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A two-wavelength interferometer (TWI) based on a sinusoidal-phase-modulation method with an acetylene stabilized laser and a second harmonic generation (SHG) was developed. The periodic non-linearity error for the TWI was estimated to be ± 0.1 µm at a dead path of 0.32 m. A long-term measurement showed that the TWI stability was ± 3 × 10−7 at a dead path of 1.00 m for 12 hours with an ambient pressure variation of 3 hPa under controlled conditions of ambient temperature and humidity. Finally, we confirmed that the TWI has substantially better stability than a single-wavelength interferometer by comparing both interferometers with large temporal and spatial temperature variations.
© 2015 Optical Society of America
1. Introduction
Laser interferometers have been widely used to measure distances or displacements in industrial and scientific fields, such as spatial distributions of air refractive indices through optical paths or relatively fast index fluctuations compared to the response rate of environmental sensors. For example, thermometer response times introduce uncertainties to measurement results. To overcome the above issues, a two-wavelength interferometer (TWI) was theoretically proposed in the 1960s as a device that can correct for the refractive indices in air for the entire optical path without environmental sensors, e.g., thermometers and barometers [1]. The principle of the proposed TWI is given by the formulas:
where D is a geometrical distance, D1 and D2 are the optical distances of individual single wavelength interferometers, n1 and n2 are the air refractive indices for D1 and D2 respectively, and A is, as it is called, the A-coefficient which is an almost independent constant of ambient temperature, pressure and carbon dioxide concentration.Since the above proposal, TWIs have been extensively studied in order to demonstrate corrections for air refractive indices along their entire optical path [2,3]. Nevertheless, practical TWIs have not yet been available. One reason why TWIs have not been put into practical use is that it is much more difficult to overcome wavelength uncertainties and cyclic errors than for conventional single wavelength interferometers. Recently, to suppress wavelengths uncertainties, a TWI consisting of an optical frequency comb achieved an uncertainty of 5 × 10−8 with pulse-to-pulse interferometry when air refractive index fluctuations were on the order of 10−6 [4–6]. Though optical frequency combs have high wavelength stability and small wavelength uncertainty, length measurements with pulse-to-pulse interferometry were limited to the positions where optical frequency comb pulses interfered within the optical path range. In another method, a phase detection technique for interference fringes by using polarizing beam splitters (PBS) and wave plates, which have been widely used in single wavelength interferometers, was adapted to a TWI to measure long-range distances continuously [7]. This method utilizes many optical components, so their thermal drifts may induce cross talks among the polarizations which can lead cyclic errors for practical long term use. Another example is a heterodyne based TWI [8]. Recently, in order to study a crustal strain by measuring small displacements at a geometrical distance of 70 m for several days, a heterodyne based TWI was developed with a relative resolution of 2 × 10−9. However, the configuration is complicated and could be expensive because the heterodyne TWI was basically for scientific research. As one of TWI applications, a temperature measurement for a long distance of 30 m was demonstrated where controlling and measuring the air temperature gets difficult due to the long distance [9].
In this study, to suppress wavelength uncertainty, an acetylene stabilized laser and a second harmonic generation (SHG) were implemented into a TWI. Furthermore, to reduce cyclic errors, a sinusoidal phase modulation (SPM) method [10–12], which has a small number of optical components, is applied to the TWI. To minimize the uncertainty associated with electrical circuit drift and to achieve fast response rate, field-programmable gate array (FPGA) boards (PXIe-7961R, National Instruments Corporation) were used in the TWI.
2. Sinusoidal phase modulation (SPM)
Equation (1) implies that when the optical distances D1 and D2 measured by individual single wavelength interferometers have cyclic errors, the TWI measurement distance must have a cyclic error that is as large as the A-coefficient times the cyclic error difference from each single wavelength interferometer. Standard PBS based interferometers have single nanometer cyclic errors [13] and their A-coefficients are generally constants ranging from approximately 50 to 150. As a consequence, TWI uncertainties derived from cyclic errors are enlarged to sub-micrometer ranges.
The principle of the SPM is described here and shown in Fig. 1. Laser diode (LD) current modulation at an angular frequency of ωm by sinusoidal wave creates an optical output frequency described as a function of time (t) by:
where fo is the optical carrier frequency and γ is an optical frequency deviation. The emitted light from the LD is branched into two parts by a beam splitter (BS). One is directed to a reference reflector and the other propagates to a probe reflector. The reflectors direct the light back to the BS where they interfere with each other. The light interference is detected by a photodetector (PD) and converted into an electrical signal. The signal intensity I is given by the following equations: where B is the amplitude of the alternate current component, Io is the direct current component, n is the refractive index of air, co is the speed of light in a vacuum, and Dm is the difference between the reference distance dr and the probe distance dp. In Eqs. (5) and (6), ϕ corresponds to the interference signal phase, mf is the modulation index and λo is the optical carrier wavelength given by co/fo. Developing the right side of Eq. (5) by a k-th order Bessel function Jk(mf) gives the equation:Equation (8) shows that the multiplication of sinusoidal waves with angular frequencies 2ωm and 3ωm by the signal I gives 2BJ2(mf)cos(ϕ) and 2BJ3(mf)sin(ϕ), respectively, by employing lock-in amplifiers. To normalize the amplifier outputs, dividing them by 2BJ2(mf) and 2BJ3(mf) results in cos(ϕ) and sin(ϕ), respectively. Finally, the optical distance 2nDm is given by the equation:
Fig. 1 Schematic image of an SPM laser interferometer. LD: laser diode, BS: beam splitter, PD: photodetector, dr: geometrical distance between the BS and the reference reflector, dp: geometrical distance between the BS and the probe reflector. Dm is the difference between dp and dr.
3. Experimental setup
3.1 The two-wavelength interferometer (TWI)
Figure 2 shows the TWI experimental setup including a distributed feedback (DFB) LD (FRL15TCWD-D66-19610-D, Furukawa Electric Co., Ltd.) with a 1539 nm fundamental wavelength, a 500 kHz linewidth, and a 40 mW output power. The DFB laser was stabilized to an acetylene isotope 13C2H2 absorption line (detailed in Section 3.2). The output light propagated through an optical fiber (OF) to a coupler, which split the light into two beams. One was connected to an SHG which was a piece of periodically poled lithium niobate (PPLN), in order to generate half wavelength light to the incident fundamental light. In this setup, 90% of the DFB output was assigned for the SHG and a 0.4 mW power was yielded as the SHG outout. Then the SHG light propagated to the wavelength division multiplex (WDM) coupler. The other beam was directly propagated to the WDM coupler where the fundamentaland the SHG beams were combined. The combined beam propagated to the end of the OF and was collimated by a parabolic mirror (PM). The collimated light propagated toward a BS where it was divided into a reference reflector and a probe reflector. The probe reflector was fixed on a piezo stage (P-621.1CD, Physik Instrumente GmbH). The beams were reflected back to the BS where they interfered with each other. The recombined beam was focused into another OF by a second parabolic mirror. The light travelled through the OF to a third distant parabolic mirror in order to prevent thermal effect by the PD to the interference. A dichroic mirror (DM) separated the light into the fundamental wavelength beam and the SHG beam. Finally, the two beams were converted into electrical signals by PDs via lenses (L). Conversion of the electrical signals to displacement information was carried out by the procedure described in the previous section.
Fig. 2 Configuration of the TWI. Coup.: coupler, OF: optical fiber, WDM: WDM coupler, PM: parabolic mirror, BS: beam splitter, DM: dichroic mirror, L: lens, PD: photodetector, M: mirror.
Regarding the displacement calculations, a 10 MHz sinusoidal wave was generated in the FPGA unit and applied to the LD via a digital to analog (DA) converter (NI 5782, National Instruments Corporation) in order to modulate the optical frequency for the SPM. Then, 20 and 30 MHz sinusoidal waves synchronized to the 10 MHz wave were generated in the FPGA unit for the lock-in amplifiers.
3.2 Acetylene stabilized laser
The optical carrier frequency fo for the laser was stabilized to the P(12) liner absorption line for the acetylene isotope 13C2H2 [14–16]. This realized that a standard deviation of fo was 3.3 × 10−9. Figure 3 shows the assembled acetylene stabilized laser configuration. In this setup, the 10 MHz sinusoidal current wave was transmitted to the DFB laser to stabilize fo [16], as well as perform the SPM for the TWI as descried in the previous section. The feedback signal was created by a proportional-integral-derivative (PID) feedback circuit based on the FPGA unit, and was transported through the current driver which limited the signal bandwidth to 1.2 kHz.
Fig. 3 Configuration of the acetylene stabilized laser. The interior of the dotted frame indicates the acetylene stabilized laser. Coup.: optical coupler, OF: optical fiber, WDM: WDM coupler, AD converter: digital to analog converter, AD converter: analog to digital converter.
4. Results and discussion
4.1 Linearity
To evaluate the TWI measurement displacement linearity, the probe reflector on the piezo stage was moved back and forth with a velocity of 50 nm/s at a displacement of 4 µm with an optical probe path of 0.32 m. The probe optical path was covered with a 0.30 m long and 35 mm inner-diameter pipe in order to avoid ambient air refractive index fluctuations for the single interferometers. Measurements were carried out under an ambient temperature of 19.8 ~20.0 °C, an ambient pressure of 1022.1 ~1022.4 hPa, and a relative humidity of 46 ~48%.
Figure 4(a) shows the probe reflector displacement measurement results for the SHG interferometer (SHGI), the fundamental wavelength interferometer (FWI) and the TWI, where the SHGI operates at a wavelength of about 769.5 nm. To make the plots visible, the FWI data (black plot) and the TWI data (red plot) are shifted by + 0.5 µm and − 0.5 µm respectively from the actual values. The TWI displacement Dt in the plot was obtained by
where Ds and Df are displacements derived from the SHGI and the FWI, respectively. The A-coefficient is a constant equal to approximately 138 calculated from Eq. (2) and the Ciddor equation [17]. As shown in Fig. 4(b), the difference plot of Ds – Df has oscillating behavior with an amplitude of less than 1 nm. This behavior can be attributed to so-called “cyclic error”as discussed in the next paragraph. Since cyclic errors for the single wavelength interferometers by SPM can be suppressed to sub-nanometer ranges, the oscillating amplitude of Ds – Df is less than 1 nm as a result. A differences plot of Dt – Ds shown in Fig. 4(c) has similar behavior to Fig. 4(b) except for the amplitude of 0.1 µm and the sign. The amplified oscillating behavior can be attributed to the error of – A multiplied by Ds – Df. It can be confirmed by subtracting Ds from both sides of Eq. (10) as written in the equation:On the other hand, regarding Ds as the true value of the reflector displacement because of the SHGI’s small cyclic error, Dt has the error which equals to – A multiplied by Ds – Df. Since the A-coefficient is approximately 138 and the oscillating amplitude of Ds – Df is less than 1 nm, the oscillating error component of Dt which means a non-linearity error is estimated to be about 0.1 µm. This estimated amount can be seen as the amplitude of the plot in Fig. 4(c).Fig. 4 Measurement results for 4 µm probe reflector displacements on the piezo stage with a velocity of 50 nm/s. (a) Displacement measurements for the SHGI (green line) and the FWI (black line) without an environmental correction, and TWI (red line) are plotted as a function of time. To make the plots visible, the red line is shifted by − 0.5 µm from the actual values. (b) Displacement differences between the SHGI and FWII as a function of time. (C) Displacement differences between the TWI and SHGI as a function of time.
Figure 5 is obtained by exchanging the horizontal axis in Fig. 4(c) with Ds. The single data points and the blue line represent individual Dt – Ds data points and a 5-point moving average, respectively. The calculation of the moving average reduces the random fluctuation and shows clearer oscillation with a period of 0.77 µm, which closely corresponds to the fundamental light half wavelength. In other words, the oscillation is derived from the single wavelength interferometers “cyclic error.” This specific period can be also obtained for Ds – Df as a function of Ds if it is plotted. Consequently, Dt and Ds – Df oscillating behaviors originate from the single wavelength interferometers cyclic errors.
4.2 Stability under ambient pressure variations
Figure 6 shows stability measurement results over a 12 hour period. The results were obtained under the following conditions. The reference reflector was fixed near the BS, and the probe reflector was placed at a specific position to set a dead path of 1.00 m. The TWI was fixed on a ceramic board with a low thermal expansion coefficient. The probe optical path was covered with a 0.70 m long pipe to reduce airflow disturbance. During the measurements, the relative humidity was controlled to 45 ~47% and ambient temperature variations near the pipe were controlled to 20.2 ± 0.3 °C.
Fig. 6 Stability measurement results for (a) Ambient pressure. (b) Ambient temperature. (c) SHGI displacement measurements with no environmental corrections (green line), with environmental corrections (blue line), with only an ambient pressure correction (gray line), and TWI displacement measurements (red line). (d) Displacement differences between the TWI and SHGI with only a pressure correction.
Figure 6(a), 6(b), and 6(c) show the ambient pressure, the ambient temperature, and the displacement measurement results, respectively, as a function of time. In Fig. 6(c), the red, blue, green, and gray lines correspond to the displacements measured by the TWI, SHGI with environmental (pressure, temperature and humidity) corrections, the SHGI without environmental corrections and the SHGI with only a pressure correction, respectively. The environmental and pressure corrections were calculated using the Ciddor equation. To clearly see the results in Fig. 6(c), the green and red lines are respectively shifted by − 0.5 µm and + 0.35 µm from the actual values. The gray line indicates higher stability than the blue curve because the ambient temperature measurement uncertainty affects the temperature correction calculation. Figure 6(d) shows the displacement differences between the TWI and SHGI with only a pressure correction in order to eliminate measurement uncertainty from the temperature change. From the data variations, the TWI measurement error is ± 0.3 µm with a dead path of 1.00 m for 12 hours. The measurement stability can be estimated to be 3.0 × 10−7 by dividing the error amount by the dead path. In this estimation, the measured displacement by the SHGI, which includes the instability of the barometer and the uncertainty by the temperature variation of the optical path, is used as the true value for the comparison with the TWI. That is to say, the displacement measured by the SHGI still has a tiny error and is not ideal value. Therefore, the estimated stability could be larger and the actual TWI stability is expected to be better than that. Main limiting factor of the actual stability is considered to be the TWI cyclic error as described in section 4.1. Although it is technologically difficult to decrease the cyclic error further, by suppressing the noise components for 2BJ2(mf) and 2BJ3(mf) in Eq. (8) and calculating arctan{sin(ϕ)/cos(ϕ)}in Eq. (9) more precisely the TWI stability will improve.
4.3 Stability under temperature variations
To evaluate the influence of temperature variations on the optical path for length measurements, we covered the 1.00 m probe optical path with a 0.70 m long pipe and wound a rubber heater around the pipe to vary the temperature of part of the optical path. The interferometers were fixed on a ceramic board with a low thermal expansion coefficient. Figure 7 shows a schematic diagram of the experimental configuration. To monitor the temperature at different positions, three thermometers were employed. One (thermometer 1) was placed on the center of the external pipe surface. Another (thermometer 2) was placed close to the pipe to measure air temperatures near the optical path. The other (thermometer 3) was attached to the ceramic surface (not shown in Fig. 7). During measurements, the relative humidity was controlled to 45 ~47%, and a barometer measured an ambient pressure of 1010 ~1011 hPa.
Fig. 7 Experimental setup for the optical path temperature variation. The 1.00 m probe optical path was covered with a 0.70 m long pipe and wound with a rubber heater to vary the temperature of part of the path. The interferometers were fixed on a ceramic board with a low thermal expansion coefficient.
The measurement uncertainty due to the 2% humidity variation is negligibly small because it induces an error of no more than 0.02 µm for the SHGI even if no humidity correction is applied. The ambient pressure variation of 1 hPa leads to a measurement error of 0.3 µm for the SHGI as estimated by the Ciddor equation. However, the ambient pressure is generally spatially uniform and the temporal variation is corrected using the barometer so that the SHGI measurement uncertainties with environmental corrections, including ambient pressure, are sufficiently small when compared to the temperature variation influence.
Figure 8 shows measurement results while the optical path temperature was changing. Figures 8(a)-8(c) show thermometer 1, 2, and 3 outputs, respectively, as a function of time. In Fig. 8(c), the temperature of the ceramic surface varies from 21.05 to 21.12 °C. With this 0.07 °C temperature change, ceramic thermal extension displacement is estimated to be at most + 0.3 µm with an extension coefficient of 4 × 10−6 /°C for the optical path length.
Fig. 8 Measurement results while the optical path temperature was changing. (a) Temperature of the external pipe surface center. (b) Ambient temperature close to the pipe. (c) Ceramic surface temperature. (d) Measured SHGI displacements without environmental corrections (green line), SHGI with barometer, hydrometer and thermometer 1 corrections (blue line), SHGI with barometer, hydrometer and thermometer 2 corrections (gray line), and the TWI (the red line) as a function of time.
Figure 8(d) shows the displacement measurement results for each interferometer. The green line represents SHGI measurement displacement without any environmental corrections. By comparing the pipe temperature curve in Fig. 8(a) to the green line, it can be seen that the temporal temperature variations for the pipe strongly affect the SHGI measurement results. This is because the temperature variation induces a refractive index change in the air inside the pipe. The gray and blue curves are the SHGI outputs with environmental corrections from the Ciddor equation utilizing the output data from a barometer, a hydrometer and thermometers 1 and 2. The red curve shows the TWI measurement displacement results. The gray and blue curves show that the refractive index correction for the entire optical path using thermometer 1 or 2 overestimates or underestimates the ceramic thermal expansion of + 0.3 µm, respectively. This is because the thermometers cannot measure the spatial temperature average of the entire optical path. On the other hand, the TWI measurement results remain nearly constant compared to the SHGI results. This fact signifies that the TWI can correct for the air refractive indices throughout the entire optical path. A more detailed data analysis that shows the results derived from the TWI still has a small variation at around 0.5 hours and 1.25 hours in Fig. 8(d). One possibility for the contributing factors of the small variations is heat radiation from the rubber heater to the interferometer body, for example, between the BS and the reference reflector. The displacement of the TWI ranges from − 0.4 to 0 µm, which is nearly the same as the estimated ceramic thermal expansion, while the single wavelength interferometer displacement is from − 0.9 to 0 µm or from 0 to + 1.2 µm. This shows that the TWI is substantially better than the interferometer with conventional environmental corrections when temperature distributions are extensively large.
5. Conclusion
We developed a TWI based on a SPM with an acetylene stabilized laser and a SHG, and estimated the periodic non-linearity of the TWI to be ± 0.1 µm at a dead path of 0.32 m. By performing long term measurements, the stability of ± 3 × 10−7 was achieved for a dead path of 1.00 m for 12 hours with an ambient pressure variations of 3 hPa under controlled conditions of ambient temperature and humidity. Finally, we performed a comparison of the TWI measurement results with typical single wavelength interferometers by changing the optical path temperature. The measured displacement range from the TWI was − 0.4 to 0 µm while the single wavelength interferometers with environmental sensors ranged from − 0.8 to 0 µm or from 0 to + 1.2 µm. These results show that the TWI is superior to the interferometer with conventional environmental corrections under large temporal and spatial temperature changes, and could be a powerful tool for measuring displacements with extensive refractive index fluctuations in air.
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