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Abstract
A high-resolution radar ranging scheme is proposed and demonstrated based on the ultra-wideband chaotic optoelectronic oscillator (OEO). Through biasing the electro-optic intensity modulator near its minimum transmission point, high-dimensional chaotic signals with flat spectra and low time-delayed signatures can be generated in the OEO, which are favorable for increasing the ranging resolution and the confidentiality. In the experiment, the optimized broadband OEO generates a high-dimensional chaotic signal with a flat spectrum in the frequency range of 2 GHz to 16 GHz and a high permutation entropy of 0.9754. This chaotic signal is used to achieve multiple target ranging, where a ranging resolution of 1.4 cm is realized.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Chaotic signals, which are characterized by noise-like waveform, wide spectrum and low power spectral density, are promising for radar and communication applications due to their benefits for security, anti-interference and suppression of range-Doppler coupling effect [1–3]. The common methods for generating chaotic signals are using electrical circuits [4–6] or function iterations [7–9]. Nevertheless, due to the limited instantaneous bandwidth of purely electronic devices, such as microwave amplifiers, the bandwidth of the generated chaotic signal is generally below a few gigahertz, which cannot meet the ever-increasing requirement of high-resolution radars and broadband secure communications.
Nonlinear feedback systems based on semiconductor lasers or optoelectronic oscillators (OEOs) are powerful candidates to solve the bandwidth limitation problem of chaotic signal generation [10]. Broadband chaotic signals can be generated through disturbing the active layer of a semiconductor laser via the delayed optical signals [11]. The main problem of this scheme lies in that the spectrum flatness is poor, which is attributed to the relaxation oscillation effect in the laser resonant cavity [12,13]. Continuous-wave (CW) optical injection is an effective method to enhance the spectrum flatness, which has been widely researched in past years [14,15]. However, the chaotic source based on a semiconductor laser is sensitive to external disturbance [16], where a slight perturbation of working temperature or injected light intensity will result in the instability of the chaotic state. Nonlinear feedback systems based on broadband OEOs exhibit rich dynamic behaviors due to the nonlinearity induced by the optoelectronic devices in the cavity, which can be used to generate various broadband and complex signals such as periodic signals, pulse packages, chaotic breathers and hyper-chaos [17–22]. For example, a hyperchaotic signal with a Lyapunov dimension of about 3700 and a bandwidth over 10 GHz has been generated in a broadband OEO with a high cavity gain and a large loop delay [23]. Compared with the chaotic source based on a semiconductor laser, the nonlinear dynamic characteristics of the OEO-based chaotic source depend on the loop structure and the devices outside of the laser source. Hence, it is insensitive to the complex laser properties, which is more stable and controllable. These advantages make the OEO-based chaotic source a promising candidate for high-precision radar and broadband secure communication applications.
In this paper, a high-resolution radar ranging scheme is proposed and demonstrated based on using an OEO to generate a broadband chaotic signal with a flat spectrum and a low time-delayed signature (TDS). Through optimizing the OEO, a chaotic signal with a flat spectrum in the frequency range of 2 GHz to 16 GHz and a permutation entropy of 0.9754 is generated. In the radar ranging experiment, multiple target detection with a ranging resolution of 1.4 cm is achieved by using this chaotic signal.
2. Operation principle
Figure 1 shows the schematic diagram of the proposed chaotic radar ranging system. High-dimensional chaotic signals are generated by a broadband OEO as shown in the upper part of Fig. 1. In the OEO loop, there is no bandpass filter. Hence, the bandwidth of the OEO cavity is determined by the operation bandwidth of the employed microwave and optoelectronic devices. A variable optical attenuator (VOA) is used to tune the OEO loop gain, which guarantees that the broadband OEO works in chaotic signal generation state. The generated chaotic signals are divided into two parts by using an electrical splitter, i.e., ES2 in Fig. 1, where one part is used as the reference signal, and the other part is sent to the transmitting antenna. The echo signals received by the receiving antenna are amplified by an electrical amplifier, and are then sent to a high-speed oscilloscope together with the reference signals for digitization. After computing the cross-correlation between the echo signals and the reference signals, the round-trip times between the antenna and the targets can be precisely measured. Then, the distance of the target from the antenna can be calculated by using the formula D = c/(2TR), where D is the distance, c is the speed of light in vacuum, and TR is the round-trip time.
Fig. 1. Schematic diagram of the proposed chaotic radar ranging system. LD: laser diode; MZM: Mach-Zehnder modulator; SMF: single-mode fiber; VOA: variable optical attenuator; PD: photodetector; EA: electrical amplifier; ES: electrical splitter.
The kernel of the proposed scheme is the generation of high-dimensional chaotic signals by using a broadband OEO. Although there is no electrical bandpass filter in the cavity, the OEO still has a wide bandpass characteristic due to the low-frequency cut-off characteristic of the electrical amplifier. Hence, the dynamic process in the OEO cavity can be mathematically described by Ikeda equation [23–25] as
For simplicity, Eq. (1) can be rewritten as
By normalizing the time-related parameters through τ, Eq. (2) can be written in a more convenient way as
3. Chaotic signal generation optimization
Numerical simulation is implemented to optimize the chaotic signal generation in the OEO. In the simulation, the loop delay τ is set to be 1.15 µs, which corresponds to a spool of SMF with a length of 300 m. The output optical power of the laser diode (LD) at 1550 nm is 16 dBm. The responsivity and the output matching resistance of the PD are 0.8 A/W and 50 Ω, respectively. In addition, the low cut-off frequency and the high cut-off frequency of the 2nd-order bandpass filter are set to be 200 MHz and 10 GHz, respectively. Hence, the ratio of the low cut-off frequency to the high cut-off frequency is calculated to be α=0.02. The two half-wave voltages of the MZM are set to be Vπ=6 V and Vπ0 = 6 V. Moreover, an initial Gaussian white noise with a power spectral density of -160 dBm/Hz is added after a single-loop transmission.
The bias voltage of the MZM has a great influence on the dynamic characteristics of the broadband OEO. Figure 2(a), (c) and (e) present the bifurcation diagram, the Lyapunov components and the 0-1 test diagram under different Ikeda gain β when the MZM is biased at its linear transmission point (LTP), i.e., Vbias = Vπ0/2, respectively. For comparison, Fig. 2(b), (d) and (f) exhibit the bifurcation diagram, the Lyapunov components and the 0-1 test diagram under different Ikeda gain β when the MZM is biased near its minimum transmission point (MITP), i.e., Vbias = 21Vπ0/20, respectively. The Lyapunov exponent is an effective index to evaluate the complexity of the chaotic signal, where a larger positive Lyapunov exponent corresponds to a higher chaotic complexity. If the Lyapunov exponent is larger than 0, chaotic oscillation is built. On the contrary, the dynamic system enters into periodic oscillation state. Similarly, the 0-1 test diagram is also used to judge whether the signal is a chaotic signal or not, where Kc≈0 and Kc≈1 represent periodic and chaotic dynamics, respectively [26]. It can be seen from Fig. 2(a) that, when the MZM is biased at its LTP, the broadband OEO exhibits obvious period doubling process as the Ikeda gain β increases. This dynamic process is similar to that in the optical feedback chaotic lasers. Nevertheless, when the MZM is biased near its MITP, the period doubling process with the increasing Ikeda gain β suppresses. This irregular dynamic process is favorable for generating high-dimensional chaotic signals, i.e., chaotic signals with large correlation dimensions. In addition, it should be pointed out that the dynamic process for the MZM biased near its maximum transmission point (MATP) is similar to that in Fig. 2(b), (d) and (f).
Fig. 2. Simulation results of the generated chaotic signals when the MZM in the broadband OEO is biased at its LTP (left column) and near its MITP (right column). (a)-(b) Bifurcation diagrams. (c)-(d) Lyapunov spectra. (e)-(f) 0-1 test diagrams.
For a nonlinear feedback system, there is an inevitable TDS in the autocorrelation diagram, since the feedback signal is a delayed replica of the output. The TDS not only leads to ranging ambiguity for inducing false side-lobes in the cross-correlation between the reference signals and echo signals [1], but also reduces the confidentiality of the chaotic radar system. Apart from the TDS, the bandwidth of the generated chaotic signal also plays a significant role in radar ranging since it directly determines the ranging resolution. Therefore, it is of great importance to optimize these characteristics for radar ranging. Figure 3(a) and (b) show the autocorrelation diagram and the spectrum when the MZM is biased at its LTP, i.e., Vbias = Vπ0/2, respectively. Figure 3(c) and (d) present the autocorrelation diagram and the spectrum when the MZM is biased near its MITP, i.e., Vbias = 21Vπ0/20, respectively. For both cases, the Ikeda gain β is set to be 5.5. It can be seen from Fig. 3 that the generated chaotic signal for the MZM biased near its MITP has a flatter spectrum and a lower TDS, which is more favorable for radar ranging.
Fig. 3. Simulation results of the generated chaotic signals when the broadband OEO is biased at its LTP (upper row, blue line) and near its MITP (bottom row, red line). (a)-(b) Autocorrelation diagrams (The insets are autocorrelation peaks). (c)-(d) Spectra.
The reason that the chaotic signal generated via biasing the MZM near its MITP has a flatter spectrum and a lower TDS than that generated via biasing the MZM at its LTP can be explained by using the bifurcation diagram in Fig. 2. When the MZM is biased at its LTP, the broadband OEO exhibits obvious Hopf bifurcations. On this condition, limited cycle oscillation is built, which is similar to the relaxation oscillation in the chaotic lasers. The output spectrum bulges near the oscillation frequency, which reduces the spectral flatness and the effective bandwidth. The oscillation frequency is proportional to the free spectra range (FSR) of the OEO loop. Hence, the existence of the limited cycle will expose the time delay of the OEO cavity, which corresponds to the peak locations in the correlation diagram as shown in Fig. 3. When the MZM is biased near its MITP, the OEO cavity is with a larger bifurcation threshold and a smaller bifurcation interval. Hence, the limited-cycle bifurcation is greatly suppressed. This transient dynamic process indicates that the OEO cavity exhibits a weak periodicity, which leads to a flatter spectrum and a lower TDS.
4. Experimental results and discussion
A proof-of-concept experiment is carried out to demonstrate the proposed scheme for high-resolution radar ranging. In the broadband OEO, continuous-wave (CW) light from a tunable narrow-linewidth laser source (Teraxion PS-TNL) with a power of 10.03 dBm and a center wavelength of 1545.6 nm propagates through a 25 Gb/s electro-optic MZM (FUJITSU 7938EZ). The MZM is biased near its MITP, where the bias voltage for the MITP and the applied DC bias voltage are 8.50 V and 8.65 V, respectively. After propagating through a spool of SMF (YOFC) with a length of 300 m and a VOA, the modulated optical signals are detected by using a home-made PD whose operation bandwidth is up to 20 GHz. An electrical amplifier (GT-HLNA-0022 G) with an operation bandwidth from 34.25 MHz to 22 GHz and a small-signal gain of 28 dB is employed to amplify the electrical signal from the PD. Besides, an electrical power divider (GTPD-COMB50G) with an operation bandwidth from DC to 50 GHz is used to extract the chaotic signal out of the OEO. Another electrical power divider (MARKI PD-0030) with an operation bandwidth from DC to 30 GHz divides the extracted chaotic signals into two parts. One part is transmitted by the transmitting antenna (HD-10200DRHA10S, 1-20 GHz), and the other part is recorded as the reference signal by a high-speed real-time oscilloscope (Tektronix DPO75002SX) with a sampling rate of 50 GSa/s. The echo signal from the receiving antenna (HD-10200DRHA10S, 1-20 GHz) is amplified by an electrical amplifier (TLPA50K20G-28-20) with an operation bandwidth from 698 MHz to 25 GHz and a small-signal gain of 31 dB, and then recorded by the high-speed real-time oscilloscope. Figure 4 shows the experimental setup for target ranging. In the single target ranging experiment, target A is placed at distances of 106 cm, 178 cm, and 240 cm from the antennas, respectively. In the multiple target ranging experiment, target C is fixed at the distance of 180 cm, while target B is placed at distances of 148 cm, 160 cm and 170 cm, respectively.
Fig. 4. Experimental setup for target ranging. (a) Single target ranging. (b) Multiple target ranging. The testing distance refers to the vertical distance from the target to the antenna.
Figure 5 shows the experimental results of the OEO-based chaotic source. In Fig. 5(a), the temporal sequences of the generated chaotic signal within 5 µs and 5 ns are exhibited. In order to quantify the complexity of the chaotic signal, the temporal sequences with a length of 30000 are adopted for evaluating the permutation entropy (PE), which can be used to evaluate the complexity of the generated chaotic signal [27]. The PE is calculated to be 0.9754 (the maximum value is 1) with an embedding dimension of 5 and an embedding delay of 1. The large PE value indicates that a high-dimensional chaotic signal is successfully generated. Figure 5(b) exhibits the spectrum of the generated chaotic signal. It can be seen that the spectrum is with an excellent flatness in the frequency range of 2 GHz to 16 GHz. The effective bandwidth of the chaotic signal is about 10 GHz, which fits in with the full-width at half-maximum (FWHM) of the autocorrelation peak shown in the inset of Fig. 5(c). In addition, as shown in Fig. 3(c), the maximum TDS in the autocorrelation diagram is 0.18. Although this value is larger than that in Fig. 3(b), it is much smaller than the TDS for the MZM biased at its LTP as shown in Fig. 3(a).
Fig. 5. Experimental results of the OEO-based chaotic source. (a) Temporal sequences within 5 µs (Left) and 5 ns (Right), (b) spectrum, and (c) autocorrelation diagram (The inset is the autocorrelation peak).
Figure 6 shows the experimental results of chaotic radar ranging. Thereinto, Fig. 6(a) presents the results for detecting target A at different distance. As target A moves away from the antenna, the strength of the echo signal is gradually attenuated, which makes the strength of the clutter relatively increase. It can be seen from the inset in Fig. 6(a) that the ranging resolution is about 1.4 cm. This value fits in with the theoretical resolution Δr = c/(2B) = 1.5 cm, where Δr is the ranging resolution, c is the speed of light in vacuum, and B is the bandwidth of the chaotic signal. Figure 6(b) exhibits the experimental results of chaotic radar ranging for target B and target C with different intervals. Since target B and target C are with different sizes and are made of different materials, they have different electromagnetic wave absorption characteristics. Hence, the cross-correlation peak corresponding to target B is lower and wider than that corresponding to target C.
Fig. 6. Experimental results of chaotic radar ranging. (a) Ranging results of target A (The inset is the FWHM of the cross-correlation peak). (b) Ranging results of target B and target C.
The limitations of the broadband chaotic radar system can be summarized into the following two points. Firstly, broadband power amplifiers are required in the transmitting end to achieve long-distance ranging. In general, the operation bandwidth of a commercial power amplifier is below 10 GHz, which puts a restriction on the effective bandwidth of transmitted chaotic signals as well as the ranging resolution. Secondly, a broadband digital receiver with a high sampling rate (up to tens of GS/s) is required to achieve analog-to-digital conversion of the echo signal and the reference signal before digital correlation processing, which is difficult to implement real-time detection. A new method to correlate ultra-wideband signals in the analog domain in [28] is beneficial for improving the signal processing speed. Last but not the least, in order to expand the application of this broadband chaotic radar ranging system, such as autonomously driven vehicle [29], the broadband OEO should be miniaturized. Fortunately, the fiber length has little effect on the performance of the broadband OEO. Hence, the proposed system can be monolithic integrated [30].
5. Conclusion
In summary, we have proposed and demonstrated a high-resolution radar ranging scheme based on ultra-wideband chaotic OEO. Numerical simulation is implemented to optimize the chaotic signal. The results indicate that high-dimensional chaotic signals with flat spectra and low TDSs can be easily generated when the MZM in the OEO loop is biased near its MITP or MATP. In the experiment, chaotic signal with a flat spectrum in the frequency range of 2 GHz to 16 GHz and a PE of 0.9754 is generated by biasing the MZM near its MITP. This chaotic signal is used to achieve multiple target ranging, where a ranging resolution of 1.4 cm is realized.
Funding
National Key Research and Development Program of China (2019YFB2203800); National Natural Science Foundation of China (61927821); Fundamental Research Funds for the Central Universities (ZYGX2020ZB012).
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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