Suppression of Cavity Time-Delay Signature Using Noise-Phase-Modulated Feedback

We propose reconfiguration of the conventional feedback scheme to suppress the external-cavity time-delay signature (TDS) by adding noise phase modulation in the feedback of semiconductor laser. Noise-phase-modulated feedback introduces broad band noise frequencies into the feedback light and enables the suppression of the TDS. In this work, by means of simulations, the effect on TDS suppression of phase modulation (PM) index is explored. In addition, the influence of noise bandwidth variation is investigated. Using the auto-correlation function to quantity the TDS, we find that, for a large range of operating parameters, the TDS is significantly suppressed to the noise level and even submerged into the base noise. It is shown that suppression TDS is achievable over a wide operating parameter provided the noise generator bandwidth is of order 10 GHz and the PM index is greater than about 3. The proposed configuration will have widespread applications in contexts where suppression of the TDS is required.

The associate editor coordinating the review of this manuscript and approving it for publication was Chao-Yang Chen . features into the dynamics which threaten the security of chaos communication [8], limit the key space [9], weaken anti-jamming capabilities [10] and reduce the randomness of bit sequences [11].
As means to suppress or even eliminate TDS, much work has been done in respect of the configuration of the optical feedback, including double feedback [12], FBG feedback [13], [14], chirped FBG feedback [15], random feedback [16], phased-array feedback [17], as well as phase-modulated feedback [18], [19]. With phase-modulated feedback, the bandwidth of chaos can be enhanced which is desirable in realizing high-speed communication.
Some driving signals of phase modulator based on self-delay or chaos in phase-modulated feedback have been proposed in recent research on chaos communications. Lavrov et al. reported delayed electo-optic phase modulation feedback and realized 10Gb/s transmission of a message hidden in chaos [20]. Jiang et al. used common chaotic phase-modulated optical feedback as an external-cavity structure to improve the communications performance [21].
More recently, he proposed a physical WDM transmission by using chaotic spectral phase encryption where the phase modulator was driven by chaos from a chaos optical injection and a mirror feedback [22]. It was shown that the security of transmission arises partly from the complication of the driving signal of phase modulator, since any small difference from the authorized scheme prevents unauthorized attacks from decrypting the transmitted messages [22]. Also, the weak correlation between driving chaos synchronization signals and transmission carriers can improve the security performance in chaos communication, since such weaker correlation inhibits eavesdroppers illegal attacks from replicating the scheme of communication [21].
In this paper, we propose using noise as the driving signal of the phase modulator to achieve the suppression of the TDS. In comparison with the above mentioned driving signals of phase modulator, noise is a naturally stochastic signal, which has no correlation with the generated chaos in the system. Therefore, if noise-phase-modulated feedback were used in chaotic communication, it would require an enormous effort on the part of an unauthorized user to decrypt the transmitted messages, so the security will be further enhanced. One knows that the bandwidth of any photonic device is limited, so the bandwidth of noise generator containing an amplified spontaneous emission source, a photodetector and an electronic amplifier is considered in the simulation. The effects on the TDS of the bandwidth of noise generator are revealed. This bandwidth determines the phase modulation frequency.
The paper is structured as follows. A theoretical model of noise-phase-modulated feedback semiconductor laser is introduced in Section II. Mechanism for adding noise phase modulation is presented in Section III. The results given in Section IV delineate both phase modulation index and bandwidth of noise generator, that is phase modulation frequency, effects on suppression of the TDS. Finally, in section V, conclusions are drawn based on the results obtained.

II. THEORETICAL MODEL
The configuration of noise-phase-modulated feedback of semiconductor laser is illustrated in Fig.1. A phase modulator is deployed in the feedback of distributed feedback laser (DFB laser). The driving signals of phase modulator are from the noise generator which is comprised of an ASE source, a photodetector and an electronic amplifier.
Numerical simulations are based on the Lang-Kobayashi equations [23]. By adding the feedback term, the modified Lang-Kobayashi equations for the noise-phase-modulated feedback single-model semiconductor laser can be written as E(t) denotes the slow-varying complex electric field, and N (t) denotes the carrier number in the laser cavity. The additional phase from the PM is presented in Eq. (3), where χ PM (t) denotes the Gaussian noise (which is added at the phase modulator), which has zero mean and an amplitude ranging from −1 to 1. β PM is the phase modulation index which is β PM = A/V π , where A is the amplitude of driving signal, and V π is the half-wave voltage of phase modulator. As a common method of quantifying the time-delaysignature, the auto-correlation function (ACF) is calculated. The ACF measures how well the time series matches its time-shifted replica, which is defined as [18], [21] where I (t) = |E(t)| 2 is the intensity time series, I (t + t) contains the time shift t with respect to I (t), and · stands for the time averaging. The integration step used in the simulations is dt = 1 ps. The definition and value of parameters in simulation are listed in Table 1, which are mainly ref. [21]. Considering the bandwidth of the photodetector and electronic amplifier, that is bandwidth of noise source generator, in the simulations we set the Gaussian noise χ PM (t) through a low pass filter, which is used to control the bandwidth of the noise signal. Also in practice, the response modulation frequency of the phase modulator affects the output chaos.
Here the response modulation frequency of the phase modulator is supposed 100 GHz, the maximum bandwidth of the noise source generator is no larger than 100 GHz. Therefore, only the bandwidth of the noise source generator, that is 3dB cut-off frequency of low pass filter, which also determines the modulation frequency, needs to be considered in simulation. In the simulations, the Gaussian noise χ PM (t) has zero mean and variance from −1 to 1; when it passes the low pass filter, its amplitude will be reduced. In practice, the use of  an electronic amplifier, can enable amplification of the noise signals. Therefore, in order to demonstrate TDS suppression, the phase modulation index β PM is set in the range 0 to 20.

III. MECHANISM
Here an outline is given of the underpinning physical effects which give rise to the advantageous use of noise-phasemodulated optical feedback in the suppression of the TDS. Since different frequencies are induced due to noise-phasemodulated feedback. Then as shown in Fig. 2(a), the temporal waveforms of the optical intensities display more fluctuation components as compared to the case of no PM. Here the PM index β PM is 4 and the bandwidth of the noise generator is 40 GHz. When the feedback light contains many frequency components due to the noise PM, it is no longer a linear delayed replica of the output of the semiconductor laser. Therefore, as can be seen from the red curve of Fig. 2(b), with noise PM the TDS in ACF curve is significantly weakened.
To illustrate the mechanism of suppression TDS by noisephase-modulated feedback, in this section the comparison cases without PM and with noise PM for different bandwidths of noise generator and varying modulation index are presented. For a typical external-cavity optical feedback semiconductor laser, the time-delay signature is clearly observed from the ACF as exemplified by the black curve in Fig. 2 (b) which comes from the linear delayed replica of the output chaos. Without PM, as shown in Figure 3, two dynamical effects are discernible from the RF spectrum. Firstly, relaxation oscillations produce periodical features near the peak of the RF spectrum. Secondly, the frequency difference between those features is determined by the external cavity round trip delay τ f . It is found that the frequency difference between the features is 0.2 GHz which is 1/τ f . After adding noise phase modulation feedback, as the PM index and noise generator bandwidth are increased both these dynamical features of chaos are highly weakened.
After adding noise-phase-modulated feedback, new frequency components are induced into the feedback light, and as a result, periodical peaks in RF spectrum are generally weakened, whilst high frequency components share the energy as Fig. 4(b1)-(b3) illustrate. It has also been observed here that the RF bandwidth of the output chaos using noise PM is enhanced with a remarkable enhancement being found at high bias current. From the ACF curves shown in Fig. 4(a1)-(a3), one can observe that the TDS gradually decreases as the bandwidth of the noise generator or the modulation index increases. Here, it is found that when the  bandwidth of the noise generator reaches 40 GHz and the modulation index is 4, the periodical peaks in RF spectrum disappear as shown in Fig. 4(b3), whilst the TDS is totally suppressed as shown in Fig. 4(a3).
The above mentioned suppression process can be found from the optical spectral as shown in Fig. 5(a)-(c). When the bandwidth of the noise generator and the PM index are relatively small, the fixed mode spacing corresponding to the external-cavity (1/τ f ) can be observed in the optical spectrum shown in Fig. 5(a). This feature is weakened as the bandwidth of the noise generator increases to 40 GHz (in Fig. 5(b)) and highly suppressed by further increasing the PM index to 4 (in Fig. 5(c)).

IV. SUPPRESSION RESULTS
Having established the main principles underpinning the approach taken here, attention is now given to detailed performance aspects.

A. PM INDEX EFFECTS
Attention is first given to the impact of the phase modulation index β PM which is clearly a salient parameter of this process. Figure 6 shows results of the TDS in ACF versus PM index β PM . It is found that as β PM increases, the TDS in ACF decreasing quickly, then the TDS is suppressed to a relative small value. Even for the case of a relatively high laser bias current I = 5I th , the TDS value can decrease from 0.62 to below 0.1(green dashed line) when the phase modulation index β PM is above 8 as the blue circles shows in Fig. 6. If the index further increases to 18, the TDS will be submerged in the noise as the inset of Fig. 6 shows.

B. INFLUENCE OF THE NOISE GENERATOR BANDWIDTH
As well as the modulation index, the modulation frequency of the PM is an important factor which can be expected to affect the TDS suppression. As mentioned in Section II, we presume the modulation response frequency of PM is 100 GHz which is greater or equal to the bandwidth of the noise generator in use. Therefore, here consideration only needs to be given to the impact of the bandwidth of the noise generator, which finally determines the modulation frequency on PM. A low pass filter is used to control the bandwidth of the noise signal. By changing the 3dB bandwidth of low filter, the impact of the noise generator bandwidth can be  determined, that is the effects of the modulation frequency on the TDS in the ACF. Figures 7 and 8 show the variations of TDS value in ACF curve as a function of bandwidth of noise generator and PM index under relatively low pump current 2I th and high pump current 5I th . As has been shown in detail in Figs. 4 and 5, the larger the noise generator bandwidth, higher frequencies can be induced into the phase term, thereby increasing the frequency components in the feedback light, and thus weakening the TDS. Based on this expectation, with increasing bandwidth of noise generator the TDS in the ACF generally decreases to a low level such as 0.1, which is the blue region in the right upper region of Figs. 7 and 8. The well suppressed TDS can be found in both conditions of relative weak feedback strength 20 ns −1 as shown in Fig. 7, and strong feedback strength of 95 ns −1 seen in Fig. 8. We also notice that if the bandwidth of noise generator is not sufficiently wide e.g. below 10 GHz in these both cases, the TDS remains around 0.2, even if the PM index is as high as 10. In any case, by comparing with the case of no PM, one can observe that the TDS value is well suppressed when the noise generator bandwidth is of order 10 GHz, whilst PM index should be larger than about 3 corresponding to the blue upper regions in Figs. 7 and 8.

V. CONCLUSION
A novel noise-phase-modulation feedback chaos laser has been proposed here. Noise as the driving signal of PM, induces a broad noise frequencies range into the feedback light and thus enable TDS suppression. This work has shown that over a large parameter range of PM index and noise generator bandwidth the TDS can be significant suppressed and even submerged into noise in a large parameter region. This study provides a new method to suppress TDS in a chaotic external cavity laser, which can be exploited in several application contexts. ZHI WEI JIA received the Ph.D. degree from the Institute of Semiconductors, Chinese Academy of Sciences, in 2017. Since September 2017, he has been working as a Lecturer with the Taiyuan University of Technology. His research interests include the single-mode quantum cascade lasers and their applications such as trace gas detection and laser spectroscopy.
K. ALAN SHORE (Life Senior Member, IEEE) received the degree in mathematics from the University of Oxford, Oxford, U.K., and the Ph.D. degree from University College, Cardiff, Wales, U.K. He was a Lecturer with the University of Liverpool, from 1979 to 1983, and then at the University of Bath where he became a Senior Lecturer in 1986, a Reader in 1990, and a Professor in 1995. He was a Visiting Researcher with the