CN113847882B - Large-caliber vertical absolute inspection method based on gravity deformation and refractive index non-uniformity compensation - Google Patents
Large-caliber vertical absolute inspection method based on gravity deformation and refractive index non-uniformity compensation Download PDFInfo
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Abstract
The invention discloses a large-caliber vertical absolute inspection method based on gravity deformation and refractive index non-uniformity compensation, which comprises the following steps of: setting 3 large-caliber optical flat crystals, wherein the working surface of a flat crystal I is A in an initial state, the upper surface of a flat crystal II is B, the lower surface of the flat crystal II is C, the working surface of a flat crystal III is D, the flat crystal I is used as a reference mirror, the flat crystal II is used as a medium flat crystal, and the flat crystal III is used as a test mirror; six measurements were performed; obtaining the gravity deformation of the flat crystal II by using a finite element method, obtaining the refractive index non-uniformity of the flat crystal II by using the partial measurement combination, and compensating the influence of the two factors in the measurement; obtaining a surface shape rotational symmetry item by using the rotational symmetry function property; obtaining a surface shape rotation asymmetric term by using an error iteration method; the absolute surface shape distribution of the planes A, B, C and D is obtained. The invention compensates the influence of heavy-calibre optical flat crystal gravity deformation and refractive index non-uniformity on measurement, and can ensure the information of each surface shape at all frequencies.
Description
Technical Field
The invention belongs to the technical field of optical interference measurement, and particularly relates to a large-caliber vertical absolute inspection method based on gravity deformation and refractive index non-uniformity compensation.
Background
The large-aperture laser interferometer is a high-precision non-contact optical device and can be used for astronomical detection, chip manufacturing, high-energy laser and the like. The quality of optical elements, such as the precision of the surface, has affected the development of these fields. At present, absolute inspection is the most efficient and popular method to achieve real surface detection. When large-caliber vertical absolute inspection is carried out, particularly inspection of the caliber larger than 300mm, surface deformation of the flat crystal caused by gravity and refractive index nonuniformity during transmission of the flat crystal are two key factors influencing the detection effect of the flat crystal. Therefore, the method for analyzing the compensation of the two factors for the large-caliber vertical absolute inspection is a hot spot problem at present. When the compensation of two factors is realized, the true and effective absolute surface shape distribution can be obtained.
In 1967, G.Schulz and J.Schwider originally reported a reference-free absolute surface shape solving method, namely a three-surface mutual inspection method, which can obtain two absolute contour lines on the diameter. Then, fritz proposes a three-surface mutual inspection method based on Zernike polynomial fitting, which can effectively obtain a full-aperture absolute surface shape, but loses part of high-frequency information. Furthermore, ai and Wyant propose a three-face mutual inspection method based on an odd-even function method, divide the face shape into an odd-even term, an even-odd term, an even-even term and an odd-odd term, and respectively solve. The method can obtain complete frequency information. However, the above methods are all suitable for the absolute inspection of small caliber, and the influence of the flat crystal deformation and the refractive index non-uniformity on the test result is neglected, which cannot be avoided in the large caliber vertical absolute inspection.
Disclosure of Invention
The invention aims to provide an effective large-caliber vertical absolute inspection method based on gravity deformation and refractive index non-uniformity compensation.
The technical solution for realizing the purpose of the invention is as follows: a large-caliber vertical absolute inspection method based on gravity deformation and refractive index non-uniformity compensation comprises the following steps:
step 1, setting flat crystals: setting 3 large-caliber optical flat crystals I, II and III respectively, wherein the working surface of the flat crystal I is A in an initial state, the upper surface of the flat crystal II is B, the lower surface of the flat crystal II is C, the working surface of the flat crystal III is D, the flat crystal I is used as a reference mirror, the flat crystal II is placed in the middle and used as an intermediate flat crystal, and the flat crystal III is used as a test mirror;
and 2, performing six measurements: 1) The relative measurement of plane A and plane B, including the gravitational deformation of the plate II, is designated M 1 (ii) a 2) The relative measurement of plane A and plane C, which is transmitted through plane B and contains the gravitational deformation and refractive index non-uniformity of the flat crystal II, denoted M 2 (ii) a 3) Plane A and plane D are measured through plane B and plane C and contain refractive index non-uniformity of the slab II, but the gravitational distortion of the slab II is cancelled out in the optical path calculation and is denoted as M 3 (ii) a 4) Taking the flat crystal II away, and directly performing relative measurement between the plane A and the plane D, and recording as M 4 (ii) a 5) The flat crystal II is turned over along the y-axis and then put back to the middle position, and the plane A and the plane C are relatively measured, wherein the gravity deformation of the flat crystal II is included and is marked as M 5 (ii) a 6) Rotating the flat crystal II along the z-axis counterclockwise by a certain angle, and performing relative measurement on the plane A and the plane C, wherein the gravity deformation of the flat crystal II is included and is marked as M 6 ；
Step 3, obtaining the gravity deformation of the flat crystal II by using a finite element method, obtaining the refractive index non-uniformity of the flat crystal II by using the partial measurement combination, and compensating the influence of the gravity deformation and the refractive index non-uniformity in the measurement;
step 4, solving the rotational symmetry item of the surface shape by using the property of the rotational symmetry function;
step 5, solving a surface shape rotation asymmetric term by using an error iteration method;
and 6, obtaining the absolute surface shape distribution of the planes A, B, C and D.
Compared with the prior art, the invention has the following remarkable advantages: (1) The reference mirror is kept in place, does not need to be disassembled, and is particularly suitable for the test adjustment of a large-aperture interferometer; (2) The influence of the gravity deformation and the refractive index non-uniformity of the large-caliber optical flat crystal on the test result is compensated; (3) The reconstruction precision is high, and the information of each surface shape at all frequencies can be ensured.
The present invention is described in further detail below with reference to the attached drawing figures.
Drawings
FIG. 1 is a diagram of a vertical absolute inspection apparatus for large-aperture optical flat crystals.
Fig. 2 shows a measurement method of a vertical absolute test.
Fig. 3 shows the amount of flat crystal deformation obtained by FEM simulation.
Detailed Description
The invention relates to a large-caliber vertical absolute inspection method based on gravity deformation and refractive index non-uniformity compensation, which comprises the following steps of:
step 1, 3 large-caliber optical flat crystals are respectively I, II and III, the working surface of the flat crystal I is A in an initial state, the upper surface of the flat crystal II is B, the lower surface of the flat crystal II is C, the working surface of the flat crystal III is D, the flat crystal I is used as a reference mirror, the flat crystal II is placed in the middle to be used as an intermediate flat crystal, and the flat crystal III is used as a test mirror;
step 2, the required six measurements are performed according to the algorithm of the present invention. 1) The relative measurement of plane A and plane B, including the gravitational deformation of the plate II, is designated M 1 (ii) a 2) The relative measurements of plane A and plane C, which are transmitted through plane B and which include the gravitational deformation and refractive index non-uniformity of the plano-crystal II, denoted M 2 (ii) a 3) Plane A and plane D are measured through plane B and plane C and contain refractive index non-uniformity of plane II, but the gravitational distortion of plane II is cancelled out in the optical path calculation and is denoted as M 3 (ii) a 4) Taking off the flat crystal II, and directly carrying out the opposite between the plane A and the plane DMeasured and recorded as M 4 (ii) a 5) Turning the flat crystal II along the y-axis, returning to the middle position, and performing relative measurement on the plane A and the plane C, wherein the gravity deformation of the flat crystal II is included and is marked as M 5 (ii) a 6) Rotating the flat crystal II along the z-axis counterclockwise by a certain angle, and performing relative measurement on the plane A and the plane C, wherein the gravity deformation of the flat crystal II is included and is marked as M 6 ；
Step 3, obtaining the gravity deformation of the flat crystal II by using a Finite Element Method (FEM), obtaining the refractive index non-uniformity of the flat crystal II by using the partial measurement combination, and compensating the influence of the two factors in the measurement;
step 4, solving a surface-shaped rotational symmetry item by using the properties of the rotational symmetry function;
step 5, solving a surface shape rotation asymmetric term by using an error iteration method;
and 6, obtaining the absolute surface shape distribution of the planes A, B, C and D.
Further, the flat crystal setting in step 1 is specifically as follows:
there are 3 large-caliber optical flat crystals, I, II and III, respectively, the working surface of the flat crystal I in the initial state is a, the upper surface of the flat crystal II is B and the lower surface is C, the working surface of the flat crystal III is D, the flat crystal I is used as a reference mirror, the flat crystal II is placed in the middle as an intermediate flat crystal, and the flat crystal III is used as a test mirror.
Further, the six measurements of the present invention described in step 2 are specifically as follows:
1) The relative measurements of plane A and plane B, including the gravitational deformation of the plate II, are taken and are denoted M 1 ；
M 1 (x，y)＝A(-x，y)+B(x，y)-G(x，y) (1)
Wherein A (-x, y) represents the surface shape of A, B (x, y) represents the surface shape of B, and G (x, y) represents the gravitational deformation of the flat crystal II.
2) The relative measurements of plane A and plane C, which are transmitted through plane B and which contain the gravitational deformation and refractive index inhomogeneity of the flat crystal II, denoted M 2 ；
M 2 (x，y)＝A(-x，y)+(1-n)B(x，y)-nC(-x，y)-δ(x，y)-G(x，y) (2)
In the formula, a (-x, y) represents a surface shape of a, B (x, y) represents a surface shape of B, C (-x, y) represents a surface shape of C, n represents a refractive index constant of optical flat (n = 1.5), δ (x, y) represents refractive index non-uniformity of flat crystal II, and G (x, y) represents gravitational deformation of flat crystal II.
3) Plane A and plane D are measured through plane B and plane C and contain refractive index non-uniformity of the slab II, but the gravitational distortion of the slab II is cancelled out in the optical path calculation and is denoted as M 3 ；
M 3 (x，y)＝A(-x，y)+(1-n)B(x，y)+(1-n)C(-x，y)+D(x，y)-δ(x，y) (3)
In the formula, a (-x, y) represents a surface shape of a, B (x, y) represents a surface shape of B, C (-x, y) represents a surface shape of C, D (x, y) represents a surface shape of D, n represents a refractive index constant of the optical flat (n = 1.5), and δ (x, y) represents refractive index non-uniformity of the flat II.
4) Taking off the flat crystal II, directly performing relative measurement between the plane A and the plane D, and recording as M 4 ；
M 4 (x, y) = a (-x, y) + D (x, y) (4) wherein a (-x, y) represents the profile of a and D (x, y) represents the profile of D.
5) The flat crystal II is turned over along the y-axis and then returned to the middle position, and the plane A and the plane C are relatively measured, wherein the gravity deformation of the flat crystal II is included and is marked as M 5 ；
M 5 (x，y)＝A(-x，y)+C(x，y)-G(x，y) (5)
In the formula, A (-x, y) represents the surface shape of A, C (x, y) represents the surface shape of C, and G (x, y) represents the gravitational deformation of the flat crystal II.
6) Rotating the flat crystal II along the z-axis counterclockwise by a certain angle, and performing relative measurement on the plane A and the plane C, wherein the gravity deformation of the flat crystal II is included and is marked as M 6 ；
Wherein A (-x, y) represents the surface shape of A, and C (x, y) representsThe surface shape of the C is shown in the specification,
representing the plane C taken along the z-axis
And (3) degree rotation operation, and G (x, y) represents the gravity deformation of the flat crystal II.
Further, the influence of gravity deformation and refractive index non-uniformity in the measurement is compensated in step 3, which is specifically as follows:
1) The current elastic deformation theory and Finite Element Method (FEM) simulation have obtained better consistency, and the gravity deformation of the flat crystal II can be obtained according to the FEM. The theoretical formula for gravity deformation of the flat crystal II can be expressed as:
wherein ρ represents the density of the flat crystal, h is the thickness of the flat crystal, a is the radius of the flat crystal,
is the distance from a certain point (x, y) to the center of the plate, K is the bending stiffness, E is the Young's modulus of the plate, and μ is the Poisson's ratio of the plate.
2) The refractive index non-uniformity of the flat crystal II can be expressed as:
δ(x，y)＝(n-1)[M 2 (x，y)-M 1 (x，y)]-n[M 3 (x，y)-M 4 (x，y)] (8)
wherein n represents a refractive index constant of optical flat (n = 1.5), and M 1 (x，y)、M 2 (x，y)、M 3 (x，y)、M 4 (x, y) respectively represent the first four measurements in step 2.
3) Then, six measurements that compensate for the gravitational deformation and refractive index non-uniformity can be represented by the following equation, with W replacing the original M as the measurement result:
further, the step 4 of obtaining the rotational symmetry term of the surface shape by using the rotational symmetry function property specifically includes:
1) Selecting W 1 、W 2 、W 5 Its rotational symmetry function can be expressed as:
wherein, W 1s 、W 2s 、W 5s Is represented by W 1 、W 2 、W 5 Respective rotational symmetry function, A s (x，y)、B s (x，y)、C s (x, y) represent rotational symmetry terms of the planes a, B, C, respectively, and n represents a refractive index constant of the optical flat (n = 1.5).
2) Then, the rotational symmetry terms of the planes a, B, C are determined:
further, the step 5 of obtaining the surface shape rotation asymmetry term by using an error iteration method specifically includes:
1) Using W 5 And W 6 Establishing a rotational difference equation P (x, y):
wherein, C a (x, y) are rotationally asymmetric terms of plane C,
representing the plane C taken along the z-axis
And (4) performing rotation operation.
2) EstablishingThe error iteration relation of the rotation difference equation is used for obtaining the rotation asymmetry term [ C ] of the plane C s (x，y)] new ：
Wherein, C a The initial value of (x, y) is set to 0 and Δ P (x, y) is the known difference of the measurements [ P (x, y)] exp And the difference of the changing difference equation P (x, y), k is an iteration factor (must be greater than 1),
representing the performance of data
And (5) performing degree rotation operation.
Further, the absolute surface shape distribution of each of the planes a, B, C, and D obtained in step 6 is specifically as follows:
1) Obtaining a rotational symmetry item C by using the step 4 and the step 5 s (x, y) and rotationally asymmetric term [ C s (x，y)] new Combining to obtain the complete absolute surface shape of the plane C:
C(x，y)＝C s (x，y)+[C a (x，y)] new (14)
2) Obtaining absolute surface shapes of the planes A, B and D:
the invention is described in further detail below with reference to the figures and the embodiments.
Examples
The invention relates to a large-caliber vertical absolute inspection method based on gravity deformation and refractive index non-uniformity compensation, which comprises the following steps of:
step 1, as shown in fig. 1, performing a flat crystal setting, specifically as follows:
the optical flat crystal device comprises 3 large-caliber optical flat crystals, namely I, II and III, wherein the working surface of the flat crystal I is A in an initial state, the upper surface of the flat crystal II is B, the lower surface of the flat crystal II is C, the working surface of the flat crystal III is D, the flat crystal I is used as a reference mirror, the flat crystal II is placed in the middle to be used as a medium flat crystal, and the flat crystal III is used as a test mirror.
Step 2, as shown in fig. 2, six measurements of the present invention are performed, specifically as follows:
1) The relative measurement of plane A and plane B, including the gravitational deformation of the plate II, is designated M 1 ；
M 1 (x，y)＝A(-x，y)+B(x，y)-G(x，y) (1)
Wherein A (-x, y) represents the surface shape of A, B (x, y) represents the surface shape of B, and G (x, y) represents the gravitational deformation of the flat crystal II.
2) The relative measurements of plane A and plane C, which are transmitted through plane B and which contain the gravitational deformation and refractive index inhomogeneities of the flat crystal II, are denoted M 2 ；
M 2 (x，y)＝A(-x，y)+(1-n)B(x，y)-nC(-x，y)-δ(x，y)-G(x，y) (2)
In the formula, C (-x, y) represents a surface shape of C, n represents a refractive index constant of an optical flat, n =1.5, and δ (x, y) represents refractive index nonuniformity of flat II.
3) Plane A and plane D are measured through plane B and plane C and contain refractive index non-uniformity of the slab II, but the gravitational distortion of the slab II is cancelled out in the optical path calculation and is denoted as M 3 ；
M 3 (x，y)＝A(-x，y)+(1-n)B(x，y)+(1-n)C(-x，y)+D(x，y)-δ(x，y) (3)
In the formula, D (x, y) represents a surface shape of D.
4) Taking off the flat crystal II, directly performing relative measurement between the plane A and the plane D, and recording as M 4 ；
M 4 (x，y)＝A(-x，y)+D(x，y) (4)
5) Turning the flat crystal II along the y-axis, returning to the middle position, and performing relative measurement on the plane A and the plane C, wherein the gravity deformation of the flat crystal II is included and is marked as M 5 ；
M 5 (x，y)＝A(-x，y)+C(x，y)-G(x，y) (5)
6) Rotating the flat crystal II along the z-axis counterclockwise by a certain angle, and performing relative measurement on the plane A and the plane C, wherein the gravity deformation of the flat crystal II is included and is marked as M 6 ；
And 3, compensating the influence of gravity deformation and refractive index nonuniformity in measurement, specifically as follows:
1) The current elastic deformation theory and Finite Element Method (FEM) simulation have obtained better consistency, and as shown in fig. 3, the gravity deformation of the flat crystal II can be obtained according to the FEM. The theoretical formula of gravity deformation of the flat crystal II can be expressed as follows:
where ρ is the density of the flat crystal, h is the thickness of the flat crystal, a is the radius of the flat crystal,
is the distance from a certain point (x, y) to the center of the plate, K is the bending stiffness, E is the Young's modulus of the plate, and μ is the Poisson's ratio of the plate.
2) The refractive index non-uniformity of the flat crystal II can be expressed as:
δ(x，y)＝(n-1)[M 2 (x，y)-M 1 (x，y)]-n[M 3 (x，y)-M 4 (x，y)] (8)
wherein n represents a refractive index constant of optical flat (n = 1.5), and M 1 (x，y)、M 2 (x，y)、M 3 (x，y)、M 4 (x, y) represent the first four measurements in step 2, respectively.
3) Then, six measurements that compensate for the gravitational deformation and refractive index non-uniformity can be represented by the following equation, with W instead of M as the measurement result:
step 4, solving the rotational symmetry item of the surface shape by using the properties of the rotational symmetry function, which comprises the following steps:
1) Selecting W 1 、W 2 、W 5 Its rotational symmetry function can be expressed as:
wherein, W 1s 、W 2s 、W 5s Is represented by W 1 、W 2 、W 5 Respective rotational symmetry function, A s (x，y)、B s (x，y)、C s (x, y) represent rotational symmetry terms of the planes a, B, C, respectively, and n represents a refractive index constant of the optical flat (n = 1.5).
2) Then, the rotational symmetry terms of the planes a, B, C are determined:
step 5, solving the surface shape rotation asymmetric term by using an error iteration method, specifically as follows:
1) By means of W 5 And W 6 Establishing a rotational difference equation P (x, y):
wherein, C a (x, y) are rotationally asymmetric terms of plane C,
representing the plane C taken along the z-axis
And (4) performing rotation operation.
2) Establishing an error iteration relation of a rotation difference equation to obtain a rotation asymmetry term [ C ] of the plane C s (x，y)] new ：
Wherein, C a The initial value of (x, y) is set to 0 and Δ P (x, y) is the known difference of the measurements [ P (x, y)] exp And a varying difference equation P (x, y), k is an iteration factor (must be greater than 1),
representing data by
And (5) performing degree rotation operation.
And 6, obtaining respective absolute surface shape distribution of the planes A, B, C and D, wherein the absolute surface shape distribution is as follows:
1) Obtaining a rotational symmetry item C by using the step 4 and the step 5 s (x, y) and rotationally asymmetric term [ C s (x，y)] new Combining to obtain the complete absolute surface shape of the plane C:
C(x，y)＝C s (x，y)+[C a (x，y)] new (14)
2) Obtaining the absolute surface shapes of the planes A, B and D:
in conclusion, the invention keeps the reference mirror in place without disassembly, and is particularly suitable for the test adjustment of a large-aperture interferometer; the influence of the gravity deformation and refractive index nonuniformity of the large-caliber optical flat crystal on the measurement is compensated; the reconstruction precision is high, and the information of each surface shape at all frequencies can be ensured.
Claims (8)
1. A large-caliber vertical absolute inspection method based on gravity deformation and refractive index non-uniformity compensation is characterized by comprising the following steps:
step 1, setting a flat crystal: setting 3 large-caliber optical flat crystals I, II and III respectively, wherein the working surface of the flat crystal I is A in an initial state, the upper surface of the flat crystal II is B, the lower surface of the flat crystal II is C, the working surface of the flat crystal III is D, the flat crystal I is used as a reference mirror, the flat crystal II is placed in the middle to be used as an intermediate flat crystal, and the flat crystal III is used as a test mirror;
step 2, six measurements are carried out: 1) The relative measurements of plane A and plane B, including the gravitational deformation of the plate II, are recorded as M 1 (ii) a 2) The relative measurement of plane A and plane C, which is transmitted through plane B and contains the gravitational deformation and refractive index non-uniformity of the flat crystal II, denoted M 2 (ii) a 3) Plane A and plane D are measured through plane B and plane C and contain refractive index non-uniformities of plate II, but the gravitational distortions of plate II are cancelled out in the optical path calculation and are denoted M 3 (ii) a 4) Taking the flat crystal II away, and directly performing relative measurement between the plane A and the plane D, and recording as M 4 (ii) a 5) The flat crystal II is turned over along the y-axis and then put back to the middle position, and the plane A and the plane C are relatively measured, wherein the gravity deformation of the flat crystal II is included and is marked as M 5 (ii) a 6) Rotating the flat crystal II along the z-axis counterclockwise by a certain angle, and performing relative measurement on the plane A and the plane C, wherein the gravity deformation of the flat crystal II is included and is marked as M 6 ；
Step 3, obtaining the gravity deformation of the flat crystal II by using a finite element method, obtaining the refractive index non-uniformity of the flat crystal II by using the partial measurement combination, and compensating the influence of the gravity deformation and the refractive index non-uniformity in the measurement;
step 4, solving the rotational symmetry item of the surface shape by using the property of the rotational symmetry function;
step 5, solving a surface-shaped rotation asymmetric term by using an error iteration method;
and 6, obtaining the absolute surface shape distribution of the planes A, B, C and D.
2. The large-caliber vertical absolute inspection method based on gravity deformation and refractive index non-uniformity compensation according to claim 1, wherein the six measurements in step 2 are as follows:
1) The relative measurement of plane A and plane B, including the gravitational deformation of the plate II, is designated M 1 ；
M 1 (x,y)＝A(-x,y)+B(x,y)-G(x,y) (1)
In the formula, A (-x, y) represents the surface shape of A, B (x, y) represents the surface shape of B, and G (x, y) represents the gravity deformation of the flat crystal II;
2) The relative measurements of plane A and plane C, which are transmitted through plane B and which include the gravitational deformation and refractive index non-uniformity of the plano-crystal II, denoted M 2 ；
M 2 (x,y)＝A(-x,y)+(1-n)B(x,y)-nC(-x,y)-δ(x,y)-G(x,y) (2)
Wherein C (-x, y) represents the surface shape of C, n represents the refractive index constant of optical flat crystal, and δ (x, y) represents the refractive index nonuniformity of flat crystal II;
3) Plane A and plane D are measured through plane B and plane C and contain refractive index non-uniformities of plate II, but the gravitational distortions of plate II are cancelled out in the optical path calculation and are denoted M 3 ；
M 3 (x,y)＝A(-x,y)+(1-n)B(x,y)+(1-n)C(-x,y)+D(x,y)-δ(x,y) (3)
Wherein D (x, y) represents the surface shape of D;
4) Taking the flat crystal II away, and directly performing relative measurement between the plane A and the plane D, and recording as M 4 ；
M 4 (x,y)＝A(-x,y)+D(x,y) (4)
5) The flat crystal II is turned over along the y axis and then put back to the middle position, the plane A and the plane C carry out relative measurement, wherein the gravity deformation of the flat crystal II is included and recordedAs M 5 ；
M 5 (x,y)＝A(-x,y)+C(x,y)-G(x,y) (5)
6) Rotating the flat crystal II along the z-axis counterclockwise by a certain angle, and performing relative measurement on the plane A and the plane C, wherein the gravity deformation of the flat crystal II is included and is marked as M 6 ；
In the formula (I), the compound is shown in the specification,
representing the plane C taken along the z-axis
And (4) performing rotation operation.
3. The large-caliber vertical absolute inspection method based on gravity deformation and refractive index non-uniformity compensation of claim 1, wherein n =1.5.
4. The vertical absolute inspection method of claim 2, wherein the compensation of gravity deformation and refractive index non-uniformity in measurement in step 3 is as follows:
1) The current elastic deformation theory and finite element method simulation have obtained better consistency, and the gravity deformation of the flat crystal II can be obtained according to FEM; the theoretical formula of gravity deformation of the flat crystal II can be expressed as follows:
where ρ is the density of the flat crystal, h is the thickness of the flat crystal, a is the radius of the flat crystal,
is the distance from a point (x, y) to the center of the plate, K is the bending stiffness, E is the Young's modulus of the plate, μ is the Poisson's ratio of the plate;
2) The refractive index non-uniformity of the plano-crystal ii can be expressed as:
δ(x,y)＝(n-1)[M 2 (x,y)-M 1 (x,y)]-n[M 3 (x,y)-M 4 (x,y)] (8)
3) Six measurements that compensate for gravitational deformation and refractive index non-uniformity can be represented by the following equation, with W replacing the original M as the measurement:
5. the large-caliber vertical absolute inspection method based on gravity deformation and refractive index non-uniformity compensation according to claim 4, wherein the rotational symmetry term of the surface shape is obtained by using the rotational symmetry function property in the step 4, and the method comprises the following specific steps:
1) Selecting W 1 、W 2 、W 5 The rotational symmetry function can be expressed as:
wherein, W 1s 、W 2s 、W 5s Is represented by W 1 、W 2 、W 5 Respective rotational symmetry function, A s (x,y)、B s (x,y)、C s (x, y) represents rotational symmetry terms for planes A, B, C, respectively;
2) Then, the rotational symmetry terms of the planes a, B, C are determined:
6. the large-caliber vertical absolute inspection method based on gravity deformation and refractive index non-uniformity compensation according to claim 5, wherein the rotational asymmetry term of the surface shape is obtained by using an error iteration method in step 5, and specifically comprises the following steps:
1) By means of W 5 And W 6 Establishing a rotational difference equation P (x, y):
wherein, C a (x, y) is a rotationally asymmetric term of plane C;
2) Establishing an error iteration relation of a rotation difference equation to obtain a rotation asymmetry term [ C ] of the plane C a (x,y)] new ：
Wherein, C a The initial value of (x, y) is set to 0 and Δ P (x, y) is the known difference of the measurements [ P (x, y)] exp And a varying difference equation P (x, y), k being an iteration factor,
representing data by
And (5) performing degree rotation operation.
7. The large-caliber vertical absolute inspection method based on gravity deformation and refractive index non-uniformity compensation of claim 6, wherein k > 1.
8. The large-caliber vertical absolute inspection method based on gravity deformation and refractive index non-uniformity compensation according to claim 6, wherein the absolute surface profile distribution of each of the planes A, B, C and D is obtained in step 6, and specifically as follows:
1) Obtaining the rotation symmetry item C by using the step 4 and the step 5 s (x, y) and rotationally asymmetric term [ C a (x,y)] new And (3) combining to obtain a complete absolute surface shape of the plane C:
C(x,y)＝C s (x,y)+[C a (x,y)] new (14)
2) Obtaining the absolute surface shapes of the planes A, B and D:
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