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Abstract
A novel signal processing method based on phase shift of reference signal is proposed for heterodyne interferometer. The integer fringe counting method based on overflow judgment and compensation can realize longtime and correct integer number measurement. In order to eliminate the influence of jitter in measurement signals on combination of integer and fraction fringe counting, the reference signal with phase shift of 180° is used to obtain integer compensating number to compensate the unstable integer number in unstable phase zone, which guarantees the correct combination of integer and fraction fringe counting. The principle of the proposed signal processing was described in detail. The static and dynamic resolution of the proposed method were discussed. A signal processing board based on FPGA was developed, and three tests were performed to verify the feasibility of the proposed method. A displacement measurement experimental setup was constructed, and two experiments verified the effectiveness of proposed method in application of an interferometer to realize precision displacement and testing of a stage.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Since the first commercial heterodyne system was invented in the 1970s, due to its outstanding advantages of high signal-to-noise ratio, wide dynamic range and direct traceability to the length standard, heterodyne laser interferometric systems have been widely used in various applications, such as displacement and angle measurement [1,2], precision sensor calibration [3, 4], photolithography [5], and laser tracking of robot systems [6]. And the signal processing method of a laser heterodyne interferometric system is the key to realize high precision measurement with a large range.
In order to meet the requirement of ultra-precision measurement in engineering and science, a lot of research work on signal processing method has been done in the past few decades. Demarest described the requirements of the electronics for a heterodyne interferometer system for the high-resolution, high-speed, low data age uncertainty [7]. Liang et al analyzed the fundamental limits on the digital phase measurement based on cross-correlation for picometer level measurement [8]. And Xu et al presented a polarimetric interferometer which is capable of determining the nonlinearity of heterodyne interferometric displacement system [9]. Currently, three kinds of methods are mainly used for precision phase measurement. One kind of method is frequency-mixing technique with phase locked. For example, Hsu et al presented an all-digital phase meter for subpicometer length measurement by using digital frequency-mixing technique with phase locked loop [10]. Wang et al proposed a dynamic tracking down-conversion signal processing method for high subdivision of interference fringe by using PLL [11]. Although frequency-mixing can realize high fringe interpolation in phase measurement, the measurement speed is limited by the frequency tracking range of frequency-mixing electronics. Thus, the combination of different phase measurement methods is usually used to achieve high measurement resolution and speed [12, 13]. Besides, the effect of phase delay caused by using different measurement approaches needs to be considered to realize correct phase measurement. With the development of advanced electronic technology, another kind of method based on Lock-in amplifier and discrete Fourier transform techniques is proposed. For example, Lawall et al demonstrated a method capable of fringe interpolation accuracy of one part in 36000 with 10 pm accuracy by using a DSP lock-in amplifier [14]. The LISA group developed a phase meter by using single-bin discrete Fourier transform method with A/D converters [15]. Eom et al described a compensating method for nonlinearity to achieve sub-nanometer accuracy phase measurement [16]. Köchert et al developed an advanced phase meter to resolve displacements of a picometer by using ADC and FPGA [17]. The signal processing methods by using Lock-in amplifier or discrete Fourier transform can be implemented in analog or digital circuits. Although high measurement precision can be achieved, the complex implementing electronics and complex unwrapping algorithms are needed.
Comparing with the two kind of signal processing methods above, the third kind of method that is based on time interval analysis is a relatively simple signal processing technique for heterodyne interferometers [18, 19]. The preconditioning electronics can be implemented simply by using zero-crossing detecting, and the integer and fraction numbers can be obtained by fringe counting and filling-pulse techniques without complex unwrapping algorithms. However, the jitter in interference signals could cause false zero crossings and lead to incorrect combination of integer and fraction fringe counting. In our previous work [19], the two mode signal processing method was proposed to realize high resolution and high speed measurement. However, correct combination of integer and fraction fringe counting can be guaranteed only in the high-resolution mode at a low measurement speed, high performance electronic circuit is required to tackle the influence of phase delay between the signals processing circuits of the two modes, and the implemented electronics is still relative complicated.
In this paper, an improved signal processing method with merit of easier implementation and higher working speed of combination of integer and fraction fringe counting is proposed. In order to eliminate the influence of jitter in interference signals on combination of integer and fraction fringe counting, we propose a novel laser heterodyne interference signal processing method based on phase shift of reference signal with 180°. The integer fringe counting method and the combination principle of integer and fraction fringe counting are described in detail, and a signal processing board was developed. The experimental setup was constructed, and a series of experiments were performed to verify feasibility of the proposed signal processing method.
2. System Configuration
The proposed signal processing method is applied in our developed laser heterodyne displacement interferometer based on Faraday effect as shown in Fig. 1 [20]. In the system configuration, a stabilized dual-frequency He-Ne laser emits an orthogonally linearly polarized beam with dual frequencies, the laser beam is separated by a polarizing beam splitter (PBS1) into reference beam and measurement beam with different frequency, respectively. The reference beam is reflected back by a plane mirror (RM1), passes through a quarter-wave plate (QP1) twice, transmits PBS1 and finally projects onto a photodetector (PD). The measurement beam passes through a Faraday rotator (FR) and PBS2, and then incidents onto a corner cube prism (CC). After being reflected by CC and passing QP2, the measurement beam is reflected by PBS2 and incidents onto RM2 vertically, then this beam is reflected back along the incoming optical path. The returned measurement beam and returned reference beam recombine at PD and the measurement signal (Mea) is generated. The reference signal (Ref) is obtained from the rear of laser head. The reference and measurement signals are transmitted to the signal processing board through differential transmission. After signal processing by using the proposed method, the combination number of integer and fraction fringe counting of NReal is obtained, and then the displacement can be derived by
where, λ is the wavelength of laser, n is the refractive index of air along the path of light propagation, 4 is the interferometer fold constant.
Fig. 1 System configuration of a heterodyne interferometer for testing the proposed signal processing method.
3. Signal processing method
The proposed signal processing method is shown in Fig. 2. The reference and measurement signals are converted into square wave signals shown as Ref0.sig and Mea.sig through signal preconditioning. And then the two signals come into the phase demodulating module (PDM). In PDM, the phase shift module is used to generate the signal of Ref180.sig with phase shift of 180°. By processing the signals of Ref0.sig and Mea.sig with the proposed integer fringe counting method, the integer fringe counting number of NInt0 is obtained. And by processing the signals of Ref180.sig and Mea.sig, the compensating integer fringe counting number of NInt180 is obtained. The fraction fringe number of NFrac is obtained through fraction counting by using filling-pulse technique. Finally, the combination number of integer and fraction fringe counting can be obtained from the combination module, and the processing results are transmitted to a computer through serial transmission.
3.1 Integer fringe counting
In order to realize correct and longtime integer measurement, the integer fringe counting method based on overflow judgment and compensation is proposed as shown in Fig. 3. In each integer counting module, two high-speed counters are used to count the rising edge numbers of rectangular reference and measurement signals, respectively. Take integer counting 1 as an instance, because Ref0.sig and Mea.sig signals are continuous square wave signals with the frequency equaling to the frequency difference of laser source even at static state, thus, the two counters will overflow sooner or later. And if the two high-speed counters are overflow or not at the same time, the integer fringe counting number of NInt0 can be easily obtained by directly subtracting the two counting numbers of CRef0 and CMea with subtracting module (SM), but if only one is overflow, this method will not work and will affect normal measurement of NInt0. Although increasing the number of digits of counters can prolong their overflow time and then achieve a longer measurement time, the counting stability will reduce. Therefore, the judgment and compensation module (JC) is designed to realize the judgment of overflow state of the two counters and provide corresponding compensation. In the proposed method, CL is defined as an equivalent counting threshold which can be calculated from the maximum measurement range. For example, in displacement measurement, the CL value can be obtained through dividing the maximum measurement distance by wavelength of laser. And if the maximum counting value Cmax of the used counters is chosen larger than CL, the overflow state of the two counters can be judged precisely, the integer counting number can be compensated correspondingly and the NInt0 can be obtained correctly. When Counter 1 and Counter 2 are n-bit counters, the maximum counting number of each counter is 2n, and the NInt0 can be derived by
As is shown in Fig. 3, by processing the Ref180.sig and Mea.sig signals with the same integer counting method, the NInt180 is obtained, which will be used as a compensating value to achieve correct combination of integer and fraction fringe counting.
3.2 Fraction fringe counting
The fraction counting method based on filling-pulse is shown in Fig. 4. The pulse generating module of phase difference (POPD) is used to generate the pulse signal corresponding to the phase difference between the Ref0.sig and Mea.sig signals. The pulse generating module of period (POP) is used to generate the pulse signal corresponding to a period of Mea.sig signal. A phase locked loop (PLL) is used to generate high frequency clock signal for filling-pulse. Two pulse filling modules (PFMs) are used to measure the filling numbers as shown with nPD and nP corresponding to the phase difference pulse and period pulse signals, respectively. Then the fraction fringe counting number can be derived by
3.3 Combination of integer and fraction fringe counting
In the proposed signal processing method, it is the key to eliminate the influence of jitter in signals on correct combination of integer and fraction fringe count to obtain correct combination number of NReal. As is shown in Fig. 5, correct combination requires that the integer fringe counting number of NInt0 must add 1 or minus 1 when the phase difference between Ref0.sig and Mea.sig passes the phase difference point of 0°(NFrac = 0), and the combination number can be obtained easily through plus NInt0 to NFrac. However, because of the influence of jitter, when the phase difference is near the position where NFrac = 0, the measured NInt0 is an unstable value and its change is not synchronous with NFrac, which leads to incorrect combination of integer and fraction counting. In order to solve this problem, a combination method based phase shift of reference signal with 180° was proposed as shown in Fig. 6.
Fig. 6 Combination of integer and fraction fringe counting based on phase shift of reference signal. (a) status of signals. (b) schematic of combination principle.
As is shown in Fig. 6(a), the phase zones of [NFracL, 1) and [0, NFracR] in a phase period of 360° are called unstable phase zone (UPZ) in which the measured NInt0 is unstable. Thus, NInt0 cannot be directly used for combination of integer and fraction fringe counting. According to Fig. 6(a), it also can be seen that when phase difference between Ref0.sig and Mea.sig is 0°, the NInt180 obtained by Ref180.sig is at the phase difference of 180° between Ref180.sig and Mea.sig. And because the phase difference of 180° is not in UPZ, the measured NInt180 is a very stable number which can be used for compensating the unstable NInt0. As is shown in Fig. 6(b), the phase zone judgment module (PZJM) is used to determine the phase position of NFrac and to decide whether the NInt0 needs to be compensated with NInt180. The combination of integer and fraction module (COIF) outputs the correct combination number of NReal. And the NReal can be derived by
where, ΔN is the difference of initial integer counting numbers obtained by Ref0.sig and Ref180.sig, respectively, and the ΔN can be derived bywhere, N0Frac is initial fraction counting number, N0Int0 and N0Int180 are initial integer counting numbers obtained by Ref0.sig and Ref180.sig, respectively.4. Discussion
4.1 Influence of drift of phase difference of laser source
In the proposed signal processing method, the drift of phase difference of laser source has influence on measurement resolution. Traditional two-frequency laser heads can be used to provide two orthogonally linearly polarized beam with a frequency difference of Δf0. Although the vacuum wavelength of the laser head has a good stability, its phase difference has drift after a period of working time. In the proposed signal processing method, the static resolution can be calculated by obtaining the filling number of pulses in a period of measurement signal, and in a double pass interferometer, the static displacement resolution ΔSstatic can be derived by
where, fclk is the frequency of filling-pulse signal generated by PLL.In order to indicate the influence of drift of frequency difference more clearly, a simulation was performed. According to the actual measured data of the used Keysight 5517B laser head, the Δf would drift in a range from 2.205 MHz to 2.234 MHz, which deviates its nominal value of 2.26 MHz. The simulation result shown in Fig. 7 indicates that the static displacement resolution is reducing with the increase of frequency difference of laser source. Because of the drift influence of frequency difference, the displacement resolution changes accordingly and the increase or decrease of the resolution depends on the drifting direction. The simulation also indicates that the maximum variation of resolution is 0.006 nm in the whole drifting range, thus the drift of frequency difference has little influence on the static displacement resolution in the proposed signal processing method.
4.2 Dynamic resolution and speed
In addition to the commonly used static resolution which is used to describe the performance of a system, the dynamic resolution needs to be known sometimes and it is relatively difficult to be measured in real time, which relates to movement speed. But in the proposed signal processing method, the dynamic resolution can be easily obtained by measuring filling pulse numbers (np) in a period of measurement signal and the measurement speed can be obtained either, which can be derived by
By substituting Eq. (8) into Eq. (7), the relationship between the speed and dynamic resolution is as follows
when λ is specified with 632.991372 nm, Δf0 is specified with 2.26 MHz and the fclk is 800 MHz, the simulation results is shown in Fig. 8.
The simulation result indicates that the dynamic resolution is reducing as the movement speed increases. When the movement speed is −0.3 m/s, 0 m/s, and 1 m/s, the resolution is 0.06 nm, 0.45 nm, and 1.69 nm respectively. Therefore, in order to get a proper dynamic resolution, the measurement speed should be chosen properly.
5. Experiments
To verify the feasibility and effectiveness of the proposed signal processing method, a signal processing board is developed with a FPGA (EP2C20Q240I8, Altera Co. USA). In the signal processing board, zero-crossing detecting circuit as signal preconditioning is used to convert reference and measurement signals to square wave signals. PDM is implemented in FPGA in which the integer counting electronics is realized by using two 32-bit high speed counters, and the implemented integer counting module can realize the maximum measured distance larger than 1000 m for a single pass displacement interferometer. The signal of Ref180.sig with phase shift of 180° is generated by using a NOT logic cell. Through signal processing of the integer fringe counting number of NInt0 and NInt180 and the fraction fringe counting number of NFrac in the combination module, the final combination number of NReal is obtained. The measurement data is transmitted to computer through serial transmission circuits, the final measurement result can be achieved after processing with the designed application software. Four tests were performed to verify the feasibility of the developed signal processing board in integer fringe counting, combination of integer and faction fringe counting, measurement stability, and static and dynamic test. Two experiments were performed to demonstrate the effectiveness of the proposed method in application of precision displacement measurement.
5.1 Integer fringe counting
In this test, a function generator with a bandwidth of 100 MHz and a resolution of 0.01° (AFG3102, Tektronix, USA) is used for testing the proposed integer fringe counting method. The AFG3102 outputs two sinusoid signals with same frequency of 2.26 MHz to simulate the reference and measurement signals, and then the frequency difference between the two signals was adjusted to be 1 Hz to produce increment of integer fringe. The developed signal processing board measured the integer fringe counting number, and the application software in computer recorded the number once per second. In FPGA, since the used counters for counting rising edge of Ref0.sig and Mea.sig are 32-bits, the overflow time of each counter is 0.52 hour. And the experimental result shown in Fig. 9 indicates that in more than 16 hours, although each counter used for integer measurement has 30 times overflow, the measured integer counting number increases and decreases steadily and consecutively, and no unstable integer number occurs. This demonstrates that the proposed integer fringe counting method can achieve longtime and correct integer fringe counting effectively.
5.2 Combination of integer and fraction fringe counting
This test is to verify the ability of the proposed combination method in realizing correct combination of integer and fraction fringe counting when the measuring point is in the unstable phase zone. In the test, the AFG3102 function generator outputs two sinusoid signals with same frequency of 2.26 MHz, and the output phase difference between the two signals was adjusted to 0° which is in UPZ. The developed signal processing board measured the combination numbers, including combination numbers without compensation and with compensation by using the proposed method. The measured data are recorded per second. The experimental results are shown in Fig. 10. Figure 10(a) shows that the combination number without compensation has a maximum integer counting error of 2, actually, the integer counting error is even larger if jitter in signals is more serious. Figure 10(b) shows that the combination number with compensation has no integer counting error, and the deviations indicate that the fraction resolution of the developed signal processing board is 0.0056 when the used high frequency clock signal is 400 MHz. These experimental results demonstrate that the correct combination of integer and fraction numbers can be achieved by using the proposed combination method.
Fig. 10 Experimental results of combination of integer and fraction fringe counting. (a) combination without compensation. (b) combination by using the proposed method.
5.3 Stability test
In this test, the AFG3102 function generator outputs two sinusoid signals with same frequency of 2.26 MHz, and the output phase difference between the two signals was adjusted to 4°, 90°, 180°, 270°, and 355°, respectively, the signal processing board measured combination number of integer and fraction fringe counting correspondingly. In each state of phase difference, the combination number is recorded 500 times in 8.35 minutes. The experimental result is shown in Fig. 11 and summarized in Table 1. The experimental results indicate that the maximum peak-peak error of combination number is approximately 0.003 and the standard deviation is better than 0.0006. These experimental results demonstrate that the signal processing method can realize correct measurement of combination number and has a good stability.
Table 1. Experimental result of stability test of combination number measurement
5.4 Static and dynamic test
In the static accuracy test, the AFG3102 function generator outputs two sinusoid signals with same frequency of 2.26 MHz to simulate the laser reference and measurement signals. The function generator provides adjustable output phase in 10° step in a range of 360°. At each step, 110 position readings are recorded. The position error is calculated by converting the phase angle into equivalent position value with the interferometer fold constant of 4. For each group readings, the maximum static resolution and the standard deviation are calculated. The static resolution test shows that the minimum position resolution is 0.448 nm in all the group readings. The experimental results of static position accuracy is shown in Fig. 12(a), and the maximum standard deviation is 0.36 nm. The experimental result shows that the implemented signal processing board has a good static positioning precision.
Fig. 12 Experimental results of precision test. (a) static precision test of simulating equivalent position. (b) dynamic resolution and precision test.
In the discussion of section 4.2, the dynamic resolution is influenced by velocity, which also has an influence on position measurement accuracy. In the dynamic accuracy test, the function generator outputs a reference signal with a fixed frequency of 2.26 MHz and a measurement signal with varying frequency, which is to simulate different moving velocity of a stage. The measurement signal is adjusted to ramp up or down to the specified frequency to simulate movement velocity from −0.36 m/s to 1.17 m/s. At each velocity, 200 position readings are recorded with the sample frequency of 1 per second, and the maximum dynamic resolution and the standard deviation of the position error influenced by resolution are calculated. The experimental result shown in Fig. 12(b) shows that the dynamic resolution reduces from 0.01 nm to 1.93 nm with the increase of velocity, the dynamic accuracy reduces from 0.14 nm to 0.66 nm, and the maximum standard deviation of 0.66 nm is at the point of maximum velocity.
5.5 Application in displacement measurement
To verify the feasibility of the proposed signal processing method in application of precision displacement measurement, an experimental setup was constructed as shown in Fig. 13. The signal processing board was applied in our developed displacement interferometer based on the Faraday effect (Proposed interferometer). In the interferometer, the stabilized laser is a dual-frequency He-Ne laser (5517B, Keysight Co., USA) which emits a pair of beams with a frequency difference of 2.26 MHz and the wavelength of 632.991372 nm. A precision linear stage (XML350, Newport Co., USA) with the positioning accuracy of 0.05 μm was used as the measured object and provided displacement movement. A commercial displacement interferometer (XL-80, Renishaw Co., UK) was used to test the measured stage for comparison. Two experiments were performed, one experiment is displacement measurement comparison experiment, and the other one is bi-directional displacement measurement.
In the displacement measurement comparison experiment, the measuring mirrors of the two interferometers were fixed on the XML350 stage, they were moved with the step of 1 mm and 50 nm, respectively, and the two interferometers recorded the measured displacement simultaneously. The experimental results are shown in Fig. 14. Figure 14(a) shows that the maximum displacement deviation is 28.8 nm with the standard deviation of 7.4 nm in millimeter comparison. Figure 14(b) shows that the maximum displacement deviation is 22.7 nm and the standard deviation is 8.1 nm in nanometer comparison. The experimental results indicate that there is no integer counting error in the displacement measurement experiment, the signal processing board achieves the correct combination of integer and fraction fringe counting. And the proposed signal processing method can be applied in laser heterodyne interferometers to achieve precision displacement measurement in large range.
Fig. 14 Experimental results of displacement comparison. (a) millimeter comparison with a step increment of 1mm in range of 300 mm. (b) nanometer comparison with a step increment of 50 nm in range of 15 μm. To make the plots visible, the red dot line presenting displacement measured by Renishaw interferometer are shifted 20 mm and 2 μm from actual values in the millimeter and nanometer comparison, respectively.
In the bi-directional displacement measurement, the XML350 stage did reciprocating movement 15 times with the step increment of 1 mm, and the proposed interferometer measured the displacement each time. The experimental results shown in Fig. 15 indicate that the developed signal processing board can realize correct judgment on the direction of motion and achieve precision displacement measurement. The experimental results also indicate that the maximum forward and inverse kinematic deviation in range of 1 mm are 0.3329 μm and −0.3159 μm, respectively. This experiment shows that the proposed signal processing method can be applied in interferometer to realize testing and calibration of a stage.
Fig. 15 Experimental results of bi-directional displacement measurement. The black dot line presenting displacement provided by the XML350 stage is shifted 0.2 mm from actual values.
6. Conclusion
In this paper, a laser heterodyne interference signal processing method based on phase shift of reference signal with 180° is proposed. The principle of the signal processing method was described in detail, and a corresponding signal processing board was implemented with FPGA. A series of experiments were carried out to verify the feasibility of the proposed method. The integer counting test shows that the integer fringe counting method by utilizing overflow judgment and compensation is capable of longtime and correct integer fringe counting. The combination test shows that the proposed combination method can eliminate the influence of jitter in signals in UPZ and guarantees the correct combination of integer and fraction fringe counting. The static and dynamic test shows that the developed signal processing board has a good static accuracy, the dynamic resolution and accuracy reduces with the increase of measurement velocity. The displacement measurement comparison and bi-directional displacement measurement experiments demonstrate that the proposed method can be applied in an interferometer to realize precision displacement measurement in large range and testing of a precision stage. All these demonstrate that the proposed signal processing method can be potentially applied in heterodyne interferometry for precision displacement and angle measurement with the merits of no complex unwrapping algorithm and easy implementation.
Funding
National Natural Science Foundation of China (51605445, 51527807 and 51375461); Program for Changjiang Scholars and Innovative Research Team in University (IRT_17R98); China Postdoctoral Science Foundation (2016M601969); Young Researchers Foundation of Zhejiang Provincial Top Key Academic Discipline of Mechanical Engineering of Zhejiang Sci-tech University (ZSTUME02B07).
Acknowledgments
Authors acknowledge the Science Foundation of Zhejiang Sci-Tech University.
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