Abstract

We numerically investigated the chaos time delay signature (TDS) suppression and bandwidth enhancement by electrical heterodyning. Chaos signals generated with a semiconductor laser subject to optical feedback typically have distinct loop frequency peaks in their power spectra corresponding to the reciprocals of the time delays, which deteriorates the performance in applications including chaos radar/lidar and fast random bit generation. By electrically heterodyning the chaos signal with a single frequency local oscillator, we show that the power in the chaos spectrum can be redistributed and a smoother spectrum with a broader effective bandwidth can be obtained. Compared with the chaos directly generated from a semiconductor laser subject to optical feedback, the amplitudes of the TDS (ρTDS) measured under different feedback strengths can be suppressed up to 63% and the effective bandwidths can be enhanced up to 46% in average after the electrical heterodyning is applied.

© 2015 Optical Society of America

1. Introduction

Optical chaos generated from nonlinear dynamics of semiconductor lasers has been extensively utilized in applications including secure communication, encryption, lidar, radar, and random bit generation [15]. Compared with other perturbation schemes, such as optical injection [6] and optoelectronic feedback [7], optical feedback has the advantages of simple configuration, low cost, and having large regions of chaos states in the parameter space determined by the feedback strength and delay time [8, 9]. However, distinct loop frequency peaks corresponding to the reciprocals of the delay times are often found in the chaos spectra generated with the optical feedback scheme. The periodicity and the associated time delay signatures (TDS) in the autocorrelation function and delayed mutual information deteriorate the performance of many potential applications [1015]. For examples, the existence of the TDS will cause range ambiguity in ranging using chaos [16] and degrade the randomness in generation of the random bit signals [5, 17]. Besides the loop frequency peaks, the power spectra of the chaos generated by optical feedback usually peak higher around the relaxation oscillation frequency of the laser and drop lower in the low frequency region near dc. Together with a finite acquisition bandwidth limited by the electronic components used, only a fraction of the chaos power can be effectively utilized.

To minimize the TDS, studies showed that there is optimal feedback strengths for different delay times and bias currents [1012]. Modified feedback schemes, including double optical feedbacks [13], fiber Bragg grating feedback [14], and polarization rotated feedback [15], were also proposed to further suppress the TDS. Besides, the influence of the intrinsic laser parameters on concealing the TDS in both the amplitude and the phase of the optical chaos signal was investigated in a semiconductor ring laser [18]. To enhance the bandwidths, optical injection were used to push the upper bound of the chaos spectrum toward higher frequency [19, 20]. Based on a master-slave configuration, where a slave laser is injected by a chaos emission from a master laser subject to optical feedback, TDS suppression and bandwidth enhancement were analyzed [21]. However, the optimized operation regions for the TDS suppression and for the bandwidth enhancement in this master-slave configuration are not fully overlapped. To simultaneously achieve the TDS suppression and the bandwidth enhancement, a cascade-coupled configuration involving three semiconductor lasers (one subject to optical feedback and two subject to optical injection) was also proposed and studied [22]. However, the overall setup becomes much more complicated and many parameters have to be controlled and optimized at the same time.

To simultaneously achieve the TDS suppression and the bandwidth enhancement in chaos generated by a semiconductor laser subject to optical feedback, an electrical heterodyne technique is proposed and numerically investigated. Similar technique was also adopted in an optically injected semiconductor laser for generating multiple chaos signals in parallel [23]. In this study, the dependence of the TDS and the bandwidths on the control parameters are analyzed. Performances of two different heterodyne schemes are compared, where in both schemes only a local oscillator and an RF mixer are required. To the best of our knowledge, this is the first proposed approach to suppress the TDS and enhance the bandwidth of chaos in the electrical domain. Therefore, it can be compatible with and cascaded after any optical or electrical chaos generation systems.

2. Principle and model

The schematic setups of the electrical heterodyne technique for TDS suppression and bandwidth enhancement are shown in Fig. 1. The chaos signals are generated by a semiconductor laser subject to optical feedback, which is numerically simulated using the coupled rate equations given in Eqs. (1)(3) [14, 24, 25]. Here, a is the normalized optical field, ϕ is the optical phase, ñ is the normalized carrier density, ξ is the normalized feedback strength, and τ is the feedback delay time. For the intrinsic laser parameters, = 1.222 is the normalized bias current, γc = 5.36 × 1011 s−1 is the cavity decay rate, γn = 7.53 × 109 s−1 is the differential carrier relaxation rate, γs = 5.96 × 1010 s−1 is the nonlinear carrier relaxation rate, γp = 1.91 × 109s−1 is the spontaneous carrier relaxation rate, and b = 3.2 is the linewidth enhancement factor [26]. The feedback delay time τ is set to 1 ns, which is not critical in our analysis. The laser parameters given above were experimentally determined by using the four-wave mixing method from a single-mode distributed-feedback laser.

 figure: Fig. 1

Fig. 1 Schematic setups of chaos generation for Path O: original chaos, Path H1: heterodyned chaos, and Path H2: mixed chaos. SL: semiconductor laser; M: mirror; PD: photode-tector; SG: signal generator; A: microwave amplifier; LPF: low pass filter.

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dadt=12[γcγnγsJ˜n˜-γp(2a+a2)](1+a)+ξγc[1+a(t-τ)]cos[ϕ(t-τ)-ϕ(t)]
dϕdt=-b2[γcγnγsJ˜n˜-γp(2a+a2)]+ξγc[1+a(t-τ)]1+asin[ϕ(t-τ)-ϕ(t)]
dn˜dt=-γsn˜-γn(1+a)2n˜-γsJ˜(2a+a2)+γsγpγsJ˜(2a+a2)(1+a)2

After receiving by the photodetector, the chaos signals can be represented by their intensities I(t) = [1 + a(t)]2. For the original chaos, as shown in Path O, the chaos signal is directly acquired by devices with a finite acquisition bandwidth (a low pass filter (LPF) with a -3 dB bandwidth of 5 GHz is assumed in this study). For the proposed electrical heterodyne technique, two schemes are investigated and compared. Path H1 shows the first heterodyne scheme, where the original chaos signal is electrically heterodyned with a sinusoidal signal with a frequency of fLO generated by a signal generator as the local oscillator. In the heterodyne process, having fLO as a symmetry axis, the chaos spectrum with frequencies lower than the fLO will be flipped up to superpose with the spectrum higher than the fLO. The superposition is then down-shifted toward the dc with a frequency of fLO. As the result, heterodyned chaos signals with different spectral power distributions can be produced depending on the frequency of the sinusoidal signal fLO applied. Path H2 shows the second heterodyne scheme, where the original chaos signal is first divided into a heterodyne part and a reference part. The signal in the heterodyne part will be heterodyned with a local oscillator in the same manner as described in the Path H1, which is then recombined again with the original chaos signal in the reference part to generate the mixed chaos. Before recombining, an electric amplifier (or attenuator) with an amplification (or attenuation) factor A is used to adjust the relative amplitude of the heterodyned signal to the original signal. In these schemes, the original chaos, heterodyned chaos, and mixed chaos signals obtained from the Path O, Path H1, and Path H2 can be deduced as

SO(t)=LPF[I(t)],
SH1(t)=LPF[I(t)×cos(2πfLOt)],
and
SH2(t)=LPF[I(t)×(1+Acos(2πfLOt))],
respectively. LPF[·] is a Chebyshev Type-II low pass filter with a -3 dB bandwidth of 5 GHz to take into account the finite bandwidth in the acquisition process. For the electrical heterodyne technique to have the best performance, the photodetector, the mixers, and the amplifiers should have bandwidths that are at least twice of the LPF bandwidth.

To retrieve the time delay structures and to evaluate the performance of the TDS suppression [11, 27], the autocorrelation functions and the delayed mutual information of the chaos signals are calculated. To quantify the bandwidth enhancement, the effective bandwidths [28] (which sums up only those discrete spectral segments of the chaos power spectrum accounting for 80% of the total power) of the chaos signals are calculated. The effective bandwidth can evaluate both the flatness and the coverage of the chaos spectra to quantify their broadband characteristics.

3. Result and discussion

Figure 2(a) shows the power spectra of the original chaos signal obtained from Path O with a feedback strength of ξ = 0.09, where the dashed and solid curves are the spectra before and after filtered by the acquisition bandwidth. As can be seen, the power in the spectra is not uniformly distributed, where frequency peaks at the multiples of the loop frequency m floop (floop = 990 MHz ≈ 1/τ, m = 0,1,2,3...) with a sawtooth shape are present. Moreover, the chaos spectra also tend to peak higher near the relaxation oscillation frequency fr (fr = (2π)−1(γcγn + γsγp)1/2 = 10.25 GHz in this study [25]) and drop lower in the low frequency region. The effects from both the feedback loop and the resonance gain introduce the TDS and limit the effective bandwidths of the chaos signals generated. Figures 2(b) and 2(c) show the autocorrelation function and the delayed mutual information of the original chaos signal shown in Fig. 2(a) after filtered with the acquisition bandwidth. As can be seen, notable peaks are observed at 1.01 ns (≈τ) and its multiples, indicating the presence of the TDS. Here, the small deviations between floop and 1/τ in the power spectra and between the locations of the TDS and τ in the autocorrelation function and delayed mutual information are due to the competition between the feedback delay time and the relaxation oscillation period [1012].

 figure: Fig. 2

Fig. 2 (a) Power spectrum, (b) autocorrelation function, and (c) delayed mutual information of the original chaos generated with a feedback strength ξ = 0.09 and a feedback delay time τ = 1 ns after filtered by the acquisition bandwidth. As the reference, the dashed curve in (a) is the power spectrum of the original chaos before filtered by the acquisition bandwidth.

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By applying the electric heterodyning with different fLO on the original chaos signal, the blue solid curves in Fig. 3 show the power spectra, autocorrelation functions, and delayed mutual information of the heterodyned chaos signals obtained from the first heterodyne scheme after filtered with the acquisition bandwidth (SH1). As the reference, the power spectrum, autocorrelation function, and delayed mutual information of the original chaos are shown in back with the black dashed curves. The fLO applied are shown in Figs. 3(a1)3(a4) with the green lines. Figure 3(a1) shows the power spectrum of the heterodyned chaos with fLO = 5.94 GHz, which coincides with the 6th frequency peak (6 floop) in the power spectrum of the original chaos. In the process of heterodyning with fLO = 6 floop, the 5th and 7th frequency peaks will be superposed and down-converted to the frequency of the 1st frequency peak, the 4th and 8th frequency peaks will be superposed and down-converted to the frequency of the 2nd frequency peak, and so on. As shown in Fig. 3(a1), the heterodyned chaos still has distinct frequency peaks at the multiples of the floop in the spectrum with a separation of floop. Owing to its periodic nature, strong TDS at the multiples of τ are clearly observed in the autocorrelation function and delayed mutual information as shown in Figs. 3(b1) and 3(c1), respectively. By increasing fLO to 6.21 GHz, locating at about (6+14)floop of the original chaos spectrum, the frequency peaks converted to frequencies of (m±14)floop with a separation of 12floop after the heterodyne process. As shown in Fig. 3(a2), a heterodyned chaos with a flatter spectrum and less periodicity is generated. When compared to the original chaos, the TDS in the autocorrelation function and the delayed mutual information shown in Figs. 3(b2) and 3(c2) are notably suppressed. When the fLO is further increased to 6.44 GHz locating at one of the valley of the original chaos spectrum at (6+12)floop, the frequency peaks are converted to the frequencies of odd multiples of 12floop with a separation of floop after the heterodyne process. Since shifting the frequency peaks as shown in Fig. 3(a3) has no effect in reducing the periodicity in the heterodyned chaos, the amplitudes of the TDS in the respective autocorrelation function and the delayed mutual information shown in Figs. 3(b3) and 3(c3) are still comparable to those in the original chaos. Figures 3(a4), 3(b4), and 3(c4) show the power spectrum, autocorrelation, and delayed mutual information of the heterodyned chaos obtained with fLO = 6.53 GHz at about (6+23)floop of the original chaos spectrum. As can be seen, having the frequency peaks converted to frequencies of (m±13)floop with separations of 13floop and 23floop has little effect on the reduction of the TDS. In sum, having an fLO at (m+14)floop of the original chaos spectrum can better reduce the TDS in the heterodyned chaos generated with this heterodyne scheme. Since the spectrum of the original chaos is not really an ideal sawtooth function, adjusting the fLO at around (m+14)floop may be needed to optimize the TDS reduction. Detailed analysis on the reduction of the TDS and the enhancement of the effective bandwidth under different fLO will be discussed in Fig. 4.

 figure: Fig. 3

Fig. 3 (a) Power spectra, (b) autocorrelation functions, and (c) delayed mutual information of the heterodyned chaos (blue solid curves) obtained from the first heterodyne scheme with different fLO (green solid lines): 5.94 GHz (first row), 6.21 GHz (second row), 6.44 GHz (third row), and 6.53 GHz (fourth row). As the reference, the black dashed curves show the (a) power spectrum of the original chaos before filtering, and the (b) autocorrelation function and (c) delayed mutual information of the original chaos after filtering.

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 figure: Fig. 4

Fig. 4 (a) Absolute amplitudes of the first (bright blue) and second (dark blue) TDS peaks of the autocorrelation functions, (b) ρTDS, and (c) effective bandwidths of the heterodyned chaos under different fLO. The black solid lines in (b) and (c) show the ρTDS and effective bandwidth obtained from the original chaos for reference. The gray dashed lines show the multiples of floop/2.

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Figure 4(a) shows the absolute amplitudes (ρ) of the first (around τ) and the second (around 2τ) TDS peaks extracted from the autocorrelation functions of the heterodyned chaos obtained from the first heterodyne scheme under different fLO. As can be seen, the amplitudes of the first (ρ1) and the second (ρ2) TDS peaks have their minima in every 12floop and 14floop when the fLO is varied, respectively. While the suppression of the first TDS peak does not guarantee the suppression of the other TDS peaks (may even result in the enhancement as can be seen in Fig. 4(a)), a TDS amplitude (ρTDS) defined as the average of the absolute amplitudes from the first four primary TDS peaks

ρTDS=|ρ1|+|ρ2|+|ρ3|+|ρ4|4
at around τ, 2τ, 3τ, and 4τ is calculated to quantitatively evaluate the overall suppression of the TDS. Here, since the amplitudes of the TDS at longer lag times are relatively small and are not meaningfully affecting the results, only the amplitudes of the first four primary TDS peaks are taken into account. Figures 4(b) and 4(c) show the ρTDS and the effective bandwidths of the heterodyned chaos under different fLO, respectively. The gray dashed lines indicate the multiples of 12floop=495MHz. As the benchmarks, the black solid lines in Figs. 4(b) and 4(c) show the respective ρTDS of 0.15 and the effective bandwidth of 2.75 GHz obtained from the original chaos. As can be seen, the periodic variations in the ρTDS and the effective bandwidth are in accordance with half of the floop. When fLO=m12floop, as those shown in Figs. 3(a1) and 3(a3) as examples, the frequency peaks on the opposite sides of the fLO will be converted to frequencies coinciding with one of the original frequency peaks or valleys after the heterodyne process. The frequency peaks of the heterodyned chaos generated under such condition are preserved, where even larger ρTDS and smaller bandwidths than the original chaos are seen as shown in Figs. 4(b) and 4(c), respectively. When the fLO is applied at around (m+14)floop or (m+34)floop, as that is shown in Fig. 3(a2) as an example, the frequency peaks converted to frequencies at (m+14)floop and (m+34)floop with a closer separation of 12floop after the heterodyne process. As can be seen in Figs. 4(b) and 4(c), under such condition, smaller ρTDS and larger effective bandwidths are simultaneously achieved. With fLO = 6.21 GHz, the ρTDS and effective bandwidth of the heterodyned chaos can be suppressed and enhanced by 39% and 17% compared to the original chaos, respectively.

Note that, the effective bandwidth of a chaos is not only affected by the periodic fluctuation in the spectrum resulting from the feedback loop, but also influenced by the uneven power distribution caused by the resonance gain. Therefore, on top of increasing the effective bandwidth by eliminating the loop frequency peaks in the spectrum, the heterodyne technique further increases the effective bandwidth through redistributing the power around the resonance peak to the power dip close to dc.

Figure 5 shows the power spectra, autocorrelation functions, and delayed mutual information of the mixed chaos signals obtained from the second heterodyne scheme (SH2)(red solid curves) with different fLO at A = 0.1 dB. Similar to Fig. 3, the fLO at about 6 floop, (6+14)floop, (6+12)floop, and (6+23)floop of 5.94 GHz, 6.21 GHz, 6.44 GHz, and 6.53 GHz are chosen, respectively. In this scheme, as illustrated with the Path H2 in Fig. 1, the heterodyned chaos (having exactly the same properties as those shown in Fig. 3 with the blue solid curves) are recombined again with the original chaos (as those shown in Fig. 3 and Fig. 5 with the black dashed curves). With fLO = 5.94 GHz at the 6th frequency peak of the original chaos, the respective power spectrum, autocorrelation function, and delayed mutual information of the mixed chaos as shown in Figs. 5(a1), 5(b1), and 5(c1) remain almost unchanged compared to those of the original chaos. When fLO is increased by 14floop to 6.21 GHz, the frequency peaks of the heterodyned chaos at (m+14)floop (blue solid curve in Fig. 3(a2)) fill those valleys of the original chaos (black dashed curve in Fig. 3(a2)) when they recombine. As the result, the frequency peaks and valleys of the mixed chaos as shown in Fig. 5(a2) become shallower, and the amplitudes of the corresponding TDS in Figs. 5(b2) and 5(c2) become smaller. Further increasing fLO to (6+12)floop=6.44GHz, the frequency peaks and valleys in Fig. 5(a3) become even shallower, and the amplitudes of the TDS in Figs. 5(b3) and 5(c3) are further reduced. This reduction in the periodicity and TDS are resulting from the complement and cancelation between the frequency peaks in the heterodyned chaos and the valleys in the original chaos (blue solid and black dashed curves in Fig. 3(a3)) in the recombining process, where they coincide with each other periodically and smooth out the spectrum. For fLO at about (6+23)floop=6.53GHz, the peaks and valleys of the mixed chaos shown in Fig. 5(a4) become blurred and no meaningful frequency structure can be extracted. Compared to the original chaos, significant reduction of the TDS in the autocorrelation function and delayed mutual information as those shown in Figs. 5(b4) and 5(c4) are clearly demonstrated.

 figure: Fig. 5

Fig. 5 (a) Power spectra, (b) autocorrelation functions, and (c) delayed mutual information of the mixed chaos (red solid curves) obtained from the second heterodyne scheme for different fLO (green solid lines): 5.94 GHz (first row), 6.21 GHz (second row), 6.44 GHz (third row), and 6.53 GHz (fourth row), when A = 0.1 dB. As the reference, the black dashed curves show the (a) power spectrum of the original chaos before filtering, and the (b) autocorrelation function and (c) delayed mutual information of the original chaos after filtering.

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For the mixed chaos signals obtained in the second heterodyne scheme, Figs. 6(a) and 6(c) show the ρTDS and the effective bandwidths for different amplification factors A and fLO. Arrows next to the color bars of the ρTDS and the effective bandwidth show the respective values of the original chaos as the reference. Similar patterns with periodic U-shape areas of the low ρTDS and large effective bandwidths are seen in Figs. 6(a) and 6(c), respectively. As can be seen, low ρTDS and large effective bandwidths in general occur when the amplitudes of the heterodyned chaos and the original chaos are comparable (A ≃ 0 dB). When A is large (top edge of Figs. 6(a) and 6(c)), the results from the second heterodyne scheme converge toward the results of the first heterodyne scheme where the heterodyned chaos dominates in the mixed signal. When A is small (bottom edge of Figs. 6(a) and 6(c)), the results from the second heterodyne scheme converge toward the results of the original chaos scheme where the original chaos dominates in the mixed signal. Note that, the regions of the low ρTDS coincide with the regions of the large effective bandwidths in the parameter space shown in Figs. 6(a) and 6(c), meaning that TDS suppression and bandwidth enhancement can be simultaneously achieved.

 figure: Fig. 6

Fig. 6 (a) ρTDS and (c) effective bandwidth of the mixed chaos for different A and fLO, where the arrows on the right of the color bars indicate the ρTDS and effective bandwidth for the original chaos. (b) Minimum ρTDS and (d) maximum effective bandwidths of the mixed chaos obtained under optimal A for different fLO, where the black solid lines in (b) and (d) show the ρTDS and effective bandwidth of the original chaos for reference. The gray dashed lines show the multiples of floop/2.

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Extracted from Figs. 6(a) and 6(c) under optimal A, Figs. 6(b) and 6(d) show the minimum ρTDS and the maximum bandwidths that can be obtained at different fLO. The gray dashed lines again indicate the multiples of 12floop=495MHz. As can be seen, the periods of the variations in both the ρTDS and effective bandwidth are now in accordance with the floop. Unlike the first heterodyne scheme as shown in Figs. 4(b) and 4(c), practically for all different fLO, the ρTDS and the effective bandwidths of the mixed chaos signals from the second heterodyne scheme (under optimal A) show TDS reduction and bandwidth enhancement in relative to the original chaos. The best TDS reduction and bandwidth enhancement are found to occur at around fLO=(m+13)floop and fLO=(m+23)floop, where the frequency peaks and valleys in the heterodyned chaos and the original chaos complement each other the most. With fLO = 6.53 GHz and A = 0.1 dB, the ρTDS and effective bandwidth of the mixed chaos can be suppressed and enhanced up to 68% and 27% compared to the original chaos.

Figure 7 shows the ρTDS and the effective bandwidths obtained from the original chaos, heterodyned chaos, and the mixed chaos under different feedback strengths. For the hetero-dyned chaos, fLO are optimized to have the minimum ρTDS. For the mixed chaos, both fLO and A are optimized to have the minimum ρTDS. As can be seen in Figs. 7(a) and 7(b), the ρTDS and the effective bandwidth of the original chaos (black circles) have their respective minimum and maximum at around ξ = 0.1. By applying the electrical heterodyning proposed, the heterodyned chaos from the first heterodyne scheme (blue circles) shows TDS reduction and bandwidth enhancement for all different ξ investigated. An average reduction of 30% in ρTDS and enhancement of 28% in the effective bandwidth for different feedback strengths are achieved. For the mixed chaos from the second heterodyne scheme (red circles), even lower TDS and larger effective bandwidths are realized that an average reduction of 63% in ρTDS and enhancement of 46% in the effective bandwidth for different feedback strengths are demonstrated. At ξ = 0.09, the mixed chaos has a lowest ρTDS and an effective bandwidth of 0.05 and 3.48 GHz reduced and enhanced from 0.15 and 2.75 GHz of the original chaos, respectively. Note that the effective bandwidth of 3.48 GHz is already very close to the largest possible effective bandwidth of 4 GHz in our study (5 GHz×80% = 4 GHz), which is determined by the 5 GHz acquisition bandwidth and the definition of the effective bandwidth (that measures only those spectral segments accounting for 80% of the total power in the chaos power spectrum). In other words, with the heterodyne technique applied, the spectrum of the mixed chaos becomes comparably flat and has a relatively even power distribution.

 figure: Fig. 7

Fig. 7 (a) ρTDS and (b) effective bandwidths of the original chaos (black), the heterodyned chaos from the first heterodyne scheme (blue), and the mixed chaos from the second heterodyne scheme (red) under different feedback strengths. The fLO and A are optimized to have the minimum ρTDS in the heterodyned chaos and the mixed chaos.

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Note that, if only the first TDS peak is taken into account, average TDS reduction ratios of 80% and 93% for different feedback strengths can be achieved for the heterodyned chaos and the mixed chaos, respectively. Compared to a 55% maximal TDS reduction ratio [14] and a 30-40% average TDS reduction ratio [29] reported using other techniques, the electrical heterodyne technique proposed in this study is even more effective in suppressing just the first TDS peak.

4. Conclusion

In summary, the electrical heterodyne technique is numerically investigated for suppressing the TDS and enhancing the effective bandwidths of the chaos generated from a semiconductor laser subject to optical feedback. Two heterodyne schemes are proposed and compared. In the first heterodyne scheme, the original chaos is heterodyned with a local oscillator to produce the heterodyned chaos. In the second heterodyne scheme, the heterodyned chaos is further recombined with the original chaos to produce the mixed chaos. When the local oscillation frequency fLO and the amplification factor A are properly selected, the power in the chaos spectrum can be redistributed to elevate the dip in the low frequency region and smooth out the loop frequency peaks. By having flatter spectra, the TDS suppression and the bandwidth enhancement can be simultaneously achieved after the heterodyne process. Compared to the original chaos, the ρTDS and the effective bandwidths can be suppressed and enhanced up to 63% and 46% in average. While a 5 GHz acquisition bandwidth is assumed in this study, similar results can be obtained for acquisition devices with different bandwidths. With the possibility of achieving TDS suppression and bandwidth enhancement simultaneously, the proposed electrical heterodyne technique has the advantages of generating high-quality broadband chaos for applications including high-resolution low-ambiguity chaos ranging and fast random bit generation. It is compatible with and can be cascaded after any optical or electrical chaos generation systems.

Acknowledgments

This work is supported by the National Science Council of Taiwan under contract NSC 100-2112-M-007-012-MY3 and MOST 103-2112-M-007-019-MY3, and by the National Tsing Hua University under grant 102N2081E1.

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20. A. Wang, Y. Wang, and H. He, “Enhancing the bandwidth of the optical chaotic signal generated by a semiconductor laser with optical feedback,” IEEE Photon. Technol. Lett. 20, 1633–1635 (2008). [CrossRef]  

21. N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, X. Zou, L. Zhang, and P. Mu, “Photonic generation of wideband time-delay-signature-eliminated chaotic signals utilizing an optically injected semiconductor laser,” IEEE J. Quantum Electron. 48, 1339–1345 (2012). [CrossRef]  

22. N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, and X. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 24, 2187–2190 (2012). [CrossRef]  

23. X. Z. Li and S. C. Chan, “Heterodyne random bit generation using an optically injected semiconductor laser in chaos,” IEEE J. Quantum Electron. 49, 829–838 (2013). [CrossRef]  

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

25. J. M. Liu and T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron. 30, 957–965 (1994). [CrossRef]  

26. S. K. Hwang, J. M. Liu, and J. K. White, “35-GHz intrinsic bandwidth for direct modulation in 1.3-μ m semiconductor lasers subject to strong injection locking,” IEEE Photon. Technol. Lett.16, 972–974 (2004). [CrossRef]  

27. V. S. Udaltsov, L. Larger, J. P. Goedgebuer, A. Locquet, and D. S. Citrin, “Time delay identification in chaotic cryptosystems ruled by delay-differential equations,” J. Opt. Technol. 72, 373–377 (2005). [CrossRef]  

28. F. Y. Lin, Y. K. Chao, and T. C. Wu, “Effective bandwidths of broadband chaotic signals,” IEEE J. Quantum Electron. 48, 1010–1014 (2012). [CrossRef]  

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

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, 343–346 (2005).
    [Crossref] [PubMed]
  2. F. Y. Lin and J. M. Liu, “Chaotic lidar,” IEEE J. Sel. Top. Quantum Electron. 10, 991–997 (2004).
    [Crossref]
  3. W. T. Wu, Y. H. Liao, and F. Y. Lin, “Noise suppressions in synchronized chaos lidars,” Opt. Express 18, 26155–26162 (2010).
    [Crossref] [PubMed]
  4. F. Y. Lin and J. M. Liu, “Chaotic radar using nonlinear laser dynamics,” IEEE J. Quantum Electron. 40, 815–820 (2004).
    [Crossref]
  5. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2, 728–732 (2008).
    [Crossref]
  6. Y. H. Liao, J. M. Liu, and F. Y. Lin, “Dynamical characteristics of a dual-beam optically injected semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 19, 1500606 (2013).
    [Crossref]
  7. F. Y. Lin and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. 221, 173–180 (2003).
    [Crossref]
  8. A. Murakami, J. Ohtsubo, and Y. Liu, “Stability analysis of semiconductor laser with phase-conjugate feedback,” IEEE J. Quantum Electron. 33, 1825–1831 (1997).
    [Crossref]
  9. J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28, 93–108 (1992).
    [Crossref]
  10. D. Rontani, A. Locquet, M. Sciamanna, and D. S. Citrin, “Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback,” Opt. Lett. 32, 2960–2962 (2007).
    [Crossref] [PubMed]
  11. 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, 879–891 (2009).
    [Crossref]
  12. J. G. Wu, G. Q. Xia, X. Tang, X. D. Lin, T. Deng, L. Fan, and Z. M. Wu, “Time delay signature concealment of optical feedback induced chaos in an external cavity semiconductor laser,” Opt. Express 18, 6661–6666 (2010).
    [Crossref] [PubMed]
  13. 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, 20124–20133 (2009).
    [Crossref] [PubMed]
  14. 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, 1930–1935 (2012).
    [Crossref]
  15. J. G. Wu, G. Q. Xia, L. P. Cao, and Z. M. Wu, “Experimental investigations on the external cavity time signature in chaotic output of an incoherent optical feedback external cavity semiconductor laser,” Opt. Commun. 282, 3153–3156 (2009).
    [Crossref]
  16. F. Y. Lin and J. M. Liu, “Ambiguity functions of laser-based chaotic radar,” IEEE J. Quantum Electron. 40, 1732–1738 (2004).
    [Crossref]
  17. K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45, 1367–1379 (2009).
    [Crossref]
  18. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Loss of time-delay signature in chaotic semiconductor ring lasers,” Opt. Lett. 37, 2541–2543 (2012).
    [Crossref] [PubMed]
  19. A. Uchida, T. Heil, Y. Liu, P. Davis, and T. Aida, “High-frequency broad-band signal generation using a semiconductor laser with a chaotic optical injection,” IEEE J. Quantum Electron. 39, 1462–1467 (2003).
    [Crossref]
  20. A. Wang, Y. Wang, and H. He, “Enhancing the bandwidth of the optical chaotic signal generated by a semiconductor laser with optical feedback,” IEEE Photon. Technol. Lett. 20, 1633–1635 (2008).
    [Crossref]
  21. N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, X. Zou, L. Zhang, and P. Mu, “Photonic generation of wideband time-delay-signature-eliminated chaotic signals utilizing an optically injected semiconductor laser,” IEEE J. Quantum Electron. 48, 1339–1345 (2012).
    [Crossref]
  22. N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, and X. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 24, 2187–2190 (2012).
    [Crossref]
  23. X. Z. Li and S. C. Chan, “Heterodyne random bit generation using an optically injected semiconductor laser in chaos,” IEEE J. Quantum Electron. 49, 829–838 (2013).
    [Crossref]
  24. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347–355 (1980).
    [Crossref]
  25. J. M. Liu and T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron. 30, 957–965 (1994).
    [Crossref]
  26. S. K. Hwang, J. M. Liu, and J. K. White, “35-GHz intrinsic bandwidth for direct modulation in 1.3-μ m semiconductor lasers subject to strong injection locking,” IEEE Photon. Technol. Lett.16, 972–974 (2004).
    [Crossref]
  27. V. S. Udaltsov, L. Larger, J. P. Goedgebuer, A. Locquet, and D. S. Citrin, “Time delay identification in chaotic cryptosystems ruled by delay-differential equations,” J. Opt. Technol. 72, 373–377 (2005).
    [Crossref]
  28. F. Y. Lin, Y. K. Chao, and T. C. Wu, “Effective bandwidths of broadband chaotic signals,” IEEE J. Quantum Electron. 48, 1010–1014 (2012).
    [Crossref]
  29. A. Wang, Y. Yang, B. Wang, B. Zhang, L. Li, and Y. Wang, “Generation of wideband chaos with suppressed time-delay signature by delayed self-interference,” Opt. Express,  21, 8701–8710 (2013).
    [Crossref] [PubMed]

2013 (3)

Y. H. Liao, J. M. Liu, and F. Y. Lin, “Dynamical characteristics of a dual-beam optically injected semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 19, 1500606 (2013).
[Crossref]

X. Z. Li and S. C. Chan, “Heterodyne random bit generation using an optically injected semiconductor laser in chaos,” IEEE J. Quantum Electron. 49, 829–838 (2013).
[Crossref]

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

2012 (5)

F. Y. Lin, Y. K. Chao, and T. C. Wu, “Effective bandwidths of broadband chaotic signals,” IEEE J. Quantum Electron. 48, 1010–1014 (2012).
[Crossref]

R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Loss of time-delay signature in chaotic semiconductor ring lasers,” Opt. Lett. 37, 2541–2543 (2012).
[Crossref] [PubMed]

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, X. Zou, L. Zhang, and P. Mu, “Photonic generation of wideband time-delay-signature-eliminated chaotic signals utilizing an optically injected semiconductor laser,” IEEE J. Quantum Electron. 48, 1339–1345 (2012).
[Crossref]

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, and X. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 24, 2187–2190 (2012).
[Crossref]

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, 1930–1935 (2012).
[Crossref]

2010 (2)

2009 (4)

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, 20124–20133 (2009).
[Crossref] [PubMed]

J. G. Wu, G. Q. Xia, L. P. Cao, and Z. M. Wu, “Experimental investigations on the external cavity time signature in chaotic output of an incoherent optical feedback external cavity semiconductor laser,” Opt. Commun. 282, 3153–3156 (2009).
[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, 879–891 (2009).
[Crossref]

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45, 1367–1379 (2009).
[Crossref]

2008 (2)

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2, 728–732 (2008).
[Crossref]

A. Wang, Y. Wang, and H. He, “Enhancing the bandwidth of the optical chaotic signal generated by a semiconductor laser with optical feedback,” IEEE Photon. Technol. Lett. 20, 1633–1635 (2008).
[Crossref]

2007 (1)

2005 (2)

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, 343–346 (2005).
[Crossref] [PubMed]

V. S. Udaltsov, L. Larger, J. P. Goedgebuer, A. Locquet, and D. S. Citrin, “Time delay identification in chaotic cryptosystems ruled by delay-differential equations,” J. Opt. Technol. 72, 373–377 (2005).
[Crossref]

2004 (3)

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

F. Y. Lin and J. M. Liu, “Chaotic radar using nonlinear laser dynamics,” IEEE J. Quantum Electron. 40, 815–820 (2004).
[Crossref]

F. Y. Lin and J. M. Liu, “Ambiguity functions of laser-based chaotic radar,” IEEE J. Quantum Electron. 40, 1732–1738 (2004).
[Crossref]

2003 (2)

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

A. Uchida, T. Heil, Y. Liu, P. Davis, and T. Aida, “High-frequency broad-band signal generation using a semiconductor laser with a chaotic optical injection,” IEEE J. Quantum Electron. 39, 1462–1467 (2003).
[Crossref]

1997 (1)

A. Murakami, J. Ohtsubo, and Y. Liu, “Stability analysis of semiconductor laser with phase-conjugate feedback,” IEEE J. Quantum Electron. 33, 1825–1831 (1997).
[Crossref]

1994 (1)

J. M. Liu and T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron. 30, 957–965 (1994).
[Crossref]

1992 (1)

J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28, 93–108 (1992).
[Crossref]

1980 (1)

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

Aida, T.

A. Uchida, T. Heil, Y. Liu, P. Davis, and T. Aida, “High-frequency broad-band signal generation using a semiconductor laser with a chaotic optical injection,” IEEE J. Quantum Electron. 39, 1462–1467 (2003).
[Crossref]

Amano, K.

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45, 1367–1379 (2009).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2, 728–732 (2008).
[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, 343–346 (2005).
[Crossref] [PubMed]

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, 343–346 (2005).
[Crossref] [PubMed]

Cao, L. P.

J. G. Wu, G. Q. Xia, L. P. Cao, and Z. M. Wu, “Experimental investigations on the external cavity time signature in chaotic output of an incoherent optical feedback external cavity semiconductor laser,” Opt. Commun. 282, 3153–3156 (2009).
[Crossref]

Chan, S. C.

X. Z. Li and S. C. Chan, “Heterodyne random bit generation using an optically injected semiconductor laser in chaos,” IEEE J. Quantum Electron. 49, 829–838 (2013).
[Crossref]

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, 1930–1935 (2012).
[Crossref]

Chao, Y. K.

F. Y. Lin, Y. K. Chao, and T. C. Wu, “Effective bandwidths of broadband chaotic signals,” IEEE J. Quantum Electron. 48, 1010–1014 (2012).
[Crossref]

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, 343–346 (2005).
[Crossref] [PubMed]

Danckaert, J.

Davis, P.

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45, 1367–1379 (2009).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2, 728–732 (2008).
[Crossref]

A. Uchida, T. Heil, Y. Liu, P. Davis, and T. Aida, “High-frequency broad-band signal generation using a semiconductor laser with a chaotic optical injection,” IEEE J. Quantum Electron. 39, 1462–1467 (2003).
[Crossref]

Deng, T.

Fan, L.

Fischer, I.

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, 343–346 (2005).
[Crossref] [PubMed]

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, 343–346 (2005).
[Crossref] [PubMed]

Goedgebuer, J. P.

He, H.

A. Wang, Y. Wang, and H. He, “Enhancing the bandwidth of the optical chaotic signal generated by a semiconductor laser with optical feedback,” IEEE Photon. Technol. Lett. 20, 1633–1635 (2008).
[Crossref]

Heil, T.

A. Uchida, T. Heil, Y. Liu, P. Davis, and T. Aida, “High-frequency broad-band signal generation using a semiconductor laser with a chaotic optical injection,” IEEE J. Quantum Electron. 39, 1462–1467 (2003).
[Crossref]

Hirano, K.

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45, 1367–1379 (2009).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2, 728–732 (2008).
[Crossref]

Hwang, S. K.

S. K. Hwang, J. M. Liu, and J. K. White, “35-GHz intrinsic bandwidth for direct modulation in 1.3-μ m semiconductor lasers subject to strong injection locking,” IEEE Photon. Technol. Lett.16, 972–974 (2004).
[Crossref]

Inoue, M.

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45, 1367–1379 (2009).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2, 728–732 (2008).
[Crossref]

Kobayashi, K.

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

Kurashige, T.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2, 728–732 (2008).
[Crossref]

Lang, R.

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

Larger, L.

V. S. Udaltsov, L. Larger, J. P. Goedgebuer, A. Locquet, and D. S. Citrin, “Time delay identification in chaotic cryptosystems ruled by delay-differential equations,” J. Opt. Technol. 72, 373–377 (2005).
[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, 343–346 (2005).
[Crossref] [PubMed]

Li, L.

Li, N.

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, and X. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 24, 2187–2190 (2012).
[Crossref]

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, X. Zou, L. Zhang, and P. Mu, “Photonic generation of wideband time-delay-signature-eliminated chaotic signals utilizing an optically injected semiconductor laser,” IEEE J. Quantum Electron. 48, 1339–1345 (2012).
[Crossref]

Li, S. S.

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, 1930–1935 (2012).
[Crossref]

Li, X. Z.

X. Z. Li and S. C. Chan, “Heterodyne random bit generation using an optically injected semiconductor laser in chaos,” IEEE J. Quantum Electron. 49, 829–838 (2013).
[Crossref]

Liao, Y. H.

Y. H. Liao, J. M. Liu, and F. Y. Lin, “Dynamical characteristics of a dual-beam optically injected semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 19, 1500606 (2013).
[Crossref]

W. T. Wu, Y. H. Liao, and F. Y. Lin, “Noise suppressions in synchronized chaos lidars,” Opt. Express 18, 26155–26162 (2010).
[Crossref] [PubMed]

Lin, F. Y.

Y. H. Liao, J. M. Liu, and F. Y. Lin, “Dynamical characteristics of a dual-beam optically injected semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 19, 1500606 (2013).
[Crossref]

F. Y. Lin, Y. K. Chao, and T. C. Wu, “Effective bandwidths of broadband chaotic signals,” IEEE J. Quantum Electron. 48, 1010–1014 (2012).
[Crossref]

W. T. Wu, Y. H. Liao, and F. Y. Lin, “Noise suppressions in synchronized chaos lidars,” Opt. Express 18, 26155–26162 (2010).
[Crossref] [PubMed]

F. Y. Lin and J. M. Liu, “Chaotic radar using nonlinear laser dynamics,” IEEE J. Quantum Electron. 40, 815–820 (2004).
[Crossref]

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

F. Y. Lin and J. M. Liu, “Ambiguity functions of laser-based chaotic radar,” IEEE J. Quantum Electron. 40, 1732–1738 (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, 173–180 (2003).
[Crossref]

Lin, X. D.

Liu, J. M.

Y. H. Liao, J. M. Liu, and F. Y. Lin, “Dynamical characteristics of a dual-beam optically injected semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 19, 1500606 (2013).
[Crossref]

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

F. Y. Lin and J. M. Liu, “Chaotic radar using nonlinear laser dynamics,” IEEE J. Quantum Electron. 40, 815–820 (2004).
[Crossref]

F. Y. Lin and J. M. Liu, “Ambiguity functions of laser-based chaotic radar,” IEEE J. Quantum Electron. 40, 1732–1738 (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, 173–180 (2003).
[Crossref]

J. M. Liu and T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron. 30, 957–965 (1994).
[Crossref]

S. K. Hwang, J. M. Liu, and J. K. White, “35-GHz intrinsic bandwidth for direct modulation in 1.3-μ m semiconductor lasers subject to strong injection locking,” IEEE Photon. Technol. Lett.16, 972–974 (2004).
[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, 1930–1935 (2012).
[Crossref]

Liu, Y.

A. Uchida, T. Heil, Y. Liu, P. Davis, and T. Aida, “High-frequency broad-band signal generation using a semiconductor laser with a chaotic optical injection,” IEEE J. Quantum Electron. 39, 1462–1467 (2003).
[Crossref]

A. Murakami, J. Ohtsubo, and Y. Liu, “Stability analysis of semiconductor laser with phase-conjugate feedback,” IEEE J. Quantum Electron. 33, 1825–1831 (1997).
[Crossref]

Locquet, A.

Luo, B.

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, X. Zou, L. Zhang, and P. Mu, “Photonic generation of wideband time-delay-signature-eliminated chaotic signals utilizing an optically injected semiconductor laser,” IEEE J. Quantum Electron. 48, 1339–1345 (2012).
[Crossref]

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, and X. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 24, 2187–2190 (2012).
[Crossref]

Mark, J.

J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28, 93–108 (1992).
[Crossref]

Mirasso, C. R.

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, 343–346 (2005).
[Crossref] [PubMed]

Mork, J.

J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28, 93–108 (1992).
[Crossref]

Mu, P.

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, X. Zou, L. Zhang, and P. Mu, “Photonic generation of wideband time-delay-signature-eliminated chaotic signals utilizing an optically injected semiconductor laser,” IEEE J. Quantum Electron. 48, 1339–1345 (2012).
[Crossref]

Murakami, A.

A. Murakami, J. Ohtsubo, and Y. Liu, “Stability analysis of semiconductor laser with phase-conjugate feedback,” IEEE J. Quantum Electron. 33, 1825–1831 (1997).
[Crossref]

Naito, S.

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45, 1367–1379 (2009).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2, 728–732 (2008).
[Crossref]

Nguimdo, R. M.

Ohtsubo, J.

A. Murakami, J. Ohtsubo, and Y. Liu, “Stability analysis of semiconductor laser with phase-conjugate feedback,” IEEE J. Quantum Electron. 33, 1825–1831 (1997).
[Crossref]

Oowada, I.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2, 728–732 (2008).
[Crossref]

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, 879–891 (2009).
[Crossref]

Pan, W.

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, X. Zou, L. Zhang, and P. Mu, “Photonic generation of wideband time-delay-signature-eliminated chaotic signals utilizing an optically injected semiconductor laser,” IEEE J. Quantum Electron. 48, 1339–1345 (2012).
[Crossref]

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, and X. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 24, 2187–2190 (2012).
[Crossref]

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, 343–346 (2005).
[Crossref] [PubMed]

Rontani, D.

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, 879–891 (2009).
[Crossref]

D. Rontani, A. Locquet, M. Sciamanna, and D. S. Citrin, “Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback,” Opt. Lett. 32, 2960–2962 (2007).
[Crossref] [PubMed]

Sciamanna, M.

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, 879–891 (2009).
[Crossref]

D. Rontani, A. Locquet, M. Sciamanna, and D. S. Citrin, “Loss of time-delay signature in the chaotic output of a semiconductor laser with optical feedback,” Opt. Lett. 32, 2960–2962 (2007).
[Crossref] [PubMed]

Shiki, M.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2, 728–732 (2008).
[Crossref]

Shore, K. 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, 343–346 (2005).
[Crossref] [PubMed]

Simpson, T. B.

J. M. Liu and T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron. 30, 957–965 (1994).
[Crossref]

Someya, H.

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2, 728–732 (2008).
[Crossref]

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, 343–346 (2005).
[Crossref] [PubMed]

Tang, X.

Tromborg, B.

J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28, 93–108 (1992).
[Crossref]

Uchida, A.

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45, 1367–1379 (2009).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2, 728–732 (2008).
[Crossref]

A. Uchida, T. Heil, Y. Liu, P. Davis, and T. Aida, “High-frequency broad-band signal generation using a semiconductor laser with a chaotic optical injection,” IEEE J. Quantum Electron. 39, 1462–1467 (2003).
[Crossref]

Udaltsov, V. S.

Van der Sande, G.

Verschaffelt, G.

Wang, A.

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

A. Wang, Y. Wang, and H. He, “Enhancing the bandwidth of the optical chaotic signal generated by a semiconductor laser with optical feedback,” IEEE Photon. Technol. Lett. 20, 1633–1635 (2008).
[Crossref]

Wang, B.

Wang, Y.

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

A. Wang, Y. Wang, and H. He, “Enhancing the bandwidth of the optical chaotic signal generated by a semiconductor laser with optical feedback,” IEEE Photon. Technol. Lett. 20, 1633–1635 (2008).
[Crossref]

White, J. K.

S. K. Hwang, J. M. Liu, and J. K. White, “35-GHz intrinsic bandwidth for direct modulation in 1.3-μ m semiconductor lasers subject to strong injection locking,” IEEE Photon. Technol. Lett.16, 972–974 (2004).
[Crossref]

Wu, J. G.

Wu, T. C.

F. Y. Lin, Y. K. Chao, and T. C. Wu, “Effective bandwidths of broadband chaotic signals,” IEEE J. Quantum Electron. 48, 1010–1014 (2012).
[Crossref]

Wu, W. T.

Wu, Z. M.

Xia, G. Q.

Xiang, S.

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, and X. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 24, 2187–2190 (2012).
[Crossref]

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, X. Zou, L. Zhang, and P. Mu, “Photonic generation of wideband time-delay-signature-eliminated chaotic signals utilizing an optically injected semiconductor laser,” IEEE J. Quantum Electron. 48, 1339–1345 (2012).
[Crossref]

Yan, L.

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, X. Zou, L. Zhang, and P. Mu, “Photonic generation of wideband time-delay-signature-eliminated chaotic signals utilizing an optically injected semiconductor laser,” IEEE J. Quantum Electron. 48, 1339–1345 (2012).
[Crossref]

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, and X. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 24, 2187–2190 (2012).
[Crossref]

Yang, Y.

Yoshimori, S.

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45, 1367–1379 (2009).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2, 728–732 (2008).
[Crossref]

Yoshimura, K.

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45, 1367–1379 (2009).
[Crossref]

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2, 728–732 (2008).
[Crossref]

Zhang, B.

Zhang, L.

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, X. Zou, L. Zhang, and P. Mu, “Photonic generation of wideband time-delay-signature-eliminated chaotic signals utilizing an optically injected semiconductor laser,” IEEE J. Quantum Electron. 48, 1339–1345 (2012).
[Crossref]

Zou, X.

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, X. Zou, L. Zhang, and P. Mu, “Photonic generation of wideband time-delay-signature-eliminated chaotic signals utilizing an optically injected semiconductor laser,” IEEE J. Quantum Electron. 48, 1339–1345 (2012).
[Crossref]

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, and X. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 24, 2187–2190 (2012).
[Crossref]

IEEE J. Quantum Electron. (12)

F. Y. Lin and J. M. Liu, “Chaotic radar using nonlinear laser dynamics,” IEEE J. Quantum Electron. 40, 815–820 (2004).
[Crossref]

A. Murakami, J. Ohtsubo, and Y. Liu, “Stability analysis of semiconductor laser with phase-conjugate feedback,” IEEE J. Quantum Electron. 33, 1825–1831 (1997).
[Crossref]

J. Mork, B. Tromborg, and J. Mark, “Chaos in semiconductor lasers with optical feedback: theory and experiment,” IEEE J. Quantum Electron. 28, 93–108 (1992).
[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, 879–891 (2009).
[Crossref]

F. Y. Lin and J. M. Liu, “Ambiguity functions of laser-based chaotic radar,” IEEE J. Quantum Electron. 40, 1732–1738 (2004).
[Crossref]

K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. 45, 1367–1379 (2009).
[Crossref]

A. Uchida, T. Heil, Y. Liu, P. Davis, and T. Aida, “High-frequency broad-band signal generation using a semiconductor laser with a chaotic optical injection,” IEEE J. Quantum Electron. 39, 1462–1467 (2003).
[Crossref]

X. Z. Li and S. C. Chan, “Heterodyne random bit generation using an optically injected semiconductor laser in chaos,” IEEE J. Quantum Electron. 49, 829–838 (2013).
[Crossref]

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

J. M. Liu and T. B. Simpson, “Four-wave mixing and optical modulation in a semiconductor laser,” IEEE J. Quantum Electron. 30, 957–965 (1994).
[Crossref]

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, X. Zou, L. Zhang, and P. Mu, “Photonic generation of wideband time-delay-signature-eliminated chaotic signals utilizing an optically injected semiconductor laser,” IEEE J. Quantum Electron. 48, 1339–1345 (2012).
[Crossref]

F. Y. Lin, Y. K. Chao, and T. C. Wu, “Effective bandwidths of broadband chaotic signals,” IEEE J. Quantum Electron. 48, 1010–1014 (2012).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

Y. H. Liao, J. M. Liu, and F. Y. Lin, “Dynamical characteristics of a dual-beam optically injected semiconductor laser,” IEEE J. Sel. Top. Quantum Electron. 19, 1500606 (2013).
[Crossref]

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

IEEE Photon. Technol. Lett. (2)

A. Wang, Y. Wang, and H. He, “Enhancing the bandwidth of the optical chaotic signal generated by a semiconductor laser with optical feedback,” IEEE Photon. Technol. Lett. 20, 1633–1635 (2008).
[Crossref]

N. Li, W. Pan, S. Xiang, L. Yan, B. Luo, and X. Zou, “Loss of time delay signature in broadband cascade-coupled semiconductor lasers,” IEEE Photon. Technol. Lett. 24, 2187–2190 (2012).
[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, 1930–1935 (2012).
[Crossref]

J. Opt. Technol. (1)

Nat. Photonics (1)

A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics 2, 728–732 (2008).
[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, 343–346 (2005).
[Crossref] [PubMed]

Opt. Commun. (2)

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

J. G. Wu, G. Q. Xia, L. P. Cao, and Z. M. Wu, “Experimental investigations on the external cavity time signature in chaotic output of an incoherent optical feedback external cavity semiconductor laser,” Opt. Commun. 282, 3153–3156 (2009).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Other (1)

S. K. Hwang, J. M. Liu, and J. K. White, “35-GHz intrinsic bandwidth for direct modulation in 1.3-μ m semiconductor lasers subject to strong injection locking,” IEEE Photon. Technol. Lett.16, 972–974 (2004).
[Crossref]

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Figures (7)

Fig. 1
Fig. 1 Schematic setups of chaos generation for Path O: original chaos, Path H1: heterodyned chaos, and Path H2: mixed chaos. SL: semiconductor laser; M: mirror; PD: photode-tector; SG: signal generator; A: microwave amplifier; LPF: low pass filter.
Fig. 2
Fig. 2 (a) Power spectrum, (b) autocorrelation function, and (c) delayed mutual information of the original chaos generated with a feedback strength ξ = 0.09 and a feedback delay time τ = 1 ns after filtered by the acquisition bandwidth. As the reference, the dashed curve in (a) is the power spectrum of the original chaos before filtered by the acquisition bandwidth.
Fig. 3
Fig. 3 (a) Power spectra, (b) autocorrelation functions, and (c) delayed mutual information of the heterodyned chaos (blue solid curves) obtained from the first heterodyne scheme with different fLO (green solid lines): 5.94 GHz (first row), 6.21 GHz (second row), 6.44 GHz (third row), and 6.53 GHz (fourth row). As the reference, the black dashed curves show the (a) power spectrum of the original chaos before filtering, and the (b) autocorrelation function and (c) delayed mutual information of the original chaos after filtering.
Fig. 4
Fig. 4 (a) Absolute amplitudes of the first (bright blue) and second (dark blue) TDS peaks of the autocorrelation functions, (b) ρTDS, and (c) effective bandwidths of the heterodyned chaos under different fLO. The black solid lines in (b) and (c) show the ρTDS and effective bandwidth obtained from the original chaos for reference. The gray dashed lines show the multiples of floop/2.
Fig. 5
Fig. 5 (a) Power spectra, (b) autocorrelation functions, and (c) delayed mutual information of the mixed chaos (red solid curves) obtained from the second heterodyne scheme for different fLO (green solid lines): 5.94 GHz (first row), 6.21 GHz (second row), 6.44 GHz (third row), and 6.53 GHz (fourth row), when A = 0.1 dB. As the reference, the black dashed curves show the (a) power spectrum of the original chaos before filtering, and the (b) autocorrelation function and (c) delayed mutual information of the original chaos after filtering.
Fig. 6
Fig. 6 (a) ρTDS and (c) effective bandwidth of the mixed chaos for different A and fLO, where the arrows on the right of the color bars indicate the ρTDS and effective bandwidth for the original chaos. (b) Minimum ρTDS and (d) maximum effective bandwidths of the mixed chaos obtained under optimal A for different fLO, where the black solid lines in (b) and (d) show the ρTDS and effective bandwidth of the original chaos for reference. The gray dashed lines show the multiples of floop/2.
Fig. 7
Fig. 7 (a) ρTDS and (b) effective bandwidths of the original chaos (black), the heterodyned chaos from the first heterodyne scheme (blue), and the mixed chaos from the second heterodyne scheme (red) under different feedback strengths. The fLO and A are optimized to have the minimum ρTDS in the heterodyned chaos and the mixed chaos.

Equations (7)

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

da dt = 1 2 [ γ c γ n γ s J ˜ n ˜ - γ p ( 2 a + a 2 ) ] ( 1 + a ) + ξ γ c [ 1 + a ( t - τ ) ] cos [ ϕ ( t - τ ) - ϕ ( t ) ]
d ϕ dt = - b 2 [ γ c γ n γ s J ˜ n ˜ - γ p ( 2 a + a 2 ) ] + ξ γ c [ 1 + a ( t - τ ) ] 1 + a sin [ ϕ ( t - τ ) - ϕ ( t ) ]
d n ˜ dt = - γ s n ˜ - γ n ( 1 + a ) 2 n ˜ - γ s J ˜ ( 2 a + a 2 ) + γ s γ p γ s J ˜ ( 2 a + a 2 ) ( 1 + a ) 2
S O ( t ) = LPF [ I ( t ) ] ,
S H 1 ( t ) = LPF [ I ( t ) × cos ( 2 π f LO t ) ] ,
S H 2 ( t ) = LPF [ I ( t ) × ( 1 + A cos ( 2 π f LO t ) ) ] ,
ρ TDS = | ρ 1 | + | ρ 2 | + | ρ 3 | + | ρ 4 | 4

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