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Letter
Vol. 40, No. 19 / October 1 2015 / Optics Letters
Time-delay concealment and complexity enhancement of an external-cavity laser through optical injection NIANQIANG LI,1 WEI PAN,1 A. LOCQUET,2,3
AND
D. S. CITRIN2,3,*
1
Center for Information Photonics and Communications, Southwest Jiaotong University, Chengdu 610031, China Georgia Institute of Technology, School of Electrical and Computer Engineering, Atlanta, Georgia 30332-0250, USA 3 UMI 2958 Georgia Tech-CNRS, Georgia Tech Lorraine, 2 Rue Marconi F-57070, Metz, France *Corresponding author:
[email protected] 2
Received 13 July 2015; revised 29 August 2015; accepted 29 August 2015; posted 31 August 2015 (Doc. ID 245770); published 21 September 2015
The concealment of the time-delay signature (TDS) of chaotic external-cavity lasers is necessary to ensure the security of optical chaos-based cryptosystems. We show that this signature can be removed simply by optically injecting an external-cavity laser with a large linewidth-enhancement factor into a second, noninjection-locked, semiconductor laser. Concealment is ensured both in the amplitude and in the phase of the optical field, satisfying a soughtafter property of optical chaos-based communications. Meanwhile, enhancement of the dynamical complexity, characterized by permutation entropy, coincides with strong TDS suppression over a wide range of parameters, the area for which depends sensitively on the linewidthenhancement factor. © 2015 Optical Society of America OCIS codes: (140.3490) Lasers, distributed-feedback; (140.5960) Semiconductor lasers; (190.3100) Instabilities and chaos. http://dx.doi.org/10.1364/OL.40.004416
A vulnerability of chaotic external-cavity semiconductor laser (ECL) cryptosystems is evidence in the intensity I t of the time delay τf [1,2]. Reference [3] claimed that the time-delay signature (TDS), seen in the autocorrelation function (ACF) of It, is suppressed by choosing τf close to the relaxationoscillation time. Others [4,5] showed that the TDS is not concealed using a simple ECL; it was shown that although the TDS could be suppressed in the ACF of I t, clear evidence of τf may be in the ACF of the optical phase ϕt [5]. The linewidth-enhancement factor α accounts for coupling between amplitude and phase; α is not easily tuned but depends on laser diode (LD) material, type, and design [6,7]. α ranges from slightly negative to about 10. Usually, large α is found in quantum-dot structures (see [8] and references therein). Since α couples I t to ϕt fluctuations [9], one might expect that large α leads to TDS suppression in ϕt. We show that a significant TDS reduction can be obtained with large α, though the effect is insufficient for cryptography. Recently, it was shown [10] that broadband chaos can be generated 0146-9592/15/194416-04$15/0$15.00 © 2015 Optical Society of America
by unidirectionally injecting the output of an ECL into a second semiconductor LD [optically injected semiconductor lasers (OISLs)] and that there is also a possibility of concealing the TDS in the optical intensity of the second laser. Compared with LDs subjected to continuous-wave (CW) optical injection, OISLs present a significantly larger chaotic bandwidth [10]. Here, we provide further evidence that the OISL, in conjunction with high α, leads to a desired suppression of the TDS in both I t and ϕt. Other more complex approaches for hiding the TDS have been suggested [2,11–15]. For example, in Ref. [15], a dualinjection scheme is proposed. The cost of this and other approaches in terms of complexity, though, outweighs the benefit. As we show, OISLs provide an easily implemented approach achieving the performance that was sought. Two other aspects [16–18] are studied. One is the dynamical complexity characterized by permutation entropy (PE) [19,20] because high PE is likely connected to enhanced security, as well as higher maximum rates of extractible randomness [21–24]. The second is chaos synchronization, which is usually exploited to decode the information-bearing signal at the receiver of a chaos communication system. We find robust generalized synchronization when the output of one OISL [master (M)] is fed into another LD [slave (S)]. Figure 1(a) shows an ECL, and Fig. 1(b) shows an OISL. LD 1 (L1) has part of its output (characterized by feedback rate γ) fed back after a time delay τf . In the OISL, the output of an ECL is also unidirectionally injected into LD 2 (L2) with an injection rate σ. Latency τc is associated with coupling L1 into L2. It is of no interest here and is set to 0. The ultimate optical intensity output by the ECL or OISL is denoted It.
Fig. 1. Schematic diagram of an (a) ECL and (b) OISL.
Vol. 40, No. 19 / October 1 2015 / Optics Letters
N t 2 N_ 1;2 t Je − 1;2 τN − G 1;2 tjE 1;2 tj :
2
G j t gN j t − N 0 ∕1 ϵjE j tj2 is the optical gain (with g the differential gain and ϵ the saturation coefficient), N 0 is the carrier density at transparency, Ψ 2πf 1 τf is the feedback phase, f 1 is the optical frequency of the solitary L1, Δf is detuning between L1 and L2, τp is the photon lifetime, τN is the carrier lifetime, and J is the pump current. We take τp 2 ps, τN 2 ns, g 1.5 × 10−8 ps−1 , N 0 1.5 × 108 , Ψ 0, τf 1 ns, f 1 1.935 × 1014 Hz, and J 1.5J th (with J th 14.7 mA the threshold current of the solitary laser). Identical internal parameters are taken for L1 and L2, and we neglect spontaneous-emission noise. In the following, we study the effect on the TDS of varying α, γ, σ, and Δf . The ACF is used to identify the TDS from the chaotic time series I t or ϕt. It measures how well a time series xt matches its time shifted by s version x s t [5]: hxt − hxtix s t − hx s tii Cs pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 3 hxt − hxti2 ihx s t − hx s ti2 i with h…i being the time (i.e., t) average. For PE [19], we take the time series fx t gt1;…;T and recon fx j ; x jτ ; …; x jD−1τ g struct a D-dimensional space X D;τ j for j 1; 2; …; T − D − 1τ, with D and τ the embedding dimension and embedding time delay, respectfully. Vector X j is constructed by arranging elements of fx t gt1;…;T in increasing order x jr 1 −1τ ≤ x jr 2 −1τ ≤ … ≤ x jr D −1τ, and any X j is uniquely mapped onto an ordinal pattern π r 1 ; r 2 ; …; r D out of D! possible permutations. For the permutations π of order D, the probability distribution P pπ of the ordinal patterns is #fjjj ≤ T − D − 1τ; X j has type πg 4 ; pπ T − D − 1τ where # means “the number of.” Next, the permutation entropy H P is evaluated based Pon the permutation probability distribution P as H P − pπ log pπ. Finally, the normalized PE hP is hP H P∕ log D!, with 0 ≤ h ≤ 1, and h 0 corresponds to predictable dynamics, h 1 corresponds to fully random dynamics and where all D! permutations appear with the same probability. We take D 5, τ τf ∕Ωs 103 (Ωs 1 ps), and T 105 [20]. We first consider an ECL. The ACF peak associated with the TDS is not located at τf but at a slightly larger value; this is attributed to the internal response time of the LD [20]. As a useful tool for identifying the delay, we define the amplitude of the maximum ACF peak in time window s ∈ 0.65; 1.35 ns, which we call the peak size (PS), and that indicates the strength of the TDS. We find (not shown here) that for low injection
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To characterize the degree to which τf is evident in the output, we compute the ACF of It or ϕt and call this the TDS. Although there are other ways to extract τf , we restrict our attention to measures based on second-order statistics [3]. I t and ϕt are computed from the Lang–Kobayashi (LK) equations [25]. The complex electric field and carrier density in Lj are E j and N j [26]; the last term in Eq. (1) is present only for L2 in OISLs: 1 iα G 1;2 t − τ−1 E_ 1;2 t p E 1;2 t 2 γE 1 t − τf e −iΨ σE 1 t − τc e −i2πf 1 τc −Δf t ; (1)
Peak size
Letter
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γ (ns−1) 10
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Fig. 2. (a) PS from I t and PE as functions of γ in ECLs. I t for (b) γ 8 ns−1 , (c) 20 ns−1 , and (d) 40 ns−1 for α 5, J 1.5J th .
J 1.05J th , although the PS based on I t does not reveal the system’s delay, the latter can be seen identified on the basis of the PS identified from the phase ϕt. Of more practical interest, ECLs under high J (henceforth 1.5J th ) operate in coherence collapse with moderate γ. PS based on I t and PE as functions of γ are shown in Fig. 2(a). The PS minimum occurs near the PE maximum (the dynamical regime around these extrema has been referred to as strong chaos [21,27]). Figure 3 shows ACFs from I t and ϕt for various α for J 1.5J th with γ 30 ns−1 . The TDS is pronounced in all cases. Though increasing α leads to a PS reduction, even for quite large α ∼ 8 [Figs. 3(c1) and 3(c2)], the TDS is still evident. We shift to OISLs, which also show high-speed chaos [28,29]. ACFs of I t and ϕt in OISLs are plotted in Fig. 4, with σ 15 ns−1 and Δf 20 GHz. Parameters are the same as in Fig. 3, unless noted. In contrast to the ECL, TDS both in I t and ϕt is now strongly suppressed with increasing α. Thus, moderate σ modifies the dynamics in OISLs compared with ECLs, leading to more complex chaos [30]. Interplay with the complexity enhancement delivered by larger α results in enhanced TDS suppression compared with the ECL, as seen in Fig. 5 in which 1 ≤ α ≤ 10 [7] with the laser stable for small α and showing chaos for large values. As shown in Figs. 5(a) and 5(b), for given Δf , PS depends weakly on σ [i.e., in all cases, as α is increased, PS from both I t and ϕt follows the same trend, first decreasing sharply and then slowly approaching a small value]. Similarly, Figs. 5(c) and 5(d), for given σ, PS depends weakly on detuning; in all
Fig. 3. ACF from (left) I t and (right) ϕt in ECLs for J 1.5J th , γ 30 ns−1 and (a1) and (a2) α 3, (b1) and (b2) 5, and (c1) and (c2) 8.
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Fig. 4. ACF from (left) I t and (right) ϕt in OISLs for J 1.5J th , γ 30 ns−1 , σ 15 ns−1 , and Δf 20 GHz for (a1) and (a2) α 3, (b1) and (b2) 5, and (c1) and (c2) 8. Note the different vertical scale compared with Fig. 3.
cases, PS from both It and ϕt decreases gradually and finally is stabilized at a small value. This further confirms that α has a significant effect on TDS in OISLs, and large α significantly improves the TDS suppression, in agreement with the observation in Fig. 4. Figure 6 shows globally how PS based on I t and ϕt varies with Δf and σ for various α. For small α [Figs. 6(a1) and 6(a2)], regions with strong TDS suppression (in white) are very narrow both for I t and ϕt. As α increases, these regions broaden, indicating robustness of TDS concealment in large-α OISLs. In contrast to ECLs, for OISLs, TDS tends to be better concealed in ϕt compared with It [i.e., regions corresponding to strong TDS suppression are slightly larger and the values of low PS are slightly smaller, especially for large α values (e.g., 5 or 8 in Fig. 6)]. Hence, from these results, we conclude that for the same parameter range (σ-Δf plane), as α increases, chaotic regions and even the low-PS regions are growing rapidly (almost exponentially [31]), indicating that TDS concealment can occur for large α. We next address coexistence of high PE and TDS suppression in high-α OISLs. In Figs. 7(a1)–7(c1), PE from I t shows that the maximum PE ∼0.99 of OISLs (L2) is modestly
Fig. 5. PS from (left) It and (right) ϕt in OISLs with (upper panels) Δf 20 GHz and (bottom panel) σ 25 ns−1 .
Fig. 6. PS from I t (left) and ϕt (right) as functions of σ and Δf in OISLs for (a1) and (a2) α 3, (b1) and (b2) 5, (c1) and (c2) 8.
enhanced compared to ECLs ∼0.96 (see Fig. 2), consistent with the additional degrees of freedom due to second laser L2. Comparing Figs. 6 and 7(a1)–7(c1) shows that regions of PE enhancement coincide with those for strong TDS suppression. We note that to simultaneously achieve low PS and high PE in OISLs, one should avoid injection locking and, thus, large σ values. The cross-correlation ρmax [see
Fig. 7. (left) PE and (right) ρmax between L1 and L2 as functions of Δf and σ in OISLs for (a1) and (a2) α 3, (b1) and (b2) 5, (c1) and (c2) 8.
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Vol. 40, No. 19 / October 1 2015 / Optics Letters
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to TDS concealment, may also be easily synchronized by a receiver R. Thus, OISLs may close a security vulnerability for chaos communications.
Funding. Conseil Regional of Lorraine (GT-CNRS2958); National Natural Science Foundation of China (NSFC) (61274042); Southwest Jiaotong University.
REFERENCES
Fig. 8. I t for (a) M and (b) S, and (c) correlation with α 5, γ 30 ns−1 , σ 15 ns−1 , Δf 20 GHz, and η 150 ns−1 . (d) Correlation ρmax between M and S as η varies.
Eq. (3) in [26] for the definition] between L1 and L2 in OISLs is helpful to determine, approximately, the locking region. Figures 7(a2)–7(c2) show ρmax for the same parameters as in Figs. 7(a1)–7(c1). They confirm that OISLs generate optical chaos with low TDS and high PE only outside the injectionlocking region. For large σ, one finds strong chaos synchronization (ρmax > 0.9) between L1 and L2. In this case, the dynamics of L2 are dominated by the injection, and thus, its output exhibits properties similar to the driving signal (opticalfeedback-induced chaos) from L1. Clearly, this is precisely what we wish to avoid, as in this case, the TDS signature in L1 is directly translated to L2. Outside this region, L2 allows for the generation of dynamics distinct from, and more complex than, L1 [30]; as a result, the simultaneous realization of strong TDS suppression and PE enhancement (strong chaos) is possible in OISLs. Last, we show that strong chaos in OISLs can be used for chaos communications as a receiver (R) can be synchronized with the output of L2. Emitter (E) is the OISL, and for R, we take an open-loop configuration [26] in which R is a solitary LD similar to L2. Detuning between L2 and R is zero. The LK equations for R are similar to Eqs. (1) and (2), but without the delayed feedback term involving E 1 t − τf . In Fig. 8, generalized synchronization under strong chaos is seen, showing correlation ρMS between I m t and I s t, where η is the injection rate from E to R. Good synchronization (ρMS 0.97) is achieved for high η 150 ns−1 as seen in Figs. 8(a)–8(d). Note that although E operates in strong chaos, it is possible to achieve good synchronization and thus allows informationbearing signals to be sent, with a simple receiving scheme consisting of a solitary laser L2 subjected to optical injection. To conclude, the effect of the linewidth-enhancement factor α on the time-delay signature and the permutation entropy is investigated for ECLs and OISLs. High α is favorable to TDS suppression but in ECLs, is insufficient for concealment. In OISLs, as α is increased, TDS both from I t and ϕt can be eliminated when the second laser is not injection locked by the first one. We also demonstrate that TDS suppression coincides with high PE over a wide range of parameters. Finally, we show that the strong chaos obtained, in addition
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