## 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 (*ρ*_{TDS}) 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 [1–5]. 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 [10–15]. 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 [10–12]. 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, *J̃* = 1.222 is the normalized bias current, *γ _{c}* = 5.36 × 10

^{11}

*s*

^{−1}is the cavity decay rate,

*γ*= 7.53 × 10

_{n}^{9}

*s*

^{−1}is the differential carrier relaxation rate,

*γ*= 5.96 × 10

_{s}^{10}

*s*

^{−1}is the nonlinear carrier relaxation rate,

*γ*= 1.91 × 10

_{p}^{9}

*s*

^{−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.

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 *f*_{LO} generated by a signal generator as the local oscillator. In the heterodyne process, having *f*_{LO} as a symmetry axis, the chaos spectrum with frequencies lower than the *f*_{LO} will be flipped up to superpose with the spectrum higher than the *f*_{LO}. The superposition is then down-shifted toward the dc with a frequency of *f*_{LO}. As the result, heterodyned chaos signals with different spectral power distributions can be produced depending on the frequency of the sinusoidal signal *f*_{LO} 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

*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 f*_{loop} (*f*_{loop} = 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 *f*_{r} (*f*_{r} = (2*π*)^{−1}(*γ _{c}γ_{n}* +

*γ*)

_{s}γ_{p}^{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

*f*

_{loop}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 [10–12].

By applying the electric heterodyning with different *f*_{LO} 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 (*S*_{H1}). 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 *f*_{LO} 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 *f*_{LO} = 5.94 GHz, which coincides with the 6* ^{th}* frequency peak (6

*f*

_{loop}) in the power spectrum of the original chaos. In the process of heterodyning with

*f*

_{LO}= 6

*f*

_{loop}, the 5

*and 7*

^{th}*frequency peaks will be superposed and down-converted to the frequency of the 1*

^{th}*frequency peak, the 4*

^{st}*and 8*

^{th}*frequency peaks will be superposed and down-converted to the frequency of the 2*

^{th}*frequency peak, and so on. As shown in Fig. 3(a1), the heterodyned chaos still has distinct frequency peaks at the multiples of the*

^{nd}*f*

_{loop}in the spectrum with a separation of

*f*

_{loop}. 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

*f*

_{LO}to 6.21 GHz, locating at about $(6+\frac{1}{4}){f}_{\text{loop}}$ of the original chaos spectrum, the frequency peaks converted to frequencies of $(m\pm \frac{1}{4}){f}_{\text{loop}}$ with a separation of $\frac{1}{2}{f}_{\text{loop}}$ 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

*f*

_{LO}is further increased to 6.44 GHz locating at one of the valley of the original chaos spectrum at $(6+\frac{1}{2}){f}_{\text{loop}}$, the frequency peaks are converted to the frequencies of odd multiples of $\frac{1}{2}{f}_{\text{loop}}$ with a separation of

*f*

_{loop}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

*f*

_{LO}= 6.53 GHz at about $(6+\frac{2}{3}){f}_{\text{loop}}$ of the original chaos spectrum. As can be seen, having the frequency peaks converted to frequencies of $(m\pm \frac{1}{3}){f}_{\text{loop}}$ with separations of $\frac{1}{3}{f}_{\text{loop}}$ and $\frac{2}{3}{f}_{\text{loop}}$ has little effect on the reduction of the TDS. In sum, having an

*f*

_{LO}at $(m+\frac{1}{4}){f}_{\text{loop}}$ 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

*f*

_{LO}at around $(m+\frac{1}{4}){f}_{\text{loop}}$ 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

*f*

_{LO}will be discussed in Fig. 4.

Figure 4(a) shows the absolute amplitudes (*ρ*) of the first (around *τ*) and the second (around 2*τ*) TDS peaks extracted from the autocorrelation functions of the heterodyned chaos obtained from the first heterodyne scheme under different *f*_{LO}. As can be seen, the amplitudes of the first (*ρ*_{1}) and the second (*ρ*_{2}) TDS peaks have their minima in every
$\frac{1}{2}{f}_{\text{loop}}$ and
$\frac{1}{4}{f}_{\text{loop}}$ when the *f*_{LO} 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 (*ρ*_{TDS}) defined as the average of the absolute amplitudes from the first four primary TDS peaks

*τ*, 2

*τ*, 3

*τ*, and 4

*τ*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

*ρ*

_{TDS}and the effective bandwidths of the heterodyned chaos under different

*f*

_{LO}, respectively. The gray dashed lines indicate the multiples of $\frac{1}{2}{f}_{\text{loop}}=495\hspace{0.17em}\text{MHz}$. As the benchmarks, the black solid lines in Figs. 4(b) and 4(c) show the respective

*ρ*

_{TDS}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

*ρ*

_{TDS}and the effective bandwidth are in accordance with half of the

*f*

_{loop}. When ${f}_{\text{LO}}=m\cdot \frac{1}{2}{f}_{\text{loop}}$, as those shown in Figs. 3(a1) and 3(a3) as examples, the frequency peaks on the opposite sides of the

*f*

_{LO}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

*ρ*

_{TDS}and smaller bandwidths than the original chaos are seen as shown in Figs. 4(b) and 4(c), respectively. When the

*f*

_{LO}is applied at around $(m+\frac{1}{4}){f}_{\text{loop}}$ or $(m+\frac{3}{4}){f}_{\text{loop}}$, as that is shown in Fig. 3(a2) as an example, the frequency peaks converted to frequencies at $(m+\frac{1}{4}){f}_{\text{loop}}$ and $(m+\frac{3}{4}){f}_{\text{loop}}$ with a closer separation of $\frac{1}{2}{f}_{\text{loop}}$ after the heterodyne process. As can be seen in Figs. 4(b) and 4(c), under such condition, smaller

*ρ*

_{TDS}and larger effective bandwidths are simultaneously achieved. With

*f*

_{LO}= 6.21 GHz, the

*ρ*

_{TDS}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 (*S*_{H2})(red solid curves) with different *f*_{LO} at *A* = 0.1 dB. Similar to Fig. 3, the *f*_{LO} at about 6 *f*_{loop},
$(6+\frac{1}{4}){f}_{\text{loop}}$,
$(6+\frac{1}{2}){f}_{\text{loop}}$, and
$(6+\frac{2}{3}){f}_{\text{loop}}$ 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 *f*_{LO} = 5.94 GHz at the 6* ^{th}* 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

*f*

_{LO}is increased by $\frac{1}{4}{f}_{\text{loop}}$ to 6.21 GHz, the frequency peaks of the heterodyned chaos at $(m+\frac{1}{4}){f}_{\text{loop}}$ (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

*f*

_{LO}to $(6+\frac{1}{2}){f}_{\text{loop}}\hspace{0.17em}=\hspace{0.17em}6.44\hspace{0.17em}\text{GHz}$, 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

*f*

_{LO}at about $(6+\frac{2}{3}){f}_{\text{loop}}=6.53\hspace{0.17em}\text{GHz}$, 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.

For the mixed chaos signals obtained in the second heterodyne scheme, Figs. 6(a) and 6(c) show the *ρ*_{TDS} and the effective bandwidths for different amplification factors *A* and *f*_{LO}. Arrows next to the color bars of the *ρ*_{TDS} 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 *ρ*_{TDS} and large effective bandwidths are seen in Figs. 6(a) and 6(c), respectively. As can be seen, low *ρ*_{TDS} 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 *ρ*_{TDS} 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.

Extracted from Figs. 6(a) and 6(c) under optimal *A*, Figs. 6(b) and 6(d) show the minimum *ρ*_{TDS} and the maximum bandwidths that can be obtained at different *f*_{LO}. The gray dashed lines again indicate the multiples of
$\frac{1}{2}{f}_{\text{loop}}=495\hspace{0.17em}\text{MHz}$. As can be seen, the periods of the variations in both the *ρ*_{TDS} and effective bandwidth are now in accordance with the *f*_{loop}. Unlike the first heterodyne scheme as shown in Figs. 4(b) and 4(c), practically for all different *f*_{LO}, the *ρ*_{TDS} 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
${f}_{\text{LO}}=(m+\frac{1}{3}){f}_{\text{loop}}$ and
${f}_{\text{LO}}=(m+\frac{2}{3}){f}_{\text{loop}}$, where the frequency peaks and valleys in the heterodyned chaos and the original chaos complement each other the most. With *f*_{LO} = 6.53 GHz and *A* = 0.1 dB, the *ρ*_{TDS} 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 *ρ*_{TDS} and the effective bandwidths obtained from the original chaos, heterodyned chaos, and the mixed chaos under different feedback strengths. For the hetero-dyned chaos, *f*_{LO} are optimized to have the minimum *ρ*_{TDS}. For the mixed chaos, both *f*_{LO} and *A* are optimized to have the minimum *ρ*_{TDS}. As can be seen in Figs. 7(a) and 7(b), the *ρ*_{TDS} 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 *ρ*_{TDS} 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 *ρ*_{TDS} and enhancement of 46% in the effective bandwidth for different feedback strengths are demonstrated. At *ξ* = 0.09, the mixed chaos has a lowest *ρ*_{TDS} 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.

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 *f*_{LO} 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 *ρ*_{TDS} 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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