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Abstract
The increasing line density of the reference grating and the accelerating miniaturization of ultra-precision displacement measurement technology necessitate more stable interferometric signal processing methods for high line density gratings, particularly in low signal-to-noise ratio scenarios. This paper presents a phase demodulation method for dynamic interferometric signals for high line density gratings. The Morlet wavelet transform is utilized to obtain the instantaneous frequency of the interferometric signal, integration of which yields the relative displacement, while adding adjacent relative displacements without gaps provides the absolute displacement during dynamic motion of the grating. In simulations with a signal-to-noise ratio ranging from 40 to 70 dB, the proposed method demonstrates greater robustness compared to the traditional method. By establishing a platform for repeated experiments and comparing it with traditional methods, it was found that the maximum deviation between calculation results obtained using this method and traditional methods is 0.8 nm, further confirming its potential application.
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1. Introduction
The ultra-precision grating interferometer has garnered significant attention in the domains of ultra-precision manufacturing, micro and nano vibration measurement, and defect detection due to its inherent advantages of stability, precision, and cost-effectiveness [1–4]. Diverging from laser interferometers that employ laser wavelength as a scale, ultra-precision grating interferometers are micro and nano measuring devices that utilize high line density gratings as length Refs. [5]. Typically employed gratings can achieve a line density of up to 4700l/mm [1,2]. With an increase in the line density of the reference grating, effectively resolving the interferometric signal becomes a focal point for ongoing research [6–11]. Within this interferometric signal lies disturbances caused by hardware processes such as uneven grating morphology, ghost reflections from optical components, and fluctuations in laser wavelength; these disturbances manifest themselves as low-frequency string waves within the interferometric signal [12–16]. Consequently, an accurate and efficient phase demodulation method is crucial for achieving ultra-precise measurements with high line densities.
The existing methods for processing interferometric signals have demonstrated the favorable real-time performance of the arctangent method [5,17]. However, this approach necessitates a minimum of two interferometric signals, resulting in an increased number of optics required for optical path design. Furthermore, the augmented number of interferometric signals introduces orthogonality errors between them, which consequently distorts the arctangent method’s solution and to some extent restricts the accuracy of grating interferometer measurement as well as its further integration applications [18]. The harmonic separation method based on Wavelet Transform (WT) and Fourier Transform (FT) effectively separates the interferometric signal from other noise sources, offering high robustness and a large measurement bandwidth [19]. The Chinese National Institute of Measurement Sciences (NIM) applies the FT to separate measurement signals and noise in the frequency domain under offline conditions, reducing nonlinear errors from 2nm-3nm to 0.7nm [20–22]. Nevertheless, the FT only captures phase and frequency information of decomposed subsignals in the time domain without determining instantaneous frequency. To address this limitation, obtaining instantaneous frequency of an interferometric signal can be achieved through short-time Fourier transform (STFT) method; however, it necessitates judging window length according to prior knowledge about signal frequency transformation and is not suitable for dynamically changing interferometric signals [23,24].
The signal can be decomposed by wavelet transform to obtain the transient frequency information [25]. The WT fits the signal by a series of local waves of time period, which makes the signal decomposed by wavelet signal contain not only frequency information but also time information. In addition, WT can subdivide the signal phase in time domain by changing the shift factor and the stretch factor, which can reach the infinitesimal value in theory. Zeng H. P. and Sarac.Z et al. proposed a method to extract the instantaneous frequency and phase of interferogram from wavelet ridge, and verified the effectiveness of the above method by simulation and experiment. Sarac.Z et al. also found that WT has better repeatability and standard deviation in extracting the phase from the measurement [26–28]. Based on 1D continuous WT and STFT, Federico et al. evaluated the performance of different phase demodulation methods in noise pattern interferometry. The results show that the WT has the best performance in the presence of non-modulated noise [29]. In recent years, WT has been used in spectral interferometry of optical frequency combs, and the absolute measurement error is less than 10−5 orders of magnitude [30,31].
Therefore, the higher the density of grating etched lines, the higher the demand for high-precision and high-bandwidth demodulation of grating interferometric dynamic signals. A method of phase correction and demodulation for dynamic interferometric signal is proposed in this paper. The instantaneous frequency of the interferometric signal is obtained by Morlet wavelet transform (Morlet WT), and the absolute displacement is obtained by integrating the instantaneous frequency into the relative displacement in the time domain. The second chapter introduces the characteristics of dynamic grating signal and the necessity of WT. The third chapter elaborates the flow of the algorithm in detail. The fourth chapter proves the effectiveness of the method through simulation and comparison experiment.
2. Principle of signal demodulation for grating interferometer
2.1 Grating interferometric signal when displacement changes dynamically
Ultra-precision grating interferometer is a micro and nano scale displacement measurement device with a high line density grating as the length reference. It uses the Doppler effect generated by the movement of the grating vector direction, resulting in the frequency difference between the two diffracted light on the grating, forming a beat signal that is captured by the photodetector. The principle of the grating interferometer and the grating interferometric signal when displacement changes dynamically are shown in Fig. 1. This principle makes the displacement coefficient of the grating interferometer locked on the grating period d. The relationship between the light intensity signal detected by the photodetector and the displacement finally measured is as follows:
In the optical information detection part, two vertical beams of diffracted polarized light are divided in equal proportion by non-polarizing beam splitter (NPBS). The two beams of light are respectively emitted into the quarter-wave plate W1 at 45° and the half wave plate W2 at 22.5°, respectively. After the two equal beams of light pass through polarizing beam splitter (PBS) PBS2 and PBS3, the vertically polarized part of the light is transmitted to the photodetector PD2 and PD4 to be received. The part of the horizontal polarization is reflected to the photodetector PD1 and PD3 to be received, forming four interference light intensity signals that differ from the initial phase by 3λ/4, λ/4, λ, λ/2, respectively.
Fig. 1. Grating interferometer measurement principle diagram and grating interferometric signal processing process diagram based on wavelet phase demodulation.
When the grating moves in the time domain along the grating vector direction in the state of $x(t)$, Eq. (1) can be formed:
Equation (2) is the ideal relationship between light intensity signal and time. However, in the actual situation, the working environment or device will introduce different degrees of noise, such as laser virtual reflection noise, thermal noise caused by resistance, thermal noise of the channel of the field effect tube, photon noise, dark current noise, non-uniformity noise of the optical response and other non-periodic Gaussian noise. Therefore, Eq. (2) can be written as:
2.2 Harmonic separation principle of time domain interferometric signal
From Eq. (2), it can be seen that the grating interferometric signal is a continuously changing dynamic chordal wave signal, and $x(t)$ is the density law of chordal wave signal change. In the grating interferometer, the relationship between frequency and phase is as follows:
Then the relation between relative displacement and instantaneous frequency can be obtained as follows.
As the most basic physical quantity in nature, time is the most accurate physical quantity in the SI unit [32]. The frequency difference of the existing data acquisition card (DAQ) with high-performance atomic clock has reached 10−13 orders of magnitude, and the frequency difference of the common crystal oscillator on the market has also reached 10−6 orders of magnitude [33]. There is no technical limitation for high-precision signal subdivision in the time domain. When the signal is subdivided to a certain multiple, the frequency ${f_i}$ in the subdivided time $\Delta t$ can be set to remain unchanged, so Eq. (3) can be changed as follows.
According to Eq. (6), the signal received by the photodetector can be expressed as the linear superposition of the grating interferometric signal $\cos (2\pi \sum\limits_{j = 1}^n {{f_j}} \Delta t)$ and other chordal wave signals. The FT, STFT and WT are the methods that can comprehensively read and decompose this superposition of chord wave
3. Grating interferometric signal processing method based on wavelet phase demodulation
As shown in Fig. 1, when the displacement changes dynamically, the grating interferometric signal processing method based on WT phase demodulation mainly consists of: 1. Preprocessing, 2. The WT to solve instantaneous frequency, 3. Phase subdivision and displacement calculation, 4. Absolute displacement accumulation, these four parts are composed.
3.1 Preprocessing
Before the WT is applied to distinguish the noise from the interferometric signal in the frequency domain, the DC bias of the grating interferometric signal should be filtered first. DC bias is the excess energy generated by laser coherence, which is generally caused by incomplete circuit difference. It can be regarded as a constant value in the long-term state.
Therefore, in this paper, the original signal collected by each PD is subtracted from its own mean value in a period of time. Suppose that the voltage value collected by a photodetector by the acquisition card is PDi, i = 1,2,3,4, then the collected n interferometric signals after the DC initial screening are:
Then, the signals of the four PDs with a phase delay of 180° are differentiated to eliminate the synchronous DC bias under real-time measurement.
3.2 Principle of solving instantaneous frequency by Morlet WT
The FT decomposes a signal into a series of sine and cosine functions with different frequencies, similarly WT decomposes a signal into a series of wavelet functions with different scales and times, and these wavelet functions are obtained from a mother wavelet through translation and scale expansion.
For example, in Eq. (8), $\varphi (\frac{{t - \tau }}{a})$ is a basic wavelet, in the scale of size a, it is shifted by $\tau $, and the inner product with Eq. (6). The schematic diagram of WT is illustrated in Fig. 2. The WT can be considered as the fitting of each lattice's signal segment with its corresponding wavelet, resulting in the determination of similarity degree between the signal segment and the basic wavelet, known as the wavelet coefficient. Vertically speaking, different color grids represent different scales of basic wavelets, where scale “a” is linearly related to frequency, thus enabling vertical representation of similarity between signal and frequency. Horizontally speaking, each grid can be seen as a subdivision of the signal in time domain. Therefore, after transforming longitudinal parameters into frequency sequences, each lattice represents similarity between the signal and a specific frequency at a certain time point, allowing for extraction of corresponding instantaneous frequencies. In Fig. 1's MATLAB-processed wavelet time-domain diagram (Fig. 1's ②), the yellow part denotes the wavelet ridge line from which one can obtain instantaneous frequencies based on its position.
3.3 Frequency subdivision and displacement calculation
As can be seen from Fig. 2, although wavelet has powerful subdivision ability, the obtained instantaneous frequency is not continuous. Therefore, in this paper, the signal is uniformly interpolated at the instantaneous frequency subdivision point to obtain the calibrated instantaneous frequency shown in Fig. 3(a)’s ③.
Fig. 3. (a) Diagram of Instantaneous frequency uniform interpolation subdivision. (b) Accumulation of absolute displacement by element dislocation method.
Furthermore, the product of instantaneous frequency and subdivision time is summed up, and the result is scaled up by the period value d of the grating, and the displacement of the grating vector direction in this period is obtained, which is the displacement measurement expression of the grating interferometer:
3.4 Absolute displacement accumulation by element dislocation method
In continuous signal processing, the controller is to process the continuous signal in sections, and the result is the relative displacement of the grating in this period of time. After processing a sequence, the controller puts it into the stack or outputs, and then processes the next sequence. The two sequences will not overlap, and there is an interval of sampling time, and the grating is in continuous motion during this sampling time. Therefore, if the relative displacements in the two periods are simply added, there is a displacement error of sampling time, as shown in Fig. 3(b)’s ①. This error is determined by the moving speed of the grating and the sampling frequency of the DAQ. If the moving speed of the grating is 1mm/s and the sampling frequency is 500ks/s, there will be a displacement error of 2nm.
As shown in Fig. 3(b)’s ②, this paper adopts the method of misalignment calculation to avoid data omission caused by sampling interval. In the calculation step, after the controller calculates the previous relative displacement, the next relative displacement starts from the last element of the previous segment. In the concrete operation, in order to increase the efficiency of computer programming, we use the method of dual thread-stack processing in signal acquisition. One thread keeps pushing the grating interferometric signal into the stack, and the other thread keeps reading the signal from the stack at the same time, the last bit of the previous signal is taken as the start of the current signal reading.
4. Simulation and experiment
4.1 Simulation
We first set up model simulation in MATLAB to verify the robustness of the method. We set $x^{\prime}(t)$ as a cosine wave that has been sampled 5000 times in a period, and then multiply the cosine wave over the range to 5000 and bring it into $\cos (x)$ to obtain a harmonic signal whose instantaneous frequency changes with the cosine wave in a period of time (0∼2π). When the grating is moving, some electromagnetic white noise and low frequency noise will be introduced by the electrical devices, the grating adjusting frame and the fastener. We introduce two low-frequency signals with amplitude $\sigma$ of 0.1 and 0.2 and frequency of 1hz and 2hz respectively, and use function wgn of MATLAB to introduce Gaussian white noise with SNR of 40 dB. Therefore, the dynamic grating interference simulation signal is finally obtained, as shown in Eq. (10), and the image is shown in Fig. 1 or Fig. 4(b).
The Morlet wavelet is used to convolve $y(t)$ in the time domain, and the time-frequency graph obtained is shown in Fig. 4(a). The Z-axis is the wavelet coefficient, which describes the correlation degree between the original signal and the frequency at the current time. The part of the wavelet coefficient with the largest absolute value (red and part of the yellow region) is called the wavelet ridge, which corresponds to the instantaneous frequency with the greatest correlation with the original signal.
Fig. 4. (a) Time-frequency plot of WT. (b) Simulation of wavelet phase demodulation. (c) Instantaneous frequency after processing. (d) Transient phase of the identified direction. (e) Absolute displacement.
After extracting the frequency of the wavelet ridge and interpolating the subdivision, the instantaneous frequency map of the signal shown in Fig. 4. (c) is obtained. It is worth noting that the instantaneous frequency map at this time cannot reflect the direction of the signal, and generally we will judge the change of direction by the zero crossing of the original signal detected by PDs. In the simulation, we change the sign of the frequency at the minimum of the instantaneous frequency to obtain Fig. 4. (c). Figure 4. (c) is further accumulated according to Eq. (9) to obtain Fig. 4. (d). It can be seen from Fig. 4. (d) that it is feasible to solve the displacement signal by using the proposed method. The signal x(t) was subjected to the addition of noise ranging from 40dB to 70dB, and subsequently processed with both the proposed method and the arc-tangent method at the same level of subdivision to handle grating interferometric signals. A comparative analysis between these two methods was conducted, resulting in the generation of Fig. 5(a).
Fig. 5. (a) The trend diagram of the maximum deviation for the difference between SNR results obtained using the proposed method and the arctangent method, as well as $x^{\prime}(t)$, in a noise environment ranging from 40 dB to 70 dB;.(b) Calculation results of the proposed method and arctangent method under noise environments of 45 dB, 50 dB, and 60 dB. (c) Maximum deviation from $x^{\prime}(t)$ calculated by the method presented in this paper and the arctangent method under noise environments of 45 dB, 50 dB, and 60 dB.
Figure 5(a) is obtained by solving the maximum deviation subtracted from $x^{\prime}(t)$ of the processing results of the two methods. It can be seen from Fig. 5 that under the influence of the same SNR noise, the proposed method can show stronger stability. When the SNR is 45dB, 50dB and 60dB, the maximum relative error of the method based on WT demodulation is 0.2479µm, 0.0410µm and 0.0055µm respectively. The maximum relative errors of the arctangent method are 2.7541µm, 0.9135µm and 0.0186µm. It can be seen that the maximum relative error is reduced by at least 70.43%, which is because the WT method can distinguish the harmonic signal from the abrupt signal better than the arctangent method. Of course, in the signal processing of the existing grating interferometer, before the signal inverse cut demodulation, there will be filtering, smoothing and other preprocessing to filter out the abrupt noise, but this will also filter out some abrupt displacement information such as vibration. Therefore, the method based on WT demodulation has better applicability than arctangent in the processing of vibration signal and dynamic displacement signal.
In addition, 5000 points are collected on AMD RADEOM 4000 processor to compare the processing speed of the proposed method with that of arc-cut method. Before arctangent calculation, we smooth filter the signal with a window of 9 to filter out the noise signal of grating motion. We calculated 5 times and got the average time in Table 1.
Table 1. Comparison of processing time between WT and arctangent
It can be seen that the processing speed of this method is not as fast as arctangent method at 5000 points, which makes this method not have an advantage in real-time.
4.2 Experiment
The grating is mounted on the nano displacement stage, and the controller of the nano displacement stage outputs a sinusoidal analog signal with a frequency of 2Hz. The nano displacement stage is measured by a capacitated displacement sensor in real time and calibrated by feedback, and finally drives the displacement stage to perform harmonic motion with a maximum displacement of 5000nm.
The interferometric signal of the grating interferometer was obtained by using the Thorlabs’s Si Fixed Gain Detector (PDA10A2, Thorlabs, Newton, US) and NI’s DAQ (MCC 1608GX-2AO, National Instruments, Austin, US). Using Wavelet Analysis in Advanced Signal Processing Toolkit of NI, the signal can be processed by the method of this paper on LabVIEW 2018. We utilized a self-traceable grating interferometer, developed by Tongji University, to validate our approach. The employed grating is a self-traceable one with its period directly traceable to the chromium atomic transition frequency [1,2]. In the experiment, the line density of the grating was 4700 lines per millimeter (l/mm), with an average spacing of 212.7795 nm and a standard deviation of 0.0005 nm. The period of the interferometric signal was measured as 106.390 nm. Moreover, the average peak and valley height of the grating was approximately 30 nm, while its structural range spanned 0.25 mm × 6 mm and had a substrate size of 10 mm × 10 mm. Among them, the indexes that affect the measurement accuracy of the grating interferometer are average spacing and periodic standard deviation, the grating structure range affects the measurement range, and the average peak and valley height and substrate parameters affect the application mode of the grating.
A dual-thread programming method is used to realize the real-time solution of our algorithm. In Fig. 6(a), one thread is responsible for controlling the MCC 1608GX-2AO to continuously collect the signal of PD, and the other thread is responsible for the calculation of displacement. In the displacement calculation thread, the signal processed in 3.1 was successively processed by Heydemann correction, arctan, umwrap, and sgn to obtain the instantaneous displacement direction information. The instantaneous displacement carrying the direction information was obtained by linear quadrature with the signal after WT. Finally, the motion information of the displacement stage can be obtained by accumulating the instantaneous displacements.
Fig. 6. (a) Experimental program block diagram. (b) Frequency diagram of grating interferometric signal under the condition of grating 2Hz and amplitude 5000nm under the WT. (c) Signal processing results of grating interferometric signal by WT under the condition of grating 2Hz and amplitude 5000nm.
Figure 6(b) is the time domain diagram of WT, and Fig. 6(c) is the signal processing result in this paper. It can be seen that the instantaneous frequency of grating interferometric signal is between 100Hz and 320Hz when the displacement stage moves at a frequency of 2Hz and the maximum displacement is 5000nm.
The motion of the displacement stage was measured simultaneously using both the capacitance micrometer and the grating interferometer. A total of five displacement measurement experiments were conducted. The collected grating interferometric signals were processed using three methods: harmonic separation method, Ref. [5] method, and the method proposed in this paper. The solutions obtained from each method were compared. The harmonic separation method utilized FFT to eliminate noise signals within the frequency range of 1Hz-95Hz in the frequency domain. The Ref. [5] method involved filtering and modifying the signal prior to phase resolution. The capacitance micrometer served as a reference for our displacement measurements. Each experimental system was preheated and operated independently from one another, with their respective experimental data presented in Table 2.
Table 2. Comparison between the measurement results of five repeated experiments and the traditional method results
In terms of the trend observed in the experimental data, both the capacitance micrometer and grating interferometer exhibited consistent trends across all experiments, demonstrating that our algorithm possesses independent environmental identification capabilities similar to other methods, thus validating its feasibility. Furthermore, when comparing standard errors calculated by our new algorithm (18.80nm) with those obtained from the capacitance micrometer (3.08nm), it is evident that measurements taken by our algorithm yield greater stability for grating interferometers than traditional capacitance-based approaches.
Moreover, results obtained through our proposed method closely align with those derived from both harmonic separation and pre-processed arctangent methods for measuring grating interferometers; maximum relative error between these results (compared against harmonic separation) amounts to only 0.8nm. This further confirms that our algorithm achieves comparable accuracy to existing grating solving methods while being suitable for practical applications.
5. Conclusion
The higher the line density of the grating, the greater the requirement for high precision and high bandwidth demodulation of the grating dynamic interferometric signal. In this paper, a novel phase demodulation method for grating interferometric signals is proposed. After filtering out DC noise from the signal, Morlet wavelet transform is employed to divide the phase of the dynamic interferometric signal in minimal time domain and obtain its instantaneous frequency. The instantaneous frequency is then integrated into relative displacement in time domain, followed by seamless splicing of relative displacements at both ends. Finally, absolute displacement of the grating is obtained.
By introducing noise with a signal-to-noise ratio (SNR) of 40-70 dB into the analog interferometric signal as our original signal, we compare the performance of the proposed method and the arctangent method. It is observed that the proposed method maintains certain performance and exhibits stronger robustness, particularly under low SNR conditions. Specifically, when the SNR is 45 dB, 50 dB, and 60 dB, the maximum relative error of the proposed method is reduced by at least 70.43% compared to that obtained using the arctangent method.
To verify its feasibility, we employ a displacement stage with capacitive micrometer feedback to achieve precise harmonic motion of the grating. The results demonstrate that utilizing the proposed algorithm in a grating interferometer yields greater stability than using capacitance micrometers alone. Moreover, the maximum relative error between our method and similar signal processing methods for grating interferometers (in comparison with the harmonic separation method) is 0.8nm. This further demonstrates that the algorithm possesses equivalent accuracy to other grating solving methods and can be effectively applied in practical applications.
Although this method does not have the high real-time performance of the inverse cut method. But with the development of the semiconductor industry, a new generation of grating interferometers is highly integrated. The proposed method is accurate, stable and robust, and is not strictly dependent on optical subdivision, so it is more competitive and has the potential to be widely used in the calculation of grating interferometric signals with higher line density.
Funding
National Key Research and Development Program of China (2022YFF0605502); National Key Research and Development Program for Young Scientists (2022YFF0607600); National Natural Science Foundation of China (61925504, 62075165); Fundamental Research Funds for the Central Universities, Opening Fund of Shanghai Key Laboratory of Online Detection and Control Technology (ZX2020101); Shanghai Municipal Science and Technology Major Project (2021SHZDZX0100); Program of Shanghai Academic Research Leader (21XD1425000); Special Development Funds for Major Projects of Shanghai Zhangjiang National Independent Innovation Demonstration Zone (ZJ2021-ZD-008).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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