JP2006118987A - Systematic error measuring method of flatness measuring system for specimen surface - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To accurately identify systematic errors in a flatness measuring system. <P>SOLUTION: In this systematic error measuring method of the flatness measuring system, the systematic errors in the flatness measuring system is determined by using a first shape data acquired by measuring a specimen surface shape in the first state on a prescribed reference position by an area sensor, a second shape data acquired, by measuring by the area sensor the specimen surface shape in the second state determined by being shifted rotationally from the first state by a rotating mechanism, the third shape data acquired by measuring by the area sensor the specimen surface shape in a third state determined by being shifted linearly from the first state in a prescribed direction by a positioning stage, and a fourth shape data acquired by measuring by the area sensor the specimen surface shape in a fourth state, determined by being rotationally shifted from the third state by the rotating mechanism. In the method, when the second data and the third data are determined, a large number of times of measurements accompanied by rotational shift are performed, in order to reduce accidental errors accompanying from the rotational shift. <P>COPYRIGHT: (C)2006,JPO&NCIPI

Description

  The present invention relates to the field of accurately measuring a systematic error of a measurement system in order to obtain a flatness measurement value with improved accuracy for the flatness of the surface of a subject having a substantially planar shape. More specifically, the present invention relates to the field of systematic error measurement of a flatness measurement system using an area sensor that measures an upper and lower height of a subject surface over a predetermined region.

  Conventionally, a means for identifying a systematic error caused by a reference surface error of a measurement system for measuring the surface shape and flatness of an object or a wavefront distortion by an optical component using a measurement system including the systematic error is a non-contact method. Several optical measurement methods have been proposed, but there are no methods that have reached the practical stage.

  As one of the methods proposed so far, an object surface shape in a first state at a predetermined reference position is measured by an area sensor, and is rotated and shifted by a predetermined angle from the first state by a rotation mechanism. The object surface shape in the second state is measured by the area sensor, and the object surface shape in the third state obtained by linearly shifting the object surface shape from the first state by a predetermined amount in a predetermined direction by the positioning stage is the area. The surface shape of the subject in the fourth state measured by the sensor and shifted from the third state by a predetermined rotation angle by the rotation mechanism is measured by the area sensor, and the flatness measurement system is obtained using the data obtained there There is a method for obtaining the systematic error (see Patent Document 1).

  However, in the above-described conventional method, in the process of obtaining the systematic error, the rotational shift error when shifting from the first state to the second state is the rotational shift error when shifting from the third state to the fourth state. The first order components of the pitching error, the rolling error and the systematic error accompanying the linear shift are obtained. Hereinafter, this concept is referred to as a conventional method.

However, when considering the performance of a practical rotary table, this assumption is not realistic, and as a result, a highly accurate measurement cannot be expected. That is, if the rotational shift error before and after the linear shift is 1 second, as a result, the required pitching error is an error of about 2 seconds at the maximum. The relationship between the pitching error p and the accuracy dz (x, y) of the systematic error is expressed by dz (x, y) = p / (2α) × x ² where α is the linear shift amount. Therefore, for example, if a reference surface having a diameter of 30 mm is considered and the amount of linear shift is 10 mm and converted into an error (systematic error) of the reference surface, the maximum is about 160 μm.

JP 2003-254747 A

  The present invention has been made to solve the above-mentioned drawbacks, and includes a rotation mechanism for performing a rotational shift of a subject, a positioning stage for performing a linear shift independently of the rotational shift, In a systematic error measuring method of a flatness measuring system provided with an area sensor that measures the height of the specimen surface over a predetermined area, the primary component of pitching error, rolling error and systematic error accompanying linear shift can be accurately obtained. It is an object of the present invention to provide a systematic error measurement method for a possible flatness measurement system.

  The present invention relates to a rotation mechanism for performing a rotational shift of a subject, a positioning stage for performing a linear shift independent of the rotational shift, an area sensor for measuring the height of the subject surface over a predetermined region, In the systematic error measurement method of the flatness measurement system comprising: the first shape data obtained by measuring the surface shape of the subject in the first state at the predetermined reference position with the area sensor; The second shape data obtained by measuring the surface shape of the object in the second state rotated and rotated by the rotation mechanism from the state with the area sensor, and the linear shift from the first state by a predetermined amount in the predetermined direction by the positioning stage The third shape data obtained by measuring the subject surface shape in the third state with the area sensor and the shape of the subject surface in the fourth state that is rotationally shifted from the third state by the rotation mechanism. In the method for obtaining the systematic error of the flatness measurement system using the fourth shape data obtained by the sensor, the coincidence error caused by the rotational shift is reduced when obtaining the second data and the third data. In order to do this, it is obtained by many measurements with a rotational shift.

  That is, as the second data and the third data, a plurality of times of addition data obtained by repeatedly rotating and measuring the rotation mechanism by a predetermined angle by the rotation mechanism, or the first state and the third state When the rotation mechanism is rotated and shifted by a predetermined angle, the data obtained for each predetermined minute rotation is added and the integrated value data is added, or the second method is used to repeat the rotational shift operation by a predetermined angle. Using the obtained addition data for a plurality of times, primary components of pitching error, rolling error, and systematic error associated with the linear shift at the time of transition from the second state to the third state are obtained.

  In order to explain the difference between the conventional method and the present invention in detail, first, the coordinate system (x, y, z) of the system is determined as shown in FIG. In the two-dimensional area of the subject 1, the vertical direction (optical axis direction) of the surface height is the z-axis, the one-dimensional movement direction perpendicular to the z-axis of the one-dimensional positioning stage is the x-axis, and other perpendicular to the x-axis This is a coordinate system with the one-dimensional movement direction as the y-axis, and (x, y) also corresponds to the CCD camera coordinate system. Also, (0,0, z) represents the optical axis.

  Assuming that the shape of the subject surface 3 is sufficiently smooth, the shape of the subject surface 3 is expressed as follows:

  Now, when the subject 1 is rotationally shifted from the reference position by an angle φ around the z-axis by the rotation mechanism, the rotational shift error is the tilt component in the x-axis direction is η (φ), and the tilt is in the y-axis direction. The component is ζ (φ) and the vertical movement component is τ (φ).

This is expressed as The positive direction of rotation is the counterclockwise direction according to the definition of the polar coordinate system.
Next, when the subject 1 at the reference position is linearly shifted by α in the x-axis direction and β in the y-axis direction by the positioning stage, the shift error associated with the linear shift is a pitching error with respect to the shift amounts α and β. p (α, β), rolling error r (α, β), vertical movement error g (α, β)

Let it be expressed as
Also, z (x, y) is the true value of the shape of the subject surface, ε (x, y) is the systematic error including the reference plane error, and z (α, β, x, y) is analyzed from the interference fringe data. Measured values of the shape of the subject surface including linear shift errors ξ (α, β, x, y) and systematic errors ε (x, y) generated corresponding to the linear shift amounts α, β obtained by Define. Furthermore, after z _rot (α, β, x, y) is linearly shifted and further φ rotated, linear shift error ξ (α, β, x, y), rotational shift error I (φ, x, y) And the measured value of the shape of the subject surface including the systematic error ε (x, y).

Here, the shape z (x, y) of the subject surface and the systematic error ε (x, y) including the reference plane error are defined as follows. That is, z (0,0) = ε (0,0) = 0, and the amount of displacement from the reference height, with the upper being positive and the lower being negative.
In order to determine the coefficient of the n-th order polynomial of x for each y coordinate y = y _k (k = 1, 2,..., M), first, the subject surface including the systematic error obtained before the subject is shifted 3 shape z (0, 0, x, y), that is, the shape of the subject surface 3 in the first state is expressed by the following equation.

  Similarly, the relational expression obtained by linearly shifting the subject by α in the x-axis direction, that is, the shape of the subject surface 3 in the third state includes a linear shift error.

It is.
Here, from (Equation 6)-(Equation 5), that is, when the difference in the shape of the object surface 3 between the third state and the first state is calculated.

It becomes. Substituting (Equation 1) into (Equation 7)

It becomes.
Next, in order to express the state of rotation of the subject, the shape Z (x, y) of the subject surface 3 is represented by polar coordinates. That is, the positive direction of the x-axis is taken as the angle zero, and the coordinate is the distance from the origin l (in the formula, it means “el”, and so on)) and the angle θ from the x-axis direction (positive direction around the left) This is expressed as (l, θ).
Then, the shape z (x, y) of the subject surface 3 is expressed as z (l, θ), the systematic error ε (x, y) is expressed as ε (x, θ), and the object surface 3 is rotated by an angle Φ. The measurement data at (l, θ) is expressed as z (Φ, l, θ).

  At this time, the observation equation before rotation is

It becomes.
Also, ε (l, θ) is expressed in a Fourier series expanded state as shown in the following equation.

  Next, the rotation shift error when the subject is rotated by the angle φ

Is also expressed in polar coordinates, the observation equation after rotation is

It becomes. Here, z (l, θ) is a value representing the shape along the circumference of the shape of the subject surface 3 at the radius l, and is a 2π periodic function with respect to θ under the angle.

The coefficients a _k (y) (k =,> 3), b _j (j =,> 3) of the polynomial representing the shape of the object surface are determined as follows. That is, (Equation 8) is rewritten as follows.

It becomes. However,

It is.

Here, _{n of} variables a _n (y), ..., a ₂ (y), a ₁ (y), p (α, 0), r (α, 0), g (β, 0) to be obtained +3 unknowns to _n (y), ..., a ₃ (y), c (α, 0,0, y)), d (α, 0,0, y) n unknowns If variables are converted and a matrix and a vector are used for each of x = x ₁ , x ₂ , x ₃ ,... X _m , (Equation 13) becomes

It is expressed as However,

It is.
Therefore,

N unknowns, that is, a _n (y),..., A ₃ (y), c (α, 0,0, y)), d (α, 0,0, y)
Is determined.

The difference a _k (0) −b _k (k =,> 2) between the coefficients of the second or higher order is determined using the shape calculation data before and after the rotation of π / 2. When Φ = π / 2 is rotated,

Therefore, a _n (y), ..., a ₃ (y), c (α, 0,0, y)) and d (α, 0,0, y) were obtained from (Equation 13). Similar to the process,

Can be obtained.
Next, the pitching error p (α, 0) and rolling error r (α, 0) when α is shifted in the positive direction of the x-axis are determined.
The observation equation before rotation is expressed in polar coordinates as shown in (Equation 9).

Met. In addition, when ε (l, θ) is expanded as follows,

Let πlc (0, l), πls (0, l) be the first order components ε ^c ₁ , ε ^s ₁ , related to the cosine and sine of

It is.

Next, when the subject is rotated by an angle φ before the α shift in the positive direction of the _x axis, the observation equation after rotation, that is, the shape of the subject surface 3 in the second state is (Equation 12) Than,

It is. In addition, if the subject is α shifted in the positive direction of the x axis,

Met.

Therefore, consider a new coordinate system (x _α , y) with the origin translated to (α, 0). After α shifting the subject in the positive direction of the x axis, the surface shape of the subject is z (x _α , y) using the new coordinates (x _α , y). Here, since the shape data before and after the rotation is considered by rotating the angle φ around the origin of the new coordinate system ((α, 0) in the old coordinate system), the shape z (x _α , y) is polar coordinates And expressed as z _α (l, θ), and the systematic error is expressed as ε _α (l, θ),

It becomes.
In addition, the observation equation when the subject is rotated by the angle φ (after the subject is α shifted in the positive direction of the x axis), that is, the shape of the subject surface 3 in the fourth state is the rotational shift error

It becomes.
Similarly to the case of ε (l, θ), ε _α (l, θ) is expanded by Fourier series as shown below.

At this time, let ε _α ^c ₁ , ε _α ^s ₁ , be πlc (α, l), πls (α, l),

Let's say.
Here, when the difference in the shape of the subject surface 3 between the fourth state and the second state is calculated,

It becomes.
When the Fourier transform for the first-order cosine is applied to the above equation,

It becomes.
Similarly, when performing a Fourier transform on the first-order sine,

Therefore,

(Equation 35) and (Equation 36)

It becomes.

  Here, in the conventional method, the rotation error before and after the linear shift is assumed to be equal, that is,

and

It was. Under this assumption, in (Equation 39) and (Equation 40), Φ = 0, π / 2,

From this,

As a result, the primary component of the pitching error p (α, 0), the rolling error r (α, 0), and the systematic error was obtained.

However, the assumptions of (Equation 41) and (Equation 42) are not realistic as shown above. Therefore, in the present invention, realistic assumptions are considered in the following three cases.
(Assumption 1)
Assuming that the rotating mechanism is composed of a rotary table supported by an air bearing, it is expected that the rotating shaft will be tilted for 10 seconds or less, and that repeatability for each rotation is extremely high. Thus, it is considered that the second data and the fourth data can be repeatedly shifted and measured by the rotation mechanism by a predetermined angle from the first state and the third state, respectively. By repeating the same operation repeatedly, the addition of data reduces the chance error, so the rotation error before and after the linear shift should be more equal.

  That is,

Therefore, (Equation 39) and (Equation 40) are each replaced with the average value of n measurement results. This is the first assumption. At this time,

Thus, the pitching error p (α, 0) and the rolling error r (α, 0) can be obtained as in the conventional method.
(Assumption 2)
Next, data is obtained and added every predetermined minute rotation when the rotation mechanism is rotationally shifted by a predetermined angle from the first state and the third state, respectively. That is, consider using integral value data relating to angles.

  As in the case of Assumption 1, if it is assumed that the rotating mechanism is constituted by a rotary table supported by an air bearing, it is assumed that the reproducibility for each rotation is extremely good. Therefore, the tilt component η (Φ) in the x-axis direction and the tilt component ζ (Φ) in the y-axis direction, which are rotational shift errors, can be considered as periodic functions with a period of one rotation.

Should be true. This is the second assumption.

  Considering (Equation 51), (Equation 39) and (Equation 40) are integrated from 0 to 2π with respect to φ.

And eventually

Get.

  Therefore, in (Equation 39) and (Equation 40), if Φ = 0,

And also

Therefore, the following expression is obtained after all.

From the above equation, the pitching error p (α, 0) and the rolling error r (α, 0) can be obtained.
(Assumption 3)
Here, it is a method of further increasing accuracy by combining Assumption 1 and Assumption 2. That is, the rotary table is rotated a plurality of times to further reduce the chance error. Therefore, here

And can be put. This is the third assumption.

  In this case, from (Equation 39) and (Equation 40)

Get.

  Therefore, if (Equation 62) and (Equation 63) are regarded as (Equation 54) and (Equation 55), the pitching error p (α, 0) and the rolling error r (α, 0) are the same as before. Desired.

  As explained above with the three assumptions, if the pitching error p (α, 0), the rolling error r (α, 0) and the primary component of the systematic error are obtained, then the systematic error is obtained according to the conventional method. I can do it.

  As described above, in the present invention, the rotation mechanism for performing the rotational shift of the subject, the positioning stage for performing the linear shift independently of the rotational shift, and the height of the subject surface are set to a predetermined level. In a systematic error measurement method of a flatness measurement system provided with an area sensor that measures over an area, a first shape obtained by measuring a surface shape of a subject in a first state at a predetermined reference position with the area sensor Data, second shape data obtained by measuring the shape of the surface of the subject in the second state rotated and rotated from the first state by the rotation mechanism with the area sensor, and the first state from the first state Third shape data obtained by measuring the surface shape of the subject in the third state linearly shifted by a predetermined amount in a predetermined direction by the positioning stage with the area sensor, and from the third state before In the method for obtaining the systematic error of the flatness measurement system using the fourth shape data obtained by measuring the subject surface shape in the fourth state rotated and shifted by the rotation mechanism with the area sensor, the second data When the third data is obtained, measurement is repeated many times to reduce the accidental error associated with the rotational shift, and the rotational shift error is calculated using the result. Systematic errors can be identified.

  Next, a system configuration for determining a systematic error suitable for carrying out the present invention will be described. FIG. 2 shows a system configuration for measuring the surface shape of an object by an interferometer having a systematic error including a reference plane error.

  This system includes an interference fringe measurement interferometer system 11 having a laser light source 5, a CCD camera 7 that captures a two-dimensional image of interference fringes, a reference surface 9, and the like, and a positioning stage provided for shifting the subject 1. 13 and a subject 1 placed on a rotation mechanism 15 for rotationally shifting.

  As for the xyz coordinate system, as shown in FIG. 1, the direction parallel to the optical axis 17 of the interferometer system 11 is taken as the z-axis, and the one-dimensional movement direction of the rotating mechanism 15 fixed on the positioning stage 13. Is the x-axis, and the other one-dimensional movement direction orthogonal to the x-axis is the y-axis.

  In the interferometer system 11, the laser light emitted from the laser light source 5 is transmitted through each optical system, partially transmitted through the reference surface 9, and partially reflected. The transmission part is calculated from the analysis of interference fringes based on the optical interference in which the relative shape between the object surface 3 and the reference surface 9 is measured by causing interference with the part reflected by the object surface 3 and reflected by the reference surface 9. It is done. That is, the surface shape of the subject including the systematic error is calculated.

  The subject 1 is fixed on the positioning stage 13 and the rotation mechanism 15, and as described above, the subject 1 is linearly shifted from the positioning stage 13 in the direction indicated by the arrow 19 or rotated by the rotary table 15. The surface shape data of the subject including the above-mentioned systematic error is calculated by shifting each of them, and the surface shape of the subject is determined.

z (x, y) is the true value of the shape of the object surface, ε (x, y) is the systematic error including the reference plane error, and z (α, β, x, y) is analyzed from the interference fringe data. It is defined as a measurement value of the shape of the subject surface including linear shift error ξ (α, β, x, y) and systematic error ε (x, y) generated corresponding to the obtained linear shift amounts α, β . Furthermore, linear shift error ξ (α, β, x, y), rotation shift error I (Φ, x, y) when z _rot (α, β, x, y) is further rotated φ after linear shift And the measured value of the shape of the subject surface including systematic error ε (x, y).
First, in the reference position, that is, in the first state, measurement of interference fringes is performed at each position of the subject surface 3, and calculation of shape data of the subject surface including systematic errors calculated by known interference fringe analysis. That is, first shape data is obtained.

  The left side represents the first shape data, the first term on the right side represents the true value of the shape, and the second term represents the systematic error.

  Next, second shape data is obtained from the subject 1 using the rotary table 15. The second shape data obtained by one measurement is expressed as follows in polar coordinate display.

  The left side represents the second shape data, the first term on the right side is the true value of the shape, the second term is a systematic error, the third to fifth terms are rotational shift errors, and the tilt component in the x-axis direction is η (Φ ), The inclination component in the y-axis direction is ζ (Φ), and the vertical movement component is τ (Φ).

  Next, the rotary table 7 is returned to the reference position, that is, returned to the first state, and the subject 1 is linearly shifted from the second state to the third state by the positioning stage 13, and the subject in the third state. The third shape data that is the shape of the surface 3 is obtained.

  The left side is the third shape data, the first term on the right side is the true value of the shape, the second term is the systematic error, the third to fifth terms are the rotational shift errors, and the pitching error with respect to the shift amounts α and β p (α, β), rolling error r (α, β), and vertical movement error g (α, β).

  Finally, fourth data is obtained from the subject 8 using the rotary table 7. The shape of the subject surface 3 in the fourth state obtained by one measurement is displayed in polar coordinates.

It becomes. The left side is the fourth shape data, the first on the right side is the true value of the shape, the second term represents the systematic error, the third to seventh terms represent the linear shift error and the rotational shift error.

  Here, the Fourier transform relating to the first cosine and sine of the difference between the fourth shape data and the second shape data is set as follows.

  As Example 1, Assumption 1 is used. That is, a case will be described in which an operation of rotationally shifting by an angle of π / 2 is performed many times, and data obtained a plurality of times each time is used. That is, in the second state and the fourth state, n measurements are performed,

Calculate next,

Thus, the primary component of the pitching error p (α, 0), the rolling error r (α, 0), and the systematic error is obtained.

  In the second embodiment, assumption 2 is applied. That is, the integral data regarding the angle obtained when the rotation mechanism makes one rotation from the first state is used. in this case,

Thus, the primary component of the pitching error p (α, 0), the rolling error r (α, 0), and the systematic error is obtained.

  In the third embodiment, assumption 3 is applied. In other words, integral data relating to the angle obtained when rotating a number of times is obtained n times. in this case,

Thus, the primary component of the pitching error p (α, 0), the rolling error r (α, 0), and the systematic error is obtained.

  Once the pitching error p (α, 0), rolling error r (α, 0), and the primary component of the systematic error are obtained, the systematic error including the reference plane error is less than the primary using the conventional method. It is determined except for other components.

  Since the systematic error is a value inherent to the interference system 11, it is not necessary to shift the subject in the actual surface shape measurement of the subject, and therefore the one-dimensional positioning stage and the rotary table are not necessary, A shape measurement value including a systematic error on the surface of the subject is obtained while the specimen is fixed, and "a shape measurement value including a systematic error derived from a measurement value obtained by an interferometer"-"a system in which the first and lower order components are removed By "error", high accuracy measurement of the surface shape of the object is realized.

Illustration for explaining the coordinate system Interferometer system to which the present invention is applied

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 Subject 3 Subject surface 5 Laser light source 7 CCD camera 9 Reference surface 11 Interferometer system 13 Positioning stage 15 Rotating mechanism 17 Optical axis 19 of interference optical system Arrow

Claims (4)

A rotation mechanism for performing a rotational shift of the subject within a predetermined plane;
A positioning stage for performing a linear shift of the subject independently of the rotational shift within a predetermined plane;
A systematic error measurement method of a flatness measurement system comprising an area sensor that measures the height of the surface of the subject over a predetermined region,
First shape data obtained by measuring the surface shape of the subject in a first state at a predetermined reference position with the area sensor;
Second shape data obtained by measuring the subject surface shape in a second state rotated and rotated by the rotation mechanism from the first state with the area sensor;
Third shape data obtained by measuring the object surface shape in a third state linearly shifted by a predetermined amount in a predetermined direction from the first state by the positioning stage with the area sensor;
Using the fourth shape data obtained by measuring the surface shape of the subject in the fourth state rotated and rotated by the rotation mechanism from the third state with the area sensor, the systematic error of the flatness measurement system is determined. In the way you seek,
The second data and the fourth data are shape data based on a plurality of measurements with a predetermined rotational shift, and a systematic error measurement method for a flatness measurement system on a subject surface The shape data based on a plurality of measurements with a predetermined rotation shift according to claim 1 is an operation of measuring the rotation data by rotating the rotation mechanism by a predetermined angle from the first state and the third state, respectively. Systematic error measurement method for the flatness measurement system of the object surface, characterized in that it is a plurality of addition data obtained repeatedly The shape data based on a plurality of measurements with a predetermined rotational shift according to claim 1 is predetermined when the rotational mechanism is rotationally shifted by a predetermined angle from the first state and the third state, respectively. Systematic error measuring method for flatness measuring system of subject surface, characterized in that it is data of an integral value relating to an angle obtained by adding data for each minute rotation of the subject The shape data based on a plurality of measurements with a predetermined rotational shift according to claim 1 is predetermined when the rotational mechanism is rotationally shifted by a predetermined angle from the first state and the third state, respectively. Systematic error measurement method for flatness measurement system of subject surface, characterized in that it is integrated value data relating to the angle obtained by adding and obtaining data for every minute rotation of the subject, and further addition data obtained by repeatedly performing the same operation