High dynamic detection method of nonlinear error of grating interferometer quadrature signal based on state machine duty cycle

Abstract

The present invention relates to a high-dynamic detection method for nonlinear errors of orthogonal signals of a grating interferometer based on a state machine duty cycle, and belongs to the field of optical error detection technology. In order to further reduce the influence of nonlinear errors on phase solution results, the present invention proposes a high-dynamic detection method for nonlinear errors of orthogonal signals of a grating interferometer based on a state machine duty cycle, comprising: step 1: calculation of DC bias and amplitude error; step 2: measurement of non-orthogonal error. The high-dynamic detection method for nonlinear errors of orthogonal signals of a grating interferometer based on a state machine duty cycle of the present invention can quickly and sequentially derive the magnitude of amplitude error, DC bias, and non-orthogonal error by measuring the time ratio of the signal under different states, thereby providing a basis for the subsequent nonlinear error suppression process.

Description

Technical Field

[0001] The present invention belongs to the technical field of optical error detection, and in particular relates to a high-dynamic detection method for nonlinear errors of orthogonal signals of a grating interferometer based on a state machine duty cycle. Background Art

[0002] The rapid development of industrial sectors such as high-end manufacturing equipment, the semiconductor industry, and precision machine tools has placed higher demands on precision displacement measurement systems. Grating displacement measurement systems, particularly those based on the principle of diffraction interferometry, are becoming the preferred choice for an increasing number of modern machines due to their high accuracy, resistance to environmental influences, large measurable range, and compact size.

[0003] The grating precision displacement measurement system uses the interference diffraction and phase shift interference of the grating to convert the spatial displacement information into four trigonometric function signals with a phase difference of π / 2, and obtains two orthogonal trigonometric function signals by adding and subtracting analog signals, namely:

[0004] Signal1=sinwt(1)

[0005] Signal2=coswt(2)

[0006] Ideally, the two trigonometric function signals have equal amplitudes, a DC bias of 0, and a phase difference of π / 2. The phase change of the trigonometric function is proportional to the displacement distance, and the frequency of the trigonometric function is proportional to the speed of the object being measured. However, in practical applications, the signals are subject to nonlinear errors, including amplitude error, DC bias, and non-orthogonality, due to factors such as angular errors in the x- and y-axes of the optical system laser, light leakage from the polarization beam splitter (PBS) causing light from the reference arm to mix into the optical path of the measurement arm, readhead alignment errors, grating surface errors, poor consistency in grating diffraction efficiency, and photoelectric converter characteristics. Regardless of the method used, such as the cordic algorithm, the constructor method, or the phase shift method, these nonlinear errors will severely affect the accuracy of the subsequent phase solution to varying degrees.

[0007] To reduce the impact of non - linear errors, researchers have proposed two solutions. The first is to use a special subdivision method to reduce the influence of non - linear errors on the results. However, the traditional method in the special subdivision method has limited improvement in signal quality and complex calculations, while the special subdivision method using machine learning will significantly reduce the calculation speed and cannot meet the industrial requirements for speed. The second is to suppress non - linear errors and use a more ideal signal in subsequent subdivisions. For example, Heydemann proposed a method for calculating non - linear errors based on the least - squares method. Birch first applied the Heydemann method to the measurement of optical interference signals. However, the least - squares method has complex calculations, and most related research aims to optimize the matrix of the least - squares method. There are also studies that have improved the sampling calculation rules of the Heydemann method.

[0008] The above - mentioned Heydemann method has the following four problems. First, it is insensitive to the signal frequency change caused by the change of displacement acceleration, and there may be a problem of mismatch between the sampling frequency and the signal frequency, reducing the accuracy of non - linear error detection. Second, this method requires a large amount of signal data to obtain more accurate results, increasing the delay and offsetting the advantage of using a high - line - density grating displacement measurement. Third, after optimization, the least - squares method still has complex calculations, and the system is an open - loop system, which may cause oscillations. Fourth, this method has high requirements for sampling accuracy, otherwise it will reduce the accuracy of non - linear error detection. Summary of the Invention

[0009] The present invention aims to solve the technical problems in the prior art and provides a high - dynamic detection method for non - linear errors of orthogonal signals of a grating interferometer based on the duty cycle of a state machine. By measuring the time ratio of different states of the signal, the magnitudes of amplitude error, DC offset, and non - orthogonality error can be quickly obtained in sequence, providing a basis for the subsequent non - linear error suppression process.

[0010] To solve the above - mentioned technical problems, the technical solution of the present invention is as follows:

[0011] A high - dynamic detection method for non - linear errors of orthogonal signals of a grating interferometer based on the duty cycle of a state machine, comprising the following steps:

[0012] Record the intersection times of the signal y = A*sin(ωt)+B with two references, one reference is a positive number p1 and the other reference is a positive number p2, and p1 < p2; five times t1 to t5 divide the signal into four states;

[0013]

[0014] Step 1: Calculation of DC offset and amplitude error

[0015] The DC offset and amplitude errors of the two signals are calculated separately. For one signal, the DC offset and amplitude errors are obtained by using the difference between the interval ratio at time t1 and the duty cycle of the ideal signal.

[0016] The calculation process is as follows:

[0017] The period of the signal is:

[0018]

[0019] Among them, ω is the angular velocity of the trigonometric function signal;

[0020] When y = A*sin(ωt) + B = p1, where A is the gain error and B is the zero offset;

[0021]

[0022]

[0023]

[0024] Subtracting the three from each other yields:

[0025]

[0026]

[0027] so:

[0028]

[0029] When y=A*sin(ωt)+B=p2,

[0030]

[0031]

[0032] Similarly, we can get:

[0033]

[0034] make:

[0035]

[0036]

[0037] Substituting (13) and (14) into (9) and (12), we obtain:

[0038]

[0039]

[0040] Step 2: Measurement of non-orthogonality error

[0041] The magnitude of the non-orthogonal error needs to consider the duty cycle of the two signals passing through the same reference, and compare it with the duty cycle of the standard orthogonal signal to obtain the magnitude of the nonlinear error.

[0042] The process is as follows:

[0043] When the signal has no nonlinear error,

[0044]

[0045] but:

[0046]

[0047]

[0048] make:

[0049]

[0050] but:

[0051]

[0052] If there is only non-orthogonal error in the signal, the magnitude of the non-orthogonal error is:

[0053]

[0054] However, under the influence of amplitude error and DC offset, the measured t1, t4, t4' and t1 when only non-orthogonal error exists are different. real 、t4 real 、t4' real There are differences between them, so it is necessary to compensate the measured t1, t4, and t4' and then calculate the non-orthogonality error;

[0055] Substituting equations (15) and (16) into (4), we obtain:

[0056]

[0057] If there is no amplitude error and DC offset,

[0058]

[0059] Where k is a natural number;

[0060] so,

[0061] Substituting equations (17) and (18) into (5), we obtain:

[0062]

[0063] If there is no amplitude error and DC offset,

[0064]

[0065] Where k is a natural number;

[0066] so,

[0067] Similarly, another letter:

[0068]

[0069] Substituting (25), (28), and (29) into (20), we obtain:

[0070]

[0071]

[0072] Substituting equations (20), (30) and (31) into (22), we can obtain the magnitude of the non-orthogonal error:

[0073]

[0074] From this point on, the magnitude of the DC offset, amplitude error, and non-orthogonality error are all calculated using the duty cycle of several state machines.

[0075] In the above technical solution, p1=0, p2=0.5.

[0076] In the above technical solution, the detection method is applicable to Matlab.

[0077] The beneficial effects of the present invention are:

[0078] In order to further reduce the impact of nonlinear errors on the phase solution results, the present invention proposes a high-dynamic detection method for nonlinear errors of orthogonal signals of grating interferometers based on the duty cycle of a state machine. This detection method can quickly and sequentially obtain the magnitude of amplitude error, DC bias, and non-orthogonality error by measuring the time proportion of the signal under different states, providing a basis for the subsequent nonlinear error suppression process. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0080] Figure 1 This is an amplitude-time curve diagram when the method of the present invention calculates DC offset and amplitude error.

[0081] Figure 2 The amplitude-time curve graph of the method of the present invention when measuring non-orthogonal error.

[0082] Figure 3 The displacement deviation graphs finally caused by using and not using the method of the present invention under Gaussian noise.

[0083] Figure 4 The displacement deviation graphs caused by using and not using the method of the present invention at different signal frequencies.

[0084] Figure 5 The displacement deviation graphs caused by using and not using the method of the present invention when the numerical value of the signal non-linear error changes.

[0085] Figure 6 The measurement deviation graphs caused by using and not using the method of the present invention under actual measurement. Detailed implementation manners

[0086] The present invention will be further described in detail in the following through the detailed implementation manners part.

[0087] A high-dynamic detection method for non-linear error of orthogonal signals of a grating interferometer based on the duty cycle of a state machine according to the present invention includes the following steps:

[0088] Record the intersection times of the signal y = A*sin(ωt) + B and two references, one reference is the positive number p1, and the other reference is the positive number p2, and p1 < p2; the signal is divided into four states at 5 moments t1 to t5 (see Table 1 and Figure 1 ).

[0089] Table 1

[0090]

[0091] Step 1: Calculation of DC offset and amplitude error

[0092] The DC offset and amplitude error of the two signals are calculated separately; for one signal, the difference between the interval ratio between t1 moments and the duty cycle of the ideal signal is used to obtain the magnitudes of the DC offset and amplitude error;

[0093] The calculation process is as follows:

[0094] The period of the signal is:

[0095]

[0096] where ω is the angular velocity of the trigonometric function signal;

[0097] When y = A*sin(ωt) + B = p1, where A is the gain error and B is the zero offset;

[0098]

[0099]

[0100]

[0101] Subtracting the three from each other yields:

[0102]

[0103]

[0104] so:

[0105]

[0106] When y=A*sin(ωt)+B=p2,

[0107]

[0108]

[0109] Similarly, we can get:

[0110]

[0111] make:

[0112]

[0113]

[0114] Substituting (13) and (14) into (9) and (12), we obtain:

[0115]

[0116]

[0117] Step 2: Measurement of non-orthogonality error

[0118] The magnitude of the non-orthogonal error needs to consider the duty cycle of the two signals passing through the same reference, and compare it with the duty cycle of the standard orthogonal signal to obtain the magnitude of the nonlinear error.

[0119] The process is as follows:

[0120] When the signal has no nonlinear errors (see Figure 2 ),

[0121]

[0122] but:

[0123]

[0124]

[0125] make:

[0126]

[0127] but:

[0128]

[0129] If there is only non-orthogonal error in the signal, the magnitude of the non-orthogonal error is:

[0130]

[0131] However, under the influence of amplitude error and DC offset, the measured t1, t4, t4' and t1 when only non-orthogonal error exists are different. real 、t4 real 、t4' real There are differences between them, so it is necessary to compensate the measured t1, t4, and t4' and then calculate the non-orthogonality error;

[0132] Substituting equations (15) and (16) into (4), we obtain:

[0133]

[0134] If there is no amplitude error and DC offset,

[0135]

[0136] Where k is a natural number;

[0137] so, Substituting equations (17) and (18) into (5), we obtain:

[0138]

[0139] If there is no amplitude error and DC offset,

[0140]

[0141] Where k is a natural number;

[0142] so, Similarly, another letter:

[0143]

[0144] Substituting (25), (28), and (29) into (20), we obtain:

[0145]

[0146]

[0147] Substituting equations (20), (30) and (31) into (22), we can obtain the magnitude of the non-orthogonal error:

[0148]

[0149] From this point on, the magnitude of the DC offset, amplitude error, and non-orthogonality error are all calculated using the duty cycle of several state machines.

[0150] Error analysis of the high-dynamic detection method for nonlinear error of grating interferometer orthogonal signals based on the state machine duty cycle of the present invention:

[0151] In the process of implementing the method proposed in the present invention by software and hardware, two factors may cause errors and reduce the accuracy of calculation: the difference between the theoretical threshold and the actual threshold used to detect the state and the time sampling error.

[0152] 1. The difference between the theoretical threshold and the actual threshold used to detect the state

[0153] The difference in threshold values ​​may be caused by the deviation of the comparator threshold level and the deviation of the actual p1 and p2 values.

[0154] When the threshold is p1, the actual threshold is p1+e p1 ; When the threshold is p2, the actual threshold is p2+e p2 ;

[0155] When y=A*sin(ωt)+B=p1+e p1 hour,

[0156]

[0157]

[0158]

[0159] When y=A*sin(ωt)+B=p2+e p2 hour,

[0160]

[0161]

[0162] so:

[0163]

[0164]

[0165] make:

[0166]

[0167]

[0168] Simplifying, we get:

[0169]

[0170]

[0171] Substituting (42) and (43) into (15) and (16), we obtain:

[0172]

[0173]

[0174] In the calculation of non-orthogonal error, the change of p1 will change the size of t4 and t4', but for T p =t4-t4' has no effect, so e p1 There is no impact on non-orthogonality errors.

[0175] 2. Time sampling error

[0176] make:

[0177]

[0178] Since the relationship between the intersection moment and the calculated amplitude, DC bias, and non-orthogonal error is complex, the present invention uses partial derivatives to obtain the influence of each intersection moment with time sampling error on the amplitude, DC bias, and non-orthogonal error.

[0179]

[0180]

[0181]

[0182]

[0183]

[0184] Arrange B and we get:

[0185]

[0186] but:

[0187]

[0188]

[0189]

[0190]

[0191]

[0192]

[0193]

[0194]

[0195]

[0196] Experiments and results:

[0197] In the experiment, based on the above analysis and for ease of practical application, p1 = 0 and p2 = 0.5 were set. To verify the feasibility of the proposed method, the stage was stepped at a speed of 0.01 mm / s, two orthogonal signals were recorded, and the displacement at each moment was measured using an interferometer. After low-pass filtering, the collected signals were used in MATLAB to measure the nonlinear error using both the proposed method and the improved Heidemann method. The nonlinear error was then corrected using the same method.

[0198] Figure 3 Graphs showing the displacement deviations caused by using and not using the method of the present invention under Gaussian noise show that the method of the present invention is insensitive to Gaussian noise and can reduce the 55nm displacement measurement error caused by Gaussian noise to 0.54nm.

[0199] As mentioned above, time measurement error will affect the accuracy of the method of the present invention, so a simulation of different frequency signals at the same sampling frequency is designed. Figure 4 The figure shows the displacement deviation caused by using and not using the method of the present invention at different signal frequencies. It can be seen from the figure that the accuracy of the method of the present invention is better at low frequencies. Compared with not using the method of the present invention, the accuracy is improved by at least two orders of magnitude.

[0200] Figure 5 The displacement deviation diagram is a graph showing the numerical change of the signal nonlinear error when the method of the present invention is used and not used. The systematic error of the optical system will cause the numerical change of the signal nonlinear error. Figure 5 It can be seen that the method of the present invention can significantly reduce the displacement measurement deviation caused by the system error of the optical system.

[0201] Figure 6 The figure shows the measurement deviation caused by using and not using the method of the present invention in actual measurement. It can be seen from the figure that using the method of the present invention in the actual measurement experiment can reduce the measurement deviation from 100nm to 30nm.

[0202] Obviously, the above embodiments are merely examples for clarity of explanation and are not intended to limit the implementation methods. Those skilled in the art will readily appreciate that other variations or modifications based on the above descriptions are possible. It is not necessary and impossible to enumerate all implementation methods here. Obvious variations or modifications arising therefrom remain within the scope of protection of the present invention.

Claims

1. A high-dynamic detection method for nonlinear error of grating interferometer orthogonal signals based on state machine duty cycle, characterized in that: It includes the following steps: Record the intersection times of the signal y = A*sin(ωt) + B with two references, where one reference is the positive number p1 and the other reference is the positive number p2, and p1 < p2; the signal is divided into four states at the five moments t1 to t5; Step 1: Calculation of DC offset and amplitude error The DC offset and amplitude error of the two signals are calculated separately; for one signal, the DC offset and amplitude error are obtained by using the difference between the interval ratio between t1 moments and the duty cycle of the ideal signal; The calculation process is as follows: The period of the signal is: where ω is the angular velocity of the trigonometric function signal; When y = A*sin(ωt) + B = p1, where A is the gain error and B is the zero offset; Subtracting them pairwise gives: So: When y = A*sin(ωt) + B = p2, Similarly, we can get: Let: Substitute (13) and (14) into (9) and (12), we get: Step 2: Measurement of non-orthogonal error The magnitude of the non-orthogonal error needs to consider the duty cycle formed by the two signals passing through the same reference, and the magnitude of the non-linear error is obtained by comparing with the duty cycle of the standard orthogonal signal; The process is as follows: When there is no non-linear error in the signal, Then: Let: Then: If there is only non-orthogonal error in the signal, the magnitude of the non-orthogonal error is: However, under the influence of amplitude error and DC offset, the measured t1, t4, t4' and t1 when only non-orthogonal error exists are different. real 、t4 real 、t4' real There are differences between them, so it is necessary to compensate the measured t1, t4, and t4' and then calculate the non-orthogonality error; Substitute equations (15), (16), (13), (3) into (4), we get: If there is no amplitude error and DC offset, where k is a natural number; so, Substitute equations (15), (16), (13), (3) into (5), we get: If there is no amplitude error and DC offset, where k is a natural number; so, Similarly, another signal: Substitute (25), (28), (29) into (20), we get: Substitute equations (21), (30) and (31) into (22), we get the magnitude of the non-orthogonal error: Since then, the magnitudes of the DC offset, amplitude error, and non-orthogonal error are all calculated through the duty cycles of several state machines.

2. The high-dynamic detection method for nonlinear error of grating interferometer orthogonal signals based on state machine duty cycle according to claim 1, characterized in that: p1 = 0, p2 = 0.

5.

3. The high-dynamic detection method for nonlinear error of grating interferometer orthogonal signals based on state machine duty cycle according to claim 1 or 2, characterized in that: The detection method is applicable to matlab.

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