## Abstract

We propose an absolute distance measurement method using alternately oscillating optoelectronic oscillators (OEOs) with high speed, high precision, and long range, and describe the dynamic characteristics of the measurement system. Measurement and reference OEOs are oscillated using a 2×2 optical switch, and rough and fine measurements are achieved by low- and high-order-mode oscillation. The distance is determined by the loop length difference between the two OEOs. OEO length control is not necessary, so the system is simple and the time per measurement is only 40 ms. The maximum measurement error is 3.4 µm with an emulated distance of 7.5 km, and the relative measurement accuracy reaches 4.5×10^{−10}.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Absolute distance measurement (ADM) plays important roles in aerospace technology, academic research, and advanced manufacturing [1–3], including in satellite ranging and airplane assembly. Traditional ADM methods can be categorized as time-of-flight (TOF) and interferometric measurement methods. In the TOF methods, the time that light takes to make a round trip across the distance is measured. There is no ambiguous range, which makes these methods suitable for distance measurement over large ranges. However, the resolutions of these methods are limited to the millimeter scale due to the limited time measurement capabilities [4]. Interferometric measurement methods, including multi-wavelength interferometry (MWI), and frequency-sweeping interferometry (FSI), can achieve ultra-high precision measurement resolution by utilizing synthetic wavelengths or counting synthetic fringes. However, the ambiguous ranges of these methods are limited to several tens of meters and the systems are complex. For distances less than 25 m, MWI, FSI, or a hybrid of these methods can reduce the relative measurement uncertainty to 1×10^{−7} [5–9].

With the advent of optical frequency combs (OFCs), ADM methods in which an OFC is combined with MWI [10,11], TOF measurement [12–14], dispersive interferometry [15,16], etc. [17–20], have been reported on. These approaches further improve the measurement accuracy and extend the working range. In 2010, Lee et al. demonstrated a TOF method using femtosecond light pulses [12]. An Allan deviation of 117 nm over a 0.7 km distance in air was achieved by optical cross-correlation with the help of the frequency combs of femtosecond lasers. In 2018, Zhao et al. proposed a method using a triple-comb-based multiheterodyne interferometer [11]. An uncertainty within 750 nm over an 80 m distance was achieved with an update rate of 167 µs by detecting the phase changes of the consecutive synthetic wavelengths. In general, the aforementioned methods provide improved accuracy by increasing the resolution of the measurement system, which places high demands on the system. In other words, it is difficult to improve the measurement accuracy and working range simultaneously. The working ranges of these high-precision distance measurement systems are generally limited to hundreds of meters [10–20].

In the field of measurement, another commonly used principle is accumulative magnification. This approach improves the measurement accuracy by magnifying the quantity to be measured, providing high measurement accuracy at relatively low resolution. Distance measurement methods based on optoelectronic oscillators (OEOs) utilizing this principle have been proposed, and corresponding verification experiments have been conducted [21–23]. In these methods, the distance to be measured is placed in the loop of the OEO, and distance information is obtained based on the relationship between the loop length and oscillation frequency. As the loop length of an OEO is several kilometers, long-range distance measurement is possible. According to the accumulative magnification effect, high measurement accuracy can be obtained by measuring the high-order oscillation frequencies of the OEO. However, the oscillation frequency indicates the entire length of the OEO, which includes the inherent length as well as the length to be measured. Separating these two lengths to obtain the distance to be measured is difficult technically. In [22,23], two OEOs were established to solve this problem. The distance to be measured was placed in one OEO as the measurement OEO. Another OEO shared the length of the measurement loop not including the distance to be measured as the reference OEO. The distance to be measured was the length difference between the two OEOs. The experimental results showed a 2.5×10^{−10} relative measurement accuracy when the emulated round-trip distance was 6 km. It is worth noting that the length of the reference loop is stabilized in this method by phase-lock control of the piezoelectric ceramic transducer (PZT) to ensure measurement accuracy. To achieve control accuracy on the micrometer level, complex control algorithms are necessary to control several PZTs with different precisions, which increases the complexity of the system and prevents practical usage. In addition, the measurement and reference OEOs do not strictly share all the loop lengths except the distance to be measured, leading to slow drift and causing errors in the measurement OEO even if the loop length of the reference OEO is controlled. To eliminate the limitation of controlling the loop length as well as the influence of slow drift of the length difference, we propose an ADM method based on alternately oscillating OEOs.

The concept of this method is that the measurement and reference OEOs are separated by a 2×2 magneto-optic switch (OS). When the status of the OS changes, the measurement and reference OEOs oscillate alternately. The two OEO loops are completely identical except for the length to be measured. When the switching interval is sufficiently fast, it can be considered that the common part of the two loops does not change with time. Then, the oscillation frequencies of the measurement and reference loops are recorded by respective frequency counters to calculate the loop lengths. The distance to be measured is the length difference between the two loops.

In this method, it is not necessary to consider the error accumulation due to slow drift and the length loop does not need to be controlled because the loop lengths of the measurement and reference OEOs are measured rapidly and repeatedly for each distance measurement. Furthermore, the measurement system is simplified effectively. Theoretical analysis and experimental verification of the scheme were also performed in this study. The experimental results showed that the relative measurement accuracy could reach 4.5×10^{−10} when the emulated round-trip distance was 7.5 km and that each measurement took only 40 ms. In addition, the dynamic characteristics of the system were studied.

## 2. Experiment setup and principle

The basic structure of the ADM system based on OEOs is shown in Fig. 1. Each OEO consists of a laser diode (optical output power of 100 mW), modulator (modulation bandwidth of 20 GHz), 2×2 magneto-optic switch (switching speed of 30 µs), single-mode fiber (SMF, 5 km long), photodetector (3 dB response bandwidth of 14 GHz), amplifier (SHF 806E), microwave switch (MS), filters (9.9 GHz narrow band-pass filter with 3-dB bandwidth of 10 MHz, 100 MHz lowpass filter), and microwave couplers. The linewidth of the laser diode is 170 kHz. The bias voltage of modulator was set at the quadrature point. A motorized linear stage with a retroreflector was placed on one side of two collimators (CLs) to simulate the distance change and verify the measurement accuracy. When the OS is in the cross status, the distance to be measured in space *D* is included in the OEO1 (marked with red lines) loop, defined as measurement loop. When the OS is in the parallel status, the OEO2 (marked with black lines) loop is defined as the reference loop, which does not include *D*. *D* is half the length difference between the two loops.

According to the basic principle of OEOs [24], the fundamental frequency of an OEO *f _{b}* is determined by the total group delay time

*τ*of the loop. Then, the relationship between the optical path of the OEO

*L*and

*f*can be expressed as

_{b}*c*is the speed of light in a vacuum. According to Eq. (1), the relationship between the optical path change Δ

*L*and frequency change Δ

*f*in a short time is Δ

_{b}*L*/

*L*= –Δ

*f*/

_{b}*f*. In the experiment, the length of the SMF was about 5 km and the refractive index of fiber is around 1.5. Since the length of the SMF is much longer than that of the rest part in the OEO loop, the estimated

_{b}*L*was about 7.5 km and

*f*was about 40 kHz. To measure a 1 µm length change, the relative measurement accuracy must be on the order of 10

_{b}^{−10}; thus, the measurement accuracy of

*f*should be on the order of 10

_{b}^{−6}Hz. However, the linewidth of the oscillation frequency at

*f*is much wider than that. It is impossible to achieve high precision by measuring

_{b}*f*directly. In general, an OEO oscillates at a high-order frequency

_{b}*f*(10

_{hm}^{10}Hz level) with millihertz linewidth [24,25], which is

*N*(10

_{hm}^{5}–10

^{6}) times that of

*f*.

_{b}*N*is the integer number of the high-order oscillating mode. Theoretically, the accuracy of

_{hm}*f*can reach the 10

_{b}^{−8}–10

^{−9}Hz level according to calculations based on the measured

*f*and

_{hm}*N*. Despite the limitations of the frequency counter resolution and the effects of environmental changes, the accuracy of the calculated

_{hm}*f*can actually reach the 10

_{b}^{−6}Hz level. Consequently, the measurement accuracy of

*L*can be improved to the 10

^{−10}level with the help of the cumulative magnification effect.

*L*is given by Equation (2) implies that the measurement accuracy is determined by the accuracies of

*f*and

_{hm}*N*.

_{hm}*f*is measured by a frequency counter with a 10

_{hm}^{−10}relative measurement resolution, and

*N*is given by where [] refers to the operation of rounding a number and ${f_{b}^{\ast}}$ denotes the roughly measured

_{hm}*f*. Substituting Eq. (3) into Eq. (2),

_{b}*L*can be rewritten as As

*N*is an integer, ${f_{b}^{\ast}}$ should be in the range

_{hm}*f*/(

_{hm}*N*+0.5) < ${f_{b}^{\ast}}$ <

_{hm}*f*/(

_{hm}*N*–0.5) to ensure that

_{hm}*N*is correct. Thus, the measurement accuracy of ${f_{b}^{\ast}}$ should be within

_{hm}*f*/

_{hm}*N*–

_{hm}*f*/(

_{hm}*N*±0.5) ≈ ±

_{hm}*f*/2${N_{hm}^{2}}$. Substituting

_{hm}*N*into Eq. (2), the required accuracy of ${f_{b}^{\ast}}$ is within ±

_{hm}*c*/2

^{2}*f*. Thus, it was necessary for ${f_{b}^{\ast}}$ to be within ± 0.08 Hz in this experiment. As an OEO may oscillate in an arbitrary mode within the passband of the filter, the differences between the oscillation frequencies of various oscillation modes are multiples of

_{hm}n^{2}L^{2}*f*. Thus, ${f_{b}^{\ast}}$ can be measured in this way. Due to the limited frequency counter accuracy (1 Hz@10 GHz), it is improper to measure ${f_{b}^{\ast}}$ directly in a high-order OEO oscillation mode. Consequently, the OEO was oscillated in low-order mode first to obtain ${f_{b}^{\ast}}$ from the frequency counter with higher low-frequency resolution (0.01 Hz@100 MHz). As shown in Fig. 1, we utilized an MS to select different filters to achieve high- and low-order mode oscillation. When the MS selected port B, the OEO oscillated through a 100 MHz bandwidth low-pass filter. Under these conditions, the resolution of the frequency counter reached 0.01 Hz easily, which meets the requirements of ${f_{b}^{\ast}}$. The reason for choosing a low-pass filter in low-order mode instead of a band-pass filter is that

_{b}*f*is determined by the loop length

_{b}*L*of the OEO. For different lengths,

*f*varies from kilohertz to megahertz, frequencies that could not be filtered out by a fixed narrow band-pass filter. Considering that low-order mode oscillation frequencies (defined as

_{b}*f*) are measured at different times due to switching the OS, a series of the measured frequency differences of

_{lm}*f*, defined as $\Delta {f_{lm}^{i}}$, are not equal to $\Delta N_{lm}^i{f_b}$ because of environmental changes. $\Delta {N_{lm}^{i}}$ are defined as the oscillating mode number of $\Delta {f_{lm}^{i}}$ given by:

_{lm}_{kHz}are the rounded integers on the order of kilohertz, and

*f*

_{b}_{gcd}is the greatest common divisor of [$\Delta {f_{lm}^{i}}$]

_{kHz}. Δ

*f*

_{b}_{min }= $\Delta {f_{lm}^{i}}$/$\Delta {N_{lm}^{i}}$ is defined as the minimum frequency hopping interval. Similar to the previous analysis above, as long as Δ

*f*

_{b}_{min}is within ±

*c*/2

^{2}*f*, which was ±8 Hz in the experiment,

_{lm}n^{2}L^{2}*N*is correct. It is also equivalent that the loop length difference between two adjacent measurements should be within

_{lm}*cN*/

_{lm}*nf*–

_{lm}*c*(

*N*±0.5)/

_{lm}*nf*= ±

_{lm}*c*/2

*nf*, which was ± 1 m in our experiment. It is easy to satisfy this requirement when the measurement interval is short. Hence, the calculated ${f_{b}^{\ast}}$ is determined by Δ

_{lm}*f*

_{b}_{min},

*f*, and

_{lm}*N*, and can meet the requirements of ± 0.08 Hz.

_{lm}*N*and ${f_{b}^{\ast}}$ are given by Eqs. (6) and (7), respectively: It is worth noting that band-pass and low-pass filters selected by the MS directly may cause different group delay times in circuits. To bring ${f_{b}^{\ast}}$ calculated in the low-order mode directly into the high-order mode to calculate

_{lm}*N*, it was necessary to control the length difference between the two parts of the circuit to be within ±

_{hm}*c*/2

*nf*(±10 mm in the experiment), which could be achieved by adjusting the mechanical PS using a vector network analyzer.

_{hm}The optical path of an OEO is given by

In the complete measurement process, the OS and MS change in turn, while OEO1 and OEO2 oscillate alternately, as shown in Fig. 2(a). It is assumed that *f _{lm1}*,

*f*, and

_{hm1}*L*are the

_{1}*f*,

_{lm}*f*, and

_{hm}*L*of OEO1, respectively, while

*f*,

_{lm2}*f*, and

_{hm2}*L*are those of OEO2. When measuring ${f_{b}^{\ast}}$, the MS was set to port B, and the OS was switched in turn to measure

_{2}*f*and

_{lm1}*f*alternately. After ${f_{b}^{\ast}}$ was obtained, the MS was set to port A, and

_{lm2}*f*and

_{hm1}*f*were measured alternately. Finally,

_{hm2}*L*and

_{1}*L*were calculated.

_{2}*L _{1}* is determined by physical length of

*D*, fiber

*L*and circuits

_{fiber}*L*in OEO1, while

_{circuit}*L*is determined only by physical length of fiber

_{2}*L*and circuits

_{fiber}*L*in OEO2.

_{circuit}*L*and

_{1}*L*can be expressed as

_{2}*n*,

_{air}*n*,

_{fiber}*n*are the refractive index of air, fiber and circuits respectively. The distance to be measured is half the optical path difference between OEO1 and OEO2, when the effect of the refractive index of the air is neglected or under vacuum conditions,

_{circuit}*n*= 1 and optical path in the common part can be neglected. As the loop lengths are measured at different times, which may change during the period to affect the measurement accuracy. In our experiment, the OS switched alternately and the time interval was short. Because the instantaneous changes of the environmental conditions, such as temperature, were slow, it could be considered that the change of loop length over a short time was approximately linear. Therefore, we took the average of

_{air}*L*. The average

_{2}*L*was regarded as the loop length of OEO2 at the corresponding measurement moment of OEO1. This also eliminates the impact of environmental change to the common part between two adjacent measurements. From Eq. (9) and Eq (10), the measured distance becomes The process of measuring

_{2}*D*is shown in Fig. 2(b).

## 3. Experimental results

#### 3.1 Static characteristics

In the experiment, the performance of the system was first tested with a 5-km-long SMF. The frequency counter (Keysight 53230A) used a rubidium atomic clock as time reference to ensure accurate frequency measurement. The long-term frequency stability of the rubidium atomic clock was 5×10^{−12}. The gate time of the frequency counter was set to 10 ms. When the OEO oscillated in low-order mode, *f _{lm}* (∼100 MHz) was measured directly by the frequency counter with a resolution of 0.01 Hz. When the OEO oscillated in high-order mode,

*f*(∼9.9 GHz) mixed with a 9.91 GHz signal was generated by the microwave source to produce an intermediate frequency (∼10 MHz). The intermediate frequency could be measured faster than it could be measured directly by a frequency counter with the same resolution of 1 Hz, increasing the measurement speed.

_{hm}The switching times of the OS and MS are shown in Fig. 3(a). Between 0 ms and 10 ms, the OS was set to the cross status, the MS was set to port B, and OEO1 oscillated in low-order mode. Between 10 ms and 20 ms, *f _{lm1}* was recorded by the frequency counter. Between 20 ms and 30 ms, the OS was set to the parallel status, the MS was set to port B, and OEO2 oscillated in low-order mode. Between 30 ms and 40 ms,

*f*was recorded by the frequency counter. When mode hopping occurred in both the measured

_{lm2}*f*and

_{lm1}*f*, we obtained the roughly measured

_{lm2}*f*of OEO1 and OEO2, defined as ${f_{b1}^{\ast}}$ and ${f_{b2}^{\ast}}$ respectively. At this moment, MS was set to port A and the OS was switched. Then,

_{b}*f*and

_{hm1}*f*were measured alternately with a period of 40 ms. In other words, the time per measurement was 40 ms.

_{hm2}*f _{lm1}* and

*f*were measured by the frequency counter with the motorized linear stage in the initial position, and the results are shown in Fig. 3(b) as squares and circles, respectively.

_{lm2}*f*and

_{lm1}*f*were found to be around 110 MHz. During the measurement process, mode hopping occurred, which means that

_{lm2}*f*and

_{lm1}*f*oscillated in different modes. According to the analysis in Section 2, ${f_{b1}^{\ast}}$ and ${f_{b2}^{\ast}}$ can be calculated using Eqs. (6) and (7). In our experiment, ${f_{b1}^{\ast}}$ was about 40,130.11 Hz and ${f_{b2}^{\ast}}$ was about 40,144.23 Hz.

_{lm2}Then, we set the MS to port A and measured the high-order-mode oscillation frequencies of OEO1 and OEO2 10 times alternately. The measured values of *f _{hm1}* and

*f*are shown in Fig. 3(c). During this measurement process, mode hopping did not occur. The optical path of OEO1 and OEO2 were calculated using Eq. (8). The average loop lengths were also calculated to reduce the error caused by different measurement times and environmental changes. As can be seen from Fig. 3(d),

_{hm2}*L*decreases from 7470.514470 m to 7470.514454 m and

_{1}*L*decreases from 7467.884417 m to 7467.884402 m. The measured absolute distances, which are half the loop length differences between

_{2}*L*and

_{1}*L*at the same measurement time, tend to a fixed value. The average of nine distance measurements was calculated to be 1.315026 m with a standard deviation of 0.3 µm, which is the distance from the CL to the retroreflector.

_{2}The experimental results shown in Fig. 4 were obtained while moving the motorized linear stage over a distance of 0.8 m in steps of 50 mm. These results are compared with those obtained using a commercial laser interferometer (Renishaw XL-80) in Fig. 4(a). The left ordinate refers to the emulated distance and residuals measured by the OEO system, and the abscissa indicates the absolute distance change measured by the interferometer. The two sets of data exhibit good linear matching. The mean values of the differences are within ±1.8 µm, and the standard deviations of every measured distance are less than 1 µm. In addition, the relative measurement accuracy reaches 4.8×10^{−10}. To verify the working range of the system, we then used 1 km and 10 km SMFs to simulate 0.75 km and 7.5 km round-trip distances in space, and the results are shown in Figs. 4(b) and 4(c), respectively. With an emulated distance of 0.75 km, the residuals are within ±1 µm and the standard deviations of every measured distance are less than 1 µm. At the same time, the relative measurement accuracy is 1.3×10^{−9}. With an emulated distance of 7.5 km, the residuals are within ±3.4 µm and the standard deviations of every measured distance are less than 2 µm. The relative measurement accuracy is 4.5×10^{−10}.

#### 3.2 Dynamic characteristics

As described above, as long as the loop length difference between two adjacent measurements is less than ±*c*/2*nf _{hm}*, the correct

*N*can be obtained by substituting

_{hm}*f*from the previous measurement moment. Therefore, the target speed, defined as

_{b}*v*, is limited to ±

*c*/4

*nf*Δ

_{hm}*t*, where Δ

*t*is the time interval between two adjacent measurements. In our experiment, the SMF length was 5 km, Δ

*t*was 40 ms, and the maximum

*v*was limited to 126 mm/s. To verify the dynamic characteristics of the system, we recorded

*f*and then moved the motorized linear stage continuously. The motorized linear stage started moving at 1700ms from a position of 0 mm and stopped moving at 6060 ms at a position of 50 mm. The motorized linear stage moved with acceleration of 10 mm/s

_{hm}^{2}between 1700ms and 3680 ms. Then, it remained uniform motion with speed of 20 mm/s between 3680 ms and 4080 ms. The decelerated motion was from 4080 ms to 6060 ms. It was assumed that

*N*and

_{hm1}*f*were the

_{b1}*N*and

_{hm}*f*of OEO1, respectively, while

_{b}*N*and

_{hm2}*f*were those of OEO2. The measured values of

_{b2}*f*and

_{hm1}*f*(marked with black dots and red circles, respectively), as well as the calculated loop lengths of OEO1 and OEO2,

_{hm2}*L*and

_{1}*L*, (marked with blue triangles and magenta crosses, respectively), are shown in Fig. 5(a). As can be seen in Fig. 5(a), OEO2 oscillates in three high-order modes, for which

_{2}*N*is 246,569, 246,568, and 246,567. Although mode hopping occurred twice at 2960 ms and 4680 ms during the process of measuring

_{hm2}*f*, the calculated

_{hm2}*L*values are continuous upon substitution of the corresponding

_{2}*N*. In other words, the calculation of

_{hm2}*L*was not affected by mode hopping. Similarly,

_{2}*L*was obtained by measuring

_{1}*f*, substituting

_{hm1}*f*from the previous measurement moment, and calculating the corresponding

_{b1}*N*. During the measurement process,

_{hm1}*f*decreased with the movement of the motorized linear stage, and

_{b1}*N*increased from 246,656 to 246,658. The calculated

_{hm1}*L*increased as the motorized linear stage moved. Then, the distance was obtained by calculating half of the difference between

_{1}*L*and

_{1}*L*. The distances measured by the OEO system and interferometer shown in Fig. 5(b) are marked with black dots and red triangles, respectively. The data rate of the interferometer was set to 25 Hz (40 ms). Compared with the interferometer results, the errors of the measured distances (represented by the blue dotted line) are ±5 µm throughout the movement process.

_{2}The above results were obtained under the existing experimental conditions. The distance to be measured in space is within 0.8 m, where changes in air refractive index is neglected. In practical applications, the long-range distance to be measured in air should be placed only in OEO1 rather than in the common part of the measurement and reference loops, and refractive index in air should be considered. This approach reduces the length of the common part as well as the effects of the environment on the measurement system. In addition, the measurement speed is limited by the measurement time interval. If the frequency counter measures more rapidly with the same resolution, the measurement error of the common part will decrease and the measurement accuracy will improve further. The maximum speed of the target that can be measured will also increase.

## 4. Conclusion

We proposed an ADM method based on alternately oscillating OEOs. In this method, the distance to be measured is obtained by rough and fine measurement of the oscillation frequencies of the measurement and reference loops alternately. This approach has several advantages, including that the length drift caused by the environment is eliminated, the loop length of the OEO does not need to be controlled, and long-range measurement and high precision can be realized simultaneously. Consequently, the proposed ADM method is simple and flexible and requires only 40 ms for each measurement. The experimental results show that the relative measurement accuracy could reach 4.5×10^{−10} when the emulated round-trip distance was 7.5 km. The measurement errors were 3.4 µm and the standard deviation was 2 µm. A ±5 µm measurement error of a 20 mm/s moving target were obtained. The measurement accuracy will be improved if the measurement time interval is shorter. This method will have great potential in the applications of large-scale metrology and aerospace.

## Funding

National Natural Science Foundation of China (NSFC) (61427817, 61601321, 61775162).

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