A semiconductor laser with delayed optical feedback is an experimental implementation of a nominally infinite dimensional dynamical system. As such, time series analysis of the output power from this laser system is an excellent test of complexity analysis tools, as applied to experimental data. Additionally, the systematic characterization of the range and variation in complexity that can be obtained in the output power from the system, which is available to be used in applications like secure communication, is of interest. Output power time series from a semiconductor laser system, as a function of the optical feedback level and the laser injection current, have been analyzed for complexity using permutation entropy. High resolution maps of permutation entropy as a function of optical feedback level and injection current have been achieved for the first time. This confirms prior research that identifies a coherence collapse region which is found to be uninterrupted with respect to any embedded islands with different dynamics. The results also show new observations of low optical feedback dynamics which occur in a region below that for coherence collapse. The map of the complexity shows a strong dependence on the delay time used in the permutation entropy calculation. Short delay times, which sample information at the complete measurement bandwidth, produce maps with drastically different systematic variation in complexity throughout the coherence collapse region, compared to maps generated with a delay time that matches the optical feedback delay. Evaluating the complexity with a permutation entropy delay equal to the external cavity delay produces results consistent with the notion of weak/strong chaos, as well as categorizing the dynamics as being of high complexity where the external cavity delay time is harder to identify. These are both desirable features for secure communication applications. The results also show permutation entropy as a function of delay time can be used to detect key frequencies driving the dynamics, including any that may exist due to, or arise from, technicalities of device fabrication and/or noise. A more complete insight into complexity as measured by permutation entropy is gained by considering multiple delay times.
© 2014 Optical Society of America
The effects of external perturbations on the output power dynamics of semiconductor lasers are well known . In most cases, when operated in isolation, semiconductor lasers are inherently stable. However, they have been shown to be very sensitive to perturbations including optical injection from another device , direct injection current modulation , delayed optical feedback from an external mirror , intra-cavity electro-optic modulation of optical feedback , and combinations of these. Of these different system designs, delayed optical feedback is the simplest to implement and should produce highly complex dynamics due to the infinite degrees of freedom allowed when the optical field’s amplitude and phase are coupled together with the carrier density with the introduction of the feedback .
There has been much interest in characterizing the complexity of these chaotic signals as a means of quantifying the level of security when they are used as transmitters and receivers in chaotic secure communication schemes [7, 8]. The unpredictability of the output power time series is important from the point of view that an eavesdropper cannot forecast or generate the chaotic carrier signal . There are several other issues related to improving security and functionality of this type of system, such as the need to mask characteristic time scales inherent to the transmitter (e.g. external cavity length) [10–12], and also optimizing the conditions which provide the best synchronization and message extraction . A number of techniques for quantifying time series complexity and unpredictability have been proposed [14–16]. Most are computationally expensive, not easily applicable to experimental data, and/or may give results which are not verifiable as correct for data which is noise effected even with only small amounts of noise. One method, known as permutation entropy , is a conceptually simple idea for quantifying complexity based on ordinal patterns in time series data. The method of generating the probability distributions on which the entropy is calculated (explained in Section 2) means the result is a relative measure of time series predictability. Permutation entropy analysis is easily implemented and computationally much faster than other techniques such as Lyapunov exponents , while being robust to noise .
In this work, we analyze an experimental semiconductor laser with optical feedback system by calculating permutation entropy as a function of laser injection current and optical feedback level. Dynamic maps show, for the first time, in extremely fine detail, the variation in complexity across the coherence collapse regime. Assessing the permutation entropy at different time scales is also shown to be a very sensitive measure of frequencies present in the dynamics.
2. Permutation entropy - background
The degree of disorder or uncertainty in a system can be quantified by a measure of entropy. The uncertainty associated with a physical process described by the probability distribution is related to the Shannon entropy17]. This symbolic approach based on the relative amplitude of time series values is much more robust to noise and invariant to nonlinear monotonous transformations (e.g. measurement equipment drift) when compared with other complexity measures. This makes it particularly attractive for use on experimental data.
The process of generating the probability distribution according to the Bandt and Pompe methodology is described in detail elsewhere [17, 19]. It is briefly outlined here. To obtain the ordinal pattern distribution one must choose an appropriate ordinal pattern length D and delay τ. Since there are D! possible permutations for a vector of length D, the choice should be influenced by the length of the measured time series. It has been suggested that in order to obtain reliable statistics the length of the time series N should be much larger than D . For practical purposes it is recommended to use values of D between 3 and 7.
The delay τ is the time separation between values used to construct the vector from which the ordinal pattern is determined. Its value corresponds to a multiple of the signal sampling period. Changing this delay enables the complexity of your time series to be analyzed at different time scales. The investigation of complexity maps generated from permutation entropy calculations at different time scales is an important element of the new results reported in this paper.
For a given a time series, ordinal pattern length D, and delay τ, we consider the vector
Permutation entropy analysis has been performed previously on simulated output power time series data from a semiconductor laser with optical feedback, operated as a chaotic laser. The behavior of the simulated system was quantified for several feedback-strength and injection-current settings at the time delay of the external cavity . We extend this to maps of the dynamics of an experimental system of high resolution which show the systematics in complexity quantified by permutation entropy.
3. Experiment - semiconductor laser with delayed optical feedback
The experimental setup, shown in Fig. 1, consists of a multiple quantum well 830 nm semiconductor laser (APL 830-40) which has a free-running room temperature threshold of 48.2 mA. The output beam is collimated with an 8 mm focal length aspheric lens. The beam then passes through a 50:50 cube beamsplitter and an acousto-optic modulator (G&H 23080) before being reflected from an external mirror (R = 99%). It is the zeroth order beam from the acousto-optic modulator (AOM) which is coupled back to the semiconductor laser as the delayed optical feedback. The beam splitter directs half of the output beam to a 22 GHz photodiode. The photodiode signal was recorded for 1 µs (20 kpts) on a digital oscilloscope (Agilent Infiniium 54854A DSO) with a 4 GHz real-time bandwidth at a sampling rate of 20 GSa/s to constitute a single time series for subsequent analysis. The external cavity established by the external reflector has a round trip length of 135 cm.
A high density data set of experimental output power time series was generated that could then be analyzed using a number of different analysis tools to generate high resolution maps. These quantify different aspects of the nonlinear dynamics at any point in the parameter space. The parameter space was injection current and the fraction of delayed optical feedback. The injection current was swept from 45 mA to 70 mA in steps of 0.1 mA and the optical feedback level was varied by adjusting the 0th order transmission of the AOM from 75.5% to 6.5% (in 351 non-uniform steps sizes), giving a total of 88,101 time series covering the operating region of interest. The transmission of the AOM is used as the variable which is proportional to the optical feedback level in the maps that are presented. At the resolution achieved inspection for any small islands of differentiated dynamics was possible with the data set.
4. Results and discussion
Figure 2 shows detailed maps of the (a) RMS amplitude and (b) normalized permutation entropy (ℋ𝑆) for the laser system over the range of the injection-feedback parameter space which shows complex dynamics. All maps were generated with decreasing optical feedback level and increasing injection current. No hysteresis was observed using increasing feedback or decreasing injection current. The RMS amplitude map identifies where the laser displays fluctuations in the output power, the bulk of which occur in the coherence collapse region  bounded by distinct upper and lower boundaries. Areas of low amplitude correspond to CW operation or the laser operating below threshold, indicated by the solid white curves in Fig. 2.
The map of permutation entropy in Fig. 2(b) provides a relative quantitative measure of the complexity of the output power fluctuations in the time series. Permutation entropy was calculated for ordinal pattern length D = 5 and delay of τ = 2 points. The length of the ordinal pattern was chosen to be large enough to allow for a reasonable distribution of possible ordinal patterns (D!), yet low enough for practical computation time. The delay was selected to capture the fastest laser oscillations whilst still allowing real dynamics to be distinguished from noise. Typical time series and associated RF spectra for different parts of the map marked as (i), (ii) and (iii) in Fig. 2(b) are shown in Fig. 3.
The most complex time series (ℋ𝑆 ≈ 0.99) are those recorded when the laser is operating CW or below lasing threshold. In these regions the AC recorded time series is essentially fundamental, photodetector and oscilloscope noise. The most complex region where the laser power is dynamically fluctuating occurs at high injection currents and high feedback, see region marked (i) in Fig. 2(b). The frequency spectrum of this dynamic region reveals strong peaks at multiples of the external cavity frequency (222 MHz), right up to the bandwidth limit of the measurement equipment, and also significant power at lower frequencies (< 500 MHz). There is a gradual decrease in complexity (ℋ𝑆→0.75) as the system parameters approach intermediate injection and feedback values, see region marked (ii) in Fig. 2(b). Here the external cavity peaks in the RF spectra are broader, with not as much low frequency fluctuation and less power at higher frequencies (> 3 GHz).
The maps in Fig. 2 verify previous bifurcation diagrams for such systems, confirming there are no unexpected “islands” of differentiated dynamics within the coherence collapse region (at this level of detail). However, there is a small region below the coherence collapse boundary, marked as (iii) in Fig. 2(b), where dynamical output power variations associated with a permutation entropy value of lower complexity (ℋS < 0.7), compared to the fundamental and technical noise, are observed. Such observations have not been reported by others previously to our knowledge. In this case the lower complexity corresponds to more regular dynamics as is confirmed by frequency spectra showing much sharper peaks than those seen in the coherence collapse region. The frequency spacing of these spectral peaks is approximately 210 MHz, smaller than the 222 MHz associated with the external cavity, but very close to the multiple of half the relaxation oscillation period that is closest to the external cavity delay time (as discussed later in the paper). Repeat parameter sweeps reveal that this region of dynamics is not always captured. It may be that at low feedback levels we are seeing a nonlinear enhancement of some other background signal which is not always present as was noted previously in a similar system . More investigation of this parameter space region is planned and will be reported elsewhere.
The permutation entropy quantifies the form probability distribution of different possible ordinal patterns. Figure 4 contains plots of the probability distributions as calculated for a time series with low permutation entropy, high permutation entropy, and a noisy signal from the system below lasing threshold.
The low permutation entropy probability distribution in Fig. 4(a) clearly shows a limited number of ordinal patterns which have a much greater rate of occurrence than others. Interestingly, the most frequent ordinal permutation in this region, seen at ordinal pattern index = 1, is a monotonically decreasing one. This suggests that the laser regularly exhibits sustained power drops in this region. The high permutation entropy time series has a much more uniform distribution, see Fig. 4(b), but still shows some patterns are favored significantly more than others, especially when compared to the much more uniform distribution generated from the noisy signal in Fig. 4(c).
Permutation entropy is heavily influenced by the choice of delay τ used in the calculation. There is no universally agreed upon method for selecting this delay. Practically, it sets the scale at which the complexity is evaluated. Selection should be based upon the frequency of the dynamics to be characterized and the rate at which those dynamics are sampled. Very short delays used on highly sampled low frequency dynamics give a low value of entropy, since the vectors constructed to evaluate the ordinal permutations are not long enough to effectively sample the dynamics. Conversely, if a long delay is used on high frequency dynamics, relative to the sampling rate, then the vectors do not accurately represent the dynamic variation in the time series on the timescale of the fastest fluctuations. Example experimental laser time series, taken from different regions of the parameter space are shown in Fig. 5 along with vectors created with short and long delays.
Shorter delays capture the dynamics at the fastest timescales but provide no information about slower dynamic variations. Larger delays provide information about the complexity on these longer time scales. At the experimental sampling rate of 20 GSa/s dynamics captured in the lower frequency regions of operation (region (ii)) are almost over sampled using a delay of 1 sampling period, as seen in Fig. 5(a), and consequently will give a lower value of permutation entropy. In regions where the power fluctuates at higher frequencies (region (i)), a delay of 1 sampling period is able to sample more of the time series variation, as shown in Fig. 5(b). Obviously any calculated value of permutation entropy is inherently associated with the timescale at which it was evaluated.
It was reported in  that intrinsic timescales of a semiconductor laser with optical feedback system could be identified by using permutation entropy analysis on the chaotic time series. Calculating permutation entropy as a function of delay from a laser with optical feedback results in a local minima occurring for values of delay τ that match the external cavity roundtrip time τext and integer-fractions at τext/2, τext/3,…, τext /(D-1). These extrema were observed in our experimental data at τ = τext = 90 sampling periods (4.5 ns) along with others at τext/2, τext/3,…, τext/6 for ordinal pattern length D = 7 as shown in Fig. 6. We also observe a dip at 60 sampling periods which can be attributed to an integer-fraction of 2τext. Other features present in Fig. 6 are small peaks associated with the relaxation oscillation frequency at τRO/2 and τext + τRO/2.
It has been suggested in  that permutation entropy should be evaluated using a delay τ = τext, as it reproduces results obtained with another complexity quantifier, the Kolmogorov-Sinai (KS) entropy . A map of permutation entropy for the entire parameter space with delay set to τext = 90 = 4.5 ns is shown in Fig. 7(a) and for τ = τext/2 = 45 = 2.25 ns in Fig. 7(b).
The significant drop in complexity when the delay is set to be equal to the external cavity roundtrip time is most evident when the feedback level is high. Higher levels of optical feedback mean that the dynamics are more heavily driven by the external cavity frequency, therefore causing lower complexity dynamics to occur at time scales equal to the cavity roundtrip time (and integer-fractions). When interrogated at this time scale, the regions of the parameter space that were identified as highly complex when using a short delay, refer to Fig. 2(b), now appear much more regular with lower entropy. The map of permutation entropy for delay τ = τext/2 in Fig. 7(b) shows similar features to Fig. 7(a), with lower complexity at higher feedback levels, but also shows some additional banding structure within the low complexity region. The source of this structure is the subject of further consideration.
On this scale, the maps tend to support regimes of what have been defined ‘weak’ and ‘strong’ chaos , where weak chaos is found at small and large feedback strengths and strong chaos at intermediate feedback strengths, as experimentally identified recently in a delayed feedback system  and a bi-directionally delay-coupled laser system .
Another factor to take into consideration when these systems are to be used as chaotic signal generators in secure communication applications is identification of the external cavity delay. Standard techniques including autocorrelation function and mutual information have been shown to easily identify this delay when the feedback is strong and the cavity delay is not close to the device relaxation oscillation period . In the present system the external cavity delay is much longer than the relaxation oscillation period for the laser over the range of parameters explored. The autocorrelation function (ACF) was calculated for delays up to 300 sampling periods for each time series. The amplitude and location of the highest peaks were mapped and are shown in Fig. 8.
The maps show a significant drop in peak ACF amplitude for intermediate feedback levels. Interestingly, the location of the ACF peak increases slightly as the feedback level is decreased. This trend agrees with previously published results for simulated output power time series from a semiconductor laser with optical feedback . This suggests that even though the ACF peak may be more easily identified at very low feedback levels, it is not actually revealing the true external cavity delay time. The ACF peak amplitude trend qualitatively matches the transition from weak → strong → weak chaos regimes observed with increasing feedback level [24, 25]). Assessment of the synchronization quality, as per , using the current experimental system is an objective for future studies.
Closer inspection of the actual form of the autocorrelation functions for each experimental time series sheds some light on the features seen in Figs. 8(a) and 8(b). A sequence of plots of the ACF from the system at maximum injection current is presented in Fig. 9.
The ACF plots show a short period oscillation, representative of the relaxation oscillation period, under an envelope of the longer external cavity period. The relaxation oscillation frequency tends to lower frequencies as the feedback level is decreased. The drop in amplitude of the peak ACF, seen in Fig. 8(a), has been demonstrated theoretically and has been attributed to a chaotic regime where the intrinsic nonlinearity of the laser and the feedback both act as equivalent drivers in the dynamics . The slight shift in the location of the peak is also explained as the maximum occurring at the high order multiple of τRO/2 that is closest to τext .
Thus, the transition from high to low complexity with increasing feedback, as quantified by permutation entropy with τ = τext, in Fig. 7(a) can be explained by the system transitioning from one in which the relaxation oscillation frequency is the dominant time scale in the system, to one in which the external cavity becomes dominant.
It is also of interest to look at the structure of the permutation entropy as a function of delay for fixed levels of feedback. The maps in Fig. 10 show how the locations of the local minima in the permutation entropy shift with injection current at several different feedback levels. These drops in permutation entropy for certain values of delay represent actual dynamical features in the time series and tell us that there is more structure to the probability distribution of observed ordinal patterns at these time scales.
At high levels of optical feedback, refer to the maps in Figs. 10(a) and 10(b), the parameter space is dominated by strong narrow peaks which occur at delays corresponding to the external cavity roundtrip time and its integer-fractions (white arrows), along with smaller magnitude lines corresponding to relaxation oscillation (RO) features (blue arrows). This supports earlier results from the autocorrelation function analysis indicating that the external cavity is the dominant driving mechanism in the system for high levels of feedback. This is not unexpected since relaxation oscillations are suppressed at higher effective injection currents and the optical feedback acts as an effective injection in this context. The intermittent appearance of peaks for some low injection currents in Fig. 10(a) is due to the laser sporadically jumping into coherence collapse, probably due to technical instabilities in the system when it is operated close to coherence collapse. A subtle feature at τ = 3 sampling periods is present at all feedback levels and all injection currents. This feature, marked with a black arrow and labelled τnoise, is likely correlated with a spectral feature of the detection as it is present when the injection current is above lasing threshold, where the laser is operating CW, and also below threshold where the signal is just from detector/oscilloscope. As feedback level drops, see Fig. 10(b), curved lines associated with relaxation oscillations become more prominent. There is a slight increase in the period of the RO feature at maximum injection current due to a shift in the RO frequency at lower feedback levels. We also begin to observe cascades of lines appearing at delays related to the external cavity integer-fractions (τext/2, τext/3, etc.) plus multiples of 2 times the sampling interval. At the level of uncertainty due to the 50 ps sampling period, it is not clear to which mechanism these periods correspond. Such uncertainties will be reduced when a shorter sampling period is used. In Fig. 10(c) the RO features (blue arrows) are much more obvious and follow the expected dependence on injection current. An additional peak appears at delay τext/4 as expected for permutation entropy calculation using an ordinal pattern length of D = 5. A splitting of the external cavity peak (green arrows) is observed just above the coherence collapse threshold, marked with curly brackets in both Figs. 10(b) and 10(c), the size of this split is likely an integer-fraction of the RO period. This marked section also corresponds to a region of the coherence collapse where the permutation entropy is reduced across a larger range of delays. The delays where these low permutation entropy occur are related to period differences of τext, τRO and their integer multiples and integer-fractions. At low levels of feedback, Fig. 10(d), even stronger evidence of relaxation oscillation features are present and the external cavity peaks are much broader. This again supports the idea that as the feedback level changes, so does the relative contribution of the external cavity and relaxation oscillations as drivers of the dynamics.
Experimental measurements of output power time series from a semiconductor laser with delayed optical feedback have produced high resolution maps of the system complexity based on permutation entropy, over a range of operating parameters, for the first time. Using the symbolic complexity measure, permutation entropy, it is possible to determine a relative measure of the unpredictability in the output of this dynamical system, which is nominally infinite dimensional, and includes experimental fundamental and technical noise. The maps confirm the coherence collapse region is uninterrupted by islands of different dynamics at the resolution experimentally achieved here. A previously unidentified region of somewhat regular, low complexity dynamics were identified at low feedback levels.
The maps also demonstrate that the system complexity is highly dependent on the time scale at which the output time series are evaluated. Variation of the delay time used to calculate permutation entropy allows identification of the significant timescales contributing to the dynamics (e.g. external cavity delay and relaxation oscillation period). Complexity maps calculated using different delay times show very different systematics over the parameter space investigated. The results show that the analysis should be performed with multiple delays to gain additional insight into the complexity at different timescales.
Assessing the permutation entropy on the time scale of the external cavity delay produces a map with features compatible with the notion of weak/strong chaos. Comparison between maps of complexity and peak autocorrelation amplitude identify regions of the parameter space where high permutation entropy values and low ACF peak amplitudes are coincident, a situation which is desirable for secure communications. However, if this region is shown to be “strong” chaos, then achieving synchronization is unlikely. In this case, a more suitable region for secure communications could be at slightly lower feedback levels, just above coherence collapse threshold, where both permutation entropy and the ACF peak amplitude are high, but the actual location of the ACF peak is shifted so as to not reveal the external cavity length. If connections with the idea of weak/strong chaos are valid then this region of “weak” chaos should be able to be synchronized. This is a goal for future investigations.
Maps of permutation entropy as a function of both delay and injection current for several fixed feedback levels provide further insight into the time scales driving the nonlinear dynamics under different operating conditions. In particular, for regions of low feedback, a complicated combined influence of sum and difference frequencies associated with the external cavity, relaxation oscillations and detection system features, and their multiples and integer-fractions, is revealed and indicates the importance all can have on the system dynamics.
This research was supported by the Australian Research Council (Linkage Project LP100100312), Sirca Technology Pty Ltd and Macquarie University. We would like to thank Chetan Nichkawde for helpful discussion regarding the permutation entropy analysis.
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