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 [13], random bit generation (RBG) [46], 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: Fig. 1.

Fig. 1. (a) Schematic of chaos generation using ECSL subject to nonlinear feedback from PCRR, (b) detailed configuration of PCRR. DFB, distributed-feedback laser; PC, polarization controller; OC, optical circulator; VOA, variable optical attenuator; PD, photodetector.

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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].

$$\frac{{dE(t )}}{{dt}} = \frac{{1 + i\alpha }}{2}\left[ {\frac{{g({N(t )- {N_0}} )}}{{1 + \varepsilon {{|{E(t )} |}^2}}} - \frac{1}{{{\tau_p}}}} \right]E(t) + {k_f}E({t - {\tau_f}} )\ast {h_1}(t )$$
$$\frac{{dN(t )}}{{dt}} = \frac{I}{{qV}} - \frac{{N(t )}}{{{\tau _e}}} - \frac{{g({N(t )- {N_0}} )}}{{1 + \varepsilon {{|{E(t )} |}^2}}}{|{E(t )} |^2}$$
$$E{(t )_{out}} = E({t - {{{\tau_f}} \mathord{\left/ {\vphantom {{{\tau_f}} 2}} \right.} 2}} )\ast {h_2}(t )$$
where 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 h1(t) of the “Drop” port and the linear delayed replica of laser output E(t-τf). The output chaos is expressed by Eq. (3), it is the convolution of the response function h2(t) of the “Throughput” port and the delayed laser output E(t-τf/2). The transfer function of PCRR in the frequency domain can be described by the transfer matrix method, which is written as [24,26]
$$\left[ {\begin{array}{c} {{E_1}(\omega )}\\ {{E_2}(\omega )} \end{array}} \right] = {T_1} \cdot {T_{\varphi 1}} \cdot {T_2} \cdot {T_{\varphi 2}} \ldots \cdot {T_i} \cdot {T_{\varphi i}} \ldots \cdot {T_n}\left[ {\begin{array}{c} {{E_3}(\omega )}\\ {{E_4}(\omega )} \end{array}} \right] = \left[ {\begin{array}{cc} {{P_{11}}}&{{P_{12}}}\\ {{P_{21}}}&{{P_{22}}} \end{array}} \right]\left[ {\begin{array}{c} {{E_3}(\omega )}\\ {{E_4}(\omega )} \end{array}} \right]$$
$${T_i} = \left[ {\begin{array}{cc} {\frac{{1 - {\eta_{1i}}{\eta_{2i}}\alpha_{ri}^2{e^{ - j\Delta \omega {t_{ri}}}}}}{{{\eta_{1i}} - {\eta_{2i}}\alpha_{ri}^2{e^{ - j\Delta \omega {t_{ri}}}}}}}&{\frac{{{\kappa_{1i}}{\kappa_{2i}}{\alpha_{ri}}{e^{ - j{{\Delta \omega {t_{ri}}} \mathord{\left/ {\vphantom {{\Delta \omega {t_{ri}}} 2}} \right.} 2}}}}}{{{\eta_{1i}} - {\eta_{2i}}\alpha_{ri}^2{e^{ - j\Delta \omega {t_{ri}}}}}}}\\ {\frac{{ - {\kappa_{1i}}{\kappa_{2i}}{\alpha_{ri}}{e^{ - j{{\Delta \omega {t_{ri}}} \mathord{\left/ {\vphantom {{\Delta \omega {t_{ri}}} 2}} \right.} 2}}}}}{{{\eta_{1i}} - {\eta_{2i}}\alpha_{ri}^2{e^{ - j\Delta \omega {t_{ri}}}}}}}&{\frac{{{\eta_{1i}}{\eta_{2i}} - \alpha_{ri}^2{e^{ - j\Delta \omega {t_{ri}}}}}}{{{\eta_{1i}} - {\eta_{2i}}\alpha_{ri}^2{e^{ - j\Delta \omega {t_{ri}}}}}}} \end{array}} \right]$$
$${T_{\varphi i}} = \left[ {\begin{array}{cc} {{e^{j{\beta_b}L}}}&0\\ 0&{{e^{j{\beta_b}L}}} \end{array}} \right]$$
where Ti is the transfer matrix of the ith ring resonator, Tφi 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), к1i and к2i are the coupling coefficients of “Input” port and “Drop” port, respectively, while η1i (η1i2=1-к1i2) and η2i (η2i2=1-к2i2) denote the corresponding transmission coefficients; αri2=exp(-2αiπRi) stands for the round-trip amplitude attenuation, where Ri is the radius of single ring resonator and αi is the corresponding loss coefficient in the ring resonator; Δω=ω-ω0 denotes the detuning of operation frequency with respect to the central frequency ω0; tri=2πRineff/c is the round-trip time in single ring resonator, where neff is the effective index of ring resonator; 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 E4(ω) = 0. Subsequently, the transfer function of the “Drop” port is determined by H1(ω)=E2(ω)/E1(ω)=P21/P11, and that of the “Throughput” port is determined by H2(ω)=E3(ω)/E1(ω) = 1/P11. Here P11, P21 are determined by the cascade product of Ti and Tφi as that in Eq. (4).

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 [2123,28,29]

$$C({\Delta t} )= \frac{{\left\langle {\left( {I({t + \Delta t} )- \left\langle {I(t )} \right\rangle } \right)\left( {I(t )- \left\langle {I(t )} \right\rangle } \right)} \right\rangle }}{{\sqrt {\left\langle {{{\left( {I({t + \Delta t} )- \left\langle {I({t + \Delta t} )} \right\rangle } \right)}^2}} \right\rangle \left\langle {{{\left( {I(t )- \left\langle {I(t )} \right\rangle } \right)}^2}} \right\rangle } }}$$
where I(t) denotes the output from the “Throughput” port of PCRR, it is proportional to |E(t)|2, <·> stands for the time averaging, Δt is the delay time that I(t+Δt) is shifted with respect to I(t). Regarding the PE [3032], we take the time series {xt, t = 1, 2, …, Nx} and reconstruct a d-dimensional space Xt=[x(t), x(tτ), …, x(t+(d-1)Δτ)], where d and Δτ denote the embedding dimension and the embedding time delay, respectively. The vector Xt 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 Xt is uniquely mapped onto an ordinal pattern Ω=(r1, r2, …, rd) out of d! possible permutations. For the permutation Ω of order d, the probability distribution P = p(Ω) of the ordinal patterns is:
$$p(\Omega )= \frac{{\# \{{t|t \le {N_x} - ({d - 1} )\Delta \tau ;\textrm{ }{X_t}{\kern 1pt} {\kern 1pt} {\kern 1pt} has{\kern 1pt} {\kern 1pt} {\kern 1pt} type{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \Omega } \}}}{{{N_x} - ({d - 1} )\Delta \tau }}$$
where # means “the number of”. Subsequently, the permutation entropy H[P] is evaluated by the permutation probability distribution, in term of H[P]=-∑p(Ω)logp(Ω). Finally, the normalized PE h[P] is evaluated by h[P] = H[P]/logd!, 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 µm3, the carrier lifetime τe=5.58 ns, the photon lifetime τp=4.11 ps, the differential gain coefficient g = 0.73×10−3 µm3ns−1, the transparency carrier density N0 = 1.5×106 µm−3, the active layer volume V = 175 µm3, and the electron charge q = 1.602×10−19 C. Unless otherwise stated, the PCRR parameters are chosen as к1i=к2i=0.4, αri2=1, Ri=2mm, L = 10mm, neff = 2.6; the laser bias current I = 1.5Ith, where Ith=20 mA is threshold current; the feedback strength is fixed as kf = 20 ns−1 and the feedback delay is set as τf= 5ns. For the calculations of PE, the number Nx of the time series for space reconstruction is chosen as 20000, and the dimension 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 [2123,29,33]. 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/τf) 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.

 figure: Fig. 2.

Fig. 2. Temporal series (first row), RF spectra (second row), ACF traces (third row) and PE traces (fourth row) of chaos in the cases of (a) COF, (b) SRROF, (c) DRROF and (d) TRROF.

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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: Fig. 3.

Fig. 3. Ratio of energy in frequency band Δf as a function of feedback strength for the cases of (a) COF, (b) SRROF, (c) DRROF and (d) TRROF.

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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 Δt=τf −0.5 ns to Δt=τf +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 (Δt=τf). 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.

 figure: Fig. 4.

Fig. 4. Influences of feedback strength kf and operation current I(Ith) on TDS in ACF trace (first row) and EB (second row), in the cases of COF (first column), SRROF (second column), DRROF (third column) and TRROF (fourth column).

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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 [E2(t) in Fig. 1(b)] reentering the laser cavity and the laser output light [E1(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.

 figure: Fig. 5.

Fig. 5. Transmission spectra of “Throughput” port (first row) and “Drop” port (second row) of PCRR composed of (a) SRR, (b) DRR and (c) TRR.

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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 = 5R, 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 ≈ λ2/2πRneff 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.

 figure: Fig. 6.

Fig. 6. Influences of coupling coefficient к and ring radius R on TDS in ACF trace (first row) and EB (second row), for the cases of (a) SRROF, (b) DRROF and (c) TRROF.

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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 (к2, R2), (к2, R3), (к3, R2) and (к3, R3). The results indicate that the mismatches of coupling coefficients к2 and к3 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.

 figure: Fig. 7.

Fig. 7. Influences of the mismatches of coupling coefficient к and ring radius R on TDS in ACF (first row) and effective bandwidth (second row), in the configurations of DRROF (first column) and TRROF (second-fifth column).

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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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13. A. B. Wang, Y. B. Yang, B. J. Wang, B. B. Zhang, L. Li, and Y. C. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express 21(7), 8701–8710 (2013). [CrossRef]  

14. J. G. Wu, G. Q. Xia, and Z. M. Wu, “Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback,” Opt. Express 17(22), 20124–20133 (2009). [CrossRef]  

15. N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. 36(23), 4632–4634 (2011). [CrossRef]  

16. S. S. Li, Q. Liu, and S. C. Chan, “Distributed feedbacks for time-delay signature suppression of chaos generated from a semiconductor laser,” IEEE Photonics J. 4(5), 1930–1935 (2012). [CrossRef]  

17. D. Wang, L. Wang, T. Zhao, H. Gao, Y. Wang, X. Chen, and A. Wang, “Time delay signature elimination of chaos in a semiconductor laser by dispersive feedback from a chirped FBG,” Opt. Express 25(10), 10911–10924 (2017). [CrossRef]  

18. S. Y. Xiang, A. J. Wen, W. Pan, L. Lin, H. Zhang, H. Zhang, X. Guo, and J. Li, “Suppression of chaos time delay signature in a ring network consisting of three semiconductor lasers coupled with heterogeneous delays,” J. Lightwave Technol. 34(18), 4221–4227 (2016). [CrossRef]  

19. S. S. Li, X. Z. Li, and S. C. Chan, “Chaotic time-delay signature suppression with bandwidth broadening by fiber propagation,” Opt. Lett. 43(19), 4751–4754 (2018). [CrossRef]  

20. N. Jiang, C. Wang, C. P. Xue, G. L. Li, S. Q. Lin, and K. Qiu, “Generation of flat wideband chaos with suppressed time delay signature by using optical time lens,” Opt. Express 25(13), 14359–14367 (2017). [CrossRef]  

21. P. Mu, P. He, and N. Li, “Simultaneous chaos time-delay signature cancellation and bandwidth enhancement in cascade-coupled semiconductor ring lasers,” IEEE Access 7, 11041–11048 (2019). [CrossRef]  

22. N. Jiang, A. K. Zhao, S. Q. Liu, C. P. Xue, B. Y. Wang, and K. Qiu, “Generation of broadband chaos with perfect time delay signature suppression by using self-phase-modulated feedback and a microsphere resonator,” Opt. Lett. 43(21), 5359–5362 (2018). [CrossRef]  

23. A. K. Zhao, N. Jiang, S. Q. Liu, C. P. Xue, J. M. Tang, and K. Qiu, “Wideband complex-enhanced chaos generation using a semiconductor laser subject to delay-interfered self-phase-modulated feedback,” Opt. Express 27(9), 12336–12348 (2019). [CrossRef]  

24. R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P. T. Ho, “Parallel-cascaded semiconductor micro ring resonators for high-order and wide-FSR filters,” J. Lightwave Technol. 20(5), 900–905 (2002). [CrossRef]  

25. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]  

26. D. G. Rabus, “Integrated ring resonators,” Spring Series in Optical Sciences, Springer, New York, 2007.

27. J. Carroll, J. Whiteaway, and D. Plumb, “Distributed feedback semiconductor lasers,” IEE Circuits, Devices and Systems Series 10, 1998.

28. D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–1891 (2009). [CrossRef]  

29. F. Y. Lin and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221(1-3), 173–180 (2003). [CrossRef]  

30. S. Y. Xiang, A. J. Wen, H. Zhang, J. F. Li, H. X. Zhang, and L. Lin, “Effect of gain nonlinearity on time delay signature of chaos in external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 52(4), 1–7 (2016). [CrossRef]  

31. L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E 82(4), 046212 (2010). [CrossRef]  

32. J. P. Toomey and D. M. Kane, “Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy,” Opt. Express 22(2), 1713–1725 (2014). [CrossRef]  

33. A. Zhao, N. Jiang, S. Liu, C. Xue, and K. Qiu, “Wideband time delay signature-suppressed chaos generation using self-phase-modulated feedback semiconductor laser cascaded with dispersive component,” J. Lightwave Technol. 37(19), 5132–5139 (2019). [CrossRef]  

34. D. Rontani, E. Mercier, D. Wolfersberger, and M. Sciamanna, “Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback,” Opt. Lett. 41(20), 4637–4640 (2016). [CrossRef]  

References

  • View by:

  1. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
    [Crossref]
  2. M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
    [Crossref]
  3. N. Jiang, A. K. Zhao, C. P. Xue, J. M. Tang, and K. Qiu, “Physical secure optical communication based on private chaotic spectral phase encryption/decryption,” Opt. Lett. 44(7), 1536–1539 (2019).
    [Crossref]
  4. X. Tang, G.-Q. Xia, E. Jayaprasath, T. Deng, X.-D. Lin, L. Fan, Z.-Y. Gao, and Z.-M. Wu, “Multi-channel physical random bits generation using a vertical-cavity surface-emitting laser under chaotic optical injection,” IEEE Access 6, 3565–3572 (2018).
    [Crossref]
  5. X. Z. Li, S. S. Li, J. P. Zhuang, and S. C. Chan, “Random bit generation at tunable rates using a chaotic semiconductor laser under distributed feedback,” Opt. Lett. 40(17), 3970–3973 (2015).
    [Crossref]
  6. N. Li, B. Kim, V. N. Chizhevsky, A. Locquet, M. Bloch, D. S. Citrin, and W. Pan, “Two approaches for ultrafast random bit generation based on the chaotic dynamics of a semiconductor laser,” Opt. Express 22(6), 6634–6646 (2014).
    [Crossref]
  7. Y. C. Wang, B. J. Wang, and A. B. Wang, “Chaotic correlation optical time domain reflectometer utilizing laser diode,” IEEE Photonics Technol. Lett. 20(19), 1636–1638 (2008).
    [Crossref]
  8. F. Y. Lin and J. M. Liu, “Chaotic Lidar,” IEEE J. Sel. Top. Quantum Electron. 10(5), 991–997 (2004).
    [Crossref]
  9. R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
    [Crossref]
  10. S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. N. Li, and H. N. Zhu, “Wideband unpredictability-enhanced chaotic semiconductor lasers with dual-chaotic optical injections,” IEEE J. Quantum Electron. 48(8), 1069–1076 (2012).
    [Crossref]
  11. H. Someya, I. Oowada, H. Okumura, T. Kida, and A. Uchida, “Synchronization of bandwidth-enhanced chaos in semiconductor lasers with optical feedback and injection,” Opt. Express 17(22), 19536–19543 (2009).
    [Crossref]
  12. N. Li, W. Pan, A. Locquet, and D. S. Citrin, “Time-delay concealment and complexity enhancement of an external-cavity laser through optical injection,” Opt. Lett. 40(19), 4416–4419 (2015).
    [Crossref]
  13. A. B. Wang, Y. B. Yang, B. J. Wang, B. B. Zhang, L. Li, and Y. C. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express 21(7), 8701–8710 (2013).
    [Crossref]
  14. J. G. Wu, G. Q. Xia, and Z. M. Wu, “Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback,” Opt. Express 17(22), 20124–20133 (2009).
    [Crossref]
  15. N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. 36(23), 4632–4634 (2011).
    [Crossref]
  16. S. S. Li, Q. Liu, and S. C. Chan, “Distributed feedbacks for time-delay signature suppression of chaos generated from a semiconductor laser,” IEEE Photonics J. 4(5), 1930–1935 (2012).
    [Crossref]
  17. D. Wang, L. Wang, T. Zhao, H. Gao, Y. Wang, X. Chen, and A. Wang, “Time delay signature elimination of chaos in a semiconductor laser by dispersive feedback from a chirped FBG,” Opt. Express 25(10), 10911–10924 (2017).
    [Crossref]
  18. S. Y. Xiang, A. J. Wen, W. Pan, L. Lin, H. Zhang, H. Zhang, X. Guo, and J. Li, “Suppression of chaos time delay signature in a ring network consisting of three semiconductor lasers coupled with heterogeneous delays,” J. Lightwave Technol. 34(18), 4221–4227 (2016).
    [Crossref]
  19. S. S. Li, X. Z. Li, and S. C. Chan, “Chaotic time-delay signature suppression with bandwidth broadening by fiber propagation,” Opt. Lett. 43(19), 4751–4754 (2018).
    [Crossref]
  20. N. Jiang, C. Wang, C. P. Xue, G. L. Li, S. Q. Lin, and K. Qiu, “Generation of flat wideband chaos with suppressed time delay signature by using optical time lens,” Opt. Express 25(13), 14359–14367 (2017).
    [Crossref]
  21. P. Mu, P. He, and N. Li, “Simultaneous chaos time-delay signature cancellation and bandwidth enhancement in cascade-coupled semiconductor ring lasers,” IEEE Access 7, 11041–11048 (2019).
    [Crossref]
  22. N. Jiang, A. K. Zhao, S. Q. Liu, C. P. Xue, B. Y. Wang, and K. Qiu, “Generation of broadband chaos with perfect time delay signature suppression by using self-phase-modulated feedback and a microsphere resonator,” Opt. Lett. 43(21), 5359–5362 (2018).
    [Crossref]
  23. A. K. Zhao, N. Jiang, S. Q. Liu, C. P. Xue, J. M. Tang, and K. Qiu, “Wideband complex-enhanced chaos generation using a semiconductor laser subject to delay-interfered self-phase-modulated feedback,” Opt. Express 27(9), 12336–12348 (2019).
    [Crossref]
  24. R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P. T. Ho, “Parallel-cascaded semiconductor micro ring resonators for high-order and wide-FSR filters,” J. Lightwave Technol. 20(5), 900–905 (2002).
    [Crossref]
  25. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
    [Crossref]
  26. D. G. Rabus, “Integrated ring resonators,” Spring Series in Optical Sciences, Springer, New York, 2007.
  27. J. Carroll, J. Whiteaway, and D. Plumb, “Distributed feedback semiconductor lasers,” IEE Circuits, Devices and Systems Series 10, 1998.
  28. D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–1891 (2009).
    [Crossref]
  29. F. Y. Lin and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221(1-3), 173–180 (2003).
    [Crossref]
  30. S. Y. Xiang, A. J. Wen, H. Zhang, J. F. Li, H. X. Zhang, and L. Lin, “Effect of gain nonlinearity on time delay signature of chaos in external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 52(4), 1–7 (2016).
    [Crossref]
  31. L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E 82(4), 046212 (2010).
    [Crossref]
  32. J. P. Toomey and D. M. Kane, “Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy,” Opt. Express 22(2), 1713–1725 (2014).
    [Crossref]
  33. A. Zhao, N. Jiang, S. Liu, C. Xue, and K. Qiu, “Wideband time delay signature-suppressed chaos generation using self-phase-modulated feedback semiconductor laser cascaded with dispersive component,” J. Lightwave Technol. 37(19), 5132–5139 (2019).
    [Crossref]
  34. D. Rontani, E. Mercier, D. Wolfersberger, and M. Sciamanna, “Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback,” Opt. Lett. 41(20), 4637–4640 (2016).
    [Crossref]

2019 (4)

2018 (3)

2017 (2)

2016 (3)

2015 (3)

2014 (2)

2013 (1)

2012 (2)

S. S. Li, Q. Liu, and S. C. Chan, “Distributed feedbacks for time-delay signature suppression of chaos generated from a semiconductor laser,” IEEE Photonics J. 4(5), 1930–1935 (2012).
[Crossref]

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. N. Li, and H. N. Zhu, “Wideband unpredictability-enhanced chaotic semiconductor lasers with dual-chaotic optical injections,” IEEE J. Quantum Electron. 48(8), 1069–1076 (2012).
[Crossref]

2011 (1)

2010 (1)

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E 82(4), 046212 (2010).
[Crossref]

2009 (3)

2008 (1)

Y. C. Wang, B. J. Wang, and A. B. Wang, “Chaotic correlation optical time domain reflectometer utilizing laser diode,” IEEE Photonics Technol. Lett. 20(19), 1636–1638 (2008).
[Crossref]

2005 (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

2004 (1)

F. Y. Lin and J. M. Liu, “Chaotic Lidar,” IEEE J. Sel. Top. Quantum Electron. 10(5), 991–997 (2004).
[Crossref]

2003 (1)

F. Y. Lin and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221(1-3), 173–180 (2003).
[Crossref]

2002 (1)

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P. T. Ho, “Parallel-cascaded semiconductor micro ring resonators for high-order and wide-FSR filters,” J. Lightwave Technol. 20(5), 900–905 (2002).
[Crossref]

1998 (1)

R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[Crossref]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Absil, P. P.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P. T. Ho, “Parallel-cascaded semiconductor micro ring resonators for high-order and wide-FSR filters,” J. Lightwave Technol. 20(5), 900–905 (2002).
[Crossref]

Annovazzi-Lodi, V.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Argyris, A.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Bloch, M.

Bünner, M. J.

R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[Crossref]

Calhoun, L. C.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P. T. Ho, “Parallel-cascaded semiconductor micro ring resonators for high-order and wide-FSR filters,” J. Lightwave Technol. 20(5), 900–905 (2002).
[Crossref]

Carroll, J.

J. Carroll, J. Whiteaway, and D. Plumb, “Distributed feedback semiconductor lasers,” IEE Circuits, Devices and Systems Series 10, 1998.

Chan, S. C.

Chen, X.

Chizhevsky, V. N.

Citrin, D. S.

Colet, P.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Deng, T.

X. Tang, G.-Q. Xia, E. Jayaprasath, T. Deng, X.-D. Lin, L. Fan, Z.-Y. Gao, and Z.-M. Wu, “Multi-channel physical random bits generation using a vertical-cavity surface-emitting laser under chaotic optical injection,” IEEE Access 6, 3565–3572 (2018).
[Crossref]

Fan, L.

X. Tang, G.-Q. Xia, E. Jayaprasath, T. Deng, X.-D. Lin, L. Fan, Z.-Y. Gao, and Z.-M. Wu, “Multi-channel physical random bits generation using a vertical-cavity surface-emitting laser under chaotic optical injection,” IEEE Access 6, 3565–3572 (2018).
[Crossref]

Fischer, I.

N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. 36(23), 4632–4634 (2011).
[Crossref]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E 82(4), 046212 (2010).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Gao, H.

Gao, Z.-Y.

X. Tang, G.-Q. Xia, E. Jayaprasath, T. Deng, X.-D. Lin, L. Fan, Z.-Y. Gao, and Z.-M. Wu, “Multi-channel physical random bits generation using a vertical-cavity surface-emitting laser under chaotic optical injection,” IEEE Access 6, 3565–3572 (2018).
[Crossref]

Garcia-Ojalvo, J.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Giaquinta, A.

R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[Crossref]

Grover, R.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P. T. Ho, “Parallel-cascaded semiconductor micro ring resonators for high-order and wide-FSR filters,” J. Lightwave Technol. 20(5), 900–905 (2002).
[Crossref]

Guo, X.

He, P.

P. Mu, P. He, and N. Li, “Simultaneous chaos time-delay signature cancellation and bandwidth enhancement in cascade-coupled semiconductor ring lasers,” IEEE Access 7, 11041–11048 (2019).
[Crossref]

Hegger, R.

R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[Crossref]

Ho, P. T.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P. T. Ho, “Parallel-cascaded semiconductor micro ring resonators for high-order and wide-FSR filters,” J. Lightwave Technol. 20(5), 900–905 (2002).
[Crossref]

Hryniewicz, J. V.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P. T. Ho, “Parallel-cascaded semiconductor micro ring resonators for high-order and wide-FSR filters,” J. Lightwave Technol. 20(5), 900–905 (2002).
[Crossref]

Ibrahim, T. A.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P. T. Ho, “Parallel-cascaded semiconductor micro ring resonators for high-order and wide-FSR filters,” J. Lightwave Technol. 20(5), 900–905 (2002).
[Crossref]

Jayaprasath, E.

X. Tang, G.-Q. Xia, E. Jayaprasath, T. Deng, X.-D. Lin, L. Fan, Z.-Y. Gao, and Z.-M. Wu, “Multi-channel physical random bits generation using a vertical-cavity surface-emitting laser under chaotic optical injection,” IEEE Access 6, 3565–3572 (2018).
[Crossref]

Jiang, N.

Johnson, F. G.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P. T. Ho, “Parallel-cascaded semiconductor micro ring resonators for high-order and wide-FSR filters,” J. Lightwave Technol. 20(5), 900–905 (2002).
[Crossref]

Kane, D. M.

Kantz, H.

R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[Crossref]

Kida, T.

Kim, B.

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Larger, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Li, G. L.

Li, J.

Li, J. F.

S. Y. Xiang, A. J. Wen, H. Zhang, J. F. Li, H. X. Zhang, and L. Lin, “Effect of gain nonlinearity on time delay signature of chaos in external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 52(4), 1–7 (2016).
[Crossref]

Li, L.

Li, N.

Li, N. N.

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. N. Li, and H. N. Zhu, “Wideband unpredictability-enhanced chaotic semiconductor lasers with dual-chaotic optical injections,” IEEE J. Quantum Electron. 48(8), 1069–1076 (2012).
[Crossref]

Li, S. S.

Li, X. Z.

Lin, F. Y.

F. Y. Lin and J. M. Liu, “Chaotic Lidar,” IEEE J. Sel. Top. Quantum Electron. 10(5), 991–997 (2004).
[Crossref]

F. Y. Lin and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221(1-3), 173–180 (2003).
[Crossref]

Lin, L.

S. Y. Xiang, A. J. Wen, H. Zhang, J. F. Li, H. X. Zhang, and L. Lin, “Effect of gain nonlinearity on time delay signature of chaos in external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 52(4), 1–7 (2016).
[Crossref]

S. Y. Xiang, A. J. Wen, W. Pan, L. Lin, H. Zhang, H. Zhang, X. Guo, and J. Li, “Suppression of chaos time delay signature in a ring network consisting of three semiconductor lasers coupled with heterogeneous delays,” J. Lightwave Technol. 34(18), 4221–4227 (2016).
[Crossref]

Lin, S. Q.

Lin, X.-D.

X. Tang, G.-Q. Xia, E. Jayaprasath, T. Deng, X.-D. Lin, L. Fan, Z.-Y. Gao, and Z.-M. Wu, “Multi-channel physical random bits generation using a vertical-cavity surface-emitting laser under chaotic optical injection,” IEEE Access 6, 3565–3572 (2018).
[Crossref]

Liu, J. M.

F. Y. Lin and J. M. Liu, “Chaotic Lidar,” IEEE J. Sel. Top. Quantum Electron. 10(5), 991–997 (2004).
[Crossref]

F. Y. Lin and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221(1-3), 173–180 (2003).
[Crossref]

Liu, Q.

S. S. Li, Q. Liu, and S. C. Chan, “Distributed feedbacks for time-delay signature suppression of chaos generated from a semiconductor laser,” IEEE Photonics J. 4(5), 1930–1935 (2012).
[Crossref]

Liu, S.

Liu, S. Q.

Locquet, A.

Luo, B.

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. N. Li, and H. N. Zhu, “Wideband unpredictability-enhanced chaotic semiconductor lasers with dual-chaotic optical injections,” IEEE J. Quantum Electron. 48(8), 1069–1076 (2012).
[Crossref]

Mercier, E.

Mirasso, C. R.

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E 82(4), 046212 (2010).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Mu, P.

P. Mu, P. He, and N. Li, “Simultaneous chaos time-delay signature cancellation and bandwidth enhancement in cascade-coupled semiconductor ring lasers,” IEEE Access 7, 11041–11048 (2019).
[Crossref]

Okumura, H.

Oliver, N.

Oowada, I.

Ortin, S.

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–1891 (2009).
[Crossref]

Pan, W.

Pesquera, L.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Plumb, D.

J. Carroll, J. Whiteaway, and D. Plumb, “Distributed feedback semiconductor lasers,” IEE Circuits, Devices and Systems Series 10, 1998.

Qiu, K.

Rabus, D. G.

D. G. Rabus, “Integrated ring resonators,” Spring Series in Optical Sciences, Springer, New York, 2007.

Rontani, D.

D. Rontani, E. Mercier, D. Wolfersberger, and M. Sciamanna, “Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback,” Opt. Lett. 41(20), 4637–4640 (2016).
[Crossref]

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–1891 (2009).
[Crossref]

Rosso, O. A.

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E 82(4), 046212 (2010).
[Crossref]

Sciamanna, M.

D. Rontani, E. Mercier, D. Wolfersberger, and M. Sciamanna, “Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback,” Opt. Lett. 41(20), 4637–4640 (2016).
[Crossref]

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–1891 (2009).
[Crossref]

Shore, K. A.

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Someya, H.

Soriano, M. C.

N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. 36(23), 4632–4634 (2011).
[Crossref]

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E 82(4), 046212 (2010).
[Crossref]

Sukow, D. W.

Syvridis, D.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Tang, J. M.

Tang, X.

X. Tang, G.-Q. Xia, E. Jayaprasath, T. Deng, X.-D. Lin, L. Fan, Z.-Y. Gao, and Z.-M. Wu, “Multi-channel physical random bits generation using a vertical-cavity surface-emitting laser under chaotic optical injection,” IEEE Access 6, 3565–3572 (2018).
[Crossref]

Toomey, J. P.

Uchida, A.

Van, V.

R. Grover, V. Van, T. A. Ibrahim, P. P. Absil, L. C. Calhoun, F. G. Johnson, J. V. Hryniewicz, and P. T. Ho, “Parallel-cascaded semiconductor micro ring resonators for high-order and wide-FSR filters,” J. Lightwave Technol. 20(5), 900–905 (2002).
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Wang, A.

Wang, A. B.

A. B. Wang, Y. B. Yang, B. J. Wang, B. B. Zhang, L. Li, and Y. C. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express 21(7), 8701–8710 (2013).
[Crossref]

Y. C. Wang, B. J. Wang, and A. B. Wang, “Chaotic correlation optical time domain reflectometer utilizing laser diode,” IEEE Photonics Technol. Lett. 20(19), 1636–1638 (2008).
[Crossref]

Wang, B. J.

A. B. Wang, Y. B. Yang, B. J. Wang, B. B. Zhang, L. Li, and Y. C. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express 21(7), 8701–8710 (2013).
[Crossref]

Y. C. Wang, B. J. Wang, and A. B. Wang, “Chaotic correlation optical time domain reflectometer utilizing laser diode,” IEEE Photonics Technol. Lett. 20(19), 1636–1638 (2008).
[Crossref]

Wang, B. Y.

Wang, C.

Wang, D.

Wang, L.

Wang, Y.

Wang, Y. C.

A. B. Wang, Y. B. Yang, B. J. Wang, B. B. Zhang, L. Li, and Y. C. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express 21(7), 8701–8710 (2013).
[Crossref]

Y. C. Wang, B. J. Wang, and A. B. Wang, “Chaotic correlation optical time domain reflectometer utilizing laser diode,” IEEE Photonics Technol. Lett. 20(19), 1636–1638 (2008).
[Crossref]

Wen, A. J.

S. Y. Xiang, A. J. Wen, W. Pan, L. Lin, H. Zhang, H. Zhang, X. Guo, and J. Li, “Suppression of chaos time delay signature in a ring network consisting of three semiconductor lasers coupled with heterogeneous delays,” J. Lightwave Technol. 34(18), 4221–4227 (2016).
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S. Y. Xiang, A. J. Wen, H. Zhang, J. F. Li, H. X. Zhang, and L. Lin, “Effect of gain nonlinearity on time delay signature of chaos in external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 52(4), 1–7 (2016).
[Crossref]

Whiteaway, J.

J. Carroll, J. Whiteaway, and D. Plumb, “Distributed feedback semiconductor lasers,” IEE Circuits, Devices and Systems Series 10, 1998.

Wolfersberger, D.

Wu, J. G.

Wu, Z. M.

Wu, Z.-M.

X. Tang, G.-Q. Xia, E. Jayaprasath, T. Deng, X.-D. Lin, L. Fan, Z.-Y. Gao, and Z.-M. Wu, “Multi-channel physical random bits generation using a vertical-cavity surface-emitting laser under chaotic optical injection,” IEEE Access 6, 3565–3572 (2018).
[Crossref]

Xia, G. Q.

Xia, G.-Q.

X. Tang, G.-Q. Xia, E. Jayaprasath, T. Deng, X.-D. Lin, L. Fan, Z.-Y. Gao, and Z.-M. Wu, “Multi-channel physical random bits generation using a vertical-cavity surface-emitting laser under chaotic optical injection,” IEEE Access 6, 3565–3572 (2018).
[Crossref]

Xiang, S. Y.

S. Y. Xiang, A. J. Wen, W. Pan, L. Lin, H. Zhang, H. Zhang, X. Guo, and J. Li, “Suppression of chaos time delay signature in a ring network consisting of three semiconductor lasers coupled with heterogeneous delays,” J. Lightwave Technol. 34(18), 4221–4227 (2016).
[Crossref]

S. Y. Xiang, A. J. Wen, H. Zhang, J. F. Li, H. X. Zhang, and L. Lin, “Effect of gain nonlinearity on time delay signature of chaos in external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 52(4), 1–7 (2016).
[Crossref]

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. N. Li, and H. N. Zhu, “Wideband unpredictability-enhanced chaotic semiconductor lasers with dual-chaotic optical injections,” IEEE J. Quantum Electron. 48(8), 1069–1076 (2012).
[Crossref]

Xue, C.

Xue, C. P.

Yan, L. S.

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. N. Li, and H. N. Zhu, “Wideband unpredictability-enhanced chaotic semiconductor lasers with dual-chaotic optical injections,” IEEE J. Quantum Electron. 48(8), 1069–1076 (2012).
[Crossref]

Yang, Y. B.

Zhang, B. B.

Zhang, H.

Zhang, H. X.

S. Y. Xiang, A. J. Wen, H. Zhang, J. F. Li, H. X. Zhang, and L. Lin, “Effect of gain nonlinearity on time delay signature of chaos in external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 52(4), 1–7 (2016).
[Crossref]

Zhao, A.

Zhao, A. K.

Zhao, T.

Zhu, H. N.

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. N. Li, and H. N. Zhu, “Wideband unpredictability-enhanced chaotic semiconductor lasers with dual-chaotic optical injections,” IEEE J. Quantum Electron. 48(8), 1069–1076 (2012).
[Crossref]

Zhuang, J. P.

Zou, X. H.

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. N. Li, and H. N. Zhu, “Wideband unpredictability-enhanced chaotic semiconductor lasers with dual-chaotic optical injections,” IEEE J. Quantum Electron. 48(8), 1069–1076 (2012).
[Crossref]

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L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E 82(4), 046212 (2010).
[Crossref]

IEEE Access (2)

X. Tang, G.-Q. Xia, E. Jayaprasath, T. Deng, X.-D. Lin, L. Fan, Z.-Y. Gao, and Z.-M. Wu, “Multi-channel physical random bits generation using a vertical-cavity surface-emitting laser under chaotic optical injection,” IEEE Access 6, 3565–3572 (2018).
[Crossref]

P. Mu, P. He, and N. Li, “Simultaneous chaos time-delay signature cancellation and bandwidth enhancement in cascade-coupled semiconductor ring lasers,” IEEE Access 7, 11041–11048 (2019).
[Crossref]

IEEE J. Quantum Electron. (4)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: a dynamical point of view,” IEEE J. Quantum Electron. 45(7), 879–1891 (2009).
[Crossref]

S. Y. Xiang, A. J. Wen, H. Zhang, J. F. Li, H. X. Zhang, and L. Lin, “Effect of gain nonlinearity on time delay signature of chaos in external-cavity semiconductor lasers,” IEEE J. Quantum Electron. 52(4), 1–7 (2016).
[Crossref]

S. Y. Xiang, W. Pan, B. Luo, L. S. Yan, X. H. Zou, N. N. Li, and H. N. Zhu, “Wideband unpredictability-enhanced chaotic semiconductor lasers with dual-chaotic optical injections,” IEEE J. Quantum Electron. 48(8), 1069–1076 (2012).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

F. Y. Lin and J. M. Liu, “Chaotic Lidar,” IEEE J. Sel. Top. Quantum Electron. 10(5), 991–997 (2004).
[Crossref]

IEEE Photonics J. (1)

S. S. Li, Q. Liu, and S. C. Chan, “Distributed feedbacks for time-delay signature suppression of chaos generated from a semiconductor laser,” IEEE Photonics J. 4(5), 1930–1935 (2012).
[Crossref]

IEEE Photonics Technol. Lett. (1)

Y. C. Wang, B. J. Wang, and A. B. Wang, “Chaotic correlation optical time domain reflectometer utilizing laser diode,” IEEE Photonics Technol. Lett. 20(19), 1636–1638 (2008).
[Crossref]

J. Lightwave Technol. (3)

Nat. Photonics (1)

M. Sciamanna and K. A. Shore, “Physics and applications of laser diode chaos,” Nat. Photonics 9(3), 151–162 (2015).
[Crossref]

Nature (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref]

Opt. Commun. (1)

F. Y. Lin and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221(1-3), 173–180 (2003).
[Crossref]

Opt. Express (8)

J. P. Toomey and D. M. Kane, “Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy,” Opt. Express 22(2), 1713–1725 (2014).
[Crossref]

A. K. Zhao, N. Jiang, S. Q. Liu, C. P. Xue, J. M. Tang, and K. Qiu, “Wideband complex-enhanced chaos generation using a semiconductor laser subject to delay-interfered self-phase-modulated feedback,” Opt. Express 27(9), 12336–12348 (2019).
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N. Jiang, C. Wang, C. P. Xue, G. L. Li, S. Q. Lin, and K. Qiu, “Generation of flat wideband chaos with suppressed time delay signature by using optical time lens,” Opt. Express 25(13), 14359–14367 (2017).
[Crossref]

N. Li, B. Kim, V. N. Chizhevsky, A. Locquet, M. Bloch, D. S. Citrin, and W. Pan, “Two approaches for ultrafast random bit generation based on the chaotic dynamics of a semiconductor laser,” Opt. Express 22(6), 6634–6646 (2014).
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D. Wang, L. Wang, T. Zhao, H. Gao, Y. Wang, X. Chen, and A. Wang, “Time delay signature elimination of chaos in a semiconductor laser by dispersive feedback from a chirped FBG,” Opt. Express 25(10), 10911–10924 (2017).
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H. Someya, I. Oowada, H. Okumura, T. Kida, and A. Uchida, “Synchronization of bandwidth-enhanced chaos in semiconductor lasers with optical feedback and injection,” Opt. Express 17(22), 19536–19543 (2009).
[Crossref]

A. B. Wang, Y. B. Yang, B. J. Wang, B. B. Zhang, L. Li, and Y. C. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express 21(7), 8701–8710 (2013).
[Crossref]

J. G. Wu, G. Q. Xia, and Z. M. Wu, “Suppression of time delay signatures of chaotic output in a semiconductor laser with double optical feedback,” Opt. Express 17(22), 20124–20133 (2009).
[Crossref]

Opt. Lett. (7)

N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. 36(23), 4632–4634 (2011).
[Crossref]

N. Li, W. Pan, A. Locquet, and D. S. Citrin, “Time-delay concealment and complexity enhancement of an external-cavity laser through optical injection,” Opt. Lett. 40(19), 4416–4419 (2015).
[Crossref]

S. S. Li, X. Z. Li, and S. C. Chan, “Chaotic time-delay signature suppression with bandwidth broadening by fiber propagation,” Opt. Lett. 43(19), 4751–4754 (2018).
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N. Jiang, A. K. Zhao, C. P. Xue, J. M. Tang, and K. Qiu, “Physical secure optical communication based on private chaotic spectral phase encryption/decryption,” Opt. Lett. 44(7), 1536–1539 (2019).
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X. Z. Li, S. S. Li, J. P. Zhuang, and S. C. Chan, “Random bit generation at tunable rates using a chaotic semiconductor laser under distributed feedback,” Opt. Lett. 40(17), 3970–3973 (2015).
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N. Jiang, A. K. Zhao, S. Q. Liu, C. P. Xue, B. Y. Wang, and K. Qiu, “Generation of broadband chaos with perfect time delay signature suppression by using self-phase-modulated feedback and a microsphere resonator,” Opt. Lett. 43(21), 5359–5362 (2018).
[Crossref]

D. Rontani, E. Mercier, D. Wolfersberger, and M. Sciamanna, “Enhanced complexity of optical chaos in a laser diode with phase-conjugate feedback,” Opt. Lett. 41(20), 4637–4640 (2016).
[Crossref]

Phys. Rev. E (1)

L. Zunino, M. C. Soriano, I. Fischer, O. A. Rosso, and C. R. Mirasso, “Permutation-information-theory approach to unveil delay dynamics from time-series analysis,” Phys. Rev. E 82(4), 046212 (2010).
[Crossref]

Phys. Rev. Lett. (1)

R. Hegger, M. J. Bünner, H. Kantz, and A. Giaquinta, “Identifying and modeling delay feedback systems,” Phys. Rev. Lett. 81(3), 558–561 (1998).
[Crossref]

Other (2)

D. G. Rabus, “Integrated ring resonators,” Spring Series in Optical Sciences, Springer, New York, 2007.

J. Carroll, J. Whiteaway, and D. Plumb, “Distributed feedback semiconductor lasers,” IEE Circuits, Devices and Systems Series 10, 1998.

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Figures (7)

Fig. 1.
Fig. 1. (a) Schematic of chaos generation using ECSL subject to nonlinear feedback from PCRR, (b) detailed configuration of PCRR. DFB, distributed-feedback laser; PC, polarization controller; OC, optical circulator; VOA, variable optical attenuator; PD, photodetector.
Fig. 2.
Fig. 2. Temporal series (first row), RF spectra (second row), ACF traces (third row) and PE traces (fourth row) of chaos in the cases of (a) COF, (b) SRROF, (c) DRROF and (d) TRROF.
Fig. 3.
Fig. 3. Ratio of energy in frequency band Δf as a function of feedback strength for the cases of (a) COF, (b) SRROF, (c) DRROF and (d) TRROF.
Fig. 4.
Fig. 4. Influences of feedback strength kf and operation current I(Ith) on TDS in ACF trace (first row) and EB (second row), in the cases of COF (first column), SRROF (second column), DRROF (third column) and TRROF (fourth column).
Fig. 5.
Fig. 5. Transmission spectra of “Throughput” port (first row) and “Drop” port (second row) of PCRR composed of (a) SRR, (b) DRR and (c) TRR.
Fig. 6.
Fig. 6. Influences of coupling coefficient к and ring radius R on TDS in ACF trace (first row) and EB (second row), for the cases of (a) SRROF, (b) DRROF and (c) TRROF.
Fig. 7.
Fig. 7. Influences of the mismatches of coupling coefficient к and ring radius R on TDS in ACF (first row) and effective bandwidth (second row), in the configurations of DRROF (first column) and TRROF (second-fifth column).

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

d E ( t ) d t = 1 + i α 2 [ g ( N ( t ) N 0 ) 1 + ε | E ( t ) | 2 1 τ p ] E ( t ) + k f E ( t τ f ) h 1 ( t )
d N ( t ) d t = I q V N ( t ) τ e g ( N ( t ) N 0 ) 1 + ε | E ( t ) | 2 | E ( t ) | 2
E ( t ) o u t = E ( t τ f / τ f 2 2 ) h 2 ( t )
[ E 1 ( ω ) E 2 ( ω ) ] = T 1 T φ 1 T 2 T φ 2 T i T φ i T n [ E 3 ( ω ) E 4 ( ω ) ] = [ P 11 P 12 P 21 P 22 ] [ E 3 ( ω ) E 4 ( ω ) ]
T i = [ 1 η 1 i η 2 i α r i 2 e j Δ ω t r i η 1 i η 2 i α r i 2 e j Δ ω t r i κ 1 i κ 2 i α r i e j Δ ω t r i / Δ ω t r i 2 2 η 1 i η 2 i α r i 2 e j Δ ω t r i κ 1 i κ 2 i α r i e j Δ ω t r i / Δ ω t r i 2 2 η 1 i η 2 i α r i 2 e j Δ ω t r i η 1 i η 2 i α r i 2 e j Δ ω t r i η 1 i η 2 i α r i 2 e j Δ ω t r i ]
T φ i = [ e j β b L 0 0 e j β b L ]
C ( Δ t ) = ( I ( t + Δ t ) I ( t ) ) ( I ( t ) I ( t ) ) ( I ( t + Δ t ) I ( t + Δ t ) ) 2 ( I ( t ) I ( t ) ) 2
p ( Ω ) = # { t | t N x ( d 1 ) Δ τ ;   X t h a s t y p e Ω } N x ( d 1 ) Δ τ

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