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A two-color scheme of heterodyne laser interferometer is devised for distance measurements with the capability of real-time compensation of the refractive index of the ambient air. A fundamental wavelength of 1555 nm and its second harmonic wavelength of 777.5 nm are generated, with stabilization to the frequency comb of a femtosecond laser, to provide fractional stability of the order of 3.0 × 10−12 at 1 s averaging. Achieved uncertainty is of the order of 10−8 in measuring distances of 2.5 m without sensing the refractive index of air in adverse environmental conditions.
© 2015 Optical Society of America
1. Introduction
Realizing nano-scale optical and electronic devices as mass-produced commodities such as integrated circuits and flat-panel displays requires precision machineries equipped with the capability of sub-micrometer positioning over ranges up to a few meters [1,2]. Such long range control of precision motion is attempted preferably by making use of laser interferometers which are capable of delivering sub-wavelength resolutions over wide coherent lengths provided by well-stabilized laser sources [3–5]. However, when laser interferometers are operated in air to measure long distances of several meters, the achievable measurement uncertainty is restrained ultimately by the instability of the refractive index of the ambient air. Being affected by environmental parameters of temperature, pressure, humidity and CO2 concentration [6–9], the refractive index of air is not static inherently as it varies with sensitivity to time as well as position. Thus, stringent environment control is necessary together with accurate identification of the refractive index of air in order to maintain the measurement uncertainty better than one part per million over several meters [10–12].
The geometrical distance L measured in air by laser interferometers can be expressed in terms of the wavelength λ of the laser source as L = (m + e)λ, in which m is the integer multiple (m = 0, 1, 2…) and e denotes the excess fraction (0≤e<1). The first-order measurement error of L is then derived as
with the assumption of Δm = 0. The term Δeλ in Eq. (1) represents the subdivision error of the interference fringe, which is usually much smaller than a single wavelength, i.e., Δe<<1. The other term (Δλ/λ)L is attributed by the instability of the wavelength λ and becomes substantial in proportion to the distance L even though the fractional wavelength uncertainty Δλ/λ turns out to be smaller than Δe.The wavelength λ relates to the vacuum speed of light c0 as λ = c0/nf with n and f being the refractive index of air and the optical frequency of the laser source, respectively. The distance L is then expressed as L = (m + e)c0/nf, so Eq. (1) can be rewritten as ΔL = Δeλ − (Δn/n)L − (Δf/f)L. Based on the first-order variation relation of Eq. (1), the measurement uncertainty of L, i.e. u(L), can therefore be obtained as a sum of three major contributions as
As depicted in Fig. 1, when the distance L is not large, typically up to a few micrometers, u(L) remains constant mainly being affected by Δeλ. On the other hand, as L increases, u(L) begins to rise in proportion to L, being dominated by either Δn/n or Δf/f, i.e., the fractional uncertainty of n or f. The state-of-the-art competence for stabilizing f is better than that of identifying the integral value of n. This infers that u(L) is more vulnerable to Δn/n than Δf/f.
Fig. 1 Distance measurement uncertainty of laser interferometry dominated by three limiting factors; phase measurement error (Δe⋅λ), air refractive index (n) and source frequency (f).
The refractive index of the surrounding air can be estimated during measurement by installing a refractometer in parallel to the measurement beam [13–15], or adopting empirical dispersion formula such as the updated Edlen’s equation or Ciddor’s equation together with sensing of the ambient parameters [8,9]. These methods are known capable of estimating the true value of the refractive index of air with uncertainties near a ~10−8 level in well-controlled homogeneous laboratory conditions. However, the refractive index of air at room temperature is basically not static due to inherent turbulent air flow, making it difficult to precisely identify its integral value along all the optical paths of the interferometer in operation. An alternative solution is the method of two-color interferometry which employs two different wavelengths to compensate for the effect of the refractive index of air based on the dispersion formula of light [6,7,10,11]. In comparison to the aforementioned methods, the method of two-color interferometry is found compatible for long-distance measurements in the open air where the actual fluctuation of the refractive index of air becomes severe due to non-homogenous and dynamical air flow [16–19].
In this investigation, we implement the method of two-color interferometry using two wavelengths generated by making use of the optical comb of a femtosecond laser [20,21]. The generated wavelengths are well stabilized so that the source frequency instability makes no significant contribution to the measurement uncertainty as depicted in Fig. 1. The stabilized wavelengths allow for accurate accumulation of the optical path lengths, from which the true distance can be estimated precisely without significant disturbance of phase measurement errors. In fact, the concept of adopting femtosecond lasers for the two-color method has already been proposed by Minoshima et al together with demonstration of a remarkable progress in the measurement uncertainty of a pulse-to-pulse type interferometer [22–24]. Motivated by the pioneering work, our intention here is to devise a continual type of distance interferometer which can be employed for actual industrial machines in which the refractive index of air is precisely compensated in real time while distance is measured continuously over several meters in adverse environmental conditions.
2. Two-color interferometry
The concept of two-color interferometry uses two different wavelengths simultaneously as the light source so as to compensate for the distance error attributable to the refractive index of air [6,7]. The use of two wavelengths, i.e. λ1 and λ2, leads to determination of the geometrical distance L as
in which D1 and D2 denote the optical path lengths (OPLs) which are actually measured using λ1 and λ2, respectively. The measured OPLs relate to L as D1 = n1 L and D2 = n2 L with n1 and n2 being the refractive indices of λ1 and λ2. As the refractive index of air varies from position to position in reality, n1 and n2 can be regarded as the integral indices which are averaged all along the OPLs. The well-established empirical dispersion formula of the Edlen’s equation or Ciddor’s equation [8–11] reveals that the coefficient A defined as a combination of n1 and n2 in Eq. (3) is constant for dry air, regardless of temperature and pressure. In wet air, humidity is not a free parameter as it affects the coefficient A, but the influence under usual measurement conditions is not significant as revealed by elaborate uncertainty analysis [22]. This implies that the coefficient A may be assumed as a constant which is given solely by selection of λ1 and λ2, thus the true value of L can be estimated using Eq. (3) without knowing the values of n1 and n2 which are significantly affected by environmental conditions.Implementing the two-color method requires preparing λ1 and λ2 with much care. When λ1 and λ2 are selected too close to each other in the visible or infrared range, the coefficient A becomes overly large. This is not desirable because a small error encountered in the measured D1 or D2 is augmented by the factor of A when it is reflected in the uncertainty of L determined by Eq. (3). This aspect demands that λ1 and λ2 be far apart to reduce the coefficient A and also be highly stable to measure D1 and D2 with less dynamical errors. This requirement may be fulfilled by selecting λ1 and λ2 from well-stabilized lasers, but an alternative way is generating λ1 and λ2 directly with reference to the optical comb of a femtosecond laser which is in fact a precise wavelength ruler providing a wide range of choices collectively stabilized to the atomic clock.
Figure 2 depicts the two-color interferometer system configured in this investigation. As the two-color light source, first λ1 which is called the fundamental wavelength is generated by phase-locking a distributed feedback (DFB) laser to a particular mode of 1555 nm wavelength within the optical comb of an Er-doped fiber femtosecond laser. The optical comb is stabilized by phase-locking the repetition rate fr as well as the carrier-envelope offset frequency fceo to the Rb atomic clock. For the used femtosecond laser, fr is 100 MHz which is detected using a photo-detector while fceo is monitored by attaching an f-2f interferometer to the femtosecond fiber laser. The DFB laser is interfered with the reference comb, from which the beat note fb is monitored using a photo-detector and subsequently phase-locked to an assigned offset of 30 MHz. The output frequency of the DFB laser is obtained as fDFB = fceo + ifr ± fb with i being a positive integer accurately determined by identifying the absolute value of fDFB using a wavelength-meter. Once the fundamental wavelength λ1 is obtained, the second wavelength λ2 is produced by doubling the optical frequency of the DFB laser by means of second harmonic generation (SHG) so that λ2 = λ1/2. The generated λ1 and λ2 are delivered simultaneously to a Michelson-type distance interferometer in which D1 and D2 are measured by adopting the heterodyne phase-measuring technique.
Fig. 2 Two-color heterodyne interferometer configured for real-time compensation of the refractive index of air. OC: optical coupler, PLL: phase locked loop, C: collimator, DFB: distributed feedback, FL: focusing lens, PPLN: periodically poled LiNbO3LiNbO3, SOA: semiconductor optical amplifier, DM: dichroic mirror, BS: beam splitter, AOM: acousto-optic modulator, M: mirror, PD: photodetector, CC: corner-cube, LPF: low pass filter, T: temperature, P: pressure, H: humidity, CO2: carbon dioxide concentration, PD: photo-detector, Meas.: measurement, Ref.: reference, and Rb: rubidium.
Figure 3 illustrates the time-averaged stability of the output frequency of the DFB laser which was compared with those of the optical comb and the Rb clock used for stabilization. The fundamental wavelength λ1 was 1555 nm, so the SHG wavelength λ2 was 777.5 nm. Then, using the Ciddor’s formula, the coefficient A of Eq. (3) was calculated as 141.41. A periodically poled LiNbO3 (PPLN) crystal was used to generate λ2 with SHG conversion efficiency of 0.6%/W/cm. The SHG signal was then amplified through a semiconductor optical amplifier (SOA) by 20 dB. The signal-to-noise ratio was 57.86 dB for λ1 and 35.92 dB for λ2 as shown in the optical spectra of Fig. 3(b). The fractional stability of the beat note fb was 3.082 × 10−12 at 1 s averaging, so its absolute stability was estimated to be of the same order of 10−11 as that of the Rb clock. The phase locking state of the DFB laser to the optical comb was observed well maintained for longer than 7 hours without breakdown. Finally, λ1 and λ2 were combined through a dichroic mirror to form the two-color light source. Using the two-color interferometer, the OPL of each wavelength was measured by adopting the heterodyne phase-measuring principle using an acousto-optic modulator (AOM) proving a 40 kHz heterodyne frequency. The reference arm length for the fundamental wavelength was adjusted to match that of the SHG wavelength with a residual phase of 0.07 rad which corresponds to a ~246 μm length difference.
Fig. 3 Stabilization of two wavelengths λ1 and λ2 for two-color interferometry. (a) Fractional stability in terms of the Allan deviation of the optical comb used for stabilization of the fundamental wavelength λ1. The beat frequency of λ1 with the optical comb is shown to indicate the stability of λ1 is comparable to that of the optical comb. (b) Optical spectra and signal-to-noise ratios of λ1 (1555 nm) and its second harmonic wavelength λ2 (777.5 nm).
3. Experiments and discussion
Figure 4 presents an experimental result which verifies the performance of the two-color interferometer described so far. In this experiment, the target distance L was fixed at 2.50 m after positioning the target mirror by means of frequency-sweeping interferometry [25,26]. The OPLs D1 and D2 of Eq. (3) were measured continuously during 250 s at a 40 kHz update rate. The surrounding environment was shielded to minimize air turbulence with ambient temperature being stabilized within a maximum drift of 5 mK. In this controlled room temperature condition, the measured D1 and D2 were found to undergo a similar time-dependent variation as plotted in Fig. 4(a). The maximum drift within D1 and D2 was ~70 nm, which was equivalent to a fractional change of 2.8 × 10−8. In comparison, the difference D2 − D1 was found much smaller than the variation of D1 or D2 by about two orders of magnitude, with the standard deviation (k = 1) being 5.33 × 10−10 m. With the coefficient A being 141.41, the correction term of Eq. (3), i.e. A(D2 − D1)/L was calculated as shown in Fig. 4(b). Finally, the compensated distance ΔL of the two-color method was obtained as in Fig. 4(c), in which the slowly-varying drift apparently seen in both D1 and D2 due to the refractive index of air was well suppressed. It was also noted in Fig. 4(d) that the refractive index of air estimated by ΔD1/L agreed well with Δn calculated using the Ciddor’s equation with sensing of temperature, pressure, humidity and CO2 concentration. The environment sensing was updated at a rate of 8 s because of the relatively slow response time of the used thermometer. This experimental result demonstrates that the two-color interferometer constructed in this study is able to compensate for the refractive index of air in real time to the uncertainty level of 10−8 without actual sensing of the environmental parameters.
Fig. 4 Two-color measurement result for a given distance L of 2.5 m in well-controlled environment. (a) Optical path length (OPL) variation during 250 s for ΔD1 for the fundamental wavelength λ1 and ΔD2 for the second harmonic wavelength λ2. The difference ΔD2 - ΔD1 shows a standard deviation of σ = 5.33 × 10−10 m. The vertical plot of ΔD1 is upshifted by 5 nm to avoid overlap. (b) Calculation of one-color OPL ΔD1/L and two-color correction term AΔ(D2-D1)/L. (c) Compensated distance by two-color method shows a relative stability of 7.03 × 10−9 in standard deviation at an updated rate of 40 kHz. 100-point averaging by reducing the updated rate to 1 Hz enhances the standard deviation to σ = 3.15 × 10−9. Also shown is the base temperature measured using a thermometer. (d) Variation of the refractive index of air calculated by two-color measurement. For comparison, the refractive index of air estimated by substituting measured environmental parameters to the Ciddor’s equation is shown. The residual between the two-color method and the Ciddor’s equation is σ = 2.77 × 10−9. The residual value is downshifted by 3.0 × 10−9 to avoid overlap.
The next experimental result shown in Fig. 5 was conducted to validate the performance of the two-color interferometer in uncontrolled adverse environments. The shielding chamber surrounding the interferometer system was partially open so that the measurement beam line was disturbed by an air jet injected through a nozzle by 2 bar backing pressure during 60 s as illustrated in Fig. 5(a). The air flow was contained within a glass tube to prevent thermal influence on the granite stage supporting the interferometer. Measured D1 and D2 underwent a sudden rise with instability of the order of 10−6 as the air jet was turned on as shown in Fig. 5(b). The difference D2 − D1 was of the order of 10−9, being one order of magnitude larger compared to the previous case of well-controlled environment (See Fig. 4). Despite the dynamical measurement beam disturbance, the two-color compensation was able to suppress the air refractive index variation to a fractional standard deviation of 4.53 × 10−8 throughout the entire measurement period of ~300 s as illustrated in Fig. 5(c). On the other hand, the conventional compensation method using the Ciddor’s equation was not effective in dealing with the air jet disturbance as in Fig. 5(d) because the measured ambient parameters cannot represent the integral values averaged all along the OPLs in this case of localized disturbance of the measurement beam line.
Fig. 5 Two-color measurement in dynamic environment. (a) Air flow induced for experiment. (b) Measured temporal fluctuation of ΔD1 and ΔD2. The difference ΔD2 − ΔD1 shows a standard deviation of σ = 2.68 × 10−9 m. ΔD2 is downshifted by 0.2 μm. (c) Geometrical distance calculated by two-color interferometer shows a temporal variation of σ = 4.53 × 10−8 during air injection of 60 s. (d) Air refractive index variation and its residual. The standard deviation of the residual corresponds to 8.15 × 10−9 when the environment was stable at 0<t<135 and 275<t<325. The residual is downshifted by 1.0 × 10−7 to avoid overlap.
4. Conclusions
The two-color interferometer configured in this work was based on a fundamental wavelength of 1555 nm generated by phase-locking a DFB laser to the frequency comb of a femtosecond laser stabilized to the Rb clock. The second wavelength of 777.5 nm was produced by second harmonic generation of the fundamental wavelength. The two wavelengths showed fractional stability of the order of 3.0 × 10−12 at 1 s averaging, being able to measure optical path lengths up to 2.5 m with fractional fluctuations of the order of 10−9 using heterodyne phase measurements at an update rate of 40 kHz. This two-color scheme of using the comb-reference wavelength was capable of achieving an uncertainty of the order of 10−8 in measuring distances without sensing the refractive index of air even in adverse environmental conditions. The performance was comparable to that of the compensation method using the Ciddor’s equation which requires sensing of environmental parameters. The proposed two-color method would be useful in measuring long distances in open air conditions where the refractive index of the ambient air is not easily identified.
Acknowledgments
This work was supported by the National Honor Scientist Program funded by the National Research Foundation of the Republic of Korea (NRF-2012R1A3A1050386). Y.-J. Kim acknowledges support from the Singapore National Research Foundation under its NRF Fellowship (NRF-NRFF2015-02).
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