## Abstract

In this paper, we propose a novel combined frequency-modulated continuous wave (FMCW) ladar autofocusing system and a fast compensation method for dispersion mismatch, which could allow high-precision ranging to be performed at a long distance. By using the dual-beam laser autofocusing system based on a liquid lens, this system can quickly complete a measurement with high-precision. The experimental results showed that the precision was below 126 μm in a range up to 60m, corresponding to a relative precision of 2.1 × 10 ^{−6}, compared to a reference interferometer.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

High-precision measurement has been a topic of constant study in the field of metrology. As time goes on, the pace for seeking a method of higher precision measurement has never stopped. At present, the requirements of high-speed, multi-target, high dynamic range, and high-precision seem to be a shared vision in various fields, such as manufacturing, aerospace, biomedical etc. Frequency-Modulated Continuous Wave (FMCW) laser radar, as a promising measuring method, has exceeded expectations since its inception. In 2014, Baumann et al. [1] built a set of scanning imaging systems based on frequency-comb calibrated FMCW techniques, which enabled the precise mapping of objects such as the tread of the shoe, metal, and cactus at stand-off distances of up to 10.5 m. The accuracy of measuring distance reached sub-micrometer, and the precision was less than 10 µm. In 2015, Mateo et al. [2] obtained the two-dimensional (2D) surface profiles of aluminum plates with different shapes at a standoff of 1.5 m by using the trilateration and high resolution FMCW ladar. The coordinate precision in all dimensions was below 200 µm. In 2018, Dilazaro et al. [3, 4] presented a 3D FMCW imaging system that provided high-resolution images over depths ranging from sub-millimeter to 6 m by using a 12-stitched Distributed Feedback (DFB) laser. In addition, the measuring volume was up to 14 cubic meters. In early work, the FMCW laser radar technique, also called Frequency Scanning Interferometry (FSI), was used in Optical Frequency Domain Reflectometry (OFDR) for bad spots detection and grating characterization in fiber networks [5, 6]. Moreover, FSI has been applied to Optical Coherence Tomography (OCT) for the examination of diseased tissue [7] and Synthetic Aperture Radar (SAR) for surface asperities [8, 9]. More recently, FSI has been used in precision measurements for motion scenes. The advanced interferometer contained FSI and image sensing detectors that could scan the surrounding environment and construct a 3D image over a hundred meters [10, 11]. Furthermore, Coe et al. [12] also used the FSI to monitor shape changes of the semiconductor tracker at CERN Geneva to reduce sensitivity to misalignment.

The FMCW laser radar technique is suitable for high-speed, multi-target measurement scenarios. However, its accuracy and precision are affected by environmental disturbances and light source stability. The movement errors caused by low frequency vibration can be tracked and compensated for by Kalman filter methods [13, 14]. The influence of light source stability mainly refers to the non-linearity of laser frequency sweeps, which deteriorates the spectrum and makes it unable to obtain the absolute distance precisely. Usually, this can be solved by the active linearization technique and post-processing schemes.

One of the methods of the active linearization technique is to track the frequency sweep error using a reference interferometer, or a high-precision and high-stability frequency reference. In 2011, Barber and Baumann et al. [1, 15] used an Optical Frequency Comb (OFC) as a standard frequency ruler to measure the frequency deviation from the ideal sweep linearity. This error was decreased to 15 ppb. In 2015, Mateo et al. [16] proposed a less-expensive traceable calibration method by using the molecular frequency references HCN or CO to measure the chirped error, and the calibrated sweeping linearity was below a few ppm. Moreover, the Phase-Locked Loop (PLL) is also a feasible solution. In 2009, Roos et al. [17] proposed a delayed self-heterodyne fiber interferometer and a standard reference source to detect linear sweeps residuals, and then fed back to the frequency actuator to maintain sweep linearity. The standard deviation of sweeping errors was decreased to 34 ppb after active controlling, and the precision was improved to 86 nm within a range of 1.5 m. Xie et al. [18] proposed an ultra-short delay Mach–Zehnder Interferometer (MZI) and a digital phase compensation technique to suppress the phase error, which led to a short lock time of 8 µs and robust operation while in a disturbed environment. In 2017, Behnam et al. [19] used an integrated electronic–photonic PLL to calibrate the sweeping error of a tunable laser for 3D imaging, which made the FMCW imaging technology more compact and less expensive. The precision was less than 8 μm at distances up to 50 cm.

Using a post-processing technique is the most popular way to eliminate light source nonlinearity. In 2005, Tae-Jung Ahn et al. [20] used Hilbert transform to obtain the actual chirp curve of the light source. Then, they used interpolation for resampling the optical frequency at equal intervals to eliminate the nonlinearity-induced phase error. The chirp curve can also be obtained by using the resonant peaks of the Fabry–Perot (FP) cavity [21]. The measurement accuracy of previous post-processing methods depends on the interpolation accuracy, which makes it difficult to guarantee the actual measurement. To improve this situation, an auxiliary interferometer whose optical path difference is at least twice that of the measuring interferometer was widely used [22–24]. With the increase of measuring range, the length of the optical fiber must be increased accordingly, so that the effects of fiber jitter and dispersion will become more and more significant. The variation of jitter can be corrected with a laser Doppler velocimeter [25]. Dispersion mismatch can be eliminated through chirp decomposition [26], iteration [27], interpolation [3, 28], etc. However, in most measurement applications, especially at long distances, these compensation methods are inefficient and unable to meet the requirements of real-time measurement.In previous work, we devoted ourselves to the instrumental production of frequency-modulated continuous wave ranging systems and the improvement of post-processing algorithms [23, 24, 28–30]. In this paper, we present a novel combined FMCW ladar ranging system, which is capable of fast and automatic focusing and micro-precision measurements within 60 m. To solve the dispersion mismatch, we proposed a concise and portable algorithm that can quickly correct spectral distortions. Further details and experimental verification are presented in the later sections.

## 2. Theoretical background

#### 2.1 Measuring principle of proposed system

Figure 1 shows the proposed combined measurement system. This system consists of four parts: the automatic guiding system, the combined measurement interferometer, the auxiliary interferometer, and the wavelength monitoring system. The laser beam of the continuous-wave tunable laser passes through two 80/20 splitters and a circulator. Next, it goes through a Wavelength Division Multiplexer (WDM) and the optical fiber output lens together with the exiting beam of the visible laser, is modulated by the focusing lens and reaches the target surface by rotating the mirror pair. Traditionally, the focusing lens uses the lens groups for the manual zooming lens. This is both time consuming and laborious in the process of long-distance measurement. To solve this problem, we used a liquid lens to replace the traditional zoom lens. The liquid lens is a kind of electrically controlled zoom lens filled with optical fluid that can display different shapes under voltage control, which further changes the focus length [31]. The liquid lens embedded in the focusing system was turned to a desired value within milliseconds to obtain the maximum optical coupling efficiency in distances of up to 60 m. Before starting the distance measurement, first the visible laser was used to find the position of the target point to be measured. Then an infrared camera was placed in front of the variable focus lens to monitor the spot size, and then feedback was used to adjust the liquid lens drive current value to ensure that the laser beam was guided to achieve the best focus state with the minimal focus spot. When the focus process of the target point was stable, the distance measurement was carried out. When moving to the next position, the above mentioned focusing process was repeated. Additional details have been described in previous work [29]. The combined measurement system consists of a conventional dual interferometer near-range measurement system [23–28], and an extended distance measurement system. As the distance increases, the return optical power of the measuring arm of the measuring interferometer becomes smaller and smaller. This is so that when it is combined with the Local Oscillator (LO) signal, it is easily submerged. In addition, for distances of 40 m and beyond, the length of the delay line of the auxiliary interferometer with Optical Path Difference (OPD) as 80.3 m needs to be over 100 m. However, the effects of ambient vibration and dispersion on these long fibers can introduce large re-sampling errors and phase errors. In order to eliminate this part of the impact, a Doppler link lock technique is widely used in the laboratory to lock the fibers. However, it leads to significant increase in system complexity and reduction in measurement efficiency. To solve this problem, we introduced a stable length delay line at the local oscillator end, which was placed in the isothermal isolator box. The beat signal is detected by the Photodetector 2 (PD2) and can meet the requirements necessary for measurement scene over 100 m, as shown in Fig. 1.

The beat signal detected by the Balance Detector (BD) output from the auxiliary interferometer can be expressed as:

*E*

_{0}represents the signal amplitude,

*ω*represents the angular frequency of the instantaneous optical frequency,

*R*

_{aux}is the length of auxiliary interferometer,

*c*is the light speed in vacuum,

*n(ω)*represents the refractive index of different laser wavelengths propagating on the fiber, and

*ϕ*

_{0}represents the initial phase.

The dispersion coefficient of a single-mode fiber can be expressed as [3]:

*β*

_{0}=

*n*

_{0}

*ω*

_{0}/

*c*,

*ω*

_{0}is the initial optical frequency,

*n*

_{0}is the refractive index of

*ω*

_{0},

*β*

_{1}is the reciprocal of group velocity, and

*β*

_{2}is the dispersion coefficient of a single-mode optical fiber.

To simplify the calculation, let the initial phase be zero. Substituting Eq. (2) into Eq. (1) yields:

*k*denotes the optical frequency sampling point index, and

*N*is the total number of sampling points. The calculation result of Eq. (4) can be written as:

Notably, the plus sign indicates optical frequency scan forwards and the minus sign indicates optical frequency scan backwards. In our experiment, the direction is forward. In addition, the value of 2*β*_{2}π*k*/(*β*_{1}^{2}*R*_{aux}) is between −1 and 1, which satisfies the Taylor expansion criteria. Therefore, Eq. (5) can be rewritten as:

Similarly, the beat signal of the measurement interferometer can be expressed as:

*τ*

_{OPD}denotes the time delay between the two arms of measurement interferometer, and OPD is the optical path difference of the two arms.

For a conventional near range measurement system, whose measurable range can only be less than one half of the auxiliary interferometer, the OPD can be defined as:

Where*R*

_{fiber}is the fiber length difference between two arms,

*R*

_{air}is the distance to be measured in the air, and

*n*

_{air}is the refractive index of air.

The beat signal measured by detector PD1 can be expressed as:

Substituting Eq. (4) and Eq. (6) into Eq. (9) yields:

*n*

_{g}is the group refractive index.

After performing FFT on Eq. (10), the resolved distance can be express as:

*f*

_{res}denotes the frequency of the resampling signal, and δ

*R*

_{dis}is the distance-shift caused by dispersion. The value depends on the chirp factor.

For an extended remote measurement system, whose measurable range depends on the length of the added delay line *R*_{0}. Noteworthy, the extra fiber length was slightly longer than fiber length of the auxiliary interferometer. The OPD of the new measuring interferometer can be expressed as follows:

Correspondingly, the beat signal detected by the detector PD2 can be expressed as:

Substituting Eq. (4) and Eq. (6) into Eq. (13) yields:

Therefore, the extended resolved distance can be written as:

It should be noted that the conditions for Eq. (15) are as follows:

For the opposite conditions, the resolved distance can be rewritten as:

In summary, the measurable distance of the combined measuring system can be summarized as:

#### 2.2 Method for dispersion correction

From Eq. (18), we can see that we cannot resolve distance precisely if the distortion factor, δ*R*_{dis}, is not eliminated. In Lu’s work [26], they proposed a chirp decomposition method to eliminate the distance distortion. However, this method was not suitable for low signal-to-noise ratio measurement situations, and the compensation efficiency was very low during large bandwidth scanning. The interpolation method could also eliminate distortion [3, 27, 28], but it could introduce interpolation errors and increase the post-processing system’s burden. For this reason, we propose a compensation method that is faster and more erasable. The principle is as follows:

Taking Eq. (10) as an example, it can be simplified as:

Where *A*(*k*) is the amplitude modulation signal which may be caused by the mechanical vibration inside the laser [32]. This could add a low-frequency modulation noise on the actual resampled signal. In addition, δ_{dis} is the distortion factor, and *θ*_{0} is the constant phase. The low frequency modulation signal can be written as:

Therefore, the actual resampled signal can be expressed as:

To eliminate this modulated noise, the steps are as follows:

First, build a new signal as follows:

Where $\tilde{U}(k)$is the Hilbert transform of *U*(*k*), and (*) denotes convolution. Therefore, the envelope signal can be expressed as:

The second step is to remove the envelope. The normalized resampling signal can be expressed as:

Then, Hilbert transform is performed on Eq. (24) to obtain the phase of the complex signal, which can be expressed as:

After unwrapping the phase, we perform a least-squares fit to the phase curve to obtain the fitted curve as:

Where *a* is the second order coefficient of the fitted function, *b* is the first order coefficient, and c is the constant coefficient.

The compensation factor is calculated as:

Finally, the revised resampling signal can be expressed as:

## 3. Ladar experiment and results

#### 3.1 Experimental configuration

The configuration of our system is shown in Fig. 1. The Continuous Wave (CW) tunable laser (Luna PHOENIX 1400) was used as a sweeping source, which has a tuning range of 1515-1565 nm and chirp rate of 1-2000 nm/s. The maximum average output optical power was 8 mW. The laser beam from the source was first split into two parts. 80% of the optical power further goes through the 80/20 beam splitter, 80% of which goes through the circulator, WDM, and then to the fiber output lens together with the visible laser (ADR-1805). The power launched into free-space was 2.15 mW. To increase the efficiency, we presented an automatic guiding system. The frequency of the visible laser for guiding and the infrared laser for ranging is 658 and 1550 nm, respectively. Notably, these two lasers do not share the same focus current for liquid lens. In other words, they cannot focus at the same target point at the same time. The working process of the system is as follows: First, the visible laser beam was modulated by the liquid lens (Optotune EL-10-30-Ci) which was electrically controlled by the driver for spot location. This guiding process was completed when the red spot was focused. Noteworthy, the guiding process was only used for finding the infrared spot location of the target. Second, the infrared camera (Chameleon CMLB-13S2M) was used as a monitor to detect the current returned laser beam size. Then, feedback was given to the controller to find the best focus size. The drive current of the liquid lens was adjusted based on the spot size analysis. When the minimum spot size was acquired, no more adjustment was carried out on the drive current. When the autofocusing process of the infrared laser was over and the minimum target point was obtained, the measurement was started. The returned beam was passed through the 3dB beam splitter. One part was combined with the local oscillator without delay line and sent to the photodetector 1 (Thorlabs PDA10CS-EC) for near-range measurement. The other part was combined with the local oscillator containing the delay line (82.3927 m) into the photodetector 2 for extended remote measurement. The remaining 20% of the output of the CW tunable source was also divided into two parts, of which 10% went through the auxiliary interferometer (80.3230 m) and was used as a trigger signal for data acquisition. The remaining part went through the gas absorption cell (Wavelength References HCN-13-H(16.5)-25-FC/APC) to ensure the same scanning bandwidth at each measurement, or to monitor the laser mode-hopping situation [30]. All the PD signals were simultaneously sampled and recorded by an oscilloscope (LeCroy Wave Runner 610Zi) at 500 Mega-samples per second (MS/s) and were then fed to MATLAB for further processing. In our experiment, the tuning range and measurement speed were set to 1545-1555 nm and 1000 nm/s, respectively.

#### 3.2 Dispersion correction and resolution measurement

The process of dispersion correction has been discussed in the previous section. According to the proposed method, the dispersion compensation was performed on the resampled signal at 60 m of standoff distance. The process of the compensation coefficient is shown in Fig. 2. As previously predicted, Fig. 2(a) shows that there is a low frequency noise modulation on the resampled signal. To eliminate this effect, we used Hilbert transform to remove the envelope and obtained a new resampled signal, as shown in Fig. 2(b). Then, we performed Hilbert transform again on the new signal to obtain the instantaneous phase. The new signal was a normalized signal in the time domain, which was especially favorable for the demodulating of the chirp phase. Then, Hilbert transform was performed again on the normalized signal to obtain the phase curve, as shown in Fig. 2(c). It should be noted that there were boundary effects when using the Hilbert transform, which led to the broadening of the final distance spectrum. However, when there were large enough resampled points, these effects for the distance solution can be ignored. Figure 2(d) shows the actual instantaneous phase curve after unwrapping. The resolved phase is shown in carmine, and the result of fitted phase by using a least square method is in black. In the case where the target is located at distance up to 60 m, the final quadratic phase function using the least square method is *ϕ*(*k*) = 5.1933 × 10^{−11} *k*^{2} + 0.2522 *k* + 0.9237. Correspondingly, the compensation coefficient is 8.2654 × 10^{−12}. Figure 2(e) shows the fit phase residual versus sampling point of the Fig. 2(d). Figure 2(e) reveals that the fitted phase residual is less than 0.3 rad compared to the measured phase.

We performed FFT on the resampled signal multiplied by the compensation coefficient to verify the compensation effect of this method. The tuning range was set as 1545-1555nm, and the actual sweep wavelength was recorded by the gas absorption cell [33] certified by the National Institute of Standards and Technology (NIST). The absorption peaks were fitted by a least squares polynomial fitting method to find the center of each 3-dB depth absorption feature. The P4 branch (1545.230337 nm) and the P15 branch (1553.75562 nm) were taken as the starting wavelength and end wavelength, respectively. In this case, the bandwidth-limited resolution can be approximated as 140.75 μm. 12 positions were measured from 5 m to 60 m using the combined measuring system. Each position was measured 10 times, and each group of resampled signals was compensated using the proposed method. Among them were the distance spectra without dispersion compensation (red), and with compensation (black), at a distance up to 60 m, as shown in Fig. 3. Figure 4 shows the ratio of the measured resolution with dispersion compensation and without dispersion compensation to the theoretical resolution, respectively. The vertical axis is in dB scale. The baseline of the theoretical resolution line shown in Fig. 4 is self-owned ratio with a value of 1. The error bars are the standard deviation of 10 times measurement. Before extracting the full width at half-maximum (FWHM) of the spectrum, an FFT was performed by zero-padding to 50 M points. The resulting spectral profile is an approximate *sinc* function whose distance resolution depends on the FWHM of the main lobe. Moreover, the minimum range resolution can be approximated as Δ*Z*_{min} ≈0.829Δ*Z*_{FWHM} for a rectangular window [3]. It can be clearly seen that after compensation using proposed method the measurement resolution was very close to the theoretical resolution within 60 m, and the maximum standard deviation was less than 3 μm.

#### 3.3 Accuracy and precision evaluation

We compared the results after dispersion compensation with the reference interferometer (Agilent 5530) to verify the measurement performance of the combined measurement system. We conducted an experimental verification in the underground 80 m guideway room of the National Institute of Metrology. The corner cube of our system and the corner prism of the reference interferometer were fixed on the same trolley and moved from 0 m to 65 m along the specific marble rail. It should be noted that the measurable range the combined system could be extended to 80 m, or even further, according to Eq. (16). However, it was limited to the length of the guide rail. Therefore, a total of 12 different positions were measured and compared, and each position was measured 10 times. The distance residuals versus the reference are shown in Fig. 5. The distance was measured by our combined measurement system and calibrated by the proposed dispersion compensation method. The red dot in Fig. 5 represents the average measurement residuals of the 10 measurements, and the black represents the precision at different positions. It can be seen from Fig. 5 that the maximum measurement residuals at the 12 different locations were less than 30 μm and the precision was below 120 μm.

According to Eq. (18), the measurement uncertainty with dispersion compensation is related to the uncertainties of the group refractive index *n*_{g}, the air refractive index *n*_{air}, the fiber length of the auxiliary interferometer *R*_{aux}, the added fiber length *R*_{0}, and the frequency of the resampling signal *f*_{res}. The environmental temperature, pressure, and humidity are stable at 34 mk, 11 pa, and 2%, respectively, corresponding to the air refractive index uncertainty of 3.2 × 10 ^{−8}, 3 × 10 ^{−8}, 1.4 × 10 ^{−8}.

The first term *n*_{g} is mainly dominated by ambient temperature and external vibration, and the fiber length of the auxiliary interferometer *R*_{aux} is mainly dominated by external vibration. In our experiment, we performed vibration isolation on the fiber of the auxiliary interferometer and added fiber. Therefore, the optical path difference *n*_{g}*R*_{aux} introduces a measurement standard uncertainty of 2.5 μm. The second term *n*_{air} introduces a measurement standard uncertainty of 4.6 × 10 ^{−8} *R*_{air}. In the extended remote measurement, the optical path difference *n*_{g}*R*_{0} leads to a measurement standard uncertainty of 2.6 μm. The last term *f*_{res} is mainly dominated by the fitted phase error. Within 60 m measurement, the fitted phase residual was less than 0.3 rad, corresponding to a measurement standard uncertainty of 5.6 μm. When the distance was up to 60 m, the measurement combined uncertainty was 126.7 μm, corresponding to a relative precision of 2.1 × 10 ^{−6}, compared to a reference interferometer.

Figure 5 shows that within 60 m range, the ranging accuracy compared to the reference interferometer is less than ± 30 μm and the relative ranging accuracy is 4.9 × 10 ^{−7}, which is less than 10 ^{−6}. The target mirrors of CW interferometer and the proposed ladar system are not fixed on the same line; therefore, the Abbe error slightly affects on the measurement result. In our experiment, the distance between the reference cube of the CW interferometer and the proposed ladar system was less than 120 mm, and the Abbe error was less than 2 μm along 60 m range.

## 4. Conclusion and future work

In this paper, we proposed a novel combined FMCW measurement system and a faster dispersion compensation method to achieve high-precision measurements at a long-distance range. This system could quickly complete a measurement with high-precision with its autofocus guiding design based on liquid lens. In addition, this combined-design and compensation method could be further extended to scenes of over hundreds of meters with μm-precision but was limited by the length of the guide rail. In the experiment, the minimum measured return power is less than 1 μW within 60 m. However, for even lower signal-to-noise ratio measurement situations, addition of an erbium-doped fiber amplifier after the CW tunable laser of this system might be suitable. The experimental results revealed that the precision was below 126 μm in a range up to 60 m, corresponding to a relative precision of 2.1 × 10 ^{−6}, compared to a reference interferometer. In future work, we will use this combined FMCW autofocus system and dispersion compensation method for the 3D mapping of diffuse surfaces.

## Funding

National Natural Science Foundation of China (NSFC) (51675380); Aeronautical Science Foundation of China (20160948001).

## Acknowledgments

We would like to thank Dr. M. He is in the National Institute of Metrology providing warm help. We gratefully acknowledge J. Hong for her great encouragement. We also thank the peer reviewers for their very helpful comments.

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