## Abstract

We numerically demonstrate the generation of wide-band laser chaos with flat power spectrum in a 2D circular-side hexagonal resonator (CSHR) microlaser subject to long-cavity optical feedback. The bandwidth and flatness of the chaotic power spectrum are investigated under different bias currents and optical feedback rates. Under low bias current, the bandwidth under an optimized optical feedback rate increases obviously as raising bias current and the power spectrum flatten simultaneously. Under high bias current, the optimized bandwidth gradually tends toward stabilization, with corresponding flatness less than 5 dB. We compare the chaotic power spectra with small-signal modulation response (SSR) curves under different bias currents. It can be concluded that wide-band and flat SSR indicates wide-band and flat chaotic power spectrum. This work argues that we can enhance laser chaos by using a laser device with wide-band and flat SSR and simple optical feedback configuration, which is significantly beneficial to synchronization-based applications including chaos communication and key distribution

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Laser chaos has attracted many research interests, which can find important applications in secure communication [1–4], physical random bit generation [5–7], key distribution [8,9], lidar ranging [10], and optical time domain reflectometers [11,12]. Distributed feedback (DFB) semiconductor laser with an external-cavity optical feedback from a mirror is usually used as chaotic light source due to dynamical complexity and simple configuration capable of photonic integration [13]. However, in DFB semiconductor laser, chaos oscillation typically has a sharp power spectrum with a limited bandwidth of a few gigahertz. The non-uniform spectral density and limited bandwidth seriously restrict the encryption speed of chaos communication and key distribution speed.

Many methods have been proposed to enhance chaos bandwidth and meliorate power spectrum flatness of laser chaos. These methods can be classified into two types. The first type is introducing additional optical injection or complex feedback schemes, such as optical injection from a continuous-wave laser into a chaotic DFB laser or vice versa [14,15], optical mutual-injection between two lasers [16], a laser with phase conjugation feedback [17] or self-phase-modulated optical feedback [18]. The second type is utilizing the phase-to-intensity conversion of chaotic laser by external setup such as a fiber ring resonator with an inbuilt optical filter [19,20], optical heterodyning and or electrical heterodyning [21,22]. By above methods, chaos bandwidth from 10 GHz to 50 GHz can be achieved. However, the configuration complexity of chaotic light source is also greatly added, making chaos synchronization a tough challenge. Therefore, it is of greatly importance to generate wide-band and flat-spectrum laser chaos by simple optical feedback configuration for synchronization-based applications, such as chaos secure communication and chaos-based physical key distribution.

In DFB semiconductor laser with an external-cavity mirror feedback, the chaos oscillation with sharp power spectrum and limited bandwidth is due to the relaxation oscillation. Moreover, the relaxation oscillation also dominates the small-signal modulation response (SSR). It is noted that 2D microcavity lasers with whispering-gallery-mode resonator can obtain flat SSR [23], resulted from the weak laser response to relaxation oscillation. It can be inferred that the microcavity lasers can also generate flat-power-spectrum laser chaos because of the weak laser response to relaxation oscillation.

Recently, Xiao *et al.* reported a circular-side hexagonal resonator (CSHR) microlaser with 1.55 µm single mode emission and a flat SSR with the 3 dB bandwidth of 13 GHz [24]. In this paper, we numerically investigate the bandwidth and flatness of chaotic power spectrum from the CSHR microlaser subject to optical feedback to demonstrate the possibility and conditions of generating wide-band and flat laser chaos. The effect of bias current and optical feedback rate on laser chaos is analyzed in detail and the comparison with SSR is also given. Results support that the microlaser with wide-band and flat SSR can generate wide-band and flat-spectrum laser chaos, and the profile of power spectrum is almost identical to the SSR curve.

## 2. Theoretical model

Figure 1 shows the schematic diagram of the CSHR microlaser subject to mirror feedback, the feedback time delay *τ*_{f }= 5ns. Based on the Lang-Kobayashi equations, the rate equations of the mirror-feedback CSHR microlaser is given as below [23,25–27]:

*n*is the carrier density,

*s*is the photon density, and

*ϕ*is the phase of optical field,

*I*is the bias current,

*κ*

_{f}=

*κ/τ*

_{L}is the optical feedback rate, where

*κ*and

*τ*

_{L}represent the amplitude feedback coefficient and the round-trip time [27].

*v*

_{g}

*=*

*c*/

*n*

_{g}is the lasing mode group velocity, and

*τ*

_{pc}=

*Q*/

*ω*is the photon lifetime of the cold cavity mode. The gain coefficient

*g*of the strained quantum well material is:

Note that the Eqs. (1)–(3) are based on Lang-Kobayashi model, which only considers the time variation and ignores the spatial distribution in the cavity, so they are applicable to other semiconductor lasers, such as traditional DFB lasers and VCSELs. The special parameters for the CSHR microlasers are the small mode volume *V*_{a}/Γ and the high Q factor which are simultaneously achieved due to whispering gallery modes. According to the following expression of relaxation frequency *f*_{R}, the CSHR microlaser can achieve an enhanced *f*_{R} at a large bias current due to small mode volume and low threshold resulted from high Q factor,

*a*=∂

_{n}*g*/∂

*n*is differential gain at steady state. According to the small-signal response function [23,28]

*f*

_{R}, which usually is the peak height of |

*H*(

*f*)|

^{2}, is (2π

*f*

_{R}/

*γ*)

^{2}, where

*γ=Kf*

_{R}

^{2}+

*γ*

_{0}is the damping factor,

*K*factor describes the damping of the response and

*γ*

_{0}denotes the damping factor offset. One can derive that the relaxation oscillation response decreases as the relaxation frequency increases, so that, a flat and wide-band SSR is obtained by the CSHR microlaser.

The descriptions of other symbols and the selected values for simulation are shown in Table 1, based on Ref. [23–27]. The rate equations are numerically calculated by using the fourth-order Runge-Kutta method with a step length of 5 ps.

## 3. Numerical results

Figure 2(a) shows the SSR curves of the CSHR microlaser under different bias currents. As shown, a resonance peak resulted from relaxation oscillation can be observed. Under the bias currents of 5*I*_{th}, 8*I*_{th} and 12*I*_{th}, the resonance peak frequencies are 7.7 GHz, 9.1 GHz, and 9.7 GHz, and the peak heights are 4.8 dB, 2.9 dB, 1.35 dB, respectively. With the increase of bias current, the resonance peak frequency increases and the peak height is greatly suppressed. In theory, as the bias current increases, both the relaxation oscillation frequency and damping factor increase. Therefore, the resonance peak frequency is less than the relaxation oscillation frequency [28], as shown in Fig. 2(b). Under the bias currents of 5*I*_{th}, 8*I*_{th} and 12*I*_{th}, the relaxation oscillation frequencies are 8.8 GHz, 11.3 GHz and 13.7 GHz, and the damping factors are 27.8 GHz, 45.7 GHz and 66.7 GHz, respectively.

Here, we artificially set a critical value to distinguish whether the SSR curve is flat. When the height of the resonance peak is less than 3 dB, it is considered that the SSR curve is flat, and the corresponding bias current range is considered as high current. Otherwise it is considered as low current if the resonance peak height is greater than 3 dB. In the following, we also refer to this standard to distinguish high and low current.

Figure 3 shows simulated results of optical spectra, time series, power spectra and autocorrelation functions (ACF) obtained by the CSHR microlaser under the bias current of *I *= 5*I*_{th}, corresponding to the black curve in Fig. 2(a). The behaviors are similar to that of the laser chaos produced by traditional DFB laser with optical feedback, with the disadvantages of low frequency energy loss and insufficient effective bandwidth. When the feedback rate is 12.4 ns^{−1}, relaxation oscillation mode can be seen in the optical spectrum of chaos, relaxation oscillation signature exists in the time series, and the power spectrum has a sharp peak at relaxation oscillation frequency. Accordingly, the relaxation oscillation signature can be clearly reflected in ACF curve, which is evaluated by the height of sidelobe located at τ_{R}. With the increase of the feedback rate, the energy of the relaxation oscillation mode in the optical spectrum decreases, the relaxation oscillation frequency peak of power spectrum weakens, the complexity of time series increases, and the height of the sidelobe peak of ACF curve reduction but is always greater than the background noise. Besides, the ACF curve has a correlation peak at *τ*_{f} as depicted in Figs. 3(a4)–3(c4), which can be eliminated by grating feedback [29].

In order to quantitatively analyze the features of the chaos produced by the CSHR microlaser, the definition of bandwidth and flatness are given as follows. The bandwidth is defined as the width of the band that close to the highest peak of power spectrum, where 80% energy is contained within [30]. Two performance indexes are used to judge the flatness of power spectrum: one is the relaxation oscillation signature in the ACF curve, the other one is the filtered flatness. Owing to the external cavity oscillation is obvious, which affects the observation of the general power spectrum flatness. We use the fluctuation range of the filtered power spectrum profile curve after 0.2 GHz filtering within 80% energy band to measure the flatness, as drawn in the red line in Figs. 3(a3)–3(c3). This method can refer to the grating feedback to eliminate the external cavity characteristics [29]. As shown in Fig. 3(a3) as an example, the bandwidth of the microlaser is 8.5 GHz, the filtered flatness is 7.8 dB.

Figure 4 shows simulated results of the CSHR microlaser under the bias current of 12*I*_{th}, corresponding to the red curve in Fig. 2(a). It should be pointed that the needed optical feedback rate for same dynamic state vary with the increase of bias current. Therefore, compared with Fig. 3, higher optical feedback rate is needed. The magnification of damping leads to the suppression of relaxation oscillation, and the laser output becomes more complex. As shown in Figs. 4(a1)–4(a4), the laser is in a weak feedback state, it presents an initial chaotic state which has not been fully developed. Therefore, relaxation oscillation signature can be seen clearly in the power spectrum, and the energy distribution is unbalanced, where most of the energy is limited in the neighborhood of relaxation oscillation frequency, and the relaxation oscillation signature cannot be hidden in background noise. When the feedback rate reaches 27.5 ns^{−1}, the flat and wide-band laser chaos is observed in Figs. 4(b1)–4(b4), which has no relaxation oscillation signature. The energy of the low-frequency oscillation component is greatly increased, and the relaxation oscillation signature can be hidden in background noise. When the feedback rate increases to 38.5 ns^{−1}, the low-frequency energy increases, but the high-frequency remains unchanged, resulting in a slight decrease in bandwidth and a slight deterioration in flatness, as demonstrated in Fig. 4(c3). The possible reason is that the damping factor is excessive under the strong feedback rate, and the resonance peak frequency is different from the relaxation oscillation frequency, referring to Fig. 2(b).

Then, the influence regulation of the optical feedback rate on the bandwidth, flatness and relaxation oscillation signature under the two bias currents is compared, as shown in Fig. 5. The crosses represent the single standard deviation of the ACF background noise. Obviously, the evolution of the three parameters are corresponding to each other. Under low bias current of 5*I*_{th} (squares), with the increase of optical feedback rate, the chaos bandwidth increases and the power spectrum flattens, but the relaxation oscillation signature is always larger than the background noise. Under high bias current of 12*I*_{th} (circles), with the increase of optical feedback rate, the bandwidth increases and then decreases slightly, the flatness firstly decreases from 7.5 dB to 3.8 dB and then increases, and the relaxation oscillation signature of ACF curve are suppressed. It can be concluded that there exist an optimized range of optical feedback rate for generation of wide-band laser chaos under a certain bias current.

Figure 6 shows the influence of bias current on laser chaos. With the increase of bias current, the bandwidth widens, the power spectrum flattens, and the relaxation oscillation signature weakens and gradually hide in the background noise. But there are some special cases under higher current. At the feedback rate of 27.5 ns^{−1} (squares), the chaos is underdeveloped, resulting in the decrease of bandwidth, the deterioration of power spectrum and the enhancement of relaxation oscillation signature. At the feedback rate of 30 ns^{−1} (circles), because of the phenomenon of power spectrum roll off, the declination angle of frequency power spectrum enlargement under high current, which leads to the deterioration of power spectrum flatness.

To further analyze the laser chaos produced from the CSHR mircolaser, maps of the bandwidth and flatness in the parameter space of *I*/*I*_{th} and *κ*_{f} are displayed in Figs. 7(a)–7(b). The maximum chaos bandwidth under different bias currents is shown as the dashed line in Fig. 7(a), and it is compared with the SSR 3 dB bandwidth of a solitary laser in Fig. 7(c). It can be observed that the maximum chaos bandwidth is close to the SSR 3 dB bandwidth. Under the high bias current of *I* ≥ 8*I*_{th} (the critical bias current), the maximum chaos bandwidth is more than 14 GHz, and the corresponding flatness is less than 5 dB. This indicates that the CSHR mircolaser under high bias current can produce the wide-band and flat-spectrum laser chaos. It is believed that there is relevance between the SSR curve and the chaotic power spectrum of the microlaser.

The largest Lyapunov exponent is one of important parameter measures for the complexity of deterministic chaos. As shown in Fig. 8, with the increase of bias current, the value of largest Lyapunov exponents is about doubled, and the largest values is about 3.4 ns^{−1} at bias current of 15*I*_{th} and feedback rate of 30 ns^{−1}. Referring to Fig. 6, it can be concluded that the increase of largest Lyapunov exponent is related to the widening of bandwidth, which is the result of the suppression of relaxation oscillation signature. Meantime, the largest Lyapunov exponent can be further improved by suppressing the time-delay signature.

## 4. Conclusion

The chaos bandwidth and power spectrum flatness of the CSHR microlaser with optical feedback under different bias currents and optical feedback rates are studied. The results show that with the optimization of bias current and optical feedback rate, the low frequency energy of spectrum is compensated and the relaxation oscillation is restrained. At high bias current, the CSHR microlaser has a flat and wide-band SSR curve and the CSHR microlaser with simple optical feedback can generate wide-band and flat-spectrum laser chaos. Due to the model based on Lang-Kobayashi equations does not include spatial distribution of light in cavity, it is inferred that other semiconductor lasers having a flat and wide-band SSR can also generate a wideband and flat-spectrum chaos with optical feedback. This is of great significance in practical applications such as chaos communication, key distribution and random number generation. Huang *et al*. have carried out the experimental research on the generation of a microwave signal by microring laser with optical injection [31]. And next, we will conduct the experimental research on the generation of chaos from the CSHR microlaser with optical feedback.

## Funding

National Natural Science Foundation of China (61731014, 61822509, 61927811, 61961136002); Shanxi Talent Program (201805D211027); Shanxi "1331 Project" Key Innovative Research Team; Program for the Top Young and Middle-aged Innovative Talents of Higher Learning Institutions of Shanxi; Program for Guangdong Introducing Innovative and Enterpreneurial Teams; International Cooperation of Key R&D Program of Shanxi Province (201903D421012).

## Disclosures

The authors declare no conflicts of interest.

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