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Based on the framework of spin-flip model (SFM), the output characteristics of a 1550 nm vertical-cavity surface-emitting laser (VCSEL) subject to variable-polarization fiber Bragg grating (FBG) feedback (VPFBGF) have been investigated. With the aid of the self-correlation function (SF) and the permutation entropy (PE) function, the time-delay signature (TDS) of chaos in the VPFBGF-VCSEL is evaluated, and then the influences of the operation parameters on the TDS of chaos are analyzed. The results show that the TDS of chaos can be suppressed efficiently through selecting suitable coupling coefficient and feedback rate of the FBG, and is weaker than that of chaos generated by traditional variable-polarization mirror feedback VCSELs (VPMF-VCSELs) or polarization-preserved FBG feedback VCSELs (PPFBGF-VCSELs).
© 2014 Optical Society of America
1. Introduction
In recent years, optical chaos has been studied extensively for its potential applications in secure communications, chaotic radar and ultra-fast random bit sequences [1–6] etc. Different external perturbations such as optical feedback, optical injection and optoelectronic feedback [7–12] have been introduced to achieve optical chaos output in semiconductor lasers (SLs). Since high dimension broadband chaos can be relatively easily generated through adopting external optical feedback (EOF) [13–15], optical feedback SLs-based chaotic systems have been proposed as a good candidate source generator of high-speed long distance chaos secure communication [2, 16]. However, the chaos output from an EOF SL usually retains an obvious time delay signature (TDS), which is undesirable in some applications. The TDS of chaos may threaten the security of chaos secure communication system [17, 18], make the signal-to-noise ratio reduced or even lead to misjudgment to the targets in chaotic radar application [5], and cause the randomness and statistical performance of random bits deteriorated [6]. Therefore, how to suppress the TDS of chaos generated by an EOF chaotic system has been an important issue from both academic and applied aspects.
For EOF edge-emitting lasers (EELs), there are many schemes proposed for suppressing the TDS of chaos. Rontani et al. investigated the TDS of chaos in a SL with a single mirror feedback (SMF) theoretically [19] and demonstrated that the TDS of chaos can be suppressed when the feedback rate was low while the feedback delay time was close to the relaxation oscillation period. Lee et al. suggested that by adding another EOF to form double mirrors feedback (DMF), a chaos signal with weak TDS can be obtained [20]. We have experimentally and numerically investigated the TDS of chaos in SMF-EELs [21, 22] and DMF-EELs chaotic systems [23], and given the optimal parameter space for obtaining TDS suppressed chaos. Recently, based on EELs, Li et al. proposed a novel scheme [24], in which the distributed reflection in a fiber Brag grating (FBG) is used to take the place of localized reflection by a mirror, and demonstrated that this approach of FBG feedback performs better than the conventional approach of mirror feedback in TDS suppression.
Compared with EELs, vertical-cavity surface-emitting lasers (VCSELs) have many unique characteristics such as low manufacturing cost, low threshold current, high modulation rate, single longitudinal mode operation, high-coupling efficiency to optical fiber and easy large-scale integration into two-dimensional arrays etc [25, 26]. As a result, the generation of optical chaos in VCSELs has aroused much concern, and some approaches used in EELs-based chaotic system can also be applied in VCSELs-based chaotic systems [27, 28]. Different from EELs, the output of VCSELs generally includes two orthogonally linearly polarized components (called “X- polarized component” and “Y- polarized component”) due to the weak cavity and material anisotropies. The polarization nature in VCSELs results in some novel polarization-related feedback schemes proposed such as polarization-rotated optical feedback, polarization-preserved optical feedback, and variable-polarization optical feedback [14, 29–33]. So far, there are many reports on the TDS suppression in such polarization-related optical feedback VCSELs chaotic systems [31–33]. However, these related investigations are almost always focused on the case that the optical feedback is provided by a mirror, and pay little attention on the condition that the optical feedback is provided by a FBG.
In this work, by combining the superiority of FBG in TDS suppression and our previous experience of investigating the TDS suppression in EELs-based chaotic system, we have proposed a variable-polarization FBG feedback (VPFBGF) 1550 nm VCSEL chaotic system, and investigated theoretically the TDS of chaos output from such a system via by the self-correlation function (SF) and the permutation entropy (PE) methods.
2. System model and theory
Figure 1 is the schematic diagram of a VCSEL with variable-polarization fiber Bragg grating (FBG) feedback (VPFBGF). The output of a 1550 nm VCSEL is collimated by an aspheric lens (AL) firstly, after passing through a polarizer (P), a neutral density filter (NDF), and then coupled into a FBG with length L. The reflected beam by the FBG, after passing through NDF, P and AL once again, is re-injected into the VCSEL. The polarizer (P) is introduced to control the polarizer angle of the feedback optical beam, the NDF is used to adjust the feedback rate, and the FBG is used to provide distributed feedback.
Fig. 1 Schematic diagram of a VESEL with VPFBGF. VCSEL: vertical-cavity surface-emitting laser; AL: aspheric lens; P: polarizer; NDF: neutral density filter; FBG: fiber Bragg grating.
Based on the spin-flip model (SFM) [34, 35], after taking into account the variable-polarization feedback [31] from a FBG, the rate equations of a VCSEL under VPFBGF can be described by:
where subscripts x and y represent the X- and Y- polarized components respectively. E is the slowly varying complex amplitude of the field, N is the total carrier inversion between the conduction and valence bands, n accounts for the difference between carrier inversion of spin-up and spin-down radiation channels, k is the decay rate, α is the line-width enhancement factor, γa is the linear dispersion, γp is the linear birefringence effect of active medium, γN is the decay rate for the total carrier, γS is the spin-flip rate, η is the feedback rate, τ is the feedback delay time and μ is the normalized injection current. The polarizer angle θp changes from 0° to 90°, where 0° polarization is parallel to X- polarized component whereas 90° polarization is parallel to Y- polarized component. θ is the feedback phase, and ⊗ denotes convolution operation. r(t) denotes the impulse response of the FBG reflection, and it is evaluated from a inverse Fourier transform of the frequency response of the FBG [24].For a uniform single-mode FBG, the coupling coefficient κ is a real and the frequency response of the reflection r(Ω) is [36, 37]:
where L is the length of the FBG, Ω is the frequency detuning away from the Bragg resonance frequency, δ = -ngΩ/c (ng is the refractive index, c is the speed of light in vacuum) quantifies the phase mismatch between the counter-propagating modes with angular frequency detuning Ω, and the FBG bandwidth Δf ( = c|κ|/πng) can be approximately estimated by the modulus of κ.It is worth mentioning that the rate-equation model in Eqs. (1)–(4) is a generic model for an arbitrary feedback impulse response r(t) [24]. For FBG feedback, r(t) is the inverse Fourier transform of r(Ω) described by Eq. (5), and the VCSEL at time t is affected by its emission at and before t - τ. For mirror feedback, r(t) can be replaced by δ(t), and the VCSEL at time t is only affected by its emission at exactly t - τ.
In general, numerous methods such as filling factor analysis [38], mutual information (MI) [19], self-correlation function (SF) [19] and permutation entropy (PE) [39, 40], can be used to identify the TDS of chaos signal. In this work, we adopt both SF and PE methods. For a delay-differential system, SF is defined as follow:
where I(t) is the intensity of chaotic output, Δt is the time shift, and 〈⋅〉 denotes the time average. We set Δt ∈ [0 ns, 10 ns]. The TDS can be identified from the peak location of SF curve.The PE method, which is based on information theory, has some unique advantages such as simplicity, fast calculation and robustness to noise [39]. PE could be defined as follows [39,40]: the time series {I(m), m = 1, 2, …, N} are firstly reconstructed into a set of D-dimensional vectors after choosing an appropriate embedding dimension D and embedding delay time τe. Then we study all D! permutation π of order D. For each π, the relative frequency (# means number) is determined as:
and then the permutation entropy is defined as:In this paper, we set D = 7 after combining the unique features of VCSEL with the suggestions in [39] and [40], and the length of time series is taken as 1500 ns during calculating SF and PE.
3. Results and discussion
Equations (1)–(4) can be solved with the fourth-order Runge-Kutta method. During calculations, the parameters are set as [41]: k = 125 ns−1, α = 2.2, γa = 0.02 ns−1, γp = 192 ns−1, γN = 0.67 ns−1, γS = 1000 ns−1, the feedback phase θ is assumed to be integer multiples of 2π, and the parameters of FBG are [24]: ng = 1.45, L = 20 mm. The feedback delay time τ is fixed at 3 ns unless otherwise noted.
Figure 2 shows the polarization-resolved P-I curve of a free-running 1550 nm VCSEL, where the intensities, defined as IX,Y = |Ex,y(t)|2, are averaged over a time window of 1.5 µs. From this diagram, it can be seen that the Y- polarized component begins to oscillate when μ = 1 and no polarization switching is observed as 1 < μ < 4. In the following discussions, we set μ as 1.5. Under this case, only the Y- polarized component oscillates.
Fig. 2 Polarization-resolved P-I curve for a solitary VCSEL, where the dashed line and the solid line stand for X- polarized component and Y- polarized component, respectively.
After introducing FBG feedback, the polarization-resolved intensities as a function of polarizer angle are shown in Fig. 3 for η = 10 ns−1 and |κ| = 600 m−1. As shown in this diagram, the Y- polarized component is suppressed seriously for θp ≤ 33° while the X- polarized component is suppressed once θp ≥ 57°. Therefore, for convenience of discussion in this work, we only consider the TDS of the total output intensity IT ( = |Ex(t)|2 + |Ey(t)|2) of the VCSEL.
Firstly, we discuss the TDS in the VPFBGF-VCSEL chaotic system when the polarizer angle θp takes different values. Figure 4 shows the time series, power spectra, SF curves and PE curves under η = 12 ns−1 and |κ| = 600 m−1 for θp = 90° (first row), θp = 66° (second row) and θp = 30° (third row). It should be noted that, the case of θp = 0° corresponds to pure X- polarized component feedback and the case of θp = 90° corresponds to pure Y- polarized component feedback based on Eqs. (1) and (2). Combining the time series and power spectra, it can be determined that the VCSEL is rendered into chaotic states under above three circumstances. When θp = 90° and θp = 66°, some peaks with an equal frequency interval, which is close to the reciprocal of delay time, emerge upon the power spectra as shown in Fig. 4(a2) and Fig. 4(b2), meanwhile there are obvious peaks appearing in the SF curves at ∆t = τ = 3 ns (see Fig. 4(a3) and Fig. 4(b3)) and obvious valleys locate around the embedding delay time τe = 3 ns and its multiples of 1/2 and 1/3 in the PE curves (see Fig. 4(a4) and Fig. 4(b4)). As a result, for θp = 90° and θp = 66°, the delay time is easy to extract from these locations of SF characteristic peaks and PE valleys. By contrast, when θp is taken as 30°, the power spectra is smooth (Fig. 4(c2)), the SF characteristic peak is not obvious and the PE curve behaves flat (see Fig. 4(c3) and Fig. 4(c4)). Under this case, the TDS of the chaos has been suppressed efficiently. By this diagram, it is reasonable to infer that the TDS of chaos output from a VPFBGF-VCSEL is seriously depended on the polarizer angles.
Fig. 4 Total intensity time series (first column), power spectra (second column), SF curves (third column) and PE curves (forth column) of a VPFBGF-VCSEL under η = 12 ns−1 and |κ| = 600 m−1 for θp = 90° (first row), θp = 66° (second row) and θp = 30° (third row).
Secondly, we focus on the TDS of the chaos outputs under different values of the feedback rate η, and the corresponding results are presented in Fig. 5 under θp = 60° and |κ| = 600 m−1. Combining the SF curves and PE curves, one can observe that the TDS is suppressed gradually with the decrease of η. When the feedback rate is weakened to a certain degree as η = 10 ns−1, the SF characteristic peak (as shown in Fig. 5(c3)) is broadened and suppressed, meanwhile no sharp valley in the vicinity of τ is observed in PE curve. Under this situation, it is quite difficult to identify the delay time of this chaotic system. Therefore, the feedback rate also plays an important role in TDS suppression, and a relatively small feedback rate may be helpful for generating chaotic time series with suppressed TDS. In fact, similar trend has been observed in [22] and [31].
Fig. 5 Total intensity time series (first column), power spectra (second column), SF curves (third column) and PE curves (forth column) for a VPFBGF-VCSEL under θp = 60° and |κ| = 600 m−1 for η = 20 ns−1 (first row), 15 ns−1 (second row) and 10 ns−1 (third row), respectively.
Next, we compare the TDS of chaotic signals by a VPFBGF with those by a variable polarization mirror feedback (VPMF) in a VCSEL. Under a VPMF, the rate equations of VCSEL can be obtained by setting r(t) = δ(t) in Eqs. (1) and (2). Here, we use an amplitude σ to characterize whether the TDS is obvious or not. The amplitude σ is defined as the maximum SF peak within the Δt region of [2 ns, 4 ns]. The larger σ is, the more obvious the TDS will be. Figure 6 displays the dependence of σ on the polarizer angle θp for a VCSEL subject to optical feedback from a mirror or a FBG with |κ| = 600 m−1, where the delay time τ is varied from 3 ns to 6 ns. It should be pointed out that the curves obtained for VPMF are interrupted for middle polarizer angles since the outputs of the laser are not chaotic states. From this diagram, it can be seen that the values of σ calculated for a VPMF-VCSEL are always larger than those obtained for a VPFBGF-VCSEL, i. e., adopting FBG feedback is superior to using mirror feedback for achieving weak TDS chaos signals. A similar result has been observed in a DFB-SL based chaotic system [24]. Additionally, for both VPMF and VPFBGF, the dependence of σ on the polarizer angle follows similar trends, and the influence of the delay time τ is relative weak.
Fig. 6 σ as a function of θp under η = 10 ns−1 and different feedback delay times for a VCSEL subject to optical feedback from a mirror or a FBG with |κ| = 600 m−1.
The above results demonstrate that the polarizer angle θp and the feedback rate η are two crucial parameters to affect the TDS, and thus it should be essential to investigate the overall evolution of TDS in the parameter space of θp and η for the purpose of hunting optimal operation parameters to generate chaos with weak TDS. In Fig. 7, the left column shows the maps of σ in the parameter space of η and θp for VPFBGF with three different values of |κ| and VPMF, and the right column shows the reflection spectra (blue lines) and group delays [37] (red lines) of the used FBGs and mirror. Especially, the case of σ ≈1 corresponds to the stable state or period-one state. From this diagram, it can be seen that, with the increase of |κ|, the regions with weak TDS (blue) move towards relatively small feedback rates. The reason may be explained as follows. For a larger value of |κ|, the two polarized components of VCSEL will locate at a sideband closer to the center peak (as shown in Fig. 7(a2)–Fig. 7(c2)). In other words, r(Ω) will be larger within the chaotic bandwidth (about 4GHz for above given data of parameters), and then r(t) is also larger since r(t) is the inverse Fourier transform of r(Ω). After considering that the feedback strength depends on ηr(t) in Eqs. (1) and (2), η may decrease under a larger value of |κ| for achieving similar feedback effect. Furthermore, the larger the |κ| of FBG is, the more close to the reflection spectrum of the mirror the reflection spectrum of FBG is (as shown in the right column of this diagram). As a result, the map of σ for a VPFBGF with |κ| = 900 (Fig. 7(c1)) is close to that for a VPMF (Fig. 7(d1)). Actually, multiple factors such as the relaxation oscillation frequency of the laser [23], the feedback strength and the group delay supplied by FBG [24], etc., may affect the TDS of the chaotic signal, and therefore the evolution of the TDS of chaos will be quite complicated.
Fig. 7 Left column: maps of σ in the parameter space of η and θp for a VCSEL subject to VPFBGF with |κ| = 300 m−1 (a1), |κ| = 600 m−1 (b1), |κ| = 900 m−1 (c1) and VPMF (d1); right column: reflection spectra (blue lines) and group delays (red lines) of the used FBGs (a2, b2, and c2) and mirror (d2), where the dashed lines with arrows label the angle frequencies of the two polarized components of the solitary VCSEL.
Furthermore, we compare the TDS of chaos output from a VPFBGF-VCSEL with that from a VCSEL with a polarization-preserved FBG feedback (PPFBGF). For a PPFBGF, the X- polarized component is reflected to X- polarized component while the Y- polarized component is reflected to Y- polarized component [14, 30]. Under the case of PPFBGF, the feedback term should be replaced by ηr(t)⊗Ex(t-τ)eiθ in Eq. (1) and ηr(t)⊗Ey(t-τ)eiθ in Eq. (2), respectively. Figure 8 shows the dependence of σ on θp in a PPFBGF-VCSEL (green line) or a VPFBGF-VCSEL (blue line with asterisks) under |κ| = 600 m−1 and different values of η. Adopting similar treatment method used in Fig. 6, only the values of σ are evaluated and given under the case that the outputs from the VCSEL are chaotic states. Obviously, through selecting suitable polarizer angle, the TDS of chaos generated by a VPFBGF-VCSEL is weaker than that generated by a PPFBGF-VCSEL.
Fig. 8 Dependence of σ on θp in a PPFBGF-VCSEL or a VPFBGF-VCSEL for |κ| = 600 m−1, where (a) η = 5 ns−1, (b) η = 10 ns−1, (c) η = 15 ns−1 and (d) η = 20 ns−1, respectively. Green lines represent PPFBGF while blue lines with asterisks correspond to VPFBGF.
Finally, we will examine if the TDS of chaos can be re-appeared from a low TDS output state when one puts an additional polarizer in the output and rotates it. Three typical cases for |κ| = 600 m−1 are taken as examples. For case 1, (η, θp) = (10 ns−1, 6°), the amplitude σ is about 0.15 (see Fig. 6) and the X- polarized component predominates (see Fig. 3). For case 2, (η, θp) = (10 ns−1, 54°), σ is about 0.13 (see Fig. 6) and the X- polarized component and the Y- polarized component coexist (see Fig. 3). As for case 3, (η, θp) = (10 ns−1, 60°), σ is about 0.15 (see Fig. 6) and the Y- polarized component predominates (see Fig. 3). Figure 9 displays the dependences of the output intensities (red lines with circles) and σ (black lines with squares) on the rotating angle θ of the additional polarizer. Here, θ = 0° corresponds that the direction of the additional polarizer is along with the X- polarized component of the VCSEL. From this diagram, one can see that, for case 1 and case 3, the amplitude σ mostly maintains a low level except that the rotating angle θ is very close to 90° for case 1 and 0° (or 180°) for case 3. Although the TDS of chaos may re-appear when the rotating angle θ is very close to 90° for case 1 and 0° (or 180°) for case 3, the output intensity is very low under these circumstances as shown in this diagram. After taking into account the finite extinction ratio of polarizer and the noise of test instruments in practice, the extraction of the TDS is still very difficult. However, for case 2, there exist two regions close to 0° and 180° in which both the amplitude σ and the output intensity are relatively large. Under this case, the TDS of chaos can be extracted after putting a polarizer in the output. Above results show that the system parameters should be further optimally selected if one needs to guarantee that the TDS of chaos can always be effectively concealed in some practical applications.
Fig. 9 Output intensity (red lines with circles) and σ (black lines with squares) as a function of rotating angle of the additional polarizer, where the green lines corresponds to σ calculated from the output chaos signal before passing through the additional polarizer.
4. Conclusions
In summary, we have proposed a chaotic system based on a 1550 nm VCSEL under a VPFBGF to acquire weak TDS chaos signals. Based on the SFM, the total output intensity time series can be calculated. Through adopting the self-correlation function (SF) and the permutation entropy (PE) function, the TDS of chaotic output can be evaluated. Furthermore, the influences of the polarizer angle and the feedback parameters on the TDS in this system have been investigated. Numerically simulated results show that the chaos with suppressed TDS can be obtained through optimizing the operation parameters. Meanwhile, by comparing with results obtained by using a variable polarization mirror feedback (VPMF), adopting variable-polarization fiber Bragg grating (FBG) feedback (VPFBGF) is a better scheme to generate weak TDS chaos due to the distributed reflection in a FBG. Additionally, calculations also show that adopting a VPFBGF is superior to using a polarization-preserved FBG feedback (PPFBGF) for generating the weak TDS chaos. Finally, we have examined if the TDS of chaos can be re-appeared from a low TDS output state after putting an additional polarizer in the output and rotating it, and the results show that the TDS of chaos can always be effectively concealed by further optimally selecting the system parameters. We hope this work may be helpful for the generation of high quality chaos in some special applications.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant 61178011 and Grant 61275116, the Natural Science Foundation of Chongqing City under Grant 2012jjB40011, and the Open Fund of the State Key Lab of Millimeter Waves of China under Grant K201418.
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