SINCE the idea of chaos synchronization was first predicted

Transcription

1 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 29, NO. 14, JULY 15, Unpredictability-Enhanced Chaotic Vertical-Cavity Surface-Emitting Lasers With Variable-Polarization Optical Feedback Shui Ying Xiang, Wei Pan, Bin Luo, Lian Shan Yan, Xi Hua Zou, Ning Jiang, Lei Yang, and Hong Na Zhu Abstract Variable-polarization optical feedback (VPOF) induced unpredictability enhancement in a vertical-cavity surface-emitting laser (VCSEL) is investigated numerically based on the spin-flip model. The chaotic unpredictability is evaluated quantitatively via an information-theory-based quantifier, the permutation entropy (PE). The role of polarizer angle on the chaotic unpredictability is focused on. The influences of feedback strengths, feedback delays, and injection currents are also considered. A critical polarizer angle, at which the PE reaches its maximum, is existed for relatively high feedback strength and injection current. The representations on Poincaré sphere are further given to provide physical insight into the unpredictability enhancement. Besides, larger feedback strength leads to lower critical polarizer angle, while larger injection current contributes to higher critical polarizer angle. These results show that, by selecting critical polarizer angles, the unpredictability of chaotic signals of VCSELs with VPOF can be enhanced significantly, which is extremely useful for VCSELs-based chaotic communication systems. Index Terms Chaotic unpredictability, permutation entropy, variable-polarization optical feedback, vertical-cavity surface-emitting lasers. I. INTRODUCTION SINCE the idea of chaos synchronization was first predicted by Pecora and Carroll [1], optical chaos generated by semiconductor lasers has attracted extensive attention for their potential applications in secure communications [2] [5]. Among semiconductor lasers, vertical-cavity surface-emitting lasers (VCSELs) are regarded as promising candidates for chaotic sources in optical communication systems [6] [8], owing to their several advantages such as low threshold current, circular, weakly diverging output beam, and easy production Manuscript received March 02, 2011; revised April 28, 2011; accepted May 16, Date of publication May 23, 2011; date of current version July 20, This work is supported in part by the National Natural Science Foundation of China under Grant , in part by the Fundamental Research Foundation of Sichuan Province under Grant 2011JY0030, in part by the Fundamental Research Funds for the Central Universities under Grant 2010XS18, and in part by the Funds for the Excellent Ph.D. Dissertation of Southwest Jiaotong University, The authors are with the School of Information Science and Technology, Southwest Jiaotong University, Chengdu, Sichuan , China ( jxxsy@126.com; weipan80@sina.com; Butterfly714@sina.com; lsyan@home.swjtu.edu.cn; zouxihua@126.com; swjtu_robin@sina.com; yanglei @sohu.com; zjlzhn@163.com). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /JLT of two-dimensional arrays. Moreover, it is known that VC- SELs may exhibit polarization switching (PS) between two orthogonal linearly polarized (LP) modes by modifying bias current or device temperature or by applying additional degrees of freedom [9] [15]. VCSELs-based chaotic synchronization and communication systems have been studied intensively in recent years [6] [8], [16] [19]. For instance, Gatare et al. studied theoretically the effect of polarization mode competition on the synchronization of two unidirectionally coupled VCSELs [17]; they predicted that the synchronization quality could be enhanced when the chaotic regime in the master VCSELs involved both fundamental orthogonal LP modes, which has also been verified experimentally by Hong et al. [18]. In addition, message encoding and decoding using unidirectionally coupled chaotic VCSELs were also demonstrated experimentally [6], a GHz message was successfully encoded in the chaotic transmitter and decoded from the receiver with polarization-preserved optical injection. A crucial issue in all encryption techniques is the security and its relationship with the controllable parameters. The security of data encryption using chaotic method relies upon, to a large extent, the unpredictability of chaotic carrier [20]. To quantitatively evaluate the chaotic unpredictability, many effective quantifiers have been presented, such as the Lyapunov exponents (LE), the Kaplan-Yorke dimension, the Kolmogorov- Sinai entropy, and the correlation dimensions, etc. [20] [23]. Recently, two information-theory-based quantifiers, the permutation entropy (PE) [24], and the statistical complexity measure [25] [27], have also been adopted to quantify the complexity of time series generated by semiconductor lasers [28] [33]. Among these quantifiers, the PE proposed by Bandt and Pompe (BP) can be easily calculated for any type of time series, and behaves similar to the LE for some well-known chaotic systems [24]. The key advantages of the PE are its simplicity, extremely fast calculation, robustness to noise, etc. Moreover, it was indicated that the PE based method would be up to 100 times faster than a LE based method [34]. Upon these advantages, we adopt the PE as chaotic quantifier in this work. The growing interests in realizing chaos communication using VCSELs motivate further investigations on the unpredictability properties to select optimal parameters for chaotic carrier. Recently, the variable-polarization optical feedback (VPOF), which was obtained by including a rotating polarizer (RP) in the external cavity, was used as a feedback scheme in VCSELs [15]. It was found that, when polarizer angle was varied, the PS occurred even for fixed feedback strength and /$ IEEE 转载

2 2174 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 29, NO. 14, JULY 15, 2011 re-injected feedback lights are along the XP mode and along the YP mode, which denotes a purely YP mode feedback scheme. Then we extend the SFM to take into account the VPOF in VCSELs as follows [10], [11], [14], [15] Fig. 1. (Color online) Schematic illustration of VCSEL with RP in the external cavity. VCSEL: vertical cavity surface emitting laser; ML: microscopic lens; BS: beam splitter; RP: rotating polarizer; M: mirror; NDF: neutral density filter; ISO: optical isolator. (1) injection current, which indicated the VPOF being a new feedback scheme to obtain the controllable PS in VCSELs. However, since the quantitative analyses on the chaotic properties of VCSELs with VPOF have not been reported, it is still extremely meaningful to explore whether higher degree of chaotic unpredictability can be obtained by using the VPOF than using conventional feedback schemes. This work addresses specifically this issue. The degrees of chaotic unpredictability for VCSELs with VPOF are evaluated quantitatively by using the PE. The remainder of this paper is organized as follows. In Section II, a theoretical model that describes the VCSELs with VPOF is derived from the spin-flip model (SFM) [10], and the definition of the PE is introduced. In Section III, the chaotic unpredictability properties for VCSELs with VPOF are discussed in detail, and the effects of polarizer angle are explored. The influences of the feedback strengths, feedback delays, and injection currents are also considered. Finally, conclusions are drawn in Section IV. II. THEORY AND MODEL A. Rate Equation Model for VCSELs With VPOF The schematic diagram for VCSEL with VPOF is shown in Fig. 1. Similar to [15], [35], the VPOF is obtained by means of introducing a RP in the extended cavity. From the view of experimental devices, the piezoelectric nanopositioning stage utilized in [36] may be a promising candidate to control and measure the polarizer angles. In such a scheme, the output of VCSEL passes through the RP and then re-injects into VCSEL. Here we adopt a ring-loop model to avoid multiple-reflection occurring in the direct feedback schemes. The feedback strength can be adjusted by neutral density filter (NDF). We define 0 polarization to be the x-polarization (XP) mode of VCSEL and 90 polarization to be the y-polarization (YP) mode. Supposing that the transmission axis of the polarizer (i.e., RP) is oriented at a polarizer angle with respect to the XP mode of VCSEL which can be rotated between 0 and 90. Thus, the lights re-injected into XP and YP modes can be expressed as, and [15], where is feedback delay. Especially, when, the re-injected feedback lights are along the XP mode and along the YP mode, which represents a purely XP mode feedback scheme. When, the Where and are the slowly varying amplitudes of the XP and YP components. is the total carrier inversion between conduction and valence bands, while accounts for the difference between carrier inversions with opposite spins. is the field decay rate, is the decay rate of is the spin-flip rate, is the linewidth enhancement factor, is the normalized injection current ( is at threshold), is the center frequency of VCSEL, is the feedback strength, is the linear birefringence and is the linear dichroism. is the strength of the spontaneous emission, and are two independent Gaussian white noise sources with zero mean and unit variance. We numerically solve the equations by fourth order Runge Kutta method by using an integration step of 0.5 ps with typical VCSEL parameters [11], [15]: ns ns ns ns ns nm, ns. These parameters are chosen so that the VCSEL emits XP mode and the initial condition is also chosen in the XP mode. The time interval used for calculation is ns ns. B. Information-Theory-Based Quantifier: Permutation Entropy The PE proposed by BP [24] is performed based on the comparison of neighboring values, and can be easily calculated for any type of time series. To illustrate the idea of PE method [24], let us embed a given time series to a -dimensional space, where is the embedding dimension (delay). For practical purpose, BP suggested using with and indicated that the condition should be satisfied to obtain a reliable statistic. The can be arranged as an increasing order of, and when (2) (3) (4)

3 XIANG et al.: UNPREDICTABILITY-ENHANCED CHAOTIC VCSELS 2175 we order the quantities as, if. Hence, any vector is uniquely mapped onto an ordinal pattern, which is one of permutations of distinct symbols. For all the possible permutations of order, the probability distribution of the ordinal patterns is defined by [24] In this expression the symbol # stands for number. Then the PE is evaluated for this permutation probability distribution, and is given by The normalized PE is further presented as [28] [33] (5) (6) (7) where, and represents the uniform distribution. Obviously, large indicates high degree of unpredictability, which is a highly desired property to ensure security in a chaos encryption scheme, and we have, with corresponding to a regular, predictable dynamic, and to a fully random and unpredictable one where all permutations appear with the same probability. In this work, the sampling time is set to be ps as in [32], [33], and the permutation probability distribution is determined with as suggested in [24], [28] [33], unless otherwise stated. Fig. 2. (a) Polarization-resolved intensity as a function of, (b) FP as a function of, (c) PE for the total output as a function of, (d) PE for the XP mode and total output as functions of, (e) PE for the YP mode and total output as functions of, for VCSEL with =20ns. III. RESULTS AND DISCUSSION In this section, the roles of polarizer angle on chaotic unpredictability properties are focused on, the effects of feedback strength, feedback delay, and injection current are considered. Moreover, to obtain the statistical significance of the results, a statistical average over different windows is performed when calculating, each time series is divided into several disjoint windows with points. To begin with, we fix the feedback strength as ns, and show a first sight of the PE properties of VC- SELs with VPOF. To understand better the PE properties, a fractional polarization (FP) is also introduced to indicate the degree of polarization. For a given pair of field amplitude components and, the FP is defined by four Stokes parameters in Cartesian coordinate system as [11], [15]:, where denotes time average,. The value of FP ranges from 0 (natural unpolarized light) to 1(polarized light). The polarization- resolved intensities, FP and PE as functions of polarizer angle are shown in Fig. 2. In Fig. 2(a), the intensities are defined as and. It can be seen that, as is increased from to, an abrupt PS from XP mode to YP mode occurs, and FP reaches its minimum when the XP and YP modes having comparable intensities at Fig. 3. Outputs in the time (left column) and frequency (right column) domains for VCSEL with =20ns ; = =50, (a), (d) XP mode, (b), (e) YP mode, (c), (f) total output.. We denote the polarizer angle at which FP reaches its minimum as critical polarizer angle. Obviously, when, the YP (XP) mode is seriously suppressed, and the XP (YP) mode is the dominant polarization mode. To show better the dynamics of VCSEL at, we further present the outputs in the time and frequency domains in Fig. 3. It can be seen that, both XP and YP modes coexist with comparable intensities and exhibit chaotic dynamics as the total output. Without loss of generality, we also show the time series

4 2176 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 29, NO. 14, JULY 15, 2011 Fig. 5. (Color online) The evolutions of polarization state plotted on the normalized Poincaré sphere for VCSEL with = 20 ns, (a) = 10 (b) = =50. Fig. 4. Outputs in the time domain for VCSEL with =20ns and = 10 (left column), = 80 (right column), (a), (d) XP mode, (b), (e) YP mode, (c), (f) total output. of VCSEL with other values of. The outputs in the time domains of VCSELs with two representative polarizer angles, i.e., and are shown in Fig. 4. Obviously, when, the intensity of YP mode is much smaller than that of XP mode, while for, the XP mode is seriously suppressed. Note that, when, the intensity of YP (XP) mode is quite small, which makes the YP (XP) mode be not suitable for chaotic carrier. To convenient, in the following study we denote the PE for the XP mode, YP mode and total output of VCSEL as, and, respectively. It can be seen from Fig. 2(c) that, as the polarizer angle is increased, behaves in the reverse manner as the FP, it keeps around a constant level labeled as for the range of, then increases significantly and reaches the maximum labeled as at, while with the further increase of decreases dramatically and then keeps around the former constant level. Note that, when, the averaging intensities of XP and YP modes are comparable and no fixed dominant polarization mode exists, when XP mode reaches a relatively high intensity, the intensity of YP mode becomes extremely small, and vice versus. While for the rest, a dominant polarization mode exists, i.e., the XP (YP) mode keeps the dominant role for. Obviously, by selecting polarizer angle at the critical polarizer angle, the VCSELs with VPOF can provide higher degree of chaotic unpredictability than those with conventional polarization-selected optical feedback schemes, such as VCSELs with purely XP (YP) mode feedback which corresponds to the case of. Thus, when using the total output of VCSEL with VPOF as a chaotic carrier, it is better to choose the polarizer angle at to obtain the highest degree of chaotic unpredictability. Besides, we also present the PE for the XP mode as a function of in Fig. 2(d) for the range of, where the XP mode is the dominant polarization mode. It can be seen that, the PE values for the XP mode are close to those for the total output. Correspondingly, when the YP mode is the dominant polarization mode in the range of, the PE curve for the YP mode shown in Fig. 2(e) is almost overlapped with that for the total output. Thus, we only consider the PE properties of the total output of VCSEL in the following study. To provide more physical insights into the unpredictability enhancement at the critical polarizer angle, we further give the representations on the Poincaré sphere plot in Fig. 5. For a given pair of field amplitude components and we assign a point on the Poincaré sphere whose coordinates are [11], [15], [37]. The points on the Poincaré sphere of unit radius are in one-to-one correspondence with the different polarization states of the laser beam. The northern and southern hemispheres correspond to right- and left-elliptically polarizations. The south pole, the north pole, and the equatorial plane represent the left-circularly polarized, the right-circularly polarized, and the linearly polarized lights. In particular, the positive axis and the negative x axis represent two orthogonal polarization states, the XP and YP modes. We find that, when, only one point locates at the positive x axis on the Poincaré sphere surface, which denotes purely XP mode, and when a single point locates at the negative x axis on the Poincaré sphere surface denotes purely YP mode, these pictures are quite simple and thus we do not present them. In Fig. 5, the polarization evolutions for two representative cases of polarizer angles, e.g., and are presented. It can be seen that, when the points mainly locate on the region near the positive x axis which indicates that the XP mode is the dominant polarization mode. While for, the points distribute almost on the whole sphere, there is no dominant polarization mode, which indicates quite irregular polarization fluctuations, and leads to the highest. We also consider the case of, the points mainly locate on the region near the negative x axis and thus the YP mode is the dominant polarization mode, as the pictures are similar to Fig. 5(a) we do not present it again here. That is to say, when two polarization modes coexist with no fixed dominant mode in VCSEL, the degree of unpredictability is much higher than that of the VCSEL with fixed dominant mode. Next, we consider the PE properties for the total output of VCSELs with different cases of feedback strengths, feedback delays and injection currents. The as a function of polarizer angle for different cases of is shown in Fig. 6(a). It can be seen that, when ns, with the increase of, the almost keeps constant around in the range, then increases sharply and reaches its maximum

5 XIANG et al.: UNPREDICTABILITY-ENHANCED CHAOTIC VCSELS 2177 Fig. 7. Two dimensional maps of PE as a function of, (a) and with = 1:5, (b) and with =20ns. Fig. 6. The H as a function of, (a) for different feedback strengths, (b) for different feedback delays, (c) for different injection currents. at the critical polarizer angle, but when is further increased, the decreases dramatically and goes back to the former constant level in the range. For the rest cases of, the similar trends are observed, nevertheless, the critical polarizer angle moves to a lower value at for ns ns, moreover, the constant level of increases to for ns ns and the maximum of increases to for ns ( ns.in addition, note that, a larger does not always result in a higher in the whole range of, for example, the value of is for ns and but is for ns and, which indicates that even not large feedback strengths can also contribute to high degree of unpredictability by selecting a specific polarizer angle. On the other hand, the as a function of polarizer angle for different cases of is shown in Fig. 6(b). Obviously, the PE curves for different feedback delays overlap in the whole range of, which indicates that the PE for the total output is hardly affected by the feedback delays. Correspondingly, the as a function of polarizer angle for different cases of injection currents is shown in Fig. 6(c). It can be seen that, the trends are similar to those obtained in Fig. 5(a), nevertheless, the critical polarizer angle moves to a larger for a larger. Moreover, the increment for the constant level is larger that that of the maximum. As can be seen from Figs. 6(a) and (c), the optimal value of polarizer angle exists for different feedback strengths and injection currents. To obtain a general influence, we further present two dimensional maps of showing the dependence on both and in Fig. 7(a), and on both and in Fig. 7(b), respectively. Two representative values of are labeled by the Fig. 8. The PE as a function of for different noise levels. dashed contour lines to show more clearly the pictures. It can be seen from Fig. 7(a) that, when the feedback strengths are small, the values of are low for the whole range of, and the critical polarizer angle is not observed. However, for relatively large feedback strengths, pronounced high values of are obtained at the critical polarizer angles, and moreover, the critical polarizer angle seems to move to a slightly lower polarizer angle with the increase of. On the other hand, it can be seen from Fig. 7(b) that, when the injection current is small, the values of are also low and are hardly affected by the polarizer angle, while for larger injection currents, the high values of are observed at the critical polarizer angles, and the critical polarizer angle moves to a slightly higher polarizer angle with the increase of. At last, we take the case of ns as an example to discuss the sensitivity of the PE method to noise levels in Fig. 8, three cases of noise level, and are taken into account. It can be found that, the PE method is robust to small noise level, which is quite important for the practical chaotic communication systems.

6 2178 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 29, NO. 14, JULY 15, 2011 IV. CONCLUSION In summary, chaotic unpredictability properties of VCSELs with VPOF are evaluated quantitatively via the PE. As polarizer angle is increased, the keeps constant firstly and then increases sharply until reaches its maximum at the critical angle and, decreases again to the former constant level. For a relatively large feedback strength and injection current, the maximum is obtained at, and the critical polarizer angle moves to a slightly lower polarizer angle with the increase of, while moves to a slightly higher polarizer angle with the increase of. However, the feedback delay does not modify the PE properties for the total output of VCSEL. These results indicate that the PE is an effective tool to quantify the chaotic unpredictability of VCSELs with VPOF. For practical use, one can determine the critical angle by detecting the intensities of XP and YP modes for each polarizer angle. When the intensities of XP and YP modes are comparable at one polarizer angle, i.e., there is no fixed dominant polarization mode, then this polarizer angle is chosen as the critical angle. By setting the polarizer angle of RP at this critical angle, the unpredictability degrees of the chaotic signals generated by VCSEL with VPOF can be enhanced significantly, compared to those of the chaotic signals generated by VCSEL with conventional polarization-selected optical feedback schemes. Such unpredictability enhancement is extremely useful for the VCSELs-based chaotic communication systems. Our results motivate further study on the chaos synchronization systems based on the unpredictability- enhanced VCSELs as a future work, aiming to explore the effects of the unpredictability enhancement on the synchronization quality of a chaotic link. ACKNOWLEDGMENT The authors would like to thank all reviewers and Dr. W. L. Zhang for their enlightening suggestions and help. REFERENCES [1] L. M. Pecora and T. L. Carroll, Synchronization in chaotic systems, Phys. Rev. Lett., vol. 64, pp , [2] J. Ohtsubo, Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback, IEEE J. Quantum Electron., vol. 38, no. 9, pp , Sep [3] J. M. Liu, H. F. Chen, and S. Tang, Synchronized chaotic optical communications at high bit rates, IEEE J. Quantum Electron., vol. 38, no. 9, pp , Sep [4] A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, Chaos-based communications at high bit rates using commercial fiberoptic links, Nature, vol. 437, pp , Nov [5] N. Jiang, W. Pan, L. Yan, B. Luo, W. Zhang, S. Xiang, L. Yang, and D. Zheng, Chaos synchronization and communication in mutually coupled semiconductor lasers driven by a third laser, J. Lightw. Technol., vol. 28, no. 13, pp , Jul [6] Y. Hong, M. W. Lee, J. Paul, P. S. Spencer, and K. A. Shore, GHz bandwidth message transmission using chaotic vertical-cavity surfaceemitting lasers, J. Lightw. Technol., vol. 27, no. 22, pp , Nov [7] A. Scire, J. Mulet, C. R. Mirasso, J. Danckaert, and M. S. Miguel, Polarization message encoding through vectorial chaos synchronization in vertical-cavity surface-emitting lasers, Phys. Rev. Lett., vol. 90, no. 11, pp , [8] J. Liu, Z. M. Wu, and G. Q. Xia, Dual-channel chaos synchronization and communication based on unidirectionally coupled VCSELs with polarization-rotated optical feedback and polarization-rotated optical injection, Opt. Exp., vol. 17, no. 15, pp , [9] Z. G. Pan, S. Jiang, M. Dagenais, R. A. Morgan, K. Kojima, M. T. Asom, R. E. Leibenguth, G. D. Guth, and M. W. Focht, Optical injection induced polarization bistability in vertical-cavity surface-emitting lasers, Appl. Phys. Lett., vol. 63, no. 22, pp , [10] M. S. Miguel, Q. Feng, and J. V. Moloney, Light polarization dynamics in surface-emitting semiconductor lasers, Phys. Rev. A, vol. 52, no. 2, pp , [11] J. Martin-Regalado, F. Prati, M. S. Miguel, and N. B. Abraham, Polarization properties of vertical-cavity surface-emitting lasers, IEEE J. Quantum Electron., vol. 33, no. 5, pp , May [12] M. Sciamanna, K. Panajotov, H. Thienpont, I. Veretennicoff, P. Mégret, and M. Blondel, Optical feedback induces polarization mode hopping in vertical-cavity surface-emitting lasers, Opt. Lett., vol. 28, no. 17, pp , [13] C. Masoller and M. S. Torre, Influence of optical feedback on the polarization switching of vertical-cavity surface-emitting lasers, IEEE J. Quantum Electron., vol. 41, no. 4, pp , Apr [14] W. L. Zhang, W. Pan, B. Luo, X. H. Zuo, and M. Y. Wang, Polarization switching and hysteresis of VCSELs with time-varying optical injection, IEEE J. Sel. Topics Quantum Electron., vol. 14, no. 3, pp , May/Jun [15] S. Xiang, W. Pan, L. Yan, B. Luo, N. Jiang, K. Wen, X. Zou, and L. Yang, Polarization degree of vertical-cavity surface-emitting lasers subject to optical feedback with controllable polarization., J. Opt. Soc. Amer. B., vol. 27, no. 3, pp , [16] M. Sciamanna, I. Gatare, A. Locquet, and K. Panajotov, Polarization synchronization in unidirectionally coupled vertical-cavity surface-emitting lasers with orthogonal optical injection, Phys. Rev. E., vol. 75, pp _ _10, [17] I. Gatare, M. Sciamanna, A. Locquet, and K. Panajotov, Influence of polarization mode competition on the synchronization of two unidirectionally coupled vertical-cavity surface-emitting lasers, Opt. Lett., vol. 32, no. 12, pp , [18] Y. Hong, M. W. Lee, J. Paul, P. S. Spencer, and K. A. Shore, Enhanced chaos synchronization in unidirectionally coupled vertical-cavity surface- emitting semiconductor lasers with polarization-preserved injection, Opt. Lett., vol. 33, no. 6, pp , [19] R. Ju, P. S. Spencer, and K. A. Shore, Polarization-preserved and polarization-rotated synchronization of chaotic vertical-cavity surfaceemitting lasers, IEEE J. Quantum Electron., vol. 41, no. 12, pp , Dec [20] R. Vicente, J. Daudén, P. Colet, and R. Toral, Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop, IEEE J. Quantum Electron., vol. 41, no. 4, pp , Apr [21] J. D. Farmer, Chaotic attractors of an infinite-dimensional dynamical system, Physica D, vol. 4, pp , [22] V. Ahlers, U. Parlitz, and W. Lauterborn, Hyperchaotic dynamics and synchronization of external-cavity semiconductor lasers, Phys. Rev. E., vol. 58, no. 6, pp , [23] D. M. Kane, J. P. Toomey, M. W. Lee, and K. A. Shore, Correlation dimension signature of wideband chaos synchronization of semiconductor lasers, Opt. Lett., vol. 31, no. 1, pp , [24] C. Bandt and B. Pompe, Permutation entropy: A natural complexity measure for time series, Phys. Rev. Lett., vol. 88, no. 17, pp _ _4, [25] R. López-Ruiz, H. L. Mancini, and X. Calbet, A statistical measure of complexity, Phys. Lett. A, vol. 209, pp , [26] M. T. Martin, A. Plastino, and O. A. Rosso, Statistical complexity and disequilibrium, Phys. Lett. A, vol. 311, pp , [27] O. A. Rosso, H. A. Larrondo, M. T. Martin, A. Plastino, and M. A. Fuentes, Distinguishing noise from chaos, Phys. Rev. Lett., vol. 99, pp _ _4, [28] O. A. Rosso, R. Vicente, and C. R. Mirasso, Encryption test of pseudoaleatory messages embedded on chaotic laser signals: an information theory approach, Phys. Lett. A, vol. 372, pp , [29] M. C. Soriano, L. Zunino, O. A. Rosso, and C. R. Mirasso, Quantifying complexity of the chaotic regime of a semiconductor laser subject to feedback via information theory measures, in Proc. SPIE, 2010, vol. 7720, pp G_ G_10. [30] M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis, IEEE J. Quantum Electron., vol. 47, no. 2, pp , Feb

7 XIANG et al.: UNPREDICTABILITY-ENHANCED CHAOTIC VCSELS 2179 [31] J. Tiana-Alsina, M. C. Torrent, O. A. Rosso, C. Masoller, and J. Garcia- Ojalvo, Quantifying the statistical complexity of low-frequency fluctuations in semiconductor lasers with optical feedback, Phys. Rev. A, vol. 82, pp _ _6, [32] S. Y. Xiang, W. Pan, L. S. Yan, B. Luo, X. H. Zou, N. Jiang, and K. H. Wen, Quantifying chaotic unpredictability of vertical-cavity surface-emitting lasers with polarized optical feedback via permutation entropy, IEEE J. Sel. Topics Quantum Electron., (to be published). [33] S. Xiang, W. Pan, L. Yan, B. Luo, X. Zou, N. Jiang, and K. Wen, Influence of polarization-mode-competition on chaotic unpredictability of vertical-cavity surface-emitting lasers with polarization-rotated optical feedback, Opt. Lett., vol. 36, no. 3, pp , [34] Y. Cao, W. Tung, J. B. Gao, V. A. Protopopescu, and L. M. Hively, Detecting dynamical changes in time series using the permutation entropy, Phys. Rev. E, vol. 70, pp _ _7, [35] S. Ramanujan, G. P. Agrawal, J. M. Chwalek, and H. Winful, Elliptical polarization emission from GaAlAs laser diodes in an external cavity configuration, IEEE J. Quantum Electron., vol. 32, no. 2, pp , Feb [36] L. Chrostowski, B. Faraji, W. Hofmann, M. C. Amann, S. Wieczorek, and W. W. Chow, 40 GHz bandwidth and 64 GHz resonance frequency in injection-locked 1.55 m VCSELs, IEEE J. Sel. Topics Quantum Electron., vol. 13, no. 5, pp , [37] C. Masoller and N. B. Abraham, Low-frequency fluctuations in vertical-cavity surface-emitting semiconductor lasers with optical feedback, Phys. Rev. A, vol. 59, no. 4, pp , Shui Ying Xiang was born in Jiangxi, China, in She received the B.E. degree from Southwest Jiaotong University, China, where she is currently pursuing the Ph.D. degree in the School of Information Science and Technology. She is the author or coauthor of more than 9 research papers. Her current research interests include chaotic optical communications, semiconductor lasers, and chaotic complexity. Lian Shan Yan was born in Shandong, China, in He received the B.E. degree from Zhejiang University, Hangzhou, China, and Ph.D. degree from the University of Southern California (USC), Los Angeles. He is currently a full professor at Southwest Jiaotong University, Chengdu, Sichuan, China. Prof. Yan is a senior member of the IEEE Lasers and Electro-Optic Society. Xi Hua Zou received the B.Eng. and Ph.D. degrees from Southwest Jiaotong University, China, in 2003 and 2009, respectively. He joined the Center for Information Photonics and Communications, Southwest Jiaotong University in 2009, where he is currently an Associate Professor. He has been a joint training Ph.D. student in the School of Information Science and Technology, Southwest Jiaotong University, Chengdu, China and the School of Information Technology and Engineering, University of Ottawa, Ottawa, Canada. His current interests include microwave photonics, optical pulse generation and compression, fiber Bragg gratings, and passive fiber devices for optical fiber communication system. He has authored or coauthored over 50 academic papers in refereed journals. Ning Jiang was born in Sichuan, China, in He received the B.S. degree in University of Science and Technology of China in 2005, where he is currently working toward the Ph.D. degree at the School of Information Science and Technology. His dissertation work focuses on the nonlinear dynamics of semiconductor lasers and chaotic optical communication. Lei Yang was born in Xinjiang, China, in He received the B.S. degree from Southwest Jiaotong University, China, in 2004, where he is currently working toward the Ph.D. degree. His researching interest is the chaos communication based on fiber lasers. Wei Pan was born in Hunan, China, in He received the Ph.D. degree from Southwest Jiaotong University, China, in He is a professor and the dean of the School of Information Science and Technology, Southwest Jiaotong University, China. His research interests include semiconductor lasers, nonlinear dynamic systems, and optical communications. He has published over 100 research papers. Prof. Pan is a member of the Optical Society of America and the Chinese Optical Society. Hong Na Zhu was born in Shandong, China, in She received the B.S. degree in Ludong University of China in 2001, and received the M.S. degree in National University of Defense Technology of China in She is currently working toward the Ph.D. degree at the School of Information Science and Technology, Southwest Jiaotong University of China. Her dissertation work focuses on the nonlinear fiber optics and optical communication. Bin Luo was born in Hubei, China, in He received the Ph.D. degree from Southwest Jiaotong University, China, in He is currently a Professor in the School of Information Science and Technology, Southwest Jiaotong University, China. He is the author or coauthor of more than 80 research papers. His current research interests include semiconductor optical amplifiers and optical communications.