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Abstract
To improve the measurement accuracy of interferometer displacement measurement systems, this study analyzes the characteristics of the interference signal to identify sources of nonlinear errors and develops compensation strategies. Specifically, a model is established for the nonlinear errors of the interferometer, which can be attributed to a laser and polarizing beam splitter (PBS). Following that, the dual orthogonal lock-in amplification algorithm is used to separate and compensate for the frequency uncertainty and amplitude errors. Additionally, a real-time compensation algorithm based on ellipse fitting is proposed to compensate for errors caused by the PBS and the uncertainty of amplitude caused by the light source. Experimental results demonstrate that the peak-to-peak value of the compensated nonlinear error is reduced from 11.62 nm to 5.37 nm.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the development of the semiconductor industry, chip fabrication has increased processing requirements for micro–nano measurement technology [1–4]. Overlay is a key indicator of mask aligner, which demands precise displacement measurements of the wafer and mask stages. Interferometers are the primary sensors used in mask aligners to measure displacement and are classified into laser interferometers and grating interferometers [5–9]. As lithography process nodes have now reached 7 nm and are heading toward the 5 and 3 nm process nodes, the accuracy requirements of ultraprecise micro–nano measurements are constantly growing.
Nonlinear errors in actual interferometer displacement measuring systems are caused by factors such as the structure of the optical route and the different characteristics of optical and electronic devices [10–14]. Among the nonlinear errors, the angle error in the x and y axes of a laser as well as the light leakage in a polarizing beam splitter (PBS) will cause the light of the reference arm to be mixed into the optical path of the measurement arm [15]. After a board card collects an interference signal, the spectrum contains a reference component and a reverse component. These factors can result in a significant decrease in displacement measurement accuracy. Thus, how to adjust for nonlinear errors has become an important issue in improving displacement measurement accuracy [16]. Zhu used the alternating iterative least squares fitting procedure to estimate the fringe phase and harmonic coefficients accurately [17]. Wang proposed an effective shifted-phase histogram equalization (SHE) method for nonlinear correction [18]. Liu proposed a fast and easy phase-error compensation technique based on statistical analysis of the wrapped phase distributions [19]. Xu et al. proposed a method for nonlinear error compensation by establishing a full-field look-up table (LUT) [20]. The nonlinear phase error can be compensated to a certain extent by this method. However, the establishment of the LUT is related to the stroke of the measured object, which limits the size and accuracy of the LUT. Chen reduced the first-harmonic nonlinearity errors by adjusting the orientation error of the optical element in the heterodyne interferometer [21]. This method has a certain compensation effect, but the craft of the optical element will also lead to the emergence of nonlinear errors. Wang proposed an optical compensation method to correct the nonlinear error caused by the polarization mixing of the PBS [22]. However, the compensation of optical devices has extremely high requirements for the properties of the optical device, and the cost is high. Yin proposed a method to reduce the generation of nonlinear errors by using a dual-space structure to avoid polarization light aliasing [23]. This method avoids the generation of nonlinear errors at the optical path, but the optical path structure becomes more complex and the influencing factors will increase. Xiong established a synthetic model of nonlinear error including six error sources is established, and the influence of various nonlinear error sources is analyzed according to the phase expression of nonlinear error derived from the model [24]. Fu proposed a symmetric heterodyne interferometer with spatially separated beams [25]. Yin Y proposed a 3D measurement method based on 2D grating dual-channel and Littrow equal-optical path incidence [26]. The separation-dual-channel phase decoupling algorithm was used to obtain displacement data. The test resolution is within $\pm$ 5 nm. Yin Y, Liu Z et al. proposed a new symmetrical heterodyne displacement measurement method, based on 2D grating and single diffraction quadruple subdivision method [26]. The measurement resolution is better than 3 nm. In the previous work, the aliasing of polarized light is mainly avoided by means of optical compensation, such as using higher optical elements with performance parameters, adjusting the optical path structure, and adopting a spatially separated dual optical path structure to avoid the generation of nonlinear errors. However, when the performance of the optical device is not high, it is difficult to achieve the compensation accuracy described in the reference.
This paper proposes a signal processing method to reduce the nonlinear error. Firstly, through the modeling and separation of the nonlinear error sources of the heterodyne laser interferometer, the different nonlinear error components are compensated. The biorthogonal phase-locked amplification algorithm effectively solves the problem of frequency uncertainty. The ellipse fitting algorithm effectively solves the DC problem caused by the reference component and the signal amplitude jitter problem caused by the non-coincidence degree of the spot. More importantly, the improved design of the ellipse fitting algorithm makes the application of the algorithm shift from offline or static compensation to real-time dynamic compensation, which will be of great significance to nonlinear error compensation. First, The Gauss elimination method is redesigned from the mathematical point of view, which reduces the complexity of the algorithm and makes it more suitable for engineering applications. Secondly, the accelerated design of the algorithm not only guarantees the real-time performance of the algorithm, but also simplifies the complexity of the algorithm and the occupancy rate of resources, which is a gratifying progress in the application of the project.
2. Analysis and compensation of the nonlinear error
The displacement in the heterodyne interferometry system is generated by a high-precision moving table and converted into a measuring signal using a biaxial interferometer with four subdivisions. The reference and measurement signals are converted into phase signals after passing through the measurement platform. The structure of the reading head of the biaxial laser interferometer with four subdivisions is shown in Fig. 1. The dual-frequency laser light source comprises vectors E1 and E2. It is a linearly polarized light with a fixed frequency difference and its polarization state perpendicular to each other, as shown in Eq. (1). Where $\varphi _{01}$ and $\varphi _{02}$ are the initial phases, $f_1$ and $f_2$ are frequencies and $E_{01}$ and $E_{02}$ are the amplitudes. The light source generates a reference beam R0 and a beam R1 through the beam splitter BS.R0 is received by the detector PD0 after passing through the polarizer P0. R1 pass through the beam splitter P, the light is divided into two paths, $r1$ and $r2$, which correspond to the measurement axes c1 and c2, respectively. Further, r2 passes through the PBS to generate reflected light with frequency $f_1$ and transmitted light with frequency $f_2$. The reflected light forms c22 after being reflected by the mirror RR3 for the first time, and the polarization direction of the reflected light after passing through the quarter-wave plate (QWP) twice is perpendicular to that before. However, the transmitted light also passes through the QWP twice. Thus, the transmitted light meets c22 at point a in the PBS and is separated at point b after being reflected by the corner cube prism RR2. Further, c22 passes through the mirror RR3 again to form c21, with the frequency becoming $f_1 \pm \varDelta f$ and the polarization direction again perpendicular to that before. Finally, c21 collides with the transmitted light at point b as the measurement signal X2. X2 is received by the detector PD2 after passing through the polarizer P2.
The interferometric signals measured by the interferometer when the moving platform of the system moves at a constant speed are shown in Fig. 2.
The amplitude and DC values of the reference and measurement signals are not equal (Fig. 2(a)). Fig. 2(b) presents the Lissajous figure of the two signals, which shows that the center of the circle is not at the origin and the long and short axes are not equal. Moreover, the amplitude of the measured signal shows an uncertain change with time (Fig. 3).
The power spectrum of the measurement signal is shown in Fig. 4 after spectrum analysis. Owing to the imperfect optical device performance and installation angle errors, error components that differ from the ideal frequency components appear in the spectrum. Because other random noises are inevitably introduced into the optical path and circuit, there will be other components in the spectrum, as shown in the black part of the figure. Zeng described the analysis and suppression of this noise in detail [27].
2.1 Error model of the interference signal
The coordinate system of the PBS is used as the absolute coordinate system because of certain angular deviations between the x and y axes of the laser and the PBS (Fig. 5). Thus, angles $\alpha$ and $\beta$ exist between the dual-frequency quadrature lasers $E_{01}$ and $E_{02}$ and the x and y axes of the coordinate system of the PBS. Moreover, because of certain nonorthogonal error, the two polarization states of the dual-frequency laser, $\alpha$ and $\beta$, are not equal.
The two elliptically polarized lights produced by the laser are divided into x and y axes of the absolute coordinate system, and the vector expression is Eq. (2), where $\vec {i_x}$ and $\vec {j_y}$ are the PBS optical axis direction vectors.
Because of the installation angle error and nonorthogonal error, the measurement light with frequency $f_{1}$ for the measurement arm enters the reference arm, and the reference light with frequency $f_{2}$ for the reference arm enters the measurement arm, resulting in optical aliasing and nonlinear errors.
Owing to the light leakage phenomenon in the PBS, the two polarized lights cannot be completely separated when the light is transmitted to the PBS; thus, the P- and S-polarized lights are reflected into the reference and measurement arms, respectively, resulting in optical aliasing and nonlinear errors (Fig. 6).
In a PBS, the transmittance and reflectance of $P$ light are $T_{P}$ and $R_{P}$, the transmittance and reflectance of $S$ light are $T_{S}$ and $R_{S}$, where $T_{P}+R_{P}=1$, $T_{S} +R_{S} =1$. The light passes through the PBS twice in the structure of the reading head illustrated in Fig. 1, but both are cases of light leakage; consequently, the light leakage is only calculated once during model derivation. After the light passes through the PBS, the phase difference reflected by the reference arm angle prism to the PBS is $\varphi _{1}$ in Eq. (3) and that reflected by the measurement arm angle prism to the PBS is $\varphi _{2}$ in Eq. (4). Because the frequency component caused by the corner prism was small, it was not investigated.
To simplify the analysis, the parameters in Eq. (3) and (4) are replaced as follows:
After reaching the polarizer, Eq. (3) and Eq. (4) are divided according to the different polarization states. The expressions of the light of the two polarization states are respectively shown in Eq. (6) and Eq. (7).
The light intensity generated by the final interference is shown in Eq. (8).
As shown in Fig. 4, among the components, the ideal interference component, the reference interference component, and the reverse interference component have relatively large intensities. Only these three components are considered in the establishment of the subsequent algorithm model. For convenience of equation derivation, variable substitution like Eq. (9) was performed.
Usually, the phase difference generated by the measurement arm has a relationship such as Eq. (10).
The reference signal is directly generated by the interference of the polarizer after the dual-frequency laser emits light. The frequency difference between two lasers with different frequencies is not ideally $2\pi f_1 - 2\pi f_2$ but has an uncertainty of $f'$. The expressions for the reference and measurement signals are shown in Eqs. (11) and (12).
2.2 Design of the nonlinear error compensation algorithm
To eliminate the influence of the laser frequency uncertainty $f'$ and the error in the unequal amplitudes, a nonlinear error model in the dual orthogonal lock-in amplification algorithm is established. The block diagram of the dual orthogonal lock-in amplification algorithm [28] is shown in Fig. 7.
The reference and measurement signals are multiplied by a pair of orthogonal signals, the frequency of which is the laser frequency difference produced using direct digital frequency synthesis (DDS). The filtered signals are shown in Eqs. (13), (14), (15), and (16), where $f_d$ is the frequency of a pair of orthogonal signals generated by DDS.
By calculating Eq. (16)$\times$ Eq. (13)$-$ Eq. (15)$\times$ Eq. (14) and Eq. (16)$\times$Eq. (14)$-$Eq. (15)$\times$Eq. (13), Eq. (17) can be obtained according to the sum–difference product equation.
To simplify combining terms with the same frequency, substitutions such as Eq. (18) were performed.
The combined result is shown in Eq. (19).
In Eq. (19), $D\sin (\varphi _m- \varphi _r)$ and $D\cos (\varphi _m- \varphi _r)$ are time-independent DC components. As shown in Fig. 3, due to the uncertainty in the spot, M is shown to change with time, which is further simplified into Eq. (20).
The coordinate rotation digital computer (CORDIC) algorithm [29] is used to perform phase calculation and phase accumulation on the ideal signal and the signal containing $M_2(t)$, $M_1(t)$, $d_1$ and $d_2$ errors, respectively. The results are shown in Fig. 8.
The error component in Eq. (20) leads to a periodic nonlinear error (Fig. 8). Therefore, to further eliminate the nonlinear error, a real-time nonlinear error compensation algorithm based on ellipse fitting is proposed. Eq. (20) is substituted into the general Eq. (21) of the ellipse, and Eq. (20) solves the $d_1$, $d_2$, $M_1(t)$ and $M_2(t)$ parameters through the correction of Eq. (22) to provide nonlinear error correction compensation.
In Eq. (21), $A$, $B$, $C$, $D$ and $E$ are the ellipse parameters that need to be solved. The overall design process is shown in Fig. 9.
The overall design process starts with the interval discrimination module. The matrix coefficient–solving module reads the parameters cached in the interval discrimination module and solves the matrix coefficient. The solution module obtains the ellipse parameters by solving the row echelon matrix, and the error correction compensation module uses the ellipse parameters to correct the original signal.
When the moving platform of the interferometer measurement system moves at different speeds, the frequency $\Delta f_i$ in Eq. (20) varies as well. When estimating the parameters of an ellipse, the precision of the parameter calculation is determined by the number of sampling points in a period. However, with a fixed sampling rate and the same sampling time, the periodic integrity of the interference signal increases with increasing frequency. Therefore, to maximize the accuracy of the solution, the signal is first divided into intervals, as presented in Table 1.
Table 1. Correspondence between symbols, size comparisons, and intervals
The real-time interval discrimination module shown in Fig. 10 is designed using the judgment logic presented in Table 1.
The input signal is normalized and taken as an absolute value (Fig. 10). The absolute value and sign of the cache are established by the logical relationship presented in Table 1. Finally, the original data with different intervals are cached in the eight-unit buffers for subsequent algorithm processing. Thus, the requirements of at least five sets of data from the unknown parameters of the ellipse and the integrity of the signal cycle are met to the greatest extent. The most important aspect is that the division of intervals avoids the problem of unnecessary calculations owing to excessive data points.
Eight-unit buffers with no-interval and interval discriminations are used to estimate the parameters of elliptical fitting, and the frequency responses of the amplitude accuracy are compared, as shown in Fig. 11.
With the decreasing frequency of the interferometric signal from 150 kHz to 100 kHz, the error in the amplitude calculation accuracy of no-interval discrimination gradually increases, resulting in a maximum error of about 82536 LSB. The maximum error in the amplitude calculation accuracy with interval discrimination is $5 \times 10^{ -11}$ LSB. Moreover, with decreasing signal frequency, the error in the amplitude calculation accuracy of no-interval discrimination worsens. Therefore, interval discrimination effectively guarantees the calculation accuracy of the ellipse parameters.
The matrix coefficient–solving module extracts the data from the eight-unit buffers to form a new orthogonal signal. Based on the principle of ellipse fitting, it is important to calculate the matrix coefficients and generate the augmented matrix. The pipeline design steps are shown in Fig. 12.
In Fig. 12, due to the need to calculate the matrix coefficient, the coefficients that need to be calculated for a data point are $x^{2}y^{2}$, $xy^{3}$, $x^{3}y$, $x^{2}y$, $xy^{2}$, $xy$, $y^{4}$, $y^{3}$, $y^{2}$, $y$, $x^{3}$, $x^{2}$ and $x$. To meet the real-time update rate, the first-level pipeline multiplier calculates $xx$, $xy$ and $yy$, the next-level pipeline multiplier calculates $x^{2}y^{2}$, $xy^{3}$, $x^{3}y$, $x^{2}y$, $xy^{2}$, $y^{4}$, $y^{3}$ and $x^{3}$. Because the calculated parameters of each point are cached in the corresponding buffer and each coefficient corresponds to eight points, the cumulative operation is performed to obtain the augmented matrix shown in the figure.
To solve the ellipse parameters, first transforming the augmented matrix into a row echelon matrix is required, followed by solving the unknown parameters via back substitution. In a traditional Gaussian elimination method [30], it is required to solve the row vector for division. However, the delay in the divider output in the FPGA increases with increasing data bit width. When the fixed-point number is divided, the accuracy after the decimal point will be completely discarded, resulting in a sharp decrease in accuracy. To solve this problem, an improved Gaussian elimination module is designed, and the specific design steps are shown in Fig. 13.
Because the entire improved Gaussian elimination module performs a lot of multiplication operations, a dual-frequency acceleration process is performed to reduce the occupation of resources such as DSP (Fig. 13). The new sample rate in the upper sampling module is twice the original sampling rate. The multiplier is then used as a link, and the output result is selected by a double sampling rate. The one-to-two distributor of the signal caches the data in the buffers. Finally, the subtractor achieves the goal of elimination. Moreover, because the matrix is symmetrical about the diagonal, it is only necessary to calculate the upper triangular matrix during the calculation.
Back substitution can be used to generate the ellipse parameters A, B, C, D, and E based on the row echelon matrix shown in Fig. 13. The signal is adjusted and compensated using Eq. (22) once $d_1$, $d_2$, $M_1(t)$ and $M_2(t)$ in Eq. (23) is determined.
3. Experiments and results
3.1 Hardware-in-the-loop simulation test
During the movement of the moving table, the maximum movement speed of the moving table is 20 mm/s, thus verifying the algorithm compensation effect when the moving table moves at high speed. The algorithm builds a hardware-in-the-loop simulation platform, as shown in Fig. 14.
The whole platform is composed of a signal generator, an ADC acquisition card, an FPGA board, and a PC terminal. The signal generator is used to generate a measurement signal with a frequency of 21 MHz and a reference signal with a frequency of 20 MHz. At the same time, the amplitude of the signal changes at a certain moment. The ADC sampling module converts the analog signal into a digital signal and sends it to the FPGA for ellipse parameter calculation. The calculation result is transmitted to the host computer through JTAG. The real-time test results of the system are shown in Fig. 15.
As shown in Fig. 15, the time from the actual signal amplitude of 55432 LSB to the output of the solved amplitude of 55432 LSB is $N_{S}$= 1.884 $\mathrm {\mu }$s. The test results meet the real-time requirements.
Table 2 simulates the compensation accuracy of the real-time compensation algorithm at different movement speeds of the moving platform. Table 3 simulates the compensation accuracy of the real-time compensation algorithm within Maximum velocity of the stage. Moreover, the wavelength of the laser is 632.9907 nm. First, the displacement curve is extracted and fitted using the least squares method. Second, the difference between the original curve and the fitted curve is calculated to obtain the error curve. Finally, the standard deviation and the peak-to-peak error are calculated to obtain the error curve.
Table 2. Phase delta error for different Doppler frequency shifts
Table 3. Phase delta error for different Doppler frequency shifts
From the experimental data presented in Table 2, with increasing movement speed of the moving platform, the peak-to-peak error gradually increases. When the speed reached 4.75 m/s, the peak-to-peak error reached 17.74 nm. Moreover, the standard deviation of displacement curve increases with increasing movement speed, with the maximum being 2.04 nm. This shows that the gross error increases with the acceleration in the movement speed of the moving table. From the standard deviation, the error in the algorithm only increases by about 2 nm, ensuring stability in errors. However, in practical applications, the speed does not exceed 20 mm/s, and even high-speed motion can guarantee a standard error of 2.04 nm.
From the experimental data presented in Table 3, with increasing movement speed of the moving platform, the peak-to-peak error gradually increases. When the speed reached 18.99 mm/s, the peak-to-peak error reached 0.287 nm. The standard deviation of the displacement curve remains in the range of 0.04 nm with the change of the moving speed. Therefore, the simulation accuracy of the compensation algorithm is within the error range of 0.3 nm.Comparing Tables 2 and 3, it can be found that when the motion stage moves at a high speed, the sampling accuracy of the signal is reduced due to the fixed sampling rate, which will reduce the solution accuracy.
3.2 Practical testing
To verify the compensation effect of the algorithm, a heterodyne interferometry system is set up, as shown in Fig. 16. The heterodyne interferometer generates measurement and reference signals owing to the displacement of the moving platform. First, the measurement platform samples the interference signals before performing phase calculations. Finally, the phase calculation results are sent to the host computer.
First, the moving table is moved at a constant speed of 3 mm/s. The displacement error in real-time compensation using the ellipse fitting algorithm is compared with that without using the ellipse fitting algorithm. The experimental results are shown in Fig. 17 and the sampling time t=0.625ms.
Because the displacement curve without ellipse fitting shows a periodic error of the displacement error relative to the curve with ellipse fitting, ellipse fitting effectively reduces the nonlinear error (Fig. 17). As shown in Fig. 17(b), the peak-to-peak error without ellipse fitting and that with ellipse fitting are calculated. The results show that the real-time nonlinear error compensation algorithm based on ellipse fitting reduces the peak-to-peak error in the displacement from 11.62 nm to 5.37 nm. Therefore, the nonlinear error compensation algorithm effectively eliminates the nonlinear error caused by the laser and the PBS. However, because the nonlinear error of 5.37 nm still does not meet the measurement requirements of subnanometer precision, it is necessary to continue to improve the splitting performance and transmittance of the PBS to further suppress the nonlinear error. The moving table is moved at constant speeds of 1, 3, 5, 10, and 20 mm/s, and the peak-to-peak error and standard deviation of the displacement of the different movement speeds are measured. The results are shown in Table 4.
Table 4. Phase increment error for different movement speeds
From the analysis results presented in Table 4, with increasing movement speed of the moving platform, the peak-to-peak error increases. When the speed reaches 16 mm/s, the peak-to-peak error is approximately 8.28 nm. Moreover, the standard error increases with the increase in movement speed, with the maximum error being 1.98 nm. This shows that the error increases with the acceleration in the movement speed of the moving table. However, because the peak-to-peak error at different movement speeds has uncertainty, further optimization is needed for the noise of the signal. From the standard error, the error in the algorithm is 2 nm. Therefore, further improvement is still needed for subnanometer measurements.
4. Conclusions
Based on the frequency- and time-domain analysis of the collected interference signal and its characteristics, the source and mechanisms of the nonlinear error associated with a heterodyne laser interferometer were studied. The nonlinear error expression of the light-intensity signal is deduced, a nonlinear error model of the light-intensity signal is established, and a real-time error compensation algorithm based on ellipse fitting is proposed. First, an interval discrimination module is designed so that the performance of the algorithm can meet the dynamic range of the moving platform. Second, the improved Gaussian elimination method prevents the precision loss and delay caused by the division operation and improves real-time performance and calculation precision. The multiplexing of resource occupation can be reduced through high-frequency accelerated multiplier multiplexing. A heterodyne interferometry system is set up, and real-time error compensation for an actual signal can be performed. The test results show that the nonlinear error is kept within the range, and the real-time nonlinear error compensation algorithm based on ellipse fitting effectively reduces the nonlinear error, which is of great significance in case of the subnanometer measurement.
Funding
Guangxi Key Research and Development Program (AB22035047); Natural Science Foundation of Guangxi Province (2019GXNSFDA185010); National Natural Science Foundation of China (61965005, 62205076); National Key Research and Development Program of China (2022YFF0605502); National Science and Technology Major Project (2017ZX02101007-003).
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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