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Abstract
Based on Talbot effect of optical microgratings, we report an ultra-compact sensor for displacement and vibration measurement with resolution down to sub-nanometer level. With no need of optical components such as reflectors, splitters, polarizers, and wave plates, the proposed sensor based on a common-path structure shows a high compactness. Using gratings with period of 3 µm, displacement measurement within a range of 1 mm is demonstrated experimentally. Associated with an interpolation circuit with subdividing factor of 4096, a resolution of 0.73 nm is obtained. The experimental results also show the ability for the sensor to detect in-plane vibration with frequency below 900 Hz. With a sub-nanometer resolution and an ultra-compact structure, the miniature sensor shows potential in applications such as high-precision machinery manufacturing and semiconductor processing.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High-resolution displacement sensors have been demonstrated as important components in precision positioning systems, which are generally used in applications such as high-precision machinery manufacturing and semiconductor processing [1–3]. In past years, several approaches, based on different principles such as capacitive sensing, laser interference, Morie fringe and grating diffractive interferometry, have been reported for high-resolution displacement measurement [4–7]. Among the methods mentioned above, optical-grating-based methods have attracted continuous attention benefitting from a high resolution, a good stability and the ability for multi-degree-of-freedom measurements [8]. Morie fringe has been used for displacement measurement in past decades [9]. Using optical grating with a typical period over 10 µm, the method generally suffers from a relatively low resolution (e.g., over 10 nm) [10–12]. In contrast, based on optical diffraction of optical gratings with a much smaller period, linear encoders with resolution down to sub-nanometer level have been reported in past years [13–16]. Associated with reference codes, absolute linear encoders can be operated without initialization [3,17]. Since this method is typically based on interference between different diffracted beams of optical grating, careful alignment between different diffracted beams is generally necessary. As a result, a relatively complex system consisting of reflectors, splitters, wave plates and polarizers is generally required, which takes a fair amount of space [2]. Recent interest in developing integratable and energy-efficient precision positioning systems has created a compelling need for miniaturization of high-resolution displacement sensors [18,19]. As a result, miniaturized displacement sensor with a sub-nanometric resolution is still in highly desired.
Talbot effect is a well-known phenomenon in the near field behind optical grating. A periodical intensity distribution with a same period to that of the optical grating can be observed [20]. Talbot effect has been investigated across a broadband spectrum region from visible to THz range, showing great potential in applications across from optical communication to optical sensing [21–24]. In 2015, S. Agarwal and C. Shakher reported an in-plane displacement measurement by using circular grating Talbot interferometer [25]. By analyzing the shift of Talbot interferometric fringe patterns, a resolution at micrometer level was reported. However, based on a graphical analysis process, the response time and resolution were highly limited by the imaging equipment.
Here, we report a sub-nanometric sensor for displacement and vibration measurement based on Talbot effect of optical microgratings. Talbot images are observed using optical micrograting with period of 3 µm as illuminated by monochromatic radiation. Locating the second grating within the Talbot region of the first grating in a common-path way, the total transmission of the double-layer structure is found to be in sinusoidal relationship to the relative location between the two gratings. Using a two-quadrant structure, two sinusoidal signals with a phase difference of 90° are obtained. Associated with an interpolation circuit, displacement measurement can be carried out. A resolution down to 0.73 nm within a range of 1 mm for in-plane displacement measurement is obtained experimentally. The ability for the sensor to vibration measurement is also demonstrated. The experimental results indicate a detecting range from 0.1 to 900 Hz. It is worth to mention that, with no use of optical components including reflectors, splitters, wave plates and polarizers, the proposed sensor shows ultrahigh compactness.
2. Principle
Amplitude transmission of an optical grating can be expressed as [26]
For the diffraction amplitude distribution at the plane behind the grating with a distance of z, the corresponding propagation factor should be multiplied by the plane wave function [27]. The transfer function (H) is given by
where f is the spatial frequency of the plane wave in x direction. And λ is wavelength of the input laser beam.From Eq. (1) and Eq. (2), complex amplitude distribution can be expressed as [28]
Assuming that a second optical grating with a same grating period of d is located behind the first grating, the complex amplitude transmittance function of the second grating can be expressed as
The intensity distribution (I) behind the second grating ban be determined by
The Fourier transform (IM) is given by [30]
Equation (8) indicates a sinusoidal relationship between IM and Δx.
Talbot images behind a single optical micrograting with period of 3 µm is shown in Fig. 1(a), which is obtained by a finite-different time-domain (FDTD) method. The white dotted lines indicate the Talbot region in which periodical Talbot images are observed. The Talbot images show a same period (3 µm) to that of the micrograting along the in-plane direction. As for the out-of-plane direction, the period of the Talbot images is found to be ∼ 27 µm, according to a calculated value of 28.3 µm derived from Eq. (4). As shown in Fig. 1(b), a double-layer structure consisting of two optical microgratings with a same period of 3 µm is proposed. Putting the second grating (G2) at the Talbot positions (z = N·ZT, N = 1,2,3…) or semi-Talbot positions (z = (N+1/2)·ZT) of the first grating (G1), the total transmission of the double-layer structure is found to change sinusoidally with a different relative location of G2 to G1 in the in-plane direction. The simulation results show well agreement with the theoretical intensity distribution obtained by using Eq. (8), indicating the possibility to in-plane displacement measurement by detecting the transmission.
Fig. 1. (a) Talbot images behind a 3 µm-period grating as illuminated by 635 nm-wavelength laser beam. The white dotted lines indicate the Talbot region. (b) Schematic diagram of the double-layer structure. (c) Transmission of a double-layer structure consisting of two optical microgratings with a different location of G2. In the simulation, G2 is located at the first semi-Talbot position of G1. Corresponding normalized intensity distribution behind G2, which is obtained using Eq. (8), is shown below.
3. Experimental results and discussion
3.1 Experimental setup
Schematic diagram of the propose sensor is shown in Fig. 2. Continuous-wave laser beam at wavelength of 635 nm is introduced to G1 after passing through an optical expander. Putting G2 within the Talbot region of G1, optical transmission changes as G1 moving along x direction. Using a two-quadrant detector, two sinusoidal signals with a difference of 90°in phase can be obtained. Associated with an interpolation circuit, sinusoidal signals are transformed into square signals [19]. Displacement of G1 can be derived by counting the number of square waves. Benefitting from a common-path structure, the sensor shows ultrahigh compactness without using optical components such as reflectors, splitters and polarizers. The resolution (S) of the displacement sensor can be given by
where C is subdividing factor of the interpolation circuit. As a result, a smaller grating period as well as a larger subdividing factor leads to a better resolution.Fig. 2. Schematic diagram of the proposed sensor. The grating lines are located along y direction. In the experiment, G1 moves in x direction.
3.2 Displacement measurement
A two-quadrant structure for optical gratings (as shown in Fig. 3) is proposed to obtain sinusoidal signals. The distance between the two quadrants is set to be N•d and La+(N-1/4)•d for G1 and G2 respectively, in which La is the difference in lengths of single grating quadrant for G1 and G2. Resulting from a difference of 1/4d in locations of the second grating quadrant for G1 and G2, sinusoidal signals with a phase difference of π/2 (corresponding to 90°) can be obtained. Optical images of G1 and G2 are shown in Fig. 3(b). The gratings are made by etching an Al film with height of 150 nm located on the SiO2 substrate. The size of a single grating quadrant for G1 and G2 is 3mm×3 mm and 2mm×3 mm respectively. Scanning electron microscopy image demonstrates a grating period of 3 µm.
Fig. 3. (a) Structure diagrams of G1 and G2. (b) Optical images of G1 and G2 (scar bar, 2 mm). (c) Scanning electron microscopy image of the optical micrograting with period of 3 µm (scar bar, 10 µm).
Sinusoidal signals obtained from the two-quadrant detector with an input power of ∼0.5 mW are shown in Fig. 4(a). The results demonstrate a maximum voltage change (ΔV) of ∼1 V with a phase difference of 90.03°. Using an interpolation circuit with subdividing factor of 4096, square signals are obtained (as shown in Fig. 4(b)). A resolution of 3 µm/4096≈0.73 nm is derived by Eq. (9). A 16-bit digital signal processing interpolator (iC-TW8, iC-Haus) with automatic calibration and adaption of offset, amplitude match and phase quadrature is used in the interpolation circuit. Powered by a supply voltage of 5 V, the circuit accepts the two sinusoidal signals directly from the two-quadrant photodetector. By changing the configuration resistors, the subdividing factor of the interpolator is set to be 4096. Considering that the interpolator can be operated with a subdividing factor up to 16384, it is possible to further improve the resolution in principle. However, it typically requires a heavier filtering and lag recovery enabled.
Fig. 4. (a) Sinusoidal signals obtained by two-quadrant detector. The grating period used in the experiment is 3 µm. (b) Square signals obtained by an interpolation circuit with a subdividing factor of 4096. Resolution is calculated to be ∼0.73 nm in this case.
Experimental results of displacement measurement within a range of 1 mm is shown in Fig. 5. A motorized translation stage (MT1/M-Z8, Thorlabs) is used to offer a tunable input displacement. As shown in Fig. 5(a), experimental results show a good linear relationship and a well agreement to input displacement. An intended input displacement of 1 mm is offered by the motorized translation stage. Multiple measurement results show a good repeatability (as shown in Fig. 5(b)). The inset shows measurement results from a commercial displacement sensor (MT1281, Heidenhain). The results indicate an error (the difference between the average measured value of MT1281 and the proposed sensor) of ∼1.5 µm. Using a phase-unwrapping method, displacement can also be determined directly from the two sinusoidal signals [31,32]. An average displacement of 982.4 µm is obtained experimentally, which agrees with the value of 984.6 µm obtained by using the interpolation circuit.
Fig. 5. (a) Displacement measurement results. The black dots indicate the results obtained by the proposed sensor. The red line indicates the input displacement. (b) Multiple measurement results with input displacement of 1 mm. The black dots indicate the measurement results. The red line indicates an average value of 984.6 µm. The inset shows measurement results from MT1281, which indicates an average value of 986.1 µm.
Compared to results from MT1281, accuracy and precision of the displacement measurement is analyzed. Small displacement of 150 nm, 320 nm and 620 nm are offered by a piezoceramics respectively. The measurement results are shown in Fig. 6, indicating a good agreement between results from the proposed sensor and the commercial sensor. A small error, mainly attributing to environment vibration, within ±10 nm is observed in experiment. For example, with input displacement of 620 nm, an average error of ∼1.5 nm is obtained with a maximum error of ∼9 nm, corresponding to a relative error of ∼0.24%. In addition, relative standard deviation is calculated to be ∼0.72%.
Fig. 6. Comparison between the proposed sensor and a commercial sensor with input displacement of 150, 320, and 620 nm respectively. The black rectangles and red triangles indicate results from the proposed sensor and the commercial sensor respectively.
Considering that the structured light patterns are typically obtained as d>λ [33], the resolution is limited approximately by
Using an optical source with a smaller wavelength may be an effective approach to a better resolution. For example, with a wavelength of 635 nm for input beam, Talbot images can be observed with a grating period down to 700 nm (as shown in Fig. 7(a)). With period goes down to 400 nm, the Talbot images disappear. However, using a smaller optical wavelength of 377 nm, Talbot images can be obtained using optical grating with a same period of 400 nm (as shown in Fig. 7(b)).
Fig. 7. Talbot images with different grating period and optical wavelength. (a) Simulation results with grating period of 700 nm and wavelength of 635 nm. (b) Simulation results with grating period of 400 nm and wavelength of 377 nm.
Considering that grating period of several hundred nanometers has been reported by nanofabrication techniques such as CMOS techniques and laser-induced formation [34,35], the proposed setup shows potential for displacement measurement with a resolution down to 0.01-nm level associated with a higher subdividing factor.
3.3 Vibration measurement
Vibration measurement is also demonstrated in experiment. The piezoceramics is pumped by alternating-current signal with frequency ranging from 0.1 Hz to 900 Hz. For example, the obtained sinusoidal signal and square signal with an input vibration at 0.1 Hz are shown in Fig. 8. Resulting from a change of moving direction during vibration process, many changing points are observed periodically in both sinusoidal and square signals. As shown below, since the region between the indicated changing points corresponds to one single circle of vibration, the frequency of input vibration can be obtained by counting the changing points. Since the region across one single circle of vibration corresponds to a time duration of 10 s, the frequency of vibration is calculated to be 0.1 Hz, agreeing with the frequency of pumping signal. Some fluctuations between two neighbouring changing points are observed, which are explained by the slight environment vibration.
Fig. 8. (a) Sinusoidal signal and (b) square signal with vibration at 0.1 Hz. The black arrows indicate the changing points caused by vibration. The black lines at top of the images indicate one single circle of vibration.
As shown in Fig. 9(a), frequency measurement results show an excellent agreement with the frequency of pumping signal. At the same time, amplitude of vibration is also measured by counting the square waves between two changing points. It is found that, the amplitude is in linear relationship to pumping voltage of the piezoceramics. For example, with a pumping voltage of 2.6 V, an amplitude of 5.95 µm is obtained. The output signals are sampled by a two-quadrant photodector (OSQ100-IC, OTRON) with 3 dB cut-off frequency of 4 MHz and a 200-MHz oscilloscope (TBS2204B, Tekronix) with a sampling rate of 2 GS/s. Since the sampling is limited either by the oscilloscope’s sampling rate or the frequency response of the photodetector, the sampling rate is limited to be 4 MHz in principle. To discuss the bandwidth of the sensor, amplitude of vibration with frequency from 0.1 to 900 Hz is measured (as shown in Fig. 9(b)). The input amplitude is set to be ∼ 6 µm by changing the pumping voltage of the piezoceramics. With frequency below 900 Hz, the measurement results show good agreement to the input amplitude. The results indicate a bandwidth over 900 Hz.
Fig. 9. (a) Frequency measurement results with frequency from 0.1 to 900 Hz. (b) Measured results of vibration with different frequency. The inset shows amplitude measurement results with pumping voltage from 1.0 to 2.6 V.
4. Conclusion
Based on Talbot effect of optical microgratings, we demonstrate a miniature optical sensor for in-plane displacement and vibration measurement with a sub-nanometric resolution. Locating the second grating within the Talbot region of the first grating, the total transmission is found to be in sinusoidal relationship to the relative location of the two gratings. Using a two-quadrant detector, two sinusoidal signals with a difference of 90° in phase are obtained. Experimental results show a high resolution of 0.73 nm within a range up to 1 mm. At the same time, vibration measurement below 900 Hz is also demonstrated in experiment. Using a simple common-path structure, the miniature sensor shows ultrahigh compactness without using optical components including reflectors, splitters, wave plates and polarizers. In addition, since the period of sinusoidal signal is dominated by geometries of the optical gratings in this case, the sensor shows high immunity for wavelength fluctuation of laser source. The results indicate great potential for this miniature optical sensor with sub-nanometric resolution in developing precision positioning systems, which are highly desired for applications ranging from high-precision machinery manufacturing to semiconductor processing.
Funding
National Natural Science Foundation of China (62005252).
Acknowledgments
The authors thank Rui Zhang, Li Jin, and Bin Cao for their helpful discussion in experiment preparation.
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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