Optical plane absolute detection method based on minimum norm solution

Abstract

The application discloses an optical plane absolute detection method based on a minimum norm solution and belongs to the technical field of precision measurement, and particularly relates to an absolute detection method for a plane; the application utilizes two-by-two combination of a reference flat crystal, a reflection flat crystal and an auxiliary flat crystal, establishes a linear equation set and obtains a minimum norm solution of the linear equation set, so that the absolute overall shape of each flat crystal can be obtained, the precision of an interferometer for measuring an optical plane is effectively improved, and the influence of the reference flat crystal surface shape error is reduced; in addition, only one-time rotation of the auxiliary flat crystal by 90 degrees or 180 degrees is needed, so that the precision of the absolute detection method based on an odd-even function can be achieved, and the auxiliary flat crystal does not need to be rotated for multiple times, so that the detection process is simpler.

Description

TECHNICAL FIELD

[0001] The optical plane absolute detection method based on minimum norm solution belongs to the technical field of precision measurement, and particularly relates to an optical plane absolute detection method. BACKGROUND

[0002] Large aperture optical plane elements are widely used in optical systems of extreme ultraviolet lithography, inertial confinement fusion and other devices, and the surface shape precision of the elements is required to reach nanometer level or even sub-nanometer level. An interferometer is usually used to measure the surface shape of the optical plane element, and the precision of the relative detection method depends largely on the surface shape of the standard reference flat. When the precision requirement of the surface shape measurement is very high, an absolute detection method is needed to separate and remove the surface shape error of the reference flat from the detection result, break through the limitation of the surface shape of the reference flat of the interferometer itself, and improve the surface shape detection precision of the optical plane element.

[0003] The three-flat mutual detection method is the most commonly used method in the absolute detection technology. The earliest three-flat mutual detection method uses three flat mirrors with similar surface shape precision to detect in pairs, and obtains the absolute surface shape of each plane on a line segment passing through the center of the circle through simple addition and subtraction operation. Then, through single rotation measurement of one of the flat mirrors, the absolute surface shape on another line segment passing through the center of the circle is additionally obtained, but the complete surface shape information of the reference mirror cannot be obtained. In order to obtain the complete surface shape of the reference flat and improve the accuracy of the absolute detection, the flat mirror needs to be rotated multiple times, and the measurement steps of the flat mirror operation are usually six or more times, and the reference flat of the interferometer needs to be unloaded and replaced, which is easy to introduce operation error in the implementation process and affect the measurement precision.

[0004] An optical plane absolute detection method based on the odd-even function method does not need to change the reference flat and the reflecting flat of the interferometer, can avoid the operation error in the loading and unloading process of the flat mirror, and is more suitable for large aperture interferometers. However, this method needs to rotate the auxiliary flat mirror in the three-flat mutual detection multiple times, and the whole process includes at least nine operation and measurement steps, which is time-consuming. SUMMARY

[0005] In view of the above problems, the present application provides an optical plane absolute detection method based on minimum norm solution, which only needs to rotate the auxiliary flat mirror in the three-flat mutual detection by 90°, performs four measurements in total, establishes a linear equation set and obtains the minimum norm solution, so as to obtain the absolute complete surface shape of each flat mirror, which can effectively improve the precision of the interferometer in measuring the optical plane and reduce the influence of the surface shape error of the reference flat. Compared with the existing absolute detection method based on the odd-even function, the detection precision is equivalent, and only the auxiliary flat mirror needs to be rotated by 90° or 180° at a time, without the need for multiple rotations of the auxiliary flat mirror, so that the detection process is simpler.

[0006] The purpose of the present application is achieved as follows:

[0007] An optical flat absolute testing method based on minimum norm solution, defined as:

[0008] The reference flat of the interferometer is A flat;

[0009] The reflecting flat of the interferometer is B flat;

[0010] The auxiliary flat used in absolute testing is C flat;

[0011] Comprising the following steps:

[0012] Step a, A flat and B flat interference, the measurement value is M1(x,y)=A(-x,y)+B(x,y);

[0013] Wherein, A(-x,y) is the surface error of A flat(-x,y) position, B(x,y) is the surface error of B flat(x,y) position;

[0014] Step b, keep A flat still, A flat and C flat interference, the measurement value is M2(x,y)=A(-x,y)+C(x,y);

[0015] Wherein, C(x,y) is the surface error of C flat(x,y) position;

[0016] Step c, keep B flat still, B flat and C flat interference, the measurement value is M3(x,y)=C(-x,y)+B(x,y);

[0017] Wherein, C(-x,y) is the surface error of C flat(-x,y) position;

[0018] Step d, rotate C flat 90° or 180°, B flat and C flat combined measurement, in the case of C flat rotating 90°, the measurement value is M4(x,y)=C(-y,-x)+B(x,y), in the case of C flat rotating 180°, the measurement value is M4(x,y)=C(x,-y)+B(x,y);

[0019] Wherein, C(-y,-x) is the surface error of C flat(-y,-x) position, C(x,-y) is the surface error of C flat(x,-y) position;

[0020] Step e, use step b+step c-step a, to obtain:

[0021] C(x,y)+C(-x,y)=M2(x,y)+M3(x,y)-M1(x,y)=N1(x,y)

[0022] Step f, use step b+step d-step a, to obtain:

[0023] C(x, y) + C(-y, -x) = M2(x, y) + M4(x, y) - M1(x, y) = N2(x, y)

[0024] Step g, steps e and f are combined to construct a linear equation group:

[0025] In the case of C surface rotation 90°,

[0026]

[0027] In the case of C surface rotation 180°,

[0028]

[0029] Step h, traverse all values of x and y to form AX = N linear equation group, wherein the A matrix is a matrix composed of 0 and 1, the X matrix is a matrix composed of C surface form error, the N matrix is a matrix composed of step e and step f derivation results, and the minimum norm solution of X is X = A T (AA T ) -1 N; that is, C(x, y), C(-x, y) and C(-y, -x) are obtained;

[0030] Step i, C(x, y), C(-x, y) and C(-y, -x) are respectively brought into steps a, steps b, steps c and steps d, and the surface form errors of A surface and B surface are obtained.

[0031] Advantages:

[0032] First, the absolute detection method of the optical plane based on the minimum norm solution, the reference flat crystal, the reflection flat crystal and the auxiliary flat crystal are combined in pairs, a linear equation group is established and the minimum norm solution is solved, that is, the absolute overall shape of each flat crystal is obtained, the precision of the interferometer measuring optical plane is effectively improved, and the influence of the reference flat crystal surface form error is reduced.

[0033] Second, the absolute detection method of the optical plane based on the minimum norm solution, only the auxiliary flat crystal is rotated 90° or 180° at a time, the precision of the absolute detection method based on the odd-even function can be reached, and the auxiliary flat crystal does not need to be rotated multiple times, so that the detection process is simpler. DETAILED DESCRIPTION

[0034] Figure 1 It is a process diagram for obtaining measurement values by combining the reference flat crystal, the reflection flat crystal and the auxiliary flat crystal in pairs.

[0035] Figure 2 It is a comparison result of the method of the application and the absolute detection method of the plane based on the odd-even function. DETAILED DESCRIPTION

[0036] The specific embodiments of the present application are described in further detail below with reference to the accompanying drawings. Specific embodiment one

[0038] The optical plane absolute detection method based on the minimum norm solution in this specific embodiment defines:

[0039] The reference flat of the interferometer is A face;

[0040] The reflecting flat of the interferometer is B face;

[0041] The auxiliary flat used in the absolute detection is C face;

[0042] It is characterized by comprising the following steps:

[0043] Step a, A face and B face interference, to obtain the measurement value M1(x, y) = A(-x, y) + B(x, y);

[0044] Wherein, A(-x, y) is the surface error of A face at (-x, y) position, and B(x, y) is the surface error of B face at (x, y) position;

[0045] Step b, keep A face stationary, A face and C face interference, to obtain the measurement value M2(x, y) = A(-x, y) + C(x, y);

[0046] Wherein, C(x, y) is the surface error of C face at (x, y) position;

[0047] Step c, keep B face stationary, B face and C face interference, to obtain the measurement value M3(x, y) = C(-x, y) + B(x, y);

[0048] Wherein, C(-x, y) is the surface error of C face at (-x, y) position;

[0049] Step d, rotate C face by 90° or 180°, B face and C face combined measurement, in the case of C face rotating by 90°, to obtain the measurement value M4(x, y) = C(-y, -x) + B(x, y), in the case of C face rotating by 180°, to obtain the measurement value M4(x, y) = C(x, -y) + B(x, y);

[0050] Wherein, C(-y, -x) is the surface error of C face at (-y, -x) position, and C(x, -y) is the surface error of C face at (x, -y) position;

[0051] Taking the case of C face rotating by 180° as an example, the detection processes of steps a, b, c and d are as shown in Figure 1 ;

[0052] Step e, use step b+step c-step a, get:

[0053] C(x, y) + C(-x, y) = M2(x, y) + M3(x, y) - M1(x, y) = N1(x, y)

[0054] Step f, use step b+step d-step a, get:

[0055] C(x, y) + C(-y, -x) = M2(x, y) + M4(x, y) - M1(x, y) = N2(x, y)

[0056] Step g, step e and step f are combined to construct a linear equation group:

[0057] In the case of C surface rotation 90°,

[0058]

[0059] In the case of C surface rotation 180°,

[0060]

[0061] Step h, traverse all values of x and y, form AX=N linear equation group, where A matrix is composed of 0 and 1, X matrix is composed of C surface form error, N matrix is composed of step e and step f derivation result, the minimum norm solution of X is X=A T (AA T ) -1 N; that is, C(x, y), C(-x, y) and C(-y, -x) are obtained;

[0062] Step i, C(x, y), C(-x, y) and C(-y, -x) are brought into step a, step b, step c and step d respectively, and the surface form errors of A surface and B surface are obtained. Specific implementation two

[0064] The absolute detection method of optical plane based on the minimum norm solution in this specific implementation is based on specific implementation one, rotates the C surface by 180°, and performs experimental verification.

[0065] The experiment is carried out on a Fizeau interferometer with a caliber of 600 mm, and the experimental device uses the device CN108955532B, CN110057287B and CN107036554B patented by the Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, and the method of the application is compared with the plane absolute detection method based on the parity function. In the plane absolute detection method based on the parity function, the auxiliary flat crystal co-rotates six times, which are 60°, 120°, 180°, 240°, 300° and 360°, respectively.

[0066] The comparison of the detection results of the application and the plane absolute detection method based on the parity function is as shown in Figure 2 Figure 2 In the comparison, (a) (b) (c) are the surface shape errors of the A surface, the B surface and the C surface obtained by the absolute detection method based on the minimum norm solution, respectively; (d) (e) (f) are the surface shape errors of the A surface, the B surface and the C surface obtained by the absolute detection method based on the parity function, respectively; (g) (h) (i) are the differences of the surface shape errors of the A surface, the B surface and the C surface obtained by the two methods, and it can be seen that the errors are in the order of nm, which proves that the optical plane absolute detection method based on the minimum norm solution of the application can achieve the detection precision of the absolute detection method based on the parity function; in addition, since the auxiliary flat crystal only needs to be rotated by 90° or 180° once in the application, the precision of the absolute detection method based on the parity function can be achieved, and the auxiliary flat crystal does not need to be rotated multiple times, so the detection process is simpler.​

Claims

1. An optical plane absolute detection method based on minimum norm solution, defined as: The reference flat for the interferometer is plane A; The reflective flat of the interferometer is plane B; The auxiliary optical plane used in absolute testing is the C-plane; Its features are, Includes the following steps: Step a: Interference between plane A and plane B yields the measured value M1(x,y) = A(-x,y) + B(x,y); Where A(-x,y) is the surface shape error at position (-x,y) of surface A, and B(x,y) is the surface shape error at position (x,y) of surface B; Step b: Keep surface A stationary. Surface A and surface C interfere with each other, and the measured value is M2(x,y)=A(-x,y)+C(x,y); Where C(x,y) is the surface shape error at position (x,y) of surface C; Step c: Keep plane B stationary. Planes B and C interfere with each other, and the measured value is M3(x,y)=C(-x,y)+B(x,y). Where C(-x,y) is the surface shape error at position (-x,y) of surface C; Step d: Rotate surface C by 90° or 180°, and measure surface B and surface C together. When surface C is rotated by 90°, the measured value is M4(x,y)=C(-y,-x)+B(x,y). When surface C is rotated by 180°, the measured value is M4(x,y)=C(x,-y)+B(x,y). Where C(-y,-x) is the surface shape error at position (-y,-x) of surface C, and C(x,-y) is the surface shape error at position (x,-y) of surface C; Step e, using step b + step c - step a, yields: C(x,y)+C(-x,y)=M2(x,y)+M3(x,y)-M1(x,y)=N1(x,y) Step f, using step b + step d - step a, yields: C(x,y)+C(-y,-x)=M2(x,y)+M4(x,y)-M1(x,y)=N2(x,y) Step g: Combine steps e and f to construct a system of linear equations: With surface C rotated 90° When surface C is rotated 180° Step h: Iterate through all values ​​of x and y to form a system of linear equations AX = N, where matrix A is a matrix of 0s and 1s, matrix X is a matrix of surface shape errors of surface C, and matrix N is a matrix composed of the derivations from steps e and f. The minimum norm solution of X is then obtained as X = A. T (AA T ) -1 N; that is, to obtain C(x,y), C(-x,y) and C(-y,-x); Step i: Substitute C(x,y), C(-x,y), and C(-y,-x) into steps a, b, c, and d respectively to obtain the surface shape errors of surfaces A and B.

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