Suppression of Time Delay Signature in Chaotic Semiconductor Lasers with Filtered Optical Feedback

Abstract

We report the investigation of the time delay signature in chaotic semiconductor subject to delayed filtered optical feedback, of which a Fabry-Pérot interferometer type of filter is placed in the external cavity. The numerical results show that the suppression of time delay is dependent on the spectral width of the filter and its detuning from the solitary laser frequency. Such laser system will be helpful to acquire high quality chaotic signals.

1. Introduction

Semiconductor lasers with delay external optical feedback are known to exhibit rich nonlinear dynamical output behavior [1, 2]. Because of its unique virtues, for example, simple configuration, small size, and feasible controllability, it has been widely applied in chaos-based communication system [3], fast random bit sequences generator [4], chaotic lidar [5] and chaotic optical time-domain relectometry (chaotic-OTDR) [6], and so forth. Nevertheless, for conventional optical feedback (COF), the output waveform from the semiconductor laser subject to the mirror feedback often presents the undesirable time delay signatures. The appearance of the time delay signatures in the dynamics compromises the performances of the system. In the context of chaos-based communications, the time delay signature allows an eavesdropper to successfully retrieve the time delay, which, in some systems, is enough to reconstruct the underlying chaotic dynamics [7]. For random bit generation, the time delay signature in the dynamics will significantly affect the statistical performance of the bit sequences [8]. For chaotic lidar and chaotic-OTDR, the peak of the time delay in correlation curve deteriorates the signal-noise-rate and decreases the dynamic range. Therefore, in order to provide the high quality chaotic sources, the elimination of time delay is a critical issue.

Recently, much effort has been devoted to the study of time delay concealment. For COF, the researches show the time delay signature can be concealed by adjusting feedback strength or adding the number of feedback cavities using autocorrelation function and delay mutual information [9–14]. However, quite recently, the researches propose several new methods and present the time delay signatures that can be easily observed and reconstructed in these scenarios for COF [15–18]. Furthermore, in other generating chaos projects, the time delay signature that can be suppressed, but at the cost of increasing the setup complexity [19–21]. These research activities motivate our effort to develop a novel and simple method for generating time delay signature eliminated chaotic signals.

Like mirror feedback, semiconductor laser with filtered optical feedback (FOF) also can be brought to chaotic behaviors. FOF has become a concerned topic of interest, since it has the potential of control over the laser dynamics via two external parameters, the spectral width of the filter and its detuning from the solitary laser frequency [22–24]. But, to the best of our knowledge, the time delay signature in chaotic semiconductor lasers with FOF has not been investigated.

In this paper, we place a Fabry-Pérot interferometer type of filter feedback in the external cavity. Using autocorrelation function and delay mutual information, we find that the suppression of time delay signature is possible in chaotic semiconductor lasers with FOF. The results show that the efficient time delay concealment in FOF mainly depends on the filter linewidth and the frequency detuning from the solitary laser.

2. Model and Method

The sketch of a semiconductor laser with FOF is shown in Figure 1. Filtered feedback can be obtained from a grating or a Fabry-Pérot interferometer. In our case we consider a Fabry-Pérot interferometer reflection in FOF. The frequency dependent reflectivity 𝑟(𝜔) in the FOF is described by a Lorentzian function with filter width (half-width at half-maximum) Λ and detuning Δ𝜔 with respect to the solitary laser frequency 𝜔₀, that is, 𝑟(𝜔)=Λ().Λ+𝑖Δ𝜔−𝜔 (1) The distance between the semiconductor laser and filter is the external cavity length 𝐿, corresponding to the round-trip time of the external cavity 𝜏=2𝐿/𝑐, which is the time delay signature.

Figure 1 

Sketch of a semiconductor laser subject to filtered feedback.

The optical field in the laser is represented by 𝐸(𝑡)=𝐸(𝑡)exp{𝑖𝜔₀𝑡}+c.c., while 𝐸(𝑡) is the (complex) slowly varying amplitude. The laser dynamics in the external feedback with FOF can be described by the set of rate equations [22]: 𝑑𝐸(𝑡)=1𝑑𝑡2(𝑔𝑁(𝑡)1+𝑖𝛼)−𝑁₀1+𝜀|𝐸(𝑡)|²−𝜏_𝑝⁻¹𝐸(𝑡)+𝜅𝜏_in𝐹(𝑡),𝑑𝐹(𝑡)()𝑒𝑑𝑡=Λ𝐸𝑡−𝜏^{−𝑖𝜔0𝜏}+()𝐹(𝑡),(𝑡)𝑖Δ𝜔−Λ𝑑𝑁=𝐼𝑑𝑡−𝑁(𝑡)𝑞𝑉𝜏_𝑁−𝑔𝑁(𝑡)−𝑁₀1+𝜀|𝐸(𝑡)|²|𝐸(𝑡)|², (2) where 𝜅 denotes the feedback strength of external cavity, 𝐹(𝑡) is the (complex) field amplitude reentering from the laser cavity, 𝑁(𝑡) is carrier density, and 𝐼 is pump current. The internal parameters of the laser are considered as follows: transparency carrier density 𝑁₀=0.455×106 μm/m³, angular frequency of solitary laser 𝜔₀=1.216×10¹⁵ rad/s, threshold current 𝐼_th=22 mA, differential gain 𝑔=1.414×10⁻³ μm⁻³ ns⁻¹, carrier lifetime 𝜏_𝑁=2.5 ns, photon lifetime 𝜏_𝑝=1.17 ps, round-trip time in laser intracavity 𝜏_in=7.38 ps, line-width enhancement factor 𝛼=5.0, gain saturation parameter 𝜀=5×10⁻⁵ μm³, and active laser volume 𝑉=324 μm³.

For a delay-differential system, the time delay signature can typically be retrieved employing either the autocorrelation function (ACF) or the delay mutual information (DMI). The ACF measures how well time series matches its time-lagged version, which is defined as 𝐶(𝜏)=⟨(𝐼(𝑡+𝜏)−⟨𝐼(𝑡)⟩)(𝐼(𝑡)−⟨𝐼(𝑡)⟩)⟩(𝐼(𝑡+𝜏)−⟨𝐼(𝑡)⟩)²(𝐼(𝑡)−⟨𝐼(𝑡)⟩)^21/2, (3) where 𝐼(𝑡) represents chaotic time series and 𝜏 is the time lag. The delay mutual information (DMI) measures the information shared by two variables, which is based on information theory. The DMI between 𝐼(𝑡) and 𝐼(𝑡+𝜏) is described by 𝑀(𝜏)=_{𝐼(𝑡),𝐼(𝑡+𝜏)}𝑝(𝐼(𝑡)(𝑝(𝐼(𝑡)(,𝐼𝑡+𝜏))log,𝐼𝑡+𝜏))𝑝(𝐼(𝑡𝑝(𝐼(,))𝑡+𝜏)) (4) where 𝑝(𝐼(𝑡)) and 𝑝(𝐼(𝑡+𝜏)) are the marginal probability densities and 𝑝(𝐼(𝑡), 𝐼(𝑡+𝜏)) is the joint probability density. The time delay signature can be retrieved from the peak location of ACF or DMI curve.

3. Numerical Results

The typical results for FOF and COF with 𝜏=5 ns are comparatively displayed in Figure 2. The left column presents the chaotic time series, power spectrum, and corresponding analysis function curves for the semiconductor laser with mirror feedback for 𝜅=15% and 𝐼=1.5𝐼_th, respectively. The right column presents the corresponding results for the semiconductor laser with Fabry-Pérot interferometer feedback for 𝜅=15% and 𝐼=1.5𝐼_th, the width of filter Λ=15 GHz and frequency detuning Δ𝜈=8 GHz. From the time series (Figures 2(a1) and 2(a2)), it is very difficult to identify the time delay signature directly since chaotic oscillations behave intricately. However, from the power spectrum (Figure 2(b1)) corresponding to mirror feedback, some uniform spacing peaks emerge upon the background and reveal the external cavity characteristic frequency 0.2 GHz. Thus, the value of time delay could be estimated as 5 ns. Meanwhile, from the ACF curve and DMI curve (Figures 2(c1) and 2(d1)), the clear peaks appear at the integer multiples of 𝜏, which reveal the time delay signature. In contrast, for the FOF, the power spectrum (Figure 2(b2)) becomes smooth and has no significant frequency peaks upon background. In addition, no apparent peaks appear in the corresponding ACF and DMI curve (Figures 2(c2) and 2(d2)). These results indicate that the time delay signature of the chaotic signal for the FOF is almost completely suppressed. Compared with mirror feedback, the two special parameters for FOF, the spectral width of the filter and its detuning from the solitary laser frequency, play the critical role in the concealment of time delay signature.

Figure 2 

Chaotic time series, power spectrum, associated ACF curves, and DMI curves for 𝜅=15% with mirror feedback (left column) and Fabry-Pérot interferometer feedback (right column), respectively. We consider the width of filter Λ=15 GHz and frequency detuning Δ𝜈=8 GHz.

Next, we investigate the influence of filter width and frequency detuning on the possibility of the identification of time delay signature. Firstly, we maintain feedback strength 𝜅=15% and frequency detuning Δ𝜈=0 and adjust filter width. We show the time series, ACF curves, and DMI curves for different filter width in Figure 3. From Figures 3(b1) and 3(c1), we observe that the time delay signature is clearly exhibited when Λ=17 GHz. Combining all ACF and DMI curves, we notice the amplitudes of time delay signature in ACF and DMI are gradually suppressed as Λ decrease from 17 to 3 GHz. Particularly, when the width decreases to 3 GHz in third row (Figures 3(b3) and 3(c3)), the TD signature is almost shielded completely into the background fluctuations. Furthermore, we show the evolution tendency of time delay signature amplitude in ACF with varying filter widths Λ from 4 GHz to 41 GHz in Figure 4. We observe the amplitude gradually decrease with decreasing Λ.

Figure 3 

Time series (a1)–(a3), ACF curves (b1)–(b3), and DMI curves (c1)–(c3) for the different widths of filter, where Λ=17 GHz (first row), Λ=7 GHz (second row), Λ=3 GHz (third row) keeping time delay 𝜏=5 ns, frequency detuning Δ𝜈=0, and 𝜅=15%.

Figure 4 

The variation of amplitude in ACF with the width of filter keeping time delay 𝜏=5 ns, frequency detuning Δ𝜈=0 and, 𝜅=15%.

In what follows, we investigated the influence of the frequency detuning Δ𝜈 between the filter and the solitary laser on the suppression of the time delay signature. Keeping the width of filter Λ=17 GHz, the time series, ACF and DMI curves were presented in Figure 5 when Δ𝜈 varies from −12 GHz to 14 GHz. At first, for Δ𝜈=−12 GHz as shown in Figures 5(a1)–5(c1), obvious peaks are observed in both ACF and DMI curves, but the time location corresponding to peak is 5.5 ns, which deviates from the expected delay time 𝜏. This phenomenon may provide a pseudo time delay signature. Further adjusting Δ𝜈 to −2 GHz, obvious time delay signature can be properly identified from ACF and DMI curves. When Δ𝜈 is turned to 8 GHz, the peak amplitude is significantly suppressed close to the background in ACF and DMI curves as shown in Figures 5(a3)–5(c3). In this situation, an eavesdropper would be quite difficult to accurately identify the time delay of the semiconductor dynamical system. However, increasing Δ𝜈 to 14 GHz (Figures 5(a4)–5(c4)), the amplitudes of ACF and DMI boom again and the time location of peak also deviates again from the expected delay 𝜏. Thus, adjusting the frequency detuning Δ𝜈, not only the amplitude in ACF or DMI for 𝜏 is influenced, but also the time location of peak is affected.

(a)

(b)

(c)

(d)

Figure 5 

Time series (a1)–(a4), ACF curves (b1)–(b4), and DMI curves (c1)–(c4) for the different frequency detuning, where Δ𝜈=−12 GHz (first row), Δ𝜈=−2 GHz (second row), Δ𝜈=8 GHz, (third row), and Δ𝜈=14 GHz (fourth row) keeping time delay 𝜏=5 ns, width Λ=17 GHz, and 𝜅=15%.

To further show the evolution patter of time delay signature under different frequency detuning Δ𝜈, the variation curves of amplitude and the location of the ACF peak with Δ𝜈 in a time window of 1.0 ns around 𝜏 are shown in Figures 6(a) and 6(b), respectively. From Figure 6, the evolution of time delay could be roughly divided into three regions. For −15GHz<Δ𝜈<−4GHz, although the peak is obvious, the time location corresponding to peak deviates from the expected 𝜏, so the time delay signature cannot be properly identified. For −4GHz⩽Δ𝜈<8GHz, the time location corresponding to peak is the expected 𝜏, but the amplitude of the time delay signature attenuates significantly with increasing Δ𝜈. When the peak amplitude in ACF is less than 0.1, the time delay signature is almost shielded completely into the background fluctuations. Particularly, when Δ𝜈=8 GHz, the amplitude decreases to the minimum. For 8GHz<Δ𝜈<15GHz, although the peak booms again, the deviation between time location of the peak and the expected 𝜏 again becomes large with increasing Δ𝜈. This phenomenon may provide a pseudo time delay signature. Therefore, one can reasonably deduce that the time delay suppression owns certain tolerance to frequency detuning Δ𝜈 under specific filter width.

(a)

(b)

Figure 6 

The variation of amplitude and time location in ACF with the frequency detuning Δ𝜈 are shown in (a) and (b), respectively.

4. Conclusion

In conclusion, we have demonstrated the time delay signature elimination phenomenon in a semiconductor laser with FOF, which consists of a Fabry-Pérot interferometer type of filter in the external cavity. We find that the time delay signature can be concealed in chaotic semiconductor lasers with FOF under proper filtered width and frequency detuning in terms of autocorrelation function and delay mutual information. We found the scenarios for efficient time delay concealment that would be helpful to acquire high quality chaotic signals in some applications.

Conflict of Interests

The authors declare no competing financial interests.

Acknowledgments

This work was supported in part by the National Natural Science Foundation (NSFC) of China under Grants 61227016, 61205142, and 61240017, the Natural Science Foundation of Shanxi Province under Grants 2013011019-3 and 2012021013-2, and the Program for the Outstanding Innovative Teams of Higher Learning Institutions of Shanxi, China (OIT).