## Abstract

In this paper, evolution of time delay (TD) signature of chaos generated in a mutual delay-coupled semiconductor lasers (MDC-SLs) system is investigated experimentally and theoretically. Two statistical methods, including self-correlation function (SF) and permutation entropy (PE), are used to estimate the TD signature of chaotic time series. Through extracting the characteristic peak from the SF curve, a series of TD signature evolution maps are firstly obtained in the parameter space of coupled strength and frequency detuning. Meantime, the influences of injection current on the evolution maps of TD signature have been discussed, and the optimum scope of TD signature suppression is also specified. An overall qualitative agreement between our theoretical and experimental results is obtained.

©2012 Optical Society of America

## 1. Introduction

Optical chaos has attractive applications in various fields such as secure communications [1, 2], chaotic radar and lidar [3, 4], rainbow refractometry [5], and fast physical random bit generation [6–9] *etc*. In these applications, optical chaos generated from some delay-coupled semiconductor laser (SL) systems such as an external-cavity feedback SL (ECF-SL) system and a mutually delay-coupled SLs (MDC-SLs) system [2–18], have attracted extensive attention due to their inherent wide bandwidth and dynamic complexity. For example, the optical chaos generated from an ECF-SL system has been successfully applied in Gbit/s unidirectional chaotic secure communication over 100km in a commercial metro-network [2]. A MDC-SLs system is regarded as one of main candidates in bidirectional chaotic secure communication. Through digital sampling, the optical chaos output of an ECF-SL system can produce ultra-fast random bit sequences [6–8]. Among above mentioned applications, the time delay (TD) term in delay-coupled SL systems plays an important role to modify the solitary SL becoming an infinite-dimensional dynamical system, which can export high-dimensional chaotic signals [11].

Generally, TD signature of chaos is an important issue in nonlinear system analysis [19–39]. On one hand, many techniques such as the singular values fraction measure [22], the filling factor analysis [24], local linear models [25], models with neural networks [26], the self-correlation function (SF) and mutual information (MI) [27–31] *etc*., have been proposed to identify TD signature of chaos. Recently, permutation entropy (PE) as one of information-theory-based methods is also formulated to perform the TD signature identification [32–34]. On the other hand, obvious TD signature is not undesirable in some applications such as secure communication and random bit generation. For secure communication, the security of data encryption relies upon, to a large extent, the unpredictability of chaotic carrier. However, obvious TD signature offers a possible clue to the attackers, which may compromise the confidentiality of secure communication [35–37]. As for ultra-fast random bit generation, TD signature induces recurrence features and affects seriously the statistical performance of random bit sequences [6, 7]. Therefore, it is necessary to develop some effective strategies to suppress the TD signature.

In recent years, several TD signature suppression (TDSS) strategies have been proposed. One type of TDSS strategies is based on logical process such as exclusive-OR (XOR) operation and least significant bits (LSBs) algorithm [6–8]. These strategies are designed specifically to eliminate the TD signature in generation of random bit sequence. Another type of TDSS strategies is based on pure physical mechanism. These strategies attempt to utilize the physical interaction of nonlinear dynamics to suppress the TD signature of chaos. The pure physical TDSS has been investigated in a double ECF-SL system [31, 38], a generally coherent [28–30] and incoherent [39] ECF-SL system. Also, the pure physical TDSS has been demonstrated in an opto-electronic feedback SL system [40]. More recently, we preliminarily experimentally observed pure physical TDSS phenomenon in a MDC-SLs system [41]. However, many important issues such as the overall evolution pattern of TD signature, the detailed distribution of TDSS state in the parameter space and the possible physical mechanism *etc*., are still unclear in such a MDC-SLs system and should be further clarified.

In this paper, we propose a systematical investigation about the TD signature evolution in a MDC-SLs system. In Section 2, the structure of experimental setup is illustrated. In Section 3, with the help of two quantifiers SF and PE, the TD signature of the chaotic signals is retrieved under different operation parameter conditions. Furthermore, a series of evolution maps of TD signature are obtained in the parameter space of coupling strength and frequencies detuning between two SLs. The influence of injection current on the evolution maps of TD signature has also been investigated. In Section 4, the theoretical simulations and analysis are given. Finally, conclusions are presented in Section 5.

## 2. Experimental setup

The experimental setup is illustrated in Fig. 1
. Two similar 1550nm InGaAsP/InP distributed feedback Bragg semiconductor lasers (SL1 and SL2) are connected by a fiber to form a mutually coupling system. The two SLs are driven by two ultra-low-noise current sources (ILX-Lightwave, LDX-3620) and two temperature controllers (ILX-Lightwave, LDT-5412).The polarization states of two lasers are matched each other by using a polarization controller. The coupling strength (κ) is defined as the optical power ratio of external injection light and the solitary SL output light. κ is controlled with a variable attenuator (VA1) and monitored by an optical power meter. The optical spectra of SLs are monitored by an optical spectrum analyzer (Ando AQ6317C). The frequency detuning *∆f* is defined as *∆f* = *f _{1}*-

*f*, where

_{2}*f*and

_{1}*f*are the optical frequencies of free running SL1 and SL2, respectively. Different

_{2}*∆f*is obtained by tuning the temperature of SL2 while the temperature of SL1 keeps constant. The temperature-frequency coefficient of SL2 is measured as

*∆f*/∆T≈22GHz/K. The fiber length between SL1 and SL2 is about 5.35m, which corresponds to a coupling round-trip time τ≈53.55ns. Four optical isolators (OIs) (isolation>55dB) are inserted to prevent from unwanted external disturbances. Under proper conditions, the MDC-SLs system can route into complex dynamics and export chaotic signals [13–18]. The chaotic signals are detected by two 12 GHz bandwidth photodetectors (PDs, New Focus 1544-B) and analyzed by a digital oscilloscope (Agilent 54855A, 6 GHz bandwidth and 50ps sampling interval). For convenience of observing two time series output from SL1 and SL2 simultaneously, the amplitude difference between two chaotic signals sent into the digital oscilloscope should not be too large. Then, another VA (VA2) is introduced to control the optical power of SL1 into the PD.

## 3. Experimental results

#### 3.1Influence of coupling strength on TD signature

At first, the injection currents are respectively biased at 9.28mA for SL1 (about 1.14*I _{th}*,

*I*is the threshold current) and 9.7mA for SL2 (about 1.14

_{th}*I*). The temperatures are respectively stabilized at 19.86°C for SL1 and 19.60°C for SL2. Under these conditions, both the optical wavelengths of the two SLs are about 1553.11nm and then the corresponding

_{th}*∆f*is about 0GHz. The fist column of Fig. 2 summarizes the measured chaotic time series under different coupling strength κ. It can be observed that all time series vary intricately and don’t show directly TD signature. To identify the TD signature, various analysis functions can be employed [20–34]. Here, the self-correlation (SF) function is employed to retrieve TD signature from chaotic time series. The formula of SF can be formed as [28–31]:

*P*(

*t*) represents chaotic time series,

*Δt*is the time shift, <·> denotes time average. The TD signature can be retrieved from the location of characteristic peak in SF curve. Furthermore, we introduce PE, derived from the information theory, as another method to identify TD signature. The PE owns some unique advantages such as simplicity, fast calculation and robustness to noise. Following formulas in Ref [32–34], the calculation of PE is carried out by modifying the given time series {

*x*(t), t = 1, ···, N} into

*D*-dimensional vectors

*X*

_{t}= [

*x*(t),

*x*(t +

*τ*

_{emd}), ···,

*x*(t + (

*D*-1)

*τ*

_{emd})], where

*τ*

_{emd}is the embedding delay time and

*D*is the embedding dimension. Through changing the value of

*τ*

_{emd}and fixing the value of

*D*, different PE values can be obtained and form a function curve. The value of PE will minimize when

*τ*

_{emd}is close to the characteristic time τ. Then, the TD signature can be retrieved by identifying the minima location in such PE curve. In this paper, the value of

*D*is fixed as

*D*= 6 after taking into account the suggestion in Ref [32–34].

The second column of Fig. 2 shows the power spectra which could also be used to retrieve the TD signature by identifying the periodic ripple [41]. Among these power spectra, it can be observed that there is apparent distribution variation for different κ. For κ≈0.0003, as shown in Figs. 2 (Aa2 and Ba2), the frequency components surround intensively a specific frequency 2.6GHz, which is the relaxation oscillation frequency *f _{RO}* of SL [10]. When the coupling strength increases to κ≈0.023, the strongest frequency component of spectra moves to a much higher frequency (about 3.5GH) as given in Figs. 2 (Ad2 and Bd2). Furthermore, the third column of Fig. 2 shows the calculated SF curves of chaotic time series. As shown in Figs. 2 (Aa3-Ab3, Ad3-Ae3, Ba3-Bb3, and Bd3-Be3), clear peaks emerge around

*Δt*≈ ± 53.55ns in SF curves. Meantime, in Figs. 2 (Aa4-Ab4, Ad4-Ae4, Ba4-Bb4, and Bd4-Be4), sharp valleys are observed around

*τ*

_{emd}≈53.55ns in PE curves. All the locations of such SF peaks and PE valleys match well with the characteristic time τ≈53.55ns, and reveal the TD signature contained in chaotic time series. But for the special case κ≈0.0068, as shown in Figs. 2 (Ac3-Ac4, Bc3-Bc4), no significant SF peaks or PE valleys are observed around the characteristic time τ≈53.55ns. Almost perfect δ function profile in the SF curves indicates that the TDSS happens in such a MDC-SLs system. Based on the unique parallel scheme in MDC-SLs system, two sets of chaotic signals with weak TD signatures can be obtained simultaneously.

#### 3.2 Influence of frequency detuning on TD signature

Since the oscillation frequency of SL is sensitive to temperature fluctuation, it is valuable to investigate the influences of frequency mismatch (labeled as *∆f*) between SL1 and SL2 on the TD signature. Figure 3
collectively shows the recorded chaotic time series, power spectra, SF curves and PE curves when *∆f* varies from 9GHz to −10.8GHz and κ is fixed as 0.008. At first, for *∆f*≈9.0GHz as shown in Figs. 3(Aa3-Aa4, Ba3-Ba4), obvious TD signatures are observed in both SF curves and PE curves. Further adjusting *∆f* to 5.7GHz, the TD signatures are significantly suppressed close to the background in SF curves and in PE curves as shown in Figs. 3(Ab3-Ab4, Bb3-Bb4). Moreover, the TDSS state can be well maintained for *∆f* varies from 5.7GHz to −5.3GHz as given in Figs. 3(Ac3-Ac4, Ad3-Ad4, Bc3-Bc4, and Bd3-Bd4). However, when *∆f* is tuned to −10.8GHz, the TDSS state is destroyed and TD signatures re-appear in SF curves and PE curves as shown in Figs. 3(Ae3-Ae4, Be3-Be4). Therefore, one can reasonably deduce that the TDSS state owns certain tolerance to the frequency detuning between SL1 and SL2 under specific coupling strength.

#### 3.3 Evolution maps of TD signature in parameter space

For the purpose of hunting optimum TDSS state, it may be essential to investigate the overall evolution of TD signature in some parameter space. In Fig. 4
, we integrate all TD signatures to form two evolution maps in the parameter space of κ and *∆f*, where κ varies from 0.002 to 0.025 and *∆f* scans from −15GHz to 15GHz. In these maps, the strength of TD signature is characterized by the value of amplitude *ρ*, which is the maximum peak value of SF curve in a time window 52ns<∆t<55ns, whose size is appropriately chosen to capture only the TD related peak in SF curves. For these diagrams, different regions of TD signature can be distinguished via by different pseudo-colors, where the dark blue color denotes the TDSS state. From these diagrams, one can see that TD signature shows different evolution pattern under different coupling strengths. For low κ condition (0.004<κ<0.008), the TDSS locates at a region of −5GHz<*∆f*<5GHz, and the central position is close to the vertical axis *∆f* = 0GHz. However, for high κ values (0.008<κ<0.025), the TDSS region is split into two branches. One branch stretches towards negative ∆f side, while the other branch stretches towards positive *∆f* side. For example, when κ is about 0.019, the TD signature eliminates for *∆f*≈-9.5GHz and *∆f*≈9.0GHz, but the TD signature is obvious for *∆f*≈0GHz. Moreover, with the increase of κ values, the two branches of TDSS are further separated. Consequently, the optimum scope of TDSS exhibits a shape as the letter ‘V’ in the overall parameter space of κ and *∆f*.

#### 3.4 Influence of injection current on TD signature

To explore the influence of injection current of SLs on TD signature, Fig. 5
further show the recorded evolution maps of TD signature when the injection currents of SLs vary from about 1.2*I _{th}* to about 1.5

*I*. From these maps, one can observe that the ‘V’ shape of TDSS is maintained for different injection currents but there exist some differences under different injection currents. The higher injection current level, the wider parameter scope covered by TDSS. However, for a relatively high injection current level, the TDSS effect is slightly worse than that for a relatively low injection current level since the darkest blue color in Figs. 5(A3 and B3) is slightly brighter than that in Figs. 5(A1 and B1). Therefore, a comprehensive consideration is necessary by taking into account various performance requirements. For the case that the requirement for the effect of TDSS is not high, high injection current of SLs can be chosen for conveniently hunting TDSS state in parameter space. On the contrary, low injection current of SLs should be adopted if the effect of TDSS is highly required. Moreover, if one observes these diagrams carefully, a mirror symmetry relationship can be found between the evolution maps of SL1 and SL2. That is to say, the left part of SL1’s maps is similar to the right part of SL2’s maps meanwhile the right part of SL1’s maps is similar to the left part of SL2’s maps.

_{th}## 4. Theoretical simulation and analysis

The MDC-SLs system can be modeled by the well known Lang–Kobayashi rate equations [13]. The slowly varying complex electric field *E* and the average carrier number *N* in the active region of SLs can be expressed as:

*η*is the coupling rate and labels the coupling strength,

*τ*is the coupling round-trip time,

*β*is the linewidth-enhancement factor,

*g*is differential gain coefficient,

*ε*is the gain saturation coefficient and

*N*is the transparency carrier number.

_{0}*f*and

_{1}*f*are the optical frequencies of free running SL1 and SL2, respectively.

_{2}*Δf*is defined as

*Δf*=

*f*-

_{1}*f*, where different

_{2}*Δf*is obtained by varying

*f*while

_{2}*f*keeps a constant.

_{1}*τ*is the photon lifetime,

_{p}*F*(t) is the spontaneous-emission noise,

*J*is the injection carrier rate,

*τ*is the carrier lifetime. The relaxation oscillation period

_{N}*τ*of SLs could be estimated by

_{RO}*τ*≈2π(

_{RO}*gE*

^{2}/

*τ*)

_{p}^{-1/2}.

Above coupled nonlinear rate-Eqs. (2)-(4) can be numerically solved by using the fourth-order Runge-Kutta algorithm. During the calculations, the parameters are set as: *β* = 4, *f _{1}* = 1.9355 × e

^{14}Hz,

*τ*= 4.2ps,

_{p}*τ*= 1.6ns,

_{N}*g*= 2 × 10

^{4}s

^{−1},

*N*= 1.25 × 10

_{0}^{8},

*ε*= 1.5 × 10

^{−7}and

*τ*= 8ns. Considering the weak influence of the noise on the TD signature, the spontaneous-emission noise

*F*(

_{1}*t*) and

*F*(

_{2}*t*) are set to zero for simplicity.

*J*is set initially to 1.15

*J*. Under these circumstances,

_{th}*τ*is estimated to be about 0.4ns.

_{RO}*η*,

*Δf*and

*J*are variable for different considerations. The initial conditions of SL1 and SL2 are also set different to accord with the real physical situation. Additionally, it should be pointed out that during experiments, the detection bandwidth is limited by the used digital oscilloscope whose bandwidth is 6GHz in our experimental system. However, the observed peak frequency in power spectra of the obtained chaotic signal (shown as the second column in Figs. 2 and 3) is always smaller than 6GHz of the detection bandwidth. After considering this fact, the finite detection bandwidth of the experimental system does not be taken into account during the simulations.

To investigate the influence of coupling strength on the TD signature, Fig. 6
gives the simulated temporal traces, power spectra, SF curves and PE curves under fixed *Δf* = 0GHz and different values of *η*. Firstly, as shown in power spectra (the second column) of Fig. 6, the most significant frequency component monotonously moves to high frequency with the increase of *η*. Secondly, from the third and the forth columns of Fig. 6, it can be found that the TD signatures, labeled by SF peaks and PE valleys, gradually attenuate with the increase of *η* and are effectively suppressed closely to the background for *η* = 7.5ns^{−1}. However, continuously increasing η will result in the re-arising of TD signature. All these simulated power spectra, SF curves and PE curves qualitatively agree with the experimental results shown in Fig. 2. Additionally, the first column in Fig. 6 gives the transient temporal evolutions from the initial conditions of SLs, which may offer an interpretation about the TD signature for different conditions in a MDC-SLs system. In the first time interval 0ns<t<4ns, all temporal traces rapidly fall back to the balanced point after series of damping relaxation oscillation. The reason is that there is no disturbance in this time interval since external coupling light from counterpart SL has not reached the SL. However, in the second time interval 4ns<t<8ns, all temporal traces oscillate with strong periodicity besides that the periodicity is relatively weak for the case shown in Figs. 6(Ac1, Bc1). The recurrence related to such periodic oscillations will be inherited by the oscillations in next time intervals, and then the TD signature in chaotic outputs is finally formed. As for the special case shown in Figs. 6(Ac1, Bc1), the temporal traces immediately move into complex oscillation once external coupling light is injected into SL in the time interval 4ns<t<8ns. We guess that the recurrence of temporal traces may be washed away with the help of suitable strength of external injection, and then the TD signature is effectively suppressed.

Next, we will investigate the influence of frequency detuning on the TD signature. Figure 7
gives the simulated temporal traces, power spectra, SF curves and PE curves under fixed *η*≈7.5ns^{−1} and different values of *∆f*. As shown in Figs. 7(Ab3- Ab4, Ac3- Ac4, Ad3- Ad4, Bb3- Bb4, Bc3- Bc4, and Bd3- Bd4), the TD signatures always keep at a low level for |*∆f*|<7GHz. The TD signatures will be obvious if |*∆f*| are relatively large as shown in Figs. 7 (Aa3-Aa4, Ae3-Ae4, Ba3-Ba4, and Be3-Be4). Furthermore, the transient temporal evolutions of SL are also given in the first column of Fig. 7. Through observing these temporal traces, one could find that obvious TD signatures are always companied with periodically temporal oscillation, while the TDSS is occurred for the case that the time series rapidly moves into complex oscillation.

To further show the evolution pattern of TD signature in parameter space, Fig. 8
gives the simulated evolution maps of TD signature in parameter space of *η* and *∆f* under different *J*. As shown in this diagram, for low *η*, the optimum scope of TDSS locates at the center part, such as the white dashed lines surrounded region in Figs. 8 (A1 and B1). But for relative high *η*, the TDSS region splits into two branches. Thus, the ‘V’ shape of TDSS region is formed. Moreover, with the increase of injection current, the scope of TDSS gradually broadens while the ‘V’ shape is basically maintained. All these characteristics of simulated maps agree qualitatively with previous experimental results shown in Fig. 4 and Fig. 5. In addition, the mirror symmetry relationship can also be observed in Fig. 8. The reason about such symmetry could be found from the rate Eqs. (2-3), where there is a structural similarity between Eq. (2) and Eq. (3). The role of negative *∆f* in Eq. (2) is similar to that of positive *∆f* in Eq. (3). As a result, the left part (corresponding negative *∆f*) of evolution maps of SL1 is similar to the right part (corresponding positive *∆f*) of evolution maps of SL2. Likewise, the role of positive *∆f* in Eq. (2) is similar to that of negative *∆f* in Eq. (3). Then, the right part of evolution maps of SL1 is similar to the left part of evolution maps of SL2. Therefore, the mirror symmetry relationship is formed.

At last, let us compare the TDSS scenario of a MDC-SLs system with that of an ECF-SL system. From the physical point of review, the TD signature is originated from the recurrence of system. In an ECF-SL system, the external cavity mirror works as a passive linear reflector. Therefore, the reflected signal by the external cavity mirror is always similar to the incident signal, and results in that the current output always recurrent the previous output to some extent. However, in a MDC-SLs system, the chaotic signal output from one SL (such as SL1) is injected into the other SL (such as SL2), and the regenerated chaotic signal is re-injected into SL1. Different from the case in an ECF-SL system, SL2 is not a passive linear mirror but an active nonlinear device. Thus, it offers an opportunity to control actively the regenerated signal from SL2 which is different from the incident signal from SL1 through adjusting the parameters of SL2. Figure 9
compares the incident signal output from SL1 and the regenerated signal by SL2 under the same condition as that in Fig. 2(Ac1, Bc1). As shown in this diagram, the regenerated signal is quit different from the incident signal. Under this circumstance, the TD signature could be suppressed effectively. Additionally, in order to blurry effectively the TD signature, a short external cavity, whose delay time is close to τ_{RO}, is desirable in an ECF-SL system [30]. However, in a MDC-SLs system, above experimental results demonstrate that good TDSS is obtained even if the coupling delay time (τ≈53.55ns) is much larger than the intrinsic relaxation oscillation (τ_{RO}≈0.4ns).

By the way, we have noted that very recently, it has reported that the TD signature of the chaotic time series output from an ECF-SL system can be retrieved theoretically by analyzing the phase of electronic field even if the TD signature can be suppressed in the intensity time series [42]. This work should be paid special attention, and we will focus on this issue during our next step research.

## 5. Conclusions

In this paper, by using the SF function and PE method, the TD signature in a MDC-SLs system has been investigated experimentally and theoretically. The experimental results show that a parallel TDSS scheme can be realized and two sets of TD signature suppressed chaotic signals can be generated simultaneously in a MDC-SLs system. Both the coupling strength and frequency detuning have significant influence on the evolution of TD signature. For relative low coupling strength, the TDSS locates around the center region in the parameter space of coupling strength and frequency detuning. For relative high coupling strength, the TDSS region is split into two branches. One branch stretches towards negative *∆f* part of parameter space and the other branch stretches towards positive *∆f* part of parameter space. In the overall parameter space, the optimum scope of TDSS forms a particular shape similar to the letter ‘V’. Meantime, a large injection current is helpful to enlarge the cover scope of TDSS. Furthermore, the related theoretical simulations have been finished, and the simulated results agree qualitatively with the experimental observations.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant Nos. 60978003, 61078003, 11004161 and 61178011, the Natural Science Foundation of Chongqing City, the Fundamental Research Funds for the Central Universities under Grant No. XDJK2010C021, and the Open Fund of the State Key Lab of Millimeter Waves.

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