JP2003035529A - Systematic error measuring method in flatness measuring system on object surface to be inspected, flatness measuring method and apparatus using the same - Google Patents

Abstract

PROBLEM TO BE SOLVED: To enable a systematic error and the flatness to be measured using only the measured value of the surface form of an object to be inspected including the systematic error in the flatness measuring system. SOLUTION: The form of an object 8 to be inspected in a first state being in a predetermined reference location is measured by an area sensor 2. The form of the object to be inspected in a second state acquired by giving a predetermined angle turning shift to the first state by a rolling mechanism 7 is then measured by the area sensor. The form of the object to be inspected in a third state acquired by giving only a predetermined amount linear shift to the first state in a predetermined direction by a positioning stage 6 is further measured by the area sensor. The form of the object to be inspected in a forth state acquired by giving a predetermined angle turning shift to the third state by the rolling mechanism is then measured by the area sensor. The systematic error of the flatness measuring system is determined using the obtained data.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to the field of obtaining a flatness measurement value with improved accuracy by subtracting a systematic error of a measurement system from a measurement value including an error with respect to the flatness of an object surface having a substantially flat shape. . More specifically, the present invention relates to the field of flatness measurement of the surface of a subject using an area sensor that measures the height above and below the surface of the subject over a predetermined region.

As a concrete example, it relates to the field of measuring the surface shape of an object by an optical interferometer using a CCD camera as an area sensor.

Incidentally, in the usual interferometer measurement, the shape of the surface of the subject represents a portion invariant to the posture of the subject, that is, a second-order component or more, but in the present invention, the posture of the subject is also changed. In consideration of the measurement to be measured, the second-order and higher-order components are treated as flatness in order to distinguish the shape from the concept including the inclination in the strict sense.

[0004]

2. Description of the Related Art Conventionally, a systematic error caused by a reference surface error of a measuring system for measuring the surface shape and flatness of an object or a wavefront distortion by an optical component is identified by using a measurement value including the systematic error. Several means have been proposed centering around the non-contact optical measurement method, but no method has reached the stage of practical use at present.

As a conventional technique, in most cases, only the reference surface in the optical interferometer is taken out, the shape of the reference surface is determined by a method conforming to the three-face matching method, and then the reference surface is attached to the interferometer. However, this method uses two other surfaces to measure with the two surfaces facing each other, and it takes a lot of trouble to set the position and posture of the reference surface to be accurately attached, which is very troublesome. is there. In addition, systematic errors other than the reference plane, errors in the mounting position and orientation of the reference plane after the completion of calibration, and the influence of the bending of the support body are left untouched.

On the other hand, a method for shifting the surface of the object to be examined in a direction orthogonal to the optical axis of the interferometer (referred to as a two-sided method) with the reference surface attached to the optical interferometer has been studied. Among the errors that accompany the shift of the subject,
The pitching term, rolling term, and up-and-down movement term that give an error to the shape cannot be found, and a method for eliminating their influence has not been established.

Regarding the two-sided method, there are the following four methods.

(1) Shunji Ito, Teruhiko Hinachi, Satoru Horiuchi: 2
Precision measurement of flatness using the azimuth method and radial shift method Precision engineering, 58 (1992) 883-886 This method is a rotary stage that rotates the subject around the optical axis and shifts in the radial direction. It also uses a moving stage to do so. However, since the method does not consider the linear shift error associated with the radial shift at all, it causes an error in the secondary component of the shape. A shape like an optical flat has the largest proportion of secondary components, so
It is considered that the measurement shape has a large error.

(2) R. Mercier, M. Lamare, P Picart,
JPMarioge: Two-flat method for bi-dimensional
measurement of absolutedeparture from the b
est spherePure Appl Opt 6 (1997) pp. 117-126 This method assumes a special case where the shape of the subject surface is spherical, and only linear shift error (pitching, rolling, vertical It handles three linear shift errors of movement) and shape separation using the least squares method. However, the secondary component contained in the shape cannot be separated from the linear shift error (pointed out by Kiyono, Tohoku Univ.).

(3) S. Seino, S. Hei, G. Wei: Theoretical study on absolute measurement method of plane shape by interference fringes Precision Engineering 64-8 (1998), 1137-1145 This method covers three shifts. The shape of the sample surface can be determined. However, this method assumes that there are only two linear shift errors, a pitching term and a rolling term,
Moreover, fatal mistakes have been pointed out in the algorithms that require this.

(4) Kei Kiyono, Sun Hei, Gakkou High, Taka
Wei: Autonomous calibration of shape measuring machine by Fizeau interferometer Proceedings of 1999 Annual Meeting of Precision Engineering Autumn Conference, 457 (1999) This method belongs to method (1), and radial shift is linear shift. We are trying to eliminate this effect by assuming that the error is only the pitching term, but (3)
It contains the same problem as, but this is not properly requested.

(5) Sonozaki Shohichi, Iwata Koichi, Iwasaki Yoshihisa: Research on shape measurement method that does not require a standard (measurement on the circumference of a plane object by Fourier expansion) Precision Engineering, 63 (1999) 129- 133 pages

(6) Shohachi Sonoe, Koichi Iwata, Yoshihisa Iwasaki: Research on shape measurement method that does not require a standard (measurement experiment by Fizeau interferometer on the circumference of a plane shape) Precision Engineering, 65 (1999) In the method of pages 300-304 (5) and (6), the shape (displacement) at the circumferential portion with respect to each radius is determined by eliminating the influence of the rotational shift error due to the rotational shift. However, the obtained shape is a deviation from an independent average plane, and this cannot be expanded to a plane shape, and the surface shape of the subject cannot be obtained.

(7) Shohachi Sonoe, Koichi Iwata, Yoshihisa Iwasaki: 2
Method of converting the shape on one circumference into the shape on one plane 2001 Proceedings of Precision Engineering Spring Conference Proceedings 569 (200
1) In order to make up for the deficiencies of (5) and (6), efforts are being made to introduce a straight line reference to provide a height reference between the shapes at the circumferential portion for each determined radius. However, it is still in the planning stage, and a sensor that gives a straight line reference is required, and the measurement is not autonomous.

As described above, in the two-plane method, even if a two-dimensional positioning stage is used, a method for correctly obtaining the linear shift error due to the shift of the subject is not shown.
A method for easily determining the shape and flatness of the object surface with the reference surface attached to the interferometer by removing the effects of systematic errors, and a practical interferometer calibration method are not found. .

[0016]

On the other hand, the applicant of the present invention has found that in Japanese Patent Application No. 2000-297802 "Systematic error determination method for surface profile measuring system and surface profile measuring device", the pitching caused by the shift is , Rolling, and
Of the linear shift error whose main component is vertical movement, the pitching term, the rolling term and the coefficient of the polynomial are expressed as a function of the vertical movement term, and the vertical movement term is separately detected by using a detection means (gap sensor). Decided, from this,
A method has been proposed in which the systematic error of the shape measuring system is determined by obtaining the calculated shape of the surface of the specimen.

However, this method requires a separate detecting means, which requires specialized preparation of the detecting means, cost and labor for attachment / detachment, and the user side determines the systematic error to calibrate the measurement system. Is difficult.

Further, in Japanese Patent Application No. 2000-353660 “Method for autonomously determining systematic error of surface profile measuring system using a subject for calibration”, the determination of the vertical movement in Japanese Patent Application No. 2000-297802 is separately detected. We propose a method that can be done autonomously without using any means (gap sensor).

However, the primary 2
It is necessary to process and create a test object for calibration that has no next component, and it is difficult to obtain a systematic error with high accuracy depending on the processing accuracy. Further, even if there is no first-order component of the surface shape of the subject, the first-order component is generated due to the inclination of the subject with respect to the optical axis, so that an error remains with respect to the first-order component of the reference surface.

Further, in Japanese Patent Application No. 2000-299004 "A method for autonomously determining a surface shape measuring system using an area sensor", an area sensor for measuring the upper and lower sides of the surface height in a two-dimensional area of an object is provided. In the surface profile measuring apparatus, an arbitrary object that does not need to have a specific shape, such as Japanese Patent Application No. 2000-353660, is fixed to a two-dimensional positioning stage that can move in a direction orthogonal to the vertical axis, and the sample is moved twice. By using only the measurement values of the shape measurement system by the shift, the surface shape measurement system is separated by linear shift error of pitching, rolling, and vertical movement generated with each shift and the surface shape of the object by only algebraic calculation. Has proposed an algorithm for determining the systematic error for each detection position in the area on the surface of the subject.

This is a method of approximating and solving each linear shift error and the primary and secondary components of the shape, and it is effective when the shape of the subject is gentle and the primary and secondary components are small. Although conceivable, if a general subject is used, there remains a problem in terms of accuracy.

The present invention has been made to solve the above-mentioned conventional problems, and includes a rotation mechanism for performing a rotational shift of an object in a predetermined plane and an independent rotation shift mechanism in the plane. To provide a systematic error measuring method of a flatness measuring system including a positioning stage for performing a linear shift of an object and an area sensor for measuring the upper and lower sides of the surface of the object over a predetermined area. Let's take the first challenge.

According to the present invention, in the flatness measuring system for the surface of the object to be examined, which is equipped with an area sensor for measuring the upper and lower sides of the surface height in a two-dimensional area, it is possible to operate in a direction orthogonal to the vertical axis. By using the positioning stage and the mechanism that rotates the subject, 1
Before and after the linear shift, and using the measured values of the surface shape of the object including the systematic errors of the multiple shape measurement systems obtained before and after the rotational shift and the linear shift, the error and the rotational shift of the object are measured. By eliminating the effects of both the rotational shift error associated with it and the linear shift error of pitching and rolling caused by linear shift, systematic errors other than the first-order and lower-order components of the system for measuring the flatness of the surface of the sample are examined. A determination is made for each detection position in the area of the surface, and then, regarding the flatness measurement of another arbitrary subject, the one-dimensional positioning stage for linear shift and the mechanism for rotating the subject are not used, that is, By subtracting the systematic error determined by excluding the first-order and lower-order components from the calculated shape of the surface shape of the object including the systematic error without depending on the shift of the object. To provide a flatness measurement technique of the subject surface for obtaining the flatness of the surface of the object with high accuracy second
And the subject.

[0024]

According to the present invention, a rotating mechanism for rotationally shifting a subject within a predetermined plane and a linear shift of the subject within the plane independently of the rotational shift are provided. Of the flatness measuring system including a positioning stage and an area sensor that measures the height of the sample surface above and below over a predetermined area. The area sensor measures the shape of the object, and the area sensor measures the shape of the object in the second state, which is rotationally shifted from the first state by a predetermined angle by the rotation mechanism. The area sensor measures the shape of the object in the third state linearly shifted in the predetermined direction by the positioning stage from the position measuring stage, and the rotation mechanism shifts the predetermined angle from the third state. The object of the first aspect is solved by measuring the shape of the subject in the fourth state with the area sensor and obtaining the systematic error of the flatness measurement system using the obtained measurement data. Is.

The object in the second state is returned to the first state by the rotating mechanism, and the shape of the object when linearly shifted by the positioning stage is measured by the area sensor. The measurement accuracy is improved.

The calculation is simplified by making the rotation shift amount in the second state and the rotation shift amount in the fourth state the same.

The present invention also solves the second problem by measuring the flatness of the subject using the systematic error obtained by the above method.

The present invention also includes a rotating mechanism for performing a rotational shift of the subject within a predetermined plane, and a positioning stage for performing a linear shift of the subject within the plane independently of the rotational shift. In a flatness measuring apparatus for a surface of a subject including an area sensor that measures the height above and below the surface of the subject over a predetermined region, the shape of the subject in a first state at a predetermined reference position is defined as the area. The area sensor measures the shape of the object in the second state, which is measured by the sensor and is rotationally shifted by a predetermined angle from the first state by the rotation mechanism, and is then measured by the positioning stage in the predetermined direction from the first state. The shape of the subject in the third state linearly shifted by a predetermined amount is measured by the area sensor,
The area sensor measures the shape of the object in the fourth state, which is rotationally shifted from the third state by a predetermined angle by the rotation mechanism, and uses the obtained measurement data to determine the systematic error of the flatness measurement system. The second problem is solved by including a systematic error calculating unit that obtains the flatness calculating unit and a flatness calculating unit that calculates the flatness of the subject by using the systematic error obtained by the systematic error calculating unit. It has been resolved.

The present invention also has an origin for projecting the cross-sectional shape of the subject together with coordinate axes on the surface of the subject, and orthogonal x, y, fixed in the measurement area of the area sensor together with the origin. In the coordinate system determined by the z coordinate axis, the height z of the shape is the nth degree polynomial of x and y. In the x, y plane, the rotation mechanism and the positioning stage that independently realize the rotation shift and the linear shift of the subject, respectively, are subjected to the predetermined rotation shift and the linear shift, and the measurement obtained through the area sensor is performed. Systematic error ε
(x, y) and the surface shape of the subject, the angle φ from the reference position
Tilt error component η (φ) in the x-axis and tilt error component in the y-axis when rotated by The rotational shift error I (φ, x, y) expressed as a function of the vertical movement error component τ (φ) and the pitching error p (when linearly shifting α in the x-axis direction and β in the y-axis direction α, β), rolling error r (α, β), and shape calculation affected by shift error ξ (α, β, x, y) expressed as a function of vertical movement error g (α, β) data Therefore, the flatness of the subject is determined by eliminating the systematic error ε (x, y), the rotational shift error I (φ, x, y), and the shift error ξ (α, β, x, y). In the apparatus for measuring the flatness of the surface of the object, the shape calculation data z representing the shape of the surface of the object including the systematic error of the flatness measuring system with respect to the object in the predetermined reference position and the reference state. (0,0, x, y) ＝ z (x, y)
+ Ε (x, y) and the subject is rotated by π / 2 rotation about the origin on the xy plane with respect to the origin by the rotation mechanism, and the rotation shift error I (π / 2,
The shape calculation data including x, y) And the difference shape calculation data on the y-axis of the measurement area of the area sensor Is included in the term k ≧ 2 by specifying n values of y using
The first step of obtaining the difference a _k (0) -b _k between the n−1 unknowns a _k (0) and b _k , and the rotation mechanism to rearrange the subject to the reference state, The reference state error δ (x, y), which is the amount of deviation from the reference state caused by relocation of the subject to the reference state, is determined by a predetermined method, and the influence of the reference state error is eliminated. Difference shape calculation data obtained by subjecting the subject to linear shift α in the x-axis direction from the reference position Is included in the term of k ≧ 3 by specifying n values of x using
n-2 unknowns a _k (y), k = 2 terms, k = 1 terms ξ (α, 0,
x (y) and a ₂ (y), a ₁ (y) are arranged c (α, 0, x, y) x + d (α,
Two unknowns of linear expression of the form (0, x, y) By solving a system of equations with a total of n unknowns,
_k (y); k ≧ 3 and a second step of obtaining c (α, 0, x, y) and d (α, 0, x, y), and a linear shift α in the x-axis direction of the second step , The linear shift error ξ (α, 0, x, y), which is obtained by shifting the origin by a π / 2 rotation around the origin in a new coordinate system in which the origin is translated to (α, 0), error And the shape calculation data including the systematic error ε (x, y) From the above, the rotation shift error I (π / 2, x, due to the rotation of the subject obtained by shifting by π / 2 rotations in the first step
Shape calculation data including y) Difference shape calculation data And using the difference shape calculation data M ₀ (α, 0, x, y) obtained by performing only the linear shift α in the x-axis direction in the second step,
A third step of obtaining a pitching error p (α, 0) and a rolling error r (α, 0) that are the linear shift errors accompanying the linear shift α in the x-axis direction by a predetermined method, and a second step The obtained c (α, 0, x, y) is c (α, 0,0, y) =-2
α a ₂ (y) + p (α, 0), and since a pitching error P (α, 0) is determined in the third step, a ₂ (y) is obtained, and From a _k (y); k ≧ 3 obtained in the two steps and a _k (0) -b _k ; k ≧ 2 obtained in the first step, b _k ; k
Through the fourth step of obtaining ≧ 2, the flatness of z (x, y) representing the surface shape of the subject. The shape calculation data z representing the shape of the surface of the subject including the systematic error of the flatness measurement system.
From (0,0, x, y) the flatness The systematic error of the systematic error ε (x, y) excluding the first-order and lower order components is obtained by removing the first-order and lower-order components of the data Is solved autonomously to solve the first problem.

Further, a linear shift error ξ (α, β, expressed by a function of the pitching term p (α, β), the rolling term r (α, β), and the vertical movement term g (α, β). x, y), the linear shift error generated by the shift α, β is the error g due to the vertical movement term at the observation position (α, β).
(α, β) only, and about that point, for any observation position (x, y), the error p (α, β) (x-α) due to the pitching term, the rolling Error due to term r
As a model that (α, β) (y-β) occurs, It was given in.

Further, the rotation shift error I _k (φ, x, y) (k = 0, α) due to the φ rotation of the subject is converted into a vertical movement error τ at a certain observation position (x ₀ , y ₀ ). _k (φ) only, and about that point, for any observation position (x, y), the error due to the pitching term η _k (φ) (xx ₀ ), the error due to the rolling term As a model where It was given in.

Further, in the method of linearly and rotationally shifting the subject, a rotary table as a rotating mechanism is placed and fixed on the one-dimensional moving stage, and the subject is linearly shifted by the one-dimensional moving stage. Also, the subject is rotationally shifted using the rotary table.

The reference state error δ (x, y) in the second step is the shape calculation data of the reference position in the reference state. And each shape calculation data of the reference position including the reference state error The equation of the reference state error only in which the difference of Therefore, the standard state error Δ (x, y) = px + ry + g is calculated by the least squares method.

Shape calculation data representing the shapes of the four surfaces of the subject. Is the polar coordinate system (l, θ), where the positive direction of the x-axis is taken as an angle of zero, and the coordinates are represented by the distance l from the origin and the angle θ from the x-axis direction (clockwise is the positive direction). It is the shape calculation data in one process By setting z (φ, l, θ) (φ = 0, π / 2), these shape calculation data have the systematic error difference ε (x, y) and the rotational shift error I (φ, x, y) has been added The shape calculation data before rotation after the linear shift in the x-axis direction z ₀ (α, 0, x, y), the shape calculation data after the π / 2 rotation. In, in consideration of a new polar coordinate system in which the origin is translated to (α, 0), after the α linear shift of the subject in the positive direction of the axis, the surface shape of the subject and the systematic error are respectively Similarly, To Represents the shape calculation data before rotation Is And the shape calculation data when the subject is linearly shifted in the positive direction of the axis and the subject is rotationally shifted by an angle φ. Is the rotational shift error Added, It is meant to be expressed as.

Further, in the third step, the method of obtaining the pitching error p (α, 0) and the rolling error r (α, 0) which are the linear shift errors associated with the linear shift α in the x-axis direction is the rotation. The shift error is equal before and after the linear shift α, that is, Based on the realistic assumption that the difference between the observation equations before and after the rotation when the subject before and after the linear shift α is rotated by the angle φ in the x-axis direction is Therefore, the shape of the subject surface on the circumference of the radius l before and after the linear shift α in the x-axis direction is the same, that is, And the systematic error before and after the linear shift α in the x-axis direction. The first-order coefficient of cosine on the circumference of radius l of Where c (0, l) and c (α, l) respectively, and the difference of the observation equation after rotation when the subject before and after the linear shift α in the x-axis direction is rotated by the angle φ. By performing the Fourier cosine transform to obtain the first-order coefficient of cosine, And the same as The first-order coefficient of sine on the circumference of radius l of Respectively, s (0, l), s (α, l), for the difference in the observation equation after rotation when the subject before and after the linear shift α in the x-axis direction is rotated by an angle φ. Then, similarly, the Fourier sine transform for obtaining the primary coefficient of sine is performed, We have the relation For the equation obtained by rearranging the above two relational expressions, first, φ = 0, Then, two relations are obtained, and next, with φ = π / 2, By obtaining these two relational expressions, four unknown variables p (α, 0), r (α, 0), c (α, l) -c (0,
l) and s (α, l) -s (0, l) are simultaneously solved to solve the pitching error p (α, p) of the linear shift error accompanying the linear shift α in the x-axis direction. 0), rolling error r (α, 0) It is decided by the.

Further, a CCD is used as the area sensor, and C of the surface shape of the subject is determined by the interference fringes having different optical phases.
A mechanism for calculating the vertical amount corresponding to the CD detection position is used.

The flatness of z (x, y) representing the surface shape of the subject Is determined by determining the coefficient a _k (y) of the second or higher order of the surface shape z (x, y) of the subject obtained from the first process to the fourth process.
quadratic function determined by b _k ; k ≧ 2 , The first-order component a ₁ (y) of the cross-sectional shape group for each position y of the surface shape z (x, y) of the subject is obtained in the second step d (α, 0,0, y) = α ² a ₂ (y) -α a ₁ (y) -αp (α, 0) + r (α,
0) y + g (α, 0) Is expressed by using the surface shape z of the subject.
The influence a ₁ (y) x + b ₁ y exerts on the shape of the first-order component expressed by (x, y) in the cross-sectional shape group is Therefore, the first term There is f (y) × xy form, shape z (x, y) of the object surface because it contains secondary or more components in which the quadratic function z ₂ (x, y) to Obtained by adding Is the flatness of the shape z (x, y) of the surface of the subject. Is decided.

Further, the determined flatness of the surface of the object of z (x, y) representing the surface shape of the object is determined. And, the flatness from the shape calculation data z (0,0, x, y) representing the shape of the subject surface including the systematic error of the flatness measurement system. The systematic error of the systematic error ε (x, y) excluding the first-order and lower-order components of the data obtained by subtracting The method of calculating is the systematic error ε (x,
The shape calculation data z (0,0, x, y) = z (x, y) + ε (x, y) representing the shape z (x, y) of the subject surface including y) is the flatness. , Systematic error excluding the components below the first order And the systematic error ε (x, y) and the sum νx + ωy of the first-order or lower components of the object surface shape z (x, y), Is expressed as The first derivative with respect to x is applied to the equation
By setting = 0, As a result, a first-order coefficient ν of x is obtained, and similarly, a first derivative with respect to y is applied to set x = y = 0, By calculating the first-order coefficient ω of y, As the systematic error ε (x, y) excluding the first-order and below components Is to be asked.

Further, without performing the rotational shift and the linear shift of the subject, the calculated value of the shape of the arbitrary subject surface including the systematic error obtained from the measurement value of the area sensor and the first-order and lower-order components are obtained. Systematic error determined by excluding Using, the flatness of the arbitrary subject surface Is a systematic error determined by excluding the first-order and lower-order components from the shape calculation value of the arbitrary subject surface. Data obtained by subtracting Is the flatness of the subject surface To the sum of the difference between the arbitrary object surface shape and systematic error Therefore, by performing the first-order differentiation with respect to x and setting x = y = 0, As a result, the first-order coefficient χ of x is obtained, and similarly, the first-order differentiation with respect to y is performed to set x = y = 0, By calculating the first-order coefficient γ of y, As the flatness of the arbitrary subject surface Is to be asked.

Further, the predetermined reference position and reference state of the subject are the initial position and the initial state in which the subject is arbitrarily arranged so that its center is substantially aligned with the z axis of the flatness measuring system. It is the one pointed to.

The present invention also provides a systematic error obtained by removing the first-order and lower-order components of the systematic error by the above method. Autonomously obtain, for the flatness measurement of the arbitrary subject surface, without performing the rotational shift and the linear shift of the arbitrary subject, the arbitrary subject surface including the systematic error obtained from the measurement value of the area sensor Shape error and the systematic error excluding the first-order and lower components The second problem is solved by determining the flatness of the surface of the arbitrary subject by a predetermined method using.

Further, without performing the rotational shift and the linear shift of the object, the calculated shape value of the surface of the arbitrary object including the systematic error obtained from the measurement value of the area sensor and the first-order and lower-order components are obtained. Systematic error determined by excluding The method of calculating the flatness of the surface of an arbitrary object by using the above is a shape calculation value z (0,0, x, y) = z (x,
y) + ε (x, y) systematic error determined by excluding the first-order and lower components The least squares method is applied to the data obtained by subtracting, and the average plane a′x + b of the sum of the systematic error and the arbitrary test surface
Finding'y + c ', the flatness of the surface of the subject Means that the deviation from the mean plane z = a'x + b'y + c 'is minimum. It is something that is made to seek by.

Further, without performing the rotational shift and the linear shift of the subject, the calculated shape of the surface of the arbitrary subject including the systematic error obtained from the measurement value of the area sensor and the first or lower order components are obtained. Systematic error determined by excluding The method of calculating the flatness of the surface of an arbitrary object by using a systematic error determined by excluding the first-order or lower component from the calculated shape of the surface of the arbitrary object. The least squares method is applied to the data obtained by subtracting to determine and remove the components of the first or lower order, thereby obtaining the flatness of the surface of the arbitrary subject.

[0044]

BEST MODE FOR CARRYING OUT THE INVENTION An embodiment of the present invention will be described in detail below by taking a surface shape measuring system using an interference optical system including a reference surface as an example. The present invention can be applied to any measuring instrument that has a mechanism for scanning a measurement area on the surface of a subject, measures the height of the surface, and measures the flatness of the surface of the subject.

First, section (0) describes the system configuration and system operation of this flatness measuring apparatus, and sections (1) to (10).
In the section, an algorithm for autonomously determining the flatness measurement of the surface of the subject by the rotational shift and the linear shift of the subject will be described. (0) System configuration of flatness measuring apparatus for subject surface and operation of this system (0.1) System configuration for flatness measuring apparatus for subject surface Interferometer subject surface having systematic error including reference surface error Fig. 1 shows the overall configuration of the system for measuring the flatness of
The positioning stage portion is shown in FIG.

This system has a laser light source 1 and an interference fringe 2
An interferometer system 5 for measuring an interference fringe having a CCD camera 2 for taking a three-dimensional image, a reference surface 3, etc., a one-dimensional positioning stage 6 provided for shifting the subject in the present invention, and a rotation stage for rotation. It is composed of a subject 8 placed on a turntable 7.

Regarding the coordinate system for explaining the operation and algorithm of the present system, as shown in FIG. 2, the direction parallel to the optical axis 4 of the interferometer 5 is the z-axis, and the above-mentioned 1
The one-dimensional moving direction of the rotary table 7 fixed on the three-dimensional positioning stage 6 is defined as the x-axis, and the other one-dimensional moving direction orthogonal to the x-axis is defined as the y-axis.

In the interferometer system 5, the laser light source 1
The laser light emitted from passes through each optical system, partially passes through the reference surface 3, and partially reflects. The transmissive portion is calculated by analyzing interference fringes based on optical interference in which the relative shapes of the object surface 8 and the reference surface 3 are measured by interfering with the part reflected by the object surface 8 and reflected by the reference surface 3. To be done.

(0.2) Operation and method of the system for determining the systematic error of the interferometer including the reference surface error other than the first-order and lower-order components and measuring the flatness of the object surface with high accuracy. Outline of the subject 8 is fixed on the one-dimensional positioning stage 6 and the rotary table 7, and at the reference position,
Interference fringes are measured at each position on the surface 8 of the subject, shape data of the surface of the subject including systematic errors is calculated by a known interference fringe analysis, and the subject is rotated by a rotary table 7 to obtain φ =
By rotationally shifting about π / 2, shape data of the surface of the subject including a systematic error is similarly calculated (first step). next,
φ = 3π / 2 (or φ = −π / 2) is rotated and returned to the reference position, the object is α-shifted in the positive direction of the x-axis, the shape data is calculated, and φ = The shape data is calculated by rotationally shifting by π / 2 (second step).

Using the shape calculation data before and after the rotation of the subject before and after the linear shift and the shape calculation data before and after the rotation after the linear shift, which are obtained through these two processes, from the section (1).
By applying the algorithm in section (10), the second-order or higher-order coefficient of the polynomial that approximates the surface of the subject and the pitching term and rolling term that are linear shift errors are determined. Finally, from section (10), the systematic error including the reference surface error is
In the actual measurement of the flatness of the surface of the object after the determination is made excluding the components other than the first-order components, it is not necessary to shift the object, and thus the one-dimensional positioning stage and the rotary table are not necessary, The shape measurement value including the systematic error of the surface of the object is obtained with the object fixed, and “the shape measurement value including the systematic error derived from the measurement value by the interferometer”-“the first-order or lower components are removed The systematic error "enables a highly accurate measurement of the flatness of the surface of the subject.

(1) Practical Assumptions Regarding Rotational and Linear Shifts of the Object Mathematical Section 1 The following assumptions regarding the performance of the rotating table and the one-dimensional positioning stage used are provided, but these assumptions are realistic.

(A) Assumption 1 regarding the rotation of the subject It is assumed that the center of rotation of the subject has a small deviation in the xy plane and can be ignored without affecting the measured value (the resolution is not so great in the measurement of the plane. not high). Further, it is assumed that the deviation of the surface of the object from the measurement direction (direction perpendicular to the CCD sensor surface) caused by the rotation of the object is sufficiently small and does not affect the accuracy of the measurement value. To do.

(B) Assumption 2 regarding rotation of the subject (evaluation of rotation shift error) The rotation shift error when the surface shape of the subject is rotated by an angle φ from the reference position is given by the following equation. (Where η (φ) is the tilt component in the x-axis direction, Is a tilt component in the y-axis direction, and τ (φ) is a vertical movement component). The positive direction of rotation is the counterclockwise direction according to the definition of the polar coordinate system.

(C) Assumption regarding linear shift of the object The linear shift error associated with the linear shift is the pitching error p (α, β) and the rolling error r (α, β) with respect to the linear shift amount (α, β). ), The vertical movement error g (α, β) Express.

(D) Assumption 2 regarding the rotation of the subject (reproducibility of the rotation shift error before and after the linear shift) When the rotation is performed before and after the linear shift, each rotation shift error causes the same angle φ to rotate. Rotation shift error Is the same before and after the linear shift (see FIG. 3).

(2) Approximating the shape of the subject surface by a polynomial equation In FIG. 4, the shape of the subject surface 9 is a set of cross-sectional shapes 1
Assuming 0, the cross-sectional shape in the x-axis direction is expressed by the polynomial of degree n as follows. However, α ₀ (y) in the equation (2.1) is the constant term 11 in FIG. Can be defined as

(3) Relational expression associated with linear shift of the subject To determine the coefficient of the n-th degree polynomial of x for each y coordinate y = y _k (k = 1,2, ..., m), first, The shape z (0,0, x, y) of the surface of the subject including the systematic error obtained before the shift of the subject is represented by the following equation.

Here, z (x, y) is the true value of the shape of the surface of the object, ε (x, y) is a systematic error including the reference surface error, and z (α, β, x,
y) is the linear shift error ξ (α, β, which occurs corresponding to the linear shift amounts α, β obtained by analysis from the interference fringe data.
x, y) and systematic error ε (x, y) are defined as the measurement value of the shape of the surface of the subject.

Similarly, the relational expression obtained by linearly shifting the subject in the x-axis direction by α is the linear shift error ξ (α,
0, x, y) is included, Is.

Where p (α, 0) is the pitching term and r (α, 0)
Is the rolling term, g (α, 0) is the vertical movement term, and the linear shift error is ξ (α, 0, x, y,) = p (α, 0) (x-β) + r (α, 0 ) y + g
It is represented by (α, 0).

In the equation (3.2), the linear shift error is expressed as in the third term to the fifth term.

Here, from equation (3.2) -equation (3.1), Becomes The left side is the measured value, the right side is a polynomial (z (x-α, y) -z (x, y)) that represents the cross-sectional shape of the surface of the subject, and a term (p ( α, 0) (x-α) + r (α + 0) y + g (α,
0)).

From equations (2.1) and (3.3), Is established.

In the present algorithm, the object fixed on the one-dimensional positioning stage is subjected to one linear shift of appropriate α in the positive direction of the axis as shown in FIG. That is, by measuring the interference fringe data at each position on the subject surface by aligning the positions before the linear shift, and by determining the shape of the subject surface that includes the linear shift error and the systematic error, the systematic error and the subject surface The relational expression (Equation (3.4)) excluding the constant component of the polynomial expressing the cross-sectional shape is established.

(4) Relational expression associated with rotation shift of the subject The surface shape z (x, y) of the subject is expressed by polar coordinates in order to rotate the subject. That is, the positive direction of the x-axis is taken as an angle of zero,
The coordinates are expressed as (l, θ) by the distance l from the origin and the angle θ from the x-axis direction (the positive direction is the counterclockwise direction). Then, in constructing the algorithm associated with the rotation of the subject, the surface shape of the subject z
(x, y) is represented by z (l, θ), systematic error ε (x, y) is represented by ε (l, θ),
It is also assumed that the measured data at (l, θ) when rotated by an angle φ is represented as z (φ, l, θ). At this time, the observation equation before rotation is Becomes

At this time, επ / 2 (1, θ) is subjected to Fourier series expansion as in the following equation.

Next, a rotation shift error when the subject is rotated by an angle φ If is also expressed in polar form, the observation equation after rotation is Becomes Here, z (l, θ) is a value representing the shape along the circumference at the radius l of the surface shape of the subject, and is 2 of the angle θ.
It is a periodic function of π (see FIG. 5).

(5) Coefficient group I (from the equation (3.4) for determining coefficients a _k (y) (k ≧ 3) and b _j (j ≧ 3) of the third or higher degree of the polynomial approximating the shape of the surface of the object to be examined) , Becomes However, far.

Here, the desired variables α _n (y), ..., α ₂ (y),
_{α 1 (y), p (} α, 0), r (α, 0), g (β, 0) and n + 3 unknowns of α _n
(y), ..., α ₃ (y), c (α, 0,0, y), d (α, 0,0, y) are transformed into n unknown variables, and x = x ₁ , x ₂ , x ₃ , ..., x _m , substituting in equation (5.1) and using matrix and vector, Is expressed as However, Is.

Therefore, the following equation , N unknowns, that is, α _n (y), ……, α ₃ (y), c
(α, 0,0, y), d (α, 0,0, y) is determined.

(6) Difference α _k (0) -b between coefficients of second or higher order
The shape calculation data before and after the π / 2 rotation is used in the first step of the determination (1.2) of _k (k = 2,3 ..., n). When rotated by φ = π / 2, the shape is represented by z (y, 0) at the position (0, y) on the y-axis, and the rotation shift error I (π / 2, x, y) = η (π / 2) x + τ (π / 2) is added.

Therefore, the difference between the shape calculation data before and after the rotation is Therefore, Can be obtained. That is, each coefficient α _k (0) − of the second or higher order
The difference in b _k can be determined. From this equation, the vertical movement error τ
Clearly, (π / 2) is required.

(7) Pitching error p (α, 0) and rolling error r (α, 0) when α linearly shifted in the positive direction of the x-axis
As shown in item (4), the xy coordinate system is expressed in polar coordinates, the surface shape z (x, y) of the object is z (l, θ), and the systematic error ε (x, y) is ε.
(l, θ), and (l, θ when rotated by an angle φ
It is assumed that the measurement data in θ) is represented as z (φ, l, θ). At this time, the observation equation before rotation is Met. In addition, ε (l, θ) is Fourier series expanded as follows, The first-order components ε ^c ₁ and ε ^s ₁ related to the cosine sine are π lc (0,1), π ls
If you put (0,1), Is.

Next, before the linear shift by α in the positive direction of the x-axis, the observation equation after rotation when the subject is rotationally shifted by the angle φ is as follows from equation (4.4): Then, if the subject is α-linearly shifted in the positive direction of the x-axis, from equation (3.2) Met.

Therefore, a new coordinate system in which the origin is translated to (α, 0) think of. After the α linear shift of the object in the positive direction of the x-axis, the surface shape of the object is a new coordinate Using, Is.

Here, the origin of the new coordinate system (in the old coordinate system,
Rotation shift by φ around (α, 0)) and consider shape data before and after rotation. In the new coordinate system and polar coordinates And systematic error as well, Then, the formula (7.5) becomes Becomes

Further, the observation equation when the object is rotationally shifted by the angle φ after α linear shift of the object in the positive direction of the x-axis is as follows. Is added Becomes

[0078] Similarly to the case of ε (l, θ), the Fourier series expansion is performed as in the following equation.

At this time, Let πlc (α, l), πls (α, l) be And Here, from the assumption (1) (d), the rotational shift error is equal before and after the lateral shift, that is, Using Becomes Applying the first-order Fourier transform of cosine to equation (7.13), Becomes Similarly, Becomes

Therefore, arrange this Becomes

The right side of equations (7.16) and (7.17) is defined as a function determined by giving φ as follows: Express.

[0082]

[0083]

At this time, with φ = 0,

[0085]

Next, with φ = π / 2, Becomes Therefore, from this, Becomes

(8) Determining quadratic coefficient α ₂ (y), b _{2 From} the following equation section (7), the pitching error p (α, 0) and rolling error r (α, 0) can be obtained. did it. Therefore, from equation (5.2), the quadratic coefficient α ₂ (y) is You can ask as

Since h ₂ = α ₂ (0) -b ₂ is obtained from the equation (6.2), As a result, the quadratic coefficient b ₂ can be obtained.

(9) Flatness of subject surface The second-order or higher-order component of the shape z (x, y) of the surface of the subject was determined by the algorithm up to the item (8). Also, 1
Regarding the next coefficient, from equation (5.3) Therefore, By using that represented by the surface shape z (x, y) of the subject
Of the influence on the shape of the primary component expressed by the cross-sectional shape group α
₁ (y) x + b ₁ y is Therefore, the first term Has a shape of f (y) × yx, and includes a second-order or higher-order component in the shape z (x, y) of the surface of the subject.

The second and third terms on the right side are the shape z of the surface of the subject.
It is a first-order component of (x, y), is a term indicating a slope, and can be ignored when dealing with flatness.

Therefore, the flatness of the shape z (x, y) of the surface of the subject is Can be determined by the following equation.

Therefore, the shape z (x, y) of the surface of the subject is (Here, ν'and ω'are constants that do not depend on (x, y)).

(10) Systematic error ε (x, y) for point (x, y)
The part excluding the first-order and lower components of The relationship between the measured values z (0,0, x, y) of the subject before shifting
From (3.1) and equation (9.5), Therefore, (here Is a systematic error excluding the components of the first order and below).
By setting it to 0, As a result, the first-order coefficient ν of x is obtained, and similarly, the first-order differentiation with respect to y is performed to set x = y = 0. Then, by obtaining the first-order coefficient ω of y, As the systematic error ε (x, y) excluding the first-order and below components Can be determined.

Also, using the least squares method, To the first order component Anyway, The first-order component z = αx + by + c (x, yεs) is determined by the following equation so that

[0095]

Therefore, by subtracting this , The systematic error ε (x, y) is the shape of the systematic error in the sense that the deviation from the average plane z = αx + by + c of the sum of the systematic error and the shape of the surface of the subject is the minimum. Is determined.

(11) Measurement of flatness of arbitrary subject surface From the algorithms up to (10), the systematic error ε (x, y)
The part excluding the first-order and lower components of Was able to decide.

Flatness of arbitrary subject surface In calculating, it is not necessary to rotationally or linearly shift the subject, and the rotary table and the one-dimensional positioning stage are unnecessary.

Systematic error determined from the shape calculation value z (0,0, x, y) of the arbitrary surface of the object including the systematic error obtained from the measured values of the sensor and the components of the first or lower order Includes the first-order component of systematic error or less in the arbitrary object surface shape using On the other hand, as in equations (10.3) and (10.4), each coefficient of the linear equation χx + γy of the sum of the arbitrary object surface shape and systematic error is As the flatness of the arbitrary surface shape of the object To Can be asked as

Similarly to the item (10), the least squares method is used to obtain the first-order component a'x + b'y + c ', and the flatness is calculated. Means that the deviation from the mean plane z = a´x + b´y + c´ is minimal. Can be asked as

FIG. 6 shows a flow for determining the flatness of the surface of the subject.

If the object of application of the present invention is particularly limited to an interferometer, the surface shape of the subject generally indicates a second-order or higher-order component, and the shape can be obtained by this algorithm. In other words, if it is limited to the interferometer, even if the slope component of the systematic error is undecided, the second-order and higher-order components are determined. Since it can be sought, it is highly useful.

In the above embodiment, an example of an interference optical system including a reference surface using a CCD camera as an area sensor has been described, but the present invention is not limited to the interference optical system including the reference surface. If the "measurement value of the shape including systematic error" is obtained by the measurement, the height detection sensor in the area having the same function as the area sensor is used to flatten the surface of the object using these area sensors. Degree measurement can be performed. The angle and direction of the rotation shift are not limited to those in the above embodiment.

[0104]

According to the present invention, the systematic error of the interferometer is
Specially, it is not a calibration subject with a high precision reference plane,
A normal object is fixed to a movable one-dimensional positioning stage in the x direction and a mechanism for rotating the object, and rotational measurement of a plurality of objects and shape measurement by linear shift once in the positive direction of the x axis are performed. It is possible to obtain the flatness of the surface of the object without using any other sensor by using only the measured value of the surface shape of the object including the systematic error of the system. , Can be determined at each point on the reference plane.

In order to determine the systematic error excluding the first-order and lower-order components as described above, a one-dimensional positioning stage for a linear shift operation and a rotation mechanism for rotationally shifting are necessary.
Further, in order to measure the flatness of the surface of the object to be measured in a wide area beyond the measurement area of the area sensor, such a positioning stage and a rotating mechanism are necessary, but once the first or lower order components are excluded, systematic error After the determination, it is not necessary to use the stage and the rotation mechanism when the area sensor covers the measurement area on the surface of the subject. The true value of the flatness of the subject surface can be easily obtained by subtracting the systematic error obtained by removing the first-order and lower-order components from the measured value of the shape.

The linear shift by the one-dimensional positioning stage and the rotary shift by the rotary mechanism such as the rotary stage described in the present invention can be automatically performed with high accuracy by simply giving a shift command value by ordinary numerical control. Since the measurement can be performed, the user himself / herself can determine the systematic error of the flatness measurement system for the surface of the subject, and can always perform high-precision measurement.

[Brief description of drawings]

FIG. 1 is a configuration diagram of an embodiment of the present invention.

FIG. 2 is an enlarged perspective view of the positioning stage.

FIG. 3 is a plan view showing the premise of the present invention.

FIG. 4 is a perspective view showing a cross-sectional shape approximating the surface of a subject according to the present invention.

FIG. 5 is a diagram showing the relationship between the average height of the reference surface, the average height of the subject surface, and the rotation shift error.

FIG. 6 is a flowchart schematically showing a procedure for determining flatness according to the embodiment of the present invention.

[Explanation of symbols]

1 ... Laser light source 2 ... CCD camera 3 ... Reference plane 5 ... Interferometer system 6 ... Two-dimensional positioning stage 7 ... Rotary table 8 ... Subject

Claims (20)

[Claims] 1. A rotating mechanism for performing a rotational shift of a subject within a predetermined plane, a positioning stage for performing a linear shift of the subject independently of the rotational shift within the plane, and a subject surface. A method for measuring a continuity error of a flatness measuring system including an area sensor that measures the height above and below a predetermined area over a predetermined area, wherein the shape of a subject in a first state at a predetermined reference position is defined as the area. The area sensor measures the shape of the object in the second state, which is measured by the sensor and is rotationally shifted by the predetermined angle from the first state by the rotating mechanism, and is measured in the predetermined direction by the positioning stage from the first state. The shape of the object in the third state linearly shifted by a predetermined amount is measured by the area sensor, and the object in the fourth state is rotated from the third state by a predetermined angle by the rotation mechanism. The body shape is measured by the area sensor, by using the measurement data obtained, flatness flatness system connection error measuring method of measuring system of the measuring system of the system subject surface and obtains the connection error. 2. The area sensor measures the shape of the subject when the subject in the second state is returned to the first state by the rotation mechanism and linearly shifted by the positioning stage. The method for measuring a systematic error of a flatness measuring system for a surface of a subject according to claim 1. 3. A rotation shift amount in the second state,
The systematic error measuring method for a flatness measuring system according to claim 1 or 2, wherein the rotation shift amounts in the fourth state are the same. 4. A method for measuring the flatness of a subject surface, which comprises measuring the flatness of the subject using the systematic error obtained by the method according to claim 1. 5. A rotating mechanism for performing a rotational shift of a subject within a predetermined plane, a positioning stage for performing a linear shift of the subject within the plane independently of the rotational shift, and a surface of the subject. In a flatness measuring device for a surface of a subject equipped with an area sensor for measuring a height above and below a predetermined area, a shape of the subject in a first state at a predetermined reference position is measured by the area sensor. The area sensor measures the shape of the object in the second state that has been rotationally shifted from the first state by a predetermined angle by the rotation mechanism, and measures a predetermined amount in the predetermined direction from the first state by the positioning stage. The shape of the subject in the third state that has been linearly shifted is measured by the area sensor, and the shape of the subject in the fourth state that has been rotationally shifted from the third state by a predetermined angle by the rotation mechanism is measured. And the systematic error calculated by the systematic error calculating unit, by using the measured data obtained by measuring the shape with the area sensor, and obtaining the systematic error of the flatness measuring system. And a flatness calculating means for calculating the flatness of the subject, and a flatness measuring device for measuring the surface of the subject. 6. A cross-sectional shape of a subject is determined by orthogonal x, y, z coordinate axes which have an origin projected on the surface of the subject together with coordinate axes and are fixed together with the origin in a measurement area of an area sensor. The height z of the shape in the coordinate system described by In the x, y plane, the rotation mechanism and the positioning stage that independently realize the rotation shift and the linear shift of the subject, respectively, are subjected to the predetermined rotation shift and the linear shift, and the measurement obtained through the area sensor is performed. Using only the data, through a predetermined flatness calculation process, the systematic error ε (x, y) and the surface shape of the subject are tilted in the x-axis direction when rotationally shifted from the reference position by an angle φ. Error component η
(Φ), inclination error component in the y-axis direction The rotational shift error Ι (φ, x, y) expressed as a function of the vertical movement error component τ (φ) and the pitching error p (when linearly shifting α in the x-axis direction and β in the y-axis direction α, β), rolling error r (α, β), and shape calculation affected by shift error ξ (α, β, x, y) expressed as a function of vertical movement error g (α, β) data Therefore, the flatness of the subject is determined by eliminating the systematic error ε (x, y), the rotational shift error I (φ, x, y), and the shift error ξ (α, β, x, y). In the apparatus for measuring the flatness of the surface of the subject, the predetermined reference position and the subject in the reference state,
Shape calculation data z (0,0, x, y) = z (x, y) + ε (x, y) representing the shape of the subject surface including the systematic error of the flatness measurement system.
And, by rotating the subject on the xy plane on the xy plane about the origin by the rotation of π / 2, a rotation shift error I (π / 2, x, y) associated with the rotation of the subject. The shape calculation data including And the difference shape calculation data on the axis of the measurement area of the area sensor Is included in the term k ≧ 2 by specifying n values of y using
First of all to find the difference α _k (0) -b _k between n−1 unknowns α _k (0) and b _k
And a step of relocating the subject to the reference state by the rotating mechanism, and a reference state error δ that is a deviation amount from the reference state caused by relocation of the subject to the reference state.
(x, y) is determined by a predetermined method, the influence of the reference state error is eliminated, and the difference shape calculation data obtained by subjecting the subject to a linear shift α in the x-axis direction from the reference position. Is included in the term of k ≧ 3 by specifying n values of x using
The n−2 unknowns α _k (y), k = 2 terms, and k = 1 terms are ξ
c (α, 0, x, y) where α ₂ (y) and α ₁ (y) are arranged together with (α, 0, x, y)
Two unknowns of linear equation of the form x + d (α, 0, x, y) c (α, 0,0, y) =-2α a ₂ (y) + p (α, 0), d (α , 0,0, y) = α ² a ₂ (y) -α a ₁ (y) -αp (α, 0) + r (α, 0) y + g
By solving the simultaneous equations in which a total of n of (α, 0) are unknowns,
a _k (y); k ≧ 3 and a second step of obtaining c (α, 0, x, y), d (α, 0, x, y), and a linear shift α in the x-axis direction of the second step. After that, further, the linear shift error ξ (α, 0, x, y), the rotational shift error, which is obtained by rotationally shifting the origin in a new coordinate system in which the origin is translated to (α, 0), is obtained. And the shape calculation data including the systematic error ε (x, y) Therefore, in the first step, the rotation shift error I (π / 2,
The shape calculation data including x, y) Difference shape calculation data And using the difference shape calculation data M ₀ (α, 0, x, y) obtained by performing only the linear shift α in the axial direction in the second step,
A third step of obtaining a pitching error p (α, 0) and a rolling error r (α, 0) which are the linear shift errors associated with the linear shift α in the x-axis direction by a predetermined method, and a second step The obtained c (α, 0, x, y) is c (α, 0,0, y) =-2
α a ₂ (y) + p (α, 0), and since the pitching error p (α, 0) is determined in the third step, α ₂ (y) is obtained, and A _k (y); k ≧ 3 obtained from two processes
And a _k (0) -b _k ; k ≧ 2 obtained in the first step, b _k ;
After the fourth step of obtaining k ≧ 2, the flatness of z (x, y) representing the surface shape of the object is measured. The shape calculation data z representing the shape of the surface of the subject including the systematic error of the flatness measurement system.
From (0,0, x, y) the flatness The systematic error of the systematic error ε (x, y) excluding the first-order and lower order components is obtained by removing the first-order and lower-order components of the data A method for measuring the systematic error of a flatness measuring system for the surface of a subject, which is characterized by autonomously obtaining 7. The pitching term p (α, β) and rolling term
r (α, β), and the linear shift error ξ (α, β, x, y) represented by the function of the vertical movement term g (α, β), the linear shift generated by the shift α, β The error is only the error g (α, β) due to the vertical movement term at the observation position (α, β), and with respect to that point, for any observation position (x, y) As a model in which the error p (α, β) (x-α) due to the pitching term and the error r (α, β) (y-β) due to the rolling term occur, The method for measuring the systematic error of the flatness measuring system for the surface of the object according to claim 6, wherein 8. A rotation shift error Ι _k (φ, x, y) (k = 0, α) due to φ rotation of the subject is converted into a vertical movement error τ at a certain observation position (x ₀ , y ₀ ). _k (φ) only, and about that point, for any observation position (x, y), the error due to the pitching term η _k (φ) (xx ₀ ), the error due to the rolling term As a model where The method for measuring the systematic error of the flatness measuring system for the surface of the object according to claim 6, wherein 9. A method for linearly and rotationally shifting the subject is to fix a rotary table, which is a rotation mechanism, on a one-dimensional moving stage, and to linearly shift the subject by the one-dimensional moving stage. The systematic error measuring method for the flatness measuring system for measuring the surface of a subject according to claim 1 or 6, wherein the subject is rotationally shifted using the rotary table. 10. The reference state error δ (x, y) in the second step is the shape calculation data z (0,0, x, y) = z (x, x) of the reference position in the reference state. y) + ε (x, y), and the shape calculation data of each of the reference positions including the reference state error The equation of the reference state error only in which the difference of Therefore, the standard state error δ (x, y) = px + ry + g is calculated by the least squares method.
A method for measuring a systematic error in a system for measuring the flatness of a subject surface according to 1. 11. Shape calculation data representing four shapes of the surface of the subject. Is the polar coordinate system (l, θ), where the positive direction of the x-axis is taken as an angle of zero, and the coordinates are represented by the distance l from the origin and the angle θ from the x-axis direction (clockwise is the positive direction). It is the shape calculation data in one process By setting z (φ, l, θ) (φ = 0, π / 2), these shape calculation data have the systematic error difference ε (x, y) and the rotational shift error I (φ, x, y) has been added The shape calculation data z (α, 0, x, y) before the rotation after the linear shift α in the x-axis direction and the shape calculation data after the rotation π / 2 are expressed as In, considering a new polar coordinate system in which the origin is translated to (α, 0), after the α linear shift of the subject in the positive direction of the x-axis, the surface shape of the subject and the systematic error are respectively Similarly, To Represents the shape calculation data before rotation Is And the shape calculation data when the subject is α-shifted in the positive direction of the x-axis and the subject is rotationally shifted by an angle φ. Is the rotational shift error Added, The systematic error measuring method of the flatness measuring system for measuring the surface of a subject according to claim 6, wherein 12. A method of obtaining a pitching error p (α, 0) and a rolling error r (α, 0), which are the linear shift errors accompanying the linear shift α in the x-axis direction in the third step, is a rotation method. The shift error is equal before and after the linear shift α,
That is, Based on the realistic assumption that the difference between the observation equations before and after the rotation when the subject before and after the linear shift α is rotated by the angle φ in the x-axis direction is Therefore, the shape of the subject surface on the circumference of the radius l before and after the linear shift α in the x-axis direction is the same, that is, And the systematic error before and after the linear shift α in the x-axis direction. The first-order coefficient of Fourier cosine transform on the circumference of radius l of , Respectively, c (0, l), c (α, l), with respect to the difference in the observation equation after rotation when the subject before and after the linear shift α in the x-axis direction is rotated by an angle φ. Cosine Fourier 1
By applying the Fourier cosine transform to find the next coefficient, And the same as The first-order coefficient of Fourier sine transform on the circumference of radius l of Respectively, s (0, l), s (α, l), for the difference of the observation equation after rotation when the subject before and after the linear shift α in the x-axis direction is rotated by the angle φ. Then, similarly, the Fourier sine transform for obtaining the Fourier first-order coefficient of sine is performed, We have the relation For the equation obtained by rearranging the above two relational expressions, first, φ = 0, Then, two relations are obtained, and next, with φ = π / 2, By obtaining the following two relational expressions, four unknown variables p (α, 0), r (α, 0), c (α, l) -c (0,
l), s (α, l) -s (0, l) are simultaneously solved to solve the pitching error p (α, p) of the linear shift error associated with the linear shift α in the x-axis direction. 0), rolling error r (α, 0) 7. The systematic error measuring method of the flatness measuring system for measuring the surface of a subject according to claim 6, wherein 13. The area sensor is a mechanism that uses a CCD and calculates a vertical amount corresponding to the detection position of the CCD of the surface shape of the subject based on interference fringes having different optical phases. Or the systematic error measuring method of the flatness measuring system for the surface of the object described in 6 above. 14. The flatness of z (x, y) representing the surface shape of the subject. Is determined by determining the coefficient a _k (y) of the second or higher order of the surface shape z (x, y) of the subject obtained from the first process to the fourth process.
quadratic function determined by b _k ; k ≧ 2 , The first-order component a ₁ (y) of the cross-sectional shape group for each position y of the surface shape z (x, y) of the subject is obtained in the second step d (α, 0,0, y) = α ² a ₂ (y) -α a ₁ (y) -αp (α, 0) + r (α,
0) + g (α, 0) Is expressed by using the surface shape z of the subject.
The influence a ₁ (y) x + b ₁ y has on the shape of the first-order component that represents (x, y) in the cross-sectional shape group is Therefore, the first term There is f (y) × xy form, shape z (x, y) of the object surface because it contains secondary or more components in which the quadratic function z ₂ (x, y) to Obtained by adding Is the flatness of the shape z (x, y) of the surface of the subject. The systematic error measuring method for the flatness measuring system for measuring the surface of a subject according to claim 6, wherein 15. The determined flatness of the surface of the object of z (x, y) representing the surface shape of the object. And, the flatness from the shape calculation data z (0,0, x, y) representing the shape of the subject surface including the systematic error of the flatness measurement system. The systematic error of the systematic error ε (x, y) excluding the first-order and lower order components of the data obtained by subtracting The method of calculating is the systematic error ε (x,
The shape calculation data z (0,0, x, y) = z (x, y) + ε (x, y) representing the shape z (x, y) of the subject surface including y) is the flatness. , Systematic error excluding the components below the first order And the systematic error ε (x, y) and the sum νx + ωy of the first-order or lower components of the object surface shape z (x, y), Is expressed as The first differential with respect to x is applied to the equation
By setting y = 0, As a result, the first-order coefficient ν of x is obtained, and similarly, the first-order differentiation with respect to y is performed to set x = y = 0, By calculating the first-order coefficient ω of y, As the systematic error ε (x, y) excluding the first-order and below components The method for measuring a systematic error of a flatness measuring system for a surface of a subject according to claim 6, wherein 16. A shape calculation value of the surface of an arbitrary object including a systematic error obtained from a measurement value of the area sensor and the first-order and lower-order components without performing rotational shift and linear shift of the object. Systematic error determined by excluding Using, the flatness of the arbitrary subject surface The method of calculating the systematic error is a systematic error determined by excluding the first-order and lower-order components from the shape calculation value of the arbitrary subject surface. Data obtained by subtracting Is the flatness of the subject surface To the sum of the difference between the arbitrary object surface shape and systematic error Therefore, by applying a first derivative with respect to x and setting x = y = 0, As a result, the first-order coefficient χ of x is obtained, and similarly, the first-order differentiation with respect to y is performed to set x = y = 0, By calculating the first-order coefficient γ of y, As the flatness of the arbitrary subject surface The method for measuring a systematic error of a flatness measuring system for a surface of a subject according to claim 6, wherein 17. The predetermined reference position and reference state of the subject are an initial position and an initial state in which the subject is arbitrarily arranged so that its center is substantially aligned with the z axis of the flatness measuring system. 7. The method for measuring a systematic error of a flatness measuring system for a surface of a subject according to claim 6, wherein 18. The systematic error obtained by the method according to any one of claims 6 to 15 excluding the first-order and lower-order components of the systematic error. For autonomously measuring the flatness of an arbitrary object surface, without performing rotational shift and linear shift of the arbitrary object, an arbitrary object surface including a systematic error obtained from the measurement value of the area sensor. Shape error and the systematic error excluding the first-order and lower components Is used to determine the flatness of the arbitrary subject surface by a predetermined method. 19. The shape-calculated value of the surface of the arbitrary object including the systematic error obtained from the measurement value of the area sensor and the first-order and lower-order components without performing rotational shift and linear shift of the object. Systematic error determined by excluding The method of calculating the flatness of the surface of an arbitrary object by using the above is a shape calculation value z (0,0, x, y) = z (x,
y) + ε (x, y) systematic error determined by excluding the first-order and lower components The least squares method is applied to the data obtained by subtracting, and the average plane a′x + b of the sum of the systematic error and the arbitrary test surface
Finding'y + c ', the flatness of the surface of the subject Means that the deviation from the mean plane z = a'x + b'y + c 'is minimal. 19. The systematic error measuring method for a flatness measuring system for a surface of a subject according to claim 18, wherein 20. The shape calculation value of the surface of the arbitrary object including the systematic error obtained from the measurement value of the area sensor and the first-order or lower-order component without performing the rotational shift and the linear shift of the object. Systematic error determined by excluding The method of calculating the flatness of the surface of an arbitrary object by using a systematic error determined by excluding the first-order or lower component from the calculated shape of the surface of the arbitrary object. 19. The flatness of the arbitrary subject surface is obtained by applying a least squares method to data obtained by subtracting to determine and remove components of a first order or less. Method of flatness of surface of object.