2.1. Rate equation model The simulation model of CFBG feedback semiconductor laser is displayed in Fig. 1. By modifying the Lang-Kobayashi equations, the following rate equations for the CFBG feedback semiconductor laser model are obtained [22,23,26]: [
] 𝑡 𝜅𝑓 1 𝐸+ ℎ(𝑡 − 𝑡′ )𝐸(𝑡′ − 𝜏)𝑑𝑡′ 𝜏𝑝 𝜏𝑖𝑛 ∫𝑡−𝑇 1 + 𝜀|𝐸|2 𝑔(𝑁 − 𝑁0 ) 𝐼 𝑁 𝑑𝑁 = − − |𝐸|2 , 𝑑𝑡 𝑞𝑉 𝜏𝑁 1 + 𝜀|𝐸|2
𝑑𝐸 1 + 𝑖𝛼 = 𝑑𝑡 2
𝑔(𝑁 − 𝑁0 )
−
(1) (2)
where E is the complex amplitude of the laser field, N is the carrier density, 𝛼 is the linewidth enhancement factor, 𝜀 is the gain saturation coefficient, 𝜏𝑝 is the photon lifetime, 𝜏𝑖𝑛 is the round-trip time in laser cavity, 𝜏𝑁 is the carrier lifetime, 𝜏 is the feedback delay, and 𝜅𝑓 is the feedback strength. The second term on the right side of Eq. (1) represents the dispersive optical feedback of the chirped grating. The integral therein represents the convolution of the CFBG response function h(t ) and the delayed laser field E(𝑡 − 𝜏). The integral length T should be greater than the grating response time. The dispersion feedback term is calculated by the inverse Fourier transform of H (𝜔) ⋅ FT[E(𝑡 − 𝜏)], where FT[⋅] represents the Fourier transform and H (𝜔) represents the complex reflection spectrum of the CFBG. The H (𝜔) can be obtained by the piecewise-uniform method [25,27] in which the chirped grating is equally divided into M sub-gratings regarded as uniform gratings with different grating periods. Using the propagation matrix method, the amplitudes of the forward-going wave 𝑆𝑀 and the backward-going wave 𝑅𝑀 at the incident interface of the chirped grating can be calculated as [ ] [ ] 𝑆𝑀 1 = 𝐅𝑀 ⋅ 𝐅𝑀−1 ⋅ … ⋅ 𝐅𝑗 ⋅ … 𝐅1 (3) 𝑅𝑀 0
For all probability distributions P = {p(s), 𝑠 = 𝑗 1 ,j2 , . . . , j𝑚 }, define the normalized permutation entropy as [28]: ∑ 𝐻 [𝑃 ] = − 𝑝 (𝑠) log 𝑝 (𝑠) ∕ log (m!) (6) where the value of H [P] ranges from 0 to 1. When H [P] =0, it indicates that the time series is regular and can be predicted, and H [P] = 1 means that the sequence is random and unpredictable. Therefore, the higher the value of H [P] is, the higher the complexity of the chaotic signal is.
3. Numerical results In this section, we simulate the effects of various parameters on the chaotic complexity of the CFBG feedback semiconductor laser system, using PE and TDS. For comparative analysis, the mirror feedback semiconductor laser system is also considered. The rate equation of the mirror feedback model is consistent with that in [17]. The laser parameters of the CFBG feedback and the mirror feedback are the same. Fig. 2 shows the optical spectra, ACF curves, PE curves obtained from the simulation of mirror and CFBG feedback system, where the threshold current 𝐼th is 15.56 mA, the center wavelength of the laser is 1550 nm, and the dispersion value under CFBG feedback is 2000 ps/nm. In addition, the red (gray) and blue (dark gray) curves in Fig. 2(b1) are the delayed and reflected spectra of CFBG, respectively. For mirror feedback, in the ACF curve of Fig. 2(a2), there is a significant side lobe when the feedback delay 𝜏 = 3 ns, and there are distinct valleys in the corresponding PE curve around 𝜏𝑒 = 𝜏 and 𝜏𝑒 = 1/2𝜏 in Fig. 2(a3). In this paper, we take the valley value at 𝜏𝑒 = 𝜏 as the value of the PE [29,30]. The PE in Fig. 2(a3) is 0.968. But for CFBG feedback, in Fig. 2(b2) there is no side lobe at 𝜏 = 3 ns, the PE curve in Fig. 2(b3) becomes smooth and no valleys appear. The reason for this phenomenon is that the chirped grating provides different delays, as shown in Fig. 2(b1). The dispersion feedback results in irregular external-cavity mode separation, unlike fixed mode separation in mirror feedback [25]. Therefore, the TDS and the corresponding periodic features in the PE curve are eliminated. At this point, we take the value at 𝜏𝑒 = 𝜏 as the PE value under CFBG feedback, and the value in Fig. 2(b3) is 0.994, which is increased by 2.6% compared to the mirror feedback.
The reflection spectrum of the chirped grating is obtained by 𝑅𝑀 ∕𝑆𝑀 . In Eq. (3), 𝐅𝑗 is the propagation matrix of the 𝑗 th sub-grating with the following mathematic expression 𝜅𝑗 ⎡cosh(𝛾 𝛥𝐿) − 𝑖 𝜎̂ 𝑗 sinh(𝛾 𝛥𝐿) ⎤ −𝑖 sinh(𝛾𝑗 𝛥𝐿) 𝑗 𝑗 𝛾𝑗 ⎢ ⎥ 𝛾𝑗 𝐅𝑗 = ⎢ (4) ⎥ 𝜅𝑗 𝜎̂ 𝑗 ⎢ 𝑖 sinh(𝛾𝑗 𝛥𝐿) cosh(𝛾𝑗 𝛥𝐿) + 𝑖 sinh(𝛾𝑗 𝛥𝐿)⎥ ⎣ ⎦ 𝛾𝑗 𝛾𝑗 𝛾𝑗 =
(5)
√ 𝜅𝑗 2 − 𝜎̂ 𝑗 2 , 𝜎̂ 𝑗 ≈ 𝑛ef f (𝜔 − 𝜔𝑗 )∕𝑐, 𝜅𝑗 = 𝜔𝑗 𝜌⟨𝛿𝑛⟩∕2𝑐,
where 𝜔𝑗 = 𝜋c/𝑛ef f 𝛬𝑗 , 𝛥L and 𝛬𝑗 are the Bragg frequency, length and grating period of the 𝑗 th grating. The refractive index 𝑛𝑗 = 𝑛ef f + ⟨𝛿𝑛⟩{1 + 𝜌 cos(2𝜋𝑧∕𝛬j )}, where 𝑛ef f is the effective index, ⟨𝛿𝑛⟩ is the average index change, 𝜌 is the fringe visibility of the index change in grating, and c is the speed of light in vacuum. In simulation, a 10-cm long chirped FBG with a dispersion of 2000 ps/nm is utilized. It is divided into 200 uniform sub-gratings, and the corresponding Bragg frequencies 𝜔𝑗 are calculated according to chirp factor d𝜆/dz = 0.05 nm/cm and the CFBG center wavelength 1550 nm. The other parameters are set as 𝑛ef f = 1.46, ⟨𝛿𝑛⟩ = 5 × 10−4 , 𝜌 = 1, and integral length 𝑇 = 20.48 ns, which is larger than the response time of grating. 2
M. Chao, D. Wang, L. Wang et al.
Optics Communications 456 (2020) 124702
Fig. 2. Optical spectrum (a1, b1), ACF curves (a2, b2), PE curves (a3, b3) obtained in the simulation of mirror feedback (a1–a3) and CFBG feedback (b1–b3). The group delay spectrum and reflection spectrum of CFBG are also drawn in red (gray) and blue (dark gray) in (b1). The dispersion value of CFBG is 2000 ps/nm. 𝜏 = 3 ns, 𝐼 = 1.5𝐼th , 𝜅𝑓 = 0.3 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. TDS (a) and PE (b) as a function of 𝛼 for the mirror feedback and the CFBG feedback. 𝜏 = 3 ns, 𝐼 = 1.5𝐼th , 𝜅𝑓 = 0.3. 3𝜎 represents three times standard deviation of the ACF curve noise floor.
Fig. 4. TDS (a) and PE (b) as a function of 𝜀 for the mirror feedback and the CFBG feedback. 𝜏 = 3 ns, 𝐼 = 1.5𝐼th , 𝜅𝑓 = 0.3.
3, the CFBG system can obtain laser chaos with suppressed TDS and increased complexity. Fig. 4 shows the TDS and the PE as a function of gain saturation coefficient 𝜀 for the mirror feedback (blue circles) and the CFBG feedback (red triangles). In Fig. 4(a), in the range of 1 × 10−5 μm3 ≤ 𝜀 ≤ 1.5 × 10−4 μm3 , the TDS under mirror feedback gradually decreases, and when 𝜀 ≥ 1.5×10−4 μm3 , the TDS increases linearly with the increase of 𝜀. For CFBG feedback, TDS has different trends. In the range of 1 × 10−5 μm3 ≤ 𝜀 ≤ 4 × 10−4 μm3 , the TDS is almost hidden in the background noise of the ACF curve. When 𝜀 ≥ 4 × 10−4 μm3 , TDS shows a tendency to increase slowly due to the decrease of optical spectral linewidth. The PE curves in Fig. 4(b) have a nearly opposite trend to the TDS curves. In the range of 1 × 10−5 μm3 ≤ 𝜀 ≤ 4 × 10−4 μm3 , the curve under CFBG feedback remains stable almost at H [P] = 0.994, and tends to decrease as 𝜀 increases. The PE value under CFBG feedback is always greater than the value under mirror feedback. The results in Fig. 4 show that for a CFBG feedback system, choosing a smaller 𝜀 will eliminate TDS and obtain a more complex chaotic signal.
3.1. Influence of internal parameters of laser In this section, we mainly study the influence of laser internal parameters including the linewidth enhancement factor 𝛼 and the gain saturation coefficient 𝜀 on the complexity of chaotic signals. In a curve obtained by autocorrelation of a chaos time series (i.e., autocorrelation function), obvious side lobes appear at the feedback delay, and the value of its height is usually used to measure the TDS. In this paper, the TDS value refers to the maximum value in the time window [𝜏 − 0.05𝜏, 𝜏 + 0.05𝜏]. The autocorrelation function (ACF) can be obtained by [31]: ⟨[ ][ ]⟩ 𝑃 (𝑡 − 𝜏) − ⟨𝑃 (𝑡 − 𝜏)⟩ 𝑃 (𝑡) − ⟨𝑃 (𝑡)⟩ 𝐴 (𝜏) = √⟨ ]2 ⟩ ⟨[ ]2 ⟩ [ 𝑃 (𝑡 − 𝜏) − ⟨𝑃 (𝑡 − 𝜏)⟩ 𝑃 (𝑡) − ⟨𝑃 (𝑡)⟩
(7)
where P(t) = | E(t)|2 , which is the intensity of the laser output light, and ⟨∙⟩ represents averaging time. Fig. 3 shows the TDS and PE as a function of linewidth enhancement factor 𝛼 for the mirror feedback (blue circles) and the CFBG feedback (red triangles). The stars in Fig. 3(a) are three times the standard deviation of the ACF curve noise floor. In Fig. 3(a), for the mirror feedback, as 𝛼 increase, the TDS can be suppressed but not completely eliminated and much larger than the value under CFBG feedback. For CFBG feedback, when 𝛼 ≤ 3, TDS cannot be eliminated due to the influence of spectral linewidth. As 𝛼 increase, the TDS gradually decreases and is hidden in background noise to achieve the purpose of eliminating the delay signature. In Fig. 3(b), for CFBG feedback, the values of PE are maintained at a high value. For mirror feedback, the value of PE increases with the increase of 𝛼, but is always smaller than the value under CFBG feedback. Therefore, within the range of 𝛼 ≥
3.2. Influence of external parameters of laser This section mainly analyzes the influence of external parameters of lasers including feedback strength 𝜅𝑓 and normalized bias current I /𝐼th on the complexity of the CFBG feedback and mirror feedback systems through TDS and PE. Fig. 5 plots the TDS and the PE as a function of feedback strength 𝜅𝑓 for the mirror feedback (blue circles) and the CFBG feedback (red triangles). Fig. 5(a) displays that at 𝜅𝑓 < 0.05, the TDS decreases with increasing 𝜅𝑓 , reaching a minimum when 𝜅𝑓 is 0.05. When 𝜅𝑓 continues to increase, the TDS under mirror feedback shows a linear increase trend, which is consistent with the numerical results in [23]. 3
M. Chao, D. Wang, L. Wang et al.
Optics Communications 456 (2020) 124702
Fig. 7. Maps of (a) TDS and (b) PE in the parameter space of 𝜅𝑓 and I /𝐼th for the CFBG feedback. Fig. 5. TDS (a) and PE (b) as a function of 𝜅𝑓 for the mirror feedback and the CFBG feedback. 𝜏 = 3 ns, 𝐼 = 1.5𝐼th .
Fig. 8. TDS and PE as a function of (a) frequency detuning 𝛥𝜈 and (b) dispersion for the CFBG feedback. 𝜏 = 3 ns, 𝐼 = 1.5𝐼th , 𝜅𝑓 = 0.3. Fig. 6. TDS (a) and PE (b) as a function of I /𝐼th for the mirror feedback and the CFBG feedback. 𝜏 = 3 ns, 𝜅𝑓 = 0.3.
However, the value under CFBG feedback is still hidden in the background noise and will not increase again. Correspondingly, in the PE curves of Fig. 5(b), as the 𝜅𝑓 increases, the PE gradually reaches the maximum value. Further increasing 𝜅𝑓 , the PE value under mirror feedback gradually decreases, while the PE value under CFBG feedback is stable at 0.994 and will not decrease. Then the PE values of the CFBG feedback are higher than those of the mirror feedback. It can also be seen that the parameter region for obtaining high-complexity chaotic signals under CFBG feedback is wider. Fig. 6 plots the TDS and the PE as a function of normalized bias current I /𝐼th for the mirror feedback (blue circles) and the CFBG feedback (red triangles). In Fig. 6(a), the TDS values under mirror feedback decrease with increasing I /𝐼th , whereas for CFBG feedback, the TDS values are always stable at small value and are hidden in the background noise. In Fig. 6(b), as I /𝐼th increases, the PE values under the mirror feedback increase significantly, the values under CFBG feedback increase slightly, but its values are much larger than the values under mirror feedback. It can be seen that when I ≥ 1.5𝐼th , the value of H [P] reaches 0.994. The results of Fig. 6 show that CFBG feedback can hide the TDS and obtain more complex chaotic signals over a wider range of bias currents than mirror feedback. To further analyze the influence of external parameters of laser on the complexity of the laser chaos for CFBG feedback, maps of the TDS and PE in the parameter space of I /𝐼th and 𝜅𝑓 are displayed in Fig. 7. From the two-dimensional map of Fig. 7(a), it can be seen that lower TDS appears in the region of larger bias current and stronger feedback strength. Even at large bias currents, when the feedback strength is weak, the TDS is larger. Meanwhile, the complexity in Fig. 7(b) has the same trend as the TDS in Fig. 7(a). At higher bias currents and feedback strengths, it is easier to obtain chaotic signals with higher complexity. Furthermore, as the I /𝐼th and 𝜅𝑓 increase, the high complexity regions are much wider.
that the PE value under CFBG feedback is significantly higher than that of mirror feedback in the range of the appropriate internal and external parameters of the laser. This section discusses the unique grating parameters in the CFBG feedback system, so the mirror feedback semiconductor laser system is not considered in this section. This section mainly analyzes the effects of frequency detuning and dispersion in the CFBG feedback on the TDS and PE values. Fig. 8 shows the PE (blue circles) and TDS (red star) values as a function of frequency detuning between laser and grating 𝛥𝜈 (a) and dispersion (b) for the CFBG feedback. The difference in center frequency, between the reflection spectra of chirped grating and the spectra of laser, is defined as the frequency detuning 𝛥𝜈. In Fig. 8(a), when 𝛥𝜈 ranges from −15 GHz to 10 GHz, the PE values are all at a higher value (about 0.994). Correspondingly, the TDS curve is at a lower value within this range (even ACF background noise level). When 𝛥𝜈 ⩽ −15 GHz or 𝛥𝜈 ⩾ 10 GHz, the frequency detuning range at this time is too large, which weakens the dispersive feedback caused by the chirped grating, the complexity of the chaotic signal decreases, and the value of the TDS increases significantly, as shown in Fig. 8(a). Therefore, when the frequency detuning satisfies the range of −15 GHz ⩽ 𝛥𝜈 ⩽ 10 GHz, the CFBG feedback semiconductor laser system can hide the TDS and obtain a highly complex chaotic output. The results of Fig. 8(a) are obtained when the chirped grating has a dispersion of 2000 ps/nm. In order to further study the effect of dispersion on the complexity, the TDS and PE curves under different dispersion were made with zero frequency detuning, as show in Fig. 8(b). As the dispersion value increases, the dispersive feedback caused by the chirped grating increases. When the dispersion exceeds 2000 ps/nm, the TDS is hidden in the background noise and never rises again, and the PE value is stable at 0.994. Meantime the CFBG feedback semiconductor laser system will obtain chaotic output with high complexity.
3.3. Influence of grating parameters
4. Conclusion
The above results are based on the comparative analysis of mirror feedback and CFBG feedback semiconductor laser system. Results show
The complexity of the CFBG feedback semiconductor laser system is numerically investigated by using the PE and the TDS. The trend 4
M. Chao, D. Wang, L. Wang et al.
Optics Communications 456 (2020) 124702
of TDS always has the opposite relationship with PE. And the results of CFBG feedback are compared with the mirror feedback. Different from mirror feedback, CFBG feedback can eliminate TDS and obtain chaotic signals with high complexity. The effects of laser parameters and grating parameters are studied in detail. The results show that due to the influence of spectral linewidth, the CFBG feedback can eliminate the TDS and obtain high-complexity chaotic signal with PE value up to 0.994 under the conditions of 𝛼 ≥ 3 and 𝜀 ≤ 4 × 10−4 μm3 . Under the strong feedback strength and large bias current, it is easier to enhance the complexity of chaotic signals. Compared with the mirror feedback, the parameter region of the CFBG feedback that can obtain high-complexity signals is broadened. When the chirped grating dispersion value exceeds 2000 ps/nm, chaotic signals with high complexity can be obtained at a small frequency detuning of −15 GHz ≤ 𝛥𝜈 ≤ 10 GHz. The TDS of the chaotic signal source is eliminated and the complexity is enhanced, which greatly improves the security of the chaotic secure communication system.
[10] C. Xue, N. Jiang, K. Qiu, Y. Lv, Key distribution based on synchronization in bandwidth-enhanced random bit generators with dynamic post-processing, Opt. Express 23 (2015) 14510–14519. [11] A.E. Hajomer, X. Yang, A. Sultan, W. Hu, Key distribution based on phase fluctuation between polarization modes in optical channel, IEEE Photon. Technol. Lett. 30 (2018) 704–707. [12] S. Xiang, Y. Zhang, J. Gong, X. Guo, L. Lin, Y. Hao, STDP-based unsupervised spike pattern learning in a photonic spiking neural network with VCSELs and VCSOAs, IEEE J. Sel. Top. Quantum Electron. 25 (2019) 1–9. [13] L. Zhang, W. Pan, L. Yan, B. Luo, X. Zou, M. Xu, Cluster synchronization of coupled semiconductor lasers network with complex topology, IEEE J. Sel. Top. Quantum Electron. 25 (2019) 1–7. [14] D. Rontani, A. Locquet, M. Sciamanna, D.S. Citrin, Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback, Opt. Lett. 32 (2007) 2960–2962. [15] D. Rontani, A. Locquet, M. Sciamanna, D.S. Citrin, S. Ortin, Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view, IEEE J. Quantum Electron. 45 (2009) 879–1891. [16] J. Wu, G. Xia, Z. Wu, Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback, Opt. Express 17 (2009) 20124–20133. [17] Y. Wu, Y. Wang, P. Li, Can fixed time delay signature be concealed in chaotic semiconductor laser with optical feedback? IEEE J. Quantum Electron. 48 (2012) 1371–1378. [18] X.X. Guo, S.Y. Xiang, Y.H. Zhang, A.J. Wen, Y. Hao, Information-theory-based complexity quantifier for chaotic semiconductor laser with double time delays, IEEE J. Quantum Electron. 54 (2018) 1–8. [19] T. Wu, W. Sun, X. Zhang, Concealment of time delay signature of chaotic output in a slave semiconductor laser with chaos laser injection, Opt. Commun. 381 (2016) 174–179. [20] Y. Hong, P.S. Spencer, K.A. Shore, Wideband chaos with time-delay concealment in vertical-cavity surface-emitting lasers with optical feedback and injection, IEEE J. Quantum Electron. 50 (2014) 236–242. [21] N. Li, W. Pan, A. Locquet, D.S. Citrin, Time-delay concealment and complexity enhancement of an external-cavity laser through optical injection, Opt. Lett. 40 (2015) 4416–4419. [22] S.S. Li, Q. Liu, S.C. Chan, Distributed feedbacks for time-delay signature suppression of chaos generated from a semiconductor laser, IEEE Photon. J. 5 (2012) 1930–1935. [23] S.S. Li, S.C. Chan, Chaotic time-delay signature suppression in a semiconductor laser with frequency-detuned grating feedback, IEEE J. Sel. Top. Quantum Electron. 21 (2015) 541–552. [24] Z.Q. Zhong, S.S. Li, S.C. Chan, G.Q. Xia, Z.M. Wu, Polarization-resolved timedelay signatures of chaos induced by FBG-feedback in VCSEL, Opt. Express 23 (2015) 15459–15468. [25] D.M. Wang, L.S. Wang, T. Zhao, H. Gao, Y.C. Wang, X.F. Chen, A.B. Wang, Time delay signature elimination of chaos in a semiconductor laser by dispersive feedback from a chirped FBG, Opt. Express 25 (2017) 10911–10924. [26] R. Lang, K. Kobayashi, External optical feedback effects on semiconductor injection laser properties, IEEE J. Quantum Electron. 16 (1980) 347–355. [27] T. Erdogan, Fiber grating spectra, J. Lightwave Technol. 15 (1997) 1277–1294. [28] C. Bandt, B. Pompe, Permutation entropy: a natural complexity measure for time series, Phys. Rev. Lett. 88 (2002) 174102. [29] D. Rontani, E. Mercier, D. Wolfersberger, M. Sciamanna, Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback, Opt. Lett. 41 (2016) 4637–4640. [30] S.K. Ji, Y. Hong, Effect of bias current on complexity and time delay signature of chaos in semiconductor laser with time-delayed optical feedback, IEEE J. Sel. Top. Quantum Electron. 23 (2017) 1800706. [31] T. Perez, M. Radziunas, H.J. Wunsche, C.R. Mirasso, F. Henneberger, Synchronization properties of two coupled multisection semiconductor lasers emitting chaotic light, IEEE Photon. Technol. Lett. 18 (2006) 2135–2137.
Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grants 61822509, 61731014, 61475111, 61671316, 61805170, and 61805171, and in part by the National Cryptography Development Foundation, China under Grant MMJJ20170207, and the Fund for Shanxi ‘‘1331 Project" Key Innovative Research Team. References [1] A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C.R. Mirasso, L. Pesquera, K.A. Shore, Chaos-based communications at high bit rates using commercial fiber-optic links, Nature 437 (2005) 343–346. [2] R. Lavrov, M. Jacquot, L. Larger, Nonlocal nonlinear electro-optic phase dynamics demonstrating 10 Gb/s chaos communications, IEEE J. Quantum Electron. 46 (2010) 1430–1435. [3] N. Li, H. Susanto, B. Cemlyn, I.D. Henning, M.J. Adams, Secure communication systems based on chaos in optically pumped spin- VCSELs, Opt. Lett. 42 (2017) 3494–3497. [4] N. Jiang, A.K. Zhao, S.Q. Liu, C.P. Xue, K. Qiu, Chaos synchronization and communication in closed-loop semiconductor lasers subject to common chaotic phase-modulated feedback, Opt. Express 26 (2018) 32404–32416. [5] T. Kurashige, M. Shiki, K. Yoshimura, P. Davis, S. Naito, S. Yoshimori, M. Inoue, H. Someya, A. Uchida, K. Amano, I. Oowada, K. Hirano, Fast physical random bit generation with chaotic semiconductor lasers, Nature Photon. 2 (2008) 728–732. [6] I. Kanter, Y. Aviad, I. Reidler, E. Cohen, M. Rosenbluh, An optical ultrafast random bit generator, Nature Photon. 4 (2009) 58–61. [7] A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, D. Syvridis, Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit, Opt. Express 18 (2010) 18763–18768. [8] L. Wang, T. Zhao, D. Wang, D. Wu, Real-time 14-Gbps physical random bit generator based on time-interleaved sampling of broadband white chaos, IEEE Photon. J. 9 (2017) 1–13. [9] X.Z. Li, S.S. Li, J.P. Zhuang, S.C. Chan, Random bit generation at tunable rates using a chaotic semiconductor laser under distributed feedback, Opt. Lett. 40 (2015) 3970–3973.
5
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Permutation entropy analysis of chaotic semiconductor laser with chirped FBG feedback Meng Chao a,b , Daming Wang a,b , Longsheng Wang a,b , Yuchuan Sun a,b , Hong Han a,b , Yuanyuan Guo a,b , Zhiwei Jia a,b , Yuncai Wang a,b,c , Anbang Wang a,b ,∗ a
Key Laboratory of Advanced Transducers & Intelligent Control System, Ministry of Education, Taiyuan University of Technology, Taiyuan 030024, China Institute of Optoelectronic Engineering, College of Physics & Optoelectronics, Taiyuan University of Technology, Taiyuan 030024, China c Institute of Advanced Photonics Technology, Guangdong University of Technology, Guangzhou 510006, China b
ARTICLE
INFO
Keywords: Chaos Chirped fiber Bragg grating Complexity Permutation entropy Semiconductor laser Time delay signature
ABSTRACT We numerically analyze the chaos complexity of a semiconductor laser with chirped fiber Bragg grating (CFBG) feedback by utilizing the permutation entropy (PE). Due to dispersion-induced nonlinearity of chirped grating, the time delay signature (TDS) can be suppressed or erased and the complexity enhancement is expected. The effects of laser parameters on PE as well as the TDS are studied in detail. The results show that for CFBG feedback, the TDS can be suppressed in the range of parameters where linewidth enhancement factor exceeds 3 and gain saturation coefficient is lower than 4×10−4 μm3 , and the complexity of chaotic signals is enhanced. As the feedback strength and bias current increase, the PE value under CFBG feedback increases linearly to a maximum of 0.994 and remains stable. Compared with mirror feedback, CFBG feedback can achieve increased complexity over a wider range of parameters. In addition, when the dispersion value of the chirped grating exceeds 2000 ps/nm, the frequency detuning range of -15 GHz to 10 GHz is beneficial to obtain chaotic signals with higher PE values.
1. Introduction Semiconductor lasers have been widely used in many fields due to its enormous potential, such as chaotic secure communication [1–4], physical random bit generation [5–9], secure key distribution [10,11], photonic spiking neural network [12], delay-coupled semiconductor lasers networks [13]. Obtaining broadband and high complexity signals is a key issue in laser chaos. An external cavity feedback semiconductor laser is the most attractive choice in view of its simple and integrated setup. Unfortunately, due to the linear feedback caused by the external cavity, semiconductor laser with conventional mirror optical feedback has time delay signature (TDS), which makes the chaotic signals have a high correlation with its previous states, thus reducing its complexity [14,15]. Some methods have been proposed to eliminate the TDS, including dual-cavity optical feedback [16–18], additional optical injection [19– 21], and uniform grating feedback [22,23]. Guo et al [18] have numerically studied the complexity of a chaotic laser system with dual-cavity mirror feedback using permutation entropy (PE), and found the regions of feedback rate and bias current for high complexity and low TDS. Li et al numerically shown that for optically injected semiconductor lasers the TDS can be concealed and the dynamic complexity can be
enhanced under large linewidth enhancement factor [21]. It was also experimental demonstrated that optical feedback from uniform fiber Bragg grating (FBG) can depress TDS [23] and enhance the PE [24], which is beneficial to random number generation [9]. Note that uniform FBG feedback should have a detuned frequency relative to laser frequency in order to suppress TDS greatly due to that remarkable dispersion effect appears in sideband of uniform grating spectrum [23]. Our previous work experimentally and theoretically proved that optical feedback from a chirped fiber Bragg grating (CFBG) can eliminate the TDS without necessary frequency detuning [25]. Compared with uniform grating feedback, CFBG feedback is complete dispersive feedback without filtering effect, because of CFBG reflection spectrum is much wider than that of chaotic laser. In this paper, we quantitatively analyze the complexity of a semiconductor laser with CFBG feedback by using PE as well as TDS. Theoretically, dispersion feedback from CFBG causes irregular external cavity modes [25], and resultantly laser chaos with high complexity can be expected. By analyzing the laser parameters and grating parameters, the parameter regions that can generate high complexity chaotic signals are successfully identified.
∗ Corresponding author. E-mail address: [email protected] (A. Wang).
https://doi.org/10.1016/j.optcom.2019.124702 Received 24 June 2019; Received in revised form 18 September 2019; Accepted 2 October 2019 Available online 4 October 2019 0030-4018/© 2019 Published by Elsevier B.V.
M. Chao, D. Wang, L. Wang et al.
Optics Communications 456 (2020) 124702
2.2. Permutation entropy calculation In 2002, Bandt and Pompe proposed a permutation entropy (PE) algorithm for quantitative analysis of the time series complexity [28]. Embed an arbitrary time series {𝑥𝑡 , 𝑡 = 1,. . . , N} into the m-dimensional space [28]:
Fig. 1. The simulation model of a CFBG feedback semiconductor laser system. DFB: distributed feedback; SL: semiconductor laser; CFBG: chirped fiber Bragg grating.
𝑋𝑡 = {𝑥[𝑡], 𝑥[𝑡 + 𝜏𝑒 ], … , 𝑥[𝑡 + (𝑚 − 1)𝜏𝑒 ]}
where m and 𝜏𝑒 are two basic physical quantities in the permutation entropy algorithm, which are the embedding dimension and the embedding time delay, respectively. The 𝑋𝑡 will be rearranged in ascending order of {𝑥[𝑡 + (𝑗1 − 1)𝜏𝑒 ] ≤ 𝑥[𝑡 + (𝑗2 − 1)𝜏𝑒 ] ≤ ⋯ ≤ 𝑥[𝑡 + (𝑗𝑚 − 1)𝜏𝑒 ]}. If there are two identical numbers, they are distinguished by the 𝑗𝑚 size. Therefore, for any 𝑋𝑡 , it can be uniquely mapped to an ‘‘ordered pattern’’ s = (𝑗1 , 𝑗2 , … , 𝑗𝑚 ), where s is one of m! permutations consisting of m symbols (1, 2, . . . , m) and 𝑗𝑚 is the order in which 𝑥[𝑡 + (𝑗𝑚 − 1)𝜏𝑒 ] is placed in the original 𝑥[𝑡 + (𝑚 − 1)𝜏𝑒 ]. Bandt and Pompe suggested that the embedding dimension should be 3 ≤ m ≤ 7 and pointed out that the length of the sequence N should be large enough to satisfy N ≥ m! in order to obtain more reliable probability distribution [28]. In this paper, for the convenience of calculation, we use 𝑚 = 4 and N = 10 000 to simulate.
2. Theory and model 2.1. Rate equation model The simulation model of CFBG feedback semiconductor laser is displayed in Fig. 1. By modifying the Lang-Kobayashi equations, the following rate equations for the CFBG feedback semiconductor laser model are obtained [22,23,26]: [
] 𝑡 𝜅𝑓 1 𝐸+ ℎ(𝑡 − 𝑡′ )𝐸(𝑡′ − 𝜏)𝑑𝑡′ 𝜏𝑝 𝜏𝑖𝑛 ∫𝑡−𝑇 1 + 𝜀|𝐸|2 𝑔(𝑁 − 𝑁0 ) 𝐼 𝑁 𝑑𝑁 = − − |𝐸|2 , 𝑑𝑡 𝑞𝑉 𝜏𝑁 1 + 𝜀|𝐸|2
𝑑𝐸 1 + 𝑖𝛼 = 𝑑𝑡 2
𝑔(𝑁 − 𝑁0 )
−
(1) (2)
where E is the complex amplitude of the laser field, N is the carrier density, 𝛼 is the linewidth enhancement factor, 𝜀 is the gain saturation coefficient, 𝜏𝑝 is the photon lifetime, 𝜏𝑖𝑛 is the round-trip time in laser cavity, 𝜏𝑁 is the carrier lifetime, 𝜏 is the feedback delay, and 𝜅𝑓 is the feedback strength. The second term on the right side of Eq. (1) represents the dispersive optical feedback of the chirped grating. The integral therein represents the convolution of the CFBG response function h(t ) and the delayed laser field E(𝑡 − 𝜏). The integral length T should be greater than the grating response time. The dispersion feedback term is calculated by the inverse Fourier transform of H (𝜔) ⋅ FT[E(𝑡 − 𝜏)], where FT[⋅] represents the Fourier transform and H (𝜔) represents the complex reflection spectrum of the CFBG. The H (𝜔) can be obtained by the piecewise-uniform method [25,27] in which the chirped grating is equally divided into M sub-gratings regarded as uniform gratings with different grating periods. Using the propagation matrix method, the amplitudes of the forward-going wave 𝑆𝑀 and the backward-going wave 𝑅𝑀 at the incident interface of the chirped grating can be calculated as [ ] [ ] 𝑆𝑀 1 = 𝐅𝑀 ⋅ 𝐅𝑀−1 ⋅ … ⋅ 𝐅𝑗 ⋅ … 𝐅1 (3) 𝑅𝑀 0
For all probability distributions P = {p(s), 𝑠 = 𝑗 1 ,j2 , . . . , j𝑚 }, define the normalized permutation entropy as [28]: ∑ 𝐻 [𝑃 ] = − 𝑝 (𝑠) log 𝑝 (𝑠) ∕ log (m!) (6) where the value of H [P] ranges from 0 to 1. When H [P] =0, it indicates that the time series is regular and can be predicted, and H [P] = 1 means that the sequence is random and unpredictable. Therefore, the higher the value of H [P] is, the higher the complexity of the chaotic signal is.
3. Numerical results In this section, we simulate the effects of various parameters on the chaotic complexity of the CFBG feedback semiconductor laser system, using PE and TDS. For comparative analysis, the mirror feedback semiconductor laser system is also considered. The rate equation of the mirror feedback model is consistent with that in [17]. The laser parameters of the CFBG feedback and the mirror feedback are the same. Fig. 2 shows the optical spectra, ACF curves, PE curves obtained from the simulation of mirror and CFBG feedback system, where the threshold current 𝐼th is 15.56 mA, the center wavelength of the laser is 1550 nm, and the dispersion value under CFBG feedback is 2000 ps/nm. In addition, the red (gray) and blue (dark gray) curves in Fig. 2(b1) are the delayed and reflected spectra of CFBG, respectively. For mirror feedback, in the ACF curve of Fig. 2(a2), there is a significant side lobe when the feedback delay 𝜏 = 3 ns, and there are distinct valleys in the corresponding PE curve around 𝜏𝑒 = 𝜏 and 𝜏𝑒 = 1/2𝜏 in Fig. 2(a3). In this paper, we take the valley value at 𝜏𝑒 = 𝜏 as the value of the PE [29,30]. The PE in Fig. 2(a3) is 0.968. But for CFBG feedback, in Fig. 2(b2) there is no side lobe at 𝜏 = 3 ns, the PE curve in Fig. 2(b3) becomes smooth and no valleys appear. The reason for this phenomenon is that the chirped grating provides different delays, as shown in Fig. 2(b1). The dispersion feedback results in irregular external-cavity mode separation, unlike fixed mode separation in mirror feedback [25]. Therefore, the TDS and the corresponding periodic features in the PE curve are eliminated. At this point, we take the value at 𝜏𝑒 = 𝜏 as the PE value under CFBG feedback, and the value in Fig. 2(b3) is 0.994, which is increased by 2.6% compared to the mirror feedback.
The reflection spectrum of the chirped grating is obtained by 𝑅𝑀 ∕𝑆𝑀 . In Eq. (3), 𝐅𝑗 is the propagation matrix of the 𝑗 th sub-grating with the following mathematic expression 𝜅𝑗 ⎡cosh(𝛾 𝛥𝐿) − 𝑖 𝜎̂ 𝑗 sinh(𝛾 𝛥𝐿) ⎤ −𝑖 sinh(𝛾𝑗 𝛥𝐿) 𝑗 𝑗 𝛾𝑗 ⎢ ⎥ 𝛾𝑗 𝐅𝑗 = ⎢ (4) ⎥ 𝜅𝑗 𝜎̂ 𝑗 ⎢ 𝑖 sinh(𝛾𝑗 𝛥𝐿) cosh(𝛾𝑗 𝛥𝐿) + 𝑖 sinh(𝛾𝑗 𝛥𝐿)⎥ ⎣ ⎦ 𝛾𝑗 𝛾𝑗 𝛾𝑗 =
(5)
√ 𝜅𝑗 2 − 𝜎̂ 𝑗 2 , 𝜎̂ 𝑗 ≈ 𝑛ef f (𝜔 − 𝜔𝑗 )∕𝑐, 𝜅𝑗 = 𝜔𝑗 𝜌⟨𝛿𝑛⟩∕2𝑐,
where 𝜔𝑗 = 𝜋c/𝑛ef f 𝛬𝑗 , 𝛥L and 𝛬𝑗 are the Bragg frequency, length and grating period of the 𝑗 th grating. The refractive index 𝑛𝑗 = 𝑛ef f + ⟨𝛿𝑛⟩{1 + 𝜌 cos(2𝜋𝑧∕𝛬j )}, where 𝑛ef f is the effective index, ⟨𝛿𝑛⟩ is the average index change, 𝜌 is the fringe visibility of the index change in grating, and c is the speed of light in vacuum. In simulation, a 10-cm long chirped FBG with a dispersion of 2000 ps/nm is utilized. It is divided into 200 uniform sub-gratings, and the corresponding Bragg frequencies 𝜔𝑗 are calculated according to chirp factor d𝜆/dz = 0.05 nm/cm and the CFBG center wavelength 1550 nm. The other parameters are set as 𝑛ef f = 1.46, ⟨𝛿𝑛⟩ = 5 × 10−4 , 𝜌 = 1, and integral length 𝑇 = 20.48 ns, which is larger than the response time of grating. 2
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Optics Communications 456 (2020) 124702
Fig. 2. Optical spectrum (a1, b1), ACF curves (a2, b2), PE curves (a3, b3) obtained in the simulation of mirror feedback (a1–a3) and CFBG feedback (b1–b3). The group delay spectrum and reflection spectrum of CFBG are also drawn in red (gray) and blue (dark gray) in (b1). The dispersion value of CFBG is 2000 ps/nm. 𝜏 = 3 ns, 𝐼 = 1.5𝐼th , 𝜅𝑓 = 0.3 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. TDS (a) and PE (b) as a function of 𝛼 for the mirror feedback and the CFBG feedback. 𝜏 = 3 ns, 𝐼 = 1.5𝐼th , 𝜅𝑓 = 0.3. 3𝜎 represents three times standard deviation of the ACF curve noise floor.
Fig. 4. TDS (a) and PE (b) as a function of 𝜀 for the mirror feedback and the CFBG feedback. 𝜏 = 3 ns, 𝐼 = 1.5𝐼th , 𝜅𝑓 = 0.3.
3, the CFBG system can obtain laser chaos with suppressed TDS and increased complexity. Fig. 4 shows the TDS and the PE as a function of gain saturation coefficient 𝜀 for the mirror feedback (blue circles) and the CFBG feedback (red triangles). In Fig. 4(a), in the range of 1 × 10−5 μm3 ≤ 𝜀 ≤ 1.5 × 10−4 μm3 , the TDS under mirror feedback gradually decreases, and when 𝜀 ≥ 1.5×10−4 μm3 , the TDS increases linearly with the increase of 𝜀. For CFBG feedback, TDS has different trends. In the range of 1 × 10−5 μm3 ≤ 𝜀 ≤ 4 × 10−4 μm3 , the TDS is almost hidden in the background noise of the ACF curve. When 𝜀 ≥ 4 × 10−4 μm3 , TDS shows a tendency to increase slowly due to the decrease of optical spectral linewidth. The PE curves in Fig. 4(b) have a nearly opposite trend to the TDS curves. In the range of 1 × 10−5 μm3 ≤ 𝜀 ≤ 4 × 10−4 μm3 , the curve under CFBG feedback remains stable almost at H [P] = 0.994, and tends to decrease as 𝜀 increases. The PE value under CFBG feedback is always greater than the value under mirror feedback. The results in Fig. 4 show that for a CFBG feedback system, choosing a smaller 𝜀 will eliminate TDS and obtain a more complex chaotic signal.
3.1. Influence of internal parameters of laser In this section, we mainly study the influence of laser internal parameters including the linewidth enhancement factor 𝛼 and the gain saturation coefficient 𝜀 on the complexity of chaotic signals. In a curve obtained by autocorrelation of a chaos time series (i.e., autocorrelation function), obvious side lobes appear at the feedback delay, and the value of its height is usually used to measure the TDS. In this paper, the TDS value refers to the maximum value in the time window [𝜏 − 0.05𝜏, 𝜏 + 0.05𝜏]. The autocorrelation function (ACF) can be obtained by [31]: ⟨[ ][ ]⟩ 𝑃 (𝑡 − 𝜏) − ⟨𝑃 (𝑡 − 𝜏)⟩ 𝑃 (𝑡) − ⟨𝑃 (𝑡)⟩ 𝐴 (𝜏) = √⟨ ]2 ⟩ ⟨[ ]2 ⟩ [ 𝑃 (𝑡 − 𝜏) − ⟨𝑃 (𝑡 − 𝜏)⟩ 𝑃 (𝑡) − ⟨𝑃 (𝑡)⟩
(7)
where P(t) = | E(t)|2 , which is the intensity of the laser output light, and ⟨∙⟩ represents averaging time. Fig. 3 shows the TDS and PE as a function of linewidth enhancement factor 𝛼 for the mirror feedback (blue circles) and the CFBG feedback (red triangles). The stars in Fig. 3(a) are three times the standard deviation of the ACF curve noise floor. In Fig. 3(a), for the mirror feedback, as 𝛼 increase, the TDS can be suppressed but not completely eliminated and much larger than the value under CFBG feedback. For CFBG feedback, when 𝛼 ≤ 3, TDS cannot be eliminated due to the influence of spectral linewidth. As 𝛼 increase, the TDS gradually decreases and is hidden in background noise to achieve the purpose of eliminating the delay signature. In Fig. 3(b), for CFBG feedback, the values of PE are maintained at a high value. For mirror feedback, the value of PE increases with the increase of 𝛼, but is always smaller than the value under CFBG feedback. Therefore, within the range of 𝛼 ≥
3.2. Influence of external parameters of laser This section mainly analyzes the influence of external parameters of lasers including feedback strength 𝜅𝑓 and normalized bias current I /𝐼th on the complexity of the CFBG feedback and mirror feedback systems through TDS and PE. Fig. 5 plots the TDS and the PE as a function of feedback strength 𝜅𝑓 for the mirror feedback (blue circles) and the CFBG feedback (red triangles). Fig. 5(a) displays that at 𝜅𝑓 < 0.05, the TDS decreases with increasing 𝜅𝑓 , reaching a minimum when 𝜅𝑓 is 0.05. When 𝜅𝑓 continues to increase, the TDS under mirror feedback shows a linear increase trend, which is consistent with the numerical results in [23]. 3
M. Chao, D. Wang, L. Wang et al.
Optics Communications 456 (2020) 124702
Fig. 7. Maps of (a) TDS and (b) PE in the parameter space of 𝜅𝑓 and I /𝐼th for the CFBG feedback. Fig. 5. TDS (a) and PE (b) as a function of 𝜅𝑓 for the mirror feedback and the CFBG feedback. 𝜏 = 3 ns, 𝐼 = 1.5𝐼th .
Fig. 8. TDS and PE as a function of (a) frequency detuning 𝛥𝜈 and (b) dispersion for the CFBG feedback. 𝜏 = 3 ns, 𝐼 = 1.5𝐼th , 𝜅𝑓 = 0.3. Fig. 6. TDS (a) and PE (b) as a function of I /𝐼th for the mirror feedback and the CFBG feedback. 𝜏 = 3 ns, 𝜅𝑓 = 0.3.
However, the value under CFBG feedback is still hidden in the background noise and will not increase again. Correspondingly, in the PE curves of Fig. 5(b), as the 𝜅𝑓 increases, the PE gradually reaches the maximum value. Further increasing 𝜅𝑓 , the PE value under mirror feedback gradually decreases, while the PE value under CFBG feedback is stable at 0.994 and will not decrease. Then the PE values of the CFBG feedback are higher than those of the mirror feedback. It can also be seen that the parameter region for obtaining high-complexity chaotic signals under CFBG feedback is wider. Fig. 6 plots the TDS and the PE as a function of normalized bias current I /𝐼th for the mirror feedback (blue circles) and the CFBG feedback (red triangles). In Fig. 6(a), the TDS values under mirror feedback decrease with increasing I /𝐼th , whereas for CFBG feedback, the TDS values are always stable at small value and are hidden in the background noise. In Fig. 6(b), as I /𝐼th increases, the PE values under the mirror feedback increase significantly, the values under CFBG feedback increase slightly, but its values are much larger than the values under mirror feedback. It can be seen that when I ≥ 1.5𝐼th , the value of H [P] reaches 0.994. The results of Fig. 6 show that CFBG feedback can hide the TDS and obtain more complex chaotic signals over a wider range of bias currents than mirror feedback. To further analyze the influence of external parameters of laser on the complexity of the laser chaos for CFBG feedback, maps of the TDS and PE in the parameter space of I /𝐼th and 𝜅𝑓 are displayed in Fig. 7. From the two-dimensional map of Fig. 7(a), it can be seen that lower TDS appears in the region of larger bias current and stronger feedback strength. Even at large bias currents, when the feedback strength is weak, the TDS is larger. Meanwhile, the complexity in Fig. 7(b) has the same trend as the TDS in Fig. 7(a). At higher bias currents and feedback strengths, it is easier to obtain chaotic signals with higher complexity. Furthermore, as the I /𝐼th and 𝜅𝑓 increase, the high complexity regions are much wider.
that the PE value under CFBG feedback is significantly higher than that of mirror feedback in the range of the appropriate internal and external parameters of the laser. This section discusses the unique grating parameters in the CFBG feedback system, so the mirror feedback semiconductor laser system is not considered in this section. This section mainly analyzes the effects of frequency detuning and dispersion in the CFBG feedback on the TDS and PE values. Fig. 8 shows the PE (blue circles) and TDS (red star) values as a function of frequency detuning between laser and grating 𝛥𝜈 (a) and dispersion (b) for the CFBG feedback. The difference in center frequency, between the reflection spectra of chirped grating and the spectra of laser, is defined as the frequency detuning 𝛥𝜈. In Fig. 8(a), when 𝛥𝜈 ranges from −15 GHz to 10 GHz, the PE values are all at a higher value (about 0.994). Correspondingly, the TDS curve is at a lower value within this range (even ACF background noise level). When 𝛥𝜈 ⩽ −15 GHz or 𝛥𝜈 ⩾ 10 GHz, the frequency detuning range at this time is too large, which weakens the dispersive feedback caused by the chirped grating, the complexity of the chaotic signal decreases, and the value of the TDS increases significantly, as shown in Fig. 8(a). Therefore, when the frequency detuning satisfies the range of −15 GHz ⩽ 𝛥𝜈 ⩽ 10 GHz, the CFBG feedback semiconductor laser system can hide the TDS and obtain a highly complex chaotic output. The results of Fig. 8(a) are obtained when the chirped grating has a dispersion of 2000 ps/nm. In order to further study the effect of dispersion on the complexity, the TDS and PE curves under different dispersion were made with zero frequency detuning, as show in Fig. 8(b). As the dispersion value increases, the dispersive feedback caused by the chirped grating increases. When the dispersion exceeds 2000 ps/nm, the TDS is hidden in the background noise and never rises again, and the PE value is stable at 0.994. Meantime the CFBG feedback semiconductor laser system will obtain chaotic output with high complexity.
3.3. Influence of grating parameters
4. Conclusion
The above results are based on the comparative analysis of mirror feedback and CFBG feedback semiconductor laser system. Results show
The complexity of the CFBG feedback semiconductor laser system is numerically investigated by using the PE and the TDS. The trend 4
M. Chao, D. Wang, L. Wang et al.
Optics Communications 456 (2020) 124702
of TDS always has the opposite relationship with PE. And the results of CFBG feedback are compared with the mirror feedback. Different from mirror feedback, CFBG feedback can eliminate TDS and obtain chaotic signals with high complexity. The effects of laser parameters and grating parameters are studied in detail. The results show that due to the influence of spectral linewidth, the CFBG feedback can eliminate the TDS and obtain high-complexity chaotic signal with PE value up to 0.994 under the conditions of 𝛼 ≥ 3 and 𝜀 ≤ 4 × 10−4 μm3 . Under the strong feedback strength and large bias current, it is easier to enhance the complexity of chaotic signals. Compared with the mirror feedback, the parameter region of the CFBG feedback that can obtain high-complexity signals is broadened. When the chirped grating dispersion value exceeds 2000 ps/nm, chaotic signals with high complexity can be obtained at a small frequency detuning of −15 GHz ≤ 𝛥𝜈 ≤ 10 GHz. The TDS of the chaotic signal source is eliminated and the complexity is enhanced, which greatly improves the security of the chaotic secure communication system.
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