CN116659686A - Wavefront reconstruction method and device - Google Patents

Wavefront reconstruction method and device Download PDF

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CN116659686A
CN116659686A CN202310477232.5A CN202310477232A CN116659686A CN 116659686 A CN116659686 A CN 116659686A CN 202310477232 A CN202310477232 A CN 202310477232A CN 116659686 A CN116659686 A CN 116659686A
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wavefront
differential
measured
matrix
zernike
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CN116659686B (en
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田爱玲
任柯鑫
王红军
刘丙才
张郁文
苏媛
王思淇
王凯
朱亚辉
朱学亮
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Xian Technological University
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    • G01J—MEASUREMENT OF INTENSITY, VELOCITY, SPECTRAL CONTENT, POLARISATION, PHASE OR PULSE CHARACTERISTICS OF INFRARED, VISIBLE OR ULTRAVIOLET LIGHT; COLORIMETRY; RADIATION PYROMETRY
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    • G01J9/00—Measuring optical phase difference; Determining degree of coherence; Measuring optical wavelength
    • G01J9/02—Measuring optical phase difference; Determining degree of coherence; Measuring optical wavelength by interferometric methods
    • G01J9/0215—Measuring optical phase difference; Determining degree of coherence; Measuring optical wavelength by interferometric methods by shearing interferometric methods
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    • G01J9/00—Measuring optical phase difference; Determining degree of coherence; Measuring optical wavelength
    • G01J2009/002—Wavefront phase distribution
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    • G01J—MEASUREMENT OF INTENSITY, VELOCITY, SPECTRAL CONTENT, POLARISATION, PHASE OR PULSE CHARACTERISTICS OF INFRARED, VISIBLE OR ULTRAVIOLET LIGHT; COLORIMETRY; RADIATION PYROMETRY
    • G01J9/00—Measuring optical phase difference; Determining degree of coherence; Measuring optical wavelength
    • G01J9/02—Measuring optical phase difference; Determining degree of coherence; Measuring optical wavelength by interferometric methods
    • G01J2009/0234—Measurement of the fringe pattern

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Abstract

The invention discloses a wavefront reconstruction method and device, and relates to the field of optical measurement. The method solves the problems that the existing transverse shearing interference wavefront reconstruction accuracy is affected by the orthogonality of the shearing interference patterns, and the algorithm is complex to realize, so that the wavefront reconstruction accuracy is low. The method comprises the following steps: according to the wave fronts to be measured and the shearing wave fronts along different directions, obtaining differential wave fronts at least comprising two random directions, combining the differential wave fronts at least comprising two random directions to obtain a differential wave front matrix and a differential Zernike matrix formed by corresponding differential Zernike polynomials; fitting the differential wavefront matrix to the differential Zernike matrix through least square matrix coefficient fitting to obtain a Zernike polynomial coefficient corresponding to the wavefront to be measured and the wavefront to be measured.

Description

Wavefront reconstruction method and device
Technical Field
The invention relates to the field of optical measurement, in particular to a wavefront reconstruction method and device.
Background
The transverse shearing interference technology is an important wave front detection technology and has the advantages of simple structure, no need of independent reference, common-path interference, difficult interference by external environment and the like. The method has irreplaceable application in the fields of wavefront sensing, quality evaluation of ultraviolet and deep ultraviolet band photoetching lenses, high-power laser engineering and the like.
Compared with the traditional interference technology, the transverse shearing interference technology has the difficulty that the wave front to be measured is reconstructed, the interference fringes generated by the transverse shearing interference technology do not directly contain the information of the wave front to be measured, and the wave front reconstruction is realized through a complex wave front reconstruction algorithm. Existing wavefront reconstruction algorithms can be broadly divided into two categories: one is a Zernike polynomial based mode wavefront reconstruction algorithm and the other is a discrete point based regional wavefront reconstruction algorithm. Both types of wavefront reconstruction methods are widely used, and each wavefront reconstruction method has respective advantages and disadvantages.
In order to solve the problem of deriving a shearing matrix based on a mode method, liu2003 proposes a fitting method based on a differential Zernike polynomial, the method uses the differential Zernike polynomial as an expansion basis function of a differential wavefront, the differential wavefront is directly fitted to the differential Zernike polynomial, and the obtained coefficient is the coefficient of the Zernike polynomial of the wavefront to be measured. However, the reconstruction of the wavefront to be measured can only obtain wavefront information in the orthogonal direction, and cannot obtain wavefront information in different directions of the wavefront to be measured.
For the transverse shearing interference technology, the detection of the optical element to be detected needs to obtain much information as much as possible, and a high-precision detection result is obtained. The wavefront reconstruction at the present stage is realized based on the shearing interference patterns in the orthogonal direction, and due to the influence of factors such as environment, the complete orthogonality of the acquired shearing interference patterns cannot be ensured in the experiment, so that the reconstruction precision of the wavefront to be detected is influenced.
In summary, the existing accuracy of the transverse shearing interference wavefront reconstruction is affected by the orthogonality of the shearing interference patterns, and the problem that the accuracy of the wavefront reconstruction is low due to complex algorithm implementation exists.
Disclosure of Invention
The embodiment of the invention provides a wavefront reconstruction method and device, which solve the problems that the existing transverse shearing interference wavefront reconstruction precision is influenced by the orthogonality of a shearing interference pattern, and the algorithm is complex to realize, so that the wavefront reconstruction precision is low.
The embodiment of the invention provides a wavefront reconstruction method, which comprises the following steps:
according to the wave fronts to be measured and the shearing wave fronts along different directions, obtaining differential wave fronts at least comprising two random directions, combining the differential wave fronts at least comprising two random directions to obtain a differential wave front matrix and a differential Zernike matrix formed by corresponding differential Zernike polynomials;
fitting the differential wavefront matrix to the differential Zernike matrix through least square matrix coefficient fitting to obtain a Zernike polynomial coefficient corresponding to the wavefront to be measured and the wavefront to be measured.
Preferably, before the differential wavefront including at least two arbitrary directions is obtained according to the wavefront to be measured and the shearing wavefront along different directions, the method further includes:
obtaining a differential wavefront comprising two random directions according to the wavefront to be measured and the shearing wavefront along different directions, wherein the differential wavefront in the two random directions is as follows:
obtaining a differential wavefront comprising two random directions according to the wavefront to be measured and the shearing wavefront along different directions, wherein the differential wavefront in the two random directions is as follows:
determining a differential basis function of differential wave fronts in two arbitrary directions, wherein the differential basis function is as follows:
wherein ,representing a differential wave of the wavefront under test in any direction, W (x-Scosθ i ,y-sinθ i ) The i-th shearing wave front of the wave front to be detected along the shearing direction theta is represented, and i is more than or equal to 2, |theta i -θ i-1 I+.n pi, n=0, 1,2 …, n; s is the shearing quantity, theta is the included angle between the shearing direction and the x axis, W (x, y) is the wavefront to be measured, a j The j-th term coefficient representing the basis function of the Zernike polynomial, Z j (x, y) represents a normalized Zernike polynomial, < >>Zernike polynomials representing the differences corresponding to differential wavefronts in any direction.
Preferably, the differential wavefront including at least two arbitrary directions is as follows:
obtaining a Zernike polynomial comprising at least two differential wave fronts in any direction according to the differential wave fronts comprising at least two any directions:
wherein ,the ith differential wavefront of the wavefront to be measured along the theta direction is represented, S is the shearing quantity, W (x, y) represents the wavefront to be measured, J represents the number of terms of the Zernike polynomial used, and>zernike polynomials representing the differences corresponding to differential wavefronts in any direction.
Preferably, the differential wavefront matrix is as follows:
the differential Zernike polynomial matrix is as follows:
fitting the differential wavefront matrix to the differential Zernike matrix through least square matrix coefficient fitting to obtain a Zernike polynomial coefficient corresponding to the wavefront to be measured, wherein the Zernike polynomial coefficient corresponding to the wavefront to be measured is as follows:
the wavefront to be measured is determined by the following formula:
wherein DeltaW represents a differential wavefront matrix, deltaZ represents a differential Zernike polynomial matrix, a represents the coefficients of the Zernike polynomial, W (x, y) represents the wavefront to be measured, Z j (x, y) represents a normalized Zernike polynomial, a= [ a ] 1 ,a 2 …a j ] T ，ΔZ＝[ΔZ 1 ,ΔZ 2 …ΔZ j ] T T represents the transpose matrix.
An embodiment of the present invention provides a wavefront reconstruction device, including:
the first obtaining unit is used for obtaining differential wave fronts at least comprising two random directions according to the wave fronts to be detected and the shearing wave fronts along different directions, and combining the differential wave fronts at least comprising the two random directions to obtain a differential wave front matrix and a differential Zernike matrix formed by corresponding differential Zernike polynomials;
and the second obtaining unit is used for fitting the differential wavefront matrix to the differential Zernike matrix through least square matrix coefficient fitting to obtain a Zernike polynomial coefficient corresponding to the wavefront to be measured and the wavefront to be measured.
Preferably, the first obtaining unit is further configured to:
obtaining a differential wavefront comprising two random directions according to the wavefront to be measured and the shearing wavefront along different directions, wherein the differential wavefront in the two random directions is as follows:
determining a differential basis function of differential wave fronts in two arbitrary directions, wherein the differential basis function is as follows:
wherein ,representing differential wave of wavefront to be measured along arbitrary direction, W (x-S cos θ) i ,y-sinθ i ) The i-th shearing wave front of the wave front to be detected along the shearing direction theta is represented, and i is more than or equal to 2, |theta i -θ i-1 I+.n pi, n=0, 1,2 …, n; s is the shearing quantity, theta is the included angle between the shearing direction and the x axis, W (x, y) is the wavefront to be measured, a j The j-th term coefficient representing the basis function of the Zernike polynomial, Z j (x, y) represents a normalized Zernike polynomial, < >>Zernike polynomials representing the differences corresponding to differential wavefronts in any direction.
Preferably, the differential wavefront including at least two arbitrary directions is as follows:
obtaining a Zernike polynomial comprising at least two differential wave fronts in any direction according to the differential wave fronts comprising at least two any directions:
wherein ,the ith differential wavefront of the wavefront to be measured along the theta direction is represented, S is the shearing quantity, W (x, y) represents the wavefront to be measured, J represents the number of terms of the Zernike polynomial used, and>differential wave representing arbitrary directionThe front corresponding differential Zernike polynomials.
Preferably, the differential wavefront matrix is as follows:
the differential Zernike polynomial matrix is as follows:
fitting the differential wavefront matrix to the differential Zernike matrix through least square matrix coefficient fitting to obtain a Zernike polynomial coefficient corresponding to the wavefront to be measured, wherein the Zernike polynomial coefficient corresponding to the wavefront to be measured is as follows:
the wavefront to be measured is determined by the following formula:
wherein DeltaW represents a differential wavefront matrix, deltaZ represents a differential Zernike polynomial matrix, a represents the coefficients of the Zernike polynomial, W (x, y) represents the wavefront to be measured, Z j (x, y) represents a normalized Zernike polynomial, a= [ a ] 1 ,a 2 …a j ] T ，ΔZ＝[ΔZ 1 ,ΔZ 2 …ΔZ j ] T T represents the transpose matrix.
An embodiment of the present invention provides a computer device, where the computer device includes a memory and a processor, where the memory stores a computer program, and when the computer program is executed by the processor, the processor is caused to execute the wavefront reconstruction method described above.
An embodiment of the present invention provides a computer-readable storage medium storing a computer program that, when executed by a processor, causes the processor to perform the above-described wavefront reconstruction method.
The embodiment of the invention provides a wavefront reconstruction method and device, wherein the method comprises the following steps: according to the wave fronts to be measured and the shearing wave fronts along different directions, obtaining differential wave fronts at least comprising two random directions, combining the differential wave fronts at least comprising two random directions to obtain a differential wave front matrix and a differential Zernike matrix formed by corresponding differential Zernike polynomials; fitting the differential wavefront matrix to the differential Zernike matrix through least square matrix coefficient fitting to obtain a Zernike polynomial coefficient corresponding to the wavefront to be measured and the wavefront to be measured. The method adopts an improved wavefront reconstruction algorithm based on a differential Zernike polynomial, can realize the wavefront reconstruction of any two-direction shearing interferograms, and the reconstruction accuracy is not influenced by the orthogonal angle error of the interferograms, so that the problem that the wavefront reconstruction technology is limited by the shearing direction while the complete orthogonality of the acquired shearing interferograms cannot be ensured in the prior art is solved, and the problem that the reconstruction accuracy of the wavefront to be detected is influenced is solved; furthermore, the method can use differential wave surface information in multiple directions to jointly solve the wave front coefficient to be measured, reduce the influence of random errors on wave front reconstruction accuracy and enhance the robustness of the algorithm.
Drawings
In order to more clearly illustrate the embodiments of the invention or the technical solutions in the prior art, the drawings that are required in the embodiments or the description of the prior art will be briefly described, it being obvious that the drawings in the following description are only some embodiments of the invention, and that other drawings may be obtained according to these drawings without inventive effort for a person skilled in the art.
FIG. 1 is a schematic flow chart of a wavefront reconstruction method according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of a wavefront reconstruction algorithm according to an embodiment of the present invention;
fig. 3 is a schematic structural diagram of a shear wavefront along any direction and an included angle θ between the shear direction and the positive direction of the X-axis of the wavefront to be measured according to the embodiment of the present invention;
FIG. 4A is a schematic diagram of a wavefront reconstruction result with 0℃and 45℃directional shearing interferograms simulated by MATLAB software according to an embodiment of the present invention;
FIG. 4B is a schematic diagram of a wavefront reconstruction result with 0℃and 90℃directional shearing interferograms simulated by MATLAB software according to an embodiment of the present invention;
FIG. 4C is a schematic diagram of a wavefront reconstruction with 0℃and 135℃directional shearing interferometry simulated by MATLAB software according to an embodiment of the present invention;
fig. 4D is a schematic diagram of a wavefront reconstruction result provided by an embodiment of the present invention by MATLAB software simulation with directional shearing interferograms of 0 °, 45 °, 90 ° and 135 °;
fig. 5 is a schematic structural diagram of a wavefront reconstruction method device according to an embodiment of the present invention.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
In the embodiment of the invention, the technical terms involved are as follows:
1) Lateral shearing interference (lateral shearing interferometry LSI for short) is the most common method in shearing interference.
2) Principle of lateral shearing interference: when the light wave to be measured passes through the transverse shearing interferometer, a replica light wave with transverse dislocation is generated, and the replica light wave interferes with the light wave to be measured to form a transverse shearing interference pattern.
Fig. 1 is a schematic flow chart of a wavefront reconstruction method provided by an embodiment of the present invention, and fig. 2 is a schematic flow chart of an implementation of a wavefront reconstruction algorithm provided by an embodiment of the present invention; fig. 3 is a schematic structural diagram of a shear wavefront along any direction and an included angle θ between the shear direction and the positive direction of the X-axis of the wavefront to be measured according to the embodiment of the present invention; the following describes in detail a wavefront reconstruction method provided by an embodiment of the present invention with reference to fig. 1 to 3, specifically, the method includes the following steps:
step 101, obtaining a differential wavefront including at least two random directions according to the wavefront to be measured and the shearing wavefront along different directions, combining the differential wavefront including at least two random directions to obtain a differential wavefront matrix sum and a differential Zernike matrix formed by corresponding differential Zernike polynomials
And 102, fitting the differential wavefront matrix to the differential Zernike matrix through least square matrix coefficient fitting to obtain a Zernike polynomial coefficient corresponding to the wavefront to be measured and the wavefront to be measured.
Before introducing the wavefront reconstruction method provided by the embodiment of the invention, four-step phase shift interferograms with different angles are obtained based on a transverse shearing interferometer, specifically, four-step phase shift interferograms with different angles are obtained by using the transverse shearing interferometer, and the angles theta of the interferograms with different directions are calibrated respectively i, wherein ,θi (i=1, 2, 3, 4 … … N, i≡2); the wave front carrying the information of the measured surface passes through the birefringent crystal to obtain a shearing wave front transversely translating with the wave front, the two wave fronts interfere in a space overlapping area to form interference fringes, the interference fringes enter the focal plane splitting imaging system through the imaging lens, simultaneously, four synchronous phase shift interference patterns of a single frame are obtained, and the shearing interference patterns in different directions are obtained by rotating the birefringent crystal. Wherein the light intensity of each interferogram in the same direction can be expressed by the formula (1):
wherein ,Ia (x, y) and I b (x, y) is the intensity of the two beams of light;for measuring the phase difference between the wave surface to be measured and the wave surface which replicates itself, theta i Indicating the i-th shear direction.
Further, the position of the effective area in the interferogram is recorded and marked, the shearing amount S of the interferogram is calculated, and the value outside the shearing interference area is set to be 0.
Further, a four-step phase shift algorithm is adopted to extract the wrapping phases of the interferograms in different directionsSpecifically, the wrapping phase of the multi-azimuth four-step transverse shearing interferogram can be determined by the following formula (2) and formula (3)
In practical application, the wrapping phases of the multi-azimuth transverse shearing interferograms shown in the formula (1) in different directions can be obtained through the formulas (2) and (3)
In practical application, the method performs arctangent operation when extracting the phase, so that the extracted wrapping phase is positioned between (-pi, pi), and the phase is in a discontinuous and step distribution state, so that it can be determined that the phase does not really obtain the phase information reflecting the real surface shape of the optical element.
In the embodiment of the invention, in order to obtain the real phase information of the surface shape of the optical element, discontinuous phase information with step distribution is needed to be spliced to form a continuous phase, and the real surface containing the optical element is obtained through the continuous phasePhase information of the shape. Unwrapping operations using least squares, specifically assumingUnwrapped phases for different directions at two-dimensional mxn discrete points +.>For the corresponding wrapping phase, +.> and />The relationship of (2) can be represented by the following formula (4):
wherein the wrapping phaseBetween (-pi, pi), k is a positive integer, i=
0,1,…,M-1；j＝0,1,…,N-1。
Further, the wrapping operator F is represented by the following formula (5):
further, wrapping the phaseThe difference between adjacent elements in the x-direction is expressed by the formula (6), wrapping the phaseThe difference of adjacent elements in the y direction is expressed by the formula (7):
obtaining differential wavefront phases of the wavefront to be measured in different directions, which are represented by the following formula (8):
in the above formula, F (·) is an unwrapping operationIs the differential wavefront phase of the wavefront in different directions.
In the embodiment of the invention, as the unwrapped phase reflects the differential information of the wavefront to be measured along the shearing direction, the original information of the wavefront to be measured needs to be derived back from the differential information to realize the reconstruction of the wavefront to be measured.
Specifically, for the wavefront to be measured, the shearing wavefront and the wavefront to be measured along different directions can obtain differential wavefronts in any direction, wherein the differential wavefront in any direction can be represented by the formula (9):
the wavefront reconstruction method provided by the embodiment of the invention is not aimed at a certain transverse shearing interferometry technology, the obtained interferograms do not need to be orthogonal, and a transverse shearing interference formula is deformed in principle, wherein the transverse shearing interference along any direction can be determined by the formula (9) and the formula (9-1) in the figure 3.
Note that, the above formula (9) represents a differential wavefront in any direction, for example, when θ is equal to 90 °, both the formula (9) and the formula (9-1) are as follows:
ΔW 90° (x,y)＝W(x,y-S)-W(x,y)
alternatively, when θ is equal to 0 °, then both equation (9) and equation (9-1) are as follows:
ΔW 0 °(x,y)＝W(x-S,y)-W(x,y)
alternatively, when θ is equal to 45 °, then both equation (9) and equation (9-1) are as follows:
alternatively, when θ is equal to 30 °, then both equation (9) and equation (9-1) are as follows:
in the embodiment of the invention, after the differential wavefront in any direction is determined, the differential basis function of the differential wavefront in any direction can be obtained according to the differential wavefront in any direction, and specifically is shown as a formula (10):
correspondingly, if let s= (Δx) 2 +Δy 2 ) 1/2 ，Δy=s·cos θ, Δx=s·sin θ. Because the differential wavefront in any direction shown in equation (10) can also be expressed as in equation (10-1), equation (10) can also be expressed as in equation (10-1):
in the above formula (10) and formula (10-1), a j Representing Zernike polynomial coefficients, Z j Representing a differential basis function, Z j (x, y) represents a normalized Zernike polynomial.
Further, if the differential basis function of the differential wavefront in any direction is setEqual to->Or is additionally->Equal to->The Zernike polynomials of the differential wavefront in any direction can be obtained as shown in the following formula (11) or (11-1):
wherein ,and representing the corresponding direction difference Zernike polynomial expression corresponding to the difference wave front.
In the embodiment of the invention, two differential wave fronts in any different directions can be combined into a differential wave front matrix, and the corresponding Zernike polynomials of the differential wave fronts in any two different directions are combined into a differential Zernike matrix, which is specifically shown as a formula (12) and a formula (13):
in practical applications, equation (14) shown below can be obtained according to equation (12) and equation (13):
wherein a= [ a ] 1 ,a 2 …a j ] T ，ΔZ＝[ΔZ 1 ,ΔZ 2 …ΔZ j ] T The symbol T represents the transposed matrix, θ 1 ≠θ 2 ，i≥2,|θ i -θ i-1 I+.n pi, n=0, 1,2 …, n; in the embodiment of the invention, the differential wavefront matrix is fitted into the differential Zernike matrix through least square matrix coefficient fitting, so that the Zernike polynomial coefficient corresponding to the wavefront to be measured can be obtained, and the Zernike polynomial coefficient corresponding to the wavefront to be measured is shown in the following formula (15):
in the embodiment of the invention, after the Zernike polynomial coefficient corresponding to the wavefront to be measured is determined, the wavefront to be measured can be determined by the formula (16):
wherein W (x, y) represents the wavefront to be measured, Z j (x, y) represents a normalized Zernike polynomial, a represents the Zernike polynomial coefficients, and J is the number of terms used by the Zernike polynomial in total. In the embodiment of the present invention, the value of J is 36.
According to the wavefront reconstruction method provided by the embodiment of the invention, the Zernike polynomial coefficients corresponding to the wavefront to be measured can be obtained through formulas (12), (13), (14) and (15) according to the differential wavefront in any two different directions, so that the wavefront to be measured is obtained through formula (16). According to the method, the shearing interference differential wavefront information reconstruction in any two different directions is realized by aiming at the transverse shearing interference wavefront measurement technology in any direction, and the problem that the existing wavefront reconstruction technology is limited by the shearing interference direction is solved.
It should be noted that, according to the multidirectional transverse shearing interference wavefront reconstruction technology provided by the embodiment of the invention, differential wavefront information in different directions is used for expansion, the obtained shearing interference patterns do not need to be orthogonalized, the wavefront coefficients in all groups of orthogonalized directions do not need to be solved respectively, and finally only one group of the solved Zernike polynomial coefficients is the last wavefront coefficient to be detected, and the average is not needed.
Further, in order to improve the accuracy of wavefront reconstruction and reduce the influence of random errors on the accuracy of multi-wavefront reconstruction, the Zernike polynomial coefficients corresponding to the wavefront to be measured and the wavefront to be measured can be obtained through (12), (13), (14), (15) and (16) according to the differential wavefronts in a plurality of different directions.
Specifically, according to the differential wavefront in any direction shown in the formula (9), a plurality of sets of differential wavefronts including any direction can be obtained, specifically as shown in the formula (9-2):
correspondingly, according to the plurality of groups of differential wave fronts comprising any direction, a plurality of groups of Zernike polynomials comprising differential wave fronts comprising any direction can be obtained, and the method is specifically shown as a formula (11-2):
wherein W (x-S cos θ) i ,y-S·sinθ i ) The ith shearing wave front of the wave front to be detected along the theta direction is represented, S is the shearing quantity, theta represents the included angle between the shearing direction and the x axis, W (x, y) represents the wave front to be detected, and a is represented by the wave front to be detected j Representing Zernike polynomial coefficients, deltaZ j The basis functions of the differences are represented,zernike polynomials representing differential wavefronts in arbitrary directions.
Further, according to the plurality of sets of differential wavefront including any direction shown in the formula (9-2) and the plurality of sets of Zernike polynomials including differential wavefront including any direction shown in the formula (11-2), a differential wavefront matrix and a differential Zernike matrix can be obtained, respectively, as shown in the formula (12-1) and the formula (13-1):
accordingly, the Zernike polynomial coefficients corresponding to the wavefront to be measured can be obtained by substituting the formula (12-1) and the formula (13-1) into the formula (15), and further, the wavefront to be measured can be obtained according to the formula (16).
FIG. 4A is a schematic diagram of a wavefront reconstruction result with 0℃and 45℃directional shearing interferograms simulated by MATLAB software according to an embodiment of the present invention; FIG. 4B is a schematic diagram of a wavefront reconstruction result with 0℃and 90℃directional shearing interferograms simulated by MATLAB software according to an embodiment of the present invention; FIG. 4C is a schematic diagram of a wavefront reconstruction with 0℃and 135℃directional shearing interferometry simulated by MATLAB software according to an embodiment of the present invention; fig. 4D is a schematic diagram of a wavefront reconstruction result provided by an embodiment of the present invention by MATLAB software simulation with directional shearing interferograms of 0 °, 45 °, 90 ° and 135 °; table 1 is a comparison example of wavefront reconstruction results of any two directions and a plurality of directions simulated by MATLAB software provided in the embodiment of the present invention, and the wavefront reconstruction method provided in the embodiment of the present invention is described in detail below with reference to fig. 4A to 4D and table 1.
The wavefront reconstruction results of the shearing interferograms in two arbitrary directions are provided in fig. 4A and 4C, the wavefront reconstruction results of the shearing interferograms in one orthogonal direction are provided in fig. 4B, and the wavefront reconstruction results of the shearing interferograms in a plurality of directions are provided in fig. 4D. As can be seen from the results shown in table 1, the wavefront reconstruction accuracy in two arbitrary directions (non-orthogonal) shown in fig. 4A and 4C is substantially identical to the reconstruction accuracy in the orthogonal direction shown in fig. 4B, except that the wavefront reconstruction progress results in multiple directions shown in fig. 4D are slightly improved compared with the wavefront reconstruction results in two directions, which proves that the wavefront reconstruction method provided by the embodiment of the present invention can perform wavefront reconstruction through two sets of shearing interferograms in any directions, and also can perform wavefront reconstruction through multiple sets of shearing interferograms in different directions, and when the number of shearing waves is increased, the accuracy of the obtained wavefront reconstruction is higher.
TABLE 1 comparative examples of wavefront reconstruction results for any two and multiple directions simulated by MATLAB software
In summary, an embodiment of the present invention provides a method and an apparatus for reconstructing a wavefront, where the method includes: obtaining multiple groups of at least two groups of differential wave fronts comprising any direction according to the relation between the shearing wave fronts of at least two groups of wave fronts to be tested and the wave fronts to be tested along different directions, and combining the differential wave fronts comprising any direction to obtain a differential wave front matrix and a differential Zernike matrix; fitting the differential wavefront matrix to the differential Zernike matrix through least square matrix coefficient fitting to obtain a Zernike polynomial coefficient corresponding to the wavefront to be measured and the wavefront to be measured. The method adopts an improved wavefront reconstruction algorithm based on a differential Zernike polynomial, can realize the wavefront reconstruction of any two-direction shearing interferograms, and the reconstruction accuracy is not influenced by the orthogonal angle error of the interferograms, so that the problem that the wavefront reconstruction technology is limited by the shearing direction while the complete orthogonality of the acquired shearing interferograms cannot be ensured in the prior art is solved, and the problem that the reconstruction accuracy of the wavefront to be detected is influenced is solved; furthermore, the method can use differential wave surface information in multiple directions to jointly solve the wave front coefficient to be measured, reduce the influence of random errors on the reconstruction progress before sowing, and enhance the robustness of the algorithm.
Based on the same inventive concept, the embodiment of the invention provides a wavefront reconstruction device, and because the principle of the device for solving the technical problem is similar to that of a wavefront reconstruction method, the implementation of the device can refer to the implementation of the method, and the repetition is omitted.
As shown in fig. 5, the apparatus comprises a first deriving unit 201 and a second deriving unit 202.
A first obtaining unit 201, configured to obtain a differential wavefront including at least two random directions according to the wavefront to be measured and the shearing wavefronts along different directions, and combine the differential wavefront including at least two random directions to obtain a differential wavefront matrix and a corresponding differential Zernike polynomial to form a differential Zernike matrix;
a second obtaining unit 202, configured to fit the differential wavefront matrix to the differential Zernike matrix through least square matrix coefficient fitting, so as to obtain a Zernike polynomial coefficient corresponding to the wavefront to be measured and the wavefront to be measured.
Preferably, the first obtaining unit 201 further comprises:
obtaining a differential wavefront comprising two random directions according to the wavefront to be measured and the shearing wavefront along different directions, wherein the differential wavefront in the two random directions is as follows:
determining a differential basis function of differential wave fronts in two arbitrary directions, wherein the differential basis function is as follows:
wherein ,representing a differential wave of the wavefront under test in any direction, W (x-Scosθ i ,y-sinθ i ) The i-th shearing wave front of the wave front to be detected along the shearing direction theta is represented, and i is more than or equal to 2, |theta i -θ i-1 I not equal to n pi, n=0, 1,2 …, n, S is the shearing amount, θ represents the angle between the shearing direction and the x-axis of the wavefront to be measured, W (x, y) represents the wavefront to be measured, a j The j-th term coefficient representing the basis function of the Zernike polynomial, Z j (x, y) represents a normalized Zernike polynomial, < >>Zernike polynomials representing the differences corresponding to differential wavefronts in any direction.
Preferably, the differential wavefront including at least two arbitrary directions is as follows:
obtaining a Zernike polynomial comprising at least two differential wave fronts in any direction according to the differential wave fronts comprising at least two any directions:
wherein ,the ith differential wavefront of the wavefront to be measured along the theta direction is represented, S is the shearing quantity, W (x, y) represents the wavefront to be measured, J represents the number of terms of the Zernike polynomial used, and>zernike polynomials representing the differences corresponding to differential wavefronts in any direction.
Preferably, the differential wavefront matrix is as follows:
the differential Zernike polynomial matrix is as follows:
fitting the differential wavefront matrix to the differential Zernike matrix through least square matrix coefficient fitting to obtain a Zernike polynomial coefficient corresponding to the wavefront to be measured, wherein the Zernike polynomial coefficient corresponding to the wavefront to be measured is as follows:
the wavefront to be measured is determined by the following formula:
wherein DeltaW represents a differential wavefront matrix, deltaZ represents a differential Zernike polynomial matrix, a represents the coefficients of the Zernike polynomial, W (x, y) represents the wavefront to be measured, Z j (x, y) represents GuiNormalized Zernike polynomials, a= [ a ] 1 ,a 2 …a j ] T ，ΔZ＝[ΔZ 1 ,ΔZ 2 …ΔZ j ] T T represents the transpose matrix.
It should be understood that the above wavefront reconstruction device includes units that are only logically divided according to the functions implemented by the device, and in practical applications, the above units may be stacked or split. The functions implemented by the wavefront reconstruction device provided in this embodiment correspond to one-to-one with one wavefront reconstruction method provided in the above embodiment, and the more detailed process flow implemented by the device is described in detail in the above method embodiment one, which is not described in detail herein.
Another embodiment of the present invention also provides a computer apparatus, including: a processor and a memory; the memory is used for storing computer program codes, and the computer program codes comprise computer instructions; when the processor executes the computer instructions, the electronic device executes the steps of the wave front reconstruction method in the method flow shown in the method embodiment.
Another embodiment of the present invention further provides a computer readable storage medium, where computer instructions are stored, when the computer instructions are executed on a computer device, to cause the computer device to perform the steps of the wavefront reconstruction method in the method flow shown in the above method embodiment.
While preferred embodiments of the present invention have been described, additional variations and modifications in those embodiments may occur to those skilled in the art once they learn of the basic inventive concepts. It is therefore intended that the following claims be interpreted as including the preferred embodiments and all such alterations and modifications as fall within the scope of the invention.
It will be apparent to those skilled in the art that various modifications and variations can be made to the present invention without departing from the spirit or scope of the invention. Thus, it is intended that the present invention also include such modifications and alterations insofar as they come within the scope of the appended claims or the equivalents thereof.

Claims (10)

1. A method of wavefront reconstruction, comprising:
according to the wave fronts to be measured and the shearing wave fronts along different directions, obtaining differential wave fronts at least comprising two random directions, combining the differential wave fronts at least comprising two random directions to obtain a differential wave front matrix and a differential Zernike matrix formed by corresponding differential Zernike polynomials;
fitting the differential wavefront matrix to the differential Zernike matrix through least square matrix coefficient fitting to obtain a Zernike polynomial coefficient corresponding to the wavefront to be measured and the wavefront to be measured.
2. The method of claim 1, wherein before the differential wavefront including at least two arbitrary directions is obtained according to the wavefront to be measured and the shearing wavefront along different directions, the method further comprises:
obtaining a differential wavefront comprising two random directions according to the wavefront to be measured and the shearing wavefront along different directions, wherein the differential wavefront in the two random directions is as follows:
determining a differential basis function of differential wave fronts in two arbitrary directions, wherein the differential basis function is as follows:
wherein ,representing the differential wavefront of the wavefront to be measured along any direction, W (x-S cos θ) i ,y-sinθ i ) The i-th shearing wave front of the wave front to be detected along the shearing direction theta is represented, and i is more than or equal to 2, |theta i -θ i-1 I+.n pi, n=0, 1,2 …, n; s is the shearing quantity, theta is the included angle between the shearing direction and the x axis, W (x, y) is the wavefront to be measured, a j The j-th term coefficient representing the basis function of the Zernike polynomial, Z j (x, y) represents a normalized Zernike polynomial, < >>Zernike polynomials representing the differences corresponding to differential wavefronts in any direction.
3. The method of claim 1, wherein the differential wavefront including at least two arbitrary directions is as follows:
obtaining a Zernike polynomial comprising at least two differential wave fronts in any direction according to the differential wave fronts comprising at least two any directions:
wherein ,the ith differential wavefront of the wavefront to be measured along the theta direction is represented, S is the shearing quantity, W (x, y) represents the wavefront to be measured, J represents the number of terms of the Zernike polynomial used, and>zernike polynomials representing the differences corresponding to differential wavefronts in any direction.
4. The method of claim 1, wherein the differential wavefront matrix is as follows:
the differential Zernike polynomial matrix is as follows:
fitting the differential wavefront matrix to the differential Zernike matrix through least square matrix coefficient fitting to obtain a Zernike polynomial coefficient corresponding to the wavefront to be measured, wherein the Zernike polynomial coefficient corresponding to the wavefront to be measured is as follows:
the wavefront to be measured is determined by the following formula:
wherein DeltaW represents a differential wavefront matrix, deltaZ represents a differential Zernike polynomial matrix, a represents the coefficients of the Zernike polynomial, W (x, y) represents the wavefront to be measured, Z j (x, y) represents a normalized Zernike polynomial, a= [ a ] 1 ,a 2 …a j ] T ，ΔZ＝[ΔZ 1 ,ΔZ 2 …ΔZ j ] T T represents the transpose matrix.
5. A wavefront reconstruction device, comprising:
the first obtaining unit is used for obtaining differential wave fronts at least comprising two random directions according to the wave fronts to be detected and the shearing wave fronts along different directions, and combining the differential wave fronts at least comprising the two random directions to obtain a differential wave front matrix and a differential Zernike matrix formed by corresponding differential Zernike polynomials;
and the second obtaining unit is used for fitting the differential wavefront matrix to the differential Zernike matrix through least square matrix coefficient fitting to obtain a Zernike polynomial coefficient corresponding to the wavefront to be measured and the wavefront to be measured.
6. The apparatus of claim 5, wherein the first deriving unit is further configured to:
obtaining a differential wavefront comprising two random directions according to the wavefront to be measured and the shearing wavefront along different directions, wherein the differential wavefront in the two random directions is as follows:
determining a differential basis function of differential wave fronts in two arbitrary directions, wherein the differential basis function is as follows:
wherein ,representing differential wave of wavefront to be measured along arbitrary direction, W (x-S cos θ) i ,y-sinθ i ) The i-th shearing wave front of the wave front to be detected along the shearing direction theta is represented, and i is more than or equal to 2, |theta i -θ i-1 I+.n pi, n=0, 1,2 …, n; s is the shearing quantity, theta is the included angle between the shearing direction and the x axis, W (x, y) is the wavefront to be measured, a j The j-th term coefficient representing the basis function of the Zernike polynomial, Z j (x, y) represents a normalized Zernike polynomial, < >>Zernike polynomials representing the differences corresponding to differential wavefronts in any direction.
7. The apparatus of claim 5, wherein the differential wavefront including at least two arbitrary directions is as follows:
obtaining a Zernike polynomial comprising at least two differential wave fronts in any direction according to the differential wave fronts comprising at least two any directions:
wherein ,the ith differential wavefront of the wavefront to be measured along the theta direction is represented, S is the shearing quantity, W (x, y) represents the wavefront to be measured, J represents the number of terms of the Zernike polynomial used, and>zernike polynomials representing the differences corresponding to differential wavefronts in any direction.
8. The apparatus of claim 5, wherein the differential wavefront matrix is as follows:
the differential Zernike polynomial matrix is as follows:
fitting the differential wavefront matrix to the differential Zernike matrix through least square matrix coefficient fitting to obtain a Zernike polynomial coefficient corresponding to the wavefront to be measured, wherein the Zernike polynomial coefficient corresponding to the wavefront to be measured is as follows:
the wavefront to be measured is determined by the following formula:
wherein DeltaW represents a differential wavefront matrix, deltaZ represents a differential Zernike polynomial matrix, a represents the coefficients of the Zernike polynomial, W (x, y) represents the wavefront to be measured, Z j (x, y) represents a normalized Zernike polynomial, a= [ a ] 1 ,a 2 …a j ] T ，ΔZ＝[ΔZ 1 ,ΔZ 2 …ΔZ j ] T T represents the transpose matrix.
9. A computer device, characterized in that the computer device comprises a memory and a processor, the memory storing a computer program which, when executed by the processor, causes the processor to perform the wavefront reconstruction method according to any one of claims 1-4.
10. A computer readable storage medium, characterized in that a computer program is stored, which, when being executed by a processor, causes the processor to perform the wavefront reconstruction method according to any one of claims 1-4.
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