Periodic nonlinear errors caused by frequency mixing are serious obstacles for increasing the resolution of heterodyne grating interferometers. To eliminate the periodic nonlinear errors, a spatially separated heterodyne grating interferometer is proposed in this study. Two modulated beams with different frequencies are transferred respectively by two fibers, which form a spatially separated construction. A couple of comparison experiments in both time domain and frequency domain are designed and conducted. The results of the frequency-spectrum analysis experiment showed that the periodic nonlinear errors were no larger than 0.086 nm, which proved that the proposed system was effectual in eliminating periodic nonlinear errors.
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Ultra-precision measurement and positioning is the cornerstone of precision manufacturing, semi-conductor production, and nanoscience and technology. As research becomes deeper, the demand for measurement resolution increases. For example, in 1994, the large optics diamond turning machine (LODTM) by LLNL was equipped with seven single frequency-stabilized laser interferometers with a resolution of 1.2 nm each to measure all large-travel displacements . In the 21st century, interferometers for lithography and microlithography have achieved the sub-nanometer scale [2–4]. Undoubtedly, measurement in the picometer scale is a general tendency, and the companies in the precision instrumentation field have made great efforts and obtained high achievements. For example, the resolution of the SIOS Company’s commercial laser interferometers has reached 20 pm .
Laser interferometry and grating interferometry are the two main approaches in precision displacement measurement, and they have been widely employed in the current fields of production and research . Instead of the wavelength in laser interferometry, the grating pitch is utilized as the benchmark in grating interferometry. Using materials with low coefficient of linear expansion, the grating pitch can be less affected by fluctuations in the air refraction index and temperature than the laser wavelength. Thus, the grating interferometer is tolerant to more unstable measurement conditions, which gives it broad application prospects in ultra-precision measurement and positioning.
However, an ordinary homodyne grating is restricted by the complex reading head, influence of amplitude variation, and DC offset . In order to overcome these restrictions and further strengthen the grating interferometer, heterodyne interferometry is combined with grating interferometry . The heterodyne grating interferometer benefits from these two kinds of interferometry, measuring, as a substitute of the sinusoidal intensity, the periodical phase of the beat signal, which has better resistance to noise and surrounding disturbances and provides higher resolution. Lee and Hsieh et al. [9-11] designed several heterodyne grating interferometers, analyzed their resolution and measurement errors, and implemented multi-degree-of-freedom (multi-DOF) translational displacement and angular rotation measurements.
Nonetheless, in the heterodyne grating interferometers, some problems are induced by heterodyne interferometry. Hsieh et al.  noticed and discussed the errors from nonuniform grating, frequency mixing, polarization mixing, and polarization-frequency mixing phenomenon. Nanoscale periodic nonlinear errors caused by the frequency-mixing phenomenon are a main obstacle for achieving high measurement accuracy with the heterodyne interferometer. Therefore, to realize the goal of a higher accuracy, the barrier of periodic nonlinear errors should be removed.
In terms of heterodyne interferometry, two main solutions have emerged to overcome this barrier. One is adding the phase compensation part to the signal processing system, in order to compensate and reduce nonlinear errors [12–15]. The other is separating the reference beam and the measurement beam spatially, thus fundamentally eliminating the frequency-mixing phenomenon [16–18]. Groot et al. designed a technique for reducing the polarization mixing by using separated-beam light delivery and angled beams. The specific system employs a birefringent prism pair to avoid beam mixing.
Following the idea of spatially separating the beam, and based on our previous nonlinearity research , we propose a spatially separated heterodyne grating interferometer. In the proposed system, a fiber-delivered spatially separated construction is utilized to eliminate the periodic nonlinear errors and reduce the need of alignment. Retro-reflectors are employed to strengthen the resistance to unexpected tip and tilt of the grating.
2. Principle, method, and optical configuration
2.1 Periodic nonlinearity in heterodyne grating interferometer
In practical heterodyne laser interferometer, the demodulated phase of the interfere signal is not linear to the real displacement, this nonlinearity exhibits periodic characteristics, and thus it is called periodic nonlinear error . Similar to the heterodyne laser interferometer, the effects caused by nonorthogonality and ellipticity of the laser source and misalignment and imperfections of the polarizing beam splitter  exist in heterodyne grating interferometers. Particularly, the grating itself can also induce nonlinearity. Nonlinearity in a grating interferometer can be manifested in periodic and nonperiodic nonlinear errors. Specifically, the latter is mainly caused by the ruling error of the grating, or random fluctuations for each grating pitch. The effects of periodic and nonperiodic nonlinearity influence the measurement results at the same time. The proposed heterodyne grating interferometer is focused on the elimination of the periodic nonlinear errors. In other words, it may be ineffective for addressing nonperiodic nonlinear errors.
Figure 1 shows a typical configuration of a heterodyne grating interferometer. A polarizing beam splitter (PBS), an analyzer (ANm), and a photodetector (PDm) are employed to acquire the measurement signal. The optical configuration of the reference signal is omitted.
Ignoring the initial phase, the alternating intensity of expected measurement signal can be expressed as
In grating interferometry, it is worth to mention that the phase change of the order diffraction beam () isEq. (2), which means that the periodic nonlinearity in in-plane displacement of the heterodyne grating interferometer is only relative to the grating pitch. Thus, Eq. (1) can be revised in a grating expression
However, as Fig. 1 displays, in a common-path (COP) configuration, owing to the frequency-mixing and polarization-mixing phenomena, the actual measurement signal can be expressed as
The right chart of Fig. 1 is a MATLAB simulation based on Eq. (4). The measured displacement curves are calculated by , and their nonlinearities are determined by the transmission factor and the polarization-leakage factor.
Besides the time domain model, another approach to observe the periodic nonlinear errors is the frequency spectrum analysis. Here, it is used to prove the elimination of the periodic nonlinear errors. As reported by Badami et al. , it is a simple, visual, and effective method to measure periodic nonlinear errors. The connection between k-order Doppler frequency shift to phase change is expressed in a derivative form
Particularly, the frequencies of the three items in Eq. (4) from left to right are , , and , which represent the base signal, 1st, and 2nd harmonic signal of the spectrum, respectively.
Using a spectrum analyzer, the value of the periodic nonlinear errors can be calculated by the amplitudes of different frequencies in the interference signal. Schmitz et al.  derived the expression of displacement measurement error in the frequency spectrum analysis
2.2 Optical configuration of the proposed grating interferometer
To avoid the frequency-mixing phenomenon and eliminate periodic nonlinear errors, a spatially separated heterodyne grating interferometer is proposed and the optical configuration is illustrated in Fig. 2. It is fabricated with three parts: modulated laser source, polarization maintaining optical fibers, and grating interferometer.
As shown in the left blue dotted box of Fig. 2, a separately modulated laser source was constructed. An ordinary COP construction laser source contains only one acoustic optic modulator to generate different frequencies in different polarizations. Compared with that above, the separately modulated laser source fundamentally avoids frequency mixing, which has the advantage of eliminating the periodic nonlinear errors. In particular, a single-frequency beam emitted from a He-Ne laser (L) was equally split by a beam splitter (BS1). A mirror (M) was used to make the propagation of the two beams agree with each other. Then, two acoustic optic modulators (AOM1 and AOM2) modulated the two beams into two different frequencies, and , separately.
In the upper purple dotted box of Fig. 2, two polarization-maintaining optical fibers (PM1 and PM2) were employed to transfer the output lasers of the AOMs with the specific frequencies needed, while shielding the other unused beams. They can substitute pinholes or apertures for the same usage. The advantages of the fiber-delivered laser interferometer , such as keeping the exothermic laser source away from the optical structure and reducing the difficulties of erection and alignment, are maintained in the proposed grating interferometer. Instead of the completely modulated laser source, the four optical fiber connectors are the only components needed to align with the optical paths. Thus, the requirement of alignment was largely reduced by the fiber-delivered construction. The fibers also contribute to shorten the path length in the air and strengthen the resistance to fluctuation of the refractive index of air. Because the grating interferometer is an incremental instrument, the phase difference caused by the unequal length of fibers was calculated in the initial phase, having no effect on the measurement of displacement.
The optical reading head configuration of the grating interferometer is displayed in the right orange dotted box of Fig. 2. After transferred by the fibers, the two parallel beams were transmitted in the PBS, where they were split in different locations and divided into transmitted beams and reflected beams. The two reflected beams with p-polarization entered a beam splitter (BS2), and created a reference interference beam. The alternating intensity of the reference beam was received by a photodetector (D1), which can be expressed as
On the other side, the two transmitted beams with s-polarization passed a quarter wavelength plate (QWP) and their polarization changed. They normally entered the grating, and diffracted. We chose a + 1st order diffraction beam of one beam and a −1st order of the other beam, and placed two retro-reflectors (RR1 and RR2) in their paths, respectively. The two beams reflected by the retro-reflectors were transmitted along the opposite direction, and diffracted at the measurement grating again. Only two of all the second diffraction beams were vertical to the grating plane. These two beams passed the QWP again and changed into p-polarization beams, which were reflected at the separating surface of the PBS. Another beam splitter (BS3) was utilized to create the measurement beam, which was obtained by the photodetector (D2).
Considering a commercial PBS, no matter how large the value of extinction ratio is (usually 0.1%), it is certain that the two frequencies will not mix because of the spatial separation. Only the proportion of s- and p-polarization in each beam is influenced. The measurement beam created in BS3 is interfered by four components. They can be expressed as and , while s-polarization and p-polarization are included in both of them. Thus, the result of the interference is expressed in Eq. (8):
In contrast with Eq. (1), the measurement beam of the proposed interferometer has the same form as the ideal one, except from a factor before the phase item. The factor is the fold factor caused by the retro-reflectors and second diffraction. The retro-reflectors were utilized to compensate the tip and tilt of the grating [17,22]. Unexpected angle changes of the first diffraction beams are caused by the unexpected tip and tilt, which have great influence on the intensity of the interference signal. The retro-reflectors reflected the first diffraction beams in the opposite direction, converting the angle change of the first diffraction beams into a translational move of the second diffraction beams. It is guaranteed that the second diffraction beams are along the normal direction of the grating. The other advantage of the retro-reflectors is that the interferometer fold factor was doubled by the second diffraction, so that the resolution is folded.
The displacement of the grating can be calculated by the reference beam and the measurement beam. Similar to the factor in Eq. (8), diffracting twice at the grating makes the Doppler shift of the beams double. Thus, two beams with different frequencies of and , which contained the displacement information, met at the polarization surface of BS3, and generated the measurement interference signal. Only the low frequency part of the signal, with a frequency of , can be received by D2. Similarly, a reference interference signal without phase information was formed at the polarization surface of BS2. The signal D1 received is in frequency. Therefore, the displacement of the grating could be achieved with a signal processing board, whose function is detecting and integrating the phase difference between the sinusoidal reference and measurement signal.
It is proved by the derivation and analysis above that the spatially separated configuration is capable of eliminating periodic nonlinear errors and acquiring a more precise displacement result.
3. Experiment of system performance
To validate the effect of eliminating periodic nonlinear errors, both in time domain and frequency domain, a couple of experiments was designed and conducted.
3.1 Experimental setup
Based on the configuration of Fig. 2, the experimental setup for testing the system performance is shown below. In both experiments, some of the components remain at the same position. The separated modulated laser source, which is beyond the expression in the figure, was composed by a semiconductor laser (Sacher Lasertechnik Group TEC 500) and two AOMs (AA Opto Electronic MT80). The laser beam at the wavelength of 780 nm was split and modulated at the frequencies of 80 MHz and 85 MHz, respectively. The outputs of the AOMs, two circularly polarized beams with different frequencies, were transmitted to the interferometer configuration by optical fibers. The 1 μm custom-sized planar grating was driven by a linear guide (BOCI Company). The type of the photodetectors utilized in proposed grating interferometer is HCA-S-200M-SI.
In Fig. 3(a), a comparison experiment was built with the proposed grating interferometer and a commercial heterodyne interferometer. The interference signal of the grating interferometer was sent to a homemade signal processing board for calculation via two amplifiers (Mini-circuits ZHL-32A-S). The displacement results were uploaded and recorded in an upper monitor by USB interface.
In Fig. 3(b), a frequency spectrum analysis was conducted, and the optical configuration was adjusted slightly to be qualified for a contrast experiment. A COP heterodyne grating interferometer was created by splitting and combining the single-frequency beams. The retro-reflectors were removed to simplify the optical configuration without changing the main feature of eliminating the periodic nonlinear errors. Thus, the contrast experiment between a spatially separated heterodyne grating interferometer and a COP configuration was conducted using the experimental setup in Fig. 3(b). A signal analyzer (Agilent 9010A) acquired the interfere signals and provided the spectrum data.
3.2 Measurement results
3.2.1 Functional experiment results
A basis functional test for proving the correctness of the experimental system was carried out and presented in the chart below (Fig. 4). In this test, the linear guide rail was driven to a 25 mm displacement with different speeds. As Fig. 4 shows, in an advance and return movement, the grating interferometer and the laser interferometer agree with each other. The deviation between the two series of data is within microns, which is mainly caused by a synchronization time difference of these two independent systems. In the speed related experiment, the moving speeds were set to 0.25 m/s, 0.05 m/s, and 0.025 m/s, respectively. It is proved that the proposed system works correctly at the speed of 0.25 m/s, which is the maximum of the utilized guide rail. Moreover, a constant speed stage, from 4 mm to 24 mm, was extracted and analyzed. Nonlinearity of the system is calculated by the maximum deviation from the least square regression line, and expressed in percent of the full-scale range. It is an important characteristic to describe the accuracy of the whole system, which is affected by nonlinear errors, environmental vibration, and many other factors. Though the actual motion has nonlinearity itself, the nonlinearity of the proposed grating interferometer is lower than 0.140%. A comparison of the nonlinearity results in Table 1 shows that the proposed grating interferometer has a better performance than the laser interferometer.
3.2.2 Frequency spectrum results
The measurement results of the frequency-spectrum analysis experiment are shown in Fig. 5. When the guide rail and the grating were at rest, an instantaneous spectrum was captured and displayed in Fig. 5(a). The figure expresses that the peak of the motionless spectrum is at 5 MHz, the frequency of the beat signal, and the signal amplitude is −27.3 dB. In a continuous view, the peak amplitude fluctuated and the peak position joggled. This was caused by the surrounding vibration and nonideal grating surface.
When the grating is moving at a constant velocity, the spectra of the two kinds of interferometers were not the same anymore. The blue curve in Fig. 5(b) is the moving spectrum of the COP heterodyne grating interferometer, created by the upper photodetector in Fig. 3(b). It is easy to find that there are three frequency peaks above the maximum noise level, whose amplitude and location are −25.6 dB at 5.03 MHz, −66.5 dB at 5.00 MHz, and −77.3 dB at 4.97 MHz. Furthermore, they are the base signal, + 1st harmonic, and + 2nd harmonic, respectively. In a larger extent, suspected 3rd and 4th harmonics can be seen below the noise level. According to Eq. (6) in Section 2.1, the periodic nonlinear error of the two optical subdivision COP configuration interferometer is 0.717 nm. Regarding Fig. 5(c), the key point is that there is only one peak above the noise level. The amplitude of the peak is −26.7 dB, and its location is 5.03 MHz. The periodic nonlinear error can be calculated by the same method and the answer is no larger than 0.172 nm. A four optical fold factor configuration, as that shown in Fig. 2, can approach a 0.086 nm periodic nonlinear error with the same signal quality. The ratio of periodic nonlinear errors to grating pitch was no larger than 8.6 × 10−5. According to Eq. (6), a grating with smaller pitch may lead to a smaller periodic nonlinear error, while the wavelength is not directly relevant. The results support that the proposed grating interferometer acquires an 8 × better periodic nonlinear errors performance than the typical COP grating interferometer.
The analysis above is a strong demonstration of the effectiveness of the proposed spatially separated grating interferometer in eliminating periodic nonlinear errors. It also means that the proposed grating interferometer is effectual, but it can be improved. There are two main approaches to improve the spatially separated heterodyne grating interferometer for obtaining smaller periodic nonlinear errors: to increase the interferometer fold factor and to enhance the amplitude of the interference signal. Besides, a grating with smaller pitch may help, but altering the grating component means changes in diffraction angle and efficiency.
High order diffraction is a common and useful method to increase the fold factor. However, the factor is limited by the decreasing intensity. The configuration of the second diffraction is more familiar, especially with retro-reflectors. The reverse feature of the retro-reflector enables it to match well with gratings. A complete configuration with retro-reflectors for a double optical fold factor may halve the nonlinear error. The retro-reflector also has the advantage of converting the unexpected tip and tilt of grating to the translation of beams.
The amplitude of the interference signal is relative to the laser intensity and spatial location of the separated beams. It is obvious that, the spatially separated grating interferometer is more complex and requires a more stable optical and mechanic construction than the COP type.
Aiming at further improving the system performance, our research would proceed with the miniaturization of the optical and mechanic configuration and the expansion of the degree-of-freedom of the system.
A spatially separated heterodyne grating interferometer for eliminating periodic nonlinear errors is proposed in this study. The spatially separated configuration avoids the frequency-mixing phenomenon and has the advantage of eliminating the periodic nonlinear errors. The two frequency components are modulated, transferred, split, and diffracted independently, before they combine and interfere with each other. The retro-reflectors are utilized to increase the interferometer fold factor, improve head-to-scale tolerance, and keep the amplitude stability of the interference signal.
The elimination of the periodic nonlinear errors by the proposed grating interferometer has been demonstrated by time domain and frequency domain experiments. The frequency domain experimental results qualitatively indicate that the harmonic peaks in the spectrum have been suppressed below the background noise. At the same time, they quantitatively calculate that the periodic nonlinear errors are no larger than 0.086 nm in a fourfold factor configuration. The time domain experimental results show that the spatially separated heterodyne grating interferometer has a better linearity performance for describing an actual rectilinear motion than a commercial heterodyne laser interferometer.
National Natural Science Foundation of China (NSFC) (51675138, 61675058).
The authors cordially thank our co-workers Dr. Hongxing Yang and Dr. Haijin Fu for their helpful assistance and valuable suggestions.
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