Synchronization of complexity enhanced chaos in semiconductor lasers

In synchronized chaotic lasers based secure key distribution and other encrypted communications, presence of the time delay signature in chaos poses a threat to security. So the transmitter and receiver lasers should preferably be operated in complexity enhanced chaotic regime where the time delay signature is hidden. However, achieving good synchronization in experiments in such regime is challenging. We report experimental demonstration of achieving excellent synchronization between two semiconductor lasers even when both the lasers are operating in complexity enhanced chaotic regime with absolutely no time delay signature present in their output. This chaotic regime is ensured by evaluating the auto correlation function, permutation entropy and spectrum analysis of the time series. As a measure of synchronization, cross-correlation coefficient of 0.923 is achieved between the transmitter and receiver lasers. This results are of immense importance in chaos based secure key distribution and other encrypted communication schemes.


Introduction
Generation and synchronization of chaos [1] in semiconductor laser (SL) has exhibited tremendous application potential [2][3][4].Among those, secure key distribution using correlated SLs is the most emblematic one [5][6][7].In particular, the development of quantum computing technology along with sophisticated algorithms [8], poses a threat to the computational algorithmic based asymmetric-key schemes (like RSA) [9].An alternative of this scheme is the one-time pad (OTP) based encryption which requires a secure key distribution between two parties (i.e. the sender and the recipient) prior to the actual communication [10][11][12].Quantum key distribution (QKD), in which the security is attributed to the fundamental properties of the quantum mechanics, is ideal for this purpose [13].However, implementation of QKD in the commercial communication system is still demanding [14,15].Synchronized chaotic SLs provide an excellent alternative method to distribute the secure key.Other potential applications include high speed long distance secure communication [16,17].Two key features in such applications of synchronized chaotic SLs are the complexity of the chaos and the synchronization between transmitter laser (TL) and target receiver laser (RL).Security of the information is ensured by the complexity of chaos being used for encryption.On the other hand, recovery of the encrypted signal in receiver end depends on synchronization between TL and RL.The complexity of chaotic time series can be characterised by normalized permutation entropy (PE), autocorrelation function (ACF) and spectral analysis.PE has gained popularity as a measure of complexity of real world time series [18,19].ACF provides a robust measure of time delay information (TDI) as well as the complexity of chaotic time series [20].For less complex chaos there are repeated peaks in ACF at integral multiple of time delay.The TDI helps mapping the actual chaotic attractor to much lower dimensional attractor and hence reduce the complexity [21,22].So the presence of TDI is detrimental for security in all of the chaos based encryption techniques.Hence, the TL and RL should preferably be operated in a complexity enhanced regime where TDI is masked.In this regime the secondary peaks in ACF are disappeared [23].Recent studies have been reported about different schemes to enhance the complexity of the chaos generated in SLs and hiding the TDI [24][25][26][27][28][29].However, when both TL and RL are operating in this strong chaotic regime, meeting the other criteria of chaos based encrypted communication, namely good synchronization between TL and RL is challenging [19,30].In this report we address this point.We have demonstrated experimentally that when chaos in the TL is generated by optical injection from another chaotic laser, then not only complexity enhanced chaos with absolutely no TDI could be generated, but also excellent synchronization between TL and RL could be achieved by properly controlling the operating parameters.We have also shown that the spectral analysis provides a qualitative measure about the complexity of the chaos.In particular, for relatively less complexity of the chaos, there are repetitive and uniformly spaced spectral lines in the Fourier spectrum.In contrast, for complexity enhanced chaos, those repetitive spectral lines get merged into the continuum of the Fourier spectrum.This report is organized in the following way.In section 2, we describe in details the experimental results of reduced synchronization with increase in complexity with two lasers, TL and RL in conventional master-slave (MS) configuration under unidirectional coupling.Master laser is basically the TL, and slave laser is RL.TL is subjected to self-optical feedback from a variable back reflector (VBR).The optical feedback induces a coupling between the phase and the amplitude of the electric filed.The laser then may exhibit instabilities including chaos [31].We have shown that with the increase in the feedback strength, the complexity of chaotic output from TL is increased.This is verified from increasing PE, substantial reduction of secondary peaks in ACF, and spectral analysis.However, here TDI is not hidden completely.We have quantified the state of synchronization in terms of cross-correlation coefficient and then present our results that in MS configuration with increase in complexity, the cross correlation coefficient between TL and RL is decreased.This is consistent with the reported literature [19,30] and it limits the recovery of the signal in the receiver end in all chaos based encrypted communication schemes.In section 3, we describe experimental results with three lasers in Driver-Transmitter-Receiver (DTR) configuration.Chaos in the TL is generated by optical injection from another chaotic laser which we call driver laser (DL).Chaotic output from the TL is then injected to RL.We report that in this scheme, TDI is totally absent in the chaotic output from both the TL and RL.Also, excellent synchronization, with cross correlation coefficient of 0.923 is achieved between these two lasers by suitable adjustment of optical injection from TL to RL.

Two laser schemes 2.1. Experiments
Figure 1 shows the schematic of our experimental setup with two c-band 1550 nm semiconductor lasers under MS configuration.The lasers have free running, room temperature threshold of ∼12 mA.The experiments were carried out with injection current set at 40 mA and temperature at 25 °C in both the lasers.In these values the output power of both the laser is 4 mW.The output of the TL is divided into two equal part by a 50:50 beam splitter (BS).One arm of the BS is connected to a VBR for optical feedback.The VBR has an insertion loss ∼2 dB.The output from the other arm of the 50:50 BS is sent to the receiver end through a 10 km single mode fibre spool.In the receiving end the output from the fibre spool is divided into two parts by an 80:20 BS.The output of the 20% arm is detected by a photodetector (PD), while the output from the 80% arm is injected to the RL via a combination of a variable optical attenuator (VOA) followed by an optical circulator (OC).The light output of the RL is detected by another PD.Feedback strength is sensed by attenuation coefficient (AC) at VBR.The feedback strength decreases with increase in AC.We have set AC at 9 dB,7 dB, 5 dB, 3 dB, and 2 dB (lower limit prominent uniformly spaced spectral lines do appear at a separation of ∼20.7 MHz, the inverse of which is the delay present in the system.The corresponding time shifted ACF plot validate this information as the secondary peaks appear at ∼48.175 ns (1/20.7 MHz).With the increase in feedback strength (decrease in AC value), gradual of uniformly spaced spectral lines in FFT inside noise like continuum indicates the enhancement of complexity.Gradual decrease of secondary peaks in ACF also attest the effect.For maximum feedback in experimental configuration with AC = 2 dB, the spectral lines in FFT (figure (x)) are merged into the continuum.However, the FFT still shows some periodic profile, at least qualitatively if not quantitatively.The secondary peak in the corresponding ACF (figure (xv)) is reduced substantially, but not completely.Both these plots, indicate that chaotic complexity has been enhanced, but TDI is not masked perfectly. of the VBR).For each of these values, the attenuation coefficient at VOA in the receiver end has been adjusted to achieve best possible synchronization between TL and RL.The chaotic fluctuation of light intensity from the lasers are converted to the electronic signal by the respective PDs.The temporal waveform of PDs' output is sampled and stored simultaneously in a multi-channel digital storage oscilloscope (DSO) for a duration of 0.2 ms.The sampling frequency of the DSO is 40GS/s i.e. the sampling period is 25 ps.So each time series contains 8 M data points.These time series are then analysed using a PC (not shown in the figure).

Result and discussions
We have studied the complexity of the time series by means of fast Fourier transform (FFT) spectrum, time delayed ACF and normalized PE.The ACF at delay step τ, for a discrete time series X x x x , , N where, X s = standard deviation of X, x ̅ = average of X, N = total number of data points.Figure 2 shows the result of FFT and ACF.With increasing optical feedback (decrease in AC value), appreciable changes in the FFT and ACF have been observed.For example, at AC = 9 dB, there is dominance of repetitive and uniformly spaced spectral lines at ∼20.76 MHz separation in FFT (figure 2(vi)).This reveals a delay of ∼48.17 ns (1/20.75MHz) present in the system.The corresponding time shifted ACF plot (figure 2(xi)) validates this information as the secondary peaks appear at ∼48.175 ns separation.This delay corresponds to a total of ∼9.6 m optical path i.e. 4.8 m fibre length traversed by the reflected filed.This is in agreement with the fibre length of the fibre optic component between cavity of fibre coupled TL and the mirror in VBR through a 50:50 beam splitter in our experimental set up.So the TDI is completely disclosed here.With increase in the optical feedback we observe the following effects.In FFT, the spectral lines gradually shrink into noise like continuum (figures 2(vii)-(ix)).In ACF the height of the secondary peaks gets reduced (figures 2(xii)-(xv)).At AC = 2 dB, the spectral lines in FFT are almost merged into the continuum (figure 2(x)).However, the FFT still shows some periodic profile, at least qualitatively.The secondary peaks in the corresponding ACF (figure 2(xv)) are diminished substantially, but not completely.Both these plots, indicate that chaotic complexity has been enhanced, but TDI is still not hidden perfectly.As an alternative measure to substantiate the complexity enhanced regime, the normalised PE of TL time series for different feedback strength are calculated as the following [32][33][34].For a given TL time series x x x , ..., , which is called ordinal pattern related to time s.So this vector is basically mapped to a unique symbol r r r , , , D 0 1 1  this study, we have taken D = 4, 5, 6, 7 and t = 48.175ns (or 1927 times the sampling period of DSO), estimated from ACF at the position of the next secondary peak [32,33].The plots are given in figure 3. PE always increase with increase in optical feedback strength.For example, for D = 6, PE increases to 0.994 at AC = 2 dB, compared to PE at AC = 9 dB, which is 0.776.It shows that, in the experimental configuration, chaos based encrypted communication will provide maximum security when the feedback is at AC = 2 dB.However, for successful decryption, RL must be synchronized with the TL.We have explored the state of synchronization between the TL and RL in different chaotic regimes.The synchronization between two time series ¼ } is characterized by cross-correlation (CC) coefficient defined at a particular delay step t as x x y y where n = total points considered for the evaluation of ( ) r t which is 7.9 M for our analysis.x ̅ = average of X, y ̅ = average of Y, X s = standard deviation of X, Y s = standard deviation of Y .Figure 4 shows the results of CC analysis between TL and RL time series with increasing PE (as a measure of increasing complexity).Here we have considered PE value at t= 48.175 ns, for reason mentioned above.For the least complex chaotic regime, with AC = 9 dB, excellent time-delayed synchronization is achieved with CC coefficient of 0.92.In contrast, as the complexity increase, CC decrease.At AC = 2 dB where PE is ∼0.99,CC goes down to 0.20 practically indicating no correlation between TL and RL.This results are congruous with reported observations [19,30].

Experiments
The limitations of hiding TDI completely and achieving good synchronization between TL and RL in complexity enhanced chaotic regime can be overcome using three lasers under Driver-Transmitter-Receiver (DTR) configuration.The schematic is given in figure 5 which is similar to MS configuration (figure 1) except the chaos in TL is generated by optical injection from another chaotic laser which we call Driver Laser (DL).The experiments are carried out with injection current set at ∼40 mA and temperature set at ∼25 °C in all the three lasers.In DL itself, the chaos is generated by optical feedback from VBR.The chaotic output of DL is divided into two parts by an 80:20 BS. 20% part is converted to electronic signal by photodiode PD1.80% part is injected to TL via one VOA followed by OC.This attenuation coefficient in this VOA controls the injection strength from DL to TL.The output from TL, which is coming out through the third port of OC, is sent to receiver end through an 10 km fibre spool.In the receiver end the output from the fibre spool is divided into two parts using an 80:20 BS. 20% part is converted to the electronic signal using photodiode PD2.80% part is injected to RL via another VOA followed by OC.The chaotic output from RL is detected by the photodiode PD3.The temporal waveform from PDs for a duration of 0.2 ms are stored simultaneously in the multi-channel DSO.Each time series corresponds to 8 M data points in DSO with 40GS/S sampling frequency.The time series are then analysed by a computer.

Results and discussions
In the experiments, AC at VBR is set at 2 dB and optical injection from DL to TL and TL to RL are adjusted suitably to make the DL, TL and RL operate in complexity enhance chaotic regime.This regime is confirmed from spectrum analysis, time shifted ACF and estimation of normalized PE.There are no dominant spectral lines in the strong continuum of FFT spectra for DL, TL and RL.This is shown in figures 6(i)-(iii).In time delayed ACF of DL itself (figure 6(vii)) secondary peak is still visible at the delay τ = 47.70 ns, equivalent to 1908 times the sampling period of DSO.This is similar to the earlier TL time series in MS configuration with AC = 2 dB.However, in this configuration, the secondary peaks in the ACF of TL and RL are fully disappeared as shown in figures 6(viii)-(ix).So, TDI is totally hidden in these chaotic TL and RL.The PEs are estimated at an   For both the TL and RL the value of normalized PE is 0.99 which again asserts that these SLs are in strong chaotic regime.We then explore the other important aspect namely the synchronization between the chaotic lasers by adjusting the optical injection from DL to TL and TL to RL. Practically no synchronization is achieved between DL and TL or DL and RL.CC coefficient of ∼0.1 is observed for both the cases (plots are not shown) which is consistent with the results obtained in MS configuration.However, excellent synchronization is achieved between TL and RL.The state of the synchronization is shown in figure 8.The time shifted CC function (equation ( 3)) is shown in figure 8(i) which exhibit excellent synchronization with CC coefficient of 0.923 at a delay = 16 ns (640 times the sampling interval in DSO).This delay is due to the fact that in the receiver end, from the 80:20 splitters (figure 5) the fibre path traversed by TL output to the photodiode (PD2 in figure 5) is different than the fibre path traversed by synchronised chaotic RL output to the photodiode (PD3 in figure 5).The CC values rapidly goes down to zero within 0.5 ns.No secondary synchronization is found in this CC analyses because both the TL and RL are operating in a complexity enhanced regime.The TL time series versus RL time series (with corresponding dc value is subtracted from each of the time series) is shown in figure 8(ii).The diagonal nature of the Lissajous plot manifests that TL and RL are nicely synchronized.Figure 8(iii) compares a segment of TL time series (of a duration of 2.5 ns) with that of delay adjusted RL time series.The overlap of these two-time series segments assures good synchronization between TL and RL.
In MS configuration, in view of chaos generation mechanism, TL and RL are not under exactly identical conditions.TL is subjected to self-optical feedback while the chaos in RL is induced because of optical injection from TL.However, in DTR configuration both in TL and RL, the chaos is induced by optical injection.So TL and RL are under more identical conditions in DTR configuration than they are in MS configuration.This is possibly the reason of getting excellent synchronization between TL and RL in DTR configuration even in complexity enhanced regime.Assessment of the region in the parameter space for which TL with concealed TDI remains synchronized with RL but not with DL using the current experimental systems, and examining the effects of optical fibre length between TL and RL, in particular the effects of group velocity dispersion and self-phase modulation on the quality of synchronization are the objectives for future studies [35].

Conclusions
We have experimentally explored the generation and synchronization of complexity enhanced chaos in c-band semiconductor lasers in conventional MS configuration and in DTR configuration.Enhancement of complexity of the chaos is characterized in terms of ACF, and PE and FFT spectra analysis.The state of synchronization is quantified by means of cross correlation coefficient.It is found that in MS configuration the increase in selfoptical feedback strength in semiconductor laser tend to increase the complexity of the chaotic output from it.Depending on the feedback strength the presence of TDI may be decreased substantially, but not hidden entirely in the ACF of laser output.Also, the best achievable synchronization between TL and RL diminish rapidly with increase in complexity of the chaos.These two challenges, in particular, hiding TDI completely in the chaotic output and simultaneously achieving good synchronization between two SLs can be overcome simultaneously with three SLs under DTR configuration.In this configuration, complexity enhanced chaos in the TL is generated by optical chaotic injection from the DL.It is shown that in this method, TDI is perfectly masked in the output of TL as well as RL.Also, by suitable adjustment of optical injection, excellent synchronization with cross correlation coefficient of 0.923 is achieved between TL and RL.We believe the experimental study covering two principal features namely TDI concealment and synchronization will find importance in chaos based secure key distribution and other encrypted communication schemes.

Figure 2 .
Figure 2. (i)-(v) FFTs of TL time series at attenuation coefficient of (AC) 9 dB, 7 dB, 5 dB, 3 dB and 2 dB respectively.(vi)-(x) Corresponding enlarged FFTs between 8.0 GHz and 8.1 GHz.(xi)-(xv) Time shifted auto correlation function (ACF) for AC at 9 dB, 7 dB, 5 dB, 3 dB and 2 dB respectively.For relatively less feedback strength, for example at AC = 9 dB, in FFT plot (figure (vi)),prominent uniformly spaced spectral lines do appear at a separation of ∼20.7 MHz, the inverse of which is the delay present in the system.The corresponding time shifted ACF plot validate this information as the secondary peaks appear at ∼48.175 ns (1/20.7 MHz).With the increase in feedback strength (decrease in AC value), gradual of uniformly spaced spectral lines in FFT inside noise like continuum indicates the enhancement of complexity.Gradual decrease of secondary peaks in ACF also attest the effect.For maximum feedback in experimental configuration with AC = 2 dB, the spectral lines in FFT (figure (x)) are merged into the continuum.However, the FFT still shows some periodic profile, at least qualitatively if not quantitatively.The secondary peak in the corresponding ACF (figure (xv)) is reduced substantially, but not completely.Both these plots, indicate that chaotic complexity has been enhanced, but TDI is not masked perfectly.

Figure 3 . 1 s
Figure 3. Normalized permutation entropy (PE) of the TL time series at different level of optical-feedback strength.Embedding dimension (D) has been varied from 4 to 7 and the embedding delay is t= 48.175 ns, as estimated from the position of secondary peaks in ACF.With increasing optical feedback (decrease in attenuation coefficient) PE increase for any D which indicates enhancement in complexity of the chaos.

Figure 4 .
Figure 4. Variation of the best achieved cross correlation (CC) coefficient between TL and RL with normalized permutation entropy (PE).PE values, which are plotted here, are evaluated at an embedding dimension of D = 4,5,6,7 and embedding delay of t= 48.175 ns (details are in text).TL and RL are in the master-slave configuration.It shows that the synchronization (quantified in terms of CC coefficient) between chaotic TL and RL decreases with increase in complexity (quantified in terms of normalized PE).

Figure 6 .
Figure 6.(i)-(iii) FFTs of DL, TL and RL time series respectively when TL and RL are in synchronized state.(iv)-(vi) Corresponding enlarged FFTs between 8.0 GHz to 8.1 GHz.(vii)-(ix) Time shifted auto correlation function (ACF) for DL, TL and RL time series.Secondary peaks are still visible in the ACF of the DL time series which is subjected to self-optical feedback (just like the TL laser in earlier configuration with two lasers under master slave configuration).However, the secondary peaks are hidden thoroughly in the ACF of TL and RL time series (figure (viii) and (ix)).This indicates that TDI is absolutely masked in TL and RL in DTR configuration.For all chaos based encrypted communications, TL and RL should preferably be operated in this regime.

Figure 7 .
Figure 7. Normalized permutation entropy (PE) of DL, TL and RL for different embedding dimension (D) when TL and RL are in synchronized state.PE values, evaluated at embedding delay t= 47.70 ns are considered for this plot (details are in text).For both the TL and RL the PE is 0.99, for any D, which indicates that these lasers are in complexity enhanced chaotic regime.

Figure 8 .
Figure 8. (i) Time delayed cross correlation (CC) coefficient between TL and RL time series.It manifests excellent synchronization with a maximum CC efficient of 0.923 which is achieved at a delay = 16 ns.The reason behind this delay is explained in the text.No secondary synchronization regime i.e. no repeated peak in this CC plot is found as both the TL and RL are operating in a complexity enhanced chaotic regime where the TDI is fully masked.(ii) TL time series versus delay adjusted RL time series plot with corresponding dc value is subtracted.The diagonal nature shows that TL and RL are perfectly synchronized.(iii) Delay adjusted TL and RL time series segment of 2.5 ns duration.Very good overlapping of these two-time series segments confirms nice synchronization between TL and RL.
embedding dimension and t is embedding delay.N D t number of such vectors are generated from the time series.At each s, the elements of this vector can be sort such that x x x