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Abstract
We have demonstrated a simple method to measure high-precision absolute angular displacement using an optical frequency comb (OFC). The dispersive interferometry with parallel configuration can take advantage of its large non-ambiguity range and achieve absolute angular measurement in a large range. The influence factors of the angle accuracy, including the accuracy of optical path difference, the determination of absolute zero position and the correction of sine arm have been analyzed in detail. The angle comparison is performed with the autocollimator and multi-tooth indexing table. The angle accuracy can reach ±2 arcsec (k=2) in the range of 5°, which represents a good agreement with the Monte Carlo simulation. The proposed approach has potential to be extended to multi-degree-of-freedom measurement with a simple structure in future.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
High-precision angular measurements are essential for various scientific researches and industrial fields such as spatial positioning [1], calibration of machine tools [2], digital assembly [3,4] and precision instruments [5]. Different techniques for angle measurement have been employed including autocollimator [6,7], optical encoder [8], electronic sensor and optical interference [9 –13]. Among these methods, angular interferometer has its unique advantages, such as high precision, sensitivity and large range. According to the different optical layout and interference pattern, it can be roughly divided into self-mixing interferometry [9], parallel plate interferometer, heterodyne interferometry [10] and total internal reflection [11]. The angular displacement can be obtained by recording the continuous phase variation during the rotation.
Unfortunately, due to the existence of $2\pi $ phase-ambiguity, these methods only can measure an incremental angular displacement rather than the absolute angular position. Once the laser beam is interrupted or the system is restarted, the absolute angular position will be lost and cannot be recovery [12,13]. For this reason, the angular interferometer lacks of the capacity of measuring surface directions for alignment of segmented mirrors [14], monitoring the position of an object in a light-off environment. Meanwhile, because of the nonlinear model and lacking a fixed absolute reference, the alignment of target mirror at datum need to be performed repeatedly before each measurement. For example, Renishaw’s XR20 rotary axis calibration system, the automated calibration module is specially designed to eliminate alignment error [15]. It means that it is not a simple and compact structure in actual application. Obviously, the absolute angular position measurement instead of relative angle variation measurement is becoming increasingly important for industrial applications.
During the past decades, the invention of the optical frequency comb (OFC), also known as femtosecond pulse laser, has brought many new types of optical sensors in the field of dimensional metrology [16,17]. An OFC can emit evenly spaced ultrashort pulse trains in the temporal domain and perform uniform, discrete narrow optical modes across a broad spectrum in the frequency domain. When the tens of thousands optical modes were referenced to the microwave frequency standard, the OFC becomes an ultra-stable ruler in time, space and frequency domain [18]. The methods for measuring the absolute distance [19 –23] and thickness have been developed [24]. In recent years, the OFC has also been used in absolute angular measurement [25 –27]. For instance, angle measurement with OFC has also been achieved by the second harmonic generation in nonlinear optics [25]. The femtosecond pulse laser combined with the diffraction grating as the reflector of the autocollimator can expand absolute angular range significantly [26]. Besides, the absolute angular position measurement based on the dual-comb balanced cross-correlation technology has also been reported [27]. The above-mentioned methods employ its unique temporal and spectral characteristics to measure absolute angle. Essentially, the pulse interval of OFC is equal to its non-ambiguity range, which usually ranges from tens of millimeters to several meters. Hence, it can overcome the limitation of small non-ambiguity range of the traditional laser interference in angle measurement. Meanwhile, the excellent ranging accuracy can also offer a foundation for high-precision absolute angular measurement.
In this paper, we introduce a method for absolute angular measurement with dispersive interferometry using an optical frequency comb. The parallel configuration consisting of plane-mirror and retroreflector is employed to construct two measured arms. It can make full use of the non-ambiguity range of dispersive interferometry to achieve absolute angular measurement in a large range. The ranging performance essentially determines the angle accuracy and has been carefully evaluated. The determination of absolute zero position has been performed for achieving high-precision absolute angular measurement at arbitrary initial position. After the calibration of sine arm, a series of experiments have been done to verify the performance of the proposed approach. By comparing to the multi-tooth indexing table and autocollimator we find an agreement better than 2 arcsec in the range of 5°. Because the Fourier transform dispersive interferometry can solve multiple optical path differences (OPDs) simultaneously. By building three parallel measuring beams and optimizing the target mirror, the proposed method has potential to be applied to multi-degree of freedom measurement, such as position monitoring, geometric error measurement of linear guide rail, autocollimator calibration and calibration of machine tools in future.
2. Principle of absolute angular measurement
When the pulse trains from OFC illuminate the typical Michelson interferometer, the optical spectrum analyzer (OSA) is exploited to sample the interference spectrum. For the sake of simplicity, the optical intensity of two arms is considered as equality. The interference spectrum recorded as a function of OPD can be expressed as
In this expression, $\; I(v )$ is the interference spectrum, ${I_0}(v )$ denotes the power spectrum distribution (PSD) of the light source. v and c represent the optical frequency and the light speed in vacuum, respectively. L represents the OPD between two interference arms. From Fig. 1(a), $I(v )$ is a cosine function modulated by the PSD with a fixed oscillating period $\tau = L/c$. The interference fringe will become dense as the delay time $\tau $ increases gradually. After a Fourier transform, the result is as follows:
Here $\hat{I}(t )$ is the Fourier transform of $I(v )$, $\; \widehat {{I_0}}(t )$ is the Fourier transform of ${I_0}(v )$. $\delta (t )$ and ${\otimes}$ represent Dirac delta function and convolution calculation, respectively. Because the spectrum signal collected by the OSA is a real signal, after the Fourier transform, the three peaks can be observed shown in Fig. 1(b). For this reason, when the measured pulse being ahead of or behind the reference pulse is apart the same optical path, the same interferogram will be emerged on the OSA, which can be known as“mirror interference”. The positive peak $\tau $ represents the delay time between two interference pulses. After the band-pass filtering and the inverse Fourier-transform, the absolute distance can be derived from the slope of unwrapped phase and be expressed as
Note that ${n_g} = n + ({dn/dv} )v$ denotes the group refractive index of air and is determined by the central frequency of the optical source. And n is the refractive index of air. While the delay time $\tau $ is too small, the peak will be behind the DC peak at the origin position and cannot be extracted. Hence, it exists a minimum measurable OPD ${L_{min}} = c \cdot {\tau _{min}}$, which depends on the pulse width of optical source and is about 12 µm in our system. The maximum delay time restricted by the Nyquist sampling can be described as ${\tau _{max}}= 1/(2 \cdot v_s)$. ${v_s}$ is the actual frequency sampling interval and depends on the resolution of the used OSA. The maximum measurable OPD ${L_{max}} = c \cdot {\tau _{max}}$ is generally about tens of millimeters [21]. The features will make influences on the absolute angular measurement later.The schematic of absolute angular measurement is shown in Fig. 2. A femtosecond pulse laser (MenloSystems, C-Fiber) with spectral range of ∼160 nm and the center wavelength of 1582 nm is performed as the optical source. The output power of the light source is about 30 mW. Its repetition frequency is 100 MHz and stabilized to the Rb atomic clock (Symmetricom, 8040C). The pulse trains from the light source cross the fiber circulator and is collimated by CL (Thorlabs, RC02FC-P01) towards the BS. The output light is divided into two parallel measured beams by BS and ${M_0}$ and then transmits along measurement arm 1 (above) and measurement arm 2 (below), respectively. To take advantage of ${L_{max}}$ sufficiently, the parallel configuration is constructed by the HR (Newport, UBBR1-1S) and plane mirror (M1, M2). It should be noted that since the spectral bandwidth of the OFC is extremely broad. In order to avoid dispersive affection for absolute angular measurement, the hollow retroreflector instead of the solid corner cube is employed as the target mirror. While the diagonal face of the HR is approximately perpendicular to the probe beam, the distance between M1 and M2 is adjusted to make the OPD between two measured path $L = 2{n_g}|{({{l_A} + {l_B}} )- ({{l_C} + {l_D} + {l_E}} )} |$ be closed to zero. Then assuring that the OPD L between two measurement arms is always smaller than ${L_{max}}$ during the HRs rotation. After being reflected by a pair of HRs and plane mirrors, two measured beams recombine on BS and interfere with each other. The interference signal is coupled into the single mode fiber and delivered to the commercial OSA (Yokogawa, AQ6370C). The OPD follows a sinusoidal function of the rotation angle and the quadruple path length $L = 4l{n_g}$ is caused by the HR and plane mirror. While the target mirror takes place a rotation, the rotation angle $\theta $ can be simply expressed as:
where R is the length of the sine arm, which represents the fixed separation between the two apexes of the HRs. The OPD is defined as the positive while the measured pulse 1 is further away from measured pulse 2 relative to the BS. And the absolute angle is defined as the positive while the target mirror rotates clockwise.
Fig. 2. Schematic diagram of the absolute angular measurement. OFC, optical frequency comb; Circ., fiber circulator; CL, collimator; BS, beam splitter (50:50); M0, M1, M2, plane mirror; HR1, HR2, hollow retroreflector; OSA, optical spectrum analyzer. The red line corresponds to the light that propagates in the free space and the yellow line corresponds to the light that propagates in the single mode fiber.
3. Analysis of factors affecting the angle accuracy
3.1 Dispersive interferometry for absolute distance measurement
The precision measurement of OPD is the foundation of high-precision angular measurement. The Michelson interferometer is built to test the performance of the dispersive interferometry using an OFC. The interferogram is acquired with 32768 sampling points and wavelength ranges from 1480 nm to 1640 nm. Because the OSA separates spectral components almost linearly in wavelength, while we convert the wavelength domain into the frequency domain, as a result of the inverse relationship between wavelength and optical frequency, the frequency sampling interval will become irregular as shown in Fig. 3(a). The frequency-step variation is great from one end of the broad spectrum to the other. For this reason, as depicted in Fig. 3(b), the interference peak will be quickly broadened as the delay time $\tau $ increases in the Fourier domain. This phenomenon can be viewed as an artificial chirp. It will extremely influence the location accuracy of the interference peak and subsequently the accuracy of phase calculation. Hence, the interference spectrum is resampled using a linear interpolation prior to applying the FFT. Then the Fourier transform is performed together with an appropriate Hanning window. After the linear resampling, the interference peak always keeps sharp even if the delay time continuously increases.
Fig. 3. (a) Frequency sampling interval of the OSA. (b) Interference peaks of the different delay time in the Fourier domain.
Next, we made a linear measurement comparison with the commercial laser interferometer (Renishaw, XL-80) to test the ranging performance before and after the linear resampling. The target mirror mounted on the translation stage (Physik Instrument, M-521.DD) was moved by 5 steps at an increment of 6 mm, each position was repeatedly recorded 5 times and the whole process was accomplished within 240 s. The air temperature, relative humidity and air pressure were approximately 25.1 ℃, 68% RH, and 100.40 kPa, respectively. These environment parameters were used for the correction of the refractive index of air based on Ciddor Equation [28]. As shown in Fig. 4, when the interference spectrum undergoes FFT with a linear resampling, the precision and accuracy of the measurement results have been greatly improved. Compared to the commercial interferometer, the residuals were reduced from ±40 µm to ±1.5 µm. Immediately, the measurement uncertainty of the OPD was performed according to the Guide to the expression of uncertainty in measurement (GUM). Based on Eq. (3), the totally combined uncertainty can be calculated as:
Fig. 4. Comparison between dispersive interferometer and commercial interferometer for distances up to 30 mm. (a) Interference spectrum without linear resampling. (b) Interference spectrum with linear resampling. The error bars denote the standard uncertainty from five different measurements.
Table 1. Uncertainty evaluation of OPD using dispersive interferometry
3.2 Determination of absolute zero position
The determination of absolute zero position is the premise of high-precision absolute angular measurement. The incident beam should be perpendicular to the end face of the target at datum. Prior to determining the absolute zero position, the parallel of two probe beams is adjusted by the two position sensitive detectors (THORLABS, PDQ30C) along a 1 m rail way. Because the optical path variation of two HRs is relatively small during the rotation, the residual alignment error for angle measurement can be ignored. The angular interferometer using the double HRs and plane mirrors as the target mirror has its clear advantage. For instance, when the HRs rotates around the arbitrary point, the variation of optical path is identical. In other words, the position of rotation axis is not influence for angle measurement. On the other hand, the probe arm consisting of HR and plane mirror can effectively restrain the sensitivity of HR for the in-plane displacement. For angle measurement based on sine principle, as shown in Fig. 5(a), the position P is defined as the absolute zero position while ${l_1} = {l_2}$. Actually, it is hard to keep strictly alignment and then $P^{\prime}$ is mistaken for the absolute zero position. As illustrated in Fig. 5(b), supposing that the initial deviation angle is $\; \alpha $ and rotation angle is $\beta $, which will introduce an angle error $\; \varepsilon $ :
Fig. 5. Schematic diagram of initial angle deviation. (a) $\; \alpha $ is the initial deviation angle corresponding to path variation ${l_\alpha }$. $\beta $ is the rotation angle corresponding to path variation ${l_\beta }$. (b) $\; P$ is the absolute zero position. $P^{\prime}$ is the false zero position. $\beta ^{\prime}$ is the reading angle from the proposed interferometer.
Fig. 6. Simulation of angle error introduced by different initial angle deviation. (a) Angle error for small rotation angle with large initial offset angle. (b) Angle error for large rotation angle with small initial offset angle.
The calibrated process can be summarized in the following steps: First, as depicted in Fig. 7(a), the HRs is placed as close as possible to the rotation center of the multi-tooth indexing table (MTT). Subsequently, the position of the MTT is adjusted to make the incident beam keep roughly perpendicular with the end face of the HRs. As depicted in Fig. 7(b), assuming that the HRs has a positive angle of $\alpha $ corresponding to the output OPD ${L_\alpha }$. Immediately, because of the mirror interference, the sign of ${L_\alpha }$ cannot be determined. In order to ensure that the OPD ${L_\alpha }$ is always positive after the repeated adjustments of HRs, the M1 is moved toward the BS. As the fringe become gradually dense and then the OPD between two pulses ${L_\alpha }$ is about 10 mm, the position of M1 is fixed. Then the AZ is always on the right side of the OZ. When the HRs rotates from $\alpha $ to ${\beta _ + }$, the output OPD of dispersive interferometer is ${L_{\beta + }}$. The variation of optical path can be given by:
Fig. 7. The calibration of absolute zero position. (a) experimental setup, (b) the schematic of calibration, the horizontal axis represents the output OPD of dispersive interferometer during rotation. The vertical axis represents the position that absolute angle is zero. OZ represents the position that the OPD L is zero, AZ represents the OPD of the absolute zero angle ${L_0}$.
The difference between Eq. (7) and Eq. (8) can be expressed as:
Based on Eq. (9), the deviation angle $\alpha $ can be written as:
Then the Monte Carlo simulation (MCS) is used to evaluate the adjustment accuracy of $\; \alpha $. $\Delta L$ follows a normal distribution $N({0,{\sigma_0}} )$ and ${\sigma _0}$ is better than 3 µm. Taking into account the actual volume of HRs, the length of sine arm R is design to 140 mm and the rotation angle $\beta $ from MTT is 6°. ${\beta _6}$ is divided by the normal distribution $N({{\beta_6},\sigma } )$, the standard deviation of MTT $\sigma $ is better than 1 arcsec in the range of ±10°. Figure 8 shows that the simulation result, in the 95% confidence interval, the initial angle $\alpha $ can be well controlled within $4^\prime$. While the HRs is at the calibrated zero position, the output OPD ${L_0}$ is recorded. The Eq. (4) can be modified and rewritten as:3.3 Correction of sine arm
The length error of sine arm is another factor that affects the accuracy of absolute angular measurement. In the case of micron-level ranging accuracy, the influence on the accuracy of R for the angle measurement is studied. While ignoring the residual angular deviation ${\alpha _0}$, the angle errors $d\theta $ introduced by the length error of sine arm $dR$ can be given by
where $R = 140\; mm$ and $L = 4Rsin\theta {n_g}$. Figure 10(a) shows the simulation results that the angle errors vary with different $dR$. In order to acquire the exact value of R, the measured OPD from dispersive interferometry and the standard angle $\beta $ from MTT are employed to calibrate the sine arm. The calibrated error $dR$ can be described as:Fig. 10. (a) Different length errors of sine arm for the influence of angle measurement. (b) Different standard angle for the calibration influence of sine arm.
Figure 10(b) shows the simulation results that the $dR$ varies within different standard angle. The calibration accuracy improves gradually as the standard angle increases. In view of the maximum measurable OPD, the standard angle of 6°is used to calibrate R. The calibration formula can be described as:
3.4 Angle comparison and the uncertainty evaluation
After the calibration, the stability of the system is tested under a bad environmental condition. The HRs was mounted on a slightly vibrating table and placed at a distance of about 1.5 m away from the BS. The fan is used to impose air disturbance on the measured paths. The interference pattern was continuously collected once every 1 s over 30 minutes, and the angle variation was recorded. As shown in Fig. 11(a), the angle fluctuation was approximately 1.5 arcsec and the standard deviation is 0.31 arcsec. Since the two measured arms have a close optical path, it has a certain immunity to air disturbance. The instability of the system mainly originates from the slight vibration of the table. It can be effectively suppressed by using a high detection rate spectrometer in future. To test the resolution of the proposed system, the MTT was operated to perform forward and backward rotation with step angular displacement. From the Fig. 11(b), the system resolution was better than 4 arcsec in an ordinary laboratory condition.
Next, the angle measurement was performed with the proposed method and the autocollimator (AUTOMAT, ULTRA-2045) for comparison in a small range. The resolution and accuracy of the autocollimator is 0.1 arcsec and ± 1 arcsec, respectively. Initially, as depicted in Fig. 12, the HRs and the reflector of autocollimator were simultaneously mounted on the turntable back to back. The plane mirror was adjusted to be orthogonal to the autocollimator until the X-axis of the autocollimator reading is close to zero (Y-axis reading < 1 arcsec). Subsequently, the turntable was operated at a step of 240 arcsec, for each position the spectral interferogram was quickly recorded five times. The output $\theta $ given out by the proposed system according to Eq. (11) and the output of commercial autocollimator were recorded simultaneously. The results were plotted in Fig. 13(a), the residuals were well controlled with ±1.5 arcsec over the range of 1200 arcsec. To further validate the performance of the presented approach in a large range, the MTT was used to conduct an angle comparison. Similar to the above-mentioned measurement procedure, the HRs was mounted on top of the MTT, which were rotated by 5 steps at an increment of 1°. As shown in Fig. 13(b), the angle deviations between the measured result and MTT were well below ±2 arcsec in the range of 5°.
Fig. 13. Angle comparison experiments. The pink line represents the residuals between measurement angle and reference value, the purple error bars represents the 2 times expand uncertainty from the Monte Carlo simulation. (a) The results in comparison with an autocollimator in a small range. (b) The results in comparison with an MTT in a large range.
Finally, the measurement uncertainty of the proposed angular interferometer is evaluated. The uncertainty propagation formula based on MCS can be represented as
4. Summary and prospect
In this paper we have demonstrated a method of absolute angular measurement with dispersive interferometry using an optical frequency comb. The dispersive interferometry with parallel configuration is constructed to achieve the absolute angular measurement. The absolute zero position is precisely determined by MTT. After the calibration, high-precision angle measurement can be performed at arbitrary initial angle. In comparison to an autocollimator and the MTT, we have demonstrated an accuracy of ±2 arcsec for measured angle up to 5°. These results suggest that the proposed approach has potential towards practical applications for the industrial metrology. In future, the three parallel beams combined with three HRs will be employed to construct the novel system. Especially, the multiple OPDs can be simultaneously recorded by a single spectrometer. Then the Fourier transform dispersive interferometry has the ability to achieve the multiplexing. In addition, the parallel optical layout could provide immunity against environmental disturbances in a long distance. The OFC is considered as a broadband light source as well as a long coherent length, while the reference arm is added into the system, it can achieve simultaneous measurement of absolute distance and angle. It is almost impossible with the traditional interferometer such as homodyne and heterodyne interferometer or white light interferometer. It is of great interest in application with multi-degrees of freedom measurement. These mentioned above advantages are in the next step of our research.
Funding
National Natural Science Foundation of China (51835007, 51775380, 51721003).
Disclosures
The authors declare no conflicts of interest.
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