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Abstract
In this paper, to separate and compensate the nonlinear error in the grating interferometer, we analyze the source and generation mechanism of this error, deduce the nonlinear error model of the measured signal and the calculated phase signal, and study the characteristics of the established nonlinear error model. The reason why the frequency multiples of ideal phase signals and higher-order nonlinear errors caused by ghost reflections and angular errors of the laser’s z-axis are not integers is explained. Then, a nonlinear error separation and compensation method based on cross-correlation coefficient is proposed. Experiments show that the frequency multiplier relationship between the high-order nonlinear error and the ideal interference signal is close to but not equal to 3-fold. The peak-to-peak value of the compensated nonlinear error is reduced from 17.40 nm to 7.05 nm.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Precision measurement systems are in high demand in many fields, among which heterodyne interferometry is considered to be one of the most accurate displacement measurement methods [1–5]. In the heterodyne interferometry method, the heterodyne grating interferometer has the advantages of high resolution and stability [6–8]. However, periodic nonlinear errors caused by the unsatisfactory performance of optical devices are a major source of measurement errors in heterodyne grating interferometers [9].
Nonlinear errors can be caused by the angle error of the x and y axes of the laser, the non-orthogonal error of the dual-frequency laser, and the light leakage of the polarization beam splitter [10–13]. In laser interferometry systems, ghost reflections caused by incomplete transmission from the surface of the optics can cause nonlinear errors that are integer multiples of the ideal phase signal [14,15].
In the grating interferometry system, when the light is not vertically injected into the grating and there is vertical relative motion and parallel relative motion between the grating and the motion table, the frequency of high-order nonlinear error caused by ghost reflection is not integer times of the frequency of the ideal phase signal, which makes it difficult to separate and compensate the nonlinear error [16–18].
In order to resolve the above issue, the following work is done in this paper. The source of the nonlinear error of the grating interferometer is analyzed, and the nonlinear error model of the measured signal and the phase calculated by the bi-quadrature lock-in amplification algorithm is established. Aiming at the problem that the multiples between the high-order error caused by ghost reflection and the ideal phase signal in the grating interferometer are non-integer multiples and are difficult to separate, we propose a nonlinear error separation and compensation method based on the cross-correlation coefficient. The phase is compensated by separating the nonlinear errors.
2. Nonlinear errors in grating interferometers
In order to perform sub-nanometer precision measurements, a grating interference displacement measurement system is built, as shown in Fig. 1. In the grating interferometry platform, the displacement is generated by the high-precision moving stage, and the four-division grating interferometer converts the displacement of the moving stage into reference and measurement signals. These signals are processed as phase signals and transmitted to the host computer after passing through the signal processing platform.
The structure of the four-division grating interferometer is shown in Fig. 2. The laser generates a pair of orthogonal linearly polarized lasers with a frequency difference of 20 MHz and a center wavelength of 632.99nm , which are divided into two channels after passing through the beam splitter BS1, one of which enters the photodetector PD1 after passing through the polarizer P1 to form a reference signal. The other channel is divided into two polarization states of P and S after passing through the polarization beam splitter PBS. The $\pm$1st-order diffracted light formed by the red S light with frequency $f_1$ entering the grating is reflected by the corner cube prisms RR1 and RR2, and then it re-enters the grating for diffraction and returns to the PBS. The blue P light with frequency $f_2$ enters the reference arm, it is split by the beam splitter BS2 and is reflected by the corner cube prism RR3, and then returns to the PBS. The P and S light pass through the polarizers P2 and P3, respectively, after being combined by the PBS, and they enter the photodetectors PD2 and PD3 to form measurement signals. The schematic diagram of the grating cross section used in the experiment is shown in Fig. 3. The duty cycle of the grating is 50 %, and the grating distance $d$ = 833.33 nm. According to the grating equation, the diffraction angle $\theta _i$ of $\pm$ 1 order diffracted light is 49.5$^{\circ }$ for vertical incidence.
Ideally, both the reference signal and the measurement signal are sinusoidal signals of a single frequency. The frequency of the reference signal is the frequency difference of the dual-frequency laser, and the frequency of the measurement signal is the sum of the frequency difference between the dual-frequency laser and the Doppler frequency shift generated when the grating moves. Fig. 4 is the frequency spectrum of the measurement signal when the motion table moves at a speed of 5 mm/s. The error components that are different from the ideal frequency components appear due to the imperfect optical device performance and installation angle errors.
Since the bi-quadrature lock-in amplification algorithm can effectively eliminate the uncertainty of the laser frequency difference, this study uses it as the phase solution algorithm [19]. After the phase calculation, the phase change between the reference and measurement signals caused by the displacement of the moving stage can be calculated. Fig. 5 shows the phase change when the motion table moves at a constant speed of 5 mm/s. It can be seen that, due to the presence of other frequency components in the measurement signal, nonlinear errors that vary periodically with the phase change appear in the phase data.
3. Error source analysis and nonlinear error model of grating interferometer
In order to isolate and compensate the nonlinear errors, it is necessary to fully understand the characteristics of nonlinear errors and build nonlinear error models. We analyze the sources of nonlinear errors and simulate each error individually.
In the interferometer used in this study, the laser light source consists of a beam of frequency $f_1$ and a beam of mutually orthogonal linearly polarized light of frequency $f_2$. The expression is as in Eq. (1), where $\varphi _{01}$ and $\varphi _{02}$ represent the initial phases of the two linearly polarized lights, and $E_{01}$ and $E_{02}$ represent the amplitudes of the two linearly polarized lights.
3.1 Analysis of error sources
In this part, we analyze the causes and characteristics of the four main sources of nonlinear errors as shown in Fig. 6.
3.1.1 Angle error of the x and y axes of the laser and the non-orthogonal error of the dual-frequency laser
As shown in Fig. 7, the coordinate system of the polarizing beam splitter is used as the absolute coordinate system, because there is a certain angular deviation between the x and y axes of the laser and the polarizing beam splitter. This results in the presence of an angle $\alpha$ and $\beta$ between the dual-frequency quadrature lasers $E_{01}$ and $E_{02}$ and the x and y axes of the polarizing beam splitter’s coordinate system, respectively. Moreover, because there is a certain non-orthogonal error between the two polarization states of the dual-frequency laser, $\alpha$ and $\beta$ are not equal.
Fig. 7. Schematic diagram of the influence of the laser x, y axis angle error and the non-orthogonal error of the dual-frequency laser.
The $E_{01}$ and $E_{02}$ lasers are decomposed on the coordinate system of the polarizing beam splitter to obtain $E_{1x}$ and $E_{1y}$. The expression is as in Eq. (2), where $\vec {i_x}$ and $\vec {j_y}$ represent the direction vectors of the optical axis of the polarizing beam splitter, respectively.
Fig. 8. Spectrum of the measurement signal in the presence of laser x, y axis angle error and the non-orthogonal error of the dual-frequency laser in the measurement system.
3.1.2 Light leakage error of polarizing beam splitter
Ideally, a polarizing beam splitter would completely separate the two frequencies of light according to their polarization states. The P light of the parallel polarization state is transmitted into the measurement arm, and the S light of the perpendicular polarization state is reflected into the reference arm. In practice, due to the unsatisfactory performance of the polarizing beam splitter, the polarizing beam splitter will have light leakage and cannot completely separate the two frequencies of light, as shown in Fig. 9. The P light is reflected into the reference arm, and the S light enters the measurement arm [20].
Fig. 9. Schematic diagram of the influence of light leakage from the polarizing beam splitter on the optical path.
In the polarization beam splitter, the transmittance and reflectivity of the parallel polarization state P light is $T_p$ and $R_p$, respectively, and the transmittance and reflectivity of the parallel polarization state S light is $T_s$ and $R_s$, respectively, where $T_p$ + $R_p$ = 1, $T_s$ + $R_s$ = 1, and the expressions of the light $E_{2x}$ and $E_{2y}$ emitted from the polarizing beam splitter are shown in Eq. (3).
As shown in Fig. 10, when there is a light leakage error of the polarization beam splitter in the measurement system, the light with frequencies $f_1$ and $f_2$ entering the reference arm shows interference, and the light entering the measurement arm with frequencies $f_1$ and $f_2$ also shows interference. This produces an error component at the same frequency as the dual-frequency laser frequency difference. When $T_s$ = $R_p$ = 0.1 and the strength of the ideal frequency component is 60 dB, the strength of the error frequency component reaches 40 dB.
Fig. 10. Spectrum of the measurement signal when there is a light leakage error from the polarization beam splitter in the measurement system.
3.1.3 Ghost reflection
Ghost reflections refer to non-ideal propagations of the optics inside an interferometer. When a laser beam transmits through an optical device, it will inevitably produce weak reflection light on its surface. As shown in Fig. 11, the reflected light in the dotted line is reflected multiple times inside the interferometer and superimposed with multiple Doppler frequency shifts, and finally enters the photodetector, resulting in nonlinear errors. When light enters the PBS through the measurement arm, most of it passes through the PBS. A small portion of the light is reflected back to the grating, passes through the grating several times along the optical path shown by the dotted line, and then enters the PBS.
The expressions of the two beams $\vec {E_{2x}'}$ and $\vec {E_{2y}'}$ after the optical path has been measured by the grating interferometer are shown in Eq. (4). Among them, $\varphi _1$ represents the phase difference caused by the reflection back to the polarization beam splitter through the reference arm corner prism; $\varphi _2$ represents the phase difference caused by normal reflection back to the polarization beam splitter after passing through the measurement arm corner prism; $\varphi _g$ represents the phase difference caused by the reflection back to the polarizing beam splitter after the measurement arm corner prism is reflected by the ghost; and $g$ represents the attenuation coefficient of the ghost reflection.
As shown in Fig. 12, in the presence of a ghost reflection error in the measurement system, part of the light with frequency $f_2$ in the measurement arm will pass through the measurement optical path multiple times. This produces a frequency error component with a Doppler shift that is three times greater than the ideal signal. When $g$ = 0.1 and the strength of the ideal frequency component is 60 dB, the strength of the error frequency component reaches 40 dB.
3.1.4 Laser z-axis angle error
As shown in Fig. 13, there is an angle $\gamma$ error between the z-axis of the laser and the polarizing beam splitter, and there is also an angle $\xi$ present between the grating and the polarizing beam splitter. This causes the ghost reflection to occur at different angles across the grating multiple times. The ghost reflection hits the grating for the first time along the red light path in Fig. 13. After being diffracted by the grating, the +1st-order diffracted light is reflected by the corner cube, and then it re-enters the grating and is diffracted, but is not completely transmitted when it enters the PBS. The reflected light enters the grating again along the blue light path, producing diffracted light. The light then returns to the PBS, as shown in green. It can be seen that the entry angles of the grating along the three different optical paths 1, 2 and 3 marked in the figure are different. The expressions of the angle $\theta _{iI}$ incident to the grating along the 1st and 3rd optical paths and the angle $\theta _{iII}$ incident to the grating along the 2nd optical path are shown in Eq. (5).
As shown in Fig. 14 , light enters the grating at an angle of $\theta _i$, and the generated +1st-order diffracted light exits at an angle of $\theta _{+1}$. When the reading head produces a relative movement $\Delta x$ along the grating x direction, the resulting optical path differences are $\Delta L_{ix}$ and $\Delta L_{ix}$. Meanwhile, when the reading head produces a relative movement $\Delta y$ along the y-direction of the grating, the resulting optical path differences are $\Delta L_{iy}$ and $\Delta L_{iy}$. At the same time, this will also cause a relative displacement $\Delta xz$ along the grating direction between the grating and the reading head. Through the derivation of the geometric relationship, the expression of the optical path difference is shown in Eq. (6).
Fig. 14. Relationship between the displacement of the grating relative to the reading head and the optical path difference.
As shown in Fig. 15, during the actual movement of the moving stage, it is inevitable that there will be simultaneous movements along the two directions of the grating x and z. At this time, the expressions of the phase differences $\Delta \varphi z$ and $\Delta \varphi x$ caused by the movement of the grating and the reading head in the x and z directions relative to the grating are shown in Eq. (7).
Fig. 15. Relative position change of the grating and the moving stage when the grating has both x and z displacements.
Since light normally passes through the measurement arm, it needs to undergo two diffractions, so the relationship between the Doppler frequency shifts $\Delta fx$ and $\Delta fz$ and the phase difference generated when the grating moves relative to the reading head along the x and y directions is defined by Eq. (8).
According to the grating equation, the relationship between the incident angle and the +1st-order diffracted light output angle can be obtained, with the expression shown in Eq. (9).
Simultaneously, Eqs. (7), (8) and (9) can obtain the relationship between the Doppler frequency shifts $\Delta f_x$ and $\Delta f_z$ and the incident angle when the grating moves relative to the reading head along the x and z directions. As shown in Eq. (10), the Doppler frequency shift generated by the relative motion of the grating relative to the reading head in the x direction is independent of the incident angle, whilst that in the z direction is related to the incident angle.
As shown in Eq. (5), the angle of the first and third incident gratings of ghost reflection is different from the angle of the second incident grating. As shown in Eq. (10), the Doppler frequency shift $\Delta f_x$ generated when the grating moves relative to the reading head along the x-direction can be obtained by substituting Eq. (5) into Eq. (10). When the grating moves relative to the reading head along the z direction, the first and third incident gratings of ghost reflection produce a Doppler frequency shift $\Delta f_{1z}$, and the second incident grating of ghost reflection produces a Doppler frequency shift $\Delta f_{2z}$.
According to the performance of the device and the machining and assembly accuracy of the existing equipment, we give the possible value range of each parameter in the nonlinear error source in Table 1.
Table 1. The parameters of the nonlinear error
3.2 Establishment of nonlinear error model of grating interferometer
Since the optical path of the $\pm$1st-order diffracted light of the reading head is completely symmetrical, the source of error is the same. Therefore, the comprehensive model is only established for the +1st-order diffracted light side of the reading head.
3.2.1 Establishment of nonlinear error model of the measured signal before phase solution
The dual-frequency laser is emitted from the laser and reaches the polarization beam splitter. Since the polarization beam splitter splits light according to the polarization state of the light, the two linearly polarized lights of different frequencies emitted by the laser are decomposed into the coordinate system of the PBS. In this coordinate system, the expressions of the light waves in the x-direction and the y-direction are shown in Eq. (12) and Eq. (13), respectively.
Due to the imperfect beam splitting of the polarizing beam splitter, the two beams of light are not completely separated according to the polarization state, resulting in light leakage. The expressions of light waves entering the measurement arm and the reference arm through the polarization beam splitter are shown in Eq. (14) and Eq. (15), respectively.
The expressions after the two beams have passed through the reference arm and the measurement arm are shown in Eq. (16) and Eq. (17), respectively. Among them, $\varphi _1$ represents the phase difference generated after passing through the reference arm, $\varphi _2$ represents the phase difference generated after normally passing through the measurement arm, $\varphi _g$ represents the phase difference generated by multiple passes through the measurement arm after ghost reflection, and $g$ represents the attenuation coefficient of ghost reflection.
When measuring light $\vec {E_{2y}}$ passing through the grating, due to the unsatisfactory performance of the grating, the linearly polarized light in the measurement light will change to elliptically polarized light in different degrees. As shown in Eq. (17), $\vec {E_{2y}}$ already contains four kinds of light with two frequencies of $f_1$ and $f_2$ and two polarization states of P and S. All this extra polarization change will only affect the strength of the terms, and will not create new error terms. We do not include this effect in the model.
For the convenience of formula derivation, the variables in Eq. (18) are replaced.
After reaching the polarizer, Eq. (16) and Eq. (17) are divided according to the different polarization states. The expressions of the light of the two polarization states are respectively shown in Eq. (19) and Eq. (20).
The light intensity generated by the final interference is presented in Eq. (21).
According to the frequency difference, the light intensity signal can be divided into several terms, such as Eqs. (22)–(30).
The DC component is filtered out in the processing circuit, as shown in Eq. (22).
The low-frequency components are expressed in Eq. (23). The parameters include the phase difference $\varphi _1 - \varphi _2$ generated by the light passing through the reference arm and the measurement arm, respectively, the term including the phase difference $\varphi _1 - \varphi _g$ generated by the light passing through the reference arm and the ghost reflection, respectively, and the term containing the phase difference $\varphi _2 - \varphi _g$ produced by the light passing through the measurement arm and the ghost reflection, respectively. Their frequencies are Doppler shifts produced by the grating once or several times. Compared with the frequency difference of the dual-frequency laser, the Doppler frequency shift produced by the general moving stage is small. The low-frequency components are filtered out in the processing circuit.
The reference component is shown in Eq. (24), whose frequency is the frequency difference $2\pi f_1t - 2\pi f_2t$ of the dual-frequency laser.
The reverse component is shown in Eq. (25). Compared with the ideal component, the frequency difference $2\pi f_1t - 2\pi f_2t$ of the dual-frequency laser is opposite in sign to the phase difference $\varphi _1 - \varphi _2$ produced by the light passing through the reference arm and the measurement arm. The frequency of the reverse component and the ideal component are symmetrical with the frequency difference of the dual-frequency laser.
The ideal component is shown in Eq. (26), including the frequency difference $2\pi f_1t - 2\pi f_2t$ of the dual-frequency laser and the phase difference $\varphi _1 - \varphi _2$ generated by the light passing through the reference arm and the measurement arm.
The forward ghost reflection component is presented in Eq. (27). Compared with the ideal component, when the light passes through the measurement arm, ghost reflection occurs, and the light passes through the measurement optical path many times, resulting in a phase difference from $\varphi _2$ to $\varphi _g$. The number of times the light enters the grating at the measurement arm is three times the ideal component, but the incident angle is not exactly the same each time, so there is a slight difference in the Doppler frequency shift produced. Therefore, the frequency difference between the forward ghost reflection component and the reference component is about three times the frequency difference between the ideal component and the reference component.
The inverse ghost reflection component is shown in Eq. (28). Compared with the forward ghost reflection component, the frequency difference of the dual-frequency laser is opposite in sign to the phase difference $\varphi _1 - \varphi _g$ produced by the light passing through the reference arm and the ghost reflection. The frequencies of the reverse ghost reflection component and the forward ghost reflection component are symmetrical with the frequency difference of the dual-frequency laser.
The positive variable component is displayed in Eq. (29), which includes the frequency difference of the dual-frequency laser and the phase difference $\varphi _g - \varphi _2$ generated by the ghost reflection and the measurement arm of the light, respectively. The frequency difference between the forward variable component and the reference component is about twice the frequency difference between the ideal component and the reference component.
The inverse variable component is expressed in Eq. (30). Compared with the forward variable component, the double-frequency laser frequency difference of the reverse variable component is opposite in sign to the phase difference $\varphi _g - \varphi _2$ generated by the ghost reflection and the measurement arm, respectively. The frequency of the reverse variable component and the forward variable component are symmetrical with the frequency difference of the dual-frequency laser.
As can be seen in Fig. 4, the forward ghost reflection component, the ideal component, the reference component, the reverse component, and the reverse ghost component have greater intensities. Therefore, only these five items are considered in the establishment of the subsequent algorithm model.
In order to simplify the derivation of the formula, the variable substitution of Eq. (31) is performed.
The relationship between the phase difference generated by the normal passing of the measurement arm and the phase difference generated by the ghost reflections passing through the measurement arm many times is expressed by Eq. (32). The expressions of $\Delta f_x$, $\Delta f_{1z}$, and $\Delta f_{2z}$ are shown in Eq. (11) above. The measurement light normally passes through the measurement arm once, and the Doppler frequency shifts $\Delta f_x$ and $\Delta f_{1z}$ caused by the movement of the grating in the x and y directions are superimposed. The ghost reflection three times through the measurement arm and superimposed three times Doppler frequency shift. Doppler frequency shifts due to the threetimes passes caused by the grating moving in the x direction is $3\Delta f_x$. Doppler frequency shifts due to the first and third passes caused by the grating moving in the z direction is $2\Delta f_1z$, and the second is $\Delta f_{2z}$.
It can be seen from the Eq. (11). When $\gamma$ is 0, the Doppler frequency shift in the z-direction superposition is the same when the ghost is passes three times into the measurement arm, that is $\Delta f_{1z}=\Delta f_{2z}$. The Doppler frequency shift of the three superimposed ghost reflections is $\Delta f_g=3\Delta f_x+2\Delta f_{1z}+\Delta f_{2z}\ =3\Delta f_x+3\Delta f_{1z}$. The frequency of the phase change is $\Delta f_i=\Delta f_x+\Delta f_{1z}$. The frequency of the nonlinear error due to ghost reflections is three times the phase change.
When $\gamma$ is not 0, $\Delta f_{1z}\neq \Delta f_{2z}$. The frequency of nonlinear errors due to ghost reflections no longer has a three-fold relationship with the frequency of phase changes. Since the main moving direction of the grating is the x-axis, $\Delta f_x>>\Delta f_{1z}$, $\Delta f_x>>\Delta f_{2z}$. Therefore, the frequency of the nonlinear error caused by the ghost reflection is close to three times the frequency of the phase change.
The reference signal is directly generated by the interference of the polarizer after the laser has emitted dual-frequency laser light. In addition, the frequency difference of the dual-frequency laser is not an ideal $2\pi f_1 - 2\pi f_2$, but has an uncertainty of $f'$. The expressions of the reference signal and the measurement signal are shown in Eq. (33) and Eq. (34), respectively.
3.2.2 Establishment of a nonlinear error model of the phase signal
Once the light intensity signal has been obtained, the phase difference between the reference signal and the measurement signal needs to be solved by a bi-quadrature lock-in amplification algorithm. The block diagram of this algorithm is shown in Fig. 16. To obtain the nonlinear error characteristics in the phase, the nonlinear error model in the bi-quadrature lock-in amplification algorithm is established. In order to simplify the derivation, variable substitution is performed, as shown in Eq. (35). Among them, $\Delta f_i$ represents the frequency of the Doppler frequency shift produced by the normal measurement optical path, and $\Delta f_g$ represents the frequency of the Doppler frequency shift produced by the ghost reflection.
The reference and measurement signals are each multiplied by a pair of quadrature signals produced by direct digital frequency synthesis (DDS). The filtered signals are shown in Eqs. (36), (37), (38), and (39), where $f_d$ is the frequency of a pair of quadrature signals generated by the DDS.
The frequency $f_d$ of the signal generated by the DDS inside the FPGA is set to be equal to the ideal frequency difference $f_1-f_2$ of the laser. In the phase difference module, Eq. (39) $\times$ Eq. (36) $-$ Eq. (38) $\times$ Eq. (37) and Eq. (39) $\times$ Eq. (37) $-$ Eq. (38) $\times$ Eq. (36) are calculated separately, and Eq. (40) can be obtained according to the sum-difference product formula.
In order to simplify the merging of terms for the same frequency, a substitution such as in Eq. (41) is performed. The combined result is shown in Eq. (42).
The phase signal with error after the arctangent calculation and phase accumulation based on Coordinate Rotation Digital Computer (CORDIC) is shown in Eq. (43) [21].
The nonlinear error can be expressed by Eq. (44).
4. Separation and compensation of nonlinear error in the phase of grating interferometer during uniform motion
Different from the laser interferometer, the frequency of nonlinear error caused by ghost reflection is no longer an integer multiple of the frequency of the ideal term, which makes it difficult to separate and compensate the nonlinear error.
When the Fourier transform is used to determine the relationship between the frequency variables, since the speed of the moving stage is not an ideal uniform speed during the sampling time, the peaks corresponding to each frequency component have a certain width, as shown in Fig. 17. Therefore, it is difficult to use the Fourier transform to determine the complex relationship between the ghost reflection component and the ideal component.
Fig. 17. Schematic diagram of the broadening of each peak in the spectrum caused by the frequency change during the sampling time.
Aiming at the aforementioned problem, a nonlinear error separation algorithm based on cross-correlation coefficient is proposed to quantitatively analyze the frequency of higher-order errors. A block diagram of the algorithm is shown in Fig. 18.
Fig. 18. Block diagram of nonlinear error separation algorithm based on cross-correlation coefficient.
As can be seen from Eq. (44), the frequency of the nonlinear error is $\Delta f_g - \Delta f_i$. Because the spectral performance of the PBS, the angular error between the laser and the PBS, the non-orthogonal error of the laser, and the intensity of ghost reflections do not change in the short term, the frequency-doubling relationship between the nonlinear error and the ideal phase remains unchanged. Therefore, the phase construction errors can be used to separate and compensate for nonlinear errors.
The phase data were collected when the motion table was moving at a constant speed, and the second-order curve fitting was performed on the phase using the least squares method. The phase with the nonlinear error is subtracted from the fitted curve to obtain the nonlinear error. We multiply the phase signal by a factor n, and perform sine and cosine calculations. Then, we multiply by $k\cos \varphi _{c}$ and $k\sin \varphi _{c}$, separately, and then subtract to obtain the construction error $f_c$. The expression for the construction error is shown in Eq. (45). We multiply the construction error and nonlinear error within the sampling time and then sum up to obtain the cross-correlation coefficient between the construction error and the nonlinear error. The cross-correlation coefficient reflects the degree of similarity between the two.
The values of $n$ and $\varphi _{c}$ are used as independent variables, and the cross-correlation coefficient is used as the dependent variable. We plot $n$ and $\varphi _{c}$ as a function of the cross-correlation coefficient. With the moving table moving at a uniform speed of 5 mm/s, the relationship between the cross-correlation coefficient of nonlinear error and structural error, and $n$ and $\varphi _{c}$ is shown in Fig. 19. When the cross-correlation coefficient reaches the maximum value, this means that the construction error and the error are the most similar at this time, and $n$ and $\varphi _{c}$ are the required parameters.
Fig. 19. Relationship between the cross-correlation coefficient of nonlinear error and structural error, and n and $\varphi _c$.
The phase and fitting curves at this time are shown in Fig. 20(a), and the nonlinear error and structural error are depicted in Fig. 20(b).
At this time, it can be considered that the frequency $n_{max}\Delta f_i$ of the structural error is equal to the frequency $\Delta f_g - \Delta f_i$ of the nonlinear error. The relationship between the frequency $\Delta f_g$ of the high-order error and the ideal phase signal frequency $\Delta f_i$ can be obtained, and the expression is shown in Eq. (46).
We have carried out several experiments, and the experiment uses an analog-to-digital conversion chip with a sampling rate of 160MHz, which can meet the requirement of collecting measurement signals of 20MHz$\pm$10MHz. The movement table was set to move at a speed of 5mm/s, and data of 65,536 points were sampled. Because nonlinear errors appear periodically with the phase change, when the velocity is low, the phase change within the sampling time is less than one period, which will lead to the failure of normal separation of nonlinear errors. At 65,536 sampling points and 0.41ms sampling time, when the movement speed of the motion platform is greater than 2.01mm/s, a complete period change of phase can be sampled, and the separation effect is good.In many experiments, the value of $n_{max} + 1$, the actual average speed and the actual traveling range are shown in Table 2.
Table 2. The value of $n_{max}+1$, the actual speed and traveling range of the motion table in multiple experiments
It can be seen that, since the number of ghost reflections passing through the grating is three times that of the normal measurement light, the frequency of high-order error is close to the frequency of the ideal interference signal by a factor of three. However, since the ghost reflection in the grating interferometer is affected by the angle between the grating and the PBS, and the moving speed in the x and y directions, the Doppler frequency shift superimposed when passing through the grating is different from the ideal interference light. The multiples of frequency between the higher-order errors and the ideal phase also change, but they are close three times.
In order to achieve the effect of compensating the nonlinear error, the phase signal with the nonlinear error is subtracted from the construction error. The nonlinear error, construction error and residual nonlinear error after compensation are shown in Fig. 21. The peak-to-peak value of the nonlinear error is reduced from 17.40 nm to 7.05 nm. The nonlinear error of 7.05 nm does not yet meet the measurement requirements of sub-nanometer precision, and its further suppression still requires the improvement of the spectral performance and transmittance of the polarizing beam splitter.
5. Conclusion
In this work, based on the source and generation mechanism of nonlinear error including ghost reflection in the heterodyne grating interferometer, a nonlinear error model of measurement signal is established. Based on the bi-quadrature lock-in amplification algorithm, an expression of the calculated phase is constructed. A method of determining the coefficient based on the cross-correlation coefficient is proposed, and the relationship between the higher-order error caused by ghost reflection and the frequency multiple of the ideal interference signal is analyzed by testing this method through multiple experiments. The results show that the Doppler frequency shift superimposed when the light reflected by the ghost reflection passes through the grating many times is different from that of the ideal light. The multiples of frequency between the higher-order errors and the ideal phase are close three times.
Funding
National Science and Technology Major Project (Grant No. 2017ZX02101007-003); National Natural Science Foundation of China (Grant No: 61965005); Natural Science Foundation of Guangxi Province (Grant No: 2019GXNSFDA185010); Guangxi Key Research and Development Program (Grant No: AB22035047); Open Fund Project of Shanghai Key Laboratory of Online Test and Control Technology (Grant No: ZX2021104); Guangxi Key Laboratory of Wireless Wideband Communication and Signal Processing, Guilin University of Electronic Technology (GXKL06200130).
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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