## Abstract

We propose and demonstrate an external-feedback semiconductor laser-based chaos generation scheme supporting simultaneous bandwidth enhancement and excellent time-delay-signature (TDS) suppression, by using parallel-coupling ring resonators (PCRR) as reflector. The characteristics of effective bandwidth and TDS of chaotic signals generated in three indicative PCRR configurations are thoroughly investigated. The numerical results demonstrate that with the nonlinear feedback of PCRR, the TDS of chaos can be efficiently suppressed toward an indistinguishable level, and the bandwidth of chaos in the proposed scheme can also be enhanced, with respect to the conventional optical feedback configuration. The proposed scheme shows a flexible way to generate wideband complex chaos.

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

## 1. Introduction

Chaos generated by external cavity semiconductor laser (ECSL) has attracted a lot attention for its extensive applications in the fields of secure optical communications [1–3], random bit generation (RBG) [4–6], chaotic correlation optical time domain reflectometer [7], and chaotic radar [8]. Nevertheless, for the conventional optical feedback (COF) ECSL systems, due to the relaxation oscillation of laser, most of the energy is concentrated nearby the relaxation oscillation frequency in the radio frequency (RF) spectrum of generated chaos, this leads to the bandwidth limitation of chaotic signal. The limited bandwidth would restrict the speed of secure communications and limit the rate of RBG, as well as degrade the resolution of chaotic radar. On the other hand, in the COF-ECSL systems, the time delay signature (TDS) originating from the linear round trip between semiconductor laser and external reflector, can be easily retrieved by several analysis methods, such as the calculations of auto-correlation function (ACF), permutation entropy (PE), and digital mutual information (DMI) [9,10]. The obvious TDS would reveal the length of external cavity, which increases the risk that the eavesdropper reconstructs a similar chaos-generation system to crack the chaos-based secure communications system, and consequently degrades the system security. Moreover, the obvious TDS means the chaotic signal is periodic (or quasi-periodic), this would degrade the randomness of bit generation and the precision of radar detection for its applications in RBG and chaotic radar. Therefore, it is vital and valuable to simultaneously enhance the bandwidth and suppress the TDS of chaotic signal.

For the purpose of bandwidth enhancement and TDS suppression in ECSL-based optical chaos, several methods have been reported in recent years. It has been demonstrated that the bandwidth of chaos can be enhanced by optical injection [11,12] and delay self-interference [13]. In terms of the TDS suppression in ECSL-based chaos, it has been confirmed that the TDS can be efficiently suppressed by the ways of dual optical feedback [14], polarization-rotated feedback [15], distributed and dispersive feedback from FBG [16,17]. In particular, a few chaos generation schemes supporting simultaneous bandwidth enhancement and TDS suppression have also been demonstrated by utilizing heterodyning couplings [18], fiber propagation [19], optical time lens [20], cascaded-coupling ring lasers [21], self-phase-modulated feedback with microsphere resonator and delay-interference [22,23].

In this paper, we propose and demonstrate a bandwidth-enhanced and TDS-suppressed optical chaos generation scheme by using parallel-coupling ring resonators (PCRR) as the external reflector of ECSL. By taking three indicative PCRR configurations with single ring resonator (SRR), double ring resonators (DRR) and triple ring resonators (TRR) for instance, it is demonstrated that comparing with the COF scenario, in the proposed PCRR-based nonlinear feedback ECSL system, not only the TDS of chaos can be efficiently suppressed, but also the bandwidth of chaos can be enhanced significantly.

## 2. Principles and theoretical model

Figure 1(a) shows the schematic diagram of the proposed chaos generation scheme based on ECSL subject to nonlinear feedback from PCRR. Different from the conventional optical feedback (COF), the output light of a distributed-feedback (DFB) laser passes through *n* parallel coupling ring resonators, which act the role of external feedback reflectors [Fig. 1(b)]. The output of the “Drop” port of PCRR is sent back into the DFB laser cavity. The PCRR can be fabricated on confined waveguides in both GaAs-AlGaAs and GaInAsP-InP and directly coupled with the ECSL through optical coupling [24]. A variable optical attenuator (VOA) is adopted to adjust the feedback strength, and a polarization controller (PC) is adopted to tune the polarization of light. The generated chaotic signal is outputted from the “Throughput” port. Mathematically, the proposed PCRR-based ECSL can be modeled by the following modified Lang-Kobayashi rate equations, in which the nonlinear feedback term introduced by PCRR is considered [25].

*E*denotes the slowly-varying complex electric field and

*N*is the carrier density in the active region of ECSL. The last term in Eq. (1) stands for the nonlinear optical feedback of PCRR, it is the convolution of the response function

*h*

_{1}(

*t*) of the “Drop” port and the linear delayed replica of laser output

*E*(

*t*-

*τ*). The output chaos is expressed by Eq. (3), it is the convolution of the response function

_{f}*h*

_{2}(

*t*) of the “Throughput” port and the delayed laser output

*E*(

*t*-

*τ*/2). The transfer function of PCRR in the frequency domain can be described by the transfer matrix method, which is written as [24,26]

_{f}*T*is the transfer matrix of the

_{i}*i*ring resonator,

^{th}*T*is the transfer matrix of the waveguide jointing two adjacent ring resonators, and their transfer matrixes are given in Eqs. (5) and (6), respectively. In Eqs. (5) and (6),

_{φi}*к*

_{1i}and

*к*

_{2i}are the coupling coefficients of “Input” port and “Drop” port, respectively, while

*η*

_{1i}(

*η*

_{1i}

^{2}=1-

*к*

_{1i}

^{2}) and

*η*

_{2i}(

*η*

_{2i}

^{2}=1-

*к*

_{2i}

^{2}) denote the corresponding transmission coefficients;

*α*

_{ri}^{2}=exp(-2

*α*π

_{i}*R*) stands for the round-trip amplitude attenuation, where

_{i}*R*is the radius of single ring resonator and

_{i}*α*is the corresponding loss coefficient in the ring resonator; Δ

_{i}*ω*=

*ω*-

*ω*

_{0}denotes the detuning of operation frequency with respect to the central frequency

*ω*

_{0};

*t*=2π

_{ri}*R*/

_{i}n_{eff}*c*is the round-trip time in single ring resonator, where

*n*is the effective index of ring resonator;

_{eff}*L*is the length of the waveguide between the two adjacent ring resonators. As shown in Fig. 1(b), no light is entered into PCRR through the “Add” port, thus

*E*

_{4}(

*ω*) = 0. Subsequently, the transfer function of the “Drop” port is determined by

*H*

_{1}(

*ω*)=

*E*

_{2}(

*ω*)/

*E*

_{1}(

*ω*)=

*P*

_{21}/

*P*

_{11}, and that of the “Throughput” port is determined by

*H*

_{2}(

*ω*)=

*E*

_{3}(

*ω*)/

*E*

_{1}(

*ω*) = 1/

*P*

_{11}. Here

*P*

_{11},

*P*

_{21}are determined by the cascade product of

*T*and

_{i}*T*as that in Eq. (4).

_{φi}To quantitatively investigate the TDS characteristic of chaos, use of two methods, namely the calculations of ACF and PE, are made. The ACF is defined as [21–23,28,29]

*I*(

*t*) denotes the output from the “Throughput” port of PCRR, it is proportional to |

*E*(

*t*)|

^{2}, <·> stands for the time averaging, Δ

*t*is the delay time that

*I*(

*t+*Δ

*t*) is shifted with respect to

*I*(

*t*). Regarding the PE [30–32], we take the time series {

*x*,

_{t}*t*= 1, 2, …,

*N*} and reconstruct a

_{x}*d*-dimensional space

*X*=[

_{t}*x*(

*t*),

*x*(

*t*+Δ

*τ*), …,

*x*(

*t*+(

*d*-1)Δ

*τ*)], where

*d*and Δ

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

*X*is constructed by arranging elements of ${\{ {x_t}\} _{t = 1,\ldots ,N_{x}}}$ increasing order ${x_{t + ({r_1} - 1)\Delta \tau }} \le {x_{t + ({r_2} - 1)\Delta \tau }} \le \ldots \le {x_{t + ({r_d} - 1)\Delta \tau }}$, and any

_{t}*X*is uniquely mapped onto an ordinal pattern Ω=(

_{t}*r*

_{1},

*r*

_{2}, …,

*r*) out of

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

*d*, the probability distribution

*P*=

*p*(Ω) of the ordinal patterns is:

*H*[

*P*] is evaluated by the permutation probability distribution, in term of

*H*[

*P*]=-∑

*p*(Ω)log

*p*(Ω). Finally, the normalized PE

*h*[

*P*] is evaluated by

*h*[

*P*] =

*H*[

*P*]/log

*d*!, and it’s value ranges between 0 and 1. Specifically,

*h*= 0 corresponds to a predicable dynamics,

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

*d*! permutations appear with the same probability.

In our simulations, the rate equations are solved by the fourth-order Runge-Kutta algorithm. The intrinsic parameters of the DFB laser are chosen to be those reported in [27]: the operation wavelength *λ*=1550 nm, the linewidth enhancement factor *α*=5, the gain saturation coefficient *ε*=1×10^{−5} µm^{3}, the carrier lifetime *τ _{e}*=5.58 ns, the photon lifetime

*τ*=4.11 ps, the differential gain coefficient

_{p}*g*= 0.73×10

^{−3}µm

^{3}ns

^{−1}, the transparency carrier density

*N*

_{0 }= 1.5×10

^{6}µm

^{−3}, the active layer volume

*V*= 175 µm

^{3}, and the electron charge

*q*= 1.602×10

^{−19}C. Unless otherwise stated, the PCRR parameters are chosen as

*к*

_{1i}=

*к*

_{2i}=0.4,

*α*

_{ri}^{2}=1,

*R*=2mm,

_{i}*L*= 10mm,

*n*= 2.6; the laser bias current

_{eff}*I =*1.5

*I*

_{th}, where

*I*

_{th}=20 mA is threshold current; the feedback strength is fixed as

*k*= 20 ns

_{f}^{−1}and the feedback delay is set as

*τ*

_{f}_{ }= 5ns. For the calculations of PE, the number

*N*of the time series for space reconstruction is chosen as 20000, and the dimension

_{x}*d*of reconstructed space is set as 6.

## 3. Results and analysis

In this section, the characteristics of bandwidth enhancement and TDS suppression of the proposed scheme are thoroughly discussed. The effective bandwidth (EB) that is defined as the span between the direct current (DC) component and the frequency where 80% of the energy is contained in the RF spectrum of chaotic signal, is adopted to quantify the bandwidth of generated chaos [21–23,29,33]. The TDS is quantitatively evaluated by the maximum value nearby the position of feedback delay in the ACF and PE traces of the generated chaotic signal. For the sake of comparison, four scenarios, namely the COF and three indicative PCRR-feedback scenarios with single ring resonator (SRR), double ring resonators (DRR) and triple ring resonators (TRR) are simultaneously considered in the following discussions.

Figure 2 shows the temporal waveforms of the intensities, as well as the corresponding RF spectra, ACF traces and PE traces of the chaotic signals generated in the cases of COF, SRROF, DRROF and TRROF. For the COF case, as shown in Fig. 2(a2), the energy in the RF spectrum is mainly concentrated nearby the relaxation oscillation frequency, as such the effective bandwidth is only 5.29 GHz. Meanwhile, there are obvious periodic peaks with a frequency spacing equaling the external-cavity resonation frequency (1/*τ _{f}*) appearing in the RF spectrum, as such clear TDS is observed in the traces of ACF and PE of chaos [Figs. 2(a3) and 2(a4)]. For the SRROF case, as shown in Figs. 2(b1)-(b4), the RF spectrum is flattened to some extent, the effective bandwidth is enhanced to 6.76GHz, and the TDS is significantly suppressed. It is worth mentioning that due to the multiple propagations in the ring resonator, an additional small delay is introduced in the feedback light, as such the peak indicating the TDS appears at a position slightly deviating from the feedback delay. Even so, only a tiny peak with a value of 0.07 is observed nearby the feedback delay position in the ACF trace, and no distinguishable valley can be found in the PE trace, this means the complexity of the SRROF-based chaos is considerably enhanced, with respect to the COF case. Furthermore, for the DRROF and TRROF cases, similar bandwidth enhancement and TDS suppression phenomena are also observed. The effective bandwidths are enhanced to 8.71 GHz and 9.21 GHz, respectively. Moreover, no distinguishable peak and obvious valley can be observed at the position nearby the feedback delay in the ACF traces and PE traces, this means the TDS can be completely suppressed and the complexity of the generated chaos is greatly enhanced, with respect to the COF and the SRROF scenarios. The comparisons between the scenarios of PCRROF and COF indicate that the TDS suppression is enhanced significantly. This is because with the nonlinear effect of the ring resonators, the linearity of feedback light is degraded, this induces corresponding periodicity degradation in the generated chaos. In addition, the comparisons among the scenarios of SRROF, DRROF and TRROF indicate that as the increase of the number of the ring resonators in PCRR, the RF spectrum becomes flatter and flatter, and the TDS in ACF and PE traces get more and more indistinguishable. That is, the bandwidth enhancement and the TDS suppression properties of the proposed scheme get better and better, as the increase of the number of ring resonators in PCRR.

As mentioned above, the RF spectrum of generated chaos becomes flatter and flatter as the increase of the number of ring resonators in the PCRR. The spectrum flattening is due to that the nonlinear feedback of PCRR redistributes the energy in the generated chaotic signal. Specifically, in the proposed scheme, the energy of low-frequency components is enhanced, while that of the relaxation oscillation frequency component is reduced correspondingly. To show this improvement quantitatively, we discuss the ratio of energy distribution in different frequency bands here. Figure 3 shows the ratio of energy distribution in six frequency bands ranging from 0.02-6 GHz, as a function of the feedback strength, for the cases of COF, SRROF, DRROF and TRROF. The energy distribution ratio is defined as the energy in a frequency band with a specific bandwidth (here it is set as 1 GHz) to the total energy in the overall RF spectrum. For the COF case (Fig. 3(a)), the ratio of energy in the frequency band 2-3GHz that is nearby the relaxation-oscillation frequency is more than 20% of the total energy, while the energy in the low-frequency band 0.02-1GHz is only about 4%. That is, the energy is concentrated nearby the relaxation oscillation frequency, as such the effective bandwidth is limited. Nevertheless, for the PCRR-feedback configurations shown in Figs. 3(b)–3(d), the distribution of energy in the RF spectra is significantly more uniform, and moreover, as the increase of the number of the ring resonators in the PCRR, the energy distribution becomes more and more uniform. For the cases of DRROF and TRROF, the energy ratios of the six frequency bands gradually concentrate at about 10%, which means the flatness of RF spectrum is greatly improved.

Figure 4 shows the influences of the feedback strength and the operation current on the TDS value in ACF trace and the effective bandwidth. Here for the PCRROF case, the TDS value is evaluated as the maximum ACF in the range from Δ*t*=*τ _{f}* −0.5 ns to Δ

*t*=

*τ*+0.5 ns, in this range the additional feedback delay induced by the PCRR is considered; while for the COF case, the TDS value is the ACF value at the feedback delay position (Δ

_{f}*t*=

*τ*). Regarding the COF scenario [Fig. 4(a1)], for a fixed operation current, the TDS firstly decreases with increasing feedback strength, but then it increases quickly as the feedback strength is further enhanced. On the other hand, the effective bandwidth gradually increases as the increase of the feedback strength (for a fixed operation current) and the operation current (for a fixed feedback strength). These results agree with previously-reported theoretical and experimental results in [22,33,34] very well, which confirms the validity and feasibility of the numerical models. Regarding the three PCRR feedback scenarios, there is always a large region where the TDS is efficiently suppressed at an indistinguishable level close to 0. Moreover, as the increase of the number of the ring resonator in PCRR, the region with indistinguishable TDS would be wider and wider, as those shown in Figs. 4(b1)–4(d1). On the other hand, similar phenomena are observed in the results for the effective bandwidth [Figs. 4(b2)–4(d2)]. The bandwidth of chaos can be significantly enhanced with respect to that in the COF scenario, and the larger the number of ring resonators in PCRR, the more excellent the bandwidth enhancement. In summary, it can be concluded that perfect TDS suppression and significant bandwidth enhancement in chaos are achievable with a PCRR composed of two or more ring resonators.

_{f}The TDS suppression and the bandwidth enhancement in chaos generation are attributed to the nonlinear filtering effect of PCRR. Figure 5 shows the transmission spectra of the “Throughput” port and the “Drop” port of PCRR in the cases of SRR, DRR and TRR. It is shown that the nonlinear filtering effect of PCRR induces different transmissivities for different frequency components in the feedback light. The results are qualitatively in line with the experimental results of the parallel-cascaded ring resonators reported in [24], where the radiuses of the ring resonators are of micron scale. This guarantees the validity of the numerical simulations and also confirms the experimental implementation feasibility of the proposed scheme. With the nonlinear transmissivities of PCRR, the cross correlation between the feedback light [*E*_{2}(*t*) in Fig. 1(b)] reentering the laser cavity and the laser output light [*E*_{1}(*t*) in Fig. 1(b)] is significantly degraded, with respect to the COF case where the feedback light is a linear time-delayed replica of the laser output. For this reason, the energy distribution in the RF spectrum of laser-outputted chaos is not so concentrated nearby the relaxation oscillation frequency as that in the COF case, the laser relaxation oscillation is weakened by the nonlinear feedback of PCRR. Consequently, the RF spectrum of the generated chaos [defined by Eq. (3)] from the “Throughput” port becomes flatter, and the effective bandwidth is enhanced. On the other hand, due to the nonlinear filtering of PCRR induces different transmissivities for different components of the feedback light, the periodic external-cavity resonation modes that exist in the COF case is significantly suppressed in the proposed scheme. As such, the linearity between the feedback light and the laser output is significantly degraded, the TDS in the feedback light is efficiently suppressed, and then the periodicity in the generated chaos is correspondingly degraded and the TDS in the generated chaos is efficiently suppressed. Moreover, it is indicated that as the increase of the number of ring resonators in PCRR, both of the transmission spectra of the “Throughput” port and the “Drop” port would be more and more nonlinear and complex, and the degradation of the linearity (cross-correlation) between the ECSL output and its feedback light would be more and more significant. For this reason, the TDS suppression and the bandwidth enhancement in the configurations of SRR, DRR and TRR become more and more significant, as those shown in Figs. 2 and 4.

The nonlinear filtering effect is determined by the structure of the ring resonators consisting of the PCRR, therefore in the following discussion, we investigate the influences of the structure parameters of the ring resonators on the characteristics of TDS suppression and bandwidth enhancement. Here the influences of three critical parameters, namely the coupling coefficient *к*, the radius *R* of single ring resonator and the length *L* of the waveguide jointing two adjacent ring resonators are discussed. Figure 6 shows the influences of coupling coefficient *к* and ring radius *R* on the TDS value in ACF trace and the effective bandwidth. Here the value of *L* is fixed at *L *= 5*R*, the variation range of *к* is from 0.2 to 0.8 and that of *R* is from 0.3mm to 2.5 mm. It is observed that the TDS and the effective bandwidth gradually increase as the increase of coupling coefficient for a fixed ring radius. This is because the intensity of feedback light increases as the increase of *к*. For a fixed coupling coefficient, the TDS gradually decreases with increasing ring radius. This is because, the free spectral range (FSR) that is defined as the spacing between adjacent resonation modes and expressed as FSR ≈ *λ*^{2}/2π*Rn _{eff}* is inversely proportional to the value of

*R*[26]. As the increase of

*R*, the transmission spectra of the “Throughput” port and the “Drop” port become denser and denser, and more and more complex. Consequently, the TDS suppression and the bandwidth enhancement are correspondingly improved with increasing

*R*. Moreover, comparison among the results of SRR, DRR and TRR indicates that as the increase of the number of the ring resonators in PCRR, the efficiently-suppressed TDS region with a TDS value close to 0 becomes wider and wider, and the maximum achievable effective bandwidth is also enhanced. With proper selection of the values of

*к*,

*R*and

*L*, the effective bandwidth can be enhanced to larger than two times that in the COF case. In addition, it is worth mentioning that our repeating simulations with different values of

*L*indicate that similar phenomena are observed.

In practice, due to the nonideal manufacture precision, it is difficult to fabricate a PCRR composed of two or more identical ring resonators, therefore it is valuable to investigate the influences of the key parameters of the ring resonators in PCRR on the TDS and bandwidth properties. Figure 7 shows the influences of the mismatches of coupling coefficient and radius of the ring resonators in PCRR on the TDS and effective bandwidth characteristics. Here the coupling coefficients and radius of the first ring resonator are fixed at the aforementioned initial values, while those of the second and the third ring resonators are mismatched. The variation range of the mismatch ratio of *к* is from −5% to 5%, and that of the radius *R* is from −10% to 10%. For the DRR scenario, the results shown in Figs. 7(a1) and 7(a2) indicate that both of the TDS and the effective bandwidth characteristics in the case with mismatches are maintained similar to those with zero mismatch, the mismatches do not show obvious impacts on the effective bandwidth and TDS characteristics. For the TRR scenario, we discuss the variations of the TDS in ACF and the effective bandwidth in the mismatch spaces of (*к*_{2}, *R*_{2}), (*к*_{2}, *R*_{3}), (*к*_{3}, *R*_{2}) and (*к*_{3}, *R*_{3}). The results indicate that the mismatches of coupling coefficients *к*_{2} and *к*_{3} do not show obvious impacts on the TDS and the effective bandwidth for a fixed mismatch of radius. For a fixed mismatch of coupling coefficient, when the mismatch of radius is smaller than 4%, the TDS and bandwidth characteristics are maintained similar to those of zero mismatch, while when the mismatch is larger than 4%, the effective bandwidth is enhanced to some extent, and the TDS suppression is slightly enhanced. This phenomenon is intuitively attributed to that with a mismatch of radius larger than 4%, the FSR differences between the individual transmission spectra of the ring resonators would be non-ignorable, which induces that the transmission spectrum of PCRR gets more nonlinear than that in the zero mismatch case, as such, the bandwidth enhancement and TDS suppression properties are further improved by the mismatch.

## 4. Conclusions

In summary, we have proposed and demonstrated a simultaneous bandwidth-and-complexity-enhanced chaos generation scheme by using PCRR as the external reflector of ECSL. For the purpose of comparison, four feedback scenarios including the conventional optical feedback and three indicative PCRROF configurations (SRROF, DRROF and TRROF) are discussed. The numerical results demonstrate that comparing with the COF case, in the proposed scheme with the nonlinear filtering effect of PCRR, the bandwidth of generated chaos is significantly enhanced, and simultaneously, the TDS is efficiently suppressed at an indistinguishable level close to 0. Moreover, as the increase of the number of the ring resonators in PCRR, the TDS suppression and bandwidth enhancement can be further improved. This work provides a new way to generate wideband and complex chaos.

## Funding

National Natural Science Foundation of China (61471087, 61671119); Fundamental Research Funds for the Central Universities (ZYGX2019J003).

## Disclosures

The authors declare no conflicts of interest related to this article.

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