1. Introduction

Laser radar is generalized widely to apply in target detection, imaging, ranging, speed measurement and other fields [1–6]. The laser radars (lidars) for target ranging are mainly emitted from pulse laser and modulated continuous wave (CW) laser. In the short-pulse ranging techniques, the resolution is usually in the range of several meters, mainly determined by the pulse bandwidth [7, 8]. However, short pulse technology has many shortcomings, such as high manufacturing cost, bulky, heavy, low efficiency, and difficult to operate and maintain [9, 10]. In comparison, the modulated CW laser has many advantages such as compact size, low cost, high efficiency, easy operation and so on. However, the ranging resolution of the laser is in the range of tens of meters since that it is limited by the code rate and modulation speed [11-12]. To achieve a better resolution, an expensive external modulator has to be used for high-speed modulation. The chaotic radar ranging has many advantages over short-pulse and CW laser ranging, such as low probability of intercept, high range and velocity resolution, strong anti-interference ability, easy generation and low cost [13–24]. And it plays an important role in modern radar research.

The chaotic lidar (CLIDAR) generated by using the nonlinear dynamics of semiconductor laser has a higher resolution and is easy to generate and amplify, benefiting from the broad bandwidth of the optical chaos [13, 14]. By using the CLIDAR, the resolution of centimeter magnitude can be easily achieved. The CLIDAR for target ranging has an obvious advantage over CW lidar, such as that it does not need the generator for high-rate code and the modulator with high speed. In 2001, based on the semiconductor laser with optical feedback, Myneni K et al. firstly proposed a CLIDAR ranging scheme [4], which expands a new application of optical chaos in measuring range and velocity of target. In the next few years, Jia-Ming Liu and his associates considered a chaotic laser as a light source, and proposed a novel chaotic radar system [19]. In the system, using the correlation between the signal wave fed back from the measured target and a delayed reference wave, they realized the ranging for target, and achieved stably resolution in centimeter. In the same year, they experimentally achieved the target ranging, and the resolution of the ranging is of 9 cm [20]. In 2008, considering two plane mirrors as two-target, Bingjie Wang and his associates demonstrated the reliability of real-time ranging of the two targets by using CLIDAR, and implemented a resolution of 9 cm between two targets [21]. In 2015, they experimentally explored the target detection and ranging technology in loss media [22]. Moreover, Pengfei Du et al. experimentally realized the remote target ranging by using CLIDAR [23]. So far, as we know, most of CLIDAR ranging schemes are realized by using the correlation between the signal waveform and the reference waveform [13–24]. In these schemes, the ranging resolution can reach several centimeters, and be further improved by using the correlation principle if laser chaotic bandwidth can be further enhanced. Recently, some schemes for laser chaotic bandwidth enhancement have been experimentally and theoretically implemented [25–32]. For example, the chaos bandwidth of 14GHz is experimentally generated by optical heterodyning of two external-cavity laser diodes [27-28]. Optical chaos extending up to 26.5 GHz bandwidth has been achieved by injecting the chaos light from a DFB laser with conventional feedback into a fiber ring resonator [29]. The chaos bandwidth can be extended up to 20GHz by using optical injection [30], and further enlarged up to 32.3 GHz using dual wavelength injection [31]. Replacing conventional optical feedback with phase-conjugate feedback can improve the chaos bandwidth. In the range of achievable phase-conjugate mirror reflectivities, the bandwidth can be further enlarged up to about 12GHz, and its increase reaches 27% [32]. However, how much range resolution can be improved by enhancing the chaotic bandwidth needs to be further explored. On the other hand, due to the interference of the spontaneous emission noise and channel noise, the quality of the correlation can be weakened, which seriously affect the accuracy of the target ranging. Based on the principle of complete chaos synchronization (CCS), these problems can be solved when the chaotic radar in a system of the drive-response lasers is applied in target ranging. The reason is that the resolution of target ranging mainly depends on the CCS quality of the drive-response lasers. The distance of the target at arbitrary position can be measured under steady CCS. It is noted that the CCS quality of the drive-response lasers has very strong robustness to noise [33-34]. The complete synchronization can be used to play the role of the noise filter. Therefore, the noise has little effect on target ranging. However, in the system of the drive-response VCSELs with the spin flip mode (SFM) [33–35], since that the random anticorrelated hopping between the two orthogonal polarization components (PC) emitted by each VCSEL is induced by the feedback [36] or the injection current [37], the simultaneously chaotic synchronization for each polarization component (PC) is extremely unstable and very poor, making the resolution to be reduced seriously, and the real-time stability to be poor in target ranging. In our previous work [38-39], we proposed a novel scheme to control the CCS quality in the drive-response VCSELs by electro-optic (EO) modulation, where the CCS quality of each PC appears periodic variation and can be stably controlled simultaneously when the applied electric field is fixed at a certain value. Motivated by these, according to the principle of CCS, the theory of Hilbert phase transform and linear electro-optical (EO) effect, we present a novel scheme of real-time multi-target ranging by using chaotic polarization lidar (CPLIDAR) in the system of the drive-response VCSELs, where two PCs can be achieved stability and well CCS by using linear EO effect in periodically poled LiNbO₃ (PPLN) crystal. Under the completely synchronized condition, we further explore the resolution, the real-time stability, and the relative errors of two target ranging.

2. Theory and model

Figure 1 presents the scheme diagram of multi-target ranging by using CPLIDAR in the drive-response VCSELs. Here, the upper part is the drive system; the lower part is the response system; the master VCSEL (MCSEL) is drive laser; the slave VCSEL (SVCSEL) is response laser. In the drive system, with the bias current and the feedback strength fixed at certain value, the MVCSEL emits two chaotic PCs [see Eqs. (1) - (3)]. Since that their polarized direction along the direction of the x-axis and the y-axis, they are, respectively, defined as the x-PC_M and the y-PC_M, where the subscript M means the MVCSEL. The feedback cavity for the y-PC_M is composed of the mirror 1 (M₁), M₂ and M₆ (FC1); the feedback cavity for the x-PC_M is formed by the M₃-M₅ (FC2). The microwave signal m₁ (m₂) is modulated to the x-PC_M (y-PC_M) by the MOD₁ (MOD₂), where the MOD is EO modulator. The neutral density filter 1 (NDF₁) and the NDF₂ are used to adjust the optical feedback strength. The transmitting x-polarization optical antenna (TXPOA) is used to transmit the chaotic x-PC_M with m₁ (the probe signal 1). The chaotic y-PC with m₂ (the probe signal 2) is transmitted by the transmitting y-polarization optical antenna (TYPOA). In the response system, the chaotic x-PC_M’ and y-PC_M’ are from the receiving x-polarization optical antenna (RXPOA) and the receiving y-polarization optical antenna (RYPOA), respectively. The optical amplifier 1 (OA₁) and the OA₂ are used to compensate the loss of the probe signal 1and the probe signal 2 in the antennas and the channels. Under certain conditions, the SVCSEL can emit two PCs, which are defined as the x-PC_S and the y-PC_S [see Eqs. (4) - (6)]. Here, the subscript S denotes the SVCSEL.

 

Fig. 1 Scheme diagram of the multi-target ranging by using CPLIDAR based on the drive-response VCSELs. Here, M: mirror; PPLN: periodic poled LiNbo₃; IS: isolator; FR: Faraday rotator; HWP: half wave plate; MVCSEL: master VCSEL; SVCSEL: slave VCSEL; u_m and u_s: the bias current of MVCSEL and SVCSEL, respectively; E₁ and E₂: applied electric field; BS: beam splitter; PBS: polarization beam splitter; m₁ and m₂: modulated microwave signal;$m_1^{\prime }$ and$m_2^{\prime }$: demodulated microwave signals; MOD: modulator; TYPOA: transmitting y-polarization optical antenna; TXPOA: transmitting x-polarization optical antenna; RXPOA: receiving x-polarization optical antenna; RYPOA: receiving y-polarization optical antenna; NDF: neutral density filter; DEL: delayer; F: filter; HT: Hilbert transform; PD: photodiode; CD: distance calculation; DIV: divider; SUB: subtractor; OA: optical amplifier.

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In the system, how the CPLIDAR signals are transmitted and received in space needs to be further explained as follows. So far, typically, the CPLIDAR signals are transmitted and received by radio antenna. Based on this traditional approach, each chaotic PC should be converted to radio signal by one photodiode in the drive system. To realize the CCS, in the response system, the radio signal reflected from each target should be further converted to the CPLIDAR signal that is almost similar to that from the M-VCSEL, which is very difficultly implemented in technology. Recently, using the surface plasmon theory, several researchers have experimentally and theoretically implemented a variety of optical nanoantennas that could efficiently convert confined optical energy to free-space light, and vice versa [40–43]. Some polarized optical nanoantennas have been further proposed in experiment and theory [42-43]. Therefore, it is possible to solve the above-mentioned problem by using optical antenna. For this purpose, we choose optical antenna to transmit and receive CPLIDAR signal. On the other hand, to avoid the mutual interference between the two probe signals when they are transmitted in space, and received by the RXPOA and RYPOA simultaneously, we consider the TXPOA and the TYPOA to transmit the probe signals 1 and 2, respectively, and use the RXPOA and the RYPOA to receive two reflected probe signals from the T₁ and T₂, respectively [see Fig. 1]. Since that the polarization of the two probe signals are perpendicular to each other, the mutual interference between them transmitting in space can be suppressed. Moreover, the RXPOA and the RYPOA can isolate the probe signal 2 and the probe signal 1, respectively, owing to the reason that the polarization of the two receiving antennas are perpendicular to each other.

In the following, we consider the probe signal 1 as an example to illustrate the implementation of the target 1 (T₁) ranging. The probe signal 1 is split into two beams by the beam splitter 1 (BS₁). One of two beams is firstly fed back by the FC2, and then modulated by linear EO effect in the PPLN₁ crystal, finally injected into the MVCSEL. The other one is transmitted to the T₁ by the TXPOA. After being reflected or scattered by the T₁, it is received by the RXPOA. The probe signal 1 from the TXPOA is split into two beams by the BS₃ when the channel noise in it is filtered by the filter 1 (F₁). One beam of the probe signal 1, firstly undergoes EO modulation in the PPLN₂, then injected into the SVCSEL. The chaotic wave emitted by the SVCSEL is decomposed into the x-PC_S and the y-PC_S by polarization beam splitter (PBS). The x-PC_M’ can be completely synchronized with the x-PC_S by the control of the applied electric field. The signal m₁ can be decoded from the synchronous division between the x-PC_M’ and the x-PC_S by the divider1 (DIV₁), where the decoded signal is named as $m_1^{\prime }$. The phase of m₁ and $m_1^{\prime }$ are extracted by the Hilbert phase transform. The distance of the T₁ can be calculated based on their phase difference [see Eqs. (21)-(22)]. Similarly, the ranging for the T₂ can also be implemented. In the scheme, the propagating delay time of the probe signals 1 and 2 are defined as ${\tau }_{c1}$ and ${\tau }_{c2}$, respectively. We obtain ${\tau }_{c2}>{\tau }_{c1}$ when the distance of the T₂ is more than that of the T₁. However, under this condition, the two PCs cannot be completely synchronized. To achieve their complete synchronizations, we place a delayer between the BS₃ and the RXPOA. The probe signal 1 is delayed for time ${\tau }_T$ by the delayer. According to the theory of the CCS [33] in the drive-response lasers, ${\tau }_{c2}={\tau }_{c1}+{\tau }_T$ is considered here. And ${\tau }_c={\tau }_{c2}$ is set for the convenience of discussion.

In 1995, Miguel et al. put forward the spin flip model (SFM) of VCSEL [44], where the laser cavity is designed into cylindrical symmetry. Consequently the output of VCSEL may be two orthogonal PCs (the x-PC and the y-PC) due to the weak material and cavity anisotropies, whose dynamic behaviors are sensitive to the polarization of the laser field. In particular, the VCSEL with optical feedback or optical injection may generate the two orthogonal chaotic PCs [36-37], the chaotic behavior of the x- PC or the y-PC becomes more complex due to the linear phase anisotropy and the amplitude anisotropy. So the chaotic behaviors of each PC of VCSEL output are different from that of the single polarization mode of the edge emitting laser output. According to the SFM [44], the rate equations of the MVCSEL and the SVCSEL need to be modified when they are subject to the injection of the output two PCs from the MVCSEL modulated by linear EO effect [45]. For the MVCSEL, the rate equations are

3. Results and discussions

In the system of the drive-response VCSELs, the accuracy and the stability of target ranging mainly depends on the CCS quality between the x-PC_M’ (y-PC_M’) from -the MVCSEL output and the x-PC_S (y-PC_S) from the SVCSEL output. To describe the CCS quality, the correlation functions of the two PCs are introduced, and defined as follows.

Since that the accuracy and the real-time stability of the target ranging depends on the CCS quality of the two PCs, how to choose the reasonable ranges of the crucial parameters is key to making the synchronization of the two PCs to be high quality and strong stability. Here, the key parameters include the bias current u, the applied electric field E, the propagating delay ${\tau }_c$, the injection strength k_inj and the feedback strength k_f. For this purpose, we numerically solve Eqs. (1) - (3) and (4) - (6) by using the four-order Runge-Kutta method. The numerical values of the parameters in calculation are given in Tab. 1. We firstly calculated the maps of the correlation coefficients (${\rho }_x$ and ${\rho }_y$) evolution in the parameter space of u and ${\tau }_c$ [see Figs. 2(a) and 2(b)], as well as in that of E and ${\tau }_c$ [see Figs. 2(c) and 2(d)], where k_f = k_inj = 2 ns⁻¹. In Figs. 2 (a) and 2(b), E is fixed at 0.36kV/mm. u = 1.5 is given in Figs. 2(c) and 2(d). One sees from Figs. 2(a) and 2(b) that, with the increase of u, ${\rho }_x$ and ${\rho }_y$ appear fast oscillational change between 0 and 1 when u ranges from 1 to 1.2, denoting that the chaotic synchronization between the x-PC_M’ and the x-PC_S (CS1) and that between the y-PC_M’ and the y-PC_S (CS2) are both poor. However, with the increase of u from 1.2 to 2, ${\rho }_x$ and ${\rho }_y$ are identically equal to 1, indicating that the CS1 and CS2 can be steady and good. For example, as shown in Figs. 3(a) and 3(b), ${\rho }_x={\rho }_y=1$ when u = 1.5. At this time, the two PCs are achieved CCS; but for u = 1.18, ${\rho }_x$ oscillates between 0.06 and 0.98, and ${\rho }_y$ ranges from 0.33 to 0.98; if u is fixed at 1.15, the variation range of ${\rho }_x$ is from 0.87 to 1, and ${\rho }_y$ jitters between 0.87 and 1. These discussions indicate that the CS1 and CS2 display extremely unstable when u is less than 1.2. The reason is that the symmetry between the rate equations of the drive laser and those of the response one is destroyed because of the simultaneous existence of the two PCs [33, 36-37]. From Figs. 2(c) and 2(d), it is found that, for an arbitrary ${\tau }_c$, ρ_x and ρ_y vary periodically with E. For example, when E changes in a certain range, such as 0.1~0.2kV/mm, 0.4~0.5kV/mm and 0.7~0.8kV/mm, ${\rho }_x$ and ${\rho }_y$ oscillate violently between 0 and 1. Under these conditions, the CS1 and CS2 are also jittered violently and very poor. On the other hand, while E varies in some limited ranges, such as 0.2~0.4kV/mm, 0.5 ~0.7kV/mm and 0.8 ~1.0 kV/mm, ${\rho }_x$ and ${\rho }_y$identically equal to 1, indicating that the two PCs can be achieved CCS. Figures 3(c) and 3(d) further describe the real-time change of ${\rho }_x$ and ${\rho }_y$ under different E. It is concluded from Figs. 2 and 3 that the two PCs can be obtained stable and well synchronized quality when the bias current u and the applied electric field E are controlled in the appropriate range. This can be attributed to the case that the imbalance in the symmetry between the rate equations of the drive laser and those of the response one is offset because of the exchange of energy between the two PCs by EO modulation.

Table 1. Numerical values for the calculation of the target ranging

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Fig. 2 Maps of the correlation coefficients ${\rho }_x$and ${\rho }_y$ evolution in the parameter space of u and ${\tau }_c$, as well as in that of E and ${\tau }_c$.

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Fig. 3 Dependence of the correlation coefficients ρ_x and ρ_y on the propagating delay time ${\tau }_c$under different parameters such as E and u, where $k_f=k_{inj}=2n{s}^{-1}$ and E = 0.36kV/ mm in (a) and (b); u = 1.5 in (c) and (d).

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When ${\tau }_c$ are fixed at 5ns and 7ns, respectively, we give the chaotic behaviors of the output x-PC and y-PC from the MVCSEL and those from the SVCSEL, as shown in Figs. 4 and 5, respectively. Here, E = 0.36kV/mm, u = 1.5. Their time series and spectrums are displayed in the left column and the right column of these two figures, respectively. Their powers are, respectively, defined as follows:

 

Fig. 4 Chaotic behavior of the output x-PC and y-PC from the M-VCSEL and those from the SVCSEL with ${\tau }_c=5ns$, E = 0.36kV/mm, u = 1.5. The left column: time series; the right column: power spectrum distribution.

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Fig. 5 Chaotic behavior of the output x-PC and y-PC from the M-VCSEL and those from the VCSEL when ${\tau }_c=\text{7ns}$, E = 0.36kV/mm, u = 1.5. The left column: time series; The right column: power spectrum distribution.

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In the following discussions for chaotic radar ranging, to ensure the two PCs to be in a stable state of complete synchronization, we take E and μ as 0.36kV/mm and 1.5, respectively. Under CCS, for the verification of the reliability and accuracy of chaotic radar ranging for the targets T₁ and T₂, two sets of the actual distances of the T₁ and the T₂ are considered as follows: ${\text{L}}_{{\text{T}}_{\text{1}}}=0.45m$ and ${\text{L}}_{{\text{T}}_{\text{2}}}=0.75m$; ${\text{L}}_{{\text{T}}_{\text{1}}}=0.75m$ and ${\text{L}}_{{\text{T}}_{\text{2}}}=1.05m$. From these distances, we obtain two sets of the propagating delay times as follows: ${\tau }_{c1}=\text{3ns}$ and ${\tau }_{c2}=\text{5ns}$; ${\tau }_{c1}=\text{5ns}$ and ${\tau }_{c2}=\text{7ns}$. The delay time ${\tau }_T$in the delayer is considered as 2ns to ensure that the two PCs both have the same propagating delay. In addition, two sinusoidal signals have the same angular frequency, i.e., ${\omega }_1={\omega }_2=\omega$, where $\omega =2\pi /4({\tau }_c-\tau )$.

For the above-mentioned two sets of different target distances, Fig. 6 presents the plots of ϕ₀ and ϕ, as well as d₁ and d₂ versus time, where ϕ₀ is the initial phase of m₁ and m₂; ϕ is the phase of m₁’ and m₂’; d₁ and d₂ are the measured distance of the T₁ and T₂, respectively. From Figs. 6(a)-6(d), it is found that ϕ₀ and ϕ both show a linear increase in periodicity when the T₁ and T₂ locate in different distances, respectively. The difference (Δϕ) between ϕ₀ and ϕ almost remains unchanged at any time. By using Δϕ from Figs. 6(a) - 6(d) and Eqs. (23) - (25), we further calculate the time traces of d₁ and d₂ for two sets of different target positions, as shown in Figs. 6(e) and 6(f). One sees from Fig. 6(e) that d₁ is slightly jittered between 0.4392 ~0.4572m when $L_{{\text{T}}_{\text{1}}}=0.45m$; If$L_{{\text{T}}_{\text{1}}}=0.75m$, it varies in a small range of from 0.7488 to 0.7524m. For $L_{T_2}=0.75m$, d₂ fluctuates in small range such as 0.7392~0.7527m; if $L_{T_2}=1.05m$ its variation range is from 1.0492 to 1.0513m. These results indicate that the two chaotic PCs have very high real-time stability in target distance measurement under CCS, and the ranging resolution reaches millimeter magnitude.

 

Fig. 6 For the targets T1 and T2 at different positions, time traces of the phases ϕ₀ and ϕ, as well as the measured distances d₁ and d₂ when the output two PCs are in complete chaotic synchronization state. Here, (a): ϕ₀ and ϕ for $L_{T_1}$ = 0.45m; (b) ϕ₀ and ϕ for $L_{T_1}$ = 0.75m; (c): ϕ₀ and ϕ for $L_{T_2}$ = 0.75m; (d): ϕ₀ and ϕ for $L_{T_2}$ = 1.05m; (e): d₁ for $L_{T_1}$ = 0.45m (the blue solid line), and d₁ for $L_{T_1}$ = 0.75m (the red solid line); (f): d₂ for $L_{T_2}$ = 0.75m (the red dotted line), and d₂ for $L_{T_2}$ = 1.05m (the blue dotted line).

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In addition, the relative errors RE₁ and RE₂ are introduced to describe the accuracy of the target ranging, and defined as follows:

Figure 7 gives the maps of RE₁ and RE₂ evolutions with ${\tau }_c$ in the parameter space of μ and τ_{c .} [see Figs. 7(a) and 7(b)], as well as in the parameter space of E and ${\tau }_c$ [see Figs. 7(c) and 7(d)]. As shown in Figs. 7(a) and 7(b), if E is fixed at 0.36kV/mm, since that the two PCs have been achieved CCS [see Figs. 2(c)-2(d)], RE₁ and RE₂ are almost no more than 5% when μ≥1.5 and ${\tau }_c$ ≥4ns,. With μ fixed at 1.5, RE₁ and RE₂ vary periodically with E if ${\tau }_c$ ≥4ns. For example, RE₁ and RE₂ are more than 20% while E varies in several limited ranges, such as 0.38-0.5, and 0.68-0.8, due to the deterioration of the CS1 and CS2 [see Figs. 2(c)-2(d)]; they are almost no more than 5% if E exists in some regions, such as 0.18-0.38, 0.48-0.68, and 0.78-0.98, owing to the good quality of the CS1 and CS2 [see Figs. 2(c)-(d)]. These mean that the small relative errors can be obtained by controlling E.

 

Fig. 7 Maps of the relative errors RE₁ and RE₂ evolutions with the propagating time ${\tau }_c$ in the parameter space of $\mu$ and ${\tau }_c$ [(a) and (b)], where E = 0.36kV/mm. The maps of their evolutions in the parameter space of E and ${\tau }_c$ [(c) and (d)], where $\mu =1.5$ ; (e)-(h): the partial enlargement of (a)-(d) in turn.

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Figure 8 further present the dependences of the relative errors, the signal-to-noise ratios (SNRs), and the correlation coefficients on the propagating time under $\mu =1.5$ and $E=0kV/mm$. Here, the SNR₁ and SNR₂ are defined as

 

Fig. 8 Dependences of the relative errors (RE1 and RE2), the signal-to noise ratios (SNR1 and SNR2), and the correlation coefficients (${\rho }_x$ and ${\rho }_y$) on the propagating time ${\tau }_c$, where $\mu =1.5$, $E=0kV/mm$.

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P_n1 and P_n2 are the corresponding spontaneous radiated noise power, and written as

 

Fig. 9 Dependences of the relative errors (RE1 and RE2), the signal-to noise ratios (SNR1 and SNR2), and the correlation coefficients (${\rho }_x$ and ${\rho }_y$) on the propagating time ${\tau }_c$, where E = 0.36kV/mm and $\mu =1.5$.

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From [33–35], it is found that the CCS quality of each PC depends not only on the parameters discussed above, but also on other key parameters of the system such as ${\gamma }_P$, ${\gamma }_s$, k_f, k_ing and τ. Therefore, the variation of these four different parameters can seriously influence the relative errors (RE₁ and RE₂). Figure 10 displays the evolutions of RE₁ and RE₂ as well as ${\rho }_x$ and ${\rho }_y$ with these parameters under E = 0.36kV/mm, τ_c = 7ns, $\mu =1.5$ and k_f = k_inj. As shown from Figs. 10(a₁) and 10(a₂), there appear very low relative errors when ${\gamma }_P$ is small. But large relative errors will occurs if ${\gamma }_P$ is large. For example, if ${\gamma }_P$ varies from 1ns⁻¹ to 2.837ns⁻¹, RE₁ and RE₂ are both small and become 1.787%, due to the reason that ${\rho }_x$ and ${\rho }_y$ are identically equal to 1. With the increase of ${\gamma }_P$ from 2.837ns⁻¹ to 6.327ns⁻¹, RE₁ fluctuates between 0.1217% and 97.43%, due to the fluctuation range of ${\rho }_x$ between 0.679 and 1. RE₂ irregularly varies between 0.1615% and 109.05%, owing to the variation range of ${\rho }_y$ from 0.7569 to 1. If ${\gamma }_P$ increases from 6.327ns⁻¹ to 10 ns⁻¹, RE₁ and RE₂ both exhibit violently fluctuation since that ${\rho }_x$ and ${\rho }_y$ fluctuates over a wide range. It is found from Figs. 10(b₁) and 10(b₂) that the relative errors have less values and vary in small range when increases from 50ns⁻¹ to 100ns⁻¹. For example, RE₁ varies in a small range of from 1.177% to ${\gamma }_s$ 1.944%, and RE₂ changes from 0.6059% to 3.195% when ${\gamma }_s$ increases from 50ns⁻¹ to 100ns⁻¹, owing to the reason that ${\rho }_x$ and ${\rho }_y$ are almost close to 1. These indicate that the relative errors have strong robustness when ${\gamma }_s$ is large. From Figs. 10(c₁) and 10(c₂), one sees that if k_f varies from 1ns⁻¹ to 1.5ns⁻¹, RE₁ fluctuates between 0.4505% and 63.54%, and RE₂ oscillates between 0.4193% and 53.34% because that ${\rho }_x$ and ${\rho }_y$ change between 0.3636 and 1. When k_f increases from 1.5ns⁻¹ to 2.796 ns⁻¹, RE₁ and RE₂ almost equal to 1.787% since that ${\rho }_x$ and ${\rho }_y$ both are of 1. With k_f further increasing from 1.5ns⁻¹ to 3ns⁻¹, RE₁ and RE₂ begin to fluctuate again. These show that the large relative errors occur in the region of the smaller values of k_f or that of the larger values of k_f. As further seen from Figs. 10(d₁) and 10(d₂), RE₁ and RE₂ are small and less than 2.7% when ${\rho }_x$ and ${\rho }_y$ equal to 1 in the parameter space of τ. However, although ${\rho }_x$ and ${\rho }_y$ are slightly reduced, but also RE₁ and RE₂ have very large values if τ is no more than 4ns. For example, with ${\rho }_x$ being 0.9958 and 0.9988, RE₁ is 28.34% and 21.11% when τ is fixed at 1.102ns and 3.99ns, respectively. But if τ is fixed at 5.796ns, RE₁ is small and of 0.3281% when ${\rho }_x$ equals to 0.9985. This denotes that the relative errors for these two targets are very sensitive and weak robust to the CCS if short delay occurs in the master laser feedback loop.

 

Fig. 10 Evolutions of the correlation coefficients (${\rho }_x$ and ${\rho }_y$), and the relative errors (RE₁ and RE₂) with the four different key parameters when E = 0.36kV/mm, τ_c = 7ns, $\mu =1.5$ and k_f = k_inj. Here, (a₁): ${\rho }_x$ and ${\rho }_y\alpha {\gamma }_P$ ; (a₂): RE₁ and RE_{2$\alpha {\gamma }_P$}; (b₁): ${\rho }_x$ and ${\rho }_y\alpha {\gamma }_s$ ; (b₂): RE₁ and RE₂ $\alpha {\gamma }_s$; (c₁): ${\rho }_x$ and ${\rho }_y\alpha$ k_f; (c₂): RE₁ and RE₂ $\alpha$k_f; (d₁): ${\rho }_x$ and ${\rho }_y\alpha \tau$ ; (d₂): RE₁ and RE₂ $\alpha \tau$.

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From the above results, it is concluded that the relative errors of the ranging for the two targets are seriously dependent on the quality of the CS1 and CS2. When the CS1 and CS2 exhibit unstable and are deteriorated, the ranging accuracies for the two targets are very low and their stability are very poor. It is noted that the two PCs can be achieved steady CCS by using EO modulation and selecting the appropriate system parameters. Under this condition, the ranging accuracy of the target at any position is greatly improved, and its stability is further strengthened. Furthermore, two chaotic PCs both have high and stable SNR.

4. Conclusions

We propose a novel real-time multi-target ranging scheme by using CPLIDAR in the drive-response VCSELs, based on the EO modulation theory, complete chaotic synchronization theory and Hilbert transform principle. Here, we discuss the effects of some crucial parameters, such as the bias current, the applied electric field, and the propagating delay, the feedback or injection strength, the spin relaxation, the birefringence, the delay in the master feedback loop, on the chaotic synchronization quality and stability. By using EO modulation and selecting the system parameters reasonably, the two chaotic polarization components with arbitrary propagating delay can be achieved steady complete synchronization. Under this condition, the real-time stability for the target ranging is fairly strong and only slight jitter. The resolution of the target ranging reaches millimeter magnitude. It is important to note that the relative error of the target at any position is very small and less than 2.7%. These results have potential application in the real-time multi-target ranging with high resolution and anti-noise jamming.