Generation of polarization-resolved wideband unpredictability-enhanced chaotic signals based on vertical-cavity surface-emitting lasers subject to chaotic optical injection

Jian-Jun Chen; Zheng-Mao Wu; Xi Tang; Tao Deng; Li Fan; Zhu-Qiang Zhong; Guang-Qiong Xia

doi:10.1364/OE.23.007173

1. Introduction

In the past half century, chaos has received considerable attention in both basic science and applied technology fields, and its application has shown the tremendous vitality in many aspects of human life [1–5]. Among many species of chaos, optical chaos as a promising chaotic source has been applied extensively to chaos-based communications [6–9], chaotic lidar [10], and fast physical random bit generation [11–13] etc. External optical feedback semiconductor laser (SL) is usually employed as one kind of suitable optical chaos sources for above applications due to its simple configuration and feasible controllability [14–17]. Previous related investigations show that the chaotic bandwidth of generated optical chaos signal by an external optical feedback SL is relatively low and is typically several gigahertz (GHz) related to the relaxation oscillation frequency of SLs. It is clearly desirable to enhance the chaotic bandwidth, which is helpful to enlarge the data transmission capacity of chaotic communication systems [18], improve the spatial resolution of chaotic lidar [10], and increase the bit rates of chaos-based physical random number [19]. Accordingly, some schemes for bandwidth enhancement of chaos have been demonstrated through introducing an additional optical injection to SLs [20–24].

Besides the chaotic bandwidth, another crucial issue in a chaos-based secure system is the security of message for data transmissions. It has been demonstrated that an eavesdropper can reconstruct easily the chaotic attractor and make predictability of the message if a low dimension chaotic signal is taken as chaotic carrier [25]. Generally, the security of message in a chaos-based secure system mainly relies on unpredictability degree (UD) of chaotic signals, which is the basis for evaluating the secure level of chaotic systems [26–33]. Some efforts have been devoted to enhance the UD of chaotic carriers in SLs-based chaotic systems. Xiang et al. have demonstrated experimentally and numerically that the UD of chaotic signals in a slave SL can be enhanced significantly by dual-path injection from a single master laser [34]. Furthermore, they also investigated the UD enhancement of chaotic signals in SLs with dual-chaotic optical injections [35], and show that chaotic optical injection technique can effectively promote the UD of chaotic signals in a chaos-based secure system.

At present, related investigations on enhancing the chaotic bandwidth and UD of chaotic signals are mainly focused on conventional edge-emitting SL-based chaotic systems. As one kind of developing SLs, vertical-cavity surface-emitting lasers (VCSELs) are power efficient devices widely used in local and access communication networks, and recently regarded as promising candidates of chaotic sources owing to some unique advantages [36–39]. Moreover, their polarization-resolved properties between X polarization component (X-PC) and Y polarization component (Y-PC) have also attracted considerable research attention in some polarization-sensitive applications [40–43]. Recently, Hong et al. have investigated experimentally the chaotic bandwith and the time delay signature (inversely proportional to the UD) for four schemes of chaos generation in VCSELs [44]. The results show that the time delay signature suppression (corresponding to the enhancement of UD) and the enhanced bandwidth of the chaos can be obtained simultaneously in two schemes–a chaotic beam unidirectionally injected into a CW VCSEL and a CW VCSEL mutually coupled to a chaotic VCSEL.

In this work, a scheme based on a chaotic beam injected into a two-mode coexisting periodic VCSEL, is proposed for generating polarization-resolved wideband unpredictability-enhanced chaotic signals. In this scheme, a slave VCSEL (S-VCSEL), operated at two-mode coexisting periodic dynamics, is subject to chaotic optical injection generated by a master VCSEL (M-VCSEL) under external optical feedback. The UD and chaotic bandwidth of two polarization-components (PCs) of chaotic signals generated by M-VCSEL under optical feedback and S-VCSEL under chaotic optical injection from M-VCSEL have been studied numerically, and the influences of some related system parameters have also been analyzed.

2. System model and theory

Figure 1 is the schematic diagram of proposed VCSEL-based chaotic system for generating polarization-resolved wideband unpredictability-enhanced chaotic signals. A chaotic signal, generated by a master VCSEL (M-VCSEL) under external optical feedback from a mirror M, is unidirectionally injected into a slave VCSEL (S-VCSEL) after passing through an optical isolator (ISO) and a variable attenuator (VA). In this system, two variable attenuators VA1 and VA2 are used to adjust the feedback strength and injection strength, respectively, and ISO is used to guarantee light unidirectional transmission.

Fig. 1 Schematic diagram of a VCSELs-based chaotic system for generating polarization-resolved wideband unpredictability-enhanced chaotic signals. M-VCSEL: master vertical-cavity surface-emitting laser; S-VCSEL: slave vertical-cavity surface-emitting laser; M: mirror; VA: variable attenuator; ISO: optical isolator.

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Based on the spin-flip model (SFM) [36], which experimentally confirmed by Virte et al. recently [45], the rate equations for M-VCSEL under optical feedback and S-VCSEL under chaotic optical injection can be expressed as:

\begin{array}{l} \frac{d E_{x, y}^{M}}{d t} = k^{M} (1 + i α^{M}) [(N^{M} - 1) E_{x, y}^{M} \pm i n^{M} E_{y, x}^{M}] \mp (γ_{a}^{M} + i γ_{p}^{M}) E_{x, y}^{M} \\ + k_{f} E_{x, y}^{M} (t - τ_{f}) e^{- i 2 π v^{_{M}} τ_{f}} + \sqrt{β_{s p}} ξ_{x, y}^{M} \end{array}

\begin{array}{l} \frac{d E_{x, y}^{S}}{d t} = k^{S} (1 + i α^{S}) [(N^{S} - 1) E_{x, y}^{S} \pm i n^{S} E_{y, x}^{S}] \mp (γ_{a}^{S} + i γ_{p}^{S}) E_{x, y}^{S} \\ + η E_{x, y}^{M} (t - τ_{η}) e^{- i 2 π v^{M} τ_{η} + i 2 π Δ v t} + \sqrt{β_{s p}} ξ_{x, y}^{S} \end{array}

\frac{d N^{M, S}}{d t} = - γ_{e}^{M, S} [N^{M, S} (1 + | E_{x}^{M, S} |^{2} + | E_{y}^{M, S} |^{2}) - u^{M, S} + i n^{M, S} (E_{y}^{M, S} E_{x}^{M, S *} - E_{x}^{M, S} E_{y}^{M, S *})]

\frac{d n^{M, S}}{d t} = - γ_{s}^{M, S} n^{M, S} - γ_{e}^{M, S} [n^{M, S} (| E_{x}^{M, S} |^{2} + | E_{y}^{M, S} |^{2}) + i N^{M, S} (E_{y}^{M, S} E_{x}^{M, S *} - E_{x}^{M, S} E_{y}^{M, S *})]

where the superscripts M and S correspond to M-VCSEL and S-VCSEL, respectively, and subscripts x and y stand for X polarization component (X-PC) and Y polarization component (Y-PC), respectively. E is the slowly varied complex amplitude of the field, N is the total carrier inversions between the conduction and valence bands, n accounts for the difference between carrier inversions for the spin-up and the spin-down radiation channels, k is the field decay rate, α is the line-width enhancement factor, γ_e is the decay rate of N, γ_s is the spin-flip rate, γ_a and γ_p are the linear anisotropies representing dichroism and birefringence, respectively. k_f is the feedback strength supplied by the external cavity, and τ_f is the corresponding feedback delay time. η characterizes the injection strength from M-VCSEL to S-VCSEL, and τ_η is the injection delay time. u is the normalized bias current (u takes 1 at threshold), ν^M (ν^S) is the central frequency of M-VCSEL (S-VCSEL), Δν ( = ν^M – ν^S) is the frequency detuning between M-VCSEL and S-VCSEL, β_sp is the spontaneous emission factor, and ξ is the Gaussian white noise terms of zero mean value and unitary variance.

In this paper, we define the chaotic bandwidth as the frequency span from DC to the frequency within where 80% of the total power is contained [20–23], and the bandwidths of polarization-resolved chaos outputs from M-VCSEL and S-VCSEL can be measured by the definition.

To quantitatively characterize the UD of chaotic signals, several computational techniques such as Lyapunov exponents (LE), Kolmogorov-Sinai (KS) entropy and Correlation Dimensions (CD), have been proposed and implemented [46–48]. However, above algorithms are mostly time-consuming and have relatively low robustness to noise generated by SLs. Due to unique advantages of simplicity, fast calculation and robustness to the noise, permutation entropy (PE) proposed by Bandt and Pompe is an effective measure quantifier to evaluate quantitatively the UD of chaotic signals [26]. PE could be defined as follows: the time series {S(m), m = 1,2,…,N} are firstly reconstructed into a set of D-dimensional vectors after choosing an appropriate embedding dimension D and embedding delay time τ_e. Next, one can study all D! permutation π of order D. For each π, the relative frequency (# means number) is determined as:

p (π) = \frac{# {m | m \leq N - D, (S_{m + 1}, \dots, S_{m + D}) h a s t y p e π}}{N - D + 1}

and the PE is given by

h [p] = - \sum p (π) \log p (π)

Then, the normalized PE is further defined as follows:

H [p] = \frac{h [p]}{h_{\max}} = \frac{- \sum p (π) \log p (π)}{\log (D!)}

The normalized PE has values 0 ≤H≤ 1 with H = 0 corresponding to a completely predictable dynamics and H = 1 for a fully random stochastic dynamics. That is to say, the larger the value of H, the more irregular the time series is. More precisely, it has been suggested that the PE should be evaluated when the embedding delay τ_e coincides with the characteristic time delay τ_f of the system, to quantify the UD or randomness of chaotic signals [29,33]. In this paper, we set the embedding dimension D = 6 with τ_e = τ_f, and the length of the time series is taken as 2000 ns for the calculation of PE.

3. Results and discussion

Equations (1)-(4) can be solved numerically with the fourth-order Runge-Kutta method. During the calculations, the intrinsic parameters of solitary M-VCSEL and S-VCSEL are assumed to be identical and set as follows [39]: k = 300 ns⁻¹, α = 3, γ_e = 1 ns⁻¹, γ_s = 50 ns⁻¹, γ_a = 0.1 ns⁻¹, γ_p = 10 ns⁻¹, and β_sp = 10⁻⁶. The central frequency of the M-VCSEL ν^M is 3.52941 × 10¹⁴ Hz (corresponding to the central optical wavelength of M-VCSEL at 850 nm), and τ_η = 0 ns.

3.1 P-I curves and polarization dynamics of solitary VCSEL

Figure 2(a) shows the P-I curves for X-PC, Y-PC, and the total output from a solitary VCSEL, where the intensities are averaged over a time window of 2000 ns. For 1 <u< 1.55, the X-PC dominates meanwhile the Y-PC is strongly suppressed. Once u>1.55, the Y-PC begins to oscillate. With the further increase of u, the intensity of Y-PC increases quickly and the intensity difference between two PCs decreases gradually. Especially, for u = 2.7, the X-PC and Y-PC in the solitary VCSEL possess identical average intensity, which affords a possibility for dual-channel chaos synchronization and communication [8]. In the following discussions, we set u as 2.7. Under this circumstance, the relaxation oscillation frequency f_RO ( $= \sqrt{2 k γ_{e} (u - 1)} / 2 π$ ) of the solitary VCSEL is about 5.08 GHz. Since the dynamics of a VCSEL under injection or feedback is strongly influenced by the solitary polarization of the emitted light [44], it is necessary to analyze the dynamics of the solitary VCSEL. Referring to [36] and [45], the polarization-solved time series and optical spectra of the solitary VCSEL under u = 2.7 have been simulated and displayed in Figs. 2(b) and 2(c), respectively. From Figs. 2(b1) and 2(b2), one can see that the two PCs simultaneously exhibit periodic dynamics. Moreover, the optical spectra (shown as Fig. 2(c)) show that the central frequencies of X-PC and Y-PC are not identical. Therefore, it can be deduced that the dynamics of the solitary VCSEL under u = 2.7 is a two-mode coexistence.

Fig. 2 P-I curves (a), time series (b1, b2), and corresponding optical spectra (c1, c2) of polarization-resolved outputs from a solitary VCSEL under u = 2.7

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3.2 Bandwidths of polarization-resolved chaotic signals

Figure 3 displays the time series (the first row) of polarization-resolved outputs and their corresponding power spectra (the second row) under k_f = 15 ns⁻¹, τ_f = 3 ns, η = 50 ns⁻¹ and Δv = 20 GHz. As shown in these diagrams, both X-PC and Y-PC in two VCSELs operate at chaotic states. The chaotic bandwidths of X-PC and Y-PC in M-VCSEL are about 11.48 GHz and 10.97 GHz, respectively. After injected into the S-VCSEL, the bandwidths of the X-PC and Y-PC outputs from the S-VCSEL can be increased to 21.05 GHz and 21.50 GHz, respectively. As a result, through adopting chaotic optical injection, the bandwidths of the polarization-resolved chaos outputs can be enhanced effectively.

Fig. 3 Time series (the first row) of polarization-resolved chaos outputs and corresponding power spectra (the second row) with k_f = 15 ns⁻¹, τ_f = 3 ns, η = 50 ns⁻¹ and Δv = 20 GHz.

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To further explore the influence of the injection parameter on the chaotic bandwidth of S-VCSEL, Fig. 4 shows the bandwidths of the polarization-resolved chaotic outputs from the S-VCSEL as a function of the injection strength η under different frequency detuning Δv. The simulated results show that the chaotic optical injection will drive the S-VCSEL enter into chaotic state once the injection strength η exceeds 1 ns⁻¹. For Δv = 0 GHz, 10 GHz and 20 GHz, with the increase of η from 1 ns⁻¹ to 300 ns⁻¹, the bandwidth of the X-PC output from the S-VCSEL increases firstly, after reaches a maximal value and then decreases. When η is large enough, the X-PC of the S-VCSEL will be locked by the injected X-PC from the M-VCSEL, and then the bandwidth of the X-PC from the S-VCSEL tends to a fixed value, which is equal to the bandwidth of the injection X-PC chaotic signal. The similar variation tendency is observed for Y-PC. For Δv = −10 GHz and −20 GHz, the bandwidths of two PCs can be enhanced within a narrow injection parameter range and then rapidly decrease due to the injection locking effect. Further increasing the injection strength, the bandwidths of two PCs will increase slowly until the two PCs are locked once again.

Fig. 4 Bandwidths of X-PC and Y-PC of S-VCSEL as a function of the injection strength η under k_f = 15 ns⁻¹ and τ_f = 3 ns for different frequency detuning.

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The above results demonstrate that the bandwidths of the chaotic signals are seriously influenced not only by the injection strength but also by the frequency detuning. In Fig. 5, we give the evolution maps of bandwidths of the polarization-resolved outputs from the S-VCSEL in the parameter space of η and Δv under τ_f = 3 ns and k_f = 15 ns⁻¹. Here, some representative contour lines are marked for the case that the bandwidth of the chaotic signal is 25 GHz, and thus the parameter space is divided into three parts named as regions A, B and C, respectively. In region A, the bandwidth of the chaotic signal is less than 25 GHz. However, for regions B and C, the values of bandwidth are higher than 25 GHz, which is highly desired for generating wide bandwidth chaos. Due to the asymmetrical injection locking effect, the distributions of the bandwidth of the chaotic signal in the parameter space behave asymmetry. Obviously, for a determined driving chaotic signal, the bandwidths of two PCs outputs from the S-VCSEL can be adjusted through varying the injection strength or the frequency detuning.

Fig. 5 Evolution maps of bandwidths of the polarization-resolved outputs from S-VCSEL in the parameter space of injection strength η and frequency detuning Δv under τ_f = 3 ns and k_f = 15 ns⁻¹.

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3.3 Unpredictability degree (UD) of polarization-resolved chaotic signals

Firstly, we present the comparison of PE between the polarization-resolved outputs from M-VCSEL and S-VCSEL as functions of feedback strength k_f and feedback delay time τ_f in Fig. 6. As shown in Figs. 6(a) and 6(b) for a fixed τ_f = 3.0 ns, with the increase of k_f, PE for two PCs outputs from the M-VCSEL increase quickly and then show a slowly decreased trend, which agrees qualitatively with the results presented in [29]. However, the variation trends of PE for the polarization-resolved outputs from the S-VCSEL are quite different from those of M-VCSEL, and the values of PE for X-PC and Y-PC outputs from the S-VCSEL are always larger than that from the M-VCSEL and almost maintain a constant in the whole range of k_f. Additionally, as shown in Figs. 6(c) and 6(d), for a fixed k_f ( = 8 ns⁻¹), the influences of τ_f on the PE of two PCs outputs from the M-VCSEL and S-VCSEL are relatively weak, and the values of PE of two PCs outputs from the S-VCSEL are generally larger than that from the M-VCSEL for arbitrary τ_f. It should be pointed out that the weak influences of τ_f on the PE of two PCs outputs from the M-VCSEL and S-VCSEL are due to the variation range of τ_f, which is far from the the relaxation oscillation period τ_RO (~0.197 ns). For τ_f is near to τ_RO, the influences of τ_f will be much more obvious.

Fig. 6 PE of the polarization-resolved outputs from M-VCSEL and S-VCSEL as functions of feedback strength k_f (the first row) and feedback delay time τ_f (the second row) with Δν = 0 GHz and η = 15 ns⁻¹.

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Secondly, we simulate the evolution maps of PE of the polarization-resolved outputs from M-VCSEL and S-VCSEL in the parameter space of delay time and feedback strength in Fig. 7 under Δν = 0 GHz and η = 15 ns⁻¹. It can be obviously seen from Figs. 7(a) and 7(b) that, with the increase of the feedback strength, the transitions of PE from low to high and back to low complexity occur for the polarization-resolved outputs of the M-VCSEL, and it indicates that the high complexity chaotic signals can only be generated in some regions with relative small feedback level. However, driven by the chaotic signals with different PEs, S-VCSEL can always generate polarization-resolved outputs with high PE (shown as Figs. 7(c) and 7(d)). Therefore, the S-VCSEL under chaotic optical injection is suitable for taking as a UD-enhanced chaotic source. In addition, from this diagram, it can also be seen that, for a fixed value of the feedback strength, the variation of delay time has no obvious effect on the values of PE, which is in agreement with that presented in Figs. 6(c) and 6(d).

Fig. 7 Evolution maps of PE of the polarization-resolved outputs from M-VCSEL (the first row) and S-VCSEL (the second row) under Δν = 0 GHz and η = 15 ns⁻¹ in the parameter space of delay time τ and feedback strength k_f, where different colors represent different values of PE.

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Finally, we discuss the influences of the injection parameters on the PE of two PCs output from the S-VCSEL. Figure 8 gives the evolution maps of PE of the polarization-resolved outputs from S-VCSEL in the parameter space of injection strength η and frequency detuning ∆v under τ_f = 3.0 ns for different k_f, where different colors characterize different values of PE. The region surrounded by dash lines shows that the PE values are more than 0.98 and is named as region A, and the upper and lower of region A are named as region B and region C, respectively. Region A exhibits a clear shape as the Letter ‘V’ for both X-PC and Y-PC, but shows an asymmetrical feature to the frequency detuning. For k_f = 15 ns⁻¹ (the first row), the driving polarization-resolved chaotic signals outputs from M-VCSEL have relatively high PE (≈0.95 as shown in Figs. 6(a) and 6(b)). Under this case, for two PCs outputs from S-VCSEL, the values of PE in region B are more than 0.96. As a result, the region with high PE (including region A and region B) covers most of the parameter space. Accordingly, region C labeling the low UD is very small and located at small η meanwhile −50 GHz <∆v < −20 GHz or 25 GHz <∆v < 50 GHz. For k_f = 30 ns⁻¹, as shown in the second row, the relatively low PE (≈0.85 as shown in Figs. 6(a) and 6(b)) of two driving PCs leads to low UD within region B, in which PE of two PCs are about 0.88. However, regions A with PE more than 0.98 for two PCs are basically maintained at the same level as that for k_f = 15 ns⁻¹. For k_f = 45 ns⁻¹, lower PE (≈0.80) of the driving polarization-resolved chaotic signals results in region B with smaller PE, which is about 0.82. Through carefully observing Fig. 8, it can be seen that, with the increase of k_f, the area of region A is decreased slowly for both of two PCs, and the whole scope of “V” slightly shifts towards the positive detuning region. Combining with the mapped result presented in Fig. 5, an optimized parameter range of the injection strength η and frequency detuning ∆v for generating polarization-resolved wide bandwidth and UD-enhanced chaotic signals can be determined.

Fig. 8 Evolution maps of PE of the polarization-resolved outputs from S-VCSEL in the parameter space of injection strength η and frequency detuning Δν with τ_f = 3 ns for k_f = 15 ns⁻¹ (the first row), k_f = 30 ns⁻¹ (the second row) and k_f = 45 ns⁻¹ (the third row).

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4. Conclusions

In summary, we have proposed and numerically simulated a chaotic system for generating polarization-resolved wideband unpredictability-enhanced chaotic signals based on a slave VCSEL (S-VCSEL) subject to chaotic optical injection from another master VCSEL (M-VCSEL) with external optical feedback. Based on the SFM model, the time series of X polarization component (X-PC) and Y polarization component (Y-PC) have been calculated, and then the chaotic bandwidth and UD of the polarization-resolved chaotic signals are evaluated. The influences of feedback parameters and injection parameters on the chaotic bandwidth and UD evolution are specified in detail. The results show that under suitable system parameters, this proposed chaotic system can generate high quality polarization-resolved chaos signals with high UD (more than 0.98 of PE value) and wide bandwidth (above 50GHz). Additionally, it should be pointed out that since γ_s is a key parameter to affect the polarization dynamics of VCSELs, the above presented results are obtained under the case for γ_s = 50 ns⁻¹, and further simulation results show that similar trends can also be observed if γ_s is not more than 80 ns⁻¹.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61178011, Grant 61275116, Grant 61475127 and Grant 11204248, the Open Fund of the State Key Lab of Millimeter Waves of China under Grant K201418, the Fundamental Research Funds for the Central Universities under Grant XDJK2014C079, Grant XDJK2014C120 and Grant XDJK2014C168, and the Graduate Research and Innovation Project of Chongqing Municipality under Grant CYB14054.

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Generation of polarization-resolved wideband unpredictability-enhanced chaotic signals based on vertical-cavity surface-emitting lasers subject to chaotic optical injection

Abstract

1. Introduction

2. System model and theory

3. Results and discussion

3.1 P-I curves and polarization dynamics of solitary VCSEL

3.2 Bandwidths of polarization-resolved chaotic signals

3.3 Unpredictability degree (UD) of polarization-resolved chaotic signals

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Equations (7)

Optics Express