## Abstract

We demonstrate experimentally that flat and broadband chaotic signals can be easily generated by combining a multi-mode laser diode subject to optical feedback with a band-pass filter. Measurements are made of the RF spectra of multi-mode and single-mode outputs from an external cavity Fabry-Perot (FP) semiconductor laser before and after the filtering procedure. In this way it is found that in the chaos regime the low-frequency energy of the single-mode output is enhanced by about 25 dB comparing with that of the multi-mode output. Moreover, the associated 3-dB chaos bandwidth can reach around 6 GHz for the single mode case. Numerical demonstrations show mode competition is the physical origin of the low-frequency enhancement in the single-mode chaotic outputs.

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

## 1. Introduction

Optical chaos has been paid great attention due to its potential applications in the fields of communications and sensors. For instance, it can be widely used as the carrier signal for secure communications [1–3], the detection signal for ranging lidars [4,5] or optical time domain reflectometers (OTDR) [6], and as the entropy source of the physical random bit generators [7–19].

Laser diodes subject to optical feedback are the most commonly used systems for chaos generation due to their simplicity in construction. For example, Quay *et al*. studied the chaotic characteristics of vertical cavity surface emitting lasers (VCSELs) with strong optical feedback in 2001 [20]; Rontani *et al.* analyzed the time-delay signature and relaxation oscillation of optical feedback single-mode semiconductor lasers in 2007 [21]; Wu *et al.* observed the time-delay characteristics of a double-cavity feedback distributed feedback (DFB) semiconductor laser and analyzed the effect of relaxation oscillation on the time-delay suppression in 2010 [22]; Li *et al.* analyzed the statistical properties of the intensity in semiconductor lasers with time-delayed optical feedback in 2015 [23]; In 2018, Bayati *et al.* reported the dynamical characteristics of an AC-coupled chaotic semiconductor laser with optical feedback [24]; More recently, Huang *et al.* observed the route to chaos in optical feedback quantum dot (QD) lasers [25].

However, limited by the relaxation oscillation, nearly all the energy of the chaotic signal directly generated from optical feedback semiconductor lasers is mainly concentrated in the high-frequency band (around the relaxation oscillation frequency). Consequently the energy in low-frequency band (below the relaxation oscillation frequency) is a serious deficiency. In practical applications, the signal detection/acquisition device usually exhibits a 3-dB low-pass filtering characteristic. Therefore, the available effective bandwidth of the chaotic signal should actually be 3-dB bandwidth. The absence of low-frequency components will seriously limit the energy utilization ratio of chaotic signals and thus constrain the associated performance in applications.

In recent years, some schemes has been proposed to improve this problem of the small energy lack at low-frequencies. For instance, we have reported that combing an optical feedback chaotic DFB laser with a delayed self-interference structure [26], an optical heterodyne structure [27] or a fiber oscillation ring [28] can effectively resolve this issue. Hong *et al.* generated a flat chaos by combining an optical feedback VCSEL with a fiber ring resonator [29]. However, these schemes are very complex in structure, sensitive to polarization, and susceptible to environmental fluctuations.

In this letter, we report a simple scheme for flat chaos generation containing significant low-frequency components. We firstly use a Fabry-Perot (FP) semiconductor laser with optical feedback to generate multi-mode chaos; simply by using an optical band-pass filter, we can obtain a broadband flat single-mode chaotic signal. Our experimental results show that the low-frequency energy of the single-mode chaotic spectrum can be increased by 25 dB compared to that of the multi-mode chaos. Moreover, the 3-dB bandwidth of the single-mode chaotic laser can reach 6 GHz. Theoretical analysis shows that the significant enhancement of low-frequency band in single-mode chaotic signals is caused by the mode-competition. In other words, the low-frequency components of the multi-mode chaotic signal is offset by the mode-competition amongst the modes.

## 2. Experimental setup and results

Figure 1 is the experimental setup for the RF spectrum analysis of optical feedback FP laser diode (FP-LD). The output of an FP-LD is divided into two parts by a 60:40 optical coupler (OC1) after passing through a polarzation controller (PC). The 40% portion of the output is directed through a variable optical attenuator (VOA) onto a fiber mirror (FM), thus forming an optical feedback cavity. The multi-mode chaotic light is output from the 60% portion.

The multi-mode chaotic light passes through an erbium-doped fiber amplifier (EDFA) to amplify power. Different single-mode chaotic light can be accessed by tuning the center frequency of the optical band-pass filter (BPF). In the experiment, we record the optical spectrum using an optical spectrum analyzer (OSA, YOKOGAWA AQ6370C) and use a photodetector (PD, NEWPORT 1544B, 12 GHz) and an electrical spectrum analyzer (ESA, Agilent Techonogies N9020A, 26.5 GHz) to measure the RF spectrum of the chaos.

Figure 2 shows the optical spectrum and RF spectrum of the chaotic multi-mode FP-LD without an optical filter. In the experiment, the FP-LD with a threshold of 12 mA is biased at 18 mA, working at a feedback intensity of 20%. It can be observed from Fig. 2(a) that there are 19 longitudinal modes in the wavelength range between 1535 nm and 1555nm, where the central wavelength spacing between two adjacent modes is about 1.1 nm. For the convenience of later analysis, the longitudinal mode with the highest energy (whose center wavelength is 1549.291 nm) in Fig. 2(a) is defined as 0 mode (m = 0). Meanwhile, the two longitudinal modes close to the 0 mode are defined as + 1 mode (m = + 1) and −1 mode (m = −1), respectively. The other modes are sequentially defined as + 2, −2, + 3, −3 mode etc. It is pointed that the resolution of the optical spectrum analyzer used in the experiment was set at 0.05 nm. Figure 2(b) is the RF spectrum of the multimode chaos recorded by the ESA, whose resolution bandwidth and video bandwidth are set as 1 MHz and 1 kHz, respectively. It is obvious that the energy of the multi-mode chaotic signal is mainly concentrated around the relaxation oscillation frequency near 5 GHz. Also the energy in the low frequency range (0 - 5 GHz) is significantly absent, and the entire spectrum presents a single-peak distribution. This is similar with the chaotic signal generated by conventional optical feedback single-mode DFB or VCSEL lasers. Such a chaotic signal has the deficiencies of low frequency energy utilization ratio and insufficient effective bandwidth.

In order to solve the problem of the absence of low frequency energy, we analyze experimentally the 0, + 1, −1 mode chaotic signals filtered by the BPF with a 3-dB bandwidth of 0.27 nm. Figures 3(a1) - 3(a3) depict the optical spectra of the three single-mode chaos signals (m = −1, 0, + 1), while Figs. 3(b1) -3(b3) are their own RF spectra. As expected, the 0 mode (m = 0) at the center of the entire optical spectra [Fig. 3(a2)] has the highest energy in the power spectra [Fig. 3(b2)], while the energy of the side modes (m = −1 and m = + 1) are relatively low as shown in Figs. 3(b1) and 3(b3), respectively. However, it is beyond our expectation that the energy of all single-mode chaotic signals at low frequencies is greatly increased and the associated RF spectra becomes flat [Figs. 3(b1)-3(b3)], compared with the multi-mode chaotic signal [Fig. 2(b)]. Taking the −1 mode (m = −1) as a specific example, it is seen that its low-frequency energy is improved about 25 dB compared with the multi-mode chaos case and its 3dB bandwidth can reach 6 GHz [Fig. 3(b1)]. This means that broadband chaos without the loss of low frequency components can be easily generated by combining a multi-mode chaotic laser and an optical filter, which is a much simpler approach than previous techniques [26–29]. It should be noted that this kind of enhancement in the low-frequency energy can also be observed in the other modes. For conciseness, we only take the 0, + 1, −1 mode chaotic signals as typical examples here.

## 3. Theoretical analysis

We theoretically explored the physical mechanism underpinning the enhancement of low-frequency energy in the single-mode chaotic signals in the experimental configuration utilized here. Specifically, we have numerically simulated the chaotic dynamics of optical feedback multi-mode lasers based on the extended Lang-Kobayashi (LK) equations as below [30,31].

*M*represents the total number of modes and

*m*corresponds to the

*m-*th mode of the multi-mode laser;

*E*,

*Φ*,

*N*represent the normalized amplitude, phase and carrier number of the electric field, respectively; ${F}_{\gamma}(t)=\sqrt{2\beta \xi (t)}$is the spontaneous emission noise [note,

*ξ(t)*is Gaussian white noise term and

*β*is the spontaneous emission noise];

*Δ*is the phase difference between the laser itself and the external feedback;

*G*is the laser gain;

*N*is the carrier number at threshold which is defined as

_{th}*N*[note,

_{th}= N_{0}+ γ/g_{c}*N*

_{0}is the carrier number at transparency and

*g*is the differential gain coefficient];

_{c}*Δω*is the mode interval, where

_{L}= 2π/τ*τ*represents the internal round-trip time. The other parameters and their associated symbols and values are listed in Table 1. Note that rather typical parameter values of laser diodes are chosen in our simulation and hence the qualitative results obtained using the model are robust to reasonable changes in the parameter values. In any case, no special effort has been made to select parameter values except for choices guided by the experimental measurements. Considering the experimental condition, we set the central mode wavelength (m = 0) and the longitudinal mode interval to be 1549 nm and 1.1 nm in the simulation, respectively. Meanwhile, the level of the spontaneous emission noise is set at −30 dB.

Figure 4(a) shows the optical spectrum of a multi-mode chaotic signal obtained by simulating an optical feedback FP laser with 15 longitudinal modes. The numbers on the abscissa represents the relative wavelength of the spectral mode, respectively. When Δλ = 0 nm, the central wavelength of the corresponding mode is 1549 nm. We can clearly see from the optical spectrum that there are 15 longitudinal modes and the longitudinal mode interval is 1.1 nm, as expected. Figure 4(b) shows the RF spectrum of the corresponding multi-mode chaotic signal. Compared with Fig. 2(b), it can be observed that the simulation results are consistent with the above experimental measurements where most of the energy of the multi-mode chaotic signal is concentrated near the relaxation oscillation frequency and the low-frequency component energy is reduced.

Figure 5 shows the results of simulating single-mode chaotic signals in the three modes (m = −1, 0, + 1) after filtering. Figures 5 (a1)-5(a3) are the RF spectra in the different modes. It can be found that the center mode (m = 0) has the highest energy, and the longitudinal modes on both sides have reduced energy, which is consistent with the experimental results in Figs. 3 (b1)-3(b3). Figures 5 (b1)-5(b3) illustrate the time series of the single-mode chaotic signals in modes m = −1, 0, + 1, respectively. It can be seen that the m = −1 mode and the m = 0 mode are similar, whilst modes m = −1 and m = + 1 mode exhibit quite distinct dynamics. This shows that in the multi-mode laser the modes compete for their share of the available gain provided by the charge carriers.

We use cross-correlation to quantify the competition between longitudinal modes. The cross-correlation function, *CC*, is defined as below.

*E*and

_{i}*E*represent the complex amplitudes of the different modes. When the value of the

_{j}*CC*tends towards 1, there is a higher in-phase compatibility between the modes. Whilst

*CC*tends towards −1, the modes tend to be out of phase. Figure 5(c1)-5(c3) are the

*CC*functions between different modes. Due to the differences in the time series shown in Figs. 5(b1)-5(b3), one anticipates that the mode competition – encapsulated in the CC function – will show markedly different features when correlations between different modes are considered. This is indeed found in Figs. 5 (c1)–5(c3): Fig. 5(c1) demonstrates that modes m = −1 and m = + 1 are in anti-phase; In contrast, Fig. 5(c2) shows that modes m = −1 and mode m = 0 are in phase; From Fig. 5(c3) we have that mode m = + 1 and mode m = 0 are in anti-phase.

The anti-correlation between the laser modes like that in Figs. 5(c1) and 5(c3) causes the spectral ‘deficiency’ at low frequencies, but not for higher frequencies. To address this point more clearly, we further execute a low pass filtering (LPF) process with different cut-off frequencies (1 GHz, 3 GHz and 5 GHz) to these time series in Figs. 5(b1)-5(b3). We then calculate their associated CCs as shown in Fig. 6. As expected, it is find that the anti-correlation is enhanced with decrease of the LPF cut-off frequency in the anti-phase cases [first and third columns in Fig. 6]. In contrast, the positive correlation is greatly reduced with decrease of the LPF cut-off frequency in the in-phase case [Second column in Fig. 6]. These additional results establish that the reason for the significant increase in the low-frequency energy in the single-mode chaotic signal is the competition between the laser modes.

## 4. Conclusion

Experiments undertaken using optical feedback multi-mode semiconductor laser with simple filtering demonstrate a 25 dB enhancement of the low-frequency energy in its chaos spectrum. In addition, the RF spectrum remains relatively flat with a 3-dB effective bandwidth about 6 GHz. A numerical analysis has been performed which shows that mode competition is the physical origin of the improved performance.

The simple structure of low-cost multi-mode laser combined with a filter used to generate broadband chaotic signals with a flat RF spectrum without energy lack at low-frequencies, is beneficial to improve the energy utilization of chaos in practical applications such as chaos communications, random bit generation, ranging lidar and optical time domain reflectometry.

## Funding

National Natural Science Foundation of China (NSFC) (61775158, 61731014, 61671316, 61875147, 61771439), National Cryptography Development Fund (MMJJ20170127), China Postdoctoral Science Foundation (2018M630283), Science and Technology Commission of Shanghai Municipality (STCSM) (Grant No. SKLSFO2018-03), and Program for the Top Young Academic Leaders of High Learning Institutions of Shanxi.

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