JP2531596B2 - Connection method between divided areas and wavefront connection method between divided areas on the surface to be measured - Google Patents

Description

Detailed Description of the Invention

The present invention makes it possible to continuously connect the divided measurement areas when the measurement area of the shape measuring apparatus is divided into a plurality of measurement areas that are wider than the measurement opening and the measurement areas are divided. The present invention relates to a method of connecting between divided areas and a method of connecting a wavefront between divided areas on a surface to be measured.

A device using an interferometer or the like (hereinafter referred to as a shape measuring device) as a means for precisely measuring a state of a flat surface or a curved surface (hereinafter referred to as a measured surface) of various parts (hereinafter referred to as a measured object). Has been developed, and the shape of the measured surface of the measured object can be measured from the pattern of interference fringes formed. In this shape measuring device, the measurable area per measurement is limited to a certain value, and the entire range cannot be measured only once with respect to the surface to be measured having a certain width or more. Therefore, when a large object to be measured is to be measured, it is conceivable to divide the surface to be measured into a plurality of areas and measure the area several times.

That is, according to this measuring method, a measurement operation is performed for each divided area of a large object to be measured, and an image representing the measured surface of each divided area (hereinafter referred to as reproduction) is obtained from the data obtained by this. It is possible to obtain a composite overall image by forming a surface) and connecting and reproducing these reproduction surfaces.

By the way, according to such a partial measurement method, it is possible to obtain accurate measurement data for each of the divided regions. However, it is not enough to simply connect these in order to obtain the overall shape.
In many cases, a discontinuous surface D is generated at the connecting portion as shown in, and it is difficult to obtain an accurate overall image corresponding to the actual shape. That is, this is because when the object to be measured is sequentially moved according to the divided areas, the object to be measured causes minute displacements in the up-down direction and the front-back and left-right directions. Therefore, in view of the above-described conventional drawbacks, the present invention, based on the data obtained by the measurement for each divided area, continuously connects the reproduction surfaces calculated for each divided area, and the entire measured surface. It is an object of the present invention to provide a connection method between divided areas and a wavefront connection method between divided areas on a surface to be measured, which can obtain an accurate whole image of a reproducing surface as close as possible to an actual shape.

That is, the connecting method between the divided areas on the surface to be measured according to claim 1 of the present invention makes the peripheral side overlap at a constant rate for each of the divided areas of the surface to be measured. However, the reproduction surface corresponding to the shape of the surface to be measured is calculated for each of the divided areas based on the information obtained by the measurement, and the reproduction surface is formed on at least two kinds of reproduction surfaces including the overlapping areas. In, the error of the position for each corresponding point in the overlap area is calculated, and the overlap portion on the reproduction surface is corrected for each divided area so that the error is minimized, and the reproduction is continuously and integrally connected. A surface is formed and a reproduction surface corresponding to the shape over the entire surface to be measured is synthesized. According to a second aspect of the present invention, in the wavefront connection method between the divided areas on the measured surface, the divided light divided areas of the measured surface to be measured are overlapped at a constant ratio while the reference light from the light source is overlapped. By, by determining the function showing the wavefront according to the unique shape of the measured surface in each of the divided areas based on the measured information, for each corresponding arbitrary point in the overlapping region, respectively, Corresponding to the shape of each divided area by calculating the error between corresponding coordinate values in the function determined in each divided area including the overlap area and correcting the functions so that the error is minimized. In the above function, the overlapping portions are continuously connected, and the wavefront corresponding to the shape of the entire surface to be measured is uniquely determined over the entire area.

An embodiment of the present invention will be described below with reference to the accompanying drawings. FIG. 1 shows a method of connecting a phase wavefront between divided areas on a surface to be measured according to the present invention, which is composed of first step 1 to step 5 and has a rectangular shape and a relatively wide entire surface to be measured. Are equally divided into n rectangular divided areas S. In this embodiment,
The k-th sub-region S _k and k-th overlapped with each other by W
A case of connecting two _{+ 1st} divided areas S _{n + 1} and two divided areas will be specifically described.

Further, in this embodiment, in order to measure the surface shape of each divided area S _k , the following measuring means is used. That is, in this embodiment, a measuring device incorporating a helium-neon (He-Ne) laser emitting a laser beam of wavelength λ and a CCD (charge-coupled device) camera is used, and the measured fringes are measured from the formed interference fringes. It is designed to measure the planar state of a surface. It should be noted that this measuring means is not particularly limited to one having a configuration in which interference fringes are generated by this laser light and the degree of unevenness of the plane is measured from the interference fringes. For example, incoherent light may be projected from an ordinary light source such as a xenon lamp to form moire fringes, and thereby the flatness and the curvedness may be measured.

(I) In step 1, the surface to be measured is measured such that the edges of the adjacent divided areas overlap in each divided area, and the characteristic shape of the surface to be measured, that is, the characteristic shape of the measured surface, is obtained from the data obtained by this measurement. , A function Ψ for displaying the reproduction surface is calculated by a microprocessor (not shown) or the like. In other words, a microprocessor is used and CR is set for each divided area.
A function that can reproduce (graphically display) the surface to be measured on the screen such as T is derived. As a result, as shown in FIG. 2, for example, the k-th divided area (S
_k ) and the k + 1-th divided area (S _{k + 1} ) as a function Ψ indicating the shape of the surface to be measured, that is, (X,
It is assumed that functions Ψ _k and Ψ _{k + 1} as shown in FIG. 4 are obtained as a function Z = Ψ (X, Y) representing the amount of unevenness (Z) on the reproduction surface in the three-dimensional coordinates (Y, Z).

[0009] Each function [psi _k at this time, [psi _{k + 1} is a function of a separate coordinate system based on the respective origin, moreover divided area next order division region from (S _k) (S _k When moving to ( ₊₁ ), the object to be measured may be slightly displaced in a predetermined stage position in the up and down (Z) direction and the left and right (X, Y) directions. Therefore, in this embodiment, first, the functions Ψ _k and Ψ
_{k + 1} is converted as a function in the same coordinate system by a predetermined conversion formula described later, and both functions have a difference in the amount of unevenness at the surface height (Z) at each corresponding same point in the overlapping area ( The above-mentioned conversion formula is set to an optimum one by using the least square method so that ΔZ) becomes a minimum, which will be specifically described in the following steps.

(II) In the second step 2, both the functions of the peculiar functional formula (Ψ _k , Ψ _{k + 1} ) calculated for each of the adjacent divided areas (eg, S _k , S _{k + 1} ) After the formula is converted into the unified coordinate system, each corresponding point is calculated after the coordinate transformation, that is, in the overlapping area of the reproduction surface drawn by both functions on the unified coordinate system. Here, the unified coordinate system O-X, Y is a coordinate system that serves as a reference when the entire surface to be measured as shown in FIG. 5 is displayed at the same time, and the area S in FIG. It corresponds to the entire reproduction surface, and the S area is divided into N divided areas S ₁ , S ₂ , ..., _SN .

(1) First, with respect to the coordinate system O-x, y serving as the reference of the divided area S _k , the coordinate system O- serving as the reference of S _{k + 1.}
Assuming that x ′ and y ′ are displaced from the origin position by −x _{0 in} the x direction and −y _{0 in} the y direction,

[0012]

[Equation 1]

Both functions are coordinate-transformed by the following. That is,

[0014]

[Equation 2]

Is satisfied.

It should be noted that, in order to visualize the entire surface to be measured as one reproduction surface, it is necessary to further convert the function into a unified coordinate system. Subsequent procedures will be performed without conversion to. (2) By the way, if the overlapping areas of these two adjacent reproduction surfaces completely match, in the overlapping area,

[0017]

(Equation 3)

The above is established, and it is not necessary to add a correction to this.

However, in practice, the accuracy of the interferometric measurement,
For example, it is difficult to move the measurement operation to the next division area without causing an error of λ / 100 (λ is the wavelength of the laser beam used for measurement) or less. That is, in reality, it is inevitable that some measurement errors are involved. With respect to the factor that causes this error, when the reproduction surface indicated by the function Ψ _k is used as a reference, the reproduction surface indicated by the function Ψ _{k + 1} is inclined a in the x direction in the three-dimensional space as shown in FIG. , Due to the displacement b in the y direction and displacement c in the z direction. Therefore, it can be considered that the relationship as shown in the following expression (hereinafter referred to as a conversion expression) is established between these two functions.

[0020]

[Equation 4]

By the way, both functions Ψ _k and Ψ
_In order to connect the reproduction surfaces indicated by _{k + 1} as continuously as possible, the error of the function value (Z, Z ′) at each corresponding same point (hereinafter referred to as corresponding point) in both overlapping areas Should be minimized. Therefore, in order to minimize the value of these function values, in other words, the error σ = Z−Z ′ regarding the heights at corresponding points on both reproduction surfaces, the coefficients a, b, and c in the equation (e) are set. Is calculated by using the method of least squares.

(III) In Step 3, N in the overlapping area
The data of each reproduction surface at the corresponding points is substituted into a predetermined correction formula described below to establish a simultaneous equation for obtaining unknown coefficients a, b, and c. That is, the function value at any point (x _i , y _i ) of the N corresponding points is the function Ψ _k , Ψ
Assuming that Z _i and Z _i ′ at _{k + 1} , the error σ _i at that point is obtained by the following equation (hereinafter referred to as correction equation).

[0023]

(Equation 5)

Here, Ψ _k (x _i , y _i ) − [Ψ
_{If k + 1} (x _i −x ₀ , y _i −y ₀ )] is set to Δ (x _i , y _i ), and the sum of squares of the errors S _ij at N corresponding points is calculated,

[0025]

(Equation 6)

It becomes Then, the constants a, b, and c when this Σσ _i ² takes the minimum value may be obtained. Therefore,
Considering Σσ _i ² as a function of a, b, and c, when Σσ _i ² has an extreme value (minimum value), the value when Σσ _i ² is partially differentiated by a, b, and c is 0. Therefore,

[0027]

(Equation 7)

A simultaneous three-dimensional equation is obtained from

(IV) In step 4, a, b, c are obtained for the simultaneous equations (h) (a = a ₀ , b =
b ₀ , c = c ₀ ) is used to determine the unique continuous equation number 8 between the functions of the predetermined divided areas.

[0030]

(Equation 8)

Then, by correcting each function indicating the reproduction surface in both adjacent divided areas based on the continuous expression (i) determined in this way, the reproduction surfaces represented by these two functions are continuously obtained. It is connected. In addition,
Calculation of a, b, and c from the simultaneous equations shown in (h) can be easily performed by the following determinant. That is, the above simultaneous equations (h) are

[0032]

[Equation 9]

Since it is represented by, a, b, and c are determined from this as follows.

[0034]

[Equation 10]

It should be noted that the important factor in this equation is 3
What is included in the inverse determinant of x3 is only the constants of x _i , y _i, and n, which can be uniquely determined. Therefore, if this inverse determinant is calculated in advance, then a, b, and c can be mechanically calculated from the data Δ _i measured for each divided area S, and the continuous expression is calculated every time. Is very easy. In addition, each corresponding point (x _i ,
If y _i ) is set symmetrically with respect to a certain point (x _j , y _j ),

[0036]

[Equation 11]

Thus, it is easy to calculate each numerical value in the inverse determinant.

(V) In step 5, since the overlapping areas of the reproduction surfaces are continuously and smoothly connected from the constants a, b, and c obtained in step 4, a unique continuous expression is used for each divided area. And the function for displaying the reproduction surface for each divided area can be corrected. Although the above description has been made between the k-th divided area and the (k + 1) -th divided area, a unique continuous equation is determined in the same manner for other divided areas, and based on this, each divided area is determined. The function for displaying the reproduction surface may be corrected each time. Further, if the correction functions obtained for the first to Nth are converted into a function in the unified coordinate system by using a required coordinate transformation formula, discontinuous steps between the divided areas in the unified coordinate system. That is, it is possible to obtain the whole image F of the continuous reproducing surface as shown in FIG.

In the above embodiment, the reproduction planes are overlapped between the two divided areas and the continuous connection method between the two reproduction planes has been described. However, the present invention is not limited to this. Not a thing. For example, if there are overlapping areas for M playback surfaces adjacent to each other,
When any one of the playback surfaces is used as a reference,

[0040]

(Equation 12)

(M-1) conversion equations of are obtained, and (M-1) × 3 coefficients may be obtained from these equations. In this case, as in the case of the two surfaces of the previous embodiment, the error σ is obtained for the overlapping parts of the respective surfaces, and the a for each surface is calculated from the least squares method so that the sum of squares of all of them is minimized. , B, c, the inherent continuous equation may be determined.

For example, the case where the five planes A to E shown in FIG. 6 overlap will be described. First, the coefficients a _A , b _A , and c _A for the function Ψ _A indicating the A plane are obtained. Since the surface that does not contact the A surface does not overlap with the A surface, the square of the error Σσ
_i ² cannot be obtained. Therefore, the above Σσ _i ² is obtained for B, C, D and E. For example, for side B,

[0043]

(Equation 13)

[Mathematical formula-see original document] is obtained, and similar expressions are obtained for the C, D, and E planes. Therefore, the sum of these σ _A ²

[0045]

[Equation 14]

With respect to, it suffices to perform partial differentiation with respect to a _A , b _A , and c _A , respectively. However,

[0047]

(Equation 15)

In this way, the determined a _A , b _A , c _A
Is a correct value if the B, C, D and E surfaces are accurate,
Actually, it is necessary to correct the B, C, D, and E surfaces. Therefore, it is necessary to simultaneously correct all the measurement surfaces with respect to an appropriate reference surface. This finds the same coefficient as in the equation (15) for other planes, and thus (M−1) × 3
The original simultaneous equations are established, and each coefficient can be calculated as in the previous embodiment.

That is, as shown in FIG. 7, for the connection between 3 × 3 planes, that is, for the connection over 8 planes, the coefficient can be obtained from the determinant shown below. However, the reference surface is A, and the equations 16 and 17 are the sum of the peripheral surface (for example, the equation 1
(8, number 19, number 20, number 21 etc.)
Is established.

[0050]

[Equation 16]

[0051]

[Equation 17]

[0052]

(Equation 18)

[0053]

[Formula 19]

[0054]

(Equation 20)

[0055]

[Equation 21]

[0056]

[Equation 22]

Also in this case, a, b, and c can be easily obtained by obtaining the inverse matrix of the matrix of Expression 23.

[0058]

(Equation 23)

When a laser is used in the present invention, it is preferable to use a He—Ne laser having a wavelength of 633 nm or a semiconductor laser having a wavelength of 880 nm .

As described above, according to the connection method between the divided areas according to the present invention, the peripheral portions of the divided areas are successively measured while being overlapped with each other, and the reproduction surface is based on the information obtained by this measurement. Then, the error between the playback surfaces at each arbitrary point in the overlapping area of the playback surfaces is calculated, and the playback surface is corrected so that the error is minimized. Since the reproduction surfaces are connected to each other so as to form the reproduction surface, it is possible to obtain a reproduction surface having a phase distribution that is as close as possible to the actual measured surface.

[Brief description of drawings]

FIG. 1 is a flowchart showing a connection method between divided areas according to the present invention.

FIG. 2 is an explanatory diagram showing an overlapping state of measured surfaces in divided areas to be connected for explaining the connection method according to the present invention.

FIG. 3 is an explanatory diagram showing a concavo-convex state of a part of a reproduction surface in a three-dimensional coordinate space.

FIG. 4 is an explanatory diagram showing a state in which overlapping reproduction surfaces are connected by the connection method according to the present invention.

FIG. 5 is an explanatory diagram showing a positional relationship on the entire surface to be measured of divided areas to be connected by the connection method according to the present invention.

FIG. 6 is an explanatory diagram showing a distribution state of divided areas to be connected by another connecting method according to the present invention.

FIG. 7 is an explanatory diagram showing a distribution state of divided areas to be connected by still another connecting method according to the present invention.

FIG. 8 is a perspective view showing an entire image of a reproduction surface connected by the present invention.

FIG. 9 is a perspective view showing an overall image of a reproducing surface formed using a conventional method.

[Explanation of symbols]

F reproduction plane S _k , S _{k + 1} divided area Ψ _k , Ψ _{k + 1} function representing the reproduction plane

Claims (2)

(57) [Claims] 1. A surface to be measured is divided into a plurality of areas to be measured, and the divided areas are successively measured while overlapping the peripheral portions, and the respective divided areas are based on the information obtained by this measurement. A reproduction surface corresponding to the shape of the measured surface is calculated for each, and a position error is calculated for each corresponding point in the overlapping area on at least two kinds of reproducing surfaces including the overlapping area, A reproduction surface corresponding to a shape over the entire surface to be measured is formed by correcting the overlapping portion on the reproduction surface for each divided area so as to minimize the error and forming a reproduction surface that is continuously and integrally connected. A method for connecting between divided areas on a surface to be measured, which is characterized by combining. 2. The surface to be measured is divided into a plurality of areas to be measured, and measurement is performed while overlapping the peripheral portions of the divided areas.
Based on this measured information, the function indicating the wavefront according to the unique shape of the surface to be measured in each of the divided areas is determined, and for each corresponding arbitrary point in the overlapping area, the overlapping area is included. By calculating the error between corresponding coordinate values in the function determined in each divided area and correcting the functions so that the error is minimized, the functions overlap depending on the shape of each divided area. A wavefront connecting method between divided areas on a surface to be measured, characterized in that parts are continuously connected and a wavefront corresponding to the shape of the entire surface to be measured is determined over the entire area.