1. Introduction

High-accuracy absolute long-distance measurements are essential for many applications such as industrial sensing and monitoring of large-scale facilities such as particle accelerators. In the case of practical long-distance measurements in air, the refractive index of air is one of the most important causes of uncertainties. Conventionally, its correction is performed by using empirical equations such as the Edlen’s equation (updated version) [1] or Ciddor’s equation [2] and by using measured environmental parameters such as the air temperature, pressure, humidity, and CO₂ concentration. However, correcting the refractive index of air along the exact optical path under measurement is difficult because of its temporal variation and its distribution in space. A precise method for correcting the refractive index of air by directly using the path-length measurements, i.e., a method for self-correction of the refractive index of air would be a powerful tool.

The two-color method, which is a method for such self-correction of the refractive index of air on the basis of length measurements by using different-colored lasers, was proposed by Bender et al. [3, 4], and this method has been employed to perform several types of length measurements [5–8]. In this method, the dispersion relation for the refractive index of air is used. When a distance D is measured by simultaneously using two lasers with different wavelengths, optical distances D₁ and D₂ corresponding to the two wavelengths are obtained. The values of D₁ and D₂ are n₁D and n₂D, respectively, when n₁ and n₂ are the air refractive indices at the two wavelengths. The geometrical distance D after correcting the refractive index of air can be described as follows:

While the refractive indices n₁ and n₂ change because of environmental instability, the dispersion relation A is considerably less sensitive to the environmental changes since (n_i - 1) is nearly proportional to the density of air. According to the Edlen’s and Ciddor’s equations, in the case of dry air (humidity: 0%), A is constant since the terms that include the dependence on the air temperature and that on air pressure are cancelled out [8]. Therefore, once we know the value of A at a certain experimental condition, although the environmental parameters change, using a constant value of A in Eq. (1), we can determine the geometrical distance D only from the measurement of the objective optical distances D₁ and D₂. In other word, we can correct the refractive index of air, n_1,2, by directly using the optical distance measurements. In the case of wet air, dependences of A on the environmental parameters are not cancelled out, and this could lead some deviation in length measurements if A is considered to remain constant. However, acceptable changes in the environmental conditions needed for high-accuracy distance measurements could be significantly mitigated. For example, in order to measure a distance with a relative uncertainty of 1 × 10⁻⁷, in the case of conventional distance measurement using one-color laser, the required measurement uncertainties in environmental parameters are either an uncertainty of 0.1 K in the case of air temperature or 0.4 hPa in the case of pressure. On the other hand, in the case of the two-color method, either 1 K or 70 hPa are sufficient to determine the value of A for realizing the same uncertainty of 1 × 10⁻⁷ in the corrected distance. Within these ranges of environmental variations, the variation in the A value of 145 is less than ± 0.06. These calculations were performed using the Ciddor’s equation for phase refractive indices at wavelengths of 1568 nm and 788 nm, air temperature of 20 °C, pressure of 1013.25 hPa, humidity of 50%, and CO₂ concentration of 400 ppm. The dependence of A on the variations in the environmental parameters is similar irrespective of the wavelength and initial environmental parameters. In the two-color method, air humidity has to be measured; the required humidity uncertainties for a relative uncertainty of 1 × 10⁻⁷ in distance measurements are 12% and 4% for one- and two-color measurements, respectively. In many cases, we can practically measure humidity with sufficient precision, or further improvement is possible by measuring partial water pressure [8]. Thus, the two-color method is a powerful tool with practical applications.

However, this method has a drawback. The accuracy of the corrected distance D is significantly lower than the accuracy of the measured optical distances D₂ and D₁; this is because the small difference between the measured distances, i.e., D₂ - D₁, is used to calculate the distance, as given by Eq. (1). The degradation factor is the same as A, and its value is normally high, e.g., 145 under the experimental conditions we used in this study. Therefore, high-precision distance-measurement techniques are essential for high-accuracy refractive-index correction using the two-color method. An optical frequency comb is a powerful tool for performing such measurements. Various ultrahigh-precision length measurements ranging from picometers to hundreds of meters have been performed using optical frequency combs [7, 9–20]. Moreover, the optical frequency comb generated by a mode-locked laser is suitable for multicolor measurements using efficient optical nonlinearity. Two-color distance measurements using the electronic phase of intermode beats of frequency combs have also been reported [7].

In this study, we developed a long-path interferometric technique for two-color frequency combs generated by a mode-locked laser, and we demonstrated the efficacy of the technique for a high-accuracy two-color method. We developed a pulse-to-pulse interferometer with the fundamental and second harmonic generation (SHG) of a mode-locked laser. In the case of the pulse-to-pulse interferometer, interference fringes are observed at distances equal to the multiples of the pulse separation. Because of the narrow linewidth, i.e., long coherence length of each mode of the comb [21], interference is observed between separated pulses in a mode-locked pulse train. Therefore, an unbalanced interferometer with a long measurement path and a short stable reference path can be used in long-distance measurements. Since in the case of the pulse-to-pulse interferometer, interferometric phase delay can be precisely controlled by changing the frequencies of the frequency combs, such as mode separation [10] and offset frequencies, the phase difference between the two-color interferometers, i.e., difference between the two-color optical distances, i.e., D₂ - D₁, was successfully measured with a high stability by stabilizing one of the interference fringes by precisely controlling the comb frequency. In long term measurements, the results were in good agreement with the variations of air refractive indices calculated using environmental parameters. High-accuracy measurement of the two-color difference is essential for a high-accuracy two-color method, as mentioned above. The standard deviation of the difference between the measurement and calculation was 3×10⁻¹⁰ for 10 h of continuous measurements. Finally, by applying the two-color method, high-accuracy self-correction of the refractive index of air with an uncertainty of 5×10⁻⁸ was realized when the refractive index change was on the order of 10⁻⁶.

2. Two-color pulse-to-pulse interferometer

Figure 1 shows the experimental setup. We used a home-built diode-laser-pumped nonlinear-polarization-rotation mode-locked Er:fiber ring laser with a central wavelength of 1568 nm, spectral width 34 nm, and average power of 2 mW as a frequency comb source. The laser system is similar to the one described in reference [22], and the repetition frequency (f_rep) and the carrier-envelope-offset frequency (f_ceo) are fully stabilized or controlled referenced to the microwave frequency reference synthesized from a hydrogen maser linked to the coordinated universal time (UTC), operated by the National Metrology Institute of Japan (UTC-NMIJ). The repetition rate of the comb (f_rep) is 54.0 MHz, and it can be varied precisely up to 20 Hz by PZT-driven fiber stretching and coarsely up to 900 kHz by a fiber delay line.

 

Fig. 1 Experimental set up for the pulse-to-pulse interferometer using two-color optical combs. The output of the fiber comb is separated into three branches; one of them is used for stabilizing the comb (not shown) and other two are used for interferometric experiments for fundamental (ω) and SHG (2ω). EDFA: Er-doped fiber amplifier; FC: fiber collimator; H: half-wave plate; L: lens; C: SHG crystal; DM: dichroic mirror; BS: beam splitter of 10 mm cube; PD1,2: photodetectors. Path difference between the probe and reference is 5.55 m, which corresponds to the repetition rate of 54.0 MHz. In the interferometer, the pulses were highly chirped by propagation in fibers and a PPLN crystal. The pulse durations estimated by the coherence lengths were about 1 ps and 3 ps for fundamental and SHG, respectively.

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The output of the fiber comb is separated into three branches, and one of them is used for stabilizing the comb and other two are used for interferometric experiments. The output beam of the one of the experimental ports (average power of 0.6 mW) is introduced to an unbalanced long-path interferometer with a path difference of 5.55 m corresponding to the repetition rate of the comb. A pulse-to-pulse interferometer in which adjacent pulses in a mode-locked pulse train interfere with each other was constructed. The setup for the interferometers was placed in a thermally isolated box in a temperature controlled room. The other branch of the fiber comb output was amplified by an Er-doped fiber amplifier (EDFA), and the second harmonics of the fundamental comb were generated by using a 5-mm-thick piece of periodically poled LiNbO₃ (PPLN) heated at 120 °C with central wavelength of 788 nm, spectral width 2 nm, and average power 2 mW. To construct the SHG interferometer, the SHG beam was collimated and coaxially overlapped with the fundamental beam to be introduced into the same optical setup. Alignment was done by confirming the beam overlapping after travelling 5.55 m path length. Interference fringe signals with the two wavelengths were separated by using a dichroic mirror, and each color of the fringe signal was detected by a photodetector (response time: 100 kHz). The intensities of the signals were recorded by a digital oscilloscope simultaneously, and the interference phase was calculated using the fringe signal intensity and the fringe amplitude. Path-length change was calculated by the interference phase change and the wavelength. For further improvement in practical distance measurements, interference phase errors due to intensity fluctuations that might be caused by such as air turbulence or mechanical instabilities need to be suppressed. The use of phase measurement methods such as heterodyne detection would help in suppressing these influences.

The phase of the interference fringe can be precisely controlled and/or measured by changing either f_rep or f_ceo. Figure 2 shows the interference fringe signals for fundamental and SHG wavelengths when the repetition frequency was changed. When f_ceo is changed by changing the current to drive the pump diode laser, the intensity of the interference fringe signal is also changed. On the other hand, precise control of the interference fringe is possible by controlling f_rep, so in this experiment, f_rep was controlled to tailor the interference fringe, while f_ceo was stabilized to the frequency standards. Coherence lengths of the interference fringes were estimated as 344 μm and 936 μm for fundamental and SHG wavelengths, respectively. To extend the range of the measurements, the f_rep can be scanned up to 900 kHz, which is 2% of the f_rep. Therefore the maximum range of the length scanning is about 100 mm when the total path length is 5.55 m, and the range becomes longer for longer distance proportionally. Moreover, by changing the repetition frequency over a wide range and measuring the frequency that gives the peak of the envelope of the fringe, we can determine the absolute distance [10]. Please note that envelope follows group refractive index, while fringe phase follows phase refractive index.

 

Fig. 2 Interference fringe signals for (a) fundamental and (b) SHG wavelengths. Optical path length was changed by the scanning of the repetition frequency, f_rep. Cross: measurement results; Line: sine curve fitting for guide for eyes.

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Figure 3 shows the results of short-term measurement of the changes in the optical path length of each color interferometer. In Fig. 3 (a) and (b), the origin of each vertical axis is one of the zero phases. Each color interference fringe signal (a: fundamental and b: SHG) showed a fluctuation in the optical path length on the order of 100 nm (standard deviation of 29 nm) for 300 s because of the fluctuation in the optical path length due to air instability. The difference between the two-color measurements is shown in Fig. 3(c), and it is almost constant with a standard deviation of 3 nm. The residual sinusoidal variation seen in the plot is due to the fluctuation of the output power of the SHG signal that results from imperfect control of the oven temperature for the PPLN. Thus, we can see that interference fringe phases of the two-color wavelength are measured precisely, while phase change is small. However, when phase change is larger than π (a half fringe) or phase signal intensity is near the peak or bottom of the fringe, unwrapping of the phase signal is needed. In the case of long-distance measurements or long-term measurements, ambiguity of phase unwrapping will cause considerable uncertainty due to fluctuations. Moreover, uncertainty in each length measurement is multiplied by A value in the case of two-color method as mentioned above, when independent two length measurement results are subtracted and used in Eq. (1). In order to avoid the ambiguity and to achieve precise measurement of the two-color phase difference required for the two-color method, in the next section, we developed a method for directly measuring the phase difference between the two color fringes instead of measuring each fringe signal independently.

 

Fig. 3 Environmental variation of interference fringe signals of two-color combs ((a) fundamental, (b) SHG) and (c) their difference. In (a) and (b), the origin of each vertical axis is one of the zero phases. Residual sinusoidal variation in the plot of (c) is due to the imperfect temperature control of the SHG crystal oven.

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3. Measurement of precise difference between optical distances corresponding to two colors

3.1 Stabilization of wavelength of the fundamental comb modes in air

To do this, the fundamental interference fringe signal was stabilized so that it became constant by feedback control of the repetition frequency of the comb and so that changes in the optical path length were cancelled. Stabilizing the interference phase signal was done as follows. The interference signal was detected by a photodetector and the output voltage was stabilized to a reference voltage by a proportional–integral–derivative controller (PID controller, SIM960, Stanford Research Systems). The error signal was applied to the PZT-driven fiber stretcher in the laser oscillator. Servo bandwidth was limited by the response time of the PZT driver, which is on the order of kHz. Fringe signal intensity was kept at the signal level near one of the zero phases of the sine function. The phase zero was chosen such that it gives the steepest slope versus the path-length change. Here, f_ceo is stabilized according to time and frequency standards. Simultaneously, the SHG fringe signal was measured. All the wavelengths in air of the comb modes, which are the products of the wavelengths in vacuum and the air refractive indices, were stabilized accordingly if the dispersion of the air refractive index is negligible [13]. In other words, if an interference signal change is observed at the other wavelength, it is due to the effect of dispersion, and therefore, we can precisely measure only the difference between the air refractive indices corresponding to the two colors. In our experiment, since the variation in the fundamental interference fringe signal caused by the changes in the optical path-length at fundamental wavelength is cancelled by feedback, the detected SHG interference fringe signal shows only the optical path-length difference between the fundamental and SHG wavelengths, i.e., D₂ - D₁.

3.2 Performance of short-term measurement

Figure 4 shows the interference fringe signals at wavelengths of fundamental and SHG continuously measured for 1000 s when the fundamental fringe was stabilized as mentioned in the previous subsection. The origin of vertical axis is zero phase of the fundamental fringe. In order to investigate the effectiveness of correcting slow environmental variations, we applied the moving average to the interference signals. The time constant of 25 s that corresponds to the time constant of the oven temperature control was selected to eliminate the SHG intensity variation due to the oven temperature mentioned above. Additional sources of the fast signal fluctuations were electronic noises due to detection and feedback control, which can be further suppressed by optimizing the feedback parameters. The results are shown in Fig. 4 (a and c are without moving average and b and d are with moving average). The fundamental interference fringe signal shows a constant with the standard deviation of the change in the path length being 0.4 nm after moving averaging (Fig. 4(b)), which shows that the phase stabilization is effective. The SHG interference fringe signal also shows constant since for such a short period, the dispersion effect on the changes in the air refractive index due to changes in environmental parameters is negligible. Standard deviation of the path length change was 1.4 nm for 1000 s after moving averaging (Fig. 4(d)), and this is 10⁻¹⁰ of the measured path length of 5.55 m. The results show that both colors of the optical wavelengths were controlled simultaneously and precise phase-difference measurement on the order of 10⁻¹⁰ was achieved.

 

Fig. 4 Interferometric measurements of two-color frequency combs. The fundamental interference fringe signal (a, b) is stabilized by feedback to f_rep. Variations of the fundamental and SHG optical path lengths are shown ((a, b) and (c, d), respectively). The origin of vertical axis is zero phase of the fundamental fringe. Moving average of 25 s was applied in the cases of (b) and (d).

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4. Two-color method for self-correction of variation in refractive index of air

Next, long-term continuous measurements were carried out to investigate the efficacy of the developed method in the case of large variations in environmental parameters. By using Eq. (1), relative variation in the distance after refractive index correction ΔD/D can be represented by a simple modification of Eq. (1):

In order to test the efficacy of the method, we performed the distance measurement to a stationary target and investigate the level of stability of the ΔD/D after correcting the refractive index variation by two-color method. In the case of perfect correction, corrected distance D should be constant i.e., ΔD should be zero, therefore the stability of D will show the accuracy of the refractive index correction. In order to test the efficacy of the method by comparing to the well-established refractive index equation, the setup of the interferometer was placed in thermally isolated box in order to avoid the thermal deformation of mechanical components. Observed changes of air temperature, pressure and humidity during the long-term measurement which will be described in this section were ΔT = 0.01 K, ΔP = 3 hPa, ΔH = 1%, respectively, and main sources of air refractive index change was air pressure change. Although environmental variation causes the variation of A, i.e., ΔA, the third and fourth terms of the right hand side of Eq. (3) were negligible as will be discussed in subsection 4.3. Measurements of the second and first terms in the right hand side of Eq. (3), i.e., Δ(D₂ - D₁)/D and ΔD₁/D corresponding to the relative variations of (D₂ - D₁) and D₁, will be described in the following subsections 4.1 and 4.2, respectively.

4.1 Performances of long-term measurements

In this subsection, long-term measurements of the relative variations of the two-color optical path difference (D₂ - D₁), i.e., Δ(D₂ - D₁)/D will be discussed. Figure 5 shows the results of continuous 10 h measurements. The phase difference of the two-color interferometers, i.e., D₂ - D₁, was measured by employing the method mentioned in the previous section, and the corresponding difference between the air refractive indices, n₂ - n₁, was determined by assuming that the geometrical distance D remained constant (Fig. 5(a)). In the case of the measurement of two-color optical path difference D₂ - D₁, sensitivity to the real length change ΔD is negligible since it is a common-path measurement and it detects only small difference, (n₂ - n₁) × ΔD where n₂ - n₁ is on the order of 10⁻⁶. Therefore even though the stability of the geometrical distance D is in the level of 10⁻⁴, measurement uncertainty of 10⁻¹⁰ is possible in the variation of n₂ - n₁.

 

Fig. 5 Long-term variation in the difference between the refractive indices of air at two color wavelengths n₂-n₁ (a) obtained by the measured phase difference of the two-color interferometer, (b) calculated using environmental parameters, and (c) their difference. The vertical positions of the plots were shifted for clarity as follows. In (a) and (b), the vertical origins of the plots was chosen so that the averages of all the data in two plots were the same. In (c), the vertical origin was shifted so that the average of data was 2x10⁻⁹.

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In Fig. 5(a), the similar moving average mentioned in the previous section was applied to the interference signals. The result shows changes on the order of 10⁻⁸ over 10 h. Simultaneously, air temperature, pressure, and humidity were measured by sensors during the measurements, and the air refractive indices were calculated using the Ciddor’s equation. Variations in the difference between the calculated air refractive indices, n₂ - n₁, is shown in Fig. 5(b). The calculation and the measurement results were in agreement throughout the 10-h period, and the difference between these results showed a standard deviation of 3 × 10⁻¹⁰ (Fig. 5(c)). Here, the change in the air refractive index, n₁, was 10⁻⁶ (will be shown in the next subsection and Fig. 6(b) ), which was also calculated by environmental parameters. The obtained results demonstrated the efficacy of the developed technique for measurements of the difference between two-color optical distances, i.e., D₂ - D₁, with high consistency, which is essential for the high-accuracy two-color method.

 

Fig. 6 Long-term variation in the refractive index of air (a) obtained by the measured f_rep, (b) calculated using environmental parameters, and (c) their difference. The origin of each plot was arbitrary chosen for clarity in the similar way to Fig. 5. In (c), the vertical origin is shifted so that the average is 0.2 × 10⁻⁶.

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4.2 One-color distance measurement by frequency measurement

In order to apply the two-color method, we need to measure the one-color optical distance D₁ simultaneously. In this subsection, long-term measurements of the relative variations of the one-color optical path difference ΔD₁/D will be discussed. In the developed method for the two-color phase-difference measurement described in the section 3.1, variation in the fundamental optical distance D₁ is compensated by changing f_rep. Therefore, by simultaneously measuring f_rep during the interference measurements, the variation in D₁ is calculated as follows,

By using frequency measurement without moving the mechanical stage, which is normally used in conventional interferometric measurements, the distances can be measured precisely. The maximum range of the f_rep scanning is 900 kHz, which is 2% of the f_rep (54 MHz). In order to compensate the air refractive index fluctuation, the current frequency scanning range is sufficient. Figure 6(a) shows the long-term measurement results of D₁, which were obtained by Eq. (4) using f_rep, which were measured using a frequency counter (Agilent 53132A, gate time: 10 s) at the same time as the results shown in Fig. 5. Using the results of D₁, the refractive index of air, n₁, was calculated by assuming that the geometrical distance D remained constant, and these calculated n₁ values were plotted. Figure 6(c) shows the difference between the measured (Fig. 6(a)) n₁ values and the values calculated using the Ciddor’s equation (Fig. 6(b)). The standard deviation of this difference is 4.4 × 10⁻⁸, which means that variation of D₁ is measured accurately using f_rep. The residual deviation in the observed data in Fig. 6(c) could be caused by the real geometrical length variation such as temperature deformation of optical table. By simple estimation, temperature change of 0.01 K could cause deformation of 10⁻⁷ of a stainless steel optical table.

4.3 Correction of the variation in refractive index of air by employing two-color method

Finally, using the results of long-term optical distance measurements, the two-color method was applied to correct the changes in the air refractive index due to changes in environmental parameters. Using the results of Δ(D₂ - D₁) and ΔD₁ in previous subsections 4.1 and 4.2, relative variation in the distance during the measurement, ΔD/D, can be obtained by using Eq. (3). The results are shown in Figs. 7(a) and 7(b), which are obtained from the variation of n₁ and n₂-n₁ in Figs. 6(a) and 5(a), respectively, and A = 144.66. Using the measurement uncertainties of sensors for air temperature, pressure and humidity, δT = 0.04 K, δP = 0.05 hPa, and δH = 3%, respectively, uncertainty of the determination of the initial value of A was calculated as δA = 0.05, where the effect of δH is dominant. Note that this level of absolute uncertainty in the initial value of A = 144.66, gives only uncertainty of 10⁻¹⁰ in the two-color correction of the final length measurement A × Δ(D₂ - D₁), whose total range of variation is at most 10⁻⁶. On the other hand, the observed environmental variation causes the variation of A during the measurement, i.e., ΔA = 0.01 (standard uncertainty). As mentioned after Eq. (3), the third and fourth terms of the right hand side of Eq. (3) give uncertainty less than 2 × 10⁻⁸ and 1 × 10⁻¹⁰, respectively, at this ΔA value; thus, the effect of ΔA on the final results was negligible. Finally, by substituting these values in Eq. (3), two-color correction was achieved. The corrected distance is shown in Fig. 7(c), which shows the relative variation in the geometrical distance ΔD/D; this variation should have been constant if the correction was perfect. For clarity, the vertical origin of the plot was shifted so that the average of data is 0.2 × 10⁻⁶. The results are constant with a standard deviation of 5 × 10⁻⁸, which corresponds to 270 nm for a path length of 5.55 m, while the relative change in the optical distance itself was on the order of 10⁻⁶ over the same period, as shown in Fig. 7(a). As discussed in the previous subsection 4.2, part of the residual deviation in the geometrical distance in Fig. 7(c) could be caused by the real geometrical length variation such as temperature deformation of optical table.

 

Fig. 7 Self-correction of air-refractive index by two-color method. The variations in the relative distances to the target distance D, which are obtained from the corresponding refractive indices, are plotted; (a) one-color distance D₁ obtained by measured f_rep (Fig. 6(a)), (b) difference between optical distances corresponding to two colors, D₂ - D₁, obtained from the direct measurements of phase difference of the two-color interferometers (Fig. 5(a)), and (c) corrected geometrical distance D obtained using Eq. (3). The origin of each plot was arbitrary chosen for clarity in the similar way to Fig. 5. The vertical origin of the plot (c) was shifted so that the average of data is 0.2 × 10⁻⁶.

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5. Conclusion

In conclusion, we have developed high-accuracy two-color interferometry by using two-color frequency combs, and we have demonstrated the efficacy of the technique for high-accuracy correction of the refractive index of air. The interference fringe of the fundamental wavelength was stabilized to suppress the influence of fluctuations in the environmental parameters by changing the repetition frequency of the frequency comb. By employing this method, the wavelengths of the modes of the comb in air near the spectral region were stabilized and only the dispersion effect was measured precisely. The stability of 10⁻¹⁰ for 1000 s was achieved in the measurement of an optical path length difference between two wavelengths, and the calculation and the measurement results were in agreement with a standard deviation of 3 × 10⁻¹⁰ throughout the 10-h period. Finally, high-accuracy correction of the refractive index of 10⁻⁸ for 10 h was achieved by the two-color method. Note that the correction was realized by directly using the distance measurement results, without additional precise measurements of environmental parameters. Such a self-correction of the refractive index of air is a powerful tool since measurement of precise environmental parameters along the measurement path is difficult and causes major uncertainty. The developed technique is a key for high-accuracy length metrology in various practical applications.