## Abstract

We propose a flat wideband chaos generation scheme that shows excellent time delay signature suppression effect, by injecting the chaotic output of general external cavity semiconductor laser into an optical time lens module composed of a phase modulator and two dispersive units. The numerical results demonstrate that by properly setting the parameters of the driving signal of phase modulator and the accumulated dispersion of dispersive units, the relaxation oscillation in chaos can be eliminated, wideband chaos generation with an efficient bandwidth up to several tens of GHz can be achieved, and the RF spectrum of generated chaotic signal is nearly as flat as uniform distribution. Moreover, the periodicity of chaos induced by the external cavity modes can be simultaneously destructed by the optical time lens module, based on this the time delay signature can be completely suppressed.

© 2017 Optical Society of America

## 1. Introduction

Chaos in external-cavity semiconductor laser (ECSL) has attracted widespread attention in recent years for its potential applications in the fields of secure optical communications [1], chaotic radar [2], fast random bit generator [3,4], optical time domain reflectometer [5] and so on. Nevertheless, due to the relaxation oscillation in semiconductor laser, most of energy is concentrated nearby the relaxation oscillation frequency in the RF spectrum of chaotic signal, and thus the bandwidth of chaotic signal is limited to several GHz. The limited bandwidth would restrict the speed of secure communication, the generation rate of random bit generator and the resolution of chaotic radar [6]. On the other hand, the time delay signature of chaotic signal generated by traditional ECSL is easily identified by calculating the autocorrelation function (ACF), delayed mutual information (DMI) and permutation entropy (PE) from the intensity and phase of chaotic waveforms [7–9]. This characteristic would degrade the security of chaos-based communication using ECSLs and deteriorate the randomness and reliability of the bits generated from the chaotic signal. Therefore it is valuable to spread the bandwidth and suppress the time delay signature of chaotic signal generated by ECSL.

In the past few years, plenty of bandwidth enhancement and time delay signature suppression schemes for chaotic ECSL have been reported. Regarding the bandwidth enhancement, it has been demonstrated that the bandwidth of chaotic signal can be enhanced by optical injection [10–12], injecting chaos into a fiber ring resonator [13,14], and self-phase modulation in feedback loop [15]. Regarding the time delay signature suppression, it has been confirmed that the time delay signature can be efficiently suppressed by several methods, such as setting the feedback delay close to relaxation oscillation period [8,16], dual optical feedback [17], distributed feedback [18,19], phase-modulated feedback [20], and semiconductor laser ring configuration [21]. In addition, a few reports have pointed that simultaneous bandwidth enhancement and time delay signature suppression can be achieved by the delayed self-interference, electronic heterodyning or coupling a CW laser with a chaotic laser [6,22,23].

In this paper, we propose a novel scheme to generate flat-spectrum wideband and time delay signature suppressed chaotic signal by injecting the output of a general chaotic ECSL into an optical time lens module. Our simulation results show that a flat-spectrum chaotic signal can be generated and its efficient bandwidth can be greatly enhanced to above 100GHz, and the time delay signature in both of the intensity and phase time traces can be completely suppressed.

## 2. Principles and theoretical model

The configuration of the proposed scheme is illustrated in Fig. 1. The proposed scheme consists of a general ECSL and an optical time lens module. By properly setting the strength and time delay of feedback, the ECSL would work in a chaotic regime. The optical time lens module is composed of two sections of dispersive fiber and a phase modulator, which is similar to the time domain fractional Fourier transform (FRFT) module in [24,25]. However, differing from those in [24,25], the phase modulator is driven by an amplitude modulated signal that is the mixing of two sinusoidal signals, and the dispersion parameters are not restricted to satisfy the achievement conditions of the time domain fractional Fourier transformation. The chaotic signal generated by the ECSL is injected into the optical time lens module for spectrum expansion and time delay signature suppression.

The dynamic modeling of the ECSL is performed by the Lang-Kobayashi rate equations [26,27], and these rate equations are handled by the fourth order Runge-Kutta algorithm. The intrinsic parameters of the chaotic laser are set as the typical parameters reported in [28]: the wavelength *λ =* 1550 nm, the linewidth enhancement factor *α* = 5, the gain saturation coefficient *ε* = 5 × 10^{−7}, the spontaneous emission rate *β* = 1.5 × 10^{−6} ns^{−1}, the differential gain parameter *g* = 1.5 × 10^{−8} ps^{−1}, the photon lifetime *τ _{p}* = 2 ps, the carrier lifetime

*τ*= 2 ns and the transparent carrier number

_{e}*N*

_{0}= 1.5 × 10

^{8}. The operation current

*I*is set as

*I*= 2

*I*

_{th}, where

*I*

_{th}= 14.7 mA is the threshold current. The feedback strength is chosen as

*k*= 25 ns

^{−1}, which corresponds to 4% of the optical power of the laser diode to be fed back into the active region, and the time delay is chosen as

*τ*= 3 ns. Under such scenario, the ECSL works in a chaotic regime.

_{f}When neglecting the higher order dispersion, the transfer function of the dispersive fiber in frequency domain can be described as

where*K*

_{1}is a constant,

*L*is the length of dispersive fiber,

*β*

_{2}

*= -Dλ*2

^{2}/*πc*denotes the group velocity dispersion,

*D*is the dispersion coefficient,

*λ*is the wavelength and

*c*is the velocity of light. Taking the inverse Fourier transformation of Eq. (1), we can obtain the pulse response in time domain

*F*

^{−1}means the inverse Fourier transformation, and

*K*

_{2}is a constant associated with

*β*

_{2}

*L*.

In the time domain FRFT system, the driving signal of the phase modulator between the dispersive fibers is a parabolic signal which is usually approached by a sinusoidal signal, as those in [24,25]. While in conventional optical communication systems, the width of optical pulse is fixed usually, and then the time domain FRFT module can efficiently suppress the chromatic dispersion induced pulse broadening, by properly setting the frequency of the sinusoidal approximation driving signal of the phase modulator [25]. Nevertheless, for the purpose of enhancing the bandwidth and suppressing the time delay signature of chaotic signal, the periodicity of phase is harmful. Therefore we replace the single sinusoidal approximation driving signal with an amplitude-modulated signal that is generated by mixing two different sinusoidal signals, to degrade the periodicity of phase modulation, which can be described as

*K*is the phase modulation index which determines the maximum phase shift induced by the phase modulation,

_{PM}*f*

_{1}and

*f*

_{2}are the frequencies of the sinusoidal signals that are mixed to generate the driving signal. Unless otherwise stated,

*f*

_{1}<

*f*

_{2}. For a typical LiNbO

_{3}phase modulator, its transfer function can be given aswhere

*V*is the half-wave voltage. Combining Eqs. (3) and (4), the drive voltage

_{π}*V*(

*t*) can be expressed as

*ω*(

_{PM}*t*)|

_{max}is determined by

*K*= 5

_{PM}*π*,

*L*

_{1}=

*L*

_{2}= 2.7 km,

*D*

_{1}= 1.6 × 10

^{−5}sm

^{−2},

*D*

_{2}= 2.0 × 10

^{−4}sm

^{−2},

*f*

_{1}= 1.2 GHz,

*f*

_{2}= 10.2 GHz.

To quantitatively investigate the time delay signature characteristic of the proposed scheme, the ACF, DMI, and PE of the chaotic signal outputted from the optical time lens module are calculated. These three methods are the typical and most popular ways to observe and measure the time delay signature, for their substantial resistance to noise and high computational efficiency. For a time series *s*(*t*), the ACF [*C*(∆*t*)] that measures how well the time series matches its time-shifted replica is defined as [7,16,21]

*s*(

*t*) = |

*E*(

*t*)|

^{2}is the intensity time series,

*S*(

*t*+ ∆

*t*) contains the time shift ∆

*t*with respect to

*S*(

*t*), and <•> stands for the time averaging.

The DMI [*D*(∆*t*)] is defined as [9,16]

*P*[

*S*(

*t*)] and

*P*[

*S*(

*t +*Δ

*t*)] are the marginal probability density function of

*S*(

*t*) and

*S*(

*t*+ Δ

*t*), respectively, while

*P*[

*S*(

*t*),

*S*(

*t +*Δ

*t*)] is the joint probability distribution function of them.

Regarding the PE [29,30], we take the time series {*x _{t}*,

*t*= 1, 2, …,

*T*} and reconstruct a

*d*-dimensional space

*X*= [

_{t}*x*(

*t*),

*x*(

*t*+

*τ*), …,

*x*(

*t*+ (

*d*-1)

*τ*)], where

*d*and

*τ*denote the embedding dimension and embedding time delay, respectively. Vector

*X*is constructed by arranging elements of {

_{t}*x*}

_{t}

_{t}_{= 1,…,}

*increasing order*

_{T}_{${x}_{t+({r}_{1}-1)\tau}\le {x}_{t+({r}_{2}-1)\tau}\le \mathrm{...}\le {x}_{t+({r}_{d}-1)\tau}$}, and any

*X*is uniquely mapped onto an ordinal pattern Ω = (

_{t}*r*

_{1},

*r*

_{2}

*,…, r*) out of

_{d}*d*! possible permutations. For the permutations Ω of order

*d*, the probability distribution

*P*=

*p*(Ω) of the ordinal patterns is [7,21,29]

*H*[

*P*] is evaluated based on the permutation probability distribution

*P*as

*H*[

*P*] = -∑

*p*(Ω) log

*p*(Ω). Finally, the normalized PE

*h*[

*P*] is determined by

*h*[

*P*] =

*H*[

*P*]/log

*d*!, and its value ranges between 0 and 1 (

*h*∈[0,1]). Specifically,

*h*= 0 corresponds to a predicable dynamics,

*h*= 1 corresponds to a full random and unpredictable dynamics, and all

*d*! permutations appear with the same probability.

## 3. Results and analysis

#### 3.1 Bandwidth enhancement

First we investigate the bandwidth enhancement property of the proposed scheme. To quantify the bandwidth of the chaotic signal, we define its efficient bandwidth as the span between the DC and the frequency where 80% of the energy is contained in the RF spectrum, as those in [11–14,31]. Figure 2 shows the temporal waveforms and the corresponding RF spectra of the chaotic signals outputted from the ECSL, the optical time lens module in the proposed scheme and the time domain FRFT module in [24,25]. For the original chaotic signal shown in Fig. 2(a), due to the effect of relaxation oscillation, the dominative frequency components in the spectrum shown in Fig. 2(b) are concentrated around the relaxation oscillation frequency [6,9], and the efficient bandwidth is only about 6.8GHz. While for the proposed scheme shown in Figs. 2(c)-2(d), it is obvious that the temporal chaotic pulses are much narrower than the original chaos, the efficient bandwidth of chaotic signal is greatly enhanced to 70.4 GHz, and the spectrum is very flat. For the time domain FRFT case shown in Figs. 2(e) and 2(f), the frequency of phase modulation is set as 10.2 GHz, and the values of dispersion coefficients are chosen to satisfy the conditions of FRFT with a fractional order *p* = 1. It is indicated that the RF spectrum is broadened and flatter than that of the original chaos, but the frequency of phase modulation signal is revealed in the RF spectrum, and a periodicity can be found in the RF spectrum as the temporal waveform is composed of a succession of discrete pulses with different amplitudes. With respect to the FRFT module, the parameters of the time lens module in this work are not restricted by the FRFT achievement conditions, and a flatter and broader RF spectrum can be achieved.

The bandwidth enhancement is attributed to the spread spectrum effect of the optical time lens module. As shown in Eq. (5), the driving signal of phase modulator is an amplitude-modulated signal that is generated by mixing two sinusoidal signals. The pulse width of the driving signal is fixed and determined by the higher frequency, while its envelope varies as the lower frequency sinusoidal signal. In each pulse period of the driving signal, the phase modulation in the present scheme is similar to those in [24,25], so the chaotic pulses can be compressed and then the efficient bandwidth is enhanced and the spectrum is flattened. Moreover, due to the driving signal is a mixing signal of two sinusoidal signals, a lot of new frequency components are generated [see Eq. (6)], which also contributes to the bandwidth enhancement. In addition, it is worth mentioning that the sinks in the RF spectrum of chaotic signal are caused by the chromatic dispersion-induced power fading effect which has been theoretically analyzed in [32]. Repeating simulations indicate that the sinks can be moved towards high frequency in the RF spectrum, and a flat wide spectrum as that in Fig. 2(d) can be achieved, by reducing the length of dispersive fiber.

Figure 3 shows the influence of the phase modulation index (maximum phase shift) on the efficient bandwidth of chaotic signal outputted from the optical time lens module. It is apparent that the efficient bandwidth of the chaotic signal increases as the increase of the phase modulation index. This is intuitively because the frequency shift induced by the optical phase modulation is proportional to the modulation index *K _{PM}*, as that shown in Eq. (7). Therefore a larger phase modulation index affords a better spectrum expansion.

Furthermore, Fig. 4(a) shows the influence of the mixing frequencies of the driving signal on the bandwidth enhancement. Apparently, for a fixed value of *f*_{1} (*f*_{2}), the efficient bandwidth of chaotic signal is gradually enlarged as the increase of *f*_{2} (*f*_{1}). This is also attributed to that when one of the mixing frequencies is fixed, the frequency shift increases with the increase of the other mixing frequency, as that shown in Eq. (7). On the other hand, the influence of dispersion coefficients *D*_{1} and *D*_{2} is presented in Fig. 4(b). Since the transfer function of a dispersive fiber is determined by the value of *DL* [see Eqs. (1) and (2)], here the fiber length *L*_{1} and *L*_{2} are fixed and only the values of the dispersion coefficients *D*_{1} and *D*_{2} are varied for the sake of simplicity. It is demonstrated that the bandwidth enhancement is mainly determined by the dispersion of the second section of fiber. When the second section of dispersive fiber is fixed, the dispersion of the first section of fiber hardly affects the bandwidth enhancement. When *D*_{2} is closed to 0, the accumulated dispersion is small, the bandwidth enhancement is relatively weak, and the efficient bandwidth of chaos can only be broadened to slightly above 10 GHz. Nevertheless, when the value of *D*_{2} is larger than 4 × 10^{−5} sm^{−2}, significant bandwidth enhancement is achieved, and the efficient bandwidth of chaos can be easily expanded to above 70 GHz.

#### 3.2 Time delay signature suppression

As aforementioned, the time delay signature is a key parameter for the security of chaotic systems. An obvious time delay signature would degrade the randomness of chaotic signal for the periodicity in chaos, and induce security flaw for chaos-based communications because eavesdropper can reconstruct a feedback chaotic system by making use of the obvious time delay signature. Therefore, we focus on the time delay signature suppression characteristic of the proposed chaos generation scheme in this subsection.

Figure 5 shows the ACF traces, DMI traces and PE traces of the original chaotic signal and the spectrum spread chaotic signal. Here the time delay signature characteristics in both of intensity and phase are presented for thorough investigation. For the original chaos generated by ECSL, there are obvious and distinguishable peaks appearing around the feedback delay in the traces of ACF, DMI and PE, which means that the time delay signature is obvious and easy to be revealed. However, when the chaotic signal is propagated through the optical time lens module, all time delay signatures in all traces of ACF, DMI, and PE are erased completely. Moreover, it is indicated that the PE values of the spectrum spread chaos are a little larger than those of the original chaos, which means the complexity of chaos is also enhanced. The perfect time delay signature suppression is due to that the periodicity caused by the feedback loop of ECSL is jointly destructed by the accumulated dispersion and the phase modulation with mixing signal in the optical time lens module. On one hand, the fiber dispersion induces different group delay for different frequency components, which distorts the periodicity of chaos. Subsequently, the time delay signature is suppressed as that in [33], where a frequency-dependent group delay module is added in the feedback loop of an electro-optical chaos generation system, for the similar mechanism. On the other hand, the spectrum spread effect of the phase modulation with mixing signal would disorganize the distribution of frequency components, which also degrade the periodicity of chaotic signal. Consequently, with the optical time lens module, the time delay signature can be completely suppressed.

Furthermore, to more thoroughly investigate the time delay signature suppression property, we present the ACF traces for the cases with different *f*_{2} and *D*_{2} in Fig. 6. For the sake of simplicity, only the ACF curves are presented, while those for DMI and PE are not shown for the similar phenomena. It is apparent that there are no distinguishable peaks at the position of feedback delay in each ACF curve, even though the values of ACF for intensity series are relatively large in a certain range of *f*_{2} or *D*_{2}. That is, the feedback delay cannot be identified from the ACF curves, and the time delay signature is sufficiently suppressed. When the values of *f*_{2} and *D*_{2} are large enough (*f*_{2}>10 GHz, *D*_{2}>1.80 × 10^{−4} sm^{−2}), the autocorrelation of generated wideband chaos can be suppressed to a very low level of close to 0, as those shown in Fig. 5. In addition, it is worth mentioning that to completely suppress the time delay signature, the values of *f*_{1} and *f*_{2} should not be integral multiple of 1/*τ _{f}*, or the time delay signature may be revealed. This is because when the values of

*f*

_{1}and

*f*

_{2}are integral multiple of 1/

*τ*, the phase modulation in the optical time lens module would not destruct the periodicity induced by the external cavity feedback. Our simulations indicate that in such a case, even the dispersion coefficient

_{f}*D*is enlarged by 1000 times, the time delay signature is still distinguishable. While when

*f*

_{1}and

*f*

_{2}are not integral multiple of 1/

*τ*, it is easy to completely suppress the time delay signature. It can be concluded that by properly setting the frequency parameters and dispersion parameters, the proposed scheme can afford simultaneous time delay signature suppression and autocorrelation suppression for chaos.

_{f}## 4. Conclusions

In summary, we have proposed a scheme for generating wide bandwidth chaos with characteristics of flat RF spectrum and excellent time delay signature concealment, by using an optical time lens module to eliminate the relaxation oscillation and destruct the periodicity in the chaotic output of general external cavity semiconductor laser. In the proposed scheme, the optical time lens module is composed of two sections of dispersive fiber and one phase modulator driven by a mixing frequency signal. Simulations demonstrate that the optical time lens module shows excellent spread spectrum effect, which affords to generate wideband chaos with efficient bandwidth up to several tens of GHz and nearly uniform flat RF spectrum. Moreover, we have demonstrated that with proper selection of the mixing frequencies of the driving signal and the dispersion parameters of the optical time lens module, the time delay signatures in the curves of ACF, DMI, and PE can be completely suppressed. The broadband chaotic optical signal with excellent time delay signature suppression has great potential for improving the range resolution of chaotic lidar and enhancing the bit rate of random number generation and chaos-based communication.

## Funding

National Natural Science Foundation of China (NSFC) (61671119, 61471087, 61301156); 111 Project (B14039).

## References and links

**1. **A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature **438**(7066), 343–346 (2005). [CrossRef] [PubMed]

**2. **F. Y. Lin and J. M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. **10**(5), 991–997 (2004). [CrossRef]

**3. **K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. **45**(11), 1367–1379 (2009). [CrossRef]

**4. **I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. **103**(2), 024102 (2009). [CrossRef] [PubMed]

**5. **Y. C. Wang, B. J. Wang, and A. B. Wang, “Chaotic correlation optical time domain reflectometer utilizing laser diode,” IEEE Photonics Technol. Lett. **20**(19), 1636–1638 (2008). [CrossRef]

**6. **A. Wang, Y. Yang, B. Wang, B. Zhang, L. Li, and Y. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express **21**(7), 8701–8710 (2013). [CrossRef] [PubMed]

**7. **S. Y. Xiang, W. Pan, L. Y. Zhang, A. J. Wen, L. Shang, H. X. Zhang, and L. Lin, “Phase-modulated dual-path feedback for time delay signature suppression from intensity and phase chaos in semiconductor laser,” Opt. Commun. **324**, 38–46 (2014). [CrossRef]

**8. **M. C. Soriano, L. Zunino, O. A. Rosso, I. Fischer, and C. R. Mirasso, “Time scales of a chaotic semiconductor laser with optical feedback under the lens of a permutation information analysis,” IEEE J. Quantum Electron. **47**(2), 252–261 (2011). [CrossRef]

**9. **D. Rontani, A. Locquet, M. Sciamanna, and D. S. Citrin, “Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback,” Opt. Lett. **32**(20), 2960–2962 (2007). [CrossRef] [PubMed]

**10. **Y. Takiguchi, K. Ohyagi, and J. Ohtsubo, “Bandwidth-enhanced chaos synchronization in strongly injection-locked semiconductor lasers with optical feedback,” Opt. Lett. **28**(5), 319–321 (2003). [CrossRef] [PubMed]

**11. **A. B. Wang, Y. C. Wang, and J. F. Wang, “Route to broadband chaos in a chaotic laser diode subject to optical injection,” Opt. Lett. **34**(8), 1144–1146 (2009). [CrossRef] [PubMed]

**12. **S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. Li, and H. N. Zhu, “Wideband unpredictability-enhanced chaotic semiconductor lasers with dual-chaotic optical injections,” IEEE J. Quantum Electron. **48**(8), 1069–1076 (2012). [CrossRef]

**13. **A. B. Wang, Y. C. Wang, Y. B. Yang, M. J. Zhang, H. Xu, and B. J. Wang, “Generation of flat-spectrum wideband chaos by fiber ring resonator,” Appl. Phys. Lett. **102**(3), 031112 (2013). [CrossRef]

**14. **Y. H. Hong, X. F. Chen, P. S. Spencer, and K. A. Shore, “Enhanced flat broadband optical chaos using low-cost VCSEL and fiber ring resonator,” IEEE J. Quantum Electron. **51**(3), 1200106 (2015). [CrossRef]

**15. **S. L. Yan, “Enhancement of chaotic carrier bandwidth in a semiconductor laser transmitter using self-phase modulation in an optical fiber external round cavity,” Chin. Sci. Bull. **55**(11), 1007–1012 (2010). [CrossRef]

**16. **D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. **45**(7), 879–891 (2009). [CrossRef]

**17. **J. G. Wu, G. Q. Xia, and Z. M. Wu, “Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback,” Opt. Express **17**(22), 20124–20133 (2009). [CrossRef] [PubMed]

**18. **Z. Q. Zhong, Z. M. Wu, and G. Q. Xia, “Experimental investigation on the time-delay signature of chaotic output from a 1550nm VCSEL subject to FBG feedback,” Photonics Res. **5**(1), 6–10 (2017). [CrossRef]

**19. **S. S. Li, Q. Liu, and S. C. Chan, “Distributed feedbacks for time-delay signature suppression of chaos generated from a semiconductor laser,” IEEE Photonics J. **4**(5), 1930–1935 (2012). [CrossRef]

**20. **C. Xue, N. Jiang, Y. Lv, C. Wang, G. Li, S. Lin, and K. Qiu, “Security-enhanced chaos communication with time-delay signature suppression and phase encryption,” Opt. Lett. **41**(16), 3690–3693 (2016). [CrossRef] [PubMed]

**21. **S. Y. Xiang, A. J. Wen, W. Pan, L. Lin, H. Zhang, H. Zhang, X. Guo, and J. Li, “Suppression of chaos time delay signature in a ring network consisting of three semiconductor lasers coupled with heterogeneous delays,” J. Lighwave Technol. **34**(18), 4221–4227 (2016). [CrossRef]

**22. **C. H. Cheng, Y. C. Chen, and F. Y. Lin, “Chaos time delay signature suppression and bandwidth enhancement by electrical heterodyning,” Opt. Express **23**(3), 2308–2319 (2015). [CrossRef] [PubMed]

**23. **Y. H. Hong, P. S. Spencer, and K. A. Shore, “Wideband chaos with time-delay concealment in vertical-cavity surface-emitting lasers with optical feedback and injection,” IEEE J. Quantum Electron. **50**(5), 236–242 (2014). [CrossRef]

**24. **M. Cheng, L. Deng, H. Li, and D. Liu, “Enhanced secure strategy for electro-optic chaotic systems with delayed dynamics by using fractional Fourier transformation,” Opt. Express **22**(5), 5241–5251 (2014). [CrossRef] [PubMed]

**25. **H. Q. Shen, W. Li, and M. H. Yang, “An optical waveform pre-distortion method based on time domain fractional Fourier transformation,” Opt. Commun. **284**(2), 660–664 (2011). [CrossRef]

**26. **R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. **16**(3), 347–355 (1980). [CrossRef]

**27. **N. Jiang, C. Zhang, and K. Qiu, “Secure passive optical network based on chaos synchronization,” Opt. Lett. **37**(21), 4501–4503 (2012). [CrossRef] [PubMed]

**28. **A. Argyris and D. Syvridis, “Performance of open-loop all-optical chaotic communication systems under strong injection condition,” IEEE J. Quantum Electron. **22**(5), 1272–1279 (2004).

**29. **N. Li, W. Pan, A. Locquet, and D. S. Citrin, “Time-delay concealment and complexity enhancement of an external-cavity laser through optical injection,” Opt. Lett. **40**(19), 4416–4419 (2015). [CrossRef] [PubMed]

**30. **S. Y. Xiang, A. J. Wen, H. Zhang, J. F. Li, H. X. Zhang, and L. Lin, “Effect of gain nonlinearity on time delay signature of chaos in external-cavity semiconductor lasers,” IEEE J. Quantum Electron. **52**(4), 1–7 (2016). [CrossRef]

**31. **F. Y. Lin and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. **221**(1–3), 173–180 (2003). [CrossRef]

**32. **Y. Gao, Q. Zhuge, W. Wang, X. Xu, J. M. Buset, M. Qiu, M. Morsy-Osman, M. Chagnon, F. Li, L. Wang, C. Lu, A. P. T. Lau, and D. V. Plant, “40 Gb/s CAP32 short reach transmission over 80 km single mode fiber,” Opt. Express **23**(9), 11412–11423 (2015). [CrossRef] [PubMed]

**33. **T. T. Hou, L. L. Yi, X. L. Yang, J. X. Ke, Y. Hu, Q. Yang, P. Zhou, and W. S. Hu, “Maximizing the security of chaotic optical communications,” Opt. Express **24**(20), 23439–23449 (2016). [CrossRef] [PubMed]