WO2015156609A1

WO2015156609A1 - Method for measuring shape of free curved surface in optical type using second derivative of local area and system for measuring shape of curved surface using same

Info

Publication number: WO2015156609A1
Application number: PCT/KR2015/003545
Authority: WO
Inventors: 김병창
Original assignee: 경남대학교 산학협력단
Priority date: 2014-04-08
Filing date: 2015-04-08
Publication date: 2015-10-15
Also published as: KR101421502B1

Abstract

The present invention relates to a method for measuring the shape of a free curved surface in an optical type using a second derivative of a local area and a system for measuring the shape of a curved surface using the same. The purpose of the present invention is to provide a method and a system for measuring the shape of a curved surface, which can precisely measure the shape of a non-spherical or free curved surface using a relatively simple configuration. To this end, the method for measuring the shape of a curved surface comprises the steps of: plane-dividing a curved surface, which is to be measured, into virtual multiple micro local areas; acquiring three-dimensional shape measurement information regarding each of the micro local areas; calculating coefficients of Zernike polynomials from each of the three-dimensional shape measurement information; calculating an x-axis second derivative term, a y-axis second derivative term, and an x-y axis cross derivative term from each of the Zernike polynomials; calculating an x-axis inclination term by integrating the x-axis second derivative term and the x-y axis cross derivative term and calculating a y-axis inclination term by integrating the y-axis second derivative term and the x-y axis cross derivative term; and generating a shape of the curved surface, which is to be measured, by integrating the x-axis inclination term and the y-axis inclination term. And the system for measuring the shape of a curved surface according to the present invention comprises: a sensing unit capable of generating three-dimensional shape measurement information regarding a micro local area of a curved surface, which is to be measured; a moving unit for moving the sensing unit; and a control unit for dividing the curved surface, which is to be measured, into virtual micro local areas and controlling the sensing unit and the moving unit such that the sensing unit generates three-dimensional shape measurement information regarding the multiple local areas, the control unit having an operation unit for calculating coefficients of Zernike polynomials on the basis of the three-dimensional shape measurement information, thereby calculating an x-axis second derivative term, a y-axis second derivative term, and an x-y axis cross derivative term, calculating an x-axis inclination term and a y-axis inclination term using the x-axis second derivative term, the y-axis second derivative term, and the x-y axis cross derivative term, and generating the curved surface, which is to be measured, on the basis of the x-axis inclination term and the y-axis inclination term.

Description

Optical Freeform Shape Measurement Method Using Secondary Differential of Local Region and Curved Shape Measurement System Using It

The present invention relates to an optical free curved shape measuring method using a second derivative of a local region and a curved shape measuring system using the same.

The shape measurement on a spherical surface or a plane is relatively accurate and easy to measure the shape through the process of extracting the difference between the reflected wave surface and the reference surface (plane, sphere) by irradiating light to the measurement object. On the other hand, in the case of an aspherical surface or a free curved surface, since the light reflected from the measurement object is far from the reference plane, it is impossible to measure the overall shape by one light irradiation.

Therefore, when measuring aspherical or free curved surfaces, a correction optical system such as computer hologram (CGH) is used to overcome the wavefront optical path difference from the reference plane, or a method using a contact three-dimensional measuring instrument is used. The method has a low efficiency in terms of facilities and costs and nevertheless has low limits of measurement accuracy.

It is an object of the present invention to provide a curved shape measuring method and system capable of precisely measuring the shape of an aspherical surface or a free curved surface with a relatively simple configuration.

According to the present invention, there is provided a method of measuring a curved shape; Acquiring three-dimensional shape measurement information for each of the minute local areas; Calculating coefficients of the Zernik polynomial from the three-dimensional shape measurement information; Calculating an x-axis second derivative, a y-axis second derivative and an x-y-axis cross differential term from the respective Zernik polynomials; Calculating an x-axis gradient term by integrating the x-axis second derivative and the x-y-axis cross-differential term, and calculating the y-axis slope term by integrating the y-axis secondary differential term and the x-y-axis cross derivative term; And integrating the x-axis slope term and the y-axis slope term to generate a measurement target curved shape.

According to the present invention, there is provided a curved shape measuring system comprising: a sensing unit capable of generating three-dimensional shape measurement information about a minute local area of a curved surface to be measured; A moving unit for moving the sensing unit; The sensing target surface is divided into a plurality of virtual local regions, and the sensing unit controls the sensing unit and the moving unit to generate three-dimensional shape measurement information for each of the plurality of micro local regions, and the three-dimensional shape measurement. Based on the information, the coefficients of the Zernike polynomial are calculated to calculate the x-axis second derivative, the y-axis second derivative and the xy-axis differential derivative, and the x-axis second derivative, the y-axis second derivative and the xy-axis cross derivative And a control unit for calculating an x-axis gradient term and a y-axis gradient term and generating the measurement target surface based on the x-axis gradient term and the y-axis gradient term.

The curved shape measuring method and system according to the present invention can accurately measure the shape of an aspherical surface or a free curved surface with a relatively simple configuration.

1 is a schematic view showing a curved shape measuring system according to the present invention,

2 is a conceptual diagram briefly illustrating a plurality of fine local regions;

3 is a reference diagram showing the independent shape up to the initial sixth term of the Zernik polynomial,

4 is an integral conceptual diagram for restoring a curved shape from a second derivative term,

5 is a flowchart showing a curved shape measuring method according to the present invention;

6 is a reference diagram showing an arbitrary curved surface (left) and a curved surface (middle) and a measurement error degree (right) measured by the curved shape measuring method and system according to the present invention.

1 is a schematic diagram showing a curved shape measuring system according to the present invention, and FIG. 2 is a conceptual diagram showing a measurement object divided into a plurality of fine local regions.

Referring to FIG. 2 together with FIG. 1, the present system may generate the three-dimensional shape measurement information of the local area A by irradiating the local area A in a state close to the surface of the curved surface C. FIG. The sensing unit 10, the moving unit 15 for moving the sensing unit 10, and the measurement target surface C are divided into a plurality of virtual local regions A, and the sensing unit 10 is finely divided. The control unit 20 controls the sensing unit 10 and the moving unit 15 to generate three-dimensional shape measurement information for the local areas A. Here, the measurement information of the three-dimensional shape is a set of x, y, z values per pixel, and the sensing unit 10 is a Twyman-Green interferometer, a Mirau interferometer or a Fizeau It may be provided as an interferometer.

The control unit 20 divides the measurement target surface C into the fine local region A. FIG. At this time, the precision is improved as the distance value s between the adjacent minute local areas A is made smaller and denser.

When the surface to be measured C is divided into a plurality of minute local areas A, the controller 20 may allow the sensing unit 10 to sequentially irradiate each of the minute local areas A. Set the movement path of. The control unit 20 moves the sensing unit 10 along the movement path by using the moving unit 15, and when the sensing unit 10 is located at the center 8 of each local area A, the control unit 20 may move to the local area A 3. Create measurement information of dimensional shape. The controller 20 transmits the measurement information about the three-dimensional shape generated by the sensing unit 10 to the calculator 21. In this case, the three-dimensional shape measurement information in the local area A is a set of x, y, and z values per pixel, and the three-dimensional shape measurement information measured in each local area A is directly connected to generate the overall shape. In this case, however, a position error, a tip and tilt, or a height direction error, which occurs when the sensing unit 10 moves for each local area A, may occur. In addition, although the inclination information for each of the first-differential local areas A may be obtained based on the three-dimensional shape measurement information in the plurality of local areas A, the overall shape may be generated. ) Has a disadvantage of low precision due to the position error and tip and tilt caused by the movement.

Therefore, the calculation unit 21 according to the present invention crosses the x-axis second derivative, the y-axis second derivative and the xy-axis intersection based on three-dimensional shape measurement information in each local area A transmitted from the sensing unit 10. Extract the differential term.

In the case of generating the overall shape by acquiring the information on the second derivative based on the three-dimensional shape measurement information measured in the plurality of local areas A, the height direction (piston) or the tilt error ( It is not affected by tip and tilt, and the influence of position error is negligible so that the overall shape can be generated with the highest precision.

The basis and method for the calculation unit 21 to generate the data of the measurement object from the three-dimensional shape measurement information in the fine local region are as follows.

First, if Z (x, y) is a curved shape function to be obtained for the measurement target aspherical surface, the differential expression of Z (x, y) may be expressed as Equation 1 below.

Equation 1

Where Sx is the x-axis slope and Sy is the y-axis slope.

In addition, it can be represented by Equation 1 when the differential slope (S _y) in the x-axis tilt (S _x) and y-direction respectively, Equation 2 and Equation 3].

Equation 2

Equation 3

As shown in [Equation 2] and [Equation 3], which differentiates the inclination in the x-axis direction and the inclination in the y-axis direction, the second derivative of the curved shape Z (x, y) finally yields [Equation 4]. ] To [Equation 6] can be expressed as the x-axis second derivative, the y-axis second derivative and the xy-axis cross derivative.

Equation 4

Equation 5

Equation 6

Therefore, based on these three second derivatives, the curved shape to be measured can be restored. Based on this principle, the calculating section 21 generates three second derivatives expressed by Equations 4 to 6 from three-dimensional shape measurement information values for each local area A. Restore the surface shape function Z (x, y) of the target surface from the second derivative. The process is as follows.

First, all three-dimensional shapes of the curved surface may be expressed by Zernike polynomials as shown in Equation 7 below. The Zernik polynomial is characterized by a basis function where each term satisfies orthogonality and completeness within a unit circle. In addition, the individual terms of the Zernik polynomial are suitable for expressing optical wavefronts or shape surfaces because they represent different optical properties.

Equation 7

Here, a _n is a coefficient of the nth term, and z _n is a basis function term defined as in Equations 8 to 13. Most realistic objects to be measured, such as astronomical telescopes, have a smooth shape whose optical surface is ground, so that higher order terms of seventh order or more may be omitted. The basis function terms up to the sixth order are as shown in [Equations 8] to [Equation 13] below.

Equation 8

Equation 9

Equation 10

Equation 11

Equation 12

Equation 13

3 is a conceptual diagram showing independent shapes up to the first sixth term Z ₁ to Z ₆ of the Zernik polynomial. The measurement target surface may be generated through the overlapping of the independent shapes shown in FIG. 3. When the basis function term is applied to the Zernik polynomial, the three-dimensional shape in the local region A is expressed as shown in [Equation 14].

Equation 14

In addition, the three-dimensional shape of the local area A may be expressed by a Taylor series expansion as shown in [Equation 15].

Equation 15

[Equation 14] and [Equation 15] representing the three-dimensional shape of the local area A as described above, respectively, the second derivative with respect to the x-axis, the second derivative with respect to the y-axis, and the second derivative with respect to the xy-crossing axis. When the derivative is performed, it may be represented by Equations 16 to 18.

Equation 16

Equation 17

Equation 18

Accordingly, as shown in [Equations 16] to [Equation 18], the second derivative with respect to the x-axis, the second derivative with respect to the y-axis, and the cross derivative with respect to the xy-axis are the fourth in the Zernik polynomial. , The fourth and sixth coefficients a ₄ , a ₅ , It can be derived from a ₆ value. Where a ₄ , a ₅ , The value of a ₆ is derived by Zernike fitting using the three-dimensional shape measurement information for each local area A. The Zernike fitting method is described in Daniel malacara's book "Optical shop testing". In general, a ₄ , a ₅ , The value of a ₆ can be obtained through software associated with a three-dimensional shape measurement interferometer.

So a ₄ , a ₅ , By deriving three second derivatives for each local area A through the value of a ₆ , the entire curved shape can be restored from the series of local areas A.

4 shows an integral conceptual diagram in which the calculating unit 21 restores a curved shape from three second derivative terms. As shown in FIG. 4, the calculation unit 21 generates an x-axis gradient term by integrating the x-axis quadratic derivative and the xy-cross differential term, and integrates the y-axis quadratic derivative and the xy intersection differential term to inject the y-axis gradient. Create a term Then, the x-axis slope term and the y-axis slope term are integrated again to generate the overall shape Z (x, y). The generated shape may be transmitted to the output unit 30 connected to the control unit 20 to be represented as an image. Here, it is preferable to use the Southwell method using the Zonal approach for integration.

5 is a flowchart illustrating a curved shape measuring method according to the present invention. As can be seen in Figure 5, in order to measure the shape of the surface to be measured, the first surface to be measured is divided into a plurality of fine local regions A (S1). In addition, the controller 20 controls the sensing unit 10 to generate three-dimensional shape measurement information for each micro local area A (S2). When the sensing unit 10 completes the generation of the 3D shape measurement information for each micro local area A, all the generated 3D shape measurement information is transmitted to the calculation unit 21. The calculation unit 21 is based on the three-dimensional shape measurement information for each micro local area A transmitted from the sensing unit 10, the x-axis second derivative, the y-axis second derivative and Compute the xy-axis cross derivative term (S3).

Subsequently, the calculating unit 21 calculates the x-axis slope term by integrating the calculated x-axis second derivative and the xy-axis cross-differential term, and integrates the y-axis second derivative and the xy-axis cross derivative to the y-axis gradient term. To calculate (S4). The calculating unit 21 generates a shape of the measurement target surface by integrating a series of x-axis tilt terms and a y-axis tilt term (S5). The shape of the measurement target surface and the final equation generated in this way may be transmitted to the output unit 30 to be expressed as an image.

6 is a reference diagram showing an arbitrary curved surface (left) and a curved surface (middle) and a measurement error degree (right) measured by the curved shape measuring method and system according to the present invention. 6 is an experimental result of any curved shape in the area of 200mm diameter to verify the curved shape measuring method according to the present invention. The experiment divided the surface to be measured into a total of 501 × 501 fine local areas A, and then moved the sensing unit 10 by 0.4 mm in the x- and y-axis directions, respectively. The dimensional shape measurement information is generated, and the measurement target surface is generated according to the curved shape measurement method described above based on the generated three-dimensional shape measurement information.

As can be seen in Figure 6, the root mean square (RMS) of the measurement target surface (RMS, root mean square) is 0.080449mm, the average square root of the surface produced in accordance with the present invention is 0.080461mm with an error of less than 0.000019mm very precise measurement It can be seen that this is possible.

It is preferable that the measurement object curved surface C which concerns on this invention is an optical surface.

Claims

In the curved shape measurement method,

Plane dividing the measurement target surface into a plurality of virtual local regions;

Acquiring three-dimensional shape measurement information for each of the minute local areas;

Calculating coefficients of the Zernik polynomial from the three-dimensional shape measurement information;

Calculating an x-axis second derivative, a y-axis second derivative and an x-y-axis cross differential term from the respective Zernik polynomials;

Calculating an x-axis gradient term by integrating the x-axis second derivative and the x-y-axis cross-differential term, and calculating the y-axis slope term by integrating the y-axis secondary differential term and the x-y-axis cross derivative term;

And integrating the x-axis gradient term and the y-axis gradient term to generate a measurement target curved shape.
In the curved shape measurement system,

A sensing unit capable of generating three-dimensional shape measurement information on the minute local area of the curved surface to be measured;

A moving unit for moving the sensing unit;

The sensing target surface is divided into a plurality of virtual local regions, and the sensing unit controls the sensing unit and the moving unit to generate three-dimensional shape measurement information for each of the plurality of micro local regions, and the three-dimensional shape measurement. Based on the information, the coefficients of the Zernike polynomial are calculated to calculate the x-axis second derivative, the y-axis second derivative and the xy-axis differential derivative, and the x-axis second derivative, the y-axis second derivative and the xy-axis cross derivative And a control unit for calculating an x-axis slope term and a y-axis slope term and generating the measurement target surface based on the x-axis slope term and the y-axis slope term. Measuring system.