## Abstract

Two-color interferometry is powerful for the correction of the air refractive index especially in the turbulent air over long distance, since the empirical equations could introduce considerable measurement uncertainty if the environmental parameters cannot be measured with sufficient precision. In this paper, we demonstrate a method for absolute distance measurement with high-accuracy correction of air refractive index using two-color dispersive interferometry. The distances corresponding to the two wavelengths can be measured via the spectrograms captured by a CCD camera pair in real time. In the long-term experiment of the correction of air refractive index, the experimental results show a standard deviation of 3.3 × 10^{−8} for 12-h continuous measurement without the precise knowledge of the environmental conditions, while the variation of the air refractive index is about 2 × 10^{−6}. In the case of absolute distance measurement, the comparison with the fringe counting interferometer shows an agreement within 2.5 μm in 12 m range.

© 2016 Optical Society of America

## 1. Introduction

Since the invention of the femtosecond frequency comb, many applications both in technology and basic science have been developed with revolutionary progress. Optical frequency comb is a pulse train in space, and it can be seen as a combination of many phase-coherent single-frequency lasers with equal frequency spacing. With the well stabilization of the repetition frequency and the carrier-envelope-offset frequency to an external time reference, frequency comb becomes an ultra-stable ruler in time, frequency, and space domains [1]. As a frequency transfer, frequency comb can link an optical frequency to a microwave time reference in a single step, which represents that the frequency markers in a very wide spectral range from ultraviolet to THz regions share the same frequency precision as the frequency standard [2]. This inherent advantage has triggered a wealth of progress in the fields, such as precision spectroscopy [3], and distance metrology [4], etc. Since the first demonstration by Minoshima et al. in 2000 [5], frequency comb-based technique capable of absolute distance measurement has been developed for more than 15 years, and it is still progressing at a rapid pace. So far, various methods have been reported by the researchers all over the world, in the manners of interferometry or non-interferometry, e.g., time-of-flight [6], pulse cross correlation [7,8], pulse-to-pulse alignment [9–12], chirped pulse interferometry [13,14], heterodyne interferometry [15–17], and dispersive interferometry [18–20], etc. Thanks to the hard work of the researchers, these approaches can determine arbitrary distances with large dynamical range, high accuracy and fast responding speed, benefitting from the state-of-the-art technique of frequency comb. As mentioned by Newbury [21], however, the only limitation towards high-accuracy distance meter using frequency comb in air is the measurement uncertainty caused by the uncertainty of the air refractive index. Therefore, it is essential to correct the air refractive index with high accuracy. In general, well-established empirical equations are used for the correction of the air refractive index, with precise knowledge of the environmental conditions: temperature, air pressure, humidity, and CO_{2} concentration [22,23]. However, it is difficult to measure the exact distribution of these parameters in the total optical path. In the cases of large scale measurement out of the lab, it could become more difficult to precisely gather the real-time environmental conditions. On the other hand, the inherent uncertainty of 10^{−8} for the empirical formulas directly results in that the measurement uncertainty better than 10^{−7} is inaccessible in a range up to tens of meters. One solution can be the direct measurement of the air refractive index [24,25]. Recently, Hieta et al. presented a spectroscopic on-line system, and the environmental conditions can be measured by the absorption feature of the specific gas, which means that each single measurement can be corrected in real time [26].

Alternatively, two-color method first presented by Bender and Owens in 1965 can correct the air refractive index in real time without the need of precise knowledge of the environmental conditions [27]. Subsequently, Earnshaw and Owens demonstrated the experimental verification of this powerful method in long distance measurement, and the results show an uncertainty at 10^{−7} level over 5-km range [28,29]. The basic principle of the two-color method is that two lasers with different wavelengths simultaneously measure a geometrical distance *D*, and thus two optical paths of *D*_{1} and *D*_{2} can be obtained. *D*_{1} equals to *n*_{1}*D*, and *D*_{2} is *n*_{2}*D*, where *n*_{1} and *n*_{2} are the air refractive index corresponding to the two wavelengths, respectively. The geometrical distance *D* after correcting the refractive index of air can be expressed as:

*A*is the so-called

*A*coefficient in the two-color configuration.

As demonstrated by Meiners-Hagen et al. [30], based on Bonsch and Potulski formula [31], the dependences of temperature and pressure can be cancelled out, while the humidity still has contributions to the measurement uncertainty according to Eq. (1). However, as mentioned by Minoshima et al. [32], acceptable uncertainty of the humidity could be provided by the commercial sensors. In the case of two-color method, the required humidity uncertainty for a 10^{−7} uncertainty in distance measurement is 4%. Please note that the dependence of the humidity can be also cancelled out by using three-color method [33], but two coefficients should be used like the *A* coefficient in Eq. (1). In most practical cases, the two coefficients are too large, which makes the scheme rather difficult or even impossible. Consequently, two-color method is a powerful and alternative tool for the practical applications. A particular challenge of the two-color method is that the uncertainty of *D*_{2}-*D*_{1} will be enlarged by *A* times. In other words, for example, assuming we expect 10^{−7} uncertainty in distance measurement, the uncertainty of *D*_{2}-*D*_{1} should be at 10^{−9} level with *A* value of about 100. Such a high accuracy could be offered by the frequency comb technology.

In 2015, Hyun Jay Kang et al. proposed a two-color scheme with heterodyne interferometry by using a DFB laser, which is well stabilized to a frequency comb [34]. In this scheme, frequency comb is used as a wavelength transfer, so that the DFB laser can be traceable to the external time clock. The distance variations of the fundamental and second harmonic can be measured in real time via the phase of the heterodyne signals. After the real-time compensation of the air refractive index, the measurement uncertainty can achieve 10^{−8} in 2.5 m distance. Owing to the continuous wave laser as the working source, the correction of air refractive index can be fulfilled at arbitrary locations, but the non-ambiguity range is half of the wavelength, and the system involving two laser sources is complex. Frequency comb can serve as the direct light source to carry out the correction of the air refractive index. In 2010, Minoshima et al. presented a pulse-to-pulse interferometer based on homodyne interferometry, and the distance variation can be obtained by the interfered intensities [32]. In this scheme, a well-designed feedback loop was used to tune the repetition frequency to lock the peak of the fundamental intensity, and the non-ambiguity range of half the wavelength can be extended. During 10-h long-term measurement, the uncertainty of 5 × 10^{−8} was reached when the air refractive index was changed by up to 10^{−6}. The accuracy of the distance measurement is influenced by the intensity fluctuation due to the temperature drift of the PPLN crystal. To solve this problem, Guanhao Wu et al. investigated a heterodyne pulse-to-pulse interferometer, and the distance can be determined through the phase variations of the microwave signals for the fundamental and second harmonics [35]. Moreover, the correction over a very long path up to 61 m was carried out [36], and the uncertainty better than 7.7 × 10^{−9} for 10-h long-term measurement was achieved [37], which goes beyond the empirical formula. The non-ambiguity range was extended to be larger than half of the wavelength by using the same servo loop as that in [32]. We find that a tailored circuit is needed to overcome the half-wavelength non-ambiguity range, with careful consideration of the responding speed and the cable noise. In addition, two-color terrameter based on the amplitude modulation has also been developed [38]. More recently, scientists in PTB proposed a complete system based on frequency scanning interferometry, which is promising to realize the correction of the air refractive index. In particular, the measurement can be traceable to a stable spectral line [39]. It is possible to use method with high accuracy and larger non-ambiguity range directly, and dispersive interferometry could be a qualified candidate for the correction of the air refractive index. During the past decade, dispersive interferometry has been well investigated in absolute distance measurement with high accuracy, fast measuring speed, and sufficiently large non-ambiguity range [40], but by now there is not report about the long distance measurement with correction of the air refractive index using this powerful method. We try to investigate the scheme based on dispersive interferometry for high accuracy distance measurement with correction of air refractive index.

In this paper, we perform absolute distance measurement with the correction of air refractive index by using two-color dispersive interferometry. The distances corresponding to the fundamental and second harmonic can be measured via the spectrograms captured by a pair of CCD cameras. We first investigate the performance of the correction of the air refractive index. The experimental results show an uncertainty of 3.3 × 10^{−8} during 12-h long-term measurement while the variation of the air refractive index is 2 × 10^{−6}. For absolute distance measurement, the comparison between our scheme and the fringe counting interferometer shows an agreement within 2.5 μm over a distance of 12 m.

## 2. Experimental setup and CCD calibration

Figure 1 shows the experimental setup. The measurement system is composed of two parts: the main interferometer (frequency comb interferometer) and the reference interferometer (He-Ne interferometer).

In the main interferometer, frequency comb (Onefive Origami-15, 70 mW), whose repetition frequency is well locked to a time clock (Stanford FS725), emits a pulse light. The repetition frequency is 250.012 MHz, and the carrier-envelope-offset frequency is not actively controlled. The output of the frequency comb is then focused onto a piece of a periodically poled LiNbO_{3} (PPLN) crystal for second harmonic generation. Consequently, the fundamental (1560 nm) and the second harmonic (780 nm) are coaxially aligned into a Michelson-type interferometer. The output of the Michelson interferometer is separated by a dichroic mirror into two beams. The reflected beam is the fundamental harmonic, which is dispersed in space by a grating. A CCD camera (Chameleon CMLN-13S2M, 1296 × 964 pixels) is used to capture the spectrograms. The transmitting beam is the second harmonic, and we can picture the corresponding spectrograms in the same way as the fundamental harmonic. The CCD camera for the second harmonic is Point Grey GS3 with 2048 × 2048 pixels. Please note that, accurate synchronization between both the CCD cameras is required to measure the distance variations in real time. A signal generator is applied to synchronously trigger the CCD cameras with the rising edge. To avoid the possible thermal expansion, the experimental setup is located in a thermally isolated box in the lab. All the mechanical components are kept stationary, so that the geometrical distance *D* can be considered as a constant. The environmental conditions are measured and recorded in real time by the sensor network (USP3021 with 0.01°C temperature uncertainty, NPA201 with 10 Pa pressure uncertainty, and HIH9131 with 1.7% humidity uncertainty), and the results obtained from Ciddor formula can be applied to evaluate the results of our method.

In the case of the reference interferometer, a fringe counting interferometer (Agilent 5519B) is used to examine the measurement results of the main interferometer. Both the targets of the main interferometer and the reference interferometer are located on the same PC-controlled carriage. That is to say the main interferometer and the reference interferometer do not share the same target, and thus Abbe error could exist during the long travel. To suppress the Abbe error, the Abbe offset should be as small as possible, which is 80 mm in our cases. Both the measurement beams of the main interferometer and the reference interferometer are aligned to be parallel to the long optical rail to eliminate the possible cosine error.

Figure 2(a) shows the spectrum of the fundamental harmonic which is centered at about 1570 nm with about 50 nm spectral width. Figure 2(b) indicates the spectrum of the second harmonic which is centered at about 791 nm with about 5 nm spectral width.

#### 2.1 CCD calibration

Here, we would like to give a very short theoretical description of the dispersive interferometry, for the convenience of the explanation of the CCD calibration. The spectrogram *I*(*ω*) can be given as *Dc*(*ω*) + *Ac*(*ω*) cos(τω), where *Dc*(*ω*) is the dc offset, and *Ac*(*ω*) is the ac amplitude of the spectrogram. *ω* is the angular frequency, and *τ* is the time delay between the measurement pulse and the reference pulse. As well-known, the absolute distance can be measured via the slope of the unwrapped spectral phase, i.e., *L = cτ/*(*2n*). *L* is the unknown distance, *c* is the light speed in vacuum, and *n* is the air refractive index. In other words, the distance variation can be determined by the change of the slope of the unwrapped spectral phase. For a spectrogram captured by a CCD camera, we can also obtain a phase slope vs. the pixel number. Hence, it could be feasible to calibrate the CCD camera by using a spectrogram measured by a spectrometer and the corresponding spectrogram captured by a CCD camera in a non-ambiguity range.

Figure 3 shows the schematic of the CCD calibration for the fundamental harmonic. Please note that, the method of the CCD calibration for the second harmonic is exactly the same as that of the fundamental harmonic. The fundamental harmonic is split into two parts at BS. One part is measured by the spectrometer (YOKOGAWA AQ6370D-20), so that we can observe the spectrogram. The other part is dispersed by a grating, and a CCD camera is used to capture the spectrogram. We find that, in this case, the spectrogram is measured simultaneously by both the well-calibrated spectrometer and the CCD camera. Please note that, the spectrometer is set to measure the wavelength in vacuum. Therefore, the CCD camera can be calibrated by the spectrometer.

Figure 4 shows the calibration process. We select one line, e.g., the line at 209 pixel, and depict the spectrogram in Fig. 4(a) correspondingly. As well-demonstrated in the previous reports [18–20], the spectrogram can be processed by fast Fourier transform to get the unwrapped phase. Consequently, the slope of the unwrapped phase can be obtained, which is 0.26 rad/pixel as shown in Fig. 4(b). Simultaneously, the spectrogram measured by the spectrometer is processed to get the unwrapped phase, as shown in Fig. 4(c). It is easy to obtain the value of the distance by the slope of the unwrapped phase. The distance value is measured to be 0.461 mm. Therefore, the slope of 0.26 rad/pixel obtained by the spectrogram with CCD camera corresponds to the distance of 0.461 mm. Please note that, we can observe some discrepancy between the measured data and the fitting curve in Fig. 4(b). This is partially due to the intensity noise of the CCD camera. We select 80 lines with different pixel numbers from 169 to 249 to measure the respective slopes of the unwrapped phase. By averaging the 80 slope values, the possible error caused by the intensity distortion of the CCD camera could be well suppressed. For the evaluation of the slope of the unwrapped phase, the standard deviation of the difference between the experimental data and the fitting curve is calculated to be 3.4 × 10^{−4} rad/pixel.

We move the target mirror with a step of 20 μm for 20 times, and the experimental results are shown in Fig. 5. We find a good linearity between the displacement of the target mirror and the slope obtained from the CCD camera. The linear coefficient *k _{f}* turns to be 2.06 × 10

^{3}μm/(rad/pixel), i.e., 2.06 × 10

^{−3}m/(rad/pixel), and the standard deviation of the difference between the experimental data and the fitting curve is 2 × 10

^{−8}m/(rad/pixel).

Next, we try to calibrate the CCD camera for the second harmonic. Figure 6(a) shows a spectrogram corresponding to the second harmonic. We process Fig. 6(a) following the procedure exactly the same as that in Figs. 3 and 4. The fringe pattern in Fig. 6(a) appears slightly curved. In this case, we select 120 lines (from 1009 pixel to 1129 pixel) in the vertical direction, and eliminate the effect of the fringe distortion by averaging the 120 measurements. The target mirror is moved for 20 times with a step size of 30 μm, and meanwhile the spectrograms measured by a spectrometer are also recorded, as demonstrated above. The linear coefficient *k _{s}* is found to be 1.88 × 10

^{3}μm/(rad/pixel), i.e., 1.88 × 10

^{−3}m/(rad/pixel)., as shown in Fig. 6(b). The standard deviation of the difference between the experimental data and the fitting curve is 3.9 × 10

^{−8}m/(rad/pixel).

Therefore, we have finished the calibration of both the CCD cameras for the fundamental and second harmonic. The distance can be measured in real time through the slope of the unwrapped phase based on the spectrograms captured by the CCD camera with high accuracy. Here, it is necessary to estimate the non-ambiguity range of the presented system. As shown in Fig. 4(a), we can observe 11 fringes from 200 pixel to 450 pixel, and in this case the corresponding distance is 0.461 mm. The spectral resolution will reach the extreme when the fringe number equals to half the pixel number (one pixel for the bright fringe, and one pixel for the dark fringe), which means there are about 125 fringes in a pixel range from 200 to 450. In this case, the distance can be calculated to be 5.25 mm. This represents that the non-ambiguity range can achieve 5.25 mm, much larger than half the wavelength.

## 3. Correction of air refractive index by two-color dispersive interferometry

In this section, long-term measurements were performed to verify the performance of the correction of the air refractive index when the air refractive index changed by a larger amount. As introduced by Minoshima [32], we deduce the first-order differential of Eq. (1), and the result is as follows:

Please note that, we investigate the correction of the group refractive index of air in this work, since the slope of the unwrapped spectral phase corresponds to the group refractive index of air. Thus, *A* is obtained by using the group refractive indexes of the fundamental and second harmonics. The typical value of *A* is 47.308. The changes of temperature, air pressure, and humidity during the long-term experiment are respectively 10 mK, 8 hPa, and 3.4%, and the main source of the air refractive index change is the air pressure.

#### 3.1 Performance of long-term two-color measurement

Figure 7 shows the experimental results of variation of the air refractive index difference between the fundamental and the second harmonic, i.e., *n*_{2}-*n*_{1}, for 12-hour continuous measurements at 1.2 m distance (2.4 m optical path difference). We also calculate the values of *n*_{2}-*n*_{1} based on the environmental conditions using Ciddor formula, which can be used to evaluate the accuracy of our method.

As shown in Fig. 7, we find that, the difference of the air refractive index variation between the fundamental and second harmonic is at the order of about 4 × 10^{−8}. Meanwhile, the measured value shows a good agreement with the calculated value, and the standard deviation of 3.1 × 10^{−10} was achieved. Please note that, as mentioned in [32], we always assume the geometrical distance of *D* keeps constant, which is rather difficult in the practical applications. Therefore, it is necessary to evaluate the contribution of the drift of the geometrical distance *D*. The variation of *n*_{2}-*n*_{1} can be given by:

In Eq. (3), the second term is the contribution caused by the assumption of constant *D*. Based on the experimental results, (*D*_{2}-*D*_{1})/*D* is about 1.3 × 10^{−6}. Considering the part of ∆*D*/*D*, the fluctuation of up to 100 μm corresponds to the level of only 10^{−5} in the optical path of 2.4 m. Finally, the second term of Eq. (3) turns to be only 10^{−11}, which can be neglected in the data process [32].

#### 3.2 Performance of one-color measurement

Based on Eq. (2), we next evaluate the performance of one-color measurement, i.e., ∆*D*_{1}/*D*. The experimental results for 12-h measurements are shown in Fig. 8. We find that, the refractive index of the fundamental harmonic changed by up to about 2 × 10^{−6} in this long-term measurement. A good agreement can be observed between the results of our method and Ciddor formula, with standard deviation of 2.9 × 10^{−8} in 12-h long-term measurement. Similar with the pre-subsection, it is necessary to evaluate the contribution of the assumption of constant *D*. ∆*n*_{1} can be calculated as:

In Eq. (4), the second term in the right hand is the uncertainty caused by the drift of *D*. We consider that, the drift of *D* is mainly due to the thermal effect (e.g., 10 mK change corresponds to 10^{−7} deformation of a steel optical table) [32].

#### 3.3 Self-correction of air refractive index using two-color method of dispersive interferometry

In this subsection, we use dispersive interferometry-based two-color method to correct the air refractive index. Please let us first go back to Eq. (2), as mentioned above, (*D*_{2}-*D*_{1})/*D* is smaller than 1.3 × 10^{−6}. The standard deviation of *A* is 0.004 with typical value of 47.308. Consequently, the third term in the right hand of Eq. (2) is below 5.2 × 10^{−9}. Considering the fourth term, ∆(*D*_{2}-*D*_{1})/*D* is less than 4 × 10^{−8}, and thus ∆*A*·∆(*D*_{2}-*D*_{1})/*D* is smaller than 1.6 × 10^{−10}. We find that, both the third and the fourth terms are much smaller than ∆*D*/*D*, which is about 2 × 10^{−6}. Therefore, the third and the fourth terms are negligible. Equation (2) can be updated as:

Based on the subsection of 3.1 and 3.2, the results of ∆*D*_{1}/*D* and ∆(*D*_{2}-*D*_{1})/*D* have already been measured. The final procedure of the air refractive index correction is according to Eq. (5) with *A*-coefficient of 47.308. The result is shown in Fig. 9. Figure 9(a) shows the result of ∆*D*_{1}/*D*, Fig. 9(b) indicates the result of *A*·∆(*D*_{2}-*D*_{1})/*D*, and Fig. 9(c) represents the difference between (a) and (b). The standard deviation of 3.3 × 10^{−8} can be found based on Fig. 9(c) while the variation of air refractive index is nearly 2 × 10^{−6}. The standard deviation of ∆*D*/*D* is used to evaluate the performance of the air refractive index correction [32]. We find that, the standard deviation of the air refractive index correction using the two-color method of dispersive interferometry turns to be 3.3 × 10^{−8}, which represents 80 nm for a path length of 2.4 m. Please note that, this uncertainty value of 3.3 × 10^{−8} is the uncertainty for the correction of the air refractive index variation, not the uncertainty for the absolute distance measurement.

## 4. Absolute distance measurement with correction of air refractive index using two-color dispersive interferometry

We do long distance measurement on the optical rail in National Institute of Metrology. The environmental conditions are 21.1°C, 98.94 kPa, and 54.6% humidity. Based on Ciddor formula, the group refractive index of the fundamental harmonic and the second harmonic is 1.0002632 and 1.0002686, respectively. The *A* value is thus 48.557 with 0.004 standard deviation.

Figure 10 shows the spectrograms captured by the CCD camera pair, which we process to determine the distance following the procedure exactly the same as that shown in Fig. 4. We quickly measure each distance for five times, and the measuring time for one single measurement is about 33 ms (30 fps for our cameras). Figure 11 shows the experimental results compared with the fringe counting interferometer, where the air refractive index for the fundamental and second harmonic is corrected by Ciddor formula. The midpoints (for Fig. 11(a), the black x markers; for Fig. 11(b), the red triangle markers) are the averages of the five measurements, and the error bars are the standard deviation. We find that, the measurement scatters of the fundamental and second harmonic at each distance are nearly the same (i.e., the values of the midpoint and the standard deviation are nearly the same). The agreement with the fringe counting interferometer is within 3 μm in a distance of 12 m. Figure 12 indicates the experimental results with the correction of air refractive index using two-color method, based on Eq. (1) (i.e., the air refractive index is not corrected by Ciddor formula). Compared with the results shown in Fig. 11, the midpoints are nearly the same. However, the standard deviation (stability) is significantly improved to be within 400 nm, while the standard deviation without the correction of air refractive index using two-color method is about 1.5 μm (shown in Fig. 11). Therefore, we can consider that the contribution of the air turbulence to the total uncertainty is about 1.1 μm, and the two-color method is powerful to suppress the influence of the air turbulence. The comparison with the fringe counting interferometer shows an agreement well within 2.5 μm over 12 m distance, better than 3 μm.

## 5. Uncertainty evaluation

Based on Eq. (1) and according to the guideline of uncertainty expression [41], the measurement uncertainty is related to the measurement of *D*_{1}, the measurement of *D*_{2}, and the measurement of *A*. In the presented work, the measured distances for the fundamental and second harmonics can be given as:

*N*is an integer, which can be precisely determined with uncertainty below 0.5.

*f*is the repetition frequency of the frequency comb.

_{rep}*c*is the light speed in vacuum,

*k*is the slope of the unwrapped phase obtained by the CCD camera for the fundamental, and

_{fc}*k*is the slope of the unwrapped phase obtained by the CCD camera for the second harmonic. Please note that, the unit of

_{sc}*k*and

_{fc}*k*is rad/pixel, and the unit of

_{sc}*k*and

_{f}*k*is m/(rad/pixel).

_{s}The uncertainty can be expressed as:

In Eq. (8), the first term *u _{D}*

_{1}is the uncertainty of the measured distance of the fundamental harmonic.

*u*

_{D}_{1}is related to the repetition frequency of the frequency comb

*f*, the slope of the unwrapped spectral phase

_{rep}*k*, and the linear coefficient

_{fc}*k*in the CCD calibration, as shown in Eq. (9). The repetition frequency of the frequency comb (250.012 MHz) is well locked to the Rb clock with 10

_{f}^{−11}uncertainty. The first term in Eq. (9) is negligible. It is necessary to mention that the free-running carrier-envelope-offset frequency does not have significant impact to the uncertainty when we measure the group refractive index of air in this work. The second term in Eq. (9) is due to the measurement of the phase slope based on the spectrogram captured by the CCD camera for the fundamental. The phase slope can be affected by the intensity noise of the CCD camera, and the uncertainty of the phase slope is calculated to be 3.4 × 10

^{−4}rad/pixel, as described in Sec. 2.1. The second term can be thus calculated to be within 0.7 μm. The third term in Eq. (9) is related to the linear coefficient

*k*in the CCD calibration, and the uncertainty of

_{f}*k*is 2 × 10

_{f}^{−8}m/(rad/pixel), as shown in Sec. 2.1. The third term can be calculated to be below 20 nm, which represents that the CCD camera has been well calibrated.

Let us go back to Eq. (8). The second term in Eq. (8) is due to the term of *D*_{2}-*D*_{1} (*n*_{2}*D*-*n*_{1}*D*). Since the fundamental and the second harmonic interferometers share most of the experimental setup, including the long measurement beam, both the air turbulence and the mechanical instability can be well cancelled out by the operation of subtraction. Based on the experimental results, the values of *D*_{2}-*D*_{1} increase with increasing the distances. Compared with the values of *D*_{2}-*D*_{1} obtained by the Ciddor formula, the experimental results show an agreement within 8 nm over 12 m distance. Therefore, the uncertainty of *D*_{2}-*D*_{1} can be evaluated to be below 8 nm, while the nominal value of *A* is 48.557. Thus, the second term of Eq. (8) is within 0.4 μm. The third term is related to the *A* coefficient. The standard deviation of the *A* value is 0.004, and *D*_{2}-*D*_{1} is below 65 μm. Therefore, the third term is below 0.26 μm over 12 m distance. In our experiments, the frequency comb interferometer and the fringe counting interferometer do not use the same measurement path. Therefore, Abbe error and cosine error could exist. The Abbe offset is 80 mm, and the pitch and yaw error of the long rail is below 30 μrad. The Abbe error is thus below 2.4 μm. Along the 75 m travel, the alignment between the frequency comb interferometer and the fringe counting interferometer is better than 5 mm, which corresponds to the cosine error of 2.2 × 10^{−9}·*L* (less than 0.17 μm for 75 m). The stability of the optical rail is below 400 nm in 75 m range. Finally, the combined uncertainty with coverage factor of k = 1 turns to be 2.6 μm, which indicates a good agreement with the results shown in Fig. 12.

Please note that, we do not mention the contribution of the air refractive index in the evaluation above. However, we can find obvious effect of the air turbulence based on the data shown in Fig. 11. In our experiments, the uncertainties (stability) of the temperature, air pressure, and humidity are 82 mK, 72 Pa, and 2.7%, which corresponds to the distance uncertainty of 7.6 × 10^{−8}·*L*, 9.4 × 10^{−8}·*L*, and 2.5 × 10^{−8}·*L*, respectively. The combined uncertainty due to the air refractive index is thus 1.2 × 10^{−7}·*L* for one-color method with Ciddor formula, which is below 1.4 μm in the distance of 12 m. Consequently, with consideration of the air fluctuation, the combined uncertainty can be evaluated to be 3 μm, which shows a good agreement with the results in Fig. 11. This also shows that the two-color method is powerful to reduce the uncertainty due to the air refractive index.

## 6. Conclusion and discussion

In conclusion, we use the two-color method of dispersive interferometry to realize the high-accuracy absolute distance measurement. The distance can be obtained by the slope of the unwrapped phase based on the spectrograms captured by the CCD camera. The non-ambiguity range can be as large as several millimeters with such a simple spectrometer. We carry out 12-h long-term measurement while the variation of the air refractive index is nearly 2 × 10^{−6}. The measured values are compared with the results calculated using Ciddor formula based on the well-measured temperature, pressure, and humidity. By assuming the geometrical distance is always constant, we obtain the uncertainty of 3.3 × 10^{−8} for 12-h continuous measurements, without the precise knowledge of the environmental conditions. Our result is not as great as that in [36] and [37]. We consider that, first the floor noise of the CCD camera can make some contributions, but high-performance camera with active cooling and more pixels is truly expensive and bulky; second we do not perform data averaging in the long term experiments, and the moving averaging could be useful to suppress some random noises.

In the real absolute distance measurement, the experimental results show that the measurement stability has been significantly improved. Over 12 m distance, the agreement between the frequency comb interferometer and the fringe counting interferometer can be well within 2.5 μm, whereby the Abbe error is the main source of the measurement uncertainty. Compared with the system proposed in [20], an important difference for our system is that the fringe counting interferometer and the comb interferometer do not own the same target corner cube, which is the main reason why our results do not achieve 1-μm uncertainty.

High-accuracy correction of the air refractive index is important in the practical applications. Considering this work demonstrated in this paper, the correction of the phase refractive index has not been performed, since the individual mode of the frequency comb was not resolved due to the low resolution of the spectrometer. Therefore, we can build a mode-resolved spectrometer with techniques of e.g., fingerprinting [40], vernier spectroscopy [42] etc. in future. In this case, we can correct the phase refractive index with extensive selection of the fundamental and second harmonics. Moreover, the correction of the refractive index could be fulfilled at arbitrary locations, not just around the multiples of the laser cavity. In addition, we can also use a sufficiently long reference arm, so that the time interval between the two pulses can be arbitrarily tuned, which indicates that the air refractive index correction can be performed at arbitrary distances [19]. A fiber link could be preferable, but the stabilization of such a long fiber is truly a difficult study [43–45]. The inherent limitation of the two-color method is that the effect of the humidity cannot be well cancelled out, which mainly emerges as that the uncertainty of the *A* value is strongly dependent on the humidity. This also means that the variation of the humidity cannot be very large in the long-term experiments, and the inhomogeneity of the humidity cannot be very poor along the long optical path. Since the measurement uncertainty will be magnified by the factor of *A* (in the case of phase refractive index, often about 144; in the case of group refractive index, often about 49), it is always a challenge to measure both the distances of the fundamental and the second harmonic with high accuracy in the two-color methods.

## Funding

Natural Science Foundation of China (51327006, 51675380); Tianjin Research Program of Application Foundation and Advanced Technology (15JCZDJC39300).

## Acknowledgment

We thank Petr Balling for his very warm help and comments. We would like to thank Dr. Rujie Li for the support of the CCD camera. Thank National Institute of Metrology for the great support of the long optical rail.

## References and links

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