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Abstract
Aimed at high-precision distance measurement for noncooperative targets in free space, a spatial distance measurement method is proposed. Based on the concept of optical carrier-based microwave interferometry, this method extracts distance information from the radiofrequency domain. The interference model of broadband light beams is established, and the optical interference can be eliminated by using a broadband light source. A spatial optical system with a Cassegrain telescope as the main body is designed to effectively receive the backscattered signal without cooperative targets. A free-space distance measurement system is built to verify the feasibility of the proposed method, and the results agree well with the set distances. Long-distance measurements with a resolution of 0.033 µm can be achieved, and the errors of the ranging experiments are within 0.1 µm. The proposed method has the advantages of fast processing speed, high measurement accuracy, and high immunity to disturbances as well as the potential for measurement of other physical quantities.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. INTRODUCTION
High-precision distance measurement in free space has always been an important field of industry and frontier scientific research. For the measurement convenience in industrial applications, high-precision distance measurement for noncooperative targets has become increasingly important. Due to the influence of optical waveguides, backscattering energy, electromagnetic interference (EMI), and other factors, traditional laser or microwave interferometric measurement methods can hardly achieve noncooperative targets, high precision, and long distances at the same time. Microwave photonics is an interdisciplinary subject that combines the strengths of optics and microwaves, which has the advantages of low attenuation over the entire microwave frequency range, immunity to EMI, low dispersion, and high data-transfer capacity [1–6]. In recent years, it has been applied to the field of measurement and derives the concepts of frequency-sweeping interference (FSI) [7,8], frequency-modulated continuous wave (FMCW) [9–11], and optical carrier-based microwave interference (OCMI) [12–17]. Many distance measurement methods based on the FSI and FMCW principle have been proposed. However, the measurement accuracy and range are limited by the nonlinearity of frequency modulation, the bandwidth of swept-frequency, and instability of the environment. For example, existing high-precision interferometers based on FSI such as Nikon’s lidars have a measurement accuracy of 30 µm for noncooperative targets at 10 m [7].
The concept of optical carrier-based microwave interferometry (OCMI) was proposed and theoretically derived by Huang et al. [12,13]. Its fundamental principle is to read optical interferometers in the radiofrequency (RF) domain. By sweeping the microwave frequency, the microwave interferogram can be acquired and used to extract the information to be measured. It has unique advantages when used for sensing, including high signal-to-noise ratio, easy manipulation, and resistance to disturbances from electromagnetic and optical factors. However, the OCMI principle is currently only applied to fiber optic sensing, and the system construction is limited by the laying of fiber optics and other infrastructure, which cannot be applied flexibly in complex environments [13–17].
This paper proposed a high-precision free-space distance measurement method for noncooperative targets based on the concept of OCMI. By modulating the amplitude of the optical signal with microwaves, the nonlinear influence of the modulation process can be eliminated. Using a broadband light source as the carrier, the coherence length is much smaller than the distance to be measured, so only the microwave signals are coherently superimposed during the measurement. This method can thus avoid the disturbances of optical factors, thereby improving the signal-to-noise ratio and system stability. In order to improve the backscattered signal reception efficiency, an optical system with a Cassegrain telescope and single-mode fiber as the main body was designed and optimized by optical design software. The unique structure makes the optical system compact and free of chromatic aberration. A distance measurement system was introduced to verify the feasibility of the proposed method. By resolving the microwave interferograms, the change in the optical path difference (OPD) can be determined, from which the measured distance can be calculated. The theoretical measurement resolution of the proposed system can reach 0.033 µm and used for long-distance measurement of tens or even hundreds of meters.
Fig. 1. Concept of the high-precision free-space distance measurement method for noncooperative targets.
The presentation of this paper is as follows. Section 2 explains the concept of high-precision free-space distance measurement method for noncooperative targets and analyzes the interference characteristics of broadband light sources and the principle of the proposed spatial distance measurement. Section 3 establishes the simulation model of the proposed method, including the interference model of a broadband light source and model of an optical receiving system. Section 4 presents the implementation of the proposed method, including the schematic of the system and implementation results. The best measurement resolution achievable under the current experimental conditions is calculated and verified. The results of the ranging experiments are listed and the accuracy indicators are analyzed. The measurement performances under different noise levels are simulated. In Section 5, the research work performed is further discussed and evaluated. The application potential of the proposed method is also discussed.
2. CONCEPT
A. Concept of the High-Precision Free-Space Distance Measurement Method for Noncooperative Targets
Figure 1 illustrates the concept of the high-precision free-space distance measurement method for noncooperative targets. The broadband light source is intensity-modulated by a microwave signal to obtain the modulated signal, which takes the light wave as the carrier and the microwave as the envelope. The modulated signal is split into two beams and enters Path 1 and Path 2, respectively. The measurement signal of Path 2 is sent into the optical system in free space. After carrying the distance information of the object to be measured, it is received and interferes with the reference signal of Path 1 in fiber. The interference signal is transmitted to the signal processing unit after photoelectric conversion by a photoelectric detector. Using a broadband light source, no coherent superposition of optical carriers occurs. Therefore, by combining optical domain sensing with RF domain processing, we can extract distance information from microwave interferograms.
B. Mathematical Model of Broadband Light Interference
The broadband light source’s spectrum is approximated as a Gaussian function, as shown in Fig. 2, where ${\lambda _0}$ is the center wavelength, ${I_0}$ is the light intensity at the center wavelength, and $B$ is the bandwidth of the light source. It can be considered as being composed of multiple uncoherent monochromatic light sources. Hence, the interference of the broadband light source can be equivalent to the superposition of the interference of each monochromatic light source.
The light intensity at the wave vector $k$ can be expressed as
The light is split into two beams and propagates through different distances (${d_1}$ and ${d_2}$). According to the interference principle, the superimposed light intensity in the range of wave vector $k \sim dk$ can be expressed as
Fig. 3. Interference fringe patterns: (a) $B = {{0}}\;{\rm{nm}}$; (b) $B = {{10}}\;{\rm{nm}}$; (c) $B = {{50}}\;{\rm{nm}}$.
We define $\Delta k = k - {k_0}$:
After simplification and calculation, we can obtain
The visibility of interfering stripes can be expressed as
It can be seen that the stripe visibility decreases as the bandwidth of the light source increases. When the visibility of the interference fringes is low enough, it can be considered that there is no optical interference, only an increase in the overall light intensity.
C. Mathematical Model of the Spatial Distance Measurement Method
The modulated signal is split into the reference signal and the measurement signal by the 50/50 coupler. After going through different paths, the two-beam interference with equal amplitude. After removing the DC noise, the interference signal can be expressed as
The derivation process is described in detail in [16]. In Eq. (7), $t$ is the time; $A$ and $M$ are the amplitudes of the optical and microwave signal, respectively; $\Omega$ is the microwave angular frequencies; $c$ is the speed of light in vacuum; $W$ is the electrical length of the common microwave path; ${L_1}$ and ${L_2}$ are the optical lengths of paths 1 and 2, expressed as ${n_{\text{eff}}}z$, where ${n_{\text{eff}}}$ is the refractive index and $z$ is the length of the path; and $g$ is the modulation depth of the modulator. From Eq. (7), the amplitude of the interference signal is a function of the microwave frequency and OPD, and the phase of the interference signal is a function of the sum of the electrical and optical length. By scanning the frequency, the microwave interferogram can be obtained, from which the distance to be measured can be extracted.
3. SIMULATION MODEL
A. Interference Model of Broadband Light Source
According to the mathematical model of broadband light interference, the interference of light sources with different bandwidths was simulated. Figure 3 shows the simulated interference fringe patterns of different light sources. It can be seen that the visibility of interference fringes is significantly reduced when the light source bandwidth is 10 and 50 nm, proving that the fringe visibility decreases with increasing light source bandwidth. Therefore, using a light source with a larger bandwidth in the measurement, such as a ${\rm{C}} + {\rm{L}}$-band ASE tunable light source with a working wavelength range of 1528–1603 nm, the effect of optical domain interference will be excluded.
B. Model of Optical System
In the long-distance noncooperative target measurement, the measurement failure is mainly caused by the low receiving efficiency of the backscattered signal. Thus, the receiving optical system for a backscattered signal is an important part of the spatial distance measurement method for noncooperative targets. The schematic of the designed optical system is shown in Fig. 4. The optical transmitting lens and receiving lens are located on the same side of the system; the object and image lie on opposite sides of the imaging system. After the measurement signal is converged and shaped, it is emitted to the measured object and diffusely reflected. A telescopic system is required to receive the backscattered signal from a distance. The Cassegrain telescope system, which is widely used in modern astronomical observation due to its advantages of chromatic aberration free and compactness, can be applied to the proposed optical system. It consists of a concave primary mirror and a convex secondary mirror. The primary mirror focuses the light to the secondary mirror, which in turn focuses the light to the system’s focus point after pupil exit. Thanks to the foldable structure, the telescope is compact, and the length can be shortened to one-third of the focal length, which can effectively reduce the footprint of the measurement system. After being converged, the backscattered signal is received by the receiving lens and then interferes with the reference signal.
In order to conduct the design and analysis of the optical system, the optical design software was used to model the optical system with a Cassegrain telescope as the main body. The optical system’s parameters were determined through analyzing and optimizing the aberration and echo receiving efficiency. The detailed parameters of the Cassegrain structure are shown in Fig. 5. From the optimized system data, the numerical aperture of the image square is about 0.095, which is smaller than that of a general single-mode fiber. Therefore, the light beam passing through the designed optical system can be efficiently coupled into the fiber.
4. IMPLEMENTATION OF THE METHOD
A. System Schematic
The schematic of the free-space distance measurement system for noncooperative targets is shown in Fig. 6. The broadband light source provides optical signals. The microwave vector network analyzer (VNA) served as the microwave source and signal processor. The optical signal is amplitude modulated by the microwave at the electro-optical modulator. To stabilize the modulation process, 1% of the modulated signal is fed back to the bias voltage control system, which integrates photoelectric detection, lock-in amplifier, sinusoidal signal generator, operation circuit, A/D conversion, and other modules. Using a slope detection method, it can automatically find the operating point and efficiently stabilize the modulated signal. Reference [18] introduces the bias voltage automatic control system in detail. The remaining 99% of the modulated signal is amplified by an erbium-doped fiber amplifier (EDFA) and divided into two beams, which are used as the measurement signal and the reference signal, respectively. The measurement signal is sent into the optical system in free space and carries the distance information of the object to be measured. After being received by the receiving lens, it interferes with the reference signal at the coupler. The interferometric signal is sent to Port 2 of the VNA after photoelectric detection and conversion. By sweeping the VNA frequency, the microwave interferogram of the system can be obtained.
Fig. 7. Implementation results of the proposed system with (a) ${z_1} = {0.1}\;{\rm{m}}$, ${z_2} = {0.4}\;{\rm{m}}$; and (b) ${z_1} = {1.0}\;{\rm{m}}$, ${z_2} = {2.0}\;{\rm{m}}$.
The ${\rm{C}} + {\rm{L}}$ band ASE light source can achieve a wavelength bandwidth of 100 nm, which is wide enough to eliminate the influence of optical interference on the measurement system. The gain of EDFA can reach more than 30 dB, which can ensure that the measured signal power is high enough to be effectively received after the loss of free space transmission and diffuse reflection. Using graded-index fiber collimators as transmit and receive optical couplers for the spatial optical system, which have lower modal dispersion and less bending loss, can effectively improve signal reception efficiency. It can be seen from Eq. (7) that the interference signal is a function of the OPD of the two paths. Thus, the distance information can be extracted from the interferogram provided by VNA.
B. Implementation Results
To verify the feasibility of the proposed system, distance measurement experiments were conducted. We set several groups of the lengths of the reference path and the measurement path, which are represented as ${z_1}$ and ${z_2}$, respectively. The common electric length was set to be 1 m. The step frequency of the microwave sweep was set to be 10 Hz, which is the minimum step frequency of the VNA currently owned by our research group. By scanning the frequency in the range of 0–6 GHz, the microwave interferogram was acquired. Figure 7 shows the implementation results of the proposed system with (a) ${z_1} = {0.1}\;{\rm{m}}$, ${z_2} = {0.4}\;{\rm{m}}$ and (b) ${z_1} = {1.0}\;{\rm{m}}$, ${z_2} = {2.0}\;{\rm{m}}$, including the amplitude spectrum and the phase spectrum of the interference signal. The spectrum’s free spectral range (FSR) is also shown in Fig. 7, which is measured to be 1.18483432 and 0.56390983 GHz. Based on Eq. (7), the FSR of the spectrum is a function of the OPD, given by the formula ${\rm{OPD}} = {\rm{c}}/{\rm{FSR}}$. Thus, the calculated distances of the measurement paths were 0.399999959 and 1.999999950 m, which were in good agreement with the initial setting distance value.
The minimum step frequency of the VNA currently owned by our research group is 10 Hz, which means that the resolution of the FSR is 10 Hz, and the corresponding distance measurement resolution is calculated to be 0.033 µm. With mixed ${z_1}$ to be 0.1 m, more measurement results with different set values of ${z_2}$ are as shown in Fig. 8. The measured FSR value is represented by the blue bars, the calculated distance values by the red line, and the measurement errors by the pink line. The maximum, minimum, and average errors of the measurements are analyzed and listed in Table 1. The distance calculation results are consistent with the set lengths of the measurement path. The measurement errors are within 0.1 µm, and the minimum measurement error reaches 0.034 µm, which verifies the theoretical system resolution described above.
Table 1. Indicators of Measurement Error
Fig. 9. Amplitude spectrums of the interference signal at (a) ${\rm{SNR}} = {{5}}\;{\rm{dB}}$ and (b) ${\rm{SNR}} = {{30}}\;{\rm{dB}}$.
C. Antinoise Performance
Considering that, in harsh environments, the coupling of the space signal into single-mode fibers will be disturbed by environmental factors such as atmospheric fluctuations and shot noise, which will affect measurement accuracy. Simulations were made to explore the influence of environmental factors on the measurement accuracy under different noise levels. Since the ambient noise is independent of frequency, it was set to Gaussian white noise with signal-to-noise ratios (SNR) of 5, 10, 15, 20, 25, and 30 dB, respectively. In the case of ${z_1} = {0.1}\;{\rm{m}}$, ${z_2} = {0.4}\;{\rm{m}}$, the amplitude spectrums of the interference signal at SNR of 5 and 30 dB are as shown in Fig. 9. The measurement errors at different noise levels are as shown in Fig. 10.
Since the detected interference signals are doped with noise, the visibility of the spectrum is reduced and the interferogram appears as aberrations. Thus, the points with a visibility exceeding 40 dB are selected to calculate the value of FSR. The measured FSR values deviate, resulting in measurement errors. As can be seen from Fig. 10, the measurement error reaches tens of micrometers when the SNR is below 15 dB, but the error drops to about 1.3 µm when the SNR reaches 30 dB. Therefore, the proposed method can maintain a high ranging accuracy even under the influence of environmental factors.
5. DISCUSSION AND CONCLUSION
In this paper, a high-precision free-space distance measurement method for noncooperative targets is introduced for the first time. Based on the principle of optical-carrier-based microwave interference, combined with the utilization of a broadband light source, this method reads the interference signal in the microwave domain. With the absence of cooperative targets, the space optical system with a Cassegrain telescope as the main body ensures the effective reception of backscattered signals. By sweeping the VNA frequency, the distance information can be precisely extracted from the interferogram. Compared with traditional interferometers, it has many unique advantages, such as small footprint, fast processing speed, high immunity to disturbances, and high measurement accuracy. A spatial distance measurement system was implemented to verify the practicability of the proposed method. The space distances were resolved unambiguously through the interferogram; they also matched well with the set distance value. With the aid of a VNA with a minimum step frequency of 10 Hz, the system can achieve a measurement resolution of 0.033 µm. The errors of the distance measurement experiments are within 0.1 µm. Even under ambient noise with a 30 dB SNR, the measurement accuracy can reach about 1.3 µm.
The range of the proposed method can reach several hundreds of meters provided that the spatial optical system can effectively receive the measurement signal. The system can achieve micrometer-level measurement accuracy even at long distances with the aid of high-resolution VNAs. The proposed method enables the extension of OCMI-sensing technology to free space measurements. In addition to its use for tracking and positioning, the proposed method can also be implemented to the measurement of other physical quantities by encoding the parameters to be measured into the interferogram. Therefore, this method has broad application prospects in the field of sensing.
Funding
National Natural Science Foundation of China (62071325); Sichuan Province Science and Technology Support Program (2021YFSY0024).
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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