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Abstract
Common-signal-induced laser synchronization promoted a promising paradigm of high-speed physical key distribution. Constant-amplitude and random-phase (CARP) light was proposed as the common drive signal to enhance security by reducing the correlation between the drive and the laser response in intensity. However, the correlation in light phase is not examined. Here, we numerically reveal that the correlation coefficient of the CARP light phase and the response laser intensity (denoted as CCR-φD) can reach a value close to 0.6. Effects of parameters including optical frequency detuning, and modulation depth and noise bandwidth and transparency carrier density for CARP light generation are investigated in detail. By optimizing the optical frequency, modulation depth, and noise bandwidth, respectively, CCR-φD can be reduced to 0.32, 0.18, and 0.10. In the meantime, CCR-φD can be further reduced through secondary optimizing of parameters. CCR-φD can be further reduced by increasing transparent carrier density provided response laser synchronization is achieved. This work gives a new insight about the laser synchronization induced by common CARP light, and also contributes a suggestion of security improvement for physical key distribution based on laser synchronization.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Synchronization of laser complex dynamics has attracted extensive attention not only due to fundamental physics but also due to the significant applications in chaos secure communications [1–7] or physical key distributions [8–18]. The laser-synchronization-based secure communications and physical key distribution have advantages in high speed and compatibility with existing fiber optic communication links. Such as, Argyris et al. realized chaos secure communication over 120 km of optical fibre in the metropolitan area network of Athens, Greece [1]. Sasaki et al. controlled the chaotic synchronization state by adjusting the feedback phase of two semiconductor lasers and realized key generation at a rate of 182 kb/s [8]. Porte et al. made chaotic key generation rate increase to Mb/s by using delay-coupled semiconductor lasers [9]. Another system, proposed by Gao et al. [10] considered mode-shift keying chaos synchronization to generate the key rate of 0.75 Gb/s.
Injection-locking laser synchronization and common-signal-induced laser synchronization are the two primary methods for achieving chaotic synchronization. The drive signal is highly consistent with the response signal for the injection-locking laser synchronization. The latter generates chaotic synchronization based on the consistent response of the laser outputs, and the drive signal can be inconsistent with the response signal to avoid the leakage of information of laser dynamics from transmission link. The correlation between the drive signal and the laser response signal, denoted as drive-response correlation, is related to the kind of drive signal. For chaotic drive light, the drive-response correlation is usually about 0.46 ∼ 0.60 [11,12,19,20], and for amplified spontaneous emission (ASE) light, it is close to 0.30 [10,21]. Aida et al. proposed using constant-amplitude random-phase (CARP) light as drive signal to realize the chaotic synchronization in a closed-loop structure and reduce drive-response correlation, in which the CARP light is generated by using an electrical noise signal to modulate the phase of a continuous-wave light. A low drive-response correlation of only 0.16 was experimentally achieved [22]. Since common-signal-induced laser synchronization yields higher key rates in open-loop structures than in closed-loop structures, open-loop structures in this context have been proposed recently [10,19,21]. The security of chaotic synchronization induced by CARP light in an open-loop system has not been studied yet.
In this paper, we numerically found that under the open-loop laser synchronization induced by CARP light, there is a certain correlation between the phase of the drive signal and the intensity of the response signal, although there is nearly no correlation between the drive signal and the response signal in light intensity. It's noted that the phase of the CARP light can be extracted by optical heterodyning or coherent detection [23]. If the drive light phase has a high correlation to the response laser intensity, there is still a risk of information leakage of laser dynamins even though the drive-response correlation in light intensity is very low.
Accordingly, we numerically investigated the correlation between the CARP drive light phase and the output intensity of two distributed feedback (DFB) semiconductor lasers under common-signal-induced synchronization configuration. The effects of parameters including noise bandwidth, modulation depth, and optical frequency detuning on the correlation between the drive phase and the laser intensity are analyzed. The results show that the correlation also can reach a high value close to 0.6, and fortunately it can be reduced to near 0.1 by parameter optimization.
2. Configuration of laser synchronization induced by CARP light
Figure 1 shows the schematic diagram of the laser synchronization induced by a CARP light. The CARP light is generated by modulating the phase of a continuous-wave light with a noise signal. The noise bandwidth is denoted as Bn, and the phase modulation depth is defined as m = 2σ/Vπ, where σ is the standard deviation of the noise amplitude, and Vπ is the half-wave voltage of the phase modulator. The CARP drive light is split into two beams which are separately injected into two DFB lasers (denoted as response lasers) to induce chaos synchronization. Injection strength Kinj is defined as the power ratio of the injection light to the output power of the free-running laser. Optical frequency detuning Δυ = υcw−υ0, where υcw and υ0 are the center frequencies of the continuous-wave light and of the free-running DFB laser, respectively. By adjusting Kinj and Δυ, synchronous laser dynamics can be achieved.
Fig. 1. Schematic diagram of synchronization of DFB lasers commonly injected by constant-amplitude and random-phase (CARP) light. CW laser: continuous-wave laser; PM: phase modulator; VOA: variable optical attenuator; OC: optical coupler; OI: optical isolator; DFB SLs: distributed feedback semiconductor lasers.
In simulations, it’s used transmission line laser model (TLLM) in VPIcomponentMaker software to simulate the propagation of the electro-magnetic field in the composite optical cavity [24–26]. The travelling wave fields in the laser’s cavity are separated to the small portions of length ΔL in a longitudinal z-direction. And the travelling wave fields are calculated at small time steps Δt for each section. The propagating electric field $E(r,t)$ in each TLLM section is defined as:
A(z,t) and B(z,t) have the characteristics of slow propagation in time, but fast propagation in space. Hence, the amplitudes a(z,t) and b(z,t) which are slowly varying in both time t and z coordinates are defined as:
where β0 = 2πf0/c${\cdot}$neff (f0) is the propagation constant of the laser models. And neff is the complicated effective refractive index that varies with frequency. And δgrating = (2πfBragg/c)${\cdot}$neff (f0) is defined as the grating-induced detuning where f0 is the propagation frequency, f Bragg is the offset of the grating stop-band center frequency.Substituting Eqs. (2)-(3) to the Maxwell equations and considering the carrier dynamics, one can derive the traveling-wave equations as follows:
The variations in carrier density over time and the impact of carrier density on laser gain are considered in Eqs. (6). The constant η, I, e and V are the current injection efficiency, the bias current injected into the modeled device, the electron charge and the active region volume, respectively. And V can be calculated as a product of the active region width w, thickness d, and length L. R(N) = AN + BN2 + CN3 represents the recombination rate of carriers described by the linear, A, bimolecular, B, and the Auger, C, recombination coefficients. gpeak(N) = alin${\cdot}$(N-N0) determines the carrier dependence of the gain factor at the peak of the gain curve where alin is the linear material gain coefficient and N0 is the transparent carrier density. And ε represents gain compression factor. Its physical origin can be attributed to various phenomena like cavity standing wave dielectric grating, spectral hole burning, transient carrier heating, or to some combination of the above effects.
The average photon density S(t) within the modeled transmission-line model section of length Δz is determined from the forward and backward propagating optical fields as
And the boundary condition for the forward and backward propagating waves at the facets can be written as:
where rL and rR are the facet reflection coefficient at z = 0 and z = L, respectively. Einj(t) is the travelling waves of the drive signal. The drive signal changes the boundary conditions of the laser forward traveling wave, which further affects the output of the laser A(z = 0,t). By the way, the phase and amplitude information of the output signal can be obtained by extracting the angle and modulus of A(z = 0,t).Although the parameter settings of the two response lasers are the same, the spontaneous emission noise Qa(b) of the lasers are independent Gaussian white noise, which are generated by random number seeds. Qa(b) is defined as:
Calculations of the spontaneous emission are based on the gain due to stimulated emission, and enhanced by the inversion parameter nsp. X(z,t) and Y(z,t) are generated by a random number algorithm VPIcomponentMaker software. Each time X(z,t) and Y(z,t) are generated, the system will randomly generate algorithm seeds to ensure the randomness of X(z,t) and Y(z,t). Psp(N) is the power that measures the size of the spontaneous emission noise and depends on the carrier density within the active region of the modeled section. G(N), v and αint are the net gain in linear units, the spontaneous emission noise frequency and the loss per unit length due to waveguide scattering and FC absorption, respectively. Note that, the primary element influencing laser synchronization when the response laser's intrinsic parameters are consistent is the non-linearity caused by the spontaneous emission noise within the laser.
The parameters used in the simulation are set by the default parameter of the DFB laser model in VPIcomponentMaker software. And the laser parameters are listed in Table 1. The laser currents were set to I = 1.3Ith where the threshold current Ith is 20 mA. The optical frequencies of the free-running DFB lasers were set as υ0 = 193.1000 THz (wavelength 1553.60 nm). The laser synchronization and the correlation between the laser output and the drive signal are estimated by the correlation coefficient. The correlation coefficient of signals x(t) and y(t) is defined as the maximum of the normalized cross-correlation function [12]
Table 1. Parameter values of DFB lasers used in simulations
3. Correlation between drive light and response laser
Figure 2 demonstrates results of the CARP-light-driven laser synchronization and the drive-response correlation, which was obtained with Bn = 3 GHz, m = 1, Kinj = 0.50, and Δυ = 0 GHz. Figure 2(a) plots the intensity waveforms of the CARP drive light and the two response lasers. Obviously, the two response lasers have almost identical intensity waveforms and they are very different from the drive light intensity which is constant with a slight noisy fluctuation. The corresponding scatterplots in Figs. 2(c) and 2(e) clearly displays the high-quality synchronization with CCR1-R2 = 0.989 and a low drive-response correlation with CCR1-D = 0.110 in light intensity. This result is similar to Ref. [22]. The phase waveforms are obtained by extracting the phase angle and unwrapping the phase and normalizing by sequence maximum value. Figure 2(b) plots the phase waveforms of the CARP light and the two response lasers. Figure 2(d) shows that the response lasers also have a high-quality synchronization with CCφR1-φR2 = 0.994 in optical phase. Figure 2(f) displays a high correlation between the CARP drive light and response laser light with CCφR1-φD = 0.973 in optical phase.
Fig. 2. (a) Temporal waveforms of noise signal and the drive light intensity ID, response lasers outputs IR1 and IR2; (b) Temporal waveforms of the drive light phase φD, and the response lasers’ phase φR1 and φR2; Correlation plots between (c) IR1 and IR2, (d) φR1 and φR2, and (e) ID and IR1. (f) φR1 and φD.
Note that the laser phase dynamics is coupled with intensity dynamics under external perturbation such optical injection or feedback. As a result, the laser intensity waveform has a certain correlation with the laser phase waveform. Therefore, a high phase correlation CCφR1-φD in Fig. 2(f) indicates that the drive light phase has a correlation with the laser intensity. Figures 3(a) and 3(b) plot the scatterplot and cross-correlation curve between the drive light phase and the intensity of the response laser DFB SL1, respectively. One can find the correlation coefficient CCR1-φD is 0.521. This means that the eavesdropper can extract partial information from the drive light phase. Due to high laser synchronization, CCR1-φD is very close to CCR2-φD. Thus, in the following section, we use CCR1-φD to represent the drive-response correlation and investigate the effects of parameters to find parameter optimization for suppressing it.
Fig. 3. (a) Correlation plots and (b) cross-correlation curve between the drive light phase φD and the response laser intensity IR1.
4. Effects of parameters on correlation of response laser intensity and drive light phase (CCR-φD)
4.1 Optical frequency detuning
We first show the effects of optical frequency detuning Δυ on CCR1-φD under Bn = 3 GHz and m = 1. Figure 4(a) displays correlation coefficients CCR1-R2 and CCR1-φD as function of Δυ obtained with Kinj = 0.2. Shown in dots, high-quality laser synchronization with CCR1-R2 ≥ 0.90 is achieved within a range of Δυ from −8.2 GHz ∼ 2.3 GHz. The asymmetry of the detuning range is due to the injection-induced red shift of laser frequency. Within this detuning range, the drive-response correlation CCR1-φD shown in squares keeps larger than 0.45 for negative detuning and then reaches its maximum close to 0.60 as Δυ increases to around -2 GHz. Further increase of detuning quickly reduces CCR1-φD. The minimum value of CCR1-φD is 0.32 under the condition of laser synchronization CCR1-R2 ≥ 0.90. The corresponding frequency detuning is about 2.3 GHz, which is denoted as the optimum value Δυopt. It is readily understood that Δυopt and the corresponding CCR1-φD depend on the injection strength. As shown in Fig. 4(b), the optimum detuning Δυopt increases with growth of Kinj, and CCR1-φD fluctuates within a small range around 0.32.
Fig. 4. (a) Influences of Δυ on CCR1-R2 (red dot) and CCR1-φD (blue square) under Kinj = 0.2. Optimum detuning Δυopt is obtained by the minimum CCR1-φD within the range of Δυ making CCR1-R2 ≥ 0.9. (b) Δυopt (red inverted triangles) and the corresponding CCR1-φD (blue square) as function of injection strength Kinj. (c)-(d) Optical spectra of the CARP drive light and the response laser under Δυ = −2 GHz and 5 GHz, respectively. Bn = 3 GHz, m = 1.
To understand the physical reason of the optical frequency detuning effects, we plot the optical spectra of the CARP drive light and the response laser in Figs. 4(c) and 4(d) with Δυ = −2 GHz and 5 GHz, respectively. The corresponding CCR1-φD values are 0.57 and 0.16, respectively. By comparison, one can find that for the larger detuning the optical spectrum of the CARP drive light is less overlapped with that of the response laser, and this leads to a lower CCR1-φD. Due to the injection-induced red shift of laser spectrum, increase of positive detuning can quickly reduce the spectrum overlapping and thus suppress the correlation between the drive phase and the response laser intensity.
4.2 Modulation depth
Then we show the effects of modulation depth m on CCR1-φD under Bn = 3 GHz and Δυ = 0 GHz. Figure 5(a) displays correlation coefficients CCR1-R2 and CCR1-φD as function of m obtained with Kinj = 0.2. Shown in dots, high-quality laser synchronization with CCR1-R2 ≥ 0.90 is achieved within a range of m from 0.32 ∼ 3.20. Within this modulation depth range, the drive-response correlation CCR1-φD shown in squares first increases its maximum close to 0.60 as the modulation depth grows up to m = 1.2. Further increase of modulation depth quickly reduces CCR1-φD. The minimum value of CCR1-φD is 0.26 under the condition of laser synchronization CCR1-R2 ≥ 0.90. The corresponding modulation depth is about 2.6, which is denoted as the optimum value mopt. As shown in Fig. 5(b), the optimum modulation depth mopt increases with growth of Kinj, and CCR1-φD fluctuates within a small range around 0.18. Figure 5(c) shows the optimum mopt and the optimum CCR1-φD as function of injection strength King under Bn = 3 GHz and Δυ = 2.3 GHz. Note that, 2.3 GHz is the optimum Δυ value in Fig. 4(a). It’s clearly that with the increase of injection strength, the optimum CCR1-φD value decreases and eventually fluctuates at close to 0.1. Compare with the results in Fig. 5(b), it’s known that secondary optimization can further reduce the value of CCR1-φD and enhance the security of the system.
Figures 5(d) and 5(e) display the optical spectra of the CARP drive light and the response laser obtained with m = 1.2 and m = 4, respectively. The corresponding CCR1-φD values are 0.55 and 0.18. As m increases from 1.2 to 4, the spectrum of CARP light is obviously widened from 5.66 GHz to 14.84 GHz, and simultaneously the peak power decreases about 1.91 dB. By comparison, the sideband of the response laser spectrum is enhanced. However, the spectral main peak has little change. The peak power has a slight decrease of 0.40 dB and the 3-dB linewidth 2.34 GHz. Therefore, the laser spectrum has a lower similarity to that of the CARP light, leading to a smaller CCR1-φD value. Note that, the decrease of the spectral peak power of the CARP light is equivalent to the reduction of the injection strength to the laser mode. Thus, as m exceeds a certain value (e.g., m = 3.2 in Fig. 5(a)), the CARP light cannot drive the two lasers synchronized, due to that the effective injection strength is smaller than its critical value making laser synchronization [22].
Fig. 5. (a) Influences of m on CCR1-R2 (red dot) and CCR1-φD (blue square) under Kinj = 0.2; (b)-(c) optimum mopt (red inverted triangles) and the corresponding CCR1-φD (blue square) as function of injection strength Kinj under Δυ = 0 GHz and Δυ = 0 GHz, respectively; (d)-(e) The optical spectra of the CARP drive light and the response laser under m = 1.2 and 4, respectively. Bn = 3 GHz and Δυ = 0 GHz.
4.3 Noise bandwidth
In this subsection, the effect of noise bandwidth Bn on CCR1-φD is shown under m = 1 and Δυ = 0 GHz. Figure 6(a) displays correlation coefficients CCR1-R2 and CCR1-φD as function of Bn obtained with Kinj = 0.2. Shown in dots, laser synchronization with CCR1-R2 ≥ 0.90 is achieved as Bn ranging from 2.1 GHz to 10.5 GHz. Within this noise bandwidth range, the drive-response correlation CCR1-φD (squares) first increases with the noise bandwidth and reaches its maximum close to 0.50 as Bn = 4 GHz. Further increase of noise bandwidth results in quick decrease in CCR1-φD. The minimum value of CCR1-φD is 0.24 under the condition of laser synchronization CCR1-R2 ≥ 0.90. The corresponding optimum noise bandwidth Bn-opt is about 10.2 GHz. Figures 6(c) and 6(d) display the optical spectra of the CARP drive light and the response laser with Bn = 4 GHz and Bn = 15 GHz, respectively. The corresponding CCR1-φD values are 0.50 and 0.13. Clearly, the optical spectra in these two figures are very similar to Figs. 5(c) and 5(d). The increase of noise bandwidth drives the signal spectrum to widen, which also reduces the injection strength of the laser mode. In this case, the CCR1-φD and the synchronization of the response lasers will decrease further. Shown in Fig. 6(b), the optimum bandwidth Bn-opt increases with growth of Kinj. CCR1-φD decreases with growth of growth and finally fluctuates within a small range around 0.10.
Fig. 6. (a) Influences of Bn on CCR1-R2 (red dot) and CCR1-φD (blue square) under Kinj = 0.2; (b) optimum Bn-opt (red inverted triangles) and the corresponding CCR1-φD (blue triangles) as function of injection strength Kinj; (c)-(d) optical spectrum of the CARP drive light and the response laser under Bn = 4 GHz and 15 GHz, respectively. m = 1, and Δυ = 0 GHz.
4.4 Transparent carrier density
Next, we assessed the impact of transparency carrier density N0 mismatch on the response laser synchronization and the correlation between the drive light phase and the response laser intensity under Kinj = 0.20, Bn = 3 GHz, Δυ = 0 GHz and m = 1. The transparent carrier density of both lasers is set to 1.5 × 1024 m-3 while the transparent carrier density of DFB SL2 remain unchanged and only the carrier density of DFB SL1 is adjusted. Figure 7 shows that CCR1-φD decreases with the increase of transparent carrier density N0 and the response lasers in the negative mismatch ratio of positive mismatch is easier to realize chaotic synchronization. The increase of the transparent carrier density will decrease the peak of the laser’s gain curve, which prevents the laser from responding well to the injected signal. In this case, the reduction in the laser gain curve is approximately equivalent to the reduction in the injection strength of the laser mode. Therefore, CCR1-φD has a lower correlation coefficient value when the transparency carrier density is positively mismatched.
Fig. 7. Influences of transparent carrier density N0 mismatch of the DFB SL1 on CCR1-R2 (red inverted triangles) and CCR1-φD (blue square) and CCR2-φD (orange dots) under Kinj = 0.2, m = 1, Δυ = 0 GHz and Bn = 3 GHz; The gray rectangular area represents the synchronization range where CCR1-R2 ≥ 0.9.
CCR1-φD cannot reduced by increasing the transparent carrier density all the time, due to the synchronization of the response lasers. Figure 7 displays that in the synchronization range, by increasing the carrier density, the CCR1-φD can only be reduced by close to 0.1 compared with the zero mismatch of the transparency carrier density.
5. Conclusion
In conclusion, CARP light for the common-signal-induced laser synchronization system could enhance secure by reducing the correlation between the drive and the laser response in intensity, but we numerically demonstrated that the correlation coefficient of the CARP light phase and the response laser intensity (denoted as CCR-φD) can reach a value close to 0.6. Meanwhile, we analyzed the effects of optical frequency detuning, modulation depth, noise bandwidth and transparent carrier density on CCR-φD in detail. Reducing CCR-φD to 0.32, 0.18 and 0.10 can be achieved by optimizing the optical frequency, modulation depth, and noise bandwidth, respectively. Meanwhile, CCR-φD can be further decreased through secondary optimizing of parameters. The decrease of laser transparent carrier density can reduce CCR-φD. Under the condition of response laser synchronization, increasing transparent carrier density can reduce the CCR-φD by 0.1.This work provides a new perspective to analyze the security of common-signal-induced laser synchronization systems, which is beneficial to enhance the security of physical key distribution based on laser synchronization.
Funding
National Key Research and Development Program of China (2020YFB1806401); National Natural Science Foundation of China (61927811, 62035009, U22A2087); Program for Guangdong Introducing Innovative and Enterpreneurial Teams.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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