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Abstract
Surface plasmon lithography breaks through the diffraction limit with surface plasmon effect and achieves high-resolution imaging below sub-wavelengths. The dynamic detection and control of the working distance between the mask and the substrate is particularly important for the quality of the lithography samples. In this paper, the mask-substrate gap sensing mechanism is analyzed, and a mask-substrate gap absolute distance measurement model is established based on the principle of white light interference. We design a Fizeau interferometer demodulation system based on a hardware optical correlation method, and focus on analyzing the influence of the sub-nanometer surface profile of the optical wedge on the resolution. Finally, we propose a data processing method for extracting the dynamic gap value by SSA (singular spectrum analysis). The experimental results show that the system shows high performance with a gap detection range of 3.0∼6.0 µm, the stability of ±0.12 nm, and the resolution of better than 0.30 nm.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The concept of surface plasmon (SP) was firstly introduced by E. A. Stren and R. A. Farrel in 1960. The subwavelength structure of SP is characterized by a feature that allows coupled amplification of evanescent waves, which can be applied to super-resolution imaging that breaks the diffraction limit [1]. Likewise, this feature offers a new possible direction in the field of nanolithography, namely surface plasma lithography (SPL) [2]. In surface plasmon lithography systems, the precision of the gap control between the mask and the substrate directly affects the quality of lithography. Therefore, it is necessary to inspect the system with sub-nanometer accuracy in the micron range.
High-precision detection of mask-substrate micro-gap is essentially a distance and displacement detection. The basic means of gap detection usually include electrical methods and optical methods. Electrical methods include capacitance method [3], eddy current method [4], etc. Electrical methods have a small measurement range, low resolution, and are subject to electromagnetic influences. Basic optical methods such as a laser interferometry [5], a laser triangulation [6], and grating sensors [7], which have high requirements for auxiliary equipment and environment in order to achieve high resolution, are often used for the measurement of displacement changes and relative distances. Typical methods in recent years are as follows. In 2020, Alessandro Bertacchini and Reggio Emilia et al. proposed an ultra-low power eddy current displacement sensor (ULP-ECDS) targeting common industrial applications and designed to embed the sensors in wireless Industrial Internet of Things (IIoT) devices, which achieved the best resolution of 2.0 µm in the case of carbon steel targets [4]. In 2020, Zhao et al. proposed a high-precision micro-displacement measurement scheme based on single-layer grating, with a resolution better than 0.6 nm, a sensitivity of 0.4 nm [7]. In 2021, Lu et al. proposed an FM (frequency modulation) nonlinear kernel function-based range extraction method to improve the measurement precision of DFB(distributed feedback laser) array-based FMCW (frequency modulated continuous wave) lidar, with a resolution better than 1.9 µm, a large range of 5 m [8]. In 2021, Wang et al. proposed a high-precision micro-displacement measurement method based on alternately oscillating optoelectronic oscillators (OEOs); the measurement range is 20 mm, and the measurement precision is <300 nm [9]. The absolute distance measurement method based on white light interference has many advantages, such as high precision, less need for auxiliary devices, large range, and high resolution [10]. In 2021, Guo et al. built a white-light spectral interferometer for synchronous phase shifting based on polarization interference and used the two-step phase-shifting algorithm to retrieve phase information, with a resolution better than 1.0 µm [11]. In 2021, Sun et al. proposed a MADM (micro absolute distance measurement) method of high accuracy and frequency response based on PLCI (polarized low-coherence interferometry), with a measurement accuracy higher than 19.5 nm and a resolution better than 2.0 nm [12]. However, it is necessary to further improve the performance of the detection system to realize gap detection with a micrometer range and sub-nanometer resolution.
In this study, we establish a white light interference model of three-layer structure of mask-air gap-substrate, and propos a white light interference absolute gap detection method based on a non-scanning hardware optical correlation method. Then, the influence of the wedge shape of the Fizeau interferometer is mainly analyzed. We use SSA (singular spectrum analysis) to extract the dynamic gap value, and conduct experimental research on the detection system. Compared with the laser interferometer, this method can not only realize the high-resolution measurement of the micro-gap, but also realize the absolute measurement of the micro-gap.
2. Principles
2.1 Gap model-static correlation demodulation model
The basic structure of the white light interference-based micro-gap measurement system is shown in Fig. 1. The system uses a super continuum source. Incident light enters the mask-silicon gap structure (sensing interference unit) through the fiber coupler, where a multi-beam interference is formed. Then, the reflected light with the gap information passes through the fiber coupler and the cylindrical mirror, and then passes through the optical wedge and is received and collected by CCD (charge-coupled Device). The cylindrical mirror and the optical wedge constitute the optical path of a Fizeau interferometer (reference interference unit). The light intensity signal collected by the CCD is the optical cross-correlation of the two sets of interference signals. When the micro gap value of the sensing interference unit and the wedge thickness of the localized reference interference unit match each other, the position of the maximum value of the correlation result corresponds to the gap value.
Considering that the output intensity distribution of the optical fiber is Gaussian, ICCD(x) [13] (Luminous intensity collected by CCD) can be expressed as:
2.2 Influence of the optical wedge shape on measurement results
The structure of the Fizeau interferometer in the system is a combination of an optical wedge and a CCD. And the gap value corresponding to a single pixel of the CCD is the average gap value within the area of a single pixel. Therefore, when the wedge surface is rough, the schematic diagram of the rough surface and the ideal surface in two adjacent pixels is shown in Fig. 2. The average thickness Di1 and Di2 of the pixel in an ideal surface of the wedge increases with the increase of the thickness of the wedge (Di1< Di2). When the wedge surface type with roughness is similar to the surface type shown in Fig. 2, the corresponding CCD pixel related to the actual average thickness of the wedge will change, which will lead to errors in measurement results. If the roughness is large, it would appear that the corresponding gap value of the previous pixel is larger than that of the latter pixel (De1 > De2). Eventually, it can be concluded that the demodulation result is wrong.
Fig. 2. Schematic diagram of a wedge with large roughness. De1, De2: actual average thickness of pixel-1 and pixel-2; Di1, Di2: ideal average thickness of pixel-1and pixel-2.
In order to evaluate the influence of the roughness of the wedge surface corresponding to adjacent pixels on the measurement results, the influence of the optical wedge roughness is introduced into Eq. (1). The simulation parameters are as follows: R1 is 0.4, R2 is 0.3, R3 and R4 are both 0.5, the measurement range of the wedge is 3.0 ∼ 6.0 µm (D1∼D2), the light source wavelength is 0.55 ∼0.9 µm, the gap value is 5.0 µm. According to Eq. (1) and the micro gap demodulation principle, the ideal relative light intensity corresponding to the thickness of the optical wedge at x●tanθ is shown in Fig. 3(a). In the case of an ideal surface shape, a maximum value appears at the wedge thickness of 5.0 µm (Fig. 3(b) is an enlargement of Fig. 3(a)).
Fig. 3. Micro gap demodulation simulation diagram. The simulation result of (a) ideal surface signal, (b) details of the ideal surface signal; (c) rough surface signal, (d) detail of rough surface signal.
The characterization parameters of surface accuracy include PV (peak to valley) and RMS (root mean square) [14]. As shown in Figs. 3(c) and 3(d), when a random fluctuation with a PV value of 20 nm and an RMS value of 5.6 nm is superimposed on the ideal surface shape. The simulation results show that a large number of burrs appear in the output signal. The demodulated gap value is 4.9877µm, which has an error of 12.3 nm. Ultimately, the accuracy of the demodulation system drops sharply.
2.3 Design of shape parameters of the optical wedge
In order to obtain the optimal shape parameters of the optical wedge, and to ensure the accuracy and resolution of the measurement system, we establish the relationship between PV and RMS of optical wedge and the performance of the measurement system.
When analyzing the influence of the wedge shape in the pixel on the measurement results, the surface shape parameter RMS is used for analysis, as shown in Fig. 4(e), the included angle of the optical wedge is θ. The inner surface position of the upper and lower substrate sides of the wedge is (x1 (N1), yu1 (N1)) and (x1 (N1), yb1 (N1)) corresponding to a serial CCD pixel N1, respectively, then the average gap De is calculated by:
Fig. 4. Schematic diagram of the wedge surface. (a) Optical wedge structure, (b) ideal surface type, (c) a small fluctuation during a whole wedge, (d) a big fluctuation during a whole wedge, (e) the fluctuation in a single CCD pixel, (f) the fluctuation in a longer length (L), corresponding to CCD pixels.
When analyzing the influence of longer wedge shape on the measurement results, the surface shape parameter PV is used for analysis, as shown in Fig. 4(f). The position of the inner surface of the upper and lower substrate sides of the wedge is (x2(N2), yu2(N2)), (x2(N2), yb2(N2)), respectively. The gaps of all pixels in L can be obtained by accumulating:
2.4 Demodulation algorithm
The linear array CCD collects the output light intensity signal of the Fizeau interferometer. There are two important effects, one is the inevitable noise presented in the signal, and the other is inter-mode interference in the optical path of the fiber. As for the noises with high-frequency and low-frequency in the light intensity signal, the data processing is necessary and the designed steps are shown in Fig. 5. The noises are removed orderly by FIR (finite impulse response) filtering (step 1, Fig. 5(a)), and then the gap value demodulation processing (step 2, Fig. 5(b)) is performed. All peaks of the signal are extracted by the window comparison method. Then a three-point median moving average (step 3, Fig. 5(c)) is performed on the peaks as a whole to achieve peak correction. Finally, the main peak is subjected to Gaussian fitting interpolation (step 4, Fig. 5(d)), which achieves the addition of virtual pixels and the final determination of the peak position [15].
Fig. 5. Schematic diagram of data processing steps, (a) FIR filtering; (b) peak-finding; (c) three-point median moving average; (d) Gaussian fitting interpolation.
We know that the inter-mode interference is due to a super continuum light source used in this system. Its coherence leads to inter-mode interference in the optical path of the fiber. When the fiber is disturbed, the mode noise will cause the fluctuation of the micro-gap value. By considering this disturbance, 30 groups of simulation signals are shown in Fig. 6(a). There is a drift of the simulation signals, which causes the fluctuation of the gap values as shown in Fig. 6(b) and the calculated fluctuation of the gap values is within ∼2.0 nm (blue line).
Fig. 6. Static micro-gap calculated results. (a) Calculated 30 groups’ intensity signals and (b) calculated fluctuation of the gap values.
During the working process of the mask-silicon gap structure, its movement is in a step state or a low constant speed state. Its trajectory is a slowly varying signal, which is hidden in the signal together with random noise with a high-frequency. We propose a SSA method to obtain the gap trajectory. SSA decomposes time series (Y = [y0,y2,…,yl-1]) into random noise, periodic oscillations and slow trends. It includes the decomposition and reorganization of the signal. Firstly, we transfer time series signal Y = [y0,y2,…,yl-1]. Then, we take the embedded dimension of singular spectrum analysis as l to form the trajectory matrix X, and perform SVD (singular value decomposition) on it.
In Eq. (4), U and V are unitary matrices, whose column vectors are singular vectors. Σ is a diagonal matrix, whose diagonal elements are singular values of X. Reconstruction components can be obtained by decomposing the matrix on a diagonal average. According to the accumulation of different singular vector groups, yk can be expressed as:
3. Experiment results
Considering the accuracy requirements, the optical wedge substrate fabrication process is using CMP (chemical & mechanical polishing) [17]. A prepared photo of an optical wedge is shown in Fig. 7(a), with a range of 3 ∼ 6 µm. The surface roughness measurement result using ZYGO interferometer of the total wedge surface is shown in Fig. 7(b), with the RMS of 43.3 nm, and the PV of 32.2 nm. And the measurement results within 10×10 µm2 are shown in Fig. 7(c), with the RMS of 0.2 nm, and the PV of 0.7 nm.
Fig. 7. (a) Photo of a prepared ptical wedge, (b) the whole surface shape and (c) a localized surface (10×10 µm2) shape of the optical wedge. (d) The xperimental platform, consisting of sub-nano test parts, a laser interferometer and our designed Fizeau interferometer demodulation system, (e) the measured original signal using our designed Fizeau interferometer probe.
The experimental platform is shown in Fig. 7(d), the test system consists of sub-nano test parts, a laser interferometer and our designed Fizeau interferometer demodulation system. The sub-nano test parts are fixed at the marble base and placed in the vacuum hood for the stability of the system. The sub-nano test parts are composed of sub-nano actuator, devitrified glass, a laser interferometer probe and a designed Fizeau interferometer probe. The glass ceramic is installed on the sub-nano actuator, and the two probes are fixed by a fixed bracket. The sub-nano actuator is used to control the micro movement of the devitrified glass. By comparing the measurement results of the laser interferometer and our system, the performance of our system can be measured.
3.1 Experiment 1: static measurement
Firstly, we perform a static test on the system calibration gap value of 4.7µm. Note that this gap value is controlled by the sub-nano actuator and measured by the laser interferometer. The measured original signal using our designed Fizeau interferometer probe is shown in Fig. 7(e). Using SSA method mentioned above, the calculated gap demodulation result is 4.7241 µm.
3.2 Experiment 2: stability test
Furthermore, for stability testing, the initial gap was set to 3.8945µm using the sub-nano actuator and measured by the laser interferometer. For ease of comparison, the gap values without SSA method and using SSA method, which are shown in Fig. 8(a). A green line in Fig. 8(a) shows the inherent volatility of the working table. The calculated residuals are used to obtain the volatility of the gap detection results.
Fig. 8. (a) Gap values without SSA method (blue), using SSA method (red), an average baseline (green), (b) the calculated residuals of (b) original gap values and (c) gap values with SSA method.
The calculated residuals of the original gap values and gap values with SSA method are shown in Figs. 8(b) and 8(c), respectively. The calculated extreme volatility value of the original gap data is 2.1 nm, and the mean volatility is ± 0.3 nm. While, the extreme volatility value using SSA method is 0.9 nm, and the mean volatility is ± 0.1 nm.
3.3 Experiment 3: dynamic test
In order to verify the dynamic performance of our designed system, the required resolution is 0.5 nm, the sub-nano actuator is set up with a movement interval of 0.3 nm. With the same method mentioned above, the initial gap value is 5.3635µm. The measured gap values are shown in Fig. 9, which shows the system with a resolution of 0.3 nm.
Compared with the current reports in Table 1, our designed system has a better resolution in displacement detection.
Table 1. Comparative analysis of current methods with our work
4. Conclusion
This paper aims at a high-precision gap detection requirement with a range of 3.0 ∼6.0µm and a nanometer resolution. We focus on analyzing the influence of an optical wedge surface shape, and prepare high-quality optical wedge. The stability of micro gap detection results is improved by SSA method. The construction of the optical autocorrelation white light interference demodulation system based on Fizeau interferometer is successfully realized. The experimental results show a good performance with a static stability of ±0.10 nm, and a resolution of better than 0.30 nm.
Funding
National Natural Science Foundation of China (51875067).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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