Design, Analysis, and Implementation of the Subdivision Interpolation Technique for the Grating Interferometric Micro-Displacement Sensor

Abstract

A high-resolution grating interferometric micro-displacement sensor utilizing the subdivision interpolation technique is proposed and experimentally demonstrated. As the interference laser intensity varies sinusoidally with displacement, subdivision interpolation is a promising technique to achieve micro-displacement detection with a high resolution and linearity. However, interpolation errors occur due to the phase imbalance, offset error, and amplitude mismatch between the orthogonal signals. To address these issues, a subdivision interpolation circuit, along with 90-degree phase-shifter and high-precision DC bias-voltage techniques, converts an analog sinusoidal signal into standard incremental digital signals. This novel methodology ensures that its performance is least affected by the nonidealities induced by fabrication and assembly errors. Detailed design, analysis, and experimentation studies have been conducted to validate the proposed methodology. The experimental results demonstrate that the micro-displacement sensor based on grating interferometry achieved a displacement resolution of less than 1.9 nm, an accuracy of 99.8%, and a subdivision interpolation factor of 208. This research provides a significant guide for achieving high-precision grating interferometric displacement measurements.

1. Introduction

Micro-displacement sensors with nanometer resolution and accuracy are key components in precision positioning, aerospace, microelectronics manufacturing, and bioengineering applications [1,2,3,4]. Detection methods for micro-displacement sensing include capacitive [5,6,7,8], piezoelectric [9,10], inductive [11,12], magnetoelectric [13,14], and optical grating [15,16,17,18,19]. Capacitive micro-displacement sensors are widely used in nanoposition applications due to their excellent resolution (at the sub-nanometer level) and compatibility with IC circuits. However, the accuracy of the capacitive micro-displacement sensor is affected by parasitic capacitance and fringing field effects. Other methods are often affected by factors such as temperature fluctuations and external electromagnetic interference. To further enhance the displacement resolution, optical detection methods based on grating interferometry have gained significant attention for their high-precision, rapid response, and inherent immunity to electromagnetic interference.

Although great efforts have been made to improve the performance of the grating interferometric micro-displacement sensor, it is technically challenging to achieve nanometer resolution and accuracy for displacement measuring over a long range. Researchers have attempted to achieve high-resolution micro-displacement detection based on an image processing method [20,21]. Hai Yu et al. developed a linear displacement measurement device based on the grating projection imaging technique, with a resolution of 1 nm and measurement accuracy of 1.76 μm in the range of 250 mm [20]. As an alternative, the interpolation method also can achieve subnanometer resolution levels [22,23]. Hongzhong Liu et al. proposed an electronic interpolation interface based on the linear subdivision method with better practical accuracy, achieving a practical interpolation error of 0.94%, which is comparable with that of many high-end commercial interpolators [22]. In the interpolation method, it is crucial to generate quadrature signals as accurately as possible. However, the quadrature signals are associated with signal distortions such as amplitude mismatch, offset error, and phase imbalance, which further limit resolution and accuracy improvements.

In this paper, we present the design, analysis, and implementation of a subdivision interpolation technique for the grating interferometric micro-displacement sensor. The enhanced subdivision interpolation circuit, combined with a 90-degree phase shifter and high-precision DC bias-voltage techniques to convert the sine signal output into standard incremental digital signals. This scheme not only mitigates the adverse effects of nonidealities such as phase, offset, and amplitude errors in the interpolation circuits, but also accurately tracks micro-displacement even at higher speeds, enabling the grating interferometric micro-displacement sensor with the advantage of higher resolution and accuracy.

In the following sections, we provide a detailed introduction on the design, analysis, and implementation of the subdivision interpolation technology for grating interferometric micro-displacement sensors. Section 2 discusses the impact of errors on interpolation accuracy, Section 3 focuses on the high-resolution digital readout circuit, Section 4 presents the experimental results and analysis, and Section 5 concludes the paper.

2. Effect of Errors on Interpolation Accuracy

The schematic diagram of the grating interferometric micro-displacement measurement is depicted in Figure 1. When a coherent laser beam is vertically incident on the micro-grating, a portion of the laser beam is directly reflected and diffracted from the surface of grating, forming a series of diffraction beams, while the other part passes through the grating to reach the reflector, and a series of diffracted beams are generated after the reflection back to the grating once again. According to the diffraction theory, the same order of the two diffracted beams constructively or destructively interfere and the intensities of the 0th diffraction orders can be expressed as:

$$I_0={\displaystyle \frac{I_{in}}{2}}\left(1+\mathit{cos}{\displaystyle \frac{4\pi d_1}{\lambda }}\right)=I_{in}{cos}^2\left({\displaystyle \frac{2\pi }{\lambda }}{d}_1\right)$$

(1)

where I_in and λ represent the incident laser intensity and laser wavelength, respectively, and d₁ is the initial distance between the grating and the reflector. It can be seen that the diffraction intensity curve exhibits a sinusoidal shape as the displacement increases, with the period of λ/2.

The subdivision interpolation method can be employed to enhance the measurement of the resolution and linearity. By utilizing the two reflecting regions at different heights or two transmitting regions of gratings with varying heights to generate two orthogonal signals [24,25,26], these two signals can subsequently be further processed using phase detection or interpolation methods, as illustrated in Figure 1a. To ensure the orthogonality of the two signals, it is required to satisfy $2\pi \left(d_1+d_2\right)/\lambda -2\pi d_1/\lambda =\left(2N+1\right)\pi /4$ (where N is an integer), i.e., d₂ = (2N + 1)λ/8. However, the initial phase difference between the two signals may not be exactly 90 degrees due to the etching error $\Delta$ during the fabrication process, resulting in a phase error of approximately 90°(1 + 8$\Delta /\lambda$) between the sine and cosine signals. Additionally, amplitude errors may occur due to inconsistencies in the reflectance of the mirrors and grating diffraction efficiencies, while offset errors may arise from fluctuations in the laser intensity and photo-detector gain. Therefore, in order to suppress the phase, amplitude, and offset errors, we proposed the enhanced subdivision interpolation method illustrated in Figure 1b. This method transforms a signal into two orthogonal signals using a 90° phase-shifter. By employing high-precision DC bias voltage technology and subdivision interpolation circuits, the analog sine signal is converted into standard incremental digital signals.

To quantitatively analyze the impact of the arctangent algorithm on the phase, offset, and amplitude errors in the sine–cosine signals, a Simulink model of the arctangent algorithm was constructed. We simulated the arctangent results when the phase, offset, and amplitude errors were 20%, 10%, 5%, and 1%, respectively. The arctangent error is calculated as the difference between these error-existing arctangent results and the ideal arctangent results, as shown in Figure 2. It can be seen that phase error has the most significant influence on the arctangent error, while the amplitude error is minimal. The arctangent error increases linearly with the magnitude of the phase, offset, and amplitude errors. Specific accuracy errors resulting from different percentages of errors are detailed in

Table 1. Accuracy of the arctangent algorithm with different percentages of the phase, offset, and amplitude errors.

	Error Percentage of 1%	Error Percentage of 5%	Error Percentage of 10%
Phase error	99.00%	95.23%	90.92%
Offset error	99.11%	95.69%	91.73%
Amplitude error	99.37%	96.98%	92.07%

. It is evident that these errors need to be controlled within 5% to ensure an accuracy of 95% or higher, and within 1% to achieve an accuracy of 99% or higher.

3. High-Resolution Digital Readout Circuit

According to the optical sensing mechanism, we have developed a high-resolution digital readout circuit detailed in Figure 3. This circuit employs a high-resolution subdivision interpolation technique along with 90-degree phase-shifter and high-precision bias-voltage techniques. In the subdivision interpolation process, after constructing the tangent and cotangent signals, the phase information is extracted using the inverse tangent algorithm to achieve an 8× coarse subdivision. The digital subdivision is then completed in the DSP interpolator to produce equally spaced displacement outputs. The interpolation factor and the desired auto-adaption mode can be configured with appropriate resistors. The two analog-to-digital converters (ADCs) convert the analog sine–cosine signals into 14-bit digital values for further processing. After the ADC conversion, the DSP interpolator continuously corrects the digital gain, offset, gain match, and phase for minimal errors and jitters. Although this circuit module has the advantage of suppressing digital signal errors through initial calibration and continuous correction to some extent, the imperfect errors (such as the phase, amplitude, and offset errors) of the analog quadrature signals before the AD conversion cause a limited interpolation factor. Following correction and rectification, two differential input signals are processed using arctangent and interpolation algorithms to achieve high-resolution subdivision.

Based on the simulation results, it is evident that the phase error has the greatest impact on the measurement accuracy. The designed phase-shift circuit can convert a single input signal into two orthogonal outputs, with a phase difference of 90 degrees. The adder U₁ compares the sinusoidal input signal with the feedback signal from the integrator, and compensates for the gain of the input sinusoidal signal. Meanwhile, the integrator U₂ is utilized to realize a 90-degree phase-shift, whereas the feedback integrator U₃ can eliminate the offset voltage generated by the integrator. To quantitatively represent the function of the phase-shifter, the transfer function of the entire circuit is derived as follows:

$$H\left(s\right)=s{C}_2{R}_2{R}_4{R}_6/\left[R_1\right(s^2{C}_1{C_{2}R}_3{R}_4{R}_6+R_2\left)\right]$$

(2)

Let s = jω, and the phase angle of the transfer function is Arg[H(jω)] = 90°, ensuring that the subdivision interpolation factor remains unaffected by the phase error. Apart from the phase error, the stability of the offset error also affects the interpolation factor and the accuracy of the micro-displacement measurement due to the differential input of the subdivision interpolation circuit. The designed high-precision bias-voltage circuit utilizes a precision shunt reference with excellent temperature stability over a wide range of voltage, temperature, and operating current conditions, achieving a DC voltage stability of 10⁻⁵@1s.

To illustrate the resolution and accuracy limitation of the subdivision interpolation method, we simultaneously measured the interpolation level of the subdivision interpolation circuit over an input frequency range of 10 Hz to 200 Hz. Figure 4a presents the interpolation factor of the enhanced subdivision interpolation circuit at an input frequency of 100 Hz. The experimental results demonstrate that the output of the subdivision interpolation circuit is a standard square-wave signal at 1 MHz frequency when the frequency of the input sine and cosine signals operate at a frequency of 100 Hz, achieving a subdivision factor of 10,000, which is limited by the bits of the interpolator. Moreover, we evaluate the accuracy of the interpolation circuit by averaging 30 sets of tests over different time periods. As shown in Figure 4b, the accuracy of the interpolation circuit reaches 99.8%. The interpolation accuracy is primarily constrained by the stability of the subdivision factor, determined by the signal-to-noise ratio (SNR) and ADC distortion.

4. Experimental Results and Analysis

The experimental setup shown in Figure 5 was established to evaluate the performance of the grating interferometric micro-displacement sensor. Light emitted from the semiconductor laser (Model CPS635R, Thorlabs Inc., Shanghai, China, λ = 635 nm) is collimated by the lens onto a grating with a 1 mm spot size. The grating has dimensions of 2 mm × 2 mm, a period of 8 μm, and a duty ratio of 0.5. To enhance the interferometric sensitivity, the interference signal of the zero-order diffraction light is used as the detection signal, utilizing a polarization beam splitter (PBS) and wave plates to extract the 0th diffraction beam. A commercial displacement sensor with an accuracy of ±0.2 μm@12 mm serves as the length standard for calibrating the prototype and provides dynamic displacement. Phase modulation and demodulation techniques are employed to minimize the 1/f noise and enhance the SNR of the output signal. After a phase-shifter and bias circuit, a pair of sine and cosine signals are generated with equal amplitude, bias voltage, and a 90-degree phase difference. Subsequently, the micro-displacement signal is converted into incremental digital signals through a subdivision interpolation circuit, which determines the micro-displacement by counting the digital signals [27,28,29].

Figure 6a depicts the experimental results of the output voltage as a function of displacement, with a zoomed-in view of the signal presented in Figure 6b. The experimental data demonstrate that the light intensity detected by the photodetector varies sinusoidally with the displacement between the grating and the reflector. Upon passing through the 90-degree phase-shift module, two orthogonal signals with a phase difference of 90° are generated with a period of 0.01 s, as illustrated in Figure 6c. These orthogonal signals are then processed by the subdivision interpolation circuit to produce standard incremental orthogonal digital signals with an amplitude of 5 V and a period of 48 μs (Figure 6d). The subdivision factor is calculated to be 208. Given that the period of the photodetector signal is 317.5 nm, the theoretical displacement resolution after the subdivision interpolation circuit can reach 317.5 nm/208 = 1.52 nm. Additionally, the DSP interpolator in the signal-processing circuit plays a crucial role in determining the speed of the displacement measurement. In the pin configuration mode, the highest output frequency reaches 4 MHz. The input frequency of the displacement sensor, corresponding to this output frequency, can be calculated as 4 MHz/208 = 19,231 Hz, resulting in a speed limit of 6.11 mm/s. This makes the system highly suitable for high-speed displacement sensing.

In addition, we tested the actual resolution of the system using a piezoelectric ceramic (Model E53. B1S, Coremorrow Inc.). Under the application of a periodic voltage signal of 1 mV, the maximum displacement obtained is 1.9 nm, as shown in Figure 7. Since the signal exhibited the same vibration frequency and notable amplitude variations, these results indicate that the actual resolution of the proposed displacement sensor is less than 1.9 nm.

Additionally, the accuracy of the displacement measurement system was also evaluated. The number of the square-wave signal was recorded as the commercial displacement sensor moved by a specific displacement. The actual displacement was determined based on the displacement resolution. Figure 8a presents the experimental results of the displacement across the entire range with five measurement repetitions. As shown in Figure 8b, compared with the reference displacement sensor data, the accuracy and repeated accuracy of the displacement measurement system are less than 0.6 μm and 2.1 μm, respectively. The measurement error is primarily attributed to mechanical vibration noise during the operation of the commercial displacement sensor, which reduces the SNR of the signal, and thus affects the accuracy of displacement measurement.

Table 2. Performance comparison of the proposed method with other micro-displacement measurement techniques.

Characteristic and Abilities	This Work	M. W. Li [27] (2020)	Nuntakulkaisak [30] (2022)	X. K. Liu [8] (2021)	Q. B. Lu [31] (2015)
Methods	Interpolation	Interpolation	Quadrature detection technique	Time grating	Phase modulation
Continuous correction	Yes	No	No	No	No
Offset compensation	Yes	No	No	No	No
Phase calibration	Yes	Yes	No	No	No
Time-delay removal	Yes	No	No	No	No
Interpolation factor	208	100	/	/	/
Resolution	1.9 nm	3.175 nm (Calculation)	0.4 nm (Calculation)	/	0.017 nm (Calculation)
Accuracy	0.6 μm (99.8%)	2.2 nm (99.2%)	98.2%	0.2 μm (99.9%)	/
Displacement range	300 μm	250 nm	Nanometer scale	200 mm	/

shows the performance comparison of the proposed method with other micro-displacement measurement techniques [8,27,30,31]. In our previous work [27], we employed a polarization phase-shifting optical path to extract two sinusoidal signals with a quadrature phase shift, and combined it with the interpolation circuit, achieving a diffraction grating displacement sensor with nanometer-level resolution. Compared with the previous research results, the proposed method effectively suppress the phase, offset, and amplitude errors in the two orthogonal signals, thereby improving the subdivision interpolation factor and resolution. The accuracy of the micro-displacement system is primarily limited by the precision of the interpolation circuit affected by the ADC distortion. In terms of measurement range, it can reach up to the millimeter or even tens of millimeter levels, theoretically. However, the accuracy tends to be slightly lower than expected during actual long-range measurement, mainly due to interpolation inaccuracies caused by assembly errors. In the near future, the interpolation factor and resolution can be achieved by reducing the frequency noise and adopting other self-adaption algorithms.

5. Conclusions

This paper presents the design, signal processing, and experimental demonstration of a micro-displacement sensor based on grating interferometry. The proposed subdivision interpolation circuit converts the output signal into square-wave signals and mitigates the impact of phase, amplitude, and offset errors on the subdivision interpolation, achieving a subdivision factor of 10,000 and an accuracy exceeding 99.8%. The experimental results demonstrated that the prototype achieves a displacement resolution of less than 1.9 nm and an accuracy of 0.6 μm@300 μm. Future research will focus on suppressing vibration and light intensity noise to improve the contrast ratio of the output signal, thereby enhancing the interpolation factor for superior resolution and accuracy performance.