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Abstract
We propose a novel scheme of the real-time ranging for the six orientational targets based on the vertical cavity surface-emitting laser (VCSEL) network with three nodes. In the scheme, we explore a method to realize the globally complete chaotic synchronization (GCCS) of the network with different channel delays. Under the GCCS, we use the six chaotic polarization radars for the ranging of the six orientational targets based on Hilbert transform theory. It is found that the ranging of the six orientational targets has good performance, such as real-time stability and high accuracy, and the absolute errors of the ranging reach millimeter magnitude. Moreover, all relative errors are small and less than 11%.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Laser-based chaotic radar (CRADAR) ranging has attracted considerable attention, since it has many advantages over short-pulse and CW radar ranging, such as low probability of intercept, high ranging and velocity resolution, strong anti-interference ability, easy generation and low cost [1–6]. Moreover, unlike random noise, CRADAR are controllable and can be synchronized between different laser radar systems with the same system parameters [7–9]. Therefore, CRADAR can be generalized to application fields such as precision range finding, object tracking and locating, through-wall detection, driverless navigation system and so on [10–18].
So far, many works on CRADAR ranging have focused on its realization by the cross-correlation between the time-delayed reflected return signal and the replica of the transmitted chaotic signal [1–5,12–15]. In these works, the resolution of centimeter magnitude can be easily implemented, benefiting from the broad bandwidth of chaotic waveform generated by a nonlinear semiconductor laser [2–4]. In recent years, using the cross-correlation theory, the technology of CRADAR ranging has made some progresses by using different devices. For example, based on microwave-photonic chaotic signal generation and optical-fiber distribution, an ultra-wideband radar system for remote ranging was proposed and demonstrated experimentally in 2014 [13]. Wang et al studied the performance of chaotic radar system for target detection and ranging through lossy media in 2015 [14]. In the same year, a distributed multiple-input multiple-output chaotic radar based on wavelength-division multiplexing technology was proposed and demonstrated [16]. In 2017, using a broadband white chaos generated by optical heterodyne of two chaotic external-cavity semiconductor lasers, Wang et al theoretically explored a radar system in which ranging resolution and anti-jamming capability were enhanced [5]. Although the ranging resolution can be improved in a certain extent by the increase of chaotic bandwidth in previous works, but it is difficult to further improve its resolution due to the limitation of the cross-correlation theory. Moreover, owing to the interferences of the spontaneous emission noise and channel noise, the correlation performance can be deteriorated, which seriously affect the accuracy of the target ranging. Compared with the cross-correlation CRADAR, the synchronized CRADAR between drive laser and response one showed better detection performances in improving the resolution and anti-noise ability [6,9,17,18]. In the synchronized CRADAR system, the precision of target ranging heavily depends on the complete chaotic synchronization (CCS) quality. Due to the case that synchronization plays the role of the noise filtering, the CCS quality has very strong robustness to noise. Under stable CCS, the distance of the target at arbitrary position can be measured precisely.
In our previous work [18], we implemented the real-time ranging of two targets by using synchronized chaotic polarization radars in the drive-response vertical cavity surface-emitting lasers (VCSELs) system, where two chaotic polarization radars were used to measure the distances of two targets. Other previously reported works on CRADAR ranging also focused on one- or two-target ranging [1–6, 9–17]. It is worth noting that, in practical application fields such as driverless cars and the object tracking system with omnidirectional vision, the targets located in different orientations require to be accurately detected. In order to achieve this goal, the CRADAR sources are distributed in different nodes of the network with different channel delays. The ring network of semiconductor lasers mutually coupled with multiple delays provides potential applications for multi-orientation target ranging [19, 20]. The accuracy of multi-orientation target ranging mainly depends on the globally CCS (GCCS) quality of the laser network. However, to achieve high-quality GCCS faces new challenges under different channel delays. The mechanism of multi-orientation target ranging using the laser network needs to be further explored. Motivated by these, we investigate the way to achieve the stable GCCS with high quality. Under the GCCS, we further put forward a novel scheme to implement the six-orientation target ranging using three-node VCSEL network with different channel delays, based on the theory of Hilbert transform. Finally, we explore the precision, the real-time stability and the relative errors of the six-orientation target ranging.
2. Theory and model
Figure 1 presents the implementation of the real-time ranging for the six orientational targets by using the synchronized chaotic polarization radars in the three-node VCSEL network. Here, the optical isolators (OIs) with the subscripts 1–3 are used to ensure the unidirectional propagation of light wave. The feedback or injection strengths of all polarization components (PCs) are varied by the neutral-density filters (NDFs). The 2-VCSEL with self-feedback is considered as the drive laser. The 1-VCSEL subjected to the optical injection from the 2-VCSEL output is the response laser as well as the drive one. The 3-VCSEL is the response laser, which is subjected to the optical injection from the 1-VCSEL and the 2-VCSEL output simultaneously. Since the two polarization components (PCs) of the 2-VCSEL, respectively, are along the direction of the x-axis and that of the y-axis, they are defined as the X-PC2 and Y-PC2, respectively. Similarly, the two PCs of the 1-VCSEL are the X-PC1 and Y-PC1, and those of the 3-VCSEL are the X-PC3 and Y-PC3. The microwave signal m is modulated to the chaotic polarization radars by the electro-optic modulators. The X-PC2 with m (the probe signal X2) is used to measure the distances of the targets 1 (T1) and T4; the Y-PC2 with m (the probe signal Y2) is applied to the ranging of the T11 and T44; the X-PC1 with m (the probe signal X1) and the Y-PC1 with m (the probe signal Y1) are utilized to measure the distances of the T3 and T33, respectively. To avoid mutual interference between the same PCs, the T44 and T1 detected by different PCs are placed at the left- and right-front of the system, respectively. Similarly, the T33 and T3 are set at the left- and right-rear, respectively; the T4 and T11, respectively, are located on the left- and right-side. It is noted that the crosstalk between two different PCs can be eliminated by the x- and the y-polarization optical antenna [18]. If the distance between the targets detected by the same PCs is far enough, the crosstalk between the same PCs can be prevented since chaotic polarization radars in near infrared band are quickly attenuated in the air.
Fig. 1 Implementation of the real-time six-orientation target ranging by using the synchronized chaotic polarization radars in the three-node VCSEL network, showing (a) the system block diagram; (b) detailed light paths; (c) the real-time calculation; (d) the corresponding three-node ring topology. Here, VCSEL: vertical-cavity surface-emitting laser; OI: optical isolator; PBS: polarization beam splitter; EOM: electro-optic modulator; FC1 ∼ FC2: 1×3 fiber coupler; FC3 ∼ FC12: 1×2 fiber coupler; F: optical fiber; TX(Y)POA: transmitting x(y)-polarization optical antenna; RX(Y)POA: receiving x(y)-polarization optical antenna; T: target; NDF: neutral density filter; FR: filter; PD: photodetector; DIV: divider; HT: Hilbert transform; Phase: original phase calculation; DP: delayed phase calculation; SUB: subtracter; CD: distance calculation; m(t): modulated microwave signal; m1(t) ∼ m6(t): demodulated microwave signals.
In the following, we take the probe signal X2 as an example to illustrate the implementation of the T1 ranging. The probe signal X2 is divided into three beams by the fiber coupler 1 (FC1). One of the three beams is transmitted to the T1 by the transmitting x-polarization optical antenna 1. After being reflected or backscattered from the T1, the echo signal is received by the receiving x-polarization optical antenna 1 and then divided into two beams by the FC7 after filtering the channel noise through the filter 3. One beam of the echo signal is injected into the 1-VCSEL, the other is detected by the photodetector 1 and then received by the divider 1 (DIV1). The signal m can be decoded from the synchronous division between the X-PC1 and the X-PC2 by the DIV1. The decoded signal is denoted by m1. The phases of m and m1 are extracted by the Hilbert transform 7 (HT7) and the HT1, respectively. The distance of the T1 can be computed based on their phase difference [see Fig. 1(c)]. The distances of the T3, T4, T11, T33 and T44 can also be calculated in the same way.
The six injection delay-times in the network shown in Fig. 1(d), in turn, are defined as τ1, τ3, τ4, τ11, τ33 and τ44. The channel-delay τJ is responsible for the ranging of the target TJ (J=1, 3, 4, 11, 33 and 44, the same below). τ2 and τ22 are the round-trip time in the external-cavity of the 2-VCSEL. Based on the spin-flip mode proposed by Miguel et al. [21], the rate equations of the three node-VCSELs in the network are modified as
where i = 1, 2, 3; the subscripts 1, 2 and 3 represent the 1-VCSEL, the 2-VCSEL and the 3-VCSEL in turn; the subscripts x and y show the X-PC and the Y-PC, respectively; E is the slowly varied real amplitude of the field; φ is the real phase; N is the total carrier concentration; n is the difference between carrier inversions for the spin-up and spin-down radiation channels; κ is the decay rate of field; α is the line-width enhancement factor; γe is the nonradiative carrier relaxation rate; γs is the spin-flip rate; γa and γp are the linear dichroism and the linear birefringence, respectively; μ is the normalized injection current; the spontaneous emission rate βsp is equal to 10−6ns−1; ζx and ζy are two independent complex Gaussian white noise events with zero mean and unitary variance, their time correlation is as follows, ; K12 is the optical strength of the X-PC2 or the Y-PC2 injected into the 1-VCSEL; K32 is that of the X-PC2 or the Y-PC2 injected into the 3-VCSEL; K31 is that of the X-PC1 or the Y-PC1 injected into the 3-VCSEL; K22 is the self-feedback strength of the X-PC2 or the Y-PC2; the central angular frequencies of all node VCSELs are ω0 that equals 2πc/λ0, where λ0 is central wavelength and c is the speed of light in vacuum.3. Results and discussions
In the three-node VCSEL network, synchronization plays an important role in target ranging. According to the theory of complete synchronization, we obtain the complete synchronous solutions as follows
where and the parameter settings are required as,In the scheme, we consider cosine wave as modulated signal m for the six-orientation target ranging. It is expressed as
where ω is the angular frequency; A is the amplitude. By using the synchronous division in Fig. 1, the six demodulated signals are obtained as follows,Under Eq. (9), the six demodulated signals are derived from Eqs. (12)–(13) as follows,
According to Hilbert transform, the analytic signal of m is written as
where m̃ is the Hilbert transform of m. Based on Eq. (15), the phase of m is given as followsSimilarly, the phase of mk can be obtained by the following equation
According to Eqs. (16)–(17), the delay times for the six-orientation target ranging are described as
Combined with Eq. (10), the distances of the six orientational targets are derived as
where the subscripts 1, 3, 4, 11, 33 and 44, in turn, represent targets T1, T3, T4, T11, T33 and T44.In the following calculations, the parameter values that are common among three VCSELs are given in Table 1. Moreover, K12 = K22 = 30ns−1, K31 = 10ns−1, K32 = 20ns−1. The amplitude A and the frequency ω of m are considered as 10−3 and 2π/(5 max[|Δτk|]), respectively. The dynamic behaviors of the two PCs from the 2-VCSEL subjected to self-feedback only are displayed in Fig. 2, where τ2 = 21ns and τ22=41ns. From this diagram, one sees that the temporal waveform of the X-PC2 and that of the Y-PC2 both exhibit chaotic behaviors. The other node-VCSELs have the same chaotic behaviors as the 2-VCSEL if they can be completely synchronized with the 2-VCSEL.
Table 1. Numerical values for the calculation of the target ranging
Fig. 2 Temporal traces of the two PCs from the 2-VCSEL output. (a) the X-PC2; (b) the Y-PC2. Here, I2x(t) = |E2x(t)|2; I2y(t) = |E2y(t)|2.
According to Eqs. (9)–(19), the accuracy and the stability of six-orientation target ranging heavily depend on the GCCS quality. The GCCS can be realized under the synchronous solutions [see Eq. (9)]. In the following, the correlation coefficients are introduced to describe the GCCS quality.
where the subscript 12x(y) denotes the correlation coefficient between the X(Y)-PC1 and the X(Y)-PC2; the subscript 13x(y) represents that between the X(Y)-PC1 and the X(Y)-PC3; the subscript 23x(y) indicates that between the X(Y)-PC2 and the X(Y)-PC3; I1x,1y(t) = |E1x,1y(t)|2, I2x,2y(t) = |E2x,2y(t)|2 and I3x,3y(t) = |E3x,3y(t)|2; the symbol < > denotes the time average. Note that ρ ranges from 0 to 1. With the bigger ρ, the higher synchronization quality can be obtained. When ρ12x,y, ρ13x,y and ρ23x,y are all equal to 1, there exist the complete synchronous solutions [see Eq. (9)] in the network, denoting that the GCCS is of the highest quality.Figure 3 shows the evolutions of the correlation coefficients in the parameter space of τ3 and τ4, where τ2 = 21ns, τ22=41ns, and the other delay-times are obtained from Eq. (11). From this figure, it is found that all correlation coefficients are more than 0.97. For example, ρ12x varies from 0.992 to 1; ρ12y fluctuates between 0.9896 and 1; ρ13x ranges from 0.9812 to 1; ρ13y oscillates between 0.9845 and 1; ρ23x jitters between 0.9774 and 1; ρ23y is between 0.9782 and 1. These results indicate that the stable GCCS with high quality can be implemented in the network. Under the condition, the outputs of the 1-VCSEL and 3-VCSEL exhibit the same chaotic behaviors as the 2-VCSEL.
Fig. 3 Maps of the correlation coefficients evolution in the parameter space of τ3 and τ4. Here, (a) ρ12x; (b) ρ12y; (c) ρ13x; (d) ρ13y; (e) ρ23x; (f) ρ23y.
In the following, we use the synchronized chaotic polarization radars for the six-orientation target ranging. We consider the actual distances of the six orientational targets as follows: dT1=4.5m, dT11=7.5m, dT3=1.2m, dT33=4.2m, dT4=2.55m and dT44=5.55m. From these distances, we obtain that τ1=30ns, τ11=50ns, τ3=8ns, τ33=28ns, τ4=17ns and τ44=37ns. These values of the delay times make the network in the GCCS. For the six orientational targets, Fig. 4 presents the time traces of the phase ϕm and ϕmk by using Eqs. (16)–(17). One sees from Fig. 4 that all phases show a linear increase in periodicity and each phase difference (Δϕk) between ϕm and ϕmk almost remains unchanged at any time. By using Δϕk and Eqs. (18)–(19), we calculate the temporal traces of the six measured distances, namely d1, d3, d4, d11, d33, and d44, as shown in Fig. 5. It is found from this figure that d1 ranges from 4.4m to 4.6m if dT1=4.5m; d11 fluctuates between 7.48m and 7.53m when dT11=7.5m; d3 varies from 1.12m to 1.28m for dT3=1.2m; d33 oscillates between 4.11m and 4.24m under dT33=4.2m; d4 varies between 2.38m and 2.72m while dT4=2.55m; d44 is slightly jittered between 5.51m and 5.57m under dT44=5.55m. Moreover, we consider the expression |<dJ> −dTJ| as the absolute error (|ΔdJ|) of the Jth target ranging. From dJ and dTJ, we obtain that |Δd1| = 1.1mm, |Δd11| = 1.2mm, |Δd3| = 8.5mm, |Δd33| = 8.3mm, |Δd4| = 0.13mm and |Δd44| = 0.36mm. These results indicate that all absolute errors of the ranging reach millimeter magnitude and the target ranging has strong real-time stability under the GCCS.
Fig. 4 Time traces of the phase ϕm and ϕmk. Here, (a) ϕm and ϕm1; (b) ϕm and ϕm2; (c) ϕm and ϕm3; (d) ϕm and ϕm4; (e) ϕm and ϕm5; (f) ϕm and ϕm6.
To further describe the accuracy of the six-orientation target ranging, the relative error (REJ) of the Jth target ranging is introduced as follows.
where the symbol “| |” means absolute value. For the convenience of observation, we quantify the relative error as follows: State 1: 0 ≤ REJ ≤ 0.1%; State 2: 0.1% < REJ ≤ 1%; State 3: 1% < REJ ≤ 11%; State 4: 11% < REJ ≤ 20%; State 5: REJ > 20%. Under the GCCS, Fig. 6 presents the evolutions of six relative errors in the parameter space of τ3 and τ4. It is found from Fig. 6 that there are no sates 4 and 5, but the states 1, 2 and 3 appear in the parameter space of τ3 and τ4. Thus, all relative errors are small and less than 11%, showing that the six-orientation target ranging with high accuracy can be implemented.Fig. 6 Maps of the six relative errors evolutions in the parameter space of τ3 and τ4. Here, (a) RE1; (b) RE11; (c) RE3; (d) RE33; (e) RE4; (f) RE44. State 1: red area; State 2: green area; State 3: cyan area; State 4: blue area; State 5: yellow area.
4. Conclusions
To sum up, we propose a novel scheme of the real-time ranging for the six orientational targets based on the three-node VCSEL network with ring-topology. In the scheme, we investigate a method that makes the network with different channel delays achieve the GCCS. It is found that the stable GCCS with high quality can be realized in the parameter space of delay times. Under the condition, the synchronized chaotic polarization radars are utilized to measure the distances of the six orientational targets based on Hilbert transform theory. The results show that the ranging of the six orientational targets has strong real-time stability and high accuracy, and all absolute errors of the ranging reach millimeter magnitude. In addition, all relative errors are small and less than 11%. These results have potential applications in other fields such as precision range finding, object tracking and locating, driverless navigation system and so on.
Funding
National Natural Science Foundation of China (NSFC) (61475120); Major Projects of Basic Research and Applied Research for Natural Science in Guangdong Province (2017KZDXM086).
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