## Abstract

Optical chaos communication has advantages of high speed and long transmission distance. Unfortunately, the key space of the traditional transceiver, i.e. semiconductor laser with mirror feedback, is limited due to the time delay signature. In this paper, we propose and numerically demonstrate a key space enhancement by using semiconductor laser with optical feedback from a chirped fiber Bragg grating (FBG). The chirped FBG feedback can make feedback delay a key parameter by eliminating the time delay signature. Moreover, the grating dispersion and center frequency can also be used as new keys. As a result, the dimension of key space is increased. By taking a bidirectional communication scheme as an example, numerical results show that the key space is raised by 2^{44} times as against mirror feedback with a data rate of 2.5 Gb/s and a coupling strength of 0.447. As the coupling strength decreases, the key space increases due to the fact that chaos synchronization becomes more sensitive to parameter mismatch.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In the past two decades, optical chaos secure communication has attracted widespread attention because of its striking advantages such as high transmission rate, long transmission distance and integration with the Internet [1,2]. For example, Argyris *et al.* demonstrated the optical chaos communication with a data rate of 1-Gb/s in a 120-km fiber link in the metropolitan area network of Athens [2]. Later, Larger *et al.* and Yi *et al.* successively reported a 10-Gb/s and a 30-Gb/s optical chaos communication in a 100-km fiber link, respectively [3,4]. Semiconductor laser with optical feedback from an external mirror is a promising candidate for chaotic transceivers benefiting from simple and integratable structure. Unfortunately, its key space is limited due to the fact that the laser bias current and the feedback delay time cannot be used as key parameters. More specifically, the laser bias current is related to relaxation oscillation period, which can be extracted by the first side peak of the autocorrelation function [5]. The feedback delay time can be readily identified by the side peak of the autocorrelation function of temporal waveform [5], which is called time delay signature.

It is reported that the time delay signature can be suppressed by more complex feedback, such as double-mirror feedback [6], polarization-rotated feedback [7], and distributed feedback from uniform grating [8,9], random grating [10], or fiber Brillouin backscattering [11]. In addition, the time delay signature can also be depressed by introducing external nonlinear frequency mixing for chaotic laser, such as unlocking-injection of semiconductor lasers [12–14], delayed self-interference [15], optical heterodyning [16], electro-optic nonlinear devices [17], and optical time lens [18]. However, there are few quantitative analyses on key space. In fact, the key space consisting of all combinations of parameter values, not only depends on safety of parameter but also on chaos synchronization characteristics of system. The latter remains unknown for these schemes suppressing time-delay signature.

In this paper, we quantitatively study the key space of optical chaos communication using mirror-feedback lasers and demonstrate a key-space enhancement scheme using lasers with feedback from a chirped fiber Bragg grating (CFBG). Our previous work shows that CFBG feedback can eliminate the time delay signature in a wide parameter range due to dispersion-induced nonlinear feedback [19]. It is thus expected that the feedback delay time as well as grating dispersion and center frequency can be used as new key parameters to increase dimension of key space. Our numerically study shows that the key space of CFBG feedback increases by 2^{44} times as against mirror feedback in the bidirectional communication scheme with a data rate of 2.5Gb/s. It should be mentioned that Hou *et.al* have demonstrated a security enhanced system by using the chaotic optoelectronic oscillator with 16 cascaded fabry-perot etalons [20]. By comparison, our work focuses on chaotic semiconductor lasers with a chirped FBG as transmitter/receiver. Note that we only consider external parameters because laser internal parameters are the same for mirror-feedback and CFBG-feedback systems.

## 2. Theoretic model

The bidirectional chaos communication scheme based on mutually coupled semiconductor lasers is considered in this study due to the fact that it is more secure than unidirectional coupling scheme [21]. Figures 1(a) and 1(b) plot the schematic diagram of bidirectional communication using mirror-feedback lasers and CFBG-feedback lasers, respectively. The mutually coupled lasers are modeled by the following rate equations based on the Lang-Kobayashi equations [19,22].

*E*(

*t*) is the complex amplitude of optical field and

*N*(

*t*) represents the carrier density. The second term in the right side of Eq. (1) is the feedback term which is convolution between the response function

*h*(

*t*) of CFBG and the delayed laser field

*E*(

*t*−

*τ*

_{1,2}).

*k*

_{1,2}and

*τ*

_{1,2}represent amplitude feedback strength and feedback delay. Note the integral length

*T*should be larger than the response time of grating. The third term in the right side of Eq. (1) represents optical coupling, in which

*k*

_{c}and

*τ*

_{c}are coupling strength and delay and

*Δω*=

*ω*

_{1}-

*ω*

_{2}is the optical frequency detuning between. The chirped grating feedback is numerically calculated by inverse Fourier transform of

*H*(

*ω*)∙FT{

*E*(

*t*−

*τ*)}, where FT{} denotes Fourier transform and

*H*(

*ω*) is the grating reflection spectrum obtained by the piecewise-uniform approach [19].

In our simulation, the chirped FBG parameters are listed as follows: effective index *n*_{eff} = 1.46, average index change <*δn*> = 5 × 10^{−4}, fringe visibility of index change *ρ* = 1, grating length *L* = 10cm, grating center wavelength *λ*_{G} = 1550nm, chirp factor *dλ*/*dz* = 0.0166 nm/cm, and the integral length *T* is 20.48ns. The laser parameters are listed: transparency carrier density *N*_{0} = 1 × 10^{6}μm^{−3}, differential gain *g* = 0.5625 × 10^{−3}μm^{3}ns^{−1}, gain saturation parameter *ε* = 1 × 10^{−5}μm^{3}, carrier lifetime *τ*_{N} = 2.2ns, photon lifetime *τ*_{p} = 1.17ps, linewidth enhancement factor *α* = 6.0, round-trip time in laser cavity *τ*_{in} = 7.3ps, active layer volume *V* = 100μm^{3}, and elementary charge *q* = 1.602 × 10^{−19}C, and the laser bias current *I* = 1.5*I*_{th}, where *I*_{th} = 15.35mA is laser threshold current. The feedback and coupling parameters are *τ*_{1, 2} = 5ns, *τ*_{c} = 19ns, *k*_{1,2} = 0.3, and *k*_{c} = 0.447. The grating dispersion can be calculated as the average slope of the group delay spectrum, whose value can be evaluated as 100(dλ/dz)^{−1} [23].

Figure 2 shows a representative result of eliminating time delay signature by CFBG feedback. Figures 2(a) and 2(b) plot an optical spectrum and an intensity autocorrelation trace of laser under mirror feedback with a delay time *τ* = 5ns. Clearly, the autocorrelation trace has a side peak locating at a lag equal to the delay time. This means the delay time of mirror feedback is exposed and cannot be used as a key parameter. Figures 2(c) and 2(d) plot the laser’s optical spectrum and autocorrelation trace under CFBG feedback with a dispersion of 6000ps/nm, respectively. From Fig. 2(d), there is no side peak in the autocorrelation trace, meaning the time delay signature is eliminated. As a result, the feedback delay cannot be cracked and thus can be used as a key parameter.

In addition, we plot the reflection and group delay spectra of the CFBG in Fig. 2(c) with blue and red lines. Obviously, the values of grating parameters, i.e. dispersion (slope of group delay increasing) and center frequency ν_{G}, cannot be obtained by measuring the optical spectrum. Therefore, the grating parameters can also be employed as keys. Now, we can make a qualitative overview that the CFBG feedback can expand the key space of optical chaos communication by introducing three new dimensions: feedback delay, grating dispersion and grating center frequency.

## 3. Key space analysis

#### 3.1 Evaluation method of key space

Key space is the number of all possible values of secret parameters. For example, the Data Encryption Standard cipher using 56-bit digital key has a key space of size 2^{56}. In the optical chaos communication, a key is a vector (*p*_{1}, *p*_{2}, … *p*_{m}), where *p _{i}* is one possible value of the

*i*

^{th}parameter of chaotic laser. Denoting the number of

*p*as

_{i}*N*, the key space can be approximately evaluated as

_{i}*N*can be calculated as the following ratiowhere ∆

_{i}*p*is the parameter value range in which the laser can generate chaos and

_{i}*δp*is the critical value of parameter mismatch between two chaotic lasers. “Floor” means rounding down the result to the nearest integer. If the mismatch is smaller than this critical value, the two chaotic lasers can achieve high-quality chaos synchronization enough to decode message. Therefore, we set this critical mismatch as the minimal gap between two different keys in order to avoid possible decoding by inconsistent key.

_{i}Further, the critical mismatch *δp _{i}* is determined by a threshold of synchronization coefficient. In the transmission simulation, chaos masking [2] is adopted to encrypt the message. The message

*m*(

*t*) is made up of a pseudo-random bit sequence and encrypted into the chaotic carrier with the manner of

*E*(

*t*) (1 +

*am*(

*t*)), where

*a*is the modulation index and its value is 0.05. Figure 3 shows the simulated effects of synchronization coefficient on bit error ratio (BER) of decoded message at a rate of 2.5 Gb/s. Squares (red online) and triangles (blue online) denote synchronization coefficient caused by mismatch of feedback strength and feedback delay, respectively, and exhibit identical tendency. As synchronization coefficient decreases to 0.8, the BER increases to a limit 1.8 × 10

^{−3}under which the decoded message still can be recovered by the forward error correction (FEC) processing technique [24]. Once synchronization coefficient continues decreasing, message cannot be recovered. Therefore, we adopt 0.8 as the synchronization threshold to calculate the critical mismatch of each parameter in this paper. It should be noted that the synchronization threshold is obtained at 2.5-Gb/s rate and will increase for higher data rate. In addition, some parameters may jointly affect chaos state of semiconductor laser and thus a factor

*η*(0<

*η*≤1) is added in Eq. (3) to account the effect of correlation among parameters.

*η*= 1 means that all parameters are independent or there is only one parameter.

#### 3.2 Key space of mirror feedback

For the laser with mirror feedback, only feedback strength can be used as key parameter. In order to calculate the key space, the chaos range and critical mismatch of feedback strength have been calculated.

The chaos region is shown by the bifurcation diagram of feedback strength as plotted in Fig. 4(a). It is clear that the minimum value perturbing semiconductor laser into chaos is 0.006. Note we set the maximum feedback strength as 0.67 (the corresponding intensity feedback ratio is 45%) to avoid possible damage of semiconductor laser. Figure 4(b) shows the synchronization coefficient as a function of feedback strength mismatch. Referencing to the synchronization threshold denoted by the dashed line, the critical mismatch of feedback strength is 0.008. Hence, the key space of semiconductor laser with mirror feedback is 83 (about 2^{6}).

#### 3.3 Key space enhancement with CFBG feedback

For the laser with CFBG feedback, the key parameters include feedback strength, grating dispersion, center frequency and feedback delay. In order to ensure the feedback delay can be used as safe key parameter, we choose TDS-elimination region to calculate the key space. The elimination of time delay signature is mainly related to grating dispersion and laser spectral width determined by feedback strength. Thus, the feedback strength and grating dispersion are investigated jointly for the key space. Figure 5(a) shows effects of feedback strength and grating dispersion on TDS of CFBG feedback. Different colors represent the side-peak height at feedback delay in autocorrelation trace of chaotic temporal waveforms; the darker the color, the lower the TDS is. It is found from the bottom of Fig. 5(a) that for small dispersion less than 2000ps/nm, the TDS quickly deceases to a minimum and then grows again with increasing of feedback strength. This tendency is similar to that of mirror feedback because the grating dispersion cannot induce enough nonlinearity to feedback light. One can find a different tendency for a higher grating dispersion that the TDS decreases to the background of autocorrelation trace as the feedback strength exceeds a critical value. Therefore, the TDS elimination is achieved in the dark region with dashed boundaries. It is noticed that the critical feedback strength (labeled by the vertical dashed line) decreases slightly as the dispersion increases, which is 0.07 at 2000 ps/nm dispersion and about 0.06 at 10000 ps/nm dispersion. For convenience, we chose 0.07 as the lower boundary of feedback strength to calculate key space. Then, we estimate the region of feedback strength is [0.07, 0.67] and the dispersion region is [2000ps/nm, 10000 ps/nm].

Figures 5(b) and 5(c) display the synchronization coefficient as a function of feedback strength mismatch and grating dispersion mismatch, respectively. Indicated by the synchronization threshold, the critical mismatches of feedback strength and grating dispersion are 0.005 and 3.2ps/nm, respectively. Then the key number of feedback strength and grating dispersion are *N _{k}* = 120 and

*N*= 2500, respectively. It is marked that, the critical mismatch of CFBG feedback strength is smaller than that of mirror feedback. This means the chaos synchronization is more sensitive to the CFBG feedback.

_{d}The TDS-elimination region of grating frequency is shown in Fig. 6(a). Circles represent the TDS values and the stars denote the three standard deviation of the background noise of autocorrelation function traces. Remarkably, the TDS is lower than the background noise in a range from −25GHz to 10GHz, means TDS is eliminated. In TDS-elimination region, TDS is considered as the maximum height in (5ns, 5.5ns) of autocorrelation trace. The TDS elimination is readily achieved in negative detuning due to the laser redshift induced by feedback. Note *f _{d}* = 0 GHz denotes that the grating frequency is equal to the optical frequency of the solitary laser. Figure 6(b) depicts synchronization coefficient as a function of mismatch of grating frequency, and indicate a critical mismatch of 0.4 MHz. As a result, the key number of grating frequency is calculated as

*N*= 87500.

_{f}Last, we check the key number of feedback delay which can be used as a safe key because the time delay signature is eliminated by CFBG feedback. As shown in Fig. 7(a), the elimination of TDS is independent of time delay. From the perspective of the long-cavity feedback [25], the feedback delay should be larger than the reciprocal of the relaxation oscillation frequency to generate complex chaotic carrier. We thus take 1 ns as the minimum of feedback delay which is about two times larger than that of the relaxation oscillation period (about 0.4ns in our simulation). As for the maximum value of the time delay, because of the small size and good bending characteristics of the fiber, hundreds of meters of fiber can be bent into a small ring and the feedback delay can reach thousands of nanoseconds (hundreds of meters). It is reported that 107-meters (1070ns) fiber is used as external cavity to generate chaos [26]. Here, feedback delay of 1000ns is set as the maximum. Figure 7(b) gives synchronization coefficient as a function of feedback delay mismatch and shows a critical mismatch of 22.3ps. Thus, the key number of feedback delay is *N _{τ}* = 44798. Considering the TDS elimination area in Fig. 5(a) and the wideband reflectivity spectrum of CFBG, we approximate each parameter (grating dispersion, feedback strength and center frequency) to be independent with each other, i.e.

*η*= 1.

In order to quantitatively calculate the key space, the value of *η* is analyzed. Firstly, the parameters (feedback strength, grating dispersion and frequency) can be approximated to be independent of each other in eliminating the TDS. In case that the bias current is fixed (not used as a key parameter), the main factors affecting TDS are grating dispersion and feedback strength. As seen from Fig. 5(a), the TDS can be eliminated in a rectangular parameter region, which means the independence between feedback strength and grating dispersion. In addition, dispersion occurs inside (instead of edge) CFBG spectrum, and thus the TDS elimination can be achieved as long as that the CFBG spectrum covers that of chaotic laser. Furthermore, the parameter mismatch is mainly affected by the coupling strength and coupling method of transceiver. For example, the stronger the coupling strength is, the larger the parameter mismatch is allowed for achieving chaos synchronization. When the coupling strength is fixed, the parameter of transceiver has little influence on the synchronization. As shown in Fig. 8, it is obvious that the both feedback strength mismatch are 0.05 under different dispersion. Therefore, we approximate each parameter to be independent of each other, i.e. *η* = 1. In general, the value of *η* is difficult to obtain if the parameters are dependent. One has to go through all parameters to calculate key space precisely.

As a result, the total key space of semiconductor laser with CFBG feedback is calculated as *N*_{key} = *η N _{k} N_{d} N_{f} N_{τ}* = 1.1759475 × 10

^{15}(about 2

^{50}) under the data rate of 2.5 Gb/s and the coupling strength of 0.447. It is enhanced by 2

^{44}times compared with mirror feedback.

## 4. Discussion and conclusion

It is noted that the critical parameter mismatch is also affected by the coupling strength between two chaotic lasers, i.e. transmitter and receiver. We further analyze the key space with the coupling strength from 0.4 to 0.2. Results are shown in the Table 1. With the decreases of coupling strength, the key space of both mirror and CFBG feedback gradually increases. The key space of CFBG feedback reaches about 2^{84}, which is 2^{73} times larger than that of mirror feedback when the coupling strength is 0.2. Furthermore, as listed in the third row, the key space enhancement also gradually increases with coupling strength decreasing. The reason is that chaos synchronization in CFBG feedback system is more sensitive to the parameter mismatch under a weaker coupling strength. This result suggests that the key space can be greatly enlarged by CFBG feedback at a weaker coupling level.

It is worth mentioning that the synchronization of chaos becomes difficult as the key space increases. Actually, the mismatch tolerance of parameter will be even worse with multiple parameters mismatch. As shown in Table 1, the key space gradually decreases with the improvement of coupling strength, which indicates that the level of critical parameter mismatch generally increases with the improvement of coupling strength. It is a challenge to realize chaos synchronization of this configuration in practice. For experiments, two ways can be used to increase the possibility of the synchronization. One way is improving the coupling strength to obtain a larger parameter mismatch tolerance. Another way is using a precise temperature controller to reduce the frequency drifts. The experimental work is reserved for our next work. We think this numerical work will excite some other efforts to explore solutions in which chaos synchronization and large key space are easy to obtain simultaneously.

In conclusion, the key space enhancement for chaos communication system by CFBG feedback is numerically investigated. Compared with mirror feedback, CFBG feedback brings more safe parameters (i.e., feedback delay, grating dispersion and center frequency) which increase the dimension of key space. In addition, chaos synchronization of CFBG-feedback lasers is more sensitive to parameter mismatch. Resultantly, the key space of semiconductor laser subject to CFBG feedback is enhanced by 2^{44} times than that using mirror feedback.

## Funding

National Natural Science Foundation of China (NSFC) (61822509, 61731014, 61475111, 61671316, 61805170, 61805171); National Cryptography Development Foundation (MMJJ20170207); International Science and Technology Cooperation Program of Shanxi (201603D421008).

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