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Abstract
High-precision spatial ranging plays a significant role in both scientific research and industrial practice. However, it is difficult for existing equipment to achieve high speed, high precision, and long distance simultaneously. Inspired by the concept of optical carrier-based microwave interferometry (OCMI), this paper reports a method of high-precision spatial distance measurement. A microwave-modulated broadband optical signal is sent to the interferometer whose measuring arm is an optical echo receiving system in free space. By scanning the microwave frequency, the measured distance can be resolved from the interferogram. Since the processing of the interference spectrum is performed in the microwave domain, this method is insensitive to the types of optical waveguides and states of optical polarizations. The experimental results show that the root mean square error (RMSE) of ten repeated measurements at 0.5 m is 0.016 µm, the RMSE is 0.023 µm within a 1 m distance, which can effectively represent the length measuring capability of the proposed system.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High-precision ranging plays an important role in both scientific and industrial areas. Optical and microwave interferometers are the common methods in the field of distance measurement. Optical devices in general have higher quality factors compared with microwave devices but sensitive to optical waveguides (e.g., geometry, dispersion, modal and material characteristics). While microwave interference can be easily resolved within its fundamental oscillation frequency. However, when used for sensing, the long-distance transmission loss of microwave signals is large, and it is difficult to be manipulated with high precision. In order to bring together the strengths from both microwave and optics, microwave-photonics has been explored in recent years [1–6]. It applies functions of optical technologies in microwave systems which are very complex or even impossible to carry out directly in the radiofrequency (RF) domain. The quality and performance of many instruments such as microwave sources [7–9], microwave photonic links [10–12], phased array antennas [13–15], frequency-tunable filters [16–18] and high-speed analog-to-digital convertors [19–21] have been greatly improved. On the basis of these research advances, the concept called optical carrier-based microwave interferometry (OCMI) was proposed and experimentally verified by Huang et al [22–25].
The essence of OCMI is to read optical interferometers in the radiofrequency (RF) domain instead of in the optical domain. It inherits the advantages of laser interferometers, such as small footprint, high signal-to-noise ratio, and insensitivity to electromagnetic inference (EMI). Moreover, observing interference in microwave region, OCMI has other unique advantages over traditional laser interferometers such as insensitive to the types of optical waveguides and states of optical polarizations, and relieves instrument manufacturing requirements. However, the OCMI concept is currently only implemented in fiber optic sensing [26–27], such as the continuous distributed fiber optic sensing system [28], which has certain limitations for spatial distance measurement.
To achieve high-precision measurement of free-space distances, we propose a method of spatial distance measurement in this paper. Proved that using broadband light source as carrier, the coherence length is greatly reduced compared with a narrow-band light source. Thus it’s easy to ensure that the microwave envelope is coherently superposed and the optical signal itself does not interfere, which greatly improves the signal quality. To realize the automatic control of the electro-optic modulator, we designed a control system including driving module and a bias voltage control module. In the interferometric part, a signal shaping transmission and echo receiving system in free space is designed to ensure the effective reception of the measurement signal. By resolving the information in the microwave interferograms, the change in the optical path difference (OPD) can be determined, from which the distance to be measured can be calculated. The feasibility of the proposed measurement system is experimentally demonstrated. The results show that the measurement accuracy of this system can reach nanometer level with a 10 Hz step frequency of the microwave sweeping process. The measurement range could be extended from meters to tens or even hundreds of meters. Also, the measurement theory of the proposed method hold the potential for more optical sensing applications of other physical quantities.
The remainder of this paper is organized as follows. Section 2 explains the concept of the method of high-precision spatial distance measurement based on optical-carried microwave interference. Mathematical models of broadband light source interference and the proposed distance measurement method are established. The theoretical measurement sensitivity is calculated. Section 3 presents the experimental demonstration of the proposed method, including the system implementation, the performance tests of the modulation system, two parts of the ranging experiments and the experimental results. In section 4, final remarks are offered on the performed research work. The application prospect of the proposed measurement method is also discussed.
2. Concept
2.1. Principle of the method of high-precision spatial distance measurement
Figure 1 illustrates the fundamental concept of the method of high-precision spatial distance measurement based on optical-carried microwave interference. As shown in Fig. 1, the broadband light source is intensity-modulated by a microwave signal to obtain a modulated signal with the optical as the carrier and the microwave as the envelope. The modulated signal is then sent into the circulator as the measurement signal. After the shaping transmission and echo reception of the optical system, signal of the measuring path interferes with that of the reference path. The interference signal is detected by a high-speed photodetector and transmitted to the signal processing system. As the bandwidth of the light source increases, the visibility of interference fringes gradually decreases, and the influence of optical interference will be eliminated. Therefore, we can read optical interferometers in the radiofrequency (RF) domain and extract the distance information.
Fig. 1. Conceptual illustration of the spatial distance measurement based on optical-carried microwave interference.
2.2. Modeling of broadband light interference
The extended broadband light source can be considered as that consisting of multiple mutually non-coherent sources. Each of them generates its own interference pattern. Hence, the overall intensity is equal to the intensity sum of component patterns. The degree of light interference can be described by the visibility of interference fringes.
The spectrum of the broadband light source is approximately a Gaussian function. The light intensity at a certain wave vector can be expressed as
where ${I_0}$ is the light intensity at the center wavelength, B is the bandwidth of the light source.The light source is split into two optical paths, propagating through different paths (d1 and d2). Based on the interference principle, the superimposed light intensity can be expressed as
where k is the wave vector, n is the refractive index of air.Under the Gaussian light source model, the intensity of spectral components in the range of wave vector k∼dk in the interference field is
Hence, the overall light intensity generated by light waves of different wave vectors in the interference field is
After simplification and calculation, we can get:
The visibility of interference fringes can be determined as:
It can be seen from the above formula that the visibility of the fringes decreases as the bandwidth of the light source increases. The simulated spectral interference pattern of light sources with different bandwidths is shown in Fig. 2. When the fringe visibility is close to zero, it is manifested as an increase in the overall light intensity. Thus, the influence of optical interference will be eliminated when using broadband light source.
2.3. Modeling of the spatial distance measurement method
The intensity-modulated optical wave is split into two paths at the circulator, passing through the reference arm and the spatial measurement optical path respectively. We assume two-beam interference with equal amplitude. The mathematical model of the interference principle has been introduced in detail in Ref. [29]. After DC filtering, the interference signal obtained by the photodetector is given by:
In the above formula, t is the time; A and M are the amplitudes of the optical carrier and microwave envelope, respectively; Ω is the microwave angular frequencies; c is the speed of light in vacuum; W is the electrical length of the common microwave path; L1 and L2 are the two optical path lengths, expressed as n1z1 and n2z2, where n1 and n2 are the refractive index of fiber and air, z1 and z2 are the lengths of the reference arm and the measuring arm in free space, respectively; g is the modulation depth of the modulator. It can be seen from Eq. (7) that the amplitude of the S signal changes periodically with the change of the OPD. By scanning the frequency, the microwave amplitude spectrum and phase spectrum can be acquired. Then the measured object can be located by demodulating the spectrum.
2.4. Sensitivity
A microwave vector network analyzer (VNA) is used as the microwave source and signal processor. The minimum step frequency of the VNA used in the experiment is 10 Hz. Thus the resolution of free spectral range (FSR) is 10 Hz. Based on Eq. (7), the OPD of the interferometer can be calculated by the formula OPD = c/FSR. It can be calculated that when the measuring distance are 1 m and 10 m, the theoretical resolution value can reach 0.033 µm and 3.300 µm respectively. The comparison between our system and existing rangefinders is shown in Table 1. The proposed ranging method has good performance in the prior existing technologies.
Table 1. Comparison between our system and existing rangefinders
3. Experimental demonstration
According to the aforementioned modeling results, the free space distance can be calculated by the microwave interferogram in spectrum domain. To validate the proposed concept, we built a free space ranging system and conducted a series of ranging experiments.
3.1. System implementation
The structure of the system is shown in Fig. 3. An ASE light source with a bandwidth of 75 nm is used to provide the optical beam. A microwave vector network analyzer (VNA) is used as the microwave source and signal processor. In the modulation part, the broadband light source is intensity modulated by the microwave signal from the port 1 of the VNA. The microwave signal is amplified by a driving module to drive the electro-optic modulator (EOM) to work. The driver consists of several stages of amplification modules and adjustable attenuators to achieve automatic gain adjustment. The bias voltage control system is a feedback structure. Using the slope detection method based on the input disturbance signal, the operating point can be automatically found without initial bias scanning, and the bias voltage of the EOM can be effectively stabilized. The modulated light is then fed into an interferometer with a reference arm in optical fibers and a measuring arm in free space. The superposition of the beams of the reference and measurement paths is a function of the OPD of the two paths, as shown in Eq. (7). The interference signal is received by a photodetector with a bandwidth of 67 GHz, DC filtered and amplified, and sent back to port 2 of the VNA. By sweeping the VNA frequency, the microwave amplitude spectrum of the system is obtained.
In the interference part of this system, the measuring beam is reflected from the object to be measured, then recombined with the reference beam at the fiber coupler. A reflecting prism is placed at the position to be measured in the experiment. In order to eliminate the aberration and dispersion in the spatial optical path, we designed a shaping structure including doublet lens pairs and converging lens. The transmitting and receiving fiber optic coupler was made of graded-index (GRIN) fibers who has lower modal dispersion and smaller bending loss. They are at the same distance from the object to be measured. Thus assuming 1.0 to be the effective refractive index of the atmosphere, the value of the OPD is twice the distance to be measured. The microwave interferogram can be analyzed to determine the OPD for the purpose of distance measuring.
3.2. Performance test of the modulation part
In order to ensure that the follow-up ranging experiments can be carried out effectively, it is necessary to test the modulation part, including the performance of the driver and the bias voltage control system. Set the output power of the light source to 66.00 mW, the RF signal frequency to 1 GHz, and the amplitude to 5 dBm. The power of the output signal of the modulation part is observed, and its variation curve is shown in Fig. 4. The positions of the three special operating points can be clearly seen, and the operating point can be automatically found and kept stable after entering the control system.
To verify the long-term stability of the output signal, its power was observed for up to an hour in the same environment. The curve of output signal power versus time is shown in Fig. 5(a). As a comparison, Fig. 5(b) shows the output signal power of the modulation part without the control system. It can be seen that the operating point of the modulation part without the control system drifts continuously with the change of time, and the maximum drift reaches 5.40 mW. While the operating point using the control system quickly stabilizes around a fixed value, and the maximum drift is about 1.00 mW, which is reduced by about 81.48%. Experiments show that the designed modulation part effectively realizes the functions of automatic tracking of the operating point and stabilize the output modulated signal.
Fig. 5. One-hour output signal power of the modulation part. (a) With the designed control system. (b) Without the control system.
3.3. Ranging experiments
3.3.1. Repeatable ranging experiments for a distance of 0.5 meters
Set the VNA sweep frequency range from 1 to 6 GHz, the reference point was calibrated in the aforementioned system configuration. The spectrum when the distance to be measured is 0.5 meters is shown in the Fig. 6. As can be seen, the fringe visibility exceeds 45dB at the microwave frequency of about 5.3 GHz, indicating that the system has a high SNR. For comparison, Fig. 7. shows an ideal interferogram with high fringe visibility and its FSR is a fixed value. Considering that some of the waves with low fringe visibility are distorted in the interferograms obtained from the experiments, we selected the points with a visibility exceeding 40dB over the 6-GHz frequency span to calculate the value of FSR. The measured value of the FSR was 299.99998 MHz. Thus the OPD of the interferometer can be calculated by the formula OPD = c/FSR, which was estimated to be 1.000000050 m. The distance to be measured is 0.500000025 m and the measurement error is 0.025 µm.
The 0.5-meter ranging experiments were carried out in the same environment for ten times, the measured values and the measurement errors are shown in Fig. 8. As can be seen, the measurement errors fluctuate within 0.030 µm. The indicators of precision evaluation are calculated and listed in Table 2, including the maximum error, the minimum error, the mean absolute error (MAE) and the root mean square error (RMSE). They demonstrate the small bias and high accuracy of repeated ranging measurements under the same conditions.
Fig. 8. Experimental results of 0.5-meter ranging. Top, measured results; bottom, measurement errors of ten times
Table 2. Precision evaluation indicators of 0.5 m ranging experiments
3.3.2. Ranging experiments for a distance of 0 to 1 meter
Keep the microwave sweep range unchanged, after calibrating the reference point, the target prism was moved from 0 to 1 m with a step size of 0.05 m. The measured results are presented in Fig. 9 (top), while the residuals between the measured and reference values are presented in Fig. 9 (bottom). The calculated precision evaluation indicators are listed in Table 3. The measurement results agreed well with the displacement value set by the precision guide rail, which can effectively represent the distance measuring capability of the built system.
Fig. 9. Experimental results of distance ranging. Top, measured results versus set distance; bottom, residuals between the measured results and set distance.
Table 3. Precision evaluation indicators of 0∼1 m ranging experiments
4. Discussion and conclusion
In this paper, aiming at achieving high precision ranging in free space, the method of high-precision space distance measurement based on optical-carried microwave interference is introduced for the first time. The fundamental principle is to send the microwave-modulated broadband optical signal through an interferometer whose reference arm is in optical fibers and the measuring arm is in free space. By sweeping the VNA frequency, the microwave spectrum can be used to resolve the measured distance. It has some unique features that the conventional interferometers do not have, including high signal quality, fast processing speed, and high precision. A distance measuring system is introduced to demonstrate the effectiveness of the proposed method. The space distance were resolved unambiguously and they matched well with the distance setting of the precision guide rail. The RMSE of a fixed distance measurement reaches 0.016 µm and the RMSE reaches 0.023 µm within a 1 m distance.
For the proposed system, the maximum measurable distance is determined by the power of optical source and the spatial optical structure. Therefore, with a high-resolution VNA, the system has the potential to extend the measurement range to tens or even hundreds of meters if only the optical signal can be effectively received. Further, in addition to being used in systems such as laser positioning and tracking, this method can also be applied to the measurement of other physical quantities by encoding the parameters to be measured into the interferogram. It is envisioned that the proposed method has broad application prospects.
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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