A method for eliminating calculation errors of checkerboard grating diffraction in a transverse shearing interferometer

Abstract

This invention discloses a method for eliminating diffraction calculation errors in a checkerboard grating in a transverse shearing interferometer. The optical system of the transverse shearing interferometer includes: fixing the optical system under test on the optical platform of the transverse shearing interferometer; emitting incoherent light through an optical illumination system; the light rays sequentially passing through an object grating, the optical system under test, and an image grating, and being received by an optical detection element; uniformly shifting the phase of the object grating or the image grating to obtain shearing interference patterns in two perpendicular directions; using the least squares method to separate and solve for the shearing phases of the 0th and +1st order or 0th and -1st order diffracted light in the two perpendicular directions, and reconstructing the wavefront. This invention takes into account the different intensity coefficients of each diffraction order, accurately solves for the shearing phase, eliminates the errors introduced by the checkerboard grating in the whole-path diffraction calculation, and improves the wavefront reconstruction accuracy.

Description

Technical Field

[0001] This invention relates to the field of optical measurement technology, and in particular to a method for eliminating calculation errors of checkerboard grating diffraction in a transverse shearing interferometer. Background Technology

[0002] The transverse shearing interferometer is a typical interferometer structure that uses a grating as a beam splitter. It offers advantages such as simple structure, quasi-common optical path, and no need for an additional ideal reference wave. When performing optical measurements using a transverse shearing interferometer, only a uniformly phase-shifting object plane grating or image plane grating is needed to obtain a series of shearing interferograms in the x and y directions. The differential phase can then be calculated using methods such as the least squares method, allowing for the reconstruction of the wavefront.

[0003] Existing algorithms still suffer from inaccurate differential phase calculations, leading to low wavefront reconstruction accuracy. Due to the diffraction effect of the image plane grating, many higher-order diffracted beams exist in the shearing interference field.

[0004] Patent CN104111120B proposes a novel ten-step phase-shifting method that effectively suppresses ±3rd and ±5th order diffraction and extracts the ±1st order shear phase. However, it cannot accurately solve for the difference between the +1st or -1st order and the 0th order diffracted light. Other existing technologies can separate the phase of the corresponding diffraction order using Fourier transform, but this requires adding an extra filter to the system.

[0005] Patent CN112229604A solves for the shear phases of the 0th and -1st orders, and the 0th and +1st orders as a whole, to obtain the shear phases of the -1st and +1st order diffracted light at twice the shear rate. However, this method does not consider that each diffraction order has a different intensity coefficient, and the two interference beams cannot be simply regarded as a whole. Especially under large numerical aperture conditions, the influence of the intensity coefficient on the interference field intensity cannot be ignored.

[0006] In fact, the intensity coefficients of different pixels in the detection area are different. If their impact on the shear phase calculation is ignored, errors will be introduced into the final wavefront solution. This effect is particularly pronounced when the numerical aperture of the optical system is large.

[0007] Based on the above discussion, there is currently no algorithm for the checkerboard grating transverse shearing interferometer that has simple algorithm logic, does not require additional optical filtering components, and can accurately solve for the differential phase. Summary of the Invention

[0008] To address the shortcomings of existing technologies, this invention proposes a method for eliminating calculation errors in checkerboard grating diffraction in a transverse shearing interferometer, used to measure the wavefront aberration of an optical system. This method takes into account the different light intensity coefficients between different diffraction orders, accurately solves the differential phase based on uniform phase shift, and improves the wavefront reconstruction accuracy.

[0009] The specific technical solution of the present invention is as follows:

[0010] A method for eliminating diffraction calculation errors of a checkerboard grating in a transverse shearing interferometer includes the following steps:

[0011] Step 1: The optical system under test is fixedly arranged on the optical platform of the transverse shearing interferometer. The optical illumination system emits incoherent light, which passes sequentially through the object grating, the optical system under test, and the image grating, and is received by the optical detection element. The object grating consists of two sets of one-dimensional grating lines with mutually perpendicular directions. The optical axis is the z-axis, and according to the right-hand rule, the two directions perpendicular to the z-axis are the x-axis and y-axis, respectively. The image grating is a checkerboard grating, with its two diagonals perpendicular to the x-axis and y-axis, respectively. The ratio of the period of the image grating to that of the object grating is the same as the imaging magnification of the optical system under test, and the duty cycle of both the image grating and the object grating is 50%.

[0012] Step 2: Move the object grating so that the light rays pass through a one-dimensional grating line perpendicular to the x-axis, uniformly shifting the phase of the object or image grating to obtain a shearing interference pattern in the x-direction; use the least squares method to separate the shearing phase expression in the x-direction as follows:

[0013]

[0014] In the formula, δ1 and δ2 are parameters obtained by the least squares method, and C -1，0 C is the light intensity coefficient when the -1st and 0th order diffracted beams interfere in the x-direction. 1，0 φ is the light intensity coefficient when the +1st order and the 0th order diffracted light interfere in the x-direction; -1,0 φ represents the shear phase between the -1st and 0th order diffracted beams in the x-direction. 1,0 The shear phase between the +1st and 0th order diffracted beams in the x-direction;

[0015] For the obtained φ -1,0 or φ 1,0 Phase unwrapping is performed to obtain the shear phase in the x-direction;

[0016] Step 3: Move the object grating so that the light rays pass through a one-dimensional grating line perpendicular to the y-axis, uniformly shifting the phase of the object or image grating to obtain a shearing interference pattern in the y-direction; use the least squares method to separate the shearing phase expression in the y-direction as follows:

[0017]

[0018] In the formula, C -1，0 ′ is the light intensity coefficient when the -1st and 0th order diffracted beams interfere in the y direction, C 1，0 ′ is the light intensity coefficient when the +1st order diffracted light interferes with the 0th order diffracted light in the y direction; φ -1,0 φ' represents the shear phase between the -1st and 0th order diffracted beams in the y-direction. 1,0 ′ represents the shear phase between the +1st and 0th order diffracted beams in the y direction;

[0019] For the obtained φ -1,0 ′ or φ 1,0 The phase is unwrapped to obtain the shear phase in the y-direction;

[0020] Step 4: Reconstruct the wavefronts represented by the shear phases in the x and y directions to obtain the reconstructed wavefronts with errors eliminated.

[0021] Furthermore, the phase unwrapping method used in steps two and three employs the branch cutting method.

[0022] Furthermore, in step four, the differential Zernike method is used to reconstruct the wavefront represented by the shear phase in the x and y directions.

[0023] Furthermore, in steps two and three, the object plane grating or image plane grating is uniformly phase-shifted, with each phase shift being 2π(i-1) / Q, where Q is the number of phase shifts, and i = 1, 2, ... Q.

[0024] Furthermore, the object plane grating consists of two sets of one-dimensional Ronchi grating lines with mutually perpendicular directions.

[0025] The beneficial effects of this invention are:

[0026] This invention eliminates the influence of higher-order diffraction light without changing the existing structure of the transverse shearing interferometer. At the same time, considering that each diffraction order has a different light intensity coefficient, the shearing phase is accurately solved, eliminating the error caused by the checkerboard grating in the whole optical path diffraction calculation and improving the wavefront reconstruction accuracy. Attached Figure Description

[0027] Figure 1 This is a schematic diagram of the optical system of the transverse shearing interferometer of the present invention.

[0028] Figure 2 This is a schematic diagram of the surface grating of the present invention.

[0029] Figure 3 This is a schematic diagram of the image plane grating of the present invention.

[0030] Figure 4 This is a schematic diagram of the numerical aperture of the present invention.

[0031] Figure 5 This is a schematic diagram of the exit pupil plane of the optical system under test in this invention.

[0032] Figure 6 This is a flowchart of the method for eliminating diffraction calculation errors of checkerboard gratings according to the present invention.

[0033] Figure 7 These are comparison diagrams of the original wavefront and the reconstructed wavefront in an embodiment of the present invention, wherein (a) is a schematic diagram of the original wavefront and (b) is a schematic diagram of the reconstructed wavefront.

[0034] In the figure, 1 is the optical illumination system, 2 is the object plane grating, 3 is the optical system under test, 4 is the image plane grating, 5 is the optical detection element, 6 is the computer processing system, and 7 is the exit pupil of the optical system under test. Detailed Implementation

[0035] The present invention will be described in detail below with reference to the accompanying drawings and preferred embodiments. The objectives and effects of the present invention will become clearer as a result. The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

[0036] The transverse shearing interferometer optical system used in this invention is as follows: Figure 1 As shown, the optical system is arranged from left to right along the optical axis as follows: optical illumination system 1, object plane grating 2, optical system under test 3, image plane grating 4, and optical detection element 5. Optical illumination system 1 and object plane grating 2 are located on the object side of optical system under test 3, with object plane grating 2 located on the object plane of optical system under test 3; image plane grating 4 and optical detection element 5 are located on the image side of optical system under test 3, with image plane grating 4 located on the image plane of optical system under test 3. The optical axis direction is defined as the z-axis. According to the right-hand rule, the two directions perpendicular to the z-axis are defined as the x-axis and y-axis, respectively.

[0037] Optical illumination system 1 is used to emit incoherent light with a wavelength of λ.

[0038] like Figure 2 As shown, the object plane grating 2 consists of two sets of one-dimensional Ronchi grating lines with perpendicular directions. The left side is a one-dimensional Ronchi grating line perpendicular to the x-axis, and the right side is a one-dimensional Ronchi grating line perpendicular to the y-axis. The period of both sets of one-dimensional Ronchi grating lines is P1, and the duty cycle is 50%. They are used to perform spatial coherence modulation on the incoherent light emitted by the optical illumination system 1.

[0039] like Figure 3As shown, the image plane grating 4 is a checkerboard grating with its two diagonals perpendicular to the x-axis and y-axis directions, respectively. It has a period of P2 and a duty cycle of 50%, and is used to generate diffraction wavefronts of different orders.

[0040] The imaging magnification of the optical system under test 3 is M, and M = P2 / P1. The ratio of the periods of the image plane grating 4 to the object plane grating 2 is the same as the imaging magnification of the optical system under test 3, that is, P2:P1 = M.

[0041] The optical detection element 5 is selected from CCD or CMOS cameras, etc., and is used to receive optical images. The optical detection element 5 is connected to the computer processing system 6. With the intersection of the optical axis and the plane of the optical detection element 5 as the center, it detects a circular interference field region with a normalized radius of l. The light intensity data of the circular interference field region is transmitted to the computer processing system 6, where the shear phase is calculated and the wavefront is reconstructed.

[0042] like Figure 4 , 5 As shown, the image-side aperture angle of the optical system under test 3 is U. When the transverse shearing interferometer system is in air, the image-side numerical aperture s = sinU, and the normalized shear rate is...

[0043] like Figure 6 As shown, based on the above-described transverse shearing interferometer optical system, the present invention is realized through the following specific steps:

[0044] Step 1: Phase shift the grating along the x-axis to obtain the shear phase in the x-direction.

[0045] Specifically, this is achieved through the following sub-steps:

[0046] (1.1) Move the object plane grating 2 along the y-axis so that the light path passes through the one-dimensional Ronchi grating line perpendicular to the x-axis direction; uniformly shift the object plane grating 2 or the image plane grating 4 along the x-axis direction. The number of phase shifts Q is determined according to the accuracy of the phase shifter. Q can be 4, 8, 16, 32, 64, etc., and the phase shift amount each time is 2π(i-1) / Q, i=1,2,...Q; obtain a series of shearing interferograms in the x-direction. The light intensity data of the shearing interferogram in the circular interference field region is transmitted by the optical detection element 5 to the computer processing system 6 for calculation.

[0047] (1.2) The maximum diffraction order in the x-direction is determined by the values ​​of the normalized radius l and the normalized shear rate S as mMax = ceil(2*l / S), where ceil is the floor function.

[0048] In computer processing system 6, the expression for the interference field in the x-direction is:

[0049]

[0050] In the formula, I represents the light intensity of the interference field. A new coordinate system XOY is established on the exit pupil spherical surface of the optical system under test 3, where the direction of the X-axis is the same as that of the x-axis, and the direction of the Y-axis is the same as that of the y-axis. (X,Y) are the coordinates of the exit pupil spherical surface of the optical system under test 3, as shown below. Figure 5 The values ​​are shown, taking values ​​of [-l, l]. The checkerboard grating spectrum values ​​corresponding to the m-order diffraction in the X direction and the n-order diffraction in the Y direction are... For the wavefront to be measured, α i This is the phase shift.

[0051] The following matrix is ​​obtained using the least squares method:

[0052]

[0053]

[0054] The values ​​of parameters δ1 and δ2 are calculated according to equation (2):

[0055] A*(δ0 δ1 δ2 … … δ p δ q ) T =B (2)

[0056] If the maximum diffraction order mMax in the X direction is odd, then p = 3*mMax, q = 3*mMax + 1; if mMax is even, then p = 3*mMax - 1, q = 3*mMax.

[0057] The shear phase φ in the x-direction is obtained according to equation (3). -1,0 or φ 1,0 Value:

[0058]

[0059] In the formula, C -1，0 C is the light intensity coefficient when the -1st and 0th order diffracted beams interfere in the x-direction. 1，0 φ is the light intensity coefficient when the +1st order and the 0th order diffracted light interfere in the x-direction; -1,0 φ represents the shear phase between the -1st and 0th order diffracted beams in the x-direction. 1,0 The shear phase of the +1st and 0th order diffracted beams in the x-direction is given.

[0060] (1.3) The φ obtained in step (1.2) -1,0 or φ 1,0 By unwrapping the phase, the shear phase ΔW in the x-direction is obtained. x In this embodiment, the branch cutting method is used for phase unwrapping.

[0061] Step 2: Phase shift the grating along the y-axis to obtain the shear phase in the y-direction.

[0062] Specifically, this is achieved through the following sub-steps:

[0063] (2.1) Move the object plane grating 2 along the y-axis so that the light path passes through the one-dimensional Ronchi grating line perpendicular to the y-axis direction; uniformly shift the object plane grating 2 or the image plane grating 4 along the y-axis direction, with the number of phase shifts being Q and the phase shift amount being 2π(i-1) / Q each time, i=1,2,...Q, to obtain a series of shearing interference patterns in the y-direction. The light intensity data of the shearing interference pattern in the circular interference field region is transmitted by the optical detection element 5 to the computer processing system 6 for calculation.

[0064] (2.2) Based on the values ​​of the normalized radius l and the normalized shear rate S, the maximum diffraction order in the y-direction is determined as nMax = ceil(2 * l / S). In the computer processing system 6, the expression for the interference field in the y-direction is:

[0065]

[0066] The following matrix is ​​obtained using the least squares method:

[0067]

[0068]

[0069] The values ​​of parameters δ1′ and δ2′ are calculated according to equation (5):

[0070] A′*(δ0′ δ1′ δ2′ … … δ p ′ δ q ′) T =B' (5)

[0071] If the maximum diffraction order nMax in the Y direction is odd, then p = 3*nMax, q = 3*nMax + 1; if nMax is even, then p = 3*nMax - 1, q = 3*nMax.

[0072] The shear phase φ in the y-direction is obtained according to equation (6). -1,0 ′ or φ 1,0 The value of ′:

[0073]

[0074] In the formula, C -1，0 ′ is the light intensity coefficient when the -1st and 0th order diffracted beams interfere in the y direction, C 1，0 ′ is the light intensity coefficient when the +1st order diffracted light interferes with the 0th order diffracted light in the y direction; φ -1,0φ' represents the shear phase between the -1st and 0th order diffracted beams in the y-direction. 1,0 ′ represents the shear phase between the +1st and 0th order diffracted beams in the y direction.

[0075] (2.3) The φ obtained in step (2.2) -1,0 ′ or φ 1,0 By unwrapping the phase, we obtain the shear phase ΔW in the y-direction. y In this embodiment, the branch cutting method is used for phase unwrapping.

[0076] Step 3: Use the Zernike difference method to analyze ΔW x ΔW y The original wavefront is reconstructed to obtain the reconstructed wavefront. The original wavefront and the reconstructed wavefront are compared to obtain the reconstruction error.

[0077] In this embodiment, the original wavefront is used as the preset aberration surface of the optical system. A series of interferograms containing information from the original wavefront are generated in the x and y directions through simulation. The differential phase is then solved from the interferograms to obtain the reconstructed wavefront. For example... Figure 7 As shown, this is a comparison diagram of the original wavefront and the reconstructed wavefront in this embodiment. The calculated reconstruction accuracy reaches 99.9956%, that is, the reconstruction error is 0.0044%, which proves that the error elimination effect of the present invention is good.

[0078] This invention eliminates the influence of higher-order diffraction light without changing the existing structure of the transverse shearing interferometer, and also eliminates the calculation error of checkerboard grating diffraction caused by the different light intensity coefficients of each diffraction order, thereby improving the wavefront reconstruction accuracy.

[0079] It will be understood by those skilled in the art that the above descriptions are merely preferred examples of the invention and are not intended to limit the invention. Although the invention has been described in detail with reference to the foregoing examples, those skilled in the art can still modify the technical solutions described in the foregoing examples or make equivalent substitutions for some of the technical features. All modifications and equivalent substitutions made within the spirit and principles of the invention should be included within the scope of protection of the invention.

Claims

1. A method for eliminating calculation errors of checkerboard grating diffraction in a transverse shearing interferometer, characterized in that, Includes the following steps: Step 1: Fix the optical system under test on the optical platform of the transverse shearing interferometer. The optical illumination system emits incoherent light, which passes sequentially through the object plane grating, the optical system under test, and the image plane grating, and is received by the optical detection element. The object plane grating consists of two sets of one-dimensional grating lines with mutually perpendicular directions. The optical axis is z-axis. According to the right-hand rule, the two directions perpendicular to the z-axis are x-axis and y-axis, respectively. The image grating is a checkerboard grating, with its two diagonals perpendicular to the x-axis and y-axis, respectively. The ratio of the period of the image grating to that of the object grating is the same as the imaging magnification of the optical system under test. The duty cycle of both the image grating and the object grating is 50%. Step 2: Move the object grating so that the light rays pass through a one-dimensional grating line perpendicular to the x-axis, uniformly shifting the phase of the object or image grating to obtain a shearing interference pattern in the x-direction; use the least squares method to separate the shearing phase expression in the x-direction as follows: ； In the formula, , The parameters are obtained by the least squares method. This is the light intensity coefficient when the -1st and 0th order diffracted beams interfere in the x-direction. The intensity coefficient of light when the +1st order diffracted light interferes with the 0th order diffracted light in the x-direction; The shear phase of the -1st and 0th order diffracted beams in the x-direction is denoted as . The shear phase between the +1st and 0th order diffracted beams in the x-direction; For the obtained or Phase unwrapping is performed to obtain the shear phase in the x-direction; Step 3: Move the object grating so that the light rays pass through a one-dimensional grating line perpendicular to the y-axis, uniformly shifting the phase of the object or image grating to obtain a shearing interference pattern in the y-direction; use the least squares method to separate the shearing phase expression in the y-direction as follows: ； In the formula, , To obtain the parameters, The intensity coefficient is the light intensity coefficient when the -1st and 0th order diffracted beams interfere in the y-direction. The intensity coefficient of light when the +1st order diffracted light interferes with the 0th order diffracted light in the y direction; The shear phase of the -1st and 0th order diffracted beams in the y-direction is denoted as . The shear phase between the +1st and 0th order diffracted beams in the y direction; For the obtained or Perform phase unwrapping to obtain the shear phase in the y-direction; Step 4: Reconstruct the wavefronts represented by the shear phases in the x and y directions to obtain the reconstructed wavefronts with errors eliminated.

2. The method for eliminating diffraction calculation errors of a checkerboard grating in a transverse shearing interferometer according to claim 1, characterized in that, The phase unwrapping method used in steps two and three is the branch cutting method.

3. The method for eliminating diffraction calculation errors of a checkerboard grating in a transverse shearing interferometer according to claim 1, characterized in that, In step four, the differential Zernike method is used to reconstruct the wavefront represented by the shear phase in the x and y directions.

4. The method for eliminating diffraction calculation errors of a checkerboard grating in a transverse shearing interferometer according to claim 1, characterized in that, In steps two and three, the object plane grating or image plane grating is uniformly phase-shifted, with each phase shift amount being... Where Q is the number of phase shifts, .

5. The method for eliminating diffraction calculation errors of a checkerboard grating in a transverse shearing interferometer according to claim 1, characterized in that, The object plane grating consists of two sets of one-dimensional Ronchi grating lines with mutually perpendicular directions.

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