Polarization-resolved time-delay signatures of chaos induced by FBG-feedback in VCSEL

Polarization-resolved chaotic emission intensities from a vertical-cavity surface-emitting laser (VCSEL) subject to feedback from a fiber Bragg grating (FBG) are numerically investigated. Time-delay (TD) signatures of the feedback are examined through various means including self-correlations of intensity time-series of individual polarizations, cross-correlation of intensities time-series between both polarizations, and permutation entropies calculated for the individual polarizations. The results show that the TD signatures can be clearly suppressed by selecting suitable operation parameters such as the feedback strength, FBG bandwidth, and Bragg frequency. Also, in the operational parameter space, numerical maps of TD signatures and effective bandwidths are obtained, which show regions of chaotic signals with both wide bandwidths and weak TD signatures. Finally, by comparing with a VCSEL subject to feedback from a mirror, the VCSEL subject to feedback from the FBG generally shows better concealment of the TD signatures with similar, or even wider, bandwidths. ©2015 Optical Society of America OCIS codes: (140.5960) Semiconductor lasers; (190.3100) Instabilities and chaos. References and links 1. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). 2. J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28(1), 93–108 (1992). 3. I. Fischer, O. Hess, W. Elsa, and E. Göbel, “High-dimensional chaotic dynamics of an external cavity semiconductor laser,” Phys. Rev. Lett. 73(16), 2188–2191 (1994). 4. R. Vicente, J. Daudén, P. Colet, and R. 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Introduction
External cavity feedback (ECF) is the most commonly used technique to drive a semiconductor laser (SL) into chaotic states [1][2][3][4][5][6][7][8][9].The chaotic output from an ECF-SL typically contains obvious signatures of the time-delay (TD) originating from the optical round trip between the laser and the external feedback mirror [5,6].Such TD signatures of chaos can potentially compromise the security in optical chaos encryption systems [7][8][9].They also limit the tunability of the bit rates in high-speed random bit sequences [10,11].The suppression of the TD signature, therefore, has become very important for chaos generations in ECF-SLs.
A number of theoretical and experimental investigations have been conducted to suppress the TD signatures in edge-emitting SLs.Rontani et al. reported the elimination of TD signatures at relatively low feedback rates when the delay is close to the inverse of the relaxation resonance frequency [6,12].Lee et al. demonstrated the possibility to obscure the TD signatures by introducing another external cavity to realize a double-ECF [5].We also investigated the TD signatures in two mutually coupled SLs in which each SL can be regarded as an active nonlinear reflector to the other SL, where experimental and numerical results both show simultaneous elimination of TD signatures from the two SLs [13,14].Besides, ECF using a fiber Bragg grating (FBG) as a distributed reflector was also reported for suppressing the TD signatures [15,16].With the success of these approaches in the edge-emitting SLs, their extensions to the vertical-cavity surface-emitting lasers (VCSELs) have started to attract much attention.This is due to the many unique advantages offered by VCSELs which include being low cost, possessing low threshold currents, yielding circular output beams of narrow divergence, allowing large-scale integration into two-dimensional arrays, and supporting single longitudinal-mode operation due to the short cavity lengths [17][18][19][20].
Recently, much effort has been made to suppress the chaotic TD signatures in ECF-VCSELs [21][22][23][24][25][26].Xiang et al. proposed concealing the TD signatures in VCSELs though invoking chaos by variable-polarization optical feedback [21].Li et al. numerically investigated the TD signatures of chaos in VCSELs with optical feedback [22].Priyadarshi et al. reported experiments that verified the scheme [23].Lin et al. considered adding one more feedback to realize a double-cavity polarization-rotated optical feedback into a VCSEL [24].Hong et al. experimentally compared a number of combinations utilizing two VCSELs [25].It was found that, when an ECF-VCSEL is mutually or unidirectionally coupled to the other VCSEL, simultaneous suppression of TD signatures and enhancement of chaotic bandwidth can be obtained.We also studied similar TD signature suppression in a chaotic VCSEL that is subject to two feedbacks with variable polarizations [26] or is subject to mutual coupling from another VCSEL [27].Additionally, we investigated the TD features of a chaotic VCSEL under variable-polarization feedback from a grating [28], but only the total output intensity was considered without resolving the two polarization components (PCs).This was due to the challenge of operating the VCSEL with two co-existing PCs in chaos.However, a chaotic VCSEL with two simultaneously co-existing PCs can enable applications such as dual-channel chaos communication and a dual-polarization high-speed random bit generator [29].
In this paper, we numerically investigate the polarization-resolved TD features in a VCSEL subject to feedback from a FBG with no need for polarization rotation.When the operation parameters are carefully selected, the VCSEL is found to emit chaotic outputs with two orthogonal linear PCs.The PCs are emitted simultaneously, but the two PCs carry two different waveforms.From the intensity time-series, the TD signatures are thoroughly examined using self-correlations of individual PCs, cross-correlations between the two PCs, and permutation entropies calculated from the individual PCs.The TD signatures are shown to be much reduced as compared to using mirror instead of FBG for the feedback.Then, in the parameter space of the feedback rate and frequency deviation, numerical maps are obtained in revealing regions of chaotic signals with both wide bandwidths and small TD signatures.Lastly, when compared to the conventional ECF-VCSEL using a mirror, the feedback from the FBG is found to generally show better concealment of the TD signatures, illustrating the benefits of using FBG feedback on the VCSEL.Figure 1 shows the schematic diagram of a VCSEL subject to polarization-preserved feedback from a FBG.The output from the VCSEL is split by a beam splitter (BS) into two parts.One part is reflected by the FBG and fed back to the VCSEL with a time delay and with a feedback strength controlled by a neutral density filter (NDF).The other part is sent to a polarizing beam splitter (PBS) and then split into two PCs in x-and y-directions, respectively.In this work, we focus on the polarization-resolved nonlinear dynamics of VCSELs.Under this case, the spin-flip model (SFM) [30,31], in which the phase relationship between x-and y-PCs has been taken into account, is more suitable compared with the two-mode model [19].The rate equations for the VCSEL of Fig. 1 are: (

System model and theory
) Here, the subscripts x and y denote the x-PC and y-PC, respectively.E is the slowly-varying complex amplitude of the optical electric field.N accounts for the normalized total carrier population inversion in excess of its transparency value.n accounts for the difference in carrier inversions between opposite spins.κ is the photon decay rate.γ is the carrier decay rate.γ s is the spin homogenization rate.γ a is the linear cavity dichroism.γ p is the linear phase anisotropy.α is the linewidth enhancement factor.μ is the normalized injection current.f m ( = f 0 -f FBG , where f 0 is the mean frequency of the two PCs in the free-running VCSEL and f FBG is the Bragg frequency of the FBG) is the frequency deviation.η is the feedback rate.τ is the group delay of the round-trip feedback in exclusion of the FBG.θ specifies the phase of the optical feedback along with the impulse response r(t) of the FBG.The impulse response is given by the inverse Fourier transform of the frequency response of the FBG: where κ B is the magnitude of the coupling coefficient of the FBG, L is the length of the FBG, δ is the phase mismatch between the counterpropagating modes that equals n g Ω/c with n g and c being the group index of the fiber and the speed of light in vacuum, respectively.A uniform single-mode FBG is assumed due to its relatively simple structure.Its reflection bandwidth can be approximated by cκ B /πn g when it is highly reflective at the Bragg frequency [32].The feedback from the FBG reduces to that from a mirror when κ B goes to infinity.The rate Eqs.
The following sections focus on the output intensities of the VCSEL, which are numerically represented by I x (t) = |E x (t)| 2 and I y (t) = |E y (t)| 2 for the x-and y-PCs, respectively.Even when the VCSEL is driven into chaos, the residual information of the value of τ could be hidden in the intensities.Such signatures of the TD can possibly be extracted by different correlations of the intensity time-series.The self-correlation functions (SFs) are respectively defined for the x-and y-PCs as [6,13]: and the cross-correlation function (XF) between the x-and y-PCs is defined as: where <•> denotes time-average and Δt is the time-shift.Also, TD information can possibly be revealed by the permutation entropy (PE) [21,28,33,34].PE is evaluated versus the embedding delay time τ e by the following procedure for each polarization with intensity time series I(t).A set of D-dimensional vectors (I(mδt + τ e ), I(mδt + 2τ e ), …, I(mδt + Dτ e )) is formed for all integral index m in the simulation.Every vector belongs to one particular order permutation for the D intensities, amongst the total of D! possible permutations.For a permutation labeled as π, its number of occurrences is denoted by p(π), which is normalized to the total number of the vectors.PE is then defined as: where the summation is performed over all D! possible π.Throughout this paper, the length of the simulated time-series is kept at 1000 ns, so the total number of time steps is M = 5 × 10 5 .In order to obtain a reliable statistics, the condition D!<<M should be satisfied [34].Meanwhile, a large embedding dimension D is beneficial to include more information for estimating the quantifiers [33,34].As a result, the embedding dimension D is set at 7 during the calculation of PE in this work.

Results and discussion
The rate-equation model in Eqs. ( 1)-( 4) is expressed using the complex field amplitudes with respect to an optical frequency, in which the free-running optical frequencies for the two linear polarizations of the VCSEL equal ± (γ p −αγ a )/2π = ± 1.5 GHz when the steady-state condition is satisfied [31].The FBG coupling coefficient κ B = 100 m −1 corresponds to an approximate full width at half-maximum (FWHM) reflection bandwidth of 6.6 GHz, which is sufficiently broad for reflecting both polarizations if the detuning is set at f m = 0.The peak reflectivity of the FBG is over 0.9, while the amount of feedback light is further controlled by the feedback rate η.For η = 0 in Row 1, the VCSEL is free-running without any feedback.The output of the VCSEL is elliptically polarized due to co-existent x-PC and y-PC, which are both in periodic oscillations at about 9 GHz.The behavior is qualitatively in consistency with previous numerical results on the solitary VCSEL with a proper bias current [35].For η = 0.45 ns −1 in Row 2, the oscillations for both polarizations are further modulated by a frequency component of about 0.2 GHz, resulting in a quasi-periodic state.For η = 12 ns −1 in Row 3, both x-and y-PCs oscillate chaotically.The power spectra show the signal floors being raised much, while the time-series appear to vary erratically.Careful examination of the power spectra does give some peaks at multiples of 0.33 GHz, especially for low frequencies, as expected from the round-trip feedback delay of τ = 3 ns.
When the VCSEL is in chaos, the TD signatures are much affected by the feedback rate η. Figure 3 shows the numerical results for feedback rates η = 40 ns −1 (Row 1), 30 ns −1 (Row 2), and 20 ns −1 (Row 3).The correlation functions and permutation entropies (PEs) are computed from the intensity time-series of the two linear polarizations.Columns A and C show the SFs against Δt for x-PC and y-PC, respectively; whereas Column E show the XF between x-PC and y-PC.Columns B and D show PEs against τ e for x-PC and y-PC, respectively.The round-trip feedback delay is fixed at τ = 3 ns.The FBG is centered at the average free-running frequencies of the two polarizations with f m = 0 for the red curves.).The feedback is from the FBG with κ B = 100 m −1 for the red curves, whereas it is from a mirror for the green curves.
For comparison, the FBG is replaced by a mirror for the green curves in Fig. 3 by setting κ B to infinity in the simulation [32].The feedback from the mirror leads to clear SFs at Δt = τ in Columns A and C. Such a mirror feedback also gives anticorrelations between the polarizations as observed at Δt = ± τ in Column E, which are due to competitions between the PCs.Negative spikes in the traces of PEs are also observed at τ e = τ in Columns B and D. These features clearly unveil the information on the TD at τ in spite of the random-like chaotic waveforms emitted by the VCSEL.Upon carefully contrasting the different rows of Fig. 3, the TD signatures are seen to slightly subside when the strong feedback rate is sequentially reduced from 40 ns −1 to 30 ns −1 and then to 20 ns −1 .Observable dependence on the feedback rate is consistent with earlier works on TD signatures [6].
When the FBG is adopted for feedback, the red curves in Fig. 3 shows improved concealment of the TD information.Correlations at Δt = τ in Columns A and C are reduced.Anticorrelations at Δt = ± τ in Column E are slightly smoothed out.The negative spikes in the PEs at τ e = τ in Columns B and D become less pronounced, though they still deviate from the ideal value of unity.In general, by using the FBG instead of a mirror for feedback, all of the TD signatures presented in Fig. 3 are suppressed [15,28].Yet, it is desirable to further eliminate the residual TD signatures.
Further elimination of the residual TD signatures in FBG feedback is enabled by adjusting f m while keeping η constant.Figure 4 shows the results for the chaos outputs at η = 20 ns −1 as in the red curves in Row 3 of Fig. 3, except f m = −15 GHz (Row 1) and 15 GHz (Row 2).Contrasting Row 3 of Fig. 3 against Row 1 and Row 2 in Fig. 4 reveals that the TD signatures at Δt = ± τ and τ e = τ are nearly eliminated.In other words, the peaks at Δt = τ for the SFs are broadened and suppressed.No sharp valleys in the proximity of τ e = τ are observed in the curves.In fact, for f m = ± 15 GHz as shown in Fig. 4(F), the mean wavelength of the VCSEL is located at a sideband of the FBG reflection spectrum.The reflectivity drops to the order of 1%, which weakens the feedback and hence reduces the TD signatures.The group delay of the FBG depends on the optical frequency and leads to broadening the TD peaks in the correlation functions as well as the entropy traces [15,32], thereby effectively obscuring the TD signatures.The above results demonstrate that the feedback rate η and the frequency deviation f m both affect TD signature suppression.Besides the TD signature, the bandwidth is also an important performance index to assess a chaos signal, which is directly related to the transmission rate of messages in chaos secure communication [25] or the bit rate of the random number extracted from an SL-based chaos entropy source [10].In this work, we use the effective bandwidth to characterize the bandwidths of chaos signals after taking into account that the effective bandwidth, which is obtained by summing up only those discrete spectral segments of the power spectrum accounting for 80% of the total power, is superior to the standard bandwidth for distinguishing the broadband chaotic states from the narrowband periodic oscillations [36].Figure 5 shows the maps, in the parameter space of (η, f m ), about the TD signatures and the effective signal bandwidths of polarization-resolved outputs from the VCSEL with FBG feedback.For quantification, the TD signature σ is defined as the maximum value of the SFs at Δt in the region of [2 ns, 4 ns] in the maps in Figs.It should be pointed out that, for small η, the TD signature σ is nearly equal to 1 while the bandwidth of the power spectrum is very small.These are attributed to stable or periodic oscillatory intensities of the VCSEL for small η.As the feedback rate η increases, the effective bandwidths of the polarization-resolved outputs from the VCSEL increase rapidly, which are consistent with the dynamical development from period oscillation into chaos.Obviously, low TD signatures at around 0.1 along with bandwidth of around 4 GHz can be easily achieved simultaneously for both x-PC and y-PC, according to combining the information in the four maps in Fig. 5.In other words, the maps are useful for locating optimal operation parameters to generate chaotic signals with small TD signatures and wide bandwidths.In order to emphasize the advantages of using FBG instead of mirror for feedback, we compare VCSEL with FBG feedback and with conventional mirror feedback in Fig. 6.The TD signature σ and the effective bandwidth are, respectively, plotted in Rows 1 and 2 of Fig. 6.The results for the x-PC and y-PC are shown in Columns A and B, respectively.The feedback using FBG is examined as κ B varies, where f m is set at 0 and 15 GHz for the circles and triangles, respectively.As references, the solid lines in Fig. 6  may further conceal the TD signatures.Besides, from Row 2 of Fig. 6, the FBG feedback can even induce chaos with bandwidths exceeding that of mirror feedback.Additionally, it should be pointed out that for VCSELs with polarization rotated feedback, relevant experiments revealed that the TD signatures σ of both two PCs can reach a level of 0.1 for single cavity polarization rotated feedback or even smaller than 0.05 for double-cavity polarization rotated feedback [24].The above calculations show that adopting FBG feedback can achieve similar effect with polarization rotated feedback in TD signature concealment, and related experimental verification is possible.

Conclusions
In summary, the polarization-resolved TD signatures of the chaos outputs from the VCSEL subject to feedback from the FBG have been numerically investigated.Based on the SFM formulism, the intensity time-series for both x-PC and y-PC are calculated, where self-correlations, cross-correlations, and permutation entropies are used to quantitatively evaluate the TD signatures.The results show that, under suitable operational parameters, the TD signatures of chaos in both polarizations can be suppressed efficiently by the FBG instead of the mirror for feedbacks.With the aid of the maps of TD signatures and effective bandwidths in the parameter space of feedback rates and frequency deviations, two chaotic PCs with both small TD signatures and broad chaotic bandwidths are obtained.Direct comparison between FBG feedback and mirror feedback suggest that the former is much more useful in secure communications due to a better concealment of the TD signature in the chaotic regime.Compared with VCSELs subject to polarization rotated feedback [24] or mutually coupled VCSELs with polarization optical injection [37], in which good TD concealment can be achieved experimentally, such FBG feedback may be relatively simple since there is no need for using optics to rotate the feedback polarizations.We hope this work would be helpful for obtaining high quality dual-polarization chaos signals for some special applications.

Fig. 2 .
Fig. 2. Polarization-resolved outputs of the VCSEL under FBG feedback.Columns A and C: time-series for the individual polarizations.Columns B and D: power spectra for the individual polarizations.The feedback rate is set to η = 0 (Row 1), 0.45 ns −1 (Row 2), and 12 ns −1 (Row 3).The feedback has no detuning.

Figure 2
Figure2displays the results of emissions of the VCSEL that are polarization-resolved into the x-PC and the y-PC, as labeled.The emission intensity time series (Columns A and C) and the corresponding power spectra (Columns B and D) are shown under different feedback ratesη.For η = 0 in Row 1, the VCSEL is free-running without any feedback.The output of the VCSEL is elliptically polarized due to co-existent x-PC and y-PC, which are both in periodic oscillations at about 9 GHz.The behavior is qualitatively in consistency with previous numerical results on the solitary VCSEL with a proper bias current[35].For η = 0.45 ns −1 in Row 2, the oscillations for both polarizations are further modulated by a frequency component of about 0.2 GHz, resulting in a quasi-periodic state.For η = 12 ns −1 in Row 3, both x-and y-PCs oscillate chaotically.The power spectra show the signal floors being raised much, while the time-series appear to vary erratically.Careful examination of the power spectra does give some peaks at multiples of 0.33 GHz, especially for low frequencies, as expected from the round-trip feedback delay of τ = 3 ns.When the VCSEL is in chaos, the TD signatures are much affected by the feedback rate η.Figure3shows the numerical results for feedback rates η = 40 ns −1 (Row 1), 30 ns −1 (Row 2), and 20 ns −1 (Row 3).The correlation functions and permutation entropies (PEs) are computed from the intensity time-series of the two linear polarizations.Columns A and C show the SFs against Δt for x-PC and y-PC, respectively; whereas Column E show the XF between x-PC and y-PC.Columns B and D show PEs against τ e for x-PC and y-PC, respectively.The round-trip feedback delay is fixed at τ = 3 ns.The FBG is centered at the average free-running frequencies of the two polarizations with f m = 0 for the red curves.

#Fig. 3 .
Fig. 3. Polarization-resolved time-delay signatures at zero detuning.Columns A and C: self-correlation functions (SFs) for the individual polarizations.Columns B and D: permutation entropies (PEs) for the individual polarizations.Column E: cross-correlation function (XF)between both polarizations.The feedback rate is set to η = 40 ns −1 (Row 1), 30 ns −1 (Row 2), and 20 ns −1 (Row 3).The feedback is from the FBG with κ B = 100 m −1 for the red curves, whereas it is from a mirror for the green curves.

Fig. 5 .
Fig. 5. Polarization-resolved maps of the TD signatures σ from self-correlations (Row 1) and effective bandwidths (Row 2) of the chaotic outputs of the VCSEL under FBG feedback.Column A: x-PC.Column B: y-PC.The maps are shown in the parameter space of η and f m .
5(A1) and Fig. 5(B1).The maps for x-PC and y-PC respectively recorded in Figs.5(A1) and Fig. 5(B1) are quite similar, where regions in deep blue color at positive and negative f m show successful reduction of the TD signature to around 0.1.

Fig. 6 .
Fig. 6.TD signatures (Row 1) and effective bandwidths (Row 2) of polarization-resolved outputs from the VCSEL.Column A: x-PC.Column B: y-PC.The VCSEL is subject to feedback from mirror (solid lines), FBG under f m = 0 (circles) and f m = 15 GHz (triangles), respectively.The magnitude of the coupling coefficient κ B is varied for the FBG.The feedback rate is fixed at η = 20 ns −1 .
indicate the values when κ B is set to infinity in realizing mirror feedback into the VCSEL.Generally, from Row 1 of Fig. 6, FBG is clearly superior to mirror for suppressing chaotic TD signatures in the VCSEL.Deviating f m from zero #238400 -$15.00USD Received 20 Apr 2015; revised 29 May 2015; accepted 30 May 2015; published 3 Jun 2015 (C) 2015 OSA