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Abstract
Time-delay signature (TDS) suppression of an external-cavity semiconductor laser (ECSL) is important for chaos-based applications and has been widely studied in the literature. In this paper, the chaotic output of an ECSL is injected into a semiconductor laser and TDS suppression in the regenerated time series is revisited. The focus of the current work is the influence of parameter mismatch on the TDS evolution, which is investigated experimentally and compared systematically to simulations. The experimental results demonstrate that it is much easier to achieve desired TDS suppression in the configuration composed of mismatched laser pairs. Numerical simulations confirm the validity of the experimental results. In the experiments and simulations, the influence of the injection parameters on TDS suppression is also studied and good agreement is obtained.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical chaos generated by semiconductor lasers has been widely studied due to its potential applications in secure communications [1,2], random number generation [3,4], chaotic lidar [5], and optical time-domain reflectometry [6]. To realize chaotic operation, semiconductor lasers should be subjected to certain external perturbation including optical feedback [7], optical injection [8], optoelectronic feedback [9], and current modulation [10]. Among them, the optical feedback architecture is a simple and widely used configuration for the production of laser chaos, which is also called an external-cavity semiconductor laser (ECSL). However, optical chaos provided by an ECSL exhibits a kind of periodicity–termed time-delay signature (TDS) in the literature [11,12], which can be easily acquired by using simple statistical measures, e.g., autocorrelation function (ACF), and, thus, compromises the performance of chaos-based applications. To maintain or enhance the performance, such TDS should be suppressed or removed. Therefore, TDS suppression has been a hot area of research for more than a decade; see Refs. [13–15] and references therein.
In the literature, a lot of interesting TDS schemes have been proposed and demonstrated theoretically/experimentally [16–23]. Generally speaking, there are two types of methods for TDS suppression. In the first type, the conventional feedback for an ECSL has been replaced with several alternative schemes, such as double feedback [24], phase-modulated feedback [25], grating feedback [26], random/scattering feedback [27,28], and polarization-rotated optical feedback [29]. However, the bandwidth of optical chaos from this type is narrow and cannot afford high-speed chaos communications and ultrafast random number generation. The second mechanism for TDS suppression relies on the post-processing of chaotic ECSL outputs. For example, the optical chaos obtained from an ECSL has been injected to another slave laser (SL) and, as a result, bandwidth enhancement and TDS suppression have been realized simultaneously [30]. This approach has been generalized to injection into other active or passive components, such as two cascaded-coupled SLs [31], optical fibers [32,33], and a microsphere resonator [34]. All of these alternatives allow for broadband chaos generation and excellent TDS suppression, which demonstrate feasibility and practicality of concealing the TDS from an ECSL via optical injection. Especially, an SL subjected to chaotic optical injection, as a laser source for generating broadband chaos without TDS, is most attractive [30]. This is due to the fact that its size is relatively compact and it only adopts commercial available off-the-shelf components. Yet despite all these achievements, the influence of parameter mismatch on the TDS suppression remains unclear–an area where little research has been done [35]. In practical engineering applications, parameter mismatch between two lasers is inevitable, even though they are grown from the same wafer. We expect that certain parameter mismatch helps the SL to generate desired optical chaos in wider parameter space, where TDS is almost completely suppressed. In this paper, we carry out experiments for TDS suppression, which compare three pairs of master-slave configurations. Among these, one pair is carefully selected and best matching is expected, while the other two mismatched pairs are deliberately chosen for better TDS suppression associated with parameter mismatch. These results are compared to the simulations based on the rate equations, and consistency between experimental results and simulations is achieved.
This paper is organized as follows. The experimental setup and the theoretical model are described in Section 2, respectively. The corresponding results are given in Section 3, where ACF is used for TDS study both in experiments and simulations. Finally, we summarize the results in Section 4.
2. Experimental setup and theoretical model
As mentioned above, a semiconductor laser subjected to chaotic optical injection is used for suppressing the TDS from an ECSL, which is the same as our previous scheme [30,35], but the effect of parameter mismatch on TDS performance is addressed here. In the experiment, a commercial distributed feedback (DFB) laser is used as the master laser (ML), and three other DFB lasers are used for the SL, respectively. Therefore, three pairs of DFB lasers are used to form three similar master-slave configurations. The resulting experimental setup is shown in Fig. 1. A fiber coupler (FC), a variable attenuator (VA), and an optical fiber mirror (OFM) form a feedback loop for ML, and thus an ECSL is realized, where the resulting delay time is about 83.58 ns. The ML output is injected to SL through the same FC, another VA, and an optical circulator (OC). Note that polarization states in the fiber link are controlled and matched by the corresponding polarization controller (PC) to maximize the field interaction; the feedback power and injection power ($\def\upmu{\unicode{x00B5}}{P_i}$) are adjusted by VA. Under proper conditions, both lasers yield chaotic signals, which are sent to a photodetector (PD, HP 11982A) for photoelectric conversion, followed by an oscilloscope (OSC, LeCroy WaveMaster 820Zi-B) for data acquisition. To prevent unwanted optical feedback from PD, an optical isolator (ISO) is inserted between OC and PD.
Fig. 1. Experimental setup of a semiconductor laser subjected to chaotic optical injection for TDS suppression. ML: Master laser; SL: Slave laser; FC: Fiber coupler; PC: Polarization controller; VA: Variable attenuator; OFM: Optical fiber mirror; OC: Optical circulator; ISO: Optical isolator; PD: Photodetector; OSC: Oscilloscope. Note that three different SLs are used in turn for comparison.
In the experiment, the threshold current of the four lasers is approximately 12 mA, and they are driven by a current source and controlled by a thermoelectric controller. The ML is biased at 25 mA and its temperature is fixed at 25 ${^\circ{\textrm C}}$, which results in a lasing wavelength of 1551.173 nm. As for the SL, the bias current is set to be 30 mA, while its temperature is continuously controlled for introducing frequency detuning $\Delta f = {f_{ML}} - {f_{SL}}$ between ML and SL. Here ${f_{ML}}$ and ${f_{SL}}$ are frequencies of free-running ML and SL, respectively. It is worth noting that ML and SL1 are well matched, while their key parameters are significantly different from those of SL2 and SL3. This arrangement allows for addressing the influence of parameter mismatch on TDS suppression.
To confirm our experimental findings, we turn to simulate the theoretical model and start with modifying the Lang-Kobayashi rate equations [7] to account for the master-slave configuration, where an ML is subjected to optical feedback and its output is unidirectionally injected into an SL. This model has been successful in reproducing dynamical behaviors experimentally observed [30]. The rate equations for slowly varying complex electric field $E(t )$ and carrier density $N(t )$ are written as [30,35]
A fourth-order Runge-Kutta algorithm is used to solve Eqs. (1)–(3), where the time step is 1 ps. The parameter values of the ML used in the simulation are [30]: $\alpha = 5,$ ${N_0} = $. 1.5×108, g = 1.5×104, $\varepsilon = {\; }$5×10−7, ${\tau _p} = {\; }$2 ps, ${\tau _e} = \; $2 ns, ${\tau _c} = \; $2 ns, ${k_f} = \; $16 ns-1, ${\tau _f} = {\; }$3 ns, ${f_1} = {\; }$193.55 THz, and $q = \; $1.602×10−19 C. With these parameters, the ML operates in the chaotic state with obvious TDS. The parameter values of the SL will be varied for the consideration of parameter mismatch.
For TDS estimation, ACF is used throughout the paper due to its simplicity and efficiency. The definition of ACF is written as [36]
3. Experimental and theoretical results
The principal aim of this paper is to experimentally study the effect of parameter mismatch on the TDS concealment in the chaotic output of the SL with chaotic injection from an ECSL. We are interested in the chaos operation, so the feedback power of the ML is carefully controlled. Figures 2(a1) and 2(a2) show the experimental time series and the corresponding power spectrum of the ML, respectively. The spectrum is calculated based on Fast Fourier Transform (FFT) of the time trace. The random-like appearance of the time trace and the broadband spectrum demonstrate that the ML operates in a chaotic state. As mentioned in the last section, three SLs are prepared for the experiment: SL1 matches well with ML, while SL2 and SL3 are quite different from ML. The ML output is injected into SL1, SL2, and SL3 in three non-interfering experiments, where the TDS performance of SL1, SL2, and SL3 is carefully compared as shown below. Under proper conditions, the SL will achieve chaotic operation. For example, Figs. 2(b)–2(d) present the time series and power spectra of the three SLs under same injection parameters of $\Delta f ={-} 13.2\; \textrm{GHz}$ and$\; {P_i} = 500\; {\upmu {\rm W}}$. As can be seen clearly, all of them operate in a chaotic regime and their power spectra are greatly expanded when compared to the ML. This is expected since optical injection is a well-known approach for bandwidth enhancement [37]. However, it is worth noting that limited by the bandwidth (about 15 GHz) of PD used in the experiment, the power spectrum over 15 GHz is cut off. This issue is not the focus of the current study and the interested reader can refer to the prior work for details.
Fig. 2. (a1-d1) Experimental time series and (a2-d2) power spectra (right) of the (a1, a2) ML, (b1, b2) SL1, (c1, c2) SL2, (d1, d2) SL3, where the injection parameters are $\Delta f ={-} 13.2\; \textrm{GHz}$ and$\; {P_i} = 500{\; \upmu {\rm W}}$.
Attention is now given to the TDS performance. Firstly, the ACF of the time series shown in Fig. 2(a1) for the ML is calculated and the result is shown in Fig. 3. It can be observed that the ACF curve has a significant peak at the feedback time delay (θ ∼ 83.58 ns) and its harmonics. This is the typical result for an ECSL [11], and the concealment of such TDS has been widely explored [14–18]. In this study, we expect that parameter mismatch is beneficial to TDS suppression for the chaotic source consisting of a semiconductor laser with chaotic optical injection from an ECSL.
The ACF plots of the SL1 with different injection power are illustrated in Fig. 4 when the detuning is fixed at $\Delta f ={-} 13.2\; \textrm{GHz}$. For small values of the injection power (i.e., 50 ${\upmu }$W and 100 ${\upmu }$W), the SL1 is not injection-locked by the ML, and thus exhibits nonlinear dynamics different from the driving signal. However, one can still identify a small peak around the time delay value in the ACF plots, as shown in Figs. 4(a) and 4(b).This indicates that there exists weak correlation property (due to the feedback periodicity) in the SL1 output, which is inherited the ML. As the injection power is increased to 250 ${\upmu }$W, the autocorrelation becomes stronger and a pronounced peak could be identified from the ACF plot [Fig. 4(c)]. When the injection power is further increased, injection-locking phenomenon occurs. For example, Fig. 4(d) displays the results for ${P_i} = 500{\; \upmu {\rm W}}$, where the ACF plot of the SL1 shows a significant peak at the feedback time delay and its 2nd harmonics. In this case, the ACF plot of the SL1 is very similar to that of the ML (see Fig. 3). This observation is consistent with our previous work [30], where two matched lasers were used for a similar experiment.
Fig. 4. ACF plots of the SL1 at $\Delta f ={-} 13.2{\; \textrm{GHz}}$ for several different injection powers: (a) 50 ${\upmu }$W, (b) 100 ${\upmu }$W, (c) 250 ${\upmu }$W, and (d) 500 ${\upmu }$W.
Another similar experiment based on the SL2 subjected to chaotic optical injection from the ML has been carried out, where the parameters for the ML and SL2 are not well matched. The results of ACF estimated from the SL2 are shown in Fig. 5. As can be seen, the SL2 allows for complex chaotic signal generation when the injection power is increased from 50 ${\upmu }$W to 250 ${\upmu }$W. These signals can be termed strong chaos since there is no obvious TDS in the calculated ACF plots. Despite the mismatch, the ACF plot of the SL2 will also show a clear TDS for a strong injection power of 500 ${\upmu }$W. In comparison to Fig. 4, however, it is shown in Fig. 5 that much better performance of TDS suppression has been achieved for mismatched lasers used in the master-slave configuration.
Fig. 5. ACF plots of the SL2 at $\Delta f ={-} 13.2{\; {\rm{GHz}}}$ for several different injection powers: (a) 50 ${\upmu }$W, (b) 100 ${\upmu }$W, (c) 250 ${\upmu }$W, and (d) 500 ${\upmu }$W.
Likewise, the SL1 or SL2 is replaced by the SL3 for the experiment and the corresponding results are shown in Fig. 6. It is shown that, for all of the injection powers considered, the ACF plots of the SL3 exhibit no clear pattern for correlation property, which resembles the noisy feature. This means that the SL3 cannot be injection locked by the ML for the given conditions. This further confirms that mismatched lasers in the master-slave configuration are beneficial to generating more complex chaotic signals, which results in significant reduction in the ACF peak or TDS concealment compared to that of matched lasers.
Fig. 6. ACF plots of the SL3 at $\Delta f ={-} 13.2{\; {\rm{GHz}}}$ for several different injection powers: (a) 50 ${\upmu }$W, (b) 100 ${\upmu }$W, (c) 250 ${\upmu }$W, and (d) 500 ${\upmu }$W.
The above results are obtained by varying the injection power with a fixed frequency detuning. Actually, the chaos characteristic of the SL depends on two injection parameters including the injection power and frequency detuning. Therefore, it is important to study their effect on the TDS suppression in a wide parameter range for better understanding. In order to quantify the influence of the injection power and the frequency detuning on the TDS concealment, the following study focuses on the peak value of the ACF near the feedback time delay (θ ∼ 83.58 ns), that is, the amplitude of the maximum ACF peak in the time window [80, 85] ns, which indicates the strength of the TDS.
Figure 7 illustrates the TDS evaluation of the three SLs used above as a function of the frequency detuning for various injection powers. Three features can be identified from these plots. Firstly, in the three cases of SLs, for small injection powers, TDS is significantly suppressed or even concealed for a wide range of the frequency detuning. This is because that the ECSL only acts as the driving system and does not lock the SL output. Secondly, the ACF peak profiles for the three cases look similar when the injection power is relatively small (e.g., 50 ${\upmu }$W). This indicates that the SL output is really originated from the same ML. Thirdly, a careful comparison among the curves for large injection powers (e.g., 250 ${\upmu }$W and 500 ${\upmu }$W) shows that the detuning range for small ACF peak values is much wider in Figs. 7(b) and 7(c) when compared to Fig. 7(a). This confirms our conjecture that parameter mismatch between the ML and SL indeed contributes to TDS suppression in the SL output.
Fig. 7. ACF peak value as a function of frequency detuning for various injection powers. (a) SL1, (b) SL2, and (c) SL3.
Now we turn to theoretical analysis for confirming the experimental observations. We focus on the effect of parameter mismatch between ML and SL in this paper, so the relative mismatch ratio u is introduced to control the difference, similar to the definition in our prior work [38]. We assume that the parameters of the ML are kept unchanged, and some of the key parameters of the SL are varied according to the ratio u. These parameters are the photon lifetime ${\tau _p}({1 - u} )$, the carrier lifetime ${\tau _e}({1 + u} )$, the linewidth-enhancement factor $\alpha ({1 - u} )$, the gain coefficient g$({1 - u} )$, and the carrier density ${N_0}({1 + u} )$. Without loss of generality, the bias current of the ML and SL is set to be 22.05 mA, which is well above the threshold of 14.7 mA. With the specified parameter values, the TDS of the ACF computed from the ML time series is pronounced. The ACF plot shows a peak value of ∼0.45 located at the feedback time delay, which is not shown here for concision. Like the experiment, we extract the amplitude of the maximum ACF peak in the time window around the feedback time delay, i.e., [2.5, 3.5] ns, in order to quantify the influence of parameter mismatch on the TDS concealment in the SL.
Figure 8 shows the ACF peak of the SL as a function of the frequency detuning for various values of the injection rate. We first analyze the TDS evolution of the SL without parameter mismatch ($u$=0), as shown in Fig. 8(a). It can be observed that TDS suppression can be achieved in a wide range of the frequency detuning for low injection rates (e.g., <∼20 ns-1). However, it is difficult to suppress TDS for much higher injection rates, i.e., ${\xi \; }$=35 and 50 ns-1. It is interesting to find that these findings coincide with the experimental observations shown in Fig. 7. When the parameter mismatch is introduced deliberately, for example, $u ={-} 0.1$, the performance of TDS suppression is improved for higher injection rates; see the curve for ${\xi \; }$=35 ns-1 in Fig. 8(b). As the mismatch ratio $|u |$ continues to increase, TDS suppression becomes more impressive in a much wider range of the frequency detuning; see the trend by comparing the results in Figs. 8(c) and 8(d) with Fig. 8(b). The theoretical evidence is consistent with those experimental results shown in Fig. 7, confirming that parameter mismatch could enhance the TDS suppression in the SL subjected to chaotic optical injection from an ECSL.
Fig. 8. ACF peak value as a function of frequency detuning for various injection rates under parameter mismatch: (a) $u$=0, (b) $u$=-0.1, (c) $u$=-0.2, and (d) $u$=-0.4.
4. Conclusion
We have studied the effect of parameter mismatch on the TDS suppression in a semiconductor laser subjected to chaotic optical injection experimentally and theoretically. In the experiments, three pairs of semiconductor lasers have been used for the ML and SL in the master-slave configuration: one pair is well matched, while the other two pairs are randomly chosen. The experimental results show that better TDS suppression can be obtained for the mismatched lasers compared to the matched pair. Numerical simulations based on the modified Lang-Kobayashi equations confirm that certain parameter mismatch between the ML and SL is beneficial to TDS suppression, which is in good agreement with the experimental observation.
Funding
Soochow University (GJ15900519, GJ15900119, Q415900119).
Disclosures
The authors declare no conflicts of interest.
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