## Abstract

We experimentally and numerically demonstrate the time delay (TD) signature suppression of chaotic output in a double optical feedback semiconductor laser (DOF-SL) system. Two types of TD signature suppression are demonstrated by adjusting the lengths and the feedback power ratios of the two external cavities. One can significantly eliminate all TD signatures of the DOF-SL system and the corresponding power spectrum distribution becomes quite smooth and flat, the other suppresses one of two TD signatures and remains another one.

©2009 Optical Society of America

## 1. Introduction

In the last two decades, optical chaos has been studied extensively for its potential applications in secure communications, chaotic radar and lidar, fast physical random bit generation etc [1–7]. Various frames such as optical feedback, optoelectronic feedback and optical injection, have been presented to produce chaotic output in semiconductor lasers (SLs) [1–10]. Since high dimension broadband chaos can be relatively easily generated through external optical feedback (namely introducing time delay), optical feedback SL system has been served as one of good candidate source generators of high-speed long distance chaos secure communication [1,9,10]. Generally, obvious time delay (TD) signature can be observed in typical single optical feedback (SOF) SL system [10], which inevitably provides a possible clue to the chaos encryption attackers. Recently, some time series analysis techniques for delayed systems have been developed [11,12], and the reconstruction of the chaotic system may be realized based on the TD signature identification [13]. As a result, the security of the chaotic encryption may be partly compromised. If TD signature could be shielded, it would be quite difficult to perform such an identified projection. Therefore, it is necessary to develop some strategies to weaken or pretend the TD signature for ensuring the system security.

Previous studies have already showed that it is possible to attenuate the TD signature in a SOF-SL system. Rontani *et al.* proposed a theoretical investigation of TD signature suppression (TDSS) in a SOF-SL system [14]. We experimentally demonstrated that the TD signature can be suppressed in incoherent SOF-SL system through optimizing feedback strength and injection current of SL [15]. Based on a SOF-SL system, a double optical feedback (DOF) SL system can be constructed by introducing another external cavity. Such DOF-SL system can be used to produce rich dynamical behaviors [16] or control the dynamical states of SL system such as the low frequency fluctuation, fixed states and limit cycles [17,18]. By adopting this DOF-SL system, Lee *et al.* demonstrated the possibility to complicate the TD signature for the first time [19]. In this paper, on the basis of Ref [19], by using power spectra, self-correlation function and mutual information techniques, two methods of the TDSS are experimentally and numerically demonstrated where one can suppress all the TD signature of DIOF-SL system while the other can suppress one of two TD signatures and remain another one.

## 2. Experimental setup

Figure 1 is the schematic of experimental setup. An InGaAsP/InP DFB-SL is used in experiment. It is biased at 20.08mA (about 1.61 times of threshold) by an ultra-low-noise current source (ILX-Lightwave, LDX-3620) and stabilized at 22.05°C by a thermoelectric controller (ILX-Lightwave, LDT-5412). Under these conditions, the solitary SL output is about 2.6mW and the lasing wavelength is about 1548.3nm. The SL output is collimated by a collimator lens. After that, the beam is split into three parts by two beam splitters (BS1 and BS2) and individually incident upon two reflecting mirror (M1, M2) and an optical isolator (OI, isolation>55dB). The two reflected beams by M1 and M2 are re-injected into SL and then form the DOF configuration. By the way, the DOF configuration can be easily transformed into SOF of cavity 1 (SOF1) or SOF of cavity 2 (SOF2) configurations just by hiding M2 or M1. In the signal detection part, the optical wavelength is measured by optical spectrum analyzer (Ando AQ6317C). The optical signal is transformed into electronic signal by a wide bandwidth photodetector (PD, New Focus 1544-B, bandwidth 12 GHz) and then analyzed by a 6GHz digital oscilloscope (Agilent 54855A, sample speed 20GS/s).

## 3. Experimental results and discussion

Figure 2
compares the typical chaotic time series and corresponding power spectra under SOF and DOF configurations. During this experiment, the two external cavity lengths are taken closed values as *l _{cav1}*≈477mm,

*l*≈461mm, respectively. The reflected optical power is monitored at point “T” (see Fig. 1) by an optical power meter and controlled by a variable optical attenuator, and is set as 9.4μw for cavity 1 and 15.3μw for cavity 2 in order to obtain almost equal feedback power ratios (FPR, defined as the ratio of the feedback light power to the laser output power) for SOF1 and SOF2. Above operation conditions can assure that the SL under SOF and DOF cases can all be rendered into chaotic states. For the SOF case, both the chaotic time series of SOF1 and SOF2 as shown in Fig. 2(A1) and Fig. 2(B1) behave intricately and irregularly. However, as observed from corresponding power spectra (Fig. 2(A2) and Fig. 2(B2)), some uniform spacing frequency peaks relevant to the TD signatures obviously emerge upon continuous spectrum background. Following these clues, the TD signature of the SOF-SL chaotic system may be retrieved [13]. For the DOF case, the chaotic time series (see Fig. 2 (C1)) behave more irregularly than those for the SOF case. Meantime, the corresponding power spectrum (see Fig. 2 (C2)) becomes relatively smooth and flat, and no obvious regular peaks appear upon background.

_{cav2}Generally, numerous methods such as self-correlation function (SF) and mutual information (MI) [14,20], filling factor analysis [21], global nonlinear models with neural networks [22], local linear models [12], can be used to identify the TD signature. In this paper, we adopt the SF and MI approaches. For a delay-differential system, the SF function can be defined as:

*P*(t) represents chaotic time series, Δ

*t*is the time shift. The TD signature can be retrieved from the peak location of SF curve. The MI between

*P*(t) and

*P*(t + Δt) is described by:

*δ*(

*P*(t),

*P*(t + Δt)) is the joint probability,

*δ*(

*P*(t)) and

*δ*(

*P*(t + Δt)) are the marginal probability densities. The peak location of MI can also characterize the TD signature.

The corresponding SF and MI curves of Fig. 2(A1)-(C1) are shown in Fig. 3
, respectively. For the SOF case, the external cavity time (τ_{cav1}≈3.25ns and τ_{cav2}≈3.15ns) can be clearly extracted from the peaks location of SF and MI curves in Fig. 3(A)-(B). However, for the DOF case as shown in Fig. 3(C1) and Fig. 3(C2), both SF and MI curves show no significant peaks at τ_{cav1} and τ_{cav2}, and all the TD signatures have almost been depressed into background. Therefore, such a DOF configuration can afford a possible TD signature concealment scheme under certain conditions. In addition, there are some small troughs with a period of 0.2ns in these SF curves, which characterizes the relaxation oscillation period (τ_{RO}) of the SL as τ_{RO}≈0.2ns [14].

Above experimental results show that such a DOF configuration can effectively suppress the TD signature of a chaotic system. In the following, based on the operation conditions to obtain Fig. 2, we will further investigate the influence of the external cavity lengths and feedback powers on such TDSS performances in the DOF-SL system. Figures 4 (A1)-(E1) are experimentally recorded chaotic time series corresponding to different feedback optical powers of SOF2 while the feedback level of SOF1 keeps unchanged. Combining the according power spectra, SF and MI, one can observe that the TD signature gradually attenuates when the monitored feedback optical power of SOF2 increases from 7.83μw to 16.4μw (corresponding FPR varies from 0.15% to 0.32%). In particular, the TD signature can be completely concealed as shown in Figs. 4(D) for the feedback power of SOF2 is 15.3μw, where both two cavities have almost equal FPR (about 0.3%). Under this circumstance, an eavesdropper would be quite difficult to accurately identify the TD signature. Further increasing FPR of cavity 2 to 0.32% as shown in Fig. 4(E), the TD signature arises again.

Figure 5 further presents another type of TDSS in DOF-SL system, where the length of cavity 2 is shortened to 241mm (about 1/2 of length of cavity 1) and according feedback optical power is raised to 113μw (FPR≈2.2%) and the operation parameters of cavity 1 keep unchanged. Increasing feedback optical power of cavity 2 is mainly because more feedback is needed to obtain similar chaotic state as Fig. 2(A) when the length of cavity 2 is shortened [23]. Columns (A), (B) and (C) of this diagram correspond to SOF1, SOF2 and DOF configuration, respectively. Interestingly, the measured frequency interval under the DOF configuration (Fig. 5 (C2)) is very close to that under the SOF2 (Fig. 5(B2)). For the DOF case, the SF (Fig. 5(C3)) and MI (Fig. 5 (C4)) curves show apparent envelope peaks at Δt≈1.7ns, which is approximately the TD signature of cavity 2. However, the time peak at Δt≈3.25ns, which corresponds to the TD signature of cavity 1, has been significantly attenuated. This feature is quit different from that in Fig. 4(D), where all TD signatures are attenuated seriously. This phenomenon may provide a pseudo TD signature chaos generation scheme, and an eavesdropper would be confused whether the chaos carrier is produced from a SOF-SL system or a DOF-SL system.

Furthermore, we will investigate the influence of the feedback power on the second type TDSS corresponding to Fig. 5. Figure 6 shows chaotic time series, power spectra, SF and MI for different feedback powers of cavity 2 while the feedback power of cavity 1 keeps unchanged. When monitored feedback power of cavity 2 is about 59μw (FPR≈1.15%) as shown in the first row of Fig. 6, TD signature of cavity 1 appears distinctively while the TD signature of cavity 2 is weak. Increasing the feedback power of cavity 2 from 59μw to 113μw (corresponding FPR varies from 1.15% to 2.2%), the TD signature of cavity 1 gradually attenuates, but TD signature Δt≈1.7ns of cavity 2 enhances significantly. Particularly, for 2.2% FPR as shown in Fig. 6 (D), the TD signature of cavity 1 is suppressed seriously. Further increasing FPR to 2.63% as Fig. 6 (E), the TD signature of cavity 1 recovers again.

Above experimental investigations illustrate two types of TDSS in a DOF-SL system. For the first type of TDSS, two external cavity lengths are taken closed values and then all TD signatures of the DOF-SL system could be suppressed. For the second type of TDSS, the length of one cavity is taken about a half of length of another cavity. As a result, this system shows only one TD signature of two external cavities. Up to now, the detailed physical reasons to such TDSS are still unclear since a DOF-SL system behaves much more complex and diversified than a SOF-SL system [17]. Even so, through observing above results carefully, some information related to TDSS can be found. For the convenience of comparison, Fig. 7
superimposes the related SF cures of two types of TDSS. Figure 7 (A) corresponds to the first type of TDSS (integration of Fig. 3 (A1) and Fig. 3 (B1)). From this diagram, one can observe that the peak profile of SF curves reveals three characteristic times namely τ_{cav1}≈3.25ns, τ_{cav2}≈3.15ns and τ_{RO}≈0.2ns, respectively. Since τ_{cav2}-τ_{cav1}≈-0.1ns≈-1/2τ_{RO}, SF peaks of SOF1 approximately correspond to SF valleys of SOF2 at the region around Δt≈3.2ns. Moreover, as for the second type of TDSS shown in Fig. 7 (B) (integration of Fig. 5 (A3) and Fig. 5 (B3)), since the cavity 2 is shortened to 241mm, the 2τ_{cav2} is read about 3.35ns while the τ_{cav1} is about 3.25ns. Then there is a relation that 2τ_{cav2}-τ_{cav1} = 0.1ns≈1/2τ_{RO}, and the SF peaks of SOF1 encounter with the SF valleys of SOF2 around Δt≈3.2ns. From this diagram and above analysis, one can anticipate that the intrinsic characteristic time *τ _{RO}* may take an important role in TDSS of such DOS-SL system.

## 4. Theoretical simulation and analysis

The DOF-SL system can be modeled by well known Lang–Kobayashi rate equations. The slowly varying complex electric field *E* and the average carrier number *N* in the active region, can be expressed as:

*κ*and

_{cav1}*κ*denote the feedback strengths of external cavity 1 and external cavity 2,

_{cav1}*τ*and

_{cav1}*τ*are the delay times corresponding to the round-trip time of the external cavity 1 and the external cavity 2,

_{cav2}*β*is the linewidth-enhancement factor.

*G*(t) =

*g*(

*N*(t)−

*N*)/(1 + ε

_{0}*E*(t)

*) is the gain coefficient, where*

^{2}*g*is differential gain coefficient, ε is the gain saturation coefficient and

*N*is the transparency carrier number.

_{0}*ω*is the angular frequency of SL,

_{0}*γ*is the photon loss rate.

_{p}*γ*= 1/

_{p}*τ*, where

_{p}*τ*is the photon lifetime.

_{p}*τ*is the round-trip time in SL intracavity,

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

*J*is the injection carrier rate,

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

_{N}*τ*is an intrinsic damping time of the free-running laser and could be estimated by

_{RO}*τ*≈2π(

_{RO}*gE*

^{2}/

*τ*)

_{p}^{-1/2}. Equation (3) can be solved by using the fourth-order Runge-Kutta algorithm, and the parameters are set as:

*β*= 4,

*ω*= 1.216 × e

_{0}^{15}rad/s,

*τ*= 4.2ps,

_{p}*τ*= 8.5ps,

_{L}*τ*= 1.6ns,

_{N}*g*= 2 × 10

^{4}s

^{−1},

*F*(t) = 0,

*N*= 1.25 × 10

_{0}^{8}, and

*ε*= 1 × 10

^{−7}. For comparison with experimental results,

*J*is set as 1.6

*J*with

_{th}*τ*≈0.2ns,

_{RO}*τ*is fixed as 3.2ns and

_{cav1}*κ*= 0.04.

_{cav1}*τ*and

_{cav2}*κ*are variable for different considerations.

_{cav2}Figures 8 (A)-(C)
simulated the chaotic temporal waveforms, and corresponding power spectra, SF and MI curves of the first type TDSS where *τ _{cav1}* = 3.2ns,

*τ*= 3.12ns,

_{cav2}*κ*= 0.04,

_{cav1}*κ*= 0.05, respectively. For the SOF situation ((A) and (B)), clear TD peaks can be seen. But for the DOF configuration ((C)), the entire TD signature is attenuated significantly. Comparing this diagram with Figs. 2-3, the numerical calculations relatively successfully demonstrate the experimental results. Further calculations show that by reducing

_{cav2}*τ*and increasing

_{cav2}*κ*to suitable value, the second type TDSS as shown in Fig. 5 can also be simulated.

_{cav2}To show the influence of the external cavity delay time on TDSS, Fig. 9
integrates SF curves to form a map of TD signature distribution for *τ _{cav1}* = 3.2ns and

*κ*=

_{cav1}*κ*= 0.04 and

_{cav2}*τ*varies from 0.4ns to 4ns. As shown in this diagram, the diagonal line Δt≈

_{cav2}*τ*and the vertical line Δt≈3.2ns reveal the TD signatures of cavity 2 and of cavity 1. Meantime, there exist numerous relatively weak peaks and valleys with a period of about 0.1ns≈1/2

_{cav2}*τ*. Two obvious twisted regions can be observed around

_{RO}*τ*≈1.6ns and

_{cav2}*τ*≈3.2ns. For the region around

_{cav2}*τ*≈3.2ns, the TD signatures become very weak at several separated sub-regions such as

_{cav2}*τ*≈2.92ns,

_{cav2}*τ*≈3.12ns,

_{cav2}*τ*≈3.28ns and

_{cav2}*τ*≈3.48ns, which indicates possible TDSS sub-regions. Interestingly, these TDSS sub-regions does not locate at the point where

_{cav2}*τ*equals to

_{cav2}*τ*strictly, but periodically distribute around

_{cav1}*τ*and roughly follow relations:

_{cav1}*τ*-

_{cav2}*τ*≈-3/2

_{cav1}*τ*, −1/2

_{RO}*τ*, 1/2

_{RO}*τ*, 3/2

_{RO}*τ*, respectively. Here, we name them as type I TDSS. As for the region around

_{RO}*τ*≈1.6ns, there also appear some slight TDSS sub-regions around

_{cav2}*τ*≈1.44ns,

_{cav2}*τ*≈1.54ns,

_{cav2}*τ*≈1.64ns and

_{cav2}*τ*≈1.74ns, respectively, and these sub-regions roughly satisfy following relations: 2

_{cav2}*τ*-

_{cav2}*τ*≈-3/2

_{cav1}*τ*, −1/2

_{RO}*τ*, 1/2

_{RO}*τ*, 3/2

_{RO}*τ*. Here, we name them as type II TDSS. Further calculations show that by selecting suitable feedback strength, the TDSS effect, which suppresses the TD signature of one external cavity and remains TD signature of another external cavity, can be strengthened. Above theoretical results reveal that there exist some special regions where two types of TDSS happens just as observed in our experiments.

_{RO}To show the influence of the feedback strength on the TDSS, Figs. 10
(A1)-(B1) integrate SF curves to form two map of TD signature about type I TDSS (*τ _{cav1}* = 3.2ns and

*τ*= 3.12ns) and type II TDSS (

_{cav2}*τ*= 3.2ns and

_{cav1}*τ*= 1.64ns). From these two maps (Figs. 10 (A1)-(B1)), one can see that the theoretical results verifies previous experimental observations to certain degree. Furthermore, Figs. 10 (A2)-(B2) give the variation of the amplitude

_{cav2}*ρ*, which is the maximum of the SF peak in a time shift window around 3.2ns, with the feedback strength. These two curves also confirm experimental tendency as shown in Fig. 4 and Fig. 6.

## 5. Conclusion

In summary, chaotic dynamics of a DOF-SL system are both experimentally and numerically investigated, and the TD signatures of chaotic time series are retrieved and analyzed by using SF function and the MI technique. Two types of the TD signature suppression are demonstrated. One is that all TD signatures of two external cavities are depressed significantly and the corresponding power spectrum becomes smooth and flat when the two cavities have roughly equal lengths and feedback power ratios. The other scheme is that one of two external cavities TD signatures is attenuated seriously and another one is remained when one cavity length is about half of another cavity length. In addition, our experimental observations and simulations also illustrate that the intrinsic characteristic time *τ _{RO}* plays an important role during TDSS. We hope that this work will be helpful for enhancing security in chaotic cryptosystems. Also, this work may offer some useful insights for the nonlinear dynamical properties of such DOF-SL system.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 60978003 and the Open Fund of the State Key Lab of Millimeter Waves.

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