IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 51, NO. 9, SEPTEMBER 2015
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Simultaneous Suppression of Time-Delay Signature in Intensity and Phase of Dual-Channel Chaos Communication Amr Elsonbaty, Salem F. Hegazy, and Salah S. A. Obayya, Senior Member, IEEE
Abstract— In this paper, we propose a novel dual-channel optical chaos system with a time-delay (TD) feature simultaneously suppressed in all observables, i.e., in both intensity and phase. A hybrid optical and electro-optic feedback for a single verticalcavity surface-emitting laser (VCSEL) is verified to induce simultaneous TD suppression for the polarization-resolved components of the chaotic output. A comprehensive mathematical model is developed to incorporate the optical and electro-optic time delays into the rate equations of the VCSEL. The suppression of TD signature is then examined by means of autocorrelation function and delayed mutual information. The results show that the output chaotic signal has the TD feature well eliminated in both the intensity and phase over certain regions of parameter space identified using the peak signal to mean ratio technique. The independent evolution of the two orthogonal VCSEL modes renders the polarization-resolved output modes appropriate for the enhanced dual-channel chaos applications. To the best of our knowledge, this is the first time that a dual-channel chaos communication system is reported with simultaneous suppression of the TD feature in all the transmitted observables. Index Terms— Vertical-cavity surface-emitting laser (VCSEL), optical feedback, time delay (TD) suppression, dual-channel chaos communication.
I. I NTRODUCTION
T
HE optical chaos generated by a semiconductor laser (SL) subject to a delayed feedback has attracted considerable interest during the last two decades [1]–[3]. Its broadband spectrum, extreme sensitivity to initial conditions, and ability to establish synchronization between two remote systems make it the preferred candidate to physically obscure and encode mutual information signals in both time and frequency domains [4]–[9]. These features evoke further vital applications as ultra-fast physical random bit generation [10]–[12], Manuscript received May 13, 2015; revised July 9, 2015; accepted July 27, 2015. Date of publication August 7, 2015; date of current version August 26, 2015. A. Elsonbaty is with the Centre for Photonics and Smart Materials, Zewail City of Science and Technology, Giza 12588, Egypt, and also with the Department of Mathematics and Engineering Physics, Faculty of Engineering, Mansoura University, Mansoura 35516, Egypt (e-mail:
[email protected]). S. F. Hegazy is with the National Institute of Laser Enhanced Sciences, Cairo University, Giza 12613, Egypt, and also with the Centre for Photonics and Smart Materials, Zewail City of Science and Technology, Giza 12588, Egypt (e-mail:
[email protected]). S. S. A. Obayya is with the Centre for Photonics and Smart Materials, Zewail City of Science and Technology, Giza 12588, Egypt (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JQE.2015.2466176
chaotic radar [13], chaotic lidar [14], and optoelectronic logic gates [15]. High dimensional chaos generators are required to increase the complexity of chaotic attractor, hence to boost the security of chaos-based communication. Optical high-dimensional chaos can be generated by introducing one or more time delay (TD) terms into the rate equations describing the system or, from a physical point of view, by “actualizing” all-optical [16]–[18] and/or electro-optic [19], [20] feedbacks. However, the security of chaos encryption is not guaranteed only by the high dimensionality of the chaos [21] (Actually, the higher dimensionality does not in general lead to significantly more unpredictability [22], [23]), but also by preventing critical information about the chaos system from being leaked through the chaotic carrier. Along with the SL intrinsic parameters such as central wavelength and threshold current, the value of TD is considered one of the primary secret keys of the chaos system. An eavesdropper who could recognize the TD signature is, at least in principle, readily capable to reconstruct the chaotic system [24]–[27]. On the other hand, obvious TD signature directly diminishes the statistical performance of the random bit generation [10] as well as reduces the signal-to-noise ratio of the chaotic radar system [14]. Numerous schemes of all-optical feedback have been verified to effectively suppress TD signature in intensity time series. Among these schemes are all-optical feedback operating in short-external-cavity regime [28], double variablepolarization feedback [1], [3], double unbalanced optical feedbacks [29], [30], and distributed feedbacks [31]. However, as demonstrated by Nguimdo et al. [32], the TD information entirely hidden in the intensity time series can be still available in the phase time series. This breakthrough renders achieving effective suppression of TD signature more challenging. Recently, there have been some trials to simultaneously suppress TD information in all transmitted observables, i.e., both the intensity and phase. In a more recent paper [33], Nguimdo et al suggested some scenarios leading to simultaneous TD suppression by making use of semiconductor ring laser with a cross-feedback configuration. Also it has been shown that the chaos from two SLs with dual-path injection [34] as well as three SLs in a cascade scheme [35] exhibit simultaneous TD concealment for appropriate injection strength and frequency detuning. Relative to conventional edge-emitting SLs, vertical-cavity surface-emitting lasers (VCSELs) have a number of beneficial
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Fig. 1. Schematic diagram of the optical dual channel chaos generator. BS: beam splitter; VA: variable attenuator; HWP: half-wave plate; PBS: polarizing beam splitter; D: photodetector; G: electro-optic gain; PM: phase modulator; M: mirror; OI: optical isolator.
features such as low threshold current, low cost, on-wafer testing capability, circular output-beam profile (hence, high coupling efficiency to optical fiber) and easy large-scale integration into two-dimensional arrays [36]. It is still open and important to realize simultaneous TD concealment for the output of VCSEL subject to single external-cavity feedback. In this paper, we present a novel optical chaos system appropriate for dual-channel chaos communication with TD signature suppressed in both intensity and phase. We verify by comparison that the proper merge of optical and electro-optic feedbacks for single VCSEL induces simultaneous suppression of TD signature in both intensity and phase of the chaotic output. A mathematical model is developed for the proposed system incorporating the optical and electrooptic TDs into the rate equations of the VCSEL, taking into account the dynamics of the polarization interferometer embedded in the system. The independency of two chaotic carriers implemented by polarization-resolved outputs is also investigated. The regions of improved security are demonstrated by means of evaluating the peak signal to mean ratio (PSMR) of the resulting chaos over the parameter space. We study dual-channel synchronization and communication between transmitter and receiver systems while maintaining simultaneous TD suppression in intensity and phase over both channels. We investigate the sensitivity of the synchronization and the achievable bit error rate (BER) for some degrees of parameters mismatch. The paper is organized as follows: The optical system is described in Section II, derivation of mathematical model is introduced in Section III, the analysis of TD suppression is presented in Section IV, the description of dual-Channel communication system and the conclusion are included in Section V and Section VI, respectively. II. S YSTEM D ESCRIPTION Schematic diagram of the proposed system is illustrated in Fig. 1. Unlike most previous arrangements which adopt either optical, electro-optic, or a cascade of both feedbacks,
the proposed system fully merges the two schemes. Such hybrid feedback system combines versatile advantages like the high chaos dimension and high sensitivity to parameter mismatch by virtue of the electro-optic subsystem, as well as the difficulty of reproducing the source and the ability to mask the TD signature by virtue of the all-optical subsystem [37]. In Fig. 1, The VCSEL emission with two possible polarization modes (E x and E y ) is fed into a double-pass polarization interferometer biased by a phase modulator (PM). The angle between the y direction and the fast axes of the half-wave plates HWP1 and HWP2 is 22.5◦, therefore both HWP1 and HWP2 switch the linear polarization of the input signal from rectilinear to diagonal and vice versa. Therefore, the amplitude and the phase of the signal fed back from the polarization interferometer at time t are related to two previous portions of the input signal at different instants; In particular, the verticallypolarized signal fed at time t − 2τo2 (τo2 is optical delay period between PBS and M) and the horizontally polarized signal fed at time t − τe − τo2 (τe is the electronic delay of the electro-optic feedback). After passing the half-wave plate HWP1, the 45-deg polarized signal injects equivalent and coherent feedback signals into the two orthogonal operation modes of the VCSEL. Here, as will be shown in Section IV, the two orthogonal VCSEL signals E x and E y evolve totally independent. This renders the system appropriate to feed dualchannel chaos communication [38]. In the proposed system, a direct relation holds between the degree of interference visibility and the degree of chaos randomness. One problem that hurts the visibility of the electro-biased interferometer is the spatial and temporal distinguishability of the superimposed components due to the walkoff between the ordinary and extraordinary polarizations. Here, the output of the polarization interferometer is fully compensated for the spatial walkoff (i.e., transverse walkoff) by means of direct retroreflection into the same anisotropic medium. Further to avoid the temporal walkoff (i.e., longitudinal walkoff), the length of the nonlinear medium is considered sufficiently small, so that the group-velocity delay is far less than the coherence time of the VCSEL emission. III. M ATHEMATICAL M ODEL In previous works, either all-optical feedback [3], [16]–[18] or electro-optic feedback [19], [20] was considered in mathematical model of optical chaos generators. In [37], Hizanidis et al. presented a cascaded scheme of the two types of feedback yet they did not include the two orthogonal linear polarization (LP)-modes in rate equations of their system, and therefore the possibility of multi-channel communications are not considered. The proposed system integrates both types of feedbacks simultaneously in rate equations that model the layout of the system described in Section II and takes into account the separated two independent LP modes. Let θ pi denote the azimuthal angle of the the half-wave plate i , i = 1, 2 with the vertical direction y and assume that the electronic devices in the feedback loop have very large flat bandwidth, then we derive the following rate equations based
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TABLE I PARAMETERS OF THE M ATHEMATICAL M ODEL
on spin-flip model [39], [40]. d E x (t) = k (1 + i α) {[N(t) − 1]E x (t) + i n(t)E y (t)} dt − [ga +ig p ]E x (t)+ βsp ζx +g 1 { cos (2θ p1 ) × E y (t − τ o ) [ cos2 (2θ p2 ) sin (2θ p1 ) + ei(t −τe −2τo1 −τo2 ) sin (2θ p1 ) sin2 (2θ p2 )] + sin (2θ p1 )E x (t − τ o )[ cos2 (2θ p2 ) sin (2θ p1 ) + ei(t−τe−2τo1−τo2 ) sin (2θ p1 ) sin2 (2θ p2 )]}e−iω0 τo , d E y (t) = k (1 + i α) {[N(t) − 1]E y (t) − i n(t)E x (t) dt + [ga + ig p ]E y (t) + βsp ζ y + g2 {cos(2θ p1 )
(1)
× E y (t − τo )[cos(2θ p1 ) cos2 (2θ p2 ) + ei(t −τe −2τo1 −τo2 ) cos(2θ p1 ) sin2 (2θ p2 )] + sin(2θ p1 )E x (t − τo )[cos(2θ p1 ) cos2 (2θ p2 ) + ei(t −τe −2τo1 −τo2 ) cos(2θ p1 ) sin2 (2θ p2 )]}e−iω0 τo , (2) d N(t) 2 = g N {μ − N(t)[1 + |E x (t)|2 + E y (t) ] dt + i n(t)[E x (t) E¯ y (t) − E¯ x (t)E y (t)]}, (3) dn(t) 2 = −gs n(t) − g N {n(t)[|E x (t)|2 + E y (t) ] dt (4) + i N(t)[E y (t) E¯ x (t) − E¯ y (t)E x (t)]}, 2 (5) φ(t) = E y (t) sin(2θ p1) − E x (t) cos(2θ p1 ) , where subscripts x and y stand for horizontal and vertical LP modes, respectively, and the other parameters are described in Table 1. We solve (1)−(5) using the following VCSEL parameters values [3]: k = 300 ns−1 , α = 4, g N = 1 ns−1 , ga = 0.5 ns−1 , g p = 30 ns−1 , gs = 50 ns−1 , βsp = 10−6 ns−1 , μ = 4.5, we take and ω0 = 2.2176 × 1015 rad/s. For simplicity, g1 = g2 = g, θ pi = 22.5° , and βsp ζ y = βsp ζx = 0.
The complexity of the behavior of chaotic output time series makes it difficult to identify the TD signature directly from the time series. However, several statistical techniques can be used to identify the TD signature of chaotic output quantitatively such as the filling factor analysis [41], singular values fraction measure [42], autocorrelation function (ACF), delayed mutual information (DMI) [26]–[28], permutation entropy (PE) [43], [44], RF spectrum analyzer [24], [25], and local linear models [45]. Compared with other methods, ACF and DMI are robust, immune to noise, and highly computationally efficient [33], [46]. The ACF quantifies the tendency of time series to match its time-shifted version [46]. It can be defined for the time series of the intensity I (t) or the phase φ(t) of the chaotic signal as [r (t + t) − r (t + t)] [r (t)−r (t)] Cr ( t) = 1 1 , [r (t + t) − r (t + t)]2 2 [r (t)−r (t)]2 2 (6) where r (t) is either I (t) or φ(t), denotes the time average, and t is the lag time with 25 ps steps on [−100 ns,100 ns]. Suppose that τ ∗ is the actual time delay of SL chaotic system to be identified. Then, the absolute value of ACF in neighborhood of τ ∗ i.e. in Wg = [τ ∗ − τ ∗ × m, τ ∗ + τ ∗ × m], where m is defined as the mismatched coefficient, has obvious larger value than its values in other intervals due to the presence of TD signature. Therefore, the TD signature can be predicted from the location of the peak in ACF curve. PSMR is used to illustrate strength or weakness of TD signature mathematically. It can be computed for both types of ACFs as PSMRr =
max |Cr ( t)| t ∈Wg |Cr ( t)|
.
(7)
From the calculations of many ACFs for different cases of chaotic SL system, the authors in [3] found that the TD signature can be successfully identified when the value of PSMR is greater than 4.
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Fig. 2. The time series of (a1) the intensity and (b1) the phase of x LP mode and of (c1) the intensity and (d1) the phase of y LP mode with the corresponding (a2,b2,c2,d2) ACF and (a3,b3,c3,d3) DMI graphs. Obvious peaks (marked by arrows) are positioned at the values of τo and 2τo . Results are determined for the system in Fig. 1 in absence of the electro-optic feedback. We consider g = 50 ns−1 , τo1 = 0.5 ns, and τo2 = 1 ns.
Fig. 3. The time series of (a1) the intensity and (b1) the phase of x LP mode and of (c1) the intensity and (d1) the phase of y LP mode with the corresponding (a2,b2,c2,d2) ACF and (a3,b3,c3,d3) DMI graphs. These results are determined when employing both the optical and electro-optic feedbacks (electro-optic time delay τe = 5.94 ns, other parameters unchanged). The ACF and DMI exhibit lower peaks compared with those in Fig. 2.
Further, we calculate the DMI [46] for intensity and phase time series as follows
by arrows) which enables the extraction of values of these parameters either from the intensity or the phase time series for both x and y LP modes. As the second scenario, we investigate the performance of the system when employing both optical and electro-optic feedbacks while keeping all parameters as in the first scenario and taking τe = 5.94 ns. Figure 3 displays lower peaks positioned at τo in ACF and DMI plots whereas the TD is suppressed in the phase of x LP-mode. Further, there is no significant peaks can be identified at the particular values of τe , τo1 , or τo2 in either the intensity or phase time series. Therefore, the proposed configuration enhances the security of the system via compensating for the undesirable influences of high feedback strengths and hiding some of the key parameters of the system. To imagine the full picture of the influence that the two crucial parameters g and τo have on the system performance, we utilize PSMR as a measure parameter with m = 5% at the neighborhood of each value of TD in the system, i.e. τo , τo1 , τo2 , and τe . We then take the largest value of PSMRs corresponding to different TDs. PSMR is therefore used to deduce the optimal regions of (g, τo ) parameter space at which the TD signature of the system is concealed or maintained very weak in both intensity and phase of x and y LP-modes. This goal can be achieved via adopting threshold values of PSMR under which the TD feature is considered to be effectively concealed. Simulation results in Fig. 4 show that TD signature can be hidden in intensity while it is still obvious in phase and vice versa (as was reported in [32]). After exhaustive numerical examination of threshold values of PSMR, we set the values 4.6 and 3.5 as the thresholds for the cases of g < 12 ns−1 and 12 ns−1 < g < 30 ns−1 , respectively. Alternatively, when PSMR is calculated for phase time series of x LP-mode [see Fig. 4(b)], it is found that 2.95 is an appropriate threshold value. In Fig. 4, examples for the optimal regions where TDs cannot be retrieved using the intensity or the phase time series are (4 ns ≤ τo ≤ 7 ns, 12 ns−1 ≤ g ≤ 16 ns−1 ) and
Dr ( t) = r(t ),r(t+ t )
P [r (t), r (t + t)] ln
P [r (t), r (t + t)] , (8) P [r (t)] P [r (t + t)]
where P [r (t)] is the marginal probability distribution function of r (t) and P [r1 (t), r2 (t)] is the joint probability distribution function. The peak location of the DMI identify the TD signature. IV. TD S UPPRESSION AND C ORRELATION A NALYSIS It is known that the TD signature can be suppressed in the intensity time series when the feedback levels are moderately weak and the external cavity roundtrip time is close to the relaxation oscillation period [26], [28], where in our case 1
τ R O ≈ 2π/ [2kg N (μ − 1)] 2 = 0.13711 ns. However, it was reported that a maximum of one positive Lyapunov exponent is obtained at very small values of time delay with the number of positive Lyapunov exponents increases as the feedback delay increases [22]. For our proposed system, we find that there is at most one positive Lyapunov exponent for τo < 87 ps. Therefore, it is important to investigate the suppression of TD signature at values of τo much greater than this threshold value which leads to higher dimensional and more complex chaotic behavior of the output signal. Here, we consider several scenarios to check the possibility to retrieve information about the TD features; namely τo , τo1 , τo2 , and τe . It was reported that high feedback strength weaken the security of chaotic system [26]. Let us consider, as the first scenario, large value of feedback strength g and examine the ACF and DMI for the intensity and phase time series of x and y LP modes in absence of the electro-optic feedback. Taking g = 50 ns−1 , τo1 = 0.5 ns and τo2 = 1 ns, Fig. 2 shows clear peaks at the values of τo and 2τo (marked
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Fig. 6. Power spectrum for (a) output intensity, (b) output phase at g = 23 ns−1 , τo = 5 ns, τo1 = 0.8333 ns, τo2 = 1.666 ns, and τe = 23.25 ns. Single obvious peak appears at the relaxation oscillation frequency f RO = 7.2934 GHz. The spectrum in the zooming insets of (a) and (b) fluctuates in a random manner without regular enveloping behavior, as a result, no TD information can be concluded. Fig. 4. The PSMR computed from (a) intensity time series and (b) phase time series of x LP-mode, and from (c) intensity time series and (d) phase time series of y LP-mode. Through the computational analysis, we consider τo1 = (1/6)τo , τo2 = (1/3)τo , and τe = 23.25 ns.
Fig. 7. Chaotic phase portraits of the (a) intensity time series |E x |2 and |E y |2 , (b) phase time series Arg(E x ) and Arg(E y ). DMI graphs for (c) |E x |2 and |E y |2 , and for (d) Arg(E x ) and Arg(E y ). Fig. 5. The time series of (a1) the intensity and (b1) the phase of x LP mode and of (c1) the intensity and (d1) the phase of y LP mode with the corresponding (a2,b2,c2,d2) ACF and (a3,b3,c3,d3) DMI graphs. All results are determined at one of the regions offered by Fig. 4 with simultaneous TD suppression in intensity and phase. Here, we consider g = 23 ns−1 , τo = 5 ns, τo1 = 0.8333 ns, τo2 = 1.666 ns, and τe = 23.25 ns.
(4 ns ≤ τo ≤ 5 ns, 21 ns−1 ≤ g ≤ 25 ns−1 ). This shows that the TD signatures can be concealed or become very weak for values of TD beyond the relaxation oscillation period τ R O . As the third scenario, we modify the system parameters to be within one of the regions that exhibit simultaneous TD suppression. We consider g = 23 ns−1 , τo = 5 ns, τo1 = 0.8333 ns and τo2 = 1.666 ns while the electro-optic time delay τe = 23.25 ns. The ACF and DMI results illustrated in Fig. 5 depict well suppression of TD feature simultaneously in intensity and phase of x and y LP-modes. It was reported that the TD signature can also be recognized in the power spectrum of the chaotic signal as a peak unveiling the external-cavity resonance frequency [24]. We investigate the power spectrum of intensity and phase time series at the parameters values of the third scenario (see Fig. 6). It is noticed that a single clear peak frequency emerges in intensity spectrum at the relaxation oscillation frequency f R O = τ R−1O = 7.2934 GHz. On the other hand, the TD can also be identified by observing the fine spectral structure of the chaotic signal [24].
The insets in Fig. 6(a) and Fig. 6(b) show narrow frequency ranges in intensity and phase spectrum, respectively. It can be noticed that the spectrum in each case fluctuates in a random manner such that no regular enveloping behavior (as was observed in [24]) takes place and the frequency separation between local peaks is not at a constant value. To show that the proposed system is capable to generate two independent chaotic carriers, we illustrate in Fig. 7(a) and Fig. 7(b) the phase portraits of the intensity and the phase of the two orthogonal LP-modes. The mutual DMI is employed to ensure that each LP mode does not in any way match a delayed version of the other one for both the intensity and the phase of the two signals. The DMI plotted in Fig. 7(c) and Fig. 7(d) shows that the two LP modes have not any type of nonlocal-time dependence. It is worth observing that the mutual statistical behavior represented by the attractor in Fig. 7(a) exhibits the weak anticorrelation nature of the two orthogonal modes. Therefore, we have demonstrated that the polarizationresolved outputs E x (t), E y (t) act as if from two independent chaos sources whose TD signature suppressed in both intensity and phase. This renders the output signals appropriate for various dual-channel applications like generation of two independent random bit sequences or, as explained in the next section, secure encoding of two simultaneous and independent messages.
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stands for the output of Tx-VCSEL. Also, consider that τtr is the propagation delay time between the transmitter and receiver systems. To achieve perfect decoding, the feedback delay time between the synchronized signal at the output of R (t) and the transmitted signal E T (t) x,y the Rx-VCSEL E x,y should be compensated before the decoding process. The rate equations that describe the receiver part can be written as d E xR (t) dt = k(1 + i α){[N R (t) − 1]E xR (t) + i n R (t)E yR (t)} − [ga +ig p ]E xR (t)+ βsp ζx yT (t − τ o − τtr ) [ cos2 (2θ p2 ) sin (2θ p1 ) + g{ cos (2θ p1 ) E + ei
T (t −τ −2τ −τ −τ ) e tr 01 o2
sin (2θ p1 ) sin2 (2θ p2 )]
xT (t − τ o −τtr )[ cos2 (2θ p2 ) sin (2θ p1 ) + sin (2θ p1 ) E + ei Fig. 8. Schematic diagram of dual-channel communication system. Tx-VCSEL: transmitter VCSEL; Rx-VCSEL: receiver VCSEL; OC: optical circulator.
V. D UAL -C HANNEL C OMMUNICATION S YSTEM If two simultaneous and independent messages m x (t), m y (t) are encoded using the same chaotic carrier, the subtraction (or division) of the two encoded signals cancels out the chaotic carrier which can reveal critical information on the secret messages. Two uncorrelated chaotic carriers whose TD feature suppressed in all observables are thus indispensable to secure dual-channel communications. Figure 8 depicts a schematic diagram for proposed transmitter and receiver systems based on the chaos generator shown in Fig. 1. The transmitter encodes the two independent messages m x (t), m y (t) into the polarization-resolved components of the Tx-VCSEL emission using additive chaos modulation. In contrast to other encoding schemes, the chaos modulation does not require the message to have very small amplitude compared with the chaotic carrier [47]. After PBS2, the recombined signal passes through the optical circulator to the polarization interferometer system where it is modulated and consequently fed back into the Tx-VCSEL. A portion of the recombined signal is transmitted over the public channel to the receiver system. The structure of the receiver is identical to that of the transmitter. The Rx-VCSEL, being injected by an optical signal strictly similar to that of the Tx-VCSEL, emits chaotic signals in x and y LP modes very close to those of the Tx-VCSEL. The polarization-resolved components after PBS1 in the receiver are thus distorted versions of the signals after PBS1 in the transmitter [before adding the messages m x (t), m y (t)]. The two messages can be then decoded by subtracting the intensities of the relevant LP modes before encoding (measured by photodetectors D2 , D4 ) from those after encoding (measured by photodetectors D3 , D5 ). Let the signals transmitted on the public channel be encoded by the simultaneous signals m x (t) and m y (t) as in the form T (t) = E T (t) 1 + m x,y E x,y (t) , where the superscript T x,y
T (t −τ −2τ −τ −τ ) e tr 01 o2
sin (2θ p1 ) sin2 (2θ p2 )]}
× e−iω0 (τo +τtr ), d E yR (t)
(9)
dt = k(1 + i α){[N R (t) − 1]E yR (t) − i n R (t)E xR (t) + [ga + ig p ]E yR (t) + βsp ζ y + g{cos(2θ p1 ) yT (t − τo )[cos(2θ p1 ) cos2 (2θ p2 ) × E + ei(t −τe −2τo1 −τo2 ) cos(2θ p1 ) sin2 (2θ p2 )] xT (t − τo )[cos(2θ p1) cos2 (2θ p2 ) + sin(2θ p1 ) E + ei(t −τe −2τo1 −τo2 ) cos(2θ p1 ) sin2 (2θ p2 )]}
×e
d N R (t) dt
−iω0 (τo +τtr )
,
2 2 = g N {μ − N R (t)[1 + E xR (t) + E yR (t) ]
(10)
+ i n R (t)[E xR (t) E¯ yR (t)− E¯ xR (t)E yR (t)]}, (11) 2 2 dn R (t) = −gs n R (t) − g N {n R (t)[E xR (t) + E yR (t) ] dt (12) + i N R (t)[E yR (t) E¯ xR (t) − E¯ yR (t)E xR (t)]}, where the superscript R stands for the output of the Rx-VCSEL and τtr = 5ns is used for numerical simulation. There are various types of synchronization that have been investigated for optical chaos systems, e.g., identical, phase, and generalized synchronization [47]. Here we examine the identical synchronization which can be specified R (t) = E T (t − τ ). We use the crossby the relation E x,y tr x,y correlation of the output time series of the transmitter and receiver to demonstrate quantitatively the quality of chaos synchronization, where the peak cross-correlation value (positioned close to t = τtr ) is a good measure for the degree of chaos synchronization. To achieve identical chaos synchronization, the transmitter and receiver systems need to have fully matched parameters. However, from physical viewpoint, there may be slight mismatches in some parameters. It is therefore essential to investigate the influence of mismatching in internal and external parameters on the quality of synchronization for both x and y LP channels. Let us mismatch in relative define the any parameter ξ as ξ = ξ R − ξ T /ξ T × 100 (%). The
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Fig. 11. The BER of the proposed system for (a) x LP channel and (b) y LP channel versus mismatches in a set of system parameters. Fig. 9. Maximum cross-correlation between x−LP outputs (left column) and y−LP outputs ((right column)) of synchronized Tx-VCSEL and Rx-VCSEL versus mismatches in a set of internal (first row) and external (second row) parameters.
of the system parameters as depicted in Fig. 11. It is worth observing that the y LP channel exhibit generally higher robustness against parameters mismatch than that of x LP channel. VI. C ONCLUSION
Fig. 10. Original and recovered messages corresponding to x−LP channel (left column) and y−LP channel (right column) for 1% mismatch in the parameter k (second row) and 1% mismatch in the parameters gs , gn , k, α, τ0 , and g (third row).
numerical results in Fig. 9 emphasize that in order to maintain the quality of chaos synchronization, it is necessary for the transmitter and receiver systems to keep an appropriately small degree of parameters mismatch. Further, it is obvious that the synchronization process is less sensitive to mismatch in the parameters gs , gn , and k than in the parameters g and α. A pseudo random digital message with bit rate of 1 Gb/s is used in the numerical simulations. The decoded message is then obtained by low-pass filtration of the intensity difference between the polarization-resolved versions of the received signals and the output of the Rx-VCSEL. Figure 10 illustrates the encoded and decoded messages in two illustrative situations. In the first one, we assume 1% mismatch in the parameter k [see Fig. 10(b1), (b2)]. In the second, we address the more severe (yet realistic) situation when applying 1% mismatch in all internal and external parameters considered [see Fig. 10(c1), (c2)]. Finally, we examine the bit error rate (BER) of the proposed communication system assuming that the imperfect synchronization between transmitter and receiver is the only source of bit errors. In our calculations, we treat the synchronization error (which is mainly due to the parameter mismatch) as an additive noise perturbing the decoded messages. The BER can be then determined as a function of the individual mismatches
In this paper, simultaneous loss of TD signature in the intensity and phase time series of the chaotic output of single VCSEL subject to merged optical and electro-optic feedbacks has been reported. It has been shown that the suppression of TD feature in intensity and phase can be achieved for values of TD greater than the relaxation oscillation period over certain regions of parameter space. We have used the PSMRs values as an indicator for the strength and the weakness of TD signature where specified threshold values identify the optimal operation regions of parameter space. The numerical simulation results illustrate that TD features are not related to the feedback strength in a monotonic way, as shown in Fig. 4, which agree with recent experimental results [1]. The local and nonlocal time independence of the two orthogonal LP modes verifies that the proposed system is readily appropriate for various dual-channel chaos applications. We have presented a communication system utilizing the orthogonal VCSEL modes to establish TD-signature-suppressed secure dualchannel communication between two parties. A number of quantitative measures relevant to such issue have been investigated like the sensitivity of the synchronization and the achievable BER for some degrees of parameters mismatch. R EFERENCES [1] H. Lin, Y. Hong, and K. A. Shore, “Experimental study of time-delay signatures in vertical-cavity surface-emitting lasers subject to doublecavity polarization-rotated optical feedback,” J. Lightw. Technol., vol. 32, no. 9, pp. 1829–1836, May 1, 2014. [2] Y. Hong, P. S. Spencer, and K. A. Shore, “Wideband chaos with timedelay concealment in vertical-cavity surface-emitting lasers with optical feedback and injection,” IEEE J. Quantum Electron., vol. 50, no. 4, pp. 236–242, Apr. 2014. [3] P. Xiao et al., “Time-delay signature concealment of chaotic output in a vertical-cavity surface-emitting laser with double variable-polarization optical feedback,” Opt. Commun., vol. 286, pp. 339–343, Jan. 2013. [4] Z. Kang, J. Sun, L. Ma, Y. Qi, and S. Jian, “Multimode synchronization of chaotic semiconductor ring laser and its potential in chaos communication,” IEEE J. Quantum Electron., vol. 50, no. 3, pp. 148–157, Mar. 2014. [5] V. Annovazzi-Lodi, G. Aromataris, and M. Benedetti, “Multi-user private transmission with chaotic lasers,” IEEE J. Quantum Electron., vol. 48, no. 8, pp. 1095–1101, Aug. 2012.
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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 51, NO. 9, SEPTEMBER 2015
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Amr Elsonbaty was born in Egypt in 1984. He received the B.Sc. degree in electronics and communications engineering and the M.Sc. degree in engineering mathematics from Mansoura University, Egypt, in 2006 and 2011, respectively, and the Ph.D. degree in engineering mathematics from Mansoura University, in 2015. Since then, he has been an Assistant Professor with the Faculty of Engineering, Mansoura University, and also a Post-Doctoral Researcher with the Centre for Photonics and Smart Materials, Zewail City of Science and Technology. His current research interests include nonlinear dynamics of electronic circuits, chaos control and synchronization, chaos generation utilizing lasers, and chaotic optical communication systems.
ELSONBATY et al.: SIMULTANEOUS SUPPRESSION OF TD SIGNATURE IN INTENSITY AND PHASE
Salem F. Hegazy was born in Hulwan, Egypt, in 1980. He received the B.Sc. degree in electronics and communications engineering, the Diploma degree in optical systems and optical communications, and the M.Sc. degree in freespace communications with entangled photons from Cairo University, Egypt, in 2002, 2006, and 2011, respectively, where he is currently pursuing the Ph.D. degree in generation and compensation of photonic entanglement and in quantum cryptography. He is currently a Researcher with the National Institute of Laser Enhanced Sciences, Cairo University, and also with the Centre for Photonics and Smart Materials, Zewail City of Science and Technology. His research interests include entanglement generation and manipulation, optical chaos systems, optical delay control systems, quantum cryptography, and quantum optics experiments.
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Salah S. A. Obayya (SM’05) received the B.Sc. (Hons.) and M.Sc. degrees in electronics and communications engineering from Mansoura University, Egypt, in 1991 and 1994, respectively, and the Ph.D. degree from Mansoura University, in 1999. From 1997 to 1999, he was with the Department of Electrical, Electronic, and Information Engineering, City University London, London, U.K., to carry out the research part of his Ph.D. under the joint supervision scheme between the City University London and Mansoura University. In his Ph.D. dissertation, he developed a novel finite element- based full-vectorial-beampropagation algorithm for the analysis of various photonic devices. From 2000 to 2003, he was a Senior Research Fellow with the School of Engineering, City University London. From 2003 to 2006, he was with the School of Engineering and Design, Brunel University, London, as a Senior Lecturer, and a Reader with The School of Electronic and Electrical Engineering., University of Leeds. Since 2008, he has been a Full Professorial Chair of Photonics and led the establishment of the Nano-Photonics Research Centre with The University of Glamorgan, U.K., while he is currently the Vice Chair for Academic Affairs, Zewail City of Science and Technology, Egypt. He has built an outstanding international reputation in the area of green nanophotonics with a focus on the intelligent computational modeling of modern nanophotonic devices enabling technologies for efficient generation, distribution, and utilization of sustainable energy toward low-carbon green society. He has a track record of 123 journal publications mostly in the IEEE and IET/IEE, authored two books by the world-leading scientific publisher Wiley in 2010, and 165 conference presentations with many of them invited talks, presentations and keynote lectures. In five years, he has successfully supervised 20 Ph.D. students to completion with Leeds University, U.K., South Wales University, U.K., and Mansoura University. The team led by Prof. Obayya has developed one of the world best comprehensive numerical package for the analysis, design and optimization of nanophotonic devices, and subsystems for low-carbon sustainable energy issues, where a number of world-first numerical approaches have been developed. He has served the International Photonics Society through the active contribution to the organization and technical committees of a number of international conferences. Since 2007, he has served as the Associate Editor-in-Chief of the IEEE P HOTONICS T ECHNOLOGY L ETTERS , the Guest Editor-in-Chief of the Journal of Optical and Quantum Electronics of the Special Issues on 18th International Workshop on Optical Waveguide Theory and Numerical Modeling (University of Cambridge 2010), and the Associate Editor-in-Chief of the ISRN Journal of Applied Mathematics. He was elected as a fellow of the Institute of Engineering and Technology in 2010, and the Institute of Physics in U.K. in 2010. He is a member of the Wales Institute of Mathematical and Computational Sciences of the Computational Modeling Cluster and a member and Promotion Panel to Full Professor with Frederick University, Cyprus, and King AbulAziz City for Science and Technology, Saudi Arabia. He has a record of research funding (in the region of £2M) from EPSRC, the Royal Society, EU-FP7, and industrial partners, such as QinetiQ (Malvern), Fujitsu, Filtronics, RFMD Ltd., NDT Consultants, National Telecommunication Regulatory Authority, Egypt, and Vodafone, Egypt.
