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SECTION I

Optical chaos has attracted much attention for its potential applications in optical chaotic encryption communication, fast physical random-bit generation, chaotic radar, etc. [1], [2], [3], [4], [5]. Among various methods for generating optical chaos, the scheme based on a semiconductor laser (SL) under external cavity feedback (ECF) is usually adopted [6]. Generally, in an ECF-SL system, time-delay (TD) signature of the chaos output is obvious due to optical roundtrip in the external cavity. Such chaotic signal with obvious TD signature is undesirable in some applications. For chaotic secure communication, the TD signature inevitably provides an available clue to the encryption attackers. Recently, some time-series analysis techniques for delayed systems have been developed [7], [8], [9], [10], [11], and the reconstruction of the chaotic system may be realized based on the TD signature identification. Therefore, the security of the chaotic encryption may be partly compromised. For fast random-bit generation, the TD signature of the chaotic signal will induce recurrence feature and then affect the statistical performance of the generated random-bit sequences. Hence, it is necessary to develop some strategies to suppress the TD signature of chaotic signal.

So far, much effort for suppressing TD signature of SL-based chaos output has been made. Lee *et al.* investigated the possibility to suppress the TD signature by introducing a second cavity into an ECF system to form a double ECF (DECF) SL system [12]. Rontani *et al.* investigated the TD signature elimination in a single ECF (SECF) SL system [8], [13]. Shahverdiev *et al.* demonstrated the TD signature suppression in a laser with modulated feedback delay time [14]. More recently, through adopting a FBG to provide distributed feedback, Li *et al.* proposed a SL-based novel scheme to generate a chaotic signal without obvious TD signature [15]. In addition, we have also experimentally and numerically demonstrated the TD signature evolution in the SECF-SL system, DECF-SL system, and incoherent optical feedback SL system [16], [17], [18]. Moreover, we have proposed a strategy to obtain chaotic signal without obvious TD signature based on mutually coupled SLs (MC-SLs) [19], [20]. Compared with an ECF-SL system, the MC-SL system possesses some unique advantages. First, for one SL in the MC-SL system, the other SL can be taken as an active nonlinear reflector to provide nonlinear optical feedback. Such an active nonlinear feedback, which is different from a linear feedback provided by a mirror in ECF-SL system, will weaken the recurrence of the chaotic signal to a certain extent. As a result, TD signature of the chaotic signal generated by the MC-SL system can be effectively suppressed. Second, higher dimension chaotic signal can be obtained in the MC system due to higher degrees of freedom in the MC-SL system than that in an ECF-SL system [21], [22]. Finally, the MC structure can simultaneously produced two sets of chaotic signals [19].

At present, the investigations on the TD signature suppression are mainly focused on edge-emitting SL-based chaotic system. Compared with edge-emitting SLs, vertical-cavity surface-emitting lasers (VCSELs) own some unique advantages [23], [24], [25]. Particularly, due to weak material and cavity anisotropies, the output of a VCSEL generally includes two orthogonal linear polarization modes ($x$ and $y$ modes), which can be used as two chaotic carriers after being separated by a polarization beam splitter (PBS) [26]. In 2011, Xiang *et al.* numerically investigated the TD signature concealment of chaotic VCSELs by single variable-polarization optical feedback [27]. Recently, we have investigated the TD signature suppression of chaotic output in a VCSEL with double variable-polarization optical feedback [28].

In this paper, after combining our previous experience with reported related investigations by other scholars in TD concealment of edge-emitting SL/VCSEL-based chaotic systems, via by self-correlation function (SF) and permutation entropy (PE) analytical methods, the TD signature performances of polarization-resolved chaos outputs from MC-VCSELs have been numerically investigated, and the optimal system parameters have been determined for generating chaotic signal without obvious TD signature.

SECTION II

The system schematic is illustrated in Fig. 1. Two VCSELs (named as VCSEL1 and VCSEL2) are MC, and the coupling strength can be controlled by a variable attenuator (VA). For VCSEL1, its output is split by a beam splitter (BS). The transmission part of light is injected into VCSEL2. The reflection part of light is split further into two polarization-resolved outputs (named as $x$ and $y$ polarization modes) by a PBS, and then, $x$ and $y$ modes can be recorded, respectively. Similarly, for VCSEL2, its output can also be polarization resolved by the PBS and can be then recorded, respectively.

Based on the spin-flip mode (SFM) [29], [30], the rate equations for MC-VCSELs can be described by: TeX Source $$\eqalignno{{dE_{1, 2}^{x} \over dt} =&\, k (1 + i\alpha) \left(N_{1, 2}E_{1, 2}^{x} - E_{1, 2}^{x} + in_{1, 2}E_{1, 2}^{y} \right) - (\gamma_{a} + i\gamma_{p}) E_{1, 2}^{x} \cr& + \eta_{2, 1}E_{2, 1}^{x} (t - \tau) e^{-i\omega_{0}\tau} \mp i\Delta \omega E_{1, 2}^{x} + F_{1, 2}^{x}&\hbox{(1)}\cr {dE_{1, 2}^{y} \over dt} =&\, k (1 + i\alpha) \left(N_{1, 2}E_{1, 2}^{y} - E_{1, 2}^{y} - in_{1, 2}E_{1, 2}^{x} \right) + (\gamma_{a} + i\gamma_{p}) E_{1, 2}^{y}\cr& + \eta_{2, 1}E_{2, 1}^{y} (t - \tau) e^{- i\omega_{0}\tau} \mp i\Delta \omega E_{1, 2}^{y} + F_{1, 2}^{y}&\hbox{(2)}\cr {dN_{1, 2} \over dt} =&\, -\gamma_{e}N_{1, 2}\left(1 + \left\vert E_{1, 2}^{x} \right\vert^{2} + \left\vert E_{1, 2}^{y} \right \vert^{2}\right) + \gamma_{e}\mu - i\gamma_{e}n_{1, 2}\left(E_{1, 2}^{y}E_{1, 2}^{x\ast} - E_{1, 2}^{x}E_{1, 2}^{y \ast}\right)&\hbox{(3)}\cr {dn_{1, 2} \over dt} =&\, -\gamma_{s}n_{1, 2} - \gamma_{e}n_{1, 2}\left(\left \vert E_{1, 2}^{x} \right\vert^{2} + \left\vert E_{1, 2}^{y} \right\vert^{2}\right) - i\gamma_{e}N_{1, 2}\left(E_{1, 2}^{y}E_{1, 2}^{x\ast} - E_{1, 2}^{x}E_{1, 2}^{y\ast}\right)&\hbox{(4)}}$$ where the superscripts $x$ and $y$ represent $x$ and $y$ polarization modes, respectively; subscripts 1 and 2 stand for VCSEL1 and VCSEL2, respectively. $E$ is the slowly varying complex amplitude of the field, $N$ is the total carrier inversion between the conduction and valence bands, $n$ accounts for the difference between carrier inversions for the spin-up and spin-down radiation channels, $k$ is the decay rate of field, $\alpha$ is the linewidth enhancement factor, $\gamma_{e}$ is the decay rate of total carrier population, $\gamma_{s}$ is the spin-flip rate, and $\gamma_{a}$ and $\gamma_{p}$ are the linear anisotropies representing dichroism and birefringence, respectively. $\eta$ characterizes the coupling strength between two VCSELs and can be described by $\eta = \sqrt{p}/\tau_{in}$, where $p$ is the percentage of coupled power and $\tau_{in}$ is the roundtrip time of light in laser cavity. $\mu$ is the normalized injection current, $\tau$ is the coupling delay time, namely, the TD signature. $\omega_{0} = (\omega_{1} + \omega_{2})/2$ is the average angular frequency in symmetrical reference system, where $\omega_{1}$ and $\omega_{2}$ are the angular frequencies of VCSEL1 and VCSEL2, respectively. $\Delta \omega = (\omega_{2} - \omega_{1})/2$ is the angular frequency detuning. $F$ is the Langevin noise.

Several methods can be used to evaluate quantitatively the TD signature, such as SF [8], [12], mutual information (MI) [20], and PE [31], [32]. Here, we employ SF and PE to evaluate TD signature. SF can be defined as [8], [12]: TeX Source $$C (\Delta t) = {\left\langle \left(I (t + \Delta t) - \left\langle I (t) \right\rangle \right)\left(I (t) - \left\langle I (t) \right\rangle \right) \right\rangle \over \left\langle I (t) - \left\langle I (t) \right\rangle \right\rangle^{2}}\eqno{\hbox{(5)}}$$ where $I(t)$ represents chaotic output time series, $\langle \cdot \rangle$ denotes time average, and $\Delta t$ is the time shift. In this paper, we set $\Delta t \in (-15\ \hbox{ns}, 15\ \hbox{ns})$ with a step of 2 ps. During calculating the SF, the length of the time series is taken as 1500 ns. The TD signature can be retrieved from the peak location of SF curve.

The PE method, which is based on the information theory, owns some unique advantages such as simplicity, fast calculation, and robustness to noise [31]. PE could be defined as follows [31], [32]: the time series $\{I(m), m = 1, 2, \ldots, N\}$ are first reconstructed into a set of D-dimensional vectors after choosing an appropriate embedding dimension D and embedding delay time $\tau_{e}$. Then, we study all D! permutation $\pi$ of order D. For each $\pi$, the relative frequency (# means number) is determined as: TeX Source $$p (\pi) = {\# \left\{m\vert m \leq N - D, (I_{m + 1}, \ldots, I_{m + D})\ has\ types\ \pi \right\} \over N - D + 1}\eqno{\hbox{(6)}}$$ and then the PE is defined as: TeX Source $$H (D) = -\sum p (\pi)\, \log\, p (\pi).\eqno{\hbox{(7)}}$$

In this paper, we set ${\rm D} = 9$ after combining the unique features of VCSEL with the suggestions in [31] and [32], and the length of the time series is also taken as 1500 ns for the calculation of PE.

SECTION III

The rate equations (1)–(4) can be solved numerically by using fourth-order Runge–Kutta algorithm. The parameters are chosen as [33]: $k = 300\ \hbox{ns}^{-1}$, $\alpha = 3$, $\gamma_{e} = 1\ \hbox{ns}^{-1}$, $\gamma_{s} = 50\ \hbox{ns}^{-1}$, $\gamma_{a} = 0.1\ \hbox{ns}^{-1}$, $\gamma_{p} = 10\ \hbox{rad ns}^{-1}$, $\omega_{0} = 2.2176 \times 10^{15}\ \hbox{rad/s}$ (corresponding optical wavelength is 850 nm), and $\tau$ is fixed at 3 ns.

Fig. 2 shows the P–I curve for each polarization mode and the total output of a solitary VCSEL. During the numerical simulation, the normalized injection current $\mu$ is varied from 0 to 3.5. As shown in this diagram, for $1\ <\ \mu\ <\ 1.7$, $x$ mode (open circles) oscillates but $y$ mode (asterisks) is suppressed. When increasing $\mu$ to 1.7, $y$ mode begins to oscillate. With the further increase of $\mu$, the intensity of $y$ mode increases quickly; meanwhile, the intensity of $x$ mode increases slowly. Once $\mu$ is larger than 2.7, the intensity of $y$ mode will exceed that of $x$ mode, and $y$ mode becomes the dominant mode. From this diagram, one can observe that there exist kinks for both $x$ and $y$ modes under the case of $\mu \sim 2.2$. Under this circumstance, calculations show that there exists a conversion of dynamical state from periodic state to chaos. In the following, $\mu$ is set as 2.7. Under this case, the intensity of $x$ mode is similar with that of $y$ mode.

Next, we will investigate the TD signature characteristics of polarization-resolved chaos outputs in the MC-VCSELs system. Fig. 3 displays calculated time series (the first row), power spectra (the second row), SF curves (the third row), and PE curves (the fourth row) when $\eta$ is fixed at 50 $\hbox{ns}^{-1}$ and $\Delta f (= \Delta \omega/2\pi)$ is fixed at 0 GHz. As shown in the first row, both two modes ($x$ and $y$ modes) exhibit intricate oscillation with similar power level. It is difficult to identify the TD signature directly from the chaotic time series since they are quite complex. However, from the second row of this diagram, some peaks with equal frequency interval can be observed upon the power spectra, and the frequency interval is about 167 MHz, which is equal to the reciprocal of double coupling delay time. Therefore, the coupling delay time can be valued roughly from the power spectra of chaotic output. Furthermore, as shown by SF curves (see the third row), there exists an apparent characteristic peak at integer multiples of $\Delta t = 2\tau = 6\ \hbox{ns}$. These characteristic peaks also reveal the TD signature contained in the MC-VCSELs system. Additionally, as shown in the PE curves of chaotic time series (see the fourth row), there are some sharp valleys emerged at the integer multiples of $2\tau (= 6\ \hbox{ns})$. Based on the locations of the sharp valleys, the delay time can also be extracted. Meanwhile, the value of the sharp valley characterizes the recurrence of the chaotic signal. The deeper the sharp valley is, more obvious the TD signature will be.

Fig. 4 shows the polarization-resolved chaos outputs in MC-VCSELs system when the TD signature has been suppressed, where $\eta = 10\ \hbox{ns}^{-1}$ and $\Delta f$ is still fixed at 0 GHz. As shown in the first row of Fig. 4, the chaotic time series of four modes of two VCSELs are still intricate. However, the corresponding power spectra (as shown in the second row of Fig. 4) become much smoother, and there are no significant frequency peaks upon spectral background. Thus, TD signatures cannot be identified from these power spectra. Furthermore, as shown in the third row of Fig. 4, all SF curves exhibit $\delta$ function profile and apparent TD signature related SF peaks cannot be observed. Also, from the fourth row of Fig. 4, no apparent valleys emerge at the location of $\tau_{e} = 6\ \hbox{ns}$ in all PE curves, and the values of PE are very close to 1 except $\tau_{e} = 0\ \hbox{ns}$. Therefore, under the above given parameters, the TD signatures of polarization-resolved chaos outputs from MC-VCSELs can be suppressed effectively and simultaneously. Since we only change the coupling strength $\eta$ during obtaining Figs. 3 and 4, then $\eta$ is one of the key parameters to TD signature suppression.

To further explore the influence of coupling strength $\eta$ on TD signature, Fig. 5 simulates the evolution of SF curves of polarization-resolved chaos outputs from MC-VCSELs when $\eta$ varies from 0 $\hbox{ns}^{-1}$ to 50 $\hbox{ns}^{-1}$ and $\Delta f$ is still fixed at 0 GHz. First, when $\eta$ is low, the TD signatures are relatively obvious. As an example, the power spectra and PE curves for polarization-resolved chaos outputs from MC-VCSELs have been given in Fig. 5(a) when $\eta = 1\ \hbox{ns}^{-1}$. Second, when $\eta$ locates at the range of $2\ \hbox{ns}^{-1}\ <\ \eta\ <\ 20\ \hbox{ns}^{-1}$, TD signatures can be suppressed effectively. For example, under the case for $\eta = 10\ \hbox{ns}^{-1}$ [shown as Fig. 4(b)], the distribution of power spectra becomes smooth and the TD signature related valleys in PE curves almost disappear. Finally, when $\eta$ exceeds 20 $\hbox{ns}^{-1}$, with the increases of $\eta$, TD signatures gradually rise again and become more and more obvious. As shown in (c) $(\eta = 30\ \hbox{ns}^{-1})$ and (d) $(\eta = 50\ \hbox{ns}^{-1})$, the corresponding power spectra exhibit multipeaks with uniform frequency interval, and the PE curves show obvious valleys at the location of $\tau_{e} = 6 \ \hbox{ns}$. The larger $\eta$ is, the more obvious the TD signature becomes. Additionally, similar evolution trends can be observed for all polarization-resolved chaos outputs from MC-VCSELs.

In a MC-VCSELs system, it is necessary to investigate the influences of frequency detuning (labeled as $\Delta f$) between VCSEL1 and VCSEL2 on the TD signature since the oscillation frequency of VCSEL is sensitive to operating conditions. The above analysis ignore the frequency detuning, namely, $\Delta f$ is assumed to be 0 GHz. In the following, we will discuss the influence of the frequency detuning $\Delta f$ on the TD signature. Here, we define the amplitude $\sigma$ to characterize whether the TD signature is obvious or not. The amplitude $\sigma$ is the maximum SF peak within a $\Delta t$ region of [5 ns, 7 ns]. The larger $\sigma$ is, the more obvious the TD signature will be. Fig. 6 displays the dependence of $\sigma$ on the frequency detuning $\Delta f$ under $\eta = 30\ \hbox{ns}^{-1}$. As shown in this diagram, the maximum values of $\sigma$ for all the polarization-resolved chaos outputs from MC-VCSELs are about 0.35 and locate at $\Delta f \approx 0\ \hbox{GHz}$, which indicates that the TD signature is obvious under this case. With the increase of $\vert\Delta f\vert$, the values of $\sigma$ will be decreased. Specially, for $\Delta f \approx 30\ \hbox{GHz}$, $\sigma$ values for all modes are close to 0.05. Under this case, the TD signatures for all polarization-resolved chaos outputs from MC-VCSELs are suppressed effectively. From this diagram, one can see that the frequency detuning $\Delta f$ is another key parameter to TD signature suppression.

Combining Fig. 5 with Fig. 6, it can be concluded that the TD signatures of a MC-VCSELs system depend on the coupling strength $\eta$ and the frequency detuning $\Delta f$. In Fig. 7, we present the maps of the TD signature evolution for VCSEL1 in the parameter space of $\Delta f$ and $\eta$. As shown in this diagram, when $\eta$ is relatively low $(2\ \hbox{ns}^{-1}\ <\ \eta\ <\ 20\ \hbox{ns}^{-1})$, while $\Delta f$ varies in the range of $-20\ \hbox{GHz}\ <\ \Delta f \ <\ 20\ \hbox{GHz}$, the values of amplitude $\sigma$ are very small and the TD signatures can be well suppressed in this parameter region (dark-blue region). Also, a similar result can be obtained for VCSEL2. Physically, in a MC-VCSELs system, one VCSEL can be taken as an active nonlinear reflector of the other VCSEL. We take VCSEL1 as an example. The regenerated reflected signal from VCSEL2 can be controlled to be similar to or quit different from the incident signal from VCSEL1. Through adjusting some operation parameters (for example, $\Delta f$ and $\eta$) of the two VCSELs, when the reflected signal from VCSEL2 is quit different from the output signal of VCSEL1, the TD signature of the chaotic output may be removed.

It should be noted that the above simulated results are obtained under the case that the inner parameters of two VCSELs are assumed to be identical. Even so, further calculations show that, compared with the case mentioned above, there exist a similar TD signature evolution if the two VCSELs possess mismatched inner parameters within a relatively small range.

SECTION IV

In this paper, we has numerically investigated the TD signatures of the polarization-resolved chaos outputs from two MC VCSELs via by SF and PE method. Under suitable parameters, the TD signatures for all polarization-resolved chaos outputs from MC-VCSELs have similar characteristics and can be suppressed simultaneously. Through introducing the amplitude $(\sigma)$ of the maximum SF peak to characterize the TD signature, maps of TD signature evolution in the parameter space of coupling strength $\eta$ and frequency detuning $\Delta f$ have been presented. As a result, the optimal region for generating polarization-resolved chaos signal without obvious TD signature can be determined.

This work was supported in part by the National Natural Science Foundation of China under Grants 60978003, 61078003, 11004161, 61178011, and 61275116 and in part by the Natural Science Foundation of Chongqing City under Grant 2012jjB40011. Corresponding author: G. Q. Xia (e-mail: gqxia@swu.edu.cn).

The authors are with the School of Physics, Southwest University, Chongqing 400715, China.

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