A real-time compensation algorithm for PGC nonlinear error based on sextuple harmonic

Abstract

The present invention discloses a PGC nonlinear error real-time compensation algorithm based on sextuple harmonics, which multiplies an interference signal with the sixth harmonic, obtains a low-frequency component through a low-pass filter, calculates an orthogonal signal, calculates a phase delay, calculates a signal amount without a phase delay error, calculates a modulation depth and compensates for a modulation depth error, calculates an interference phase, and introduces multiple harmonic components to achieve real-time compensation for phase delay and modulation depth nonlinear errors, solves the singularity problem existing in existing algorithms, and improves the accuracy and stability of phase solution in interferometric measurement.

Description

Technical Field

[0001] The present invention relates to the field of interference measurement technology, and in particular to a PGC nonlinear error real-time compensation algorithm based on sextuple harmonics. Background Art

[0002] The phase modulated carrier (PGC) method is widely used in the field of interferometry. The phase deconvolution algorithm of the phase modulated carrier (PGC) method directly determines the accuracy of its sensing technology.

[0003] Among the existing known solution algorithms, the PGC-Atan algorithm is a widely used phase separation method; the traditional PGC-Atan algorithm first uses the frequency of the modulation signal and the frequency of the double frequency to mix, and then extracts the DC value through filtering; however, the PGC-Atan algorithm directly calculates the tangent of the DC value after the frequency of the double frequency and the frequency of the double frequency; although this process is relatively simple, the amount of calculation required for the tangent operation is large; with the emergence of modern digital signal processors and the great improvement of their performance, the PGC-Atan algorithm has also been greatly developed and gradually applied. In actual use, the PGC-Atan algorithm requires that the modulation depth of the modulation signal be stable at 2.63 rad, and that there is no phase delay between the carrier signal and the demodulated carrier signal. When the modulation depth deviates from 2.63 rad or the phase delay deviates from 2kπ (k=0,±1,±2,…), the interference phase demodulation result is seriously distorted. Existing nonlinear error compensation algorithms generally have singular points, which makes these algorithms unable to solve under specific interference phases. For example: Anton V et al. compensated for phase delay and modulation depth respectively. However, when the interference phase tends to kπ / 2 (k=0, ±1, ±2, ...), the numerator and denominator in the division step of the algorithm will tend to 0, resulting in unstable algorithm results and a large amount of noise in the solution results, which is the algorithm singularity; Xie et al. proposed a demodulation algorithm using quadruple harmonics, which can compensate for phase delay and modulation depth error at the same time. However, when the measured phase tends to kπ / 2, the modulation depth compensation module in the algorithm still has singularities; similarly, Yan only uses triple harmonics to compensate for nonlinear errors, but the algorithm uses a differential module. When the phase changes slowly, the signal-to-noise ratio of the solution result is low, and when the interference phase tends to kπ / 2, the algorithm still has singularities in the modulation depth solution process.

[0004] In summary, the existing algorithms generally have singularity problems, which leads to a large number of unstable values ​​in the phase delay and modulation depth solution results; therefore, it is urgent to propose a singularity-free nonlinear error compensation algorithm to achieve high-precision phase solution of the PGC interferometer system. Summary of the invention

[0005] In view of the above-mentioned defects or deficiencies in the prior art, it is desirable to provide a PGC nonlinear error real-time compensation algorithm based on sextuple harmonic.

[0006] The present invention provides a PGC nonlinear error real-time compensation algorithm based on sextuple harmonic, comprising the following steps:

[0007] 1) The interference signal is multiplied by the sixth harmonic and the low-frequency component is obtained through a low-pass filter

[0008] The interference signal I(t) is multiplied by the sine signal sin(nω0) and cosine signal cos(nω0) of the 1st to 6th harmonics of the modulated carrier, and then low-pass filtered to obtain the low-frequency component R n and Q n , where n = 1, 2, 3, 4, 5, 6, as shown in the following relationship, where LPF means low-pass filtering;

[0009]

[0010]

[0011] 2) Calculate the orthogonal signal

[0012] The phase compensation module of this algorithm adopts R 1～4 , Q 1～4 Calculate θ err (t), so that the square terms of the sine function and the cosine function appear in both the denominator and the numerator, ensuring that the numerator and the denominator will not approach 0 at any time;

[0013] Calculate the combined product of the following four signals: (R1, Q3), (R3, Q1), (R2, Q4), (R4, Q2); the new signal S1 can be obtained by combining R1Q3-R3Q1+R2Q4-R4Q2;

[0014] Similarly, calculate the combined product of the following four signals: (R1, R3), (R2, R4), (Q1, Q3), (Q2, Q4), and obtain S2 according to -R1R3-R2R4-Q1Q3-Q2Q4, as shown in the following relationship;

[0015]

[0016]

[0017] 3) Calculate phase delay

[0018] From the calculation results of the relationship in step 2), it can be seen that S1 and S2 constitute 2θ err (t) is a pair of orthogonal components, so θ can be solved using the following relationship: err(t);

[0019] Since the coefficients of the numerator and denominator are are not 0, so the algorithm can accurately solve θ err (t);

[0020]

[0021] 4) Calculate the signal quantity without phase delay error

[0022] Solve the signal quantity G without phase delay error n , where n = 1, 2, 3, 4, 5, 6, as shown in the following relationship; this step can avoid singular points in the solution process;

[0023]

[0024] 5) Calculate the modulation depth

[0025] According to the recursive relationship of the first kind of Bessel function, the phase modulation depth is calculated using the following relationship; it can be seen that the coefficients of equations c1(t) and c2(t) are orthogonal, that is, for any At least one of the equations c1(t) and c2(t) is far away from the singularity, that is, there is always a relation that can accurately calculate c(t); through the equation c(t), a relation that is farther away from the singularity can be selected for calculation;

[0026]

[0027]

[0028]

[0029] 6) Compensate for modulation depth error and calculate interference phase

[0030] Compensate for the modulation depth error in G1 and G2 to obtain a pair of strictly orthogonal signals, and use the four-quadrant inverse tangent method to obtain The relationship is shown below;

[0031]

[0032] Through the above steps, the nonlinear error in the PGC system can be compensated in real time without singularity, and the interference phase can be accurately calculated.

[0033] Compared with the prior art, the beneficial effects of the present invention are:

[0034] The PGC nonlinear error real-time compensation algorithm based on sextuple harmonics of the present invention can solve θ in real time by introducing multiple harmonic components and using six pairs of orthogonal signals to demodulate the phase.err (t) and c(t) can realize real-time compensation for phase delay and modulation depth nonlinear errors, solve the singularity problem of existing algorithms, improve the accuracy and stability of phase solution in interferometric measurement, and thus compensate for the nonlinear errors in the demodulated phase.

[0035] It should be understood that the contents described in the summary of the invention are not intended to limit the key or important features of the embodiments of the present invention, nor are they intended to limit the scope of the present invention. Other features of the present invention will become easily understood through the following description. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Other features, objects and advantages of the present invention will become more apparent from the detailed description of non-limiting embodiments made with reference to the following drawings:

[0037] Figure 1 It is a flow chart of the real-time compensation algorithm for nonlinear error of phase modulated carrier interferometry of the present invention;

[0038] Figure 2 is the correlation diagram of the Mach-Zehnder interference optical path;

[0039] Figure 3 It is a correlation diagram of the solution results of the algorithm of the present invention and the solution results of the PGC-Atan algorithm;

[0040] Figure 4 This is the correlation diagram of the error solved by the PGC-Atan algorithm;

[0041] Figure 5 This is a correlation diagram of the error solved by the algorithm of the present invention. DETAILED DESCRIPTION

[0042] The present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. It is to be understood that the specific embodiments described herein are only used to explain the relevant invention, rather than to limit the invention. It should also be noted that, for ease of description, only the parts related to the invention are shown in the accompanying drawings.

[0043] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other. The present invention will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0044] Please refer to Figures 1 to 5 An embodiment of the present invention provides a PGC nonlinear error real-time compensation algorithm based on sextuple harmonics. The algorithm realizes real-time compensation of phase delay and modulation depth nonlinear errors by introducing multi-harmonic components, solves the singularity problem existing in the existing algorithm, and improves the accuracy and stability of phase solution in interferometric measurement.

[0045] According to the PGC measurement principle, the reference arm or measurement arm of the interferometer is modulated by Ccosω0t. The obtained interference light intensity expression is shown in formula (1), where A is the DC component, B is the amplitude of the AC component, C is the carrier modulation depth, ω0 is the modulation frequency, It is the phase change caused by the combined effect of the optical path difference of the interferometer arm and the environmental disturbance.

[0046] According to the PGC-Atan algorithm, the fundamental wave and the second harmonic of the carrier are multiplied with I(t) respectively, and two phase-orthogonal signals Ix(t) and Iy(t) are obtained through low-pass filtering.

[0047] When C is 2.63 rad, the inverse tangent operation is used to obtain As shown in formula (2) to (4), where J n (where n is an integer greater than 0) are the coefficients of the Bessel function of the first kind.

[0048]

[0049] However, in actual scenarios, since the modulation device is easily affected by the environment, the modulation depth cannot be stabilized at 2.63 rad, and its true value c(t) will change over time. In addition, since the phase modulation device will introduce phase delay noise θ err (t).

[0050] c(t) and θ err (t) will result in the phase actually solved by the PGC-Atan algorithm There is a nonlinear error, as shown in equation (5).

[0051]

[0052] In order to compensate the nonlinear error in real time and without singularity, the present invention proposes a six-harmonic solution method. The algorithm uses six pairs of orthogonal signals to demodulate the phase and can solve θ in real time. err (t) and c(t), thereby compensating for the nonlinear error in the demodulated phase; the specific PGC nonlinear error real-time compensation algorithm based on sextuple harmonic includes the following steps:

[0053] 1) The interference signal is multiplied by the sixth harmonic and the low-frequency component is obtained through a low-pass filter

[0054] The interference signal I(t) is multiplied by the sine signal sin(nω0) and cosine signal cos(nω0) of the 1st to 6th harmonics of the modulated carrier, and then low-pass filtered to obtain the low-frequency component R n and Q n , where n = 1, 2, 3, 4, 5, 6, as shown in the following equation (6), where LPF represents low-pass filtering;

[0055]

[0056] 2) Calculate the orthogonal signal

[0057] The phase compensation module of this algorithm adopts R 1～4 , Q 1～4 Calculate θ err (t), so that the square terms of the sine function and the cosine function appear in both the denominator and the numerator, ensuring that the numerator and the denominator will not approach 0 at any time;

[0058] Calculate the combined product of the following four signals: (R1, Q3), (R3, Q1), (R2, Q4), (R4, Q2); the new signal S1 can be obtained by combining R1Q3-R3Q1+R2Q4-R4Q2;

[0059] Similarly, calculate the combined product of the following four signals: (R1, R3), (R2, R4), (Q1, Q3), (Q2, Q4), and obtain S2 according to -R1R3-R2R4-Q1Q3-Q2Q4, as shown in the following relations (7) and (8);

[0060]

[0061] 3) Calculate phase delay

[0062] From the calculation results of equations (7) and (8) in step 2), it can be seen that S1 and S2 constitute 2θ err (t) is a pair of orthogonal components, so we can use the following relationship (9) to solve θ err (t);

[0063] Since the coefficients of the numerator and denominator are are not 0, so the algorithm can accurately solve θ err (t);

[0064]

[0065] 4) Calculate the signal quantity without phase delay error

[0066] Solve the signal quantity G without phase delay error n , where n = 1, 2, 3, 4, 5, 6, as shown in the following equation (10); this step can avoid singular points in the solution process;

[0067]

[0068] 5) Calculate the modulation depth c(t)

[0069] According to the recursive relationship of the first kind of Bessel function, the phase modulation depth is calculated using the following relationship (11) to (13). It can be seen that the coefficients of equation (11) and equation (12) are orthogonal, that is, for any At least one of equations (11) and (12) is far away from the singularity, that is, there is always a relation that can accurately solve c(t); through equation (13), a relation that is farther away from the singularity can be selected for calculation;

[0070]

[0071] 6) Compensate for modulation depth error and calculate interference phase

[0072] Compensate for the modulation depth error in G1 and G2 to obtain a pair of strictly orthogonal signals, and use the four-quadrant inverse tangent method to obtain As shown in the following relationship (14);

[0073]

[0074] Through the above steps, the nonlinear error in the PGC system can be compensated in real time without singularity, and the interference phase can be accurately calculated.

[0075] The following is the experimental verification process and results of this application algorithm:

[0076] The Mach-Zehnder interference optical path is used to realize the PGC displacement measurement system. Figure 2 shown.

[0077] The interference signal S(t) is expressed in equation (15), through which the algorithm of this application is verified. s , B s , C s , f s ,θ s , They are the interference DC component, AC component amplitude, modulation depth, modulation frequency, phase delay error and the signal to be measured.

[0078] Set the modulation frequency to 1khz and the sampling frequency to 20khz.

[0079] In this embodiment, the electro-optic modulator modulates the actual depth C s is 3.2rad, the phase delay error θ s It is 43°, which deviates greatly from the ideal measurement condition required by the PGC-Atan algorithm. Therefore, the measurement result will contain serious nonlinear errors.

[0080] The system is used to measure 100μm / s linear motion and the experimental results show that the proposed algorithm can effectively compensate for the nonlinear error in the PGC algorithm and improve the robustness and measurement accuracy of the interference system.

[0081]

[0082] The specific solution process is as follows Figure 1 As shown:

[0083] 1. First, multiply the sine and cosine signals of the 1st to 6th harmonics of the carrier, and obtain the low-frequency signal component R through low-pass filtering n and Q n (n=1,2,...,6).

[0084] 2. Calculate θ through the phase delay error compensation module s It is 43±0.026°, and the deviation from the true value is less than 0.026°;

[0085] Solve the signal quantity G without phase delay error according to the phase delay error at each moment n (n=1,2,…,6).

[0086] 3. Then, C is calculated by the modulation depth calculation module. s It is 3.2±0.0025rad, and the deviation from the true value is less than 0.025rad;

[0087] Compensate for the modulation depth error in G1 and G2 to obtain a pair of strictly orthogonal signals, and use the four-quadrant inverse tangent method to obtain

[0088] according to The displacement to be measured can be obtained according to formula (16), where λ is the laser wavelength.

[0089] In order to compare the error compensation effect of the algorithm in this paper, the PGC-Atan algorithm is used for calculation. The settlement results of the two are as follows: Figure 3 As shown, it can be found that the PGC-Atan algorithm result contains serious nonlinear errors and the measurement results have obvious distortion, while the algorithm result of the present invention can effectively compensate for the nonlinear errors and the measurement results have good linearity.

[0090] The nonlinear error of the PGC-Atan algorithm is about 52.95nm, and the linear error of the measurement result is as follows Figure 4 As shown, the nonlinear error of the algorithm of the present invention is about 0.14nm, which can effectively compensate for the nonlinear error such as Figure 5 shown.

[0091]

[0092] The sextuple harmonic-based PGC nonlinear error real-time compensation algorithm of the present application realizes real-time compensation of phase delay and modulation depth nonlinear errors by introducing multi-harmonic components, solves the singularity problem existing in the existing algorithm, and improves the accuracy and stability of phase solution in interferometric measurement.

[0093] In the description of this specification, the terms "connection", "installation", "fixation" and the like should be understood in a broad sense. For example, "connection" can be a fixed connection, a detachable connection, or an integral connection; it can be a direct connection or an indirect connection through an intermediate medium. For ordinary technicians in this field, the specific meanings of the above terms in this application can be understood according to specific circumstances.

[0094] In the description of this specification, the description of the terms "one embodiment", "some embodiments", etc. means that the specific features, structures, materials or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present application. In this specification, the schematic representation of the above terms does not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described can be combined in any one or more embodiments or examples in a suitable manner.

[0095] The above are only preferred embodiments of the present application and are not intended to limit the present application. For those skilled in the art, the present application may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A real-time compensation algorithm for PGC nonlinear error based on sextuple harmonic, characterized in that: The steps include: 1) The interference signal is multiplied by the sixth harmonic and the low-frequency component is obtained through a low-pass filter The interference signal I(t) is multiplied by the sine signal sin(nω0) and cosine signal cos(nω0) of the 1st to 6th harmonics of the modulated carrier, and then low-pass filtered to obtain the low-frequency component R n and Q n , where n = 1, 2, 3, 4, 5, 6, as shown in the following relationship, where LPF means low-pass filtering; 2) Calculate the orthogonal signal The phase compensation module uses R1, R2, R3, R4, Q1, Q2, Q3, Q4 to calculate θ err (t), so that the square terms of the sine function and the cosine function appear in both the denominator and the numerator, ensuring that the numerator and the denominator will not approach 0 at any time; Calculate the combined product of the following four signals: (R1, Q3), (R3, Q1), (R2, Q4), (R4, Q2); the new signal S1 can be obtained by combining R1Q3-R3Q1+R2Q4-R4Q2; Similarly, calculate the combined product of the following four signals: (R1, R3), (R2, R4), (Q1, Q3), (Q2, Q4), and obtain S2 according to -R1R3-R2R4-Q1Q3-Q2Q4, as shown in the following relationship; 3) Calculate phase delay From the calculation results of the relationship in step 2), it can be seen that S1 and S2 constitute 2θ err (t) is a pair of orthogonal components, so θ can be solved using the following relationship: err (t); Since the coefficients of the numerator and denominator are are not 0, so the algorithm can accurately solve θ err (t); 4) Calculate the signal quantity without phase delay error Solve the signal quantity G without phase delay error n , where n = 1, 2, 3, 4, 5, 6, 5) Calculate the modulation depth According to the recursive relationship of the first kind of Bessel function, the phase modulation depth is calculated using the following relationship; it can be seen that the coefficients of equations c1(t) and c2(t) are orthogonal, that is, for any At least one of the equations c1(t) and c2(t) is far away from the singularity, that is, there is always a relation that can accurately calculate c(t); through the equation c(t), a relation that is farther away from the singularity can be selected for calculation; 6) Compensate for modulation depth error and calculate interference phase Compensate for the modulation depth error in G1 and G2 to obtain a pair of strictly orthogonal signals, and use the four-quadrant inverse tangent method to obtain The relationship is as follows: Through the above steps, the nonlinear error in the PGC system can be compensated in real time without singularity, and the interference phase can be accurately calculated.

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