Stage device, exposure apparatus, method of controlling stage device, exposure method, and method of producing article

Abstract

To provide a technique for acquiring a shape of a flat mirror with a high accuracy and in a short time in a stage device having an interferometer system having no redundant axis.SOLUTION: A stage device includes: a stage; a driving mechanism for driving the stage in a first direction and a second direction intersecting in a plane surface; a first interferometer and a second interferometer for measuring a position of the stage; a common flat mirror provided on the stage for measurement by the interferometers; a mark measuring instrument having a third optical axis on a plane; a control part for controlling the driving mechanism so as to locate the stage on the basis of outputs of the interferometers, where the control part causes a mark measuring instrument to measure a position while controlling the driving mechanism on the basis of outputs of the interferometers so as to locate the stage on each of a plurality of positions in such a state that a reference member having a plurality of marks is arranged on the stage so as to obtain a measurement result of positions of the plurality of marks by the mark measuring instrument, and a shape of the plane mirror on the basis of a distance between a first optical axis and the third optical axis.SELECTED DRAWING: Figure 1

Description

[Technical field]

[0001] The present invention relates to a stage apparatus, an exposure apparatus, a method for controlling a stage apparatus, an exposure method, and a method for manufacturing an article. [Background technology]

[0002] A stage device that can be incorporated in an exposure apparatus for manufacturing semiconductor devices can be equipped with a plane mirror and a laser interferometer to drive the stage in the XY directions and in small rotations. Here, a plane mirror aligned along the X direction and a plane mirror aligned along the Y direction are fixed to the stage, and at least two laser interferometers can be arranged so that measurement light is perpendicularly incident on the reflective surface of each plane mirror. It is desirable for the reflective surfaces of these plane mirrors to be perfectly flat, but in reality, there is distortion in the shape (shape error), which is a factor that reduces the movement accuracy of the stage.

[0003] Therefore, by calculating the shape of the plane mirror in advance and correcting the movement characteristics of the stage based on the calculation results, highly accurate movement characteristics can be realized. For example, in Patent Documents 1 and 3, an interferometer system is provided with a redundant axis, and the shape of the plane mirror is calculated from the measurement value of the redundant axis. However, providing a redundant axis means providing an interferometer in addition to the interferometer that controls the movement of the stage, which is disadvantageous in terms of cost.

[0004] For this reason, when the interferometer does not have a redundant axis, a common method is to use a measurement wafer with measurement marks and a measuring instrument such as a camera mounted on the apparatus, and calculate the shape of the plane mirror from the measurement results of the measuring instrument. Patent Document 2 is an example of this method. [Prior art documents] [Patent documents]

[0005] [Patent Document 1] Japanese Patent Application Publication No. 3-10105 [Patent Document 2] Japanese Patent Application Publication No. 9-79829 [Patent Document 3] Japanese Utility Model Application Publication No. 59-98446 Summary of the Invention [Problem to be solved by the invention]

[0006] In Patent Document 2, a measuring instrument (alignment sensor) is placed at the intersection of two orthogonal interferometer optical axes, and no realization method is shown for cases where the measuring instrument is not based on this premise.

[0007] The present invention aims to provide an advantageous technology for obtaining the shape of a plane mirror with high precision and in a short time in a stage device having an interferometer system that does not have a redundant axis, while eliminating the assumption that a measuring instrument is located at the intersection of the interferometer optical axes. [Means for solving the problem]

[0008] One aspect of the present invention relates to a stage device, the stage device including a stage, a drive mechanism for driving the stage in a first direction and a second direction that are orthogonal to each other in a plane and in rotation about an axis orthogonal to the plane, a first interferometer having a first optical axis parallel to the first direction and for measuring a position of the stage in the first direction, a second interferometer having a second optical axis parallel to the first direction and spaced from the first optical axis and for measuring a rotation of the stage, a common plane mirror provided on the stage for measurement by the first interferometer and the second interferometer, and a mark measurement instrument having a third optical axis orthogonal to a position shifted from the first optical axis on the plane, and a control unit that controls the driving mechanism so that the stage is positioned based on the output of the first interferometer and the second interferometer, wherein when a reference member having a plurality of marks aligned along the second direction is placed on the stage, the control unit controls the driving mechanism based on the output of the first interferometer and the second interferometer so that the stage is positioned at each of a plurality of positions corresponding to the plurality of marks, respectively, while causing the mark measuring instrument to measure the positions of the plurality of marks, and obtains the shape of the plane mirror based on the measurement results of the positions of the plurality of marks by the mark measuring instrument and the distance between the first optical axis and the third optical axis on the plane. Effect of the Invention

[0009] According to the present invention, an advantageous technique is provided for obtaining the shape of a plane mirror with high accuracy and in a short time in a stage device having an interferometer system that does not have a redundant axis, while eliminating the assumption that the measuring instrument is located at the intersection of the interferometer optical axes. [Brief description of the drawings]

[0010] [Figure 1] FIG. 2 is a diagram showing the configuration of a stage device according to an embodiment. [Diagram 2] FIG. 13 is a diagram illustrating the measurement sequence of 51 marks on a substrate serving as a reference member. [Diagram 3] FIG. 4 is a diagram illustrating measurement value data. [Figure 4] FIG. 13 is a diagram illustrating an example of an average value of the X-shift measurement value calculated for each row. [Diagram 5] 13 is a diagram illustrating an example of the relationship between the Y coordinate of the center of the substrate as a reference member, the mark measurement value, and the shape error of a plane mirror. [Figure 6] 6A and 6B are diagrams illustrating examples of the influence of rotational misalignment of a stage on a measurement value. [Figure 7] FIG. 13 is a diagram illustrating an outline of linear interpolation of a shape error of a flat mirror. [Figure 8] FIG. 11 is a diagram illustrating specific numerical values ​​of a coefficient matrix. [Figure 9]13 is a diagram illustrating an example of the relationship between the X coordinate of the center of a substrate serving as a reference member, and the mark measurement value and the shape error of a plane mirror. [Figure 10] FIG. 13 is a diagram illustrating a stage device in which a mark measurement device is arranged on the Y axis. [Figure 11] FIG. 11 is a diagram illustrating specific numerical values ​​of a coefficient matrix. [Figure 12] FIG. 1 is a diagram illustrating a stage device including a plurality of measuring instruments. [Figure 13] FIG. 13 is a diagram illustrating an example of a stage device provided with a mark measuring device that can measure only the amount of deviation in the X-axis direction. [Figure 14] FIG. 11 is a diagram illustrating specific numerical values ​​of a coefficient matrix. [Figure 15] FIG. 2 is a diagram illustrating an example of a stage device on which a measurement plate is mounted. [Figure 16] FIG. 1 is a diagram showing the arrangement of an exposure apparatus according to an embodiment. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0011] Hereinafter, the embodiments will be described in detail with reference to the accompanying drawings. Note that the following embodiments do not limit the invention according to the claims. Although the embodiments describe a plurality of features, not all of these features are essential to the invention, and the features may be combined in any manner. Furthermore, in the accompanying drawings, the same or similar configurations are given the same reference numbers, and duplicated descriptions are omitted. In the following description, directions are described according to the XYZ coordinate system. The XY plane in the XYZ coordinate system is a horizontal plane, and the Z axis can be parallel to the vertical direction. The X-axis direction is parallel to the X-axis, the Y-axis direction is parallel to the Y-axis, and the Z-axis direction is parallel to the Z-axis.

[0012] In the following description, the X-axis direction will be described as the first direction and the Y-axis direction as the second direction, but the first and second directions can be interchanged. In other words, the X-axis direction may be the second direction and the Y-axis direction may be the first direction. First Embodiment A first embodiment of the present disclosure will be described. FIG. 1(a) is a top view of a stage device ST of the first embodiment. The stage device ST can be used as a device for positioning a substrate such as a wafer in an exposure apparatus. The stage device ST can be configured to position a stage 5 as a positioning target. A circular substrate (measurement plate) 3 as a reference member can be placed on the stage 5. In the plan view, plane mirrors 2X and 2Y are fixed to the right side and the back side of the stage 5, respectively. The substrate 3 may be circular or may have other shapes such as a square or a rectangle. The outer side surfaces (surfaces) of the plane mirrors 2X and 2Y are reflective surfaces for laser light from a laser interferometer (hereinafter referred to as "interferometer") and can be mirror-finished. A first interferometer 4X1 and a second interferometer 4X2 are arranged on the sides of the plane mirror 2X with their optical axes parallel to each other, and the distance between the centers of their optical axes is s [mm]. A third interferometer 4Y is disposed to the side of the plane mirror 2Y, and the laser light emitted from the third interferometer 4Y is incident on the reflecting surface of the plane mirror 2Y. The intersection point between the optical axis of the interferometer 4X1 and the optical axis of the interferometer 4Y is set as point O, which is the origin in the XY plane. The stage 5 can be moved in the X direction (first direction) and Y direction (second direction) by a driving mechanism DM such as a linear motor (not shown), and can also be slightly rotated around the Z axis perpendicular to the XY plane.

[0013] The control of the stage 5 can be performed as follows. The translation of the stage 5 in the X-axis direction (first direction) is controlled so that the measurement value of the interferometer 4X1 becomes a suitable value calculated from the movement target position while the position of the stage 5 in the Y-axis direction (second direction) is controlled so that the measurement value of the interferometer 4Y becomes a suitable value calculated from the movement target position while the position of the stage 5 in the X-axis direction (first direction) is controlled so that the measurement value of the interferometer 4X1 becomes a suitable value calculated from the movement target position. The rotation control of the stage 5 is controlled so that the value obtained by dividing the difference between the measurement values ​​of the interferometer 4X1 and the interferometer 4X2 by the distance s becomes a suitable value calculated from the target rotation amount. Here, the position control in the first direction and the second direction and the rotation control can be performed in parallel.

[0014] The positional relationship between the interferometer 4X1 and the interferometer 4X2 may be reversed, and the average value of the measurement value of the interferometer 4X1 and the measurement value of the interferometer 4X2 may be used as the reference value of the interferometer in the drive in the first direction described above. However, in this embodiment, the reference value of the interferometer in the drive in the X-axis direction (first direction) is described as the measurement value of the interferometer 4X1. Also, the interferometer 4Y does not necessarily have to be an interferometer, and may be replaced by, for example, a linear encoder, etc., as long as it has a mechanism for referencing the current position.

[0015] A plurality of measurement marks are arranged on the substrate 3. The deviations of these measurement marks in the X-axis direction and the Y-axis direction are measured by the measuring instrument 1, which is a mark measuring instrument arranged at coordinates (g [mm], h [mm]). The measurement marks may be measurement marks capable of simultaneously measuring the X-direction and the Y-direction, as typified by a cross shape. Alternatively, the dedicated mark for the X-direction measurement and the dedicated mark for the Y-direction measurement may be arranged so close that they can be regarded as the same place. Furthermore, the dedicated mark for the X-direction measurement and the dedicated mark for the Y-direction measurement may be arranged apart, in which case the arrangement coordinates of each mark are taken into consideration. However, in this embodiment, an example will be described in which a measurement mark capable of simultaneously measuring the X-direction and the Y-direction is used.

[0016] A plurality of measurement marks are arranged at equal intervals in the X-axis and Y-axis directions on the substrate 3. The arrangement intervals in the X-axis and Y-axis directions are denoted by P X [mm],P Y The measurement marks are exposed by an exposure apparatus with a sufficiently adjusted exposure position accuracy, and the position error is small enough to be negligible. Alternatively, if the position error is accurately measured in advance, the position error can be used for correction, so the position error is not a problem. As a method for exposing marks while arranging them with high position accuracy, the methods described in JP-A-2000-299278, JP-A-2005-064268, and JP-A-2009-259966 can be adopted.

[0017] As shown in FIG. 1(b), the exposure apparatus or stage device ST may include a control unit 6 that performs operations related to driving the stage ST and calculations using the measurement values ​​of the interferometer. The control unit 6 may be configured, for example, by a PLD (abbreviation of Programmable Logic Device) such as an FPGA (abbreviation of Field Programmable Gate Array), an ASIC (abbreviation of Application Specific Integrated Circuit), a general-purpose or dedicated computer with a program installed, or a combination of all or part of these. The control unit 6 may include a calculation unit 6a and a correction processing unit 6b. The calculation unit 6a collectively calculates the shape errors of the plane mirrors 2X and 2Y from the measurement results of the measurement marks by the measuring device 1. In addition, the correction processing unit 6b performs a correction processing to correct the shape errors of the plane mirrors 2X and 2Y calculated by the calculation unit 6a by correcting the output values ​​of the interferometers 4X1, 4X2, and 4Y, and reflect them in the control of the stage 5.

[0018] The control unit 6 sequentially selects a plurality of measurement marks on the substrate 3, moves the stage 5 so that the selected measurement marks are within the field of view (measurable range) of the measuring device 1, and measures the positional deviations of the measurement marks in the X-axis direction and the Y-axis direction using the measuring device 1. Of course, it is not necessary to measure all the measurement marks, and it is sufficient to measure at least one measurement mark among the plurality of measurement marks having the same position in the X-axis direction and at least one measurement mark among the plurality of measurement marks having the same position in the Y-axis direction. However, it is preferable to use the average value for the measurement marks of the same X-coordinate or Y-coordinate in calculating the shape of the plane mirror, and from the viewpoint of reducing the measurement error, the more the number of measurement marks, the higher the calculation accuracy. In addition, the measurement order of the plurality of measurement marks on the substrate 3 is not limited to a specific order, but generally, a zigzag measurement order as shown in FIG. 2 is advantageous in terms of the required time.

[0019] The obtained measurement value data may include four items for each measurement mark: X coordinate, Y coordinate, X deviation measurement value, and Y deviation measurement value with respect to the center of the substrate 3. The control unit 6 performs averaging processing on these measurement value data as a pre-calculation process. As a result, the control unit 6 calculates the X deviation measurement value (average value) r for the Y coordinate i. X (i) [nm], Y deviation measurement value (average value) r for Y coordinate i Y (i) [nm], Y deviation measurement value (average value) r for X coordinate j Y (j) [nm] can be prepared for three groups of data.

[0020] The procedure for this pre-calculation process is as follows. For each of the 51 measurement marks shown in Fig. 2, an X-shift measurement value and a Y-shift measurement value are obtained. These are summarized in the measurement value data shown in Fig. 3(a). First, the X-shift measurement value is explained. The X-shift measurement values ​​rearranged according to the mark arrangement are shown in Fig. 3(b), and mapped in Fig. 3(c). If the average value is calculated for each row for the X-shift measurement values ​​shown in Fig. 3(b), the X-shift measurement value (average value) r for the Y coordinate i is obtained as shown in Fig. 4(a). X (i) [nm] data can be obtained, which is shown in the graph in Figure 4(b).

[0021] Similarly, the Y deviation measurement value (average value) r for Y coordinate i Y (i) [nm] data is obtained. Also, the Y deviation measurement value (average value) r for the X coordinate j is obtained. Y The data for (j) [nm] can also be obtained by appropriately swapping the X-shift measurement value and the Y-shift measurement value, as well as the rows and columns, in the above-mentioned procedure.

[0022] Since the position of the center of the substrate 3 in the Y-axis direction corresponds one-to-one to the position of the reflection point of the laser light on the plane mirror 2X in the Y-axis direction at that time, the Y coordinate i may be determined as the position in the Y-axis direction on the plane mirror 2X. Of course, the same applies to the X-axis direction.

[0023] The procedure for calculating the shape of the plane mirror 2X will be described below.

[0024] FIG. 5 shows the relationship between the Y coordinate of the center of the substrate 3 and the shape error of the plane mirror 2X. In FIG. 5, the shape error of the reflecting surface of the plane mirror 2X is exaggerated. When the Y coordinate of the center of the substrate 3 when the measurement mark is measured by the measuring device 1 is i, the shape error in the X direction at the reflection point on the plane mirror 2X of the laser light from the interferometer 4X1 is expressed as b X (i) [nm]. As mentioned above, the X-shift measurement value (average value) and the Y-shift measurement value (average value) are respectively r X (i) and r Y (i).

[0025] Here, specific values ​​are set for the above symbols as follows for explanation.

[0026] Distance between the optical axes of interferometers 4X1 and 4X2: s [mm] = 80 X coordinate of measuring device 1: g [mm] = +20 Y coordinate of measuring device 1: h [mm] = -150 Measurement mark spacing: P Y [mm]=30 Number of measurement marks in the Y direction (number of rows): N Y =9 The Y coordinates [mm] of the measurement marks relative to the center of the substrate 3 are 0, ±30, ±60, ±90, ±120. Therefore, the Y coordinates of the center of the substrate 3 when measuring these measurement marks are calculated by inverting the sign of the above Y coordinates and adding the Y coordinate h [mm] (=-150) of the measuring device 1, as follows:

[0027]

number

[0028] Next, when the Y coordinate of the center of the substrate 3 is i, a relational equation is established for when an X-shift or rotation error occurs in the stage 5 due to a shape error of the plane mirror 2X. The position of the stage 5 in the X-axis direction is controlled based on the measurement value of the interferometer 4X1, and the amount of minute rotation is controlled based on the difference between the measurement values ​​of the interferometers 4X1 and 4X2. Therefore, when a shape error b X 6(a), the presence of (i) causes an X-positional deviation and a rotational deviation with respect to the stage 5. In this case, the center of rotation of the rotational deviation is point O, which is the intersection of the optical axes of the interferometers 4X1 and 4Y.

[0029] When the Y coordinate of the center of the substrate 3 is i, the X deviation measurement value r X (i) can be expressed by the following equation, which includes the term -h*θ(i), which is the component caused by the rotation error. Note that the rotation error of the stage 5, with CCW as the positive direction, is θ(i)[urad], and the constant measurement error of the X-shift measurement value of the measuring device 1 is ε X [nm], irregular measurement error is ζ X (i) [nm].

[0030]

number

[0031] Here, the rotation error θ(i) of the stage 5 can be expressed by the following equation.

[0032]

number

[0033] Therefore, equation (2) is b X (i) and (b) X It can be expressed by equation (4) with (i+sd) as a variable.

[0034]

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[0035] Similarly, using FIG. 3(b), an equation is formulated for the Y-shift measurement value due to the Y-position shift and rotation shift occurring with respect to the stage 5. The Y-shift measurement value r Y In (i), the scale error m Y [ppm] (not shown). Therefore, the constant measurement error of the Y deviation measurement value of the measuring instrument 1 is ε Y , the irregular measurement error is ζ Y (i) can be expressed as the following equation, which includes the term g*θ(i), which is due to the rotation error. Note that the scale error m Y Although the term is not necessarily required for solving the problem further, by incorporating it into equation (5), we can expect high calculation accuracy.

[0036]

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[0037] By substituting equation (3) into equation (5), the equation can be transformed into the following equation.

[0038]

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[0039] Equation (4) and equation (6) are combined and a simultaneous linear equation is solved using a matrix. The number of equations formulated is: X (i) and Y deviation measurement value r Y Since there are two (i), the total is 18. On the other hand, the number of unknowns is b X (i) 9 pieces, b X (i+s) is 9, and the measurement error is ε X and ε Y 2, scale error m Y This totals 21, which is more than the number of equations mentioned above, so the result is indeterminate.

[0040] So, b XFor (i+s), an equation is formulated using approximation by linear interpolation at the adjacent i on both sides. Figure 7(a) shows the upper part of the plane mirror 2X (the lower part is not shown). For example, when the Y coordinate of the center of the substrate 3 is i [mm] = -270, the measured X-shift value r X (i) and Y deviation measurement value r Y b in the formula (i) X (i) is b X (-270), and b X (i+s) is b X (-190), where b X For (-190), the two adjacent i's are b X (-210) and b X Linear interpolation is performed at (-180). That is, as shown in FIG. 7(b), X (-190) is b X (-210) and b X The equation is formulated as being on the line segment connecting (-180).

[0041] b X (i+s) is quot(s,p Y ) and mod(s,p Y ) (where quot(s,p Y ) is s P Y The integer part of the quotient when divided by mod(s,p Y ) is s P Y Using the remainder when dividing by (x,y), this can be expressed as follows:

[0042]

number

[0043] Therefore, by substituting the numerical values, we obtain the following equation:

[0044]

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[0045] In addition, such linear interpolation is performed when the stage 5 is on the +Y side from the time of measurement of the measurement mark at the bottom end in FIG. 6 (i.e., i+s>(N Y -1)*P Y / 2+h=-30) is not necessary because in that region, X (i) Linear interpolation is performed using the two b X (i) does not exist (the adjacent b X (i) may exist. In this example, i+s[mm]=-10, 20, 50 corresponds to i[mm]=-90, -60, -30. Also, of course, mod(s,P Y )=0, the linear interpolation described above is not necessary.

[0046] Thus, in a given area, b X By linearly interpolating (i+s) with the adjacent i on both sides, the number of unknowns can be reduced, and ζ X (i) and ζ Y A solution that minimizes the sum of squares of (i) is obtained.

[0047] In this example, the number of unknowns is b X (i) 9 pieces, b X (i+s) is 3, and the measurement error is ε X and ε Y 2, scale error m Y This makes a total of 15, which is fewer than the 18 equations formulated, and makes it possible to avoid the equations becoming indeterminate.

[0048] Here, by substituting equation (7) into equations (4) and (6), we obtain the following two equations.

[0049]

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[0050]

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[0051] Furthermore, the shape error in the X direction at the reflection point on the plane mirror 2X is b X (i)[nm] has degrees of freedom for the average value for i, so b X The average value of (i) is not uniquely determined. To avoid this uncertainty, X Add the following constraint that the average value of (i) = 0:

[0052]

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[0053] The equations we have formulated so far can be summarized as follows, with the case divided by i, and the measured value of X deviation r X (i) and Y deviation measurement value r Y The formula for (i) can be formulated. i<(N Y -1)*P Y For / 2+hs r X Formula for (i): Equation (9) r Y Formula for (i): Equation (10) In this example, i[mm]<-110, i.e., i[mm]=-270, -240, -210, -180, -150, -120. i>(N Y -1)*P Y For / 2+hs r X Formula for (i): Equation (4) r Y Formula for (i): Equation (6) In this example, i[mm]>-110, i.e., i[mm]=-90, -60, -30.

[0054] Measurement value r X (i), r Y Let r be the column vector consisting of (i) and one 0, let A be the coefficient matrix consisting of each coefficient, let x be the column vector consisting of the unknowns to be solved, and let ζ be the random measurement error. X (i), ζ YLet e ​​be a column vector composed of (i) and one 0. Using these, the above equations can be collectively expressed as the following formula. Note that matrix A1 is a matrix composed of coefficients based on equations (4) and (9), and matrix A2 is a matrix composed of coefficients based on equations (6) and (10). Note that column vectors r, x, and e are expressed with vector symbols in the following formulas.

[0055]

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[0056]

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[0057]

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[0058]

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[0059]

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[0060] In this example, the column vector r has 19 rows and 1 column, the matrix A has 19 rows and 15 columns, the column vector x has 15 rows and 1 column, and the column vector e has 19 rows and 1 column. In addition, the specific values ​​of the right-hand sides of equations (13) to (16) calculated from the coordinates and dimensions are as shown in Figures 8(a), (b), (c), and (d), respectively.

[0061] Here, by transforming the normal equation (17) into (18), the length of the column vector e is minimized (i.e., the irregular measurement error ζ X (i),ζ Y(i) (minimizing the sum of squares) of the column vector x can be calculated.

[0062]

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[0063]

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[0064] In this way, the control unit 6 can calculate shape data of the plane mirror 2X by treating the formulated equations as one simultaneous equation and performing a collective calculation process using the least squares method.

[0065] 9 shows the relationship between the X coordinate of the center of the substrate 3 and the shape error of the plane mirror 2Y. The shape error of the reflecting surface of the plane mirror 2Y is exaggerated.

[0066] When the X coordinate of the center of the substrate 3 is j when the measurement mark is measured by the measuring instrument 1, the shape error in the Y direction at the reflection point of the laser light from the interferometer 4Y on the plane mirror 2Y is b Y (j) [nm]. Also, as mentioned above, the Y deviation measurement value (average value) is r Y (j) [nm].

[0067] For the same Y coordinate of stage 5, the rotation error is constant, so the shape error b Y (j) does not require consideration of the rotation error that was considered when calculating the shape of the plane mirror 2X, and the Y deviation measurement value r Y This can be obtained by inverting the sign of (j).

[0068] Using the above-mentioned calculation method, the calculation unit 6a collectively calculates the shape errors of the plane mirrors 2X and 2Y. In addition, the calculation unit 6b corrects the shape errors of the plane mirrors 2X and 2Y calculated by the calculation unit 6a and reflects them in the control by the control unit 6, for example, in the positioning (position correction) during the movement of the stage 5 by correcting the output values ​​of the interferometers 4X1, 4X2, and 4Y.

[0069] According to this embodiment, the shape errors of the plane mirrors 2X and 2Y can be measured with high accuracy in a short time.

[0070] To summarize the above, the control method of the stage device ST may include a measurement step, a calculation step, and a control step. The measurement step, the calculation step, and the control step may be controlled by the control unit 6. The measurement step is performed in a state where a reference member (substrate 3) having a plurality of marks aligned along the second direction is placed on the stage 5. In this state, in the measurement step, the measuring device 1 may measure the positions of the plurality of marks while controlling the driving mechanism DM based on the outputs of the interferometer 4X1 and the second interferometer 4X2 so that the stage 5 is positioned at each of a plurality of positions corresponding to the plurality of marks. In the calculation step, the shape of the plane mirror 2X may be calculated based on the measurement results of the positions of the plurality of marks by the measuring device 1 in the measurement step and the distance h between the first optical axis and the third optical axis on the plane. In the control step, the driving mechanism DM may be controlled so that the stage 5 is positioned based on the outputs of the interferometer 4X1 and the second interferometer 4X2, and the shape of the plane mirror 2X calculated in the calculation step.

[0071] <Second embodiment> A second embodiment of the present disclosure will be described below. Matters not mentioned in the second embodiment may follow the first embodiment. The second embodiment is a case where the measuring device 1 is on the Y axis as shown in FIG. 10. In this case, g=0, and in the calculation of the shape of the plane mirror 2X in equation (4), the Y deviation measurement value r Y Since (i) cannot utilize the relationship of the rotation error θ(i) of stage 5, the simultaneous equations become indefinite.

[0072] In this case, the Y deviation measurement value r Y (i) is not used in the simultaneous equations, and by adding constraints for the number of missing equations, the indeterminacy is avoided, and the irregular measurement error ζ X (i), ζ Y The simultaneous equations can be uniquely solved by assuming that (i) is zero.

[0073] For example, using the numerical values ​​in the first embodiment, the area where linear interpolation is performed (i+s>(N Y -1)*P Y The following three equations are added as equations showing the constraint that the shape error of the area (area where / 2+h=-30) is a straight line with respect to the Y coordinate i.

[0074]

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[0075]

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[0076]

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[0077] When formulas (19) to (21) are generalized, the following four formulas are obtained instead of formulas (13) to (16). However, matrix A3 is a coefficient matrix under the constraint conditions shown in formulas (19) to (21). Also, the specific values ​​of the right-hand sides of formulas (22) to (24) calculated from the coordinates and dimensions are as shown in Figures 11(a), (b), and (c).

[0078]

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[0079]

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[0080]

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[0081]

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[0082] In addition, in this embodiment, since the matrix A is regular, the column vector x may be obtained by equation (18) or the following equation.

[0083]

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[0084] <Third embodiment> A third embodiment of the present disclosure will be described below. Matters not mentioned in the third embodiment follow the first and second embodiments. In the second embodiment, simultaneous equations can be solved by adding constraint conditions, but in order to calculate the shape of the plane mirror 2X with high accuracy, a method that does not rely on adding constraint conditions is desirable. Therefore, in the third embodiment, a measuring instrument 1A is provided in addition to the measuring instrument 1, as shown in FIG. 12(a). The X-shift measurement value r X By formulating an equation relating to (i) and treating the formulated equations as one simultaneous equation and performing a collective calculation process using the least squares method, shape data of the plane mirror 2X can be calculated.

[0085] In addition, when the measuring device 1A is not on the Y axis (not shown), the X-shift measurement value r X Not only (i), but also the Y deviation measurement value r Y Since (i) can be formulated as an equation including the rotation error θ(i) of stage 5, by further increasing the number of equations, higher calculation accuracy can be expected.

[0086] Furthermore, as shown in Fig. 12(b), even if the measuring instrument 1 is not on the Y axis and the simultaneous equations are not indefinite, higher calculation accuracy using the least squares method can be expected by increasing the number of equations by adding the measuring instrument 1B. Alternatively, in Fig. 12(b), the measuring instrument 1 may be dedicated to measuring the X-axis deviation, and the measuring instrument 1B may be dedicated to measuring the Y-axis deviation, acquiring measurement values, and solving the simultaneous equations to calculate shape data of the plane mirror 2X.

[0087] In this embodiment, the number of additional measuring instruments is not limited. In addition, in the embodiment in which a plurality of measuring instruments are configured in this way, a mobile measuring instrument may be used instead of the plurality of measuring instruments to obtain measurement values ​​from the positions of the plurality of measuring instruments.

[0088] <Fourth embodiment> The fourth embodiment of the present disclosure will be described below. Matters not mentioned in the fourth embodiment follow the first embodiment. The fourth embodiment is a case where only the amount of deviation in the X direction can be measured by a plurality of mark measuring devices. As an example, the case shown in FIG. 12(b) in the third embodiment will be given.

[0089] The coordinates of the two measuring instruments are shown in FIG. 13. The Y coordinates of measuring instruments 1 and 1B are -h and -hB, respectively. The X-shift measurement values ​​obtained from each measuring instrument are expressed as r X (i), r BX (i).

[0090] The specific values ​​are set as follows.

[0091] Distance between the optical axes of interferometers 4X1 and 4X2: s [mm] = 80 Y coordinate of measuring device 1: h [mm] = -150 Y coordinate of measuring device 1B: hB [mm] = -220 Measurement mark spacing: P Y [mm]=30 Number of measurement marks in the Y direction (number of rows): N Y =9

[0092] In addition, the constant measurement error of the X-shift measurement value of measuring instrument 1 and measuring instrument 1B is ε X , ε BX [nm], and the irregular measurement error is ζ X (i), ζ BX (i) [nm].

[0093] In the fourth embodiment, the X coordinate of the measuring instrument is not used for either measuring instrument. Therefore, in FIG. 12(a), the Y coordinate is different between the two measuring instruments, so the following procedure can be applied in the same way.

[0094] The Y coordinates [mm] of the measurement marks relative to the center of the substrate 3 are 0, ±30, ±60, ±90, ±120. Therefore, the Y coordinates of the center of the substrate 3 when measuring these measurement marks can be obtained as follows by adding up the Y coordinates of each measuring instrument, h and hB [mm].

[0095] Measurement on instrument 1: i=-270,-240,-210,···,-30 Measurement with measuring instrument 1B: i=-340,-310,-280,···,-100 From this, the known quantity r X (i), r BX (i) and the unknown shape error b X The relationship with (i) is formulated as an equation. X (i) and (b) X Regarding the relationship with (i), as shown in the first embodiment, the formula formulated by dividing into two cases according to i can be used as is.

[0096] On the other hand, regarding the measurement with measuring instrument 1B, BX (i) and (b) X Regarding the relationship with (i), the cases are divided into three according to i as follows: i<-(N Y -1)*P Y For / 2+h In this example, i [mm] <-270, that is, i [mm] = -340, -310, -280. In this case, i is not used in the mark measurement by the measuring instrument 1. Therefore, the shape error b at the optical axis reflecting surface of the interferometer 4X1 X (i) does not require linear interpolation. On the other hand, the shape error b X (i + s) at the optical axis reflecting surface of the interferometer 4X2 performs linear interpolation because i related to the measurement by the measuring instrument 1 already exists on both sides of i + s. That is, r BX As the formula for (i), formula (27) established with reference to formula (9) can be established.

[0097]

Number

[0098] ·-(N Y -1)*P Y / 2 + h < i < (N Y -1)*P Y / 2 + h - s case In this example, -270 < i [mm] < -110, that is, i [mm] = -250, -220, -190, -160, -130. The shape error b X (i) at the optical axis reflecting surface of the interferometer 4X1 performs linear interpolation because i related to the measurement by the measuring instrument 1 already exists on both sides of i. Also, for the shape error b X (i + s) at the optical axis reflecting surface of the interferometer 4X2, linear interpolation is performed because i related to the measurement by the measuring instrument 1 already exists on both sides of i. Therefore, r BX (i) can be established as formula (28) established with reference to formula (7).

[0099]

Number

[0100] By substituting formula (28) into formula (27), the following formula can be obtained.

[0101]

number

[0102] i>(N Y -1)*P Y For / 2+hs In this example, i[mm]>-110, i[mm]=-100. The shape error b on the optical axis reflecting surface of the interferometer 4X1 X (i) is linearly interpolated because there are already i's related to the measurement by the measuring instrument 1 on both sides of i. Also, the shape error b X For (i+s), linear interpolation is performed because there are already i's related to the measurement by measuring device 1 on both sides of i. However, b X When performing linear interpolation on (i+s), the difference between the two adjacent i is the measurement mark placement interval P Y Since it is different from the above, care must be taken with the formula. The difference between the two adjacent i's is the interval P Y The difference is that r X This is because it relates to both of the two cases divided for (i). Specifically, when i [mm] = -100, the shape error b X (i+s) is b X (-20), and b X (-30) and b X Obtained by linear interpolation of (-10). b X The original variable of (-10), -10, is calculated using the following formula.

[0103]

number

[0104]

number

[0105] Therefore, using equations (30) and (31), r BX As the equation for (i), we can formulate equation (32) by referring to equation (29).

[0106]

number

[0107] Furthermore, when the constraint condition expressed by the formula (11) is also expressed by the matrix of the formula (12), the formulas corresponding to the formulas (13) to (16) become the formulas (33) to (36).

[0108] Measurement value r X (i), r BX Let r be the column vector consisting of (i) and one 0, let A be the coefficient matrix consisting of each coefficient, let x be the column vector consisting of the unknowns to be solved, and let ζ be the random measurement error. X (i), ζ BX Let e ​​be a column vector consisting of (i) and one 0. Then, the above equations can be collectively expressed as the following equation, where matrix A4 is a matrix consisting of coefficients based on equations (4) and (9), and matrix A5 is a matrix consisting of coefficients based on equations (27), (29), and (32).

[0109]

number

[0110]

number

[0111]

number

[0112]

number

[0113] In this example, the column vector r has 19 rows and 1 column, the matrix A has 19 rows and 17 columns, the column vector x has 17 rows and 1 column, and the column vector e has 19 rows and 1 column. The specific values ​​of the right-hand sides of equations (33) to (36), calculated from the coordinates and dimensions, are shown in Figures 14(a), (b), (c), and (d). Here, the normal equation, equation (17), can be transformed into equation (18) to minimize the length of the column vector e (i.e., to minimize the irregular measurement error ζ X (i), ζ BX (i) (minimizing the sum of squares) of the column vector x can be calculated.

[0114] <Fifth embodiment> 15(a) and 15(b), the fifth embodiment is a case where a measurement plate having measurement marks is configured as a reference member on a stage 5. In FIG. 15(a), the measurement marks are arranged at intervals P Y The measurement plate 7X arranged in the Y direction and the arrangement interval P X FIG. 15(b) shows an example in which the measurement marks are arranged at an interval P X and P Y 15(a) and 15(b), the measurement plate 7XY may be permanently fixed to the stage 5, or may have a detachable mechanism.

[0115] Sixth embodiment The sixth embodiment is an embodiment in which, in mark measurement, X-shift measurement value and Y-shift measurement value are not measured with one mark, but X-shift measurement and Y-shift measurement are performed with separate measurement marks. Even in this embodiment, the shape data of the plane mirror can be calculated by expressing the coordinates of each measurement mark in an equation and formulating it, or by approximating the two measurement marks as being close to each other and expressing them in an equation as being the same coordinates.

[0116] <Exposure equipment> 16 shows a schematic configuration of an exposure apparatus EXP according to one embodiment. The exposure apparatus EXP is configured to transfer the pattern of the original R onto the substrate S, and includes a stage device ST configured to position the substrate S as a substrate stage device. The exposure apparatus EXP may also include an original stage device RST that positions or drives the original R, an illumination optical system IL that illuminates the original R, and a projection optical system PO that projects the pattern of the original R onto the substrate S.

[0117] The stage device ST configured as a substrate stage device may include a stage 5 and a driving mechanism MA that drives the stage ST in a first direction and a second direction that are orthogonal to each other in a plane (XY plane) and in rotation around an axis orthogonal to the plane. Although not shown in FIG. 16, the stage device ST may include an interferometer 4X1 as a first interferometer and an interferometer 4X2 as a second interferometer. The interferometer 4X1 has a first optical axis parallel to the first direction (for example, the X-axis direction) and may be used to measure the position of the stage ST in the first direction. The interferometer 4X2 has a second optical axis parallel to the first direction and spaced apart from the first optical axis and may be used to measure the rotation of the stage ST. Although not shown in FIG. 16, the stage device ST may include a common plane mirror 2X provided on the stage ST for measurement by the interferometers 4X1 and 4X2. 16, the stage apparatus ST may include a measuring instrument 1 having a third optical axis that is orthogonal to the first optical axis at a position shifted from the first optical axis on the plane. The measuring instrument 1 may be understood as a component of the exposure apparatus EXP.

[0118] In one embodiment, the control method of the exposure apparatus EXP may include a measurement step, a calculation step, and a control step. The measurement step and the calculation step may be understood as part of the control method of the stage apparatus ST. The measurement step may be controlled by the control unit 6 or a control device (not shown). The measurement step is performed in a state where a reference member (substrate 3) having a plurality of marks arranged along the second direction is placed on the stage 5. In this state, in the measurement step, the measuring device 1 may measure the positions of the plurality of marks while controlling the driving mechanism DM based on the outputs of the interferometer 4X1 and the second interferometer 4X2 so that the stage 5 is positioned at each of a plurality of positions corresponding to the plurality of marks. The calculation step may be controlled by the control unit 6 or a control device (not shown). In the calculation step, the shape of the plane mirror 2X may be calculated based on the measurement results of the positions of the plurality of marks by the measuring device 1 in the measurement step and the distance h between the first optical axis and the third optical axis on the plane. The control step may be controlled by a control device (not shown). In the control step, the exposure of the substrate S can be controlled while controlling the drive mechanism DM so that the stage 5 is positioned based on the outputs of the interferometer 4X1 and the second interferometer 4X2 and the shape of the plane mirror 2X calculated in the calculation step.

[0119] <Product manufacturing method> Hereinafter, an article manufacturing method for manufacturing an article using the above-mentioned exposure apparatus will be described. The article manufacturing method may include an exposure step of exposing a substrate using the exposure apparatus, a development step of developing the substrate that has undergone the exposure step, and a step of obtaining an article from the substrate that has undergone the development step. A photosensitive material (photoresist) is applied to the substrate provided to the exposure apparatus. The pattern of the original is transferred to the photosensitive material as a latent image pattern by the exposure step. In the development step, this latent image pattern is converted into a physical device pattern. The step of obtaining an article from the substrate that has undergone the development step may include, for example, a step of patterning an underlying layer using the device pattern. The step of obtaining an article from the substrate that has undergone the development step may also include a step of dicing the substrate.

[0120] <Disclosures> The disclosure of the present specification includes the following. (Item 1) Stage and a drive mechanism for driving the stage in first and second directions perpendicular to each other in a plane and in rotation about an axis perpendicular to the plane; a first interferometer having a first optical axis parallel to the first direction, the first interferometer being configured to measure a position of the stage in the first direction; a second interferometer having a second optical axis parallel to the first direction and spaced apart from the first optical axis, for measuring rotation of the stage; and a common plane mirror provided on the stage for measurement by the first interferometer and the second interferometer; a mark measuring device having a third optical axis perpendicular to the first optical axis at a position shifted from the first optical axis on the plane; a control unit that controls the drive mechanism so that the stage is positioned based on outputs of the first interferometer and the second interferometer, The control unit is a reference member having a plurality of marks aligned along the second direction is disposed on the stage, the mark measurement device measures positions of the plurality of marks while controlling the driving mechanism based on outputs of the first interferometer and the second interferometer so that the stage is positioned at each of a plurality of positions corresponding to the plurality of marks, respectively; obtaining a shape of the plane mirror based on a measurement result of the positions of the plurality of marks by the mark measuring device and a distance between the first optical axis and the third optical axis on the plane; A stage apparatus comprising: (Item 2) the control unit further obtains a shape of the plane mirror based on an arrangement interval of the plurality of marks in the second direction and a distance between the first optical axis and the second optical axis. 2. The stage apparatus according to item 1, (Item 3) the third optical axis is perpendicular to the first optical axis and the second optical axis at a position shifted from the first optical axis and the second optical axis on the plane; 3. The stage apparatus according to item 1 or 2, (Item 4) the third optical axis is perpendicular to a position shifted from the first optical axis, the second optical axis, and the third optical axis on the plane; 3. The stage apparatus according to item 1 or 2, (Item 5) the control unit controls the stage based on measurement results obtained by the first interferometer and the second interferometer and a shape of the plane mirror. 5. The stage device according to claim 1, (Item 6) the control unit corrects measurement results obtained by the first interferometer and the second interferometer based on a shape of the plane mirror, and controls the stage based on the corrected measurement results. 5. The stage device according to claim 1, (Item 7) An exposure apparatus for transferring a pattern of an original onto a substrate, comprising: 7. A stage apparatus according to claim 1, configured to position the substrate. An exposure apparatus comprising: (Item 8) a drive mechanism for driving the stage in a first direction and a second direction that are orthogonal to each other in a plane and in rotation about an axis orthogonal to the plane; a first interferometer having a first optical axis parallel to the first direction and for measuring a position of the stage in the first direction; a second interferometer having a second optical axis that is parallel to the first direction and spaced from the first optical axis and for measuring a rotation of the stage; a common plane mirror provided on the stage for measurement by the first interferometer and the second interferometer; and a mark measurement instrument having a third optical axis orthogonal to a position shifted from the first optical axis on the plane, the method comprising: a measuring step of measuring positions of the plurality of marks with the mark measuring device while controlling the driving mechanism based on outputs of the first interferometer and the second interferometer so that the stage is positioned at each of a plurality of positions corresponding to the plurality of marks, with a reference member having a plurality of marks aligned along the second direction being placed on the stage; a calculating step of calculating a shape of the plane mirror based on a measurement result of the positions of the plurality of marks by the mark measuring instrument and a distance between the first optical axis and the third optical axis on the plane; a control step of controlling the driving mechanism so that the stage is positioned based on outputs of the first interferometer and the second interferometer and the shape of the plane mirror; A method for controlling a stage apparatus comprising the steps of: (Item 9) In the calculation step, a shape of the plane mirror is obtained based on an arrangement interval of the plurality of marks in the second direction and a distance between the first optical axis and the second optical axis. 9. The control method according to item 8, characterized in that: (Item 10) the third optical axis is perpendicular to the first optical axis and the second optical axis at a position shifted from the first optical axis and the second optical axis on the plane; 10. The control method according to item 8 or 9, (Item 11) the third optical axis is perpendicular to a position shifted from the first optical axis, the second optical axis, and the third optical axis on the plane; 10. The control method according to item 8 or 9, (Item 12) In the control step, the stage is controlled based on the measurement results obtained by the first interferometer and the second interferometer and the shape of the plane mirror. 12. The control method according to any one of items 8 to 11, (Item 13) In the control step, measurement results obtained by the first interferometer and the second interferometer are corrected based on a shape of the plane mirror, and the stage is controlled based on the corrected measurement results. 12. The control method according to any one of items 8 to 11, (Item 14) an exposure method for exposing a substrate using an exposure apparatus comprising: a stage; a drive mechanism for driving the stage in a first direction and a second direction that are orthogonal to each other in a plane and in rotation about an axis orthogonal to the plane; a first interferometer having a first optical axis parallel to the first direction and for measuring a position of the stage in the first direction; a second interferometer having a second optical axis that is parallel to the first direction and spaced from the first optical axis and for measuring a rotation of the stage; a common plane mirror provided on the stage for measurement by the first interferometer and the second interferometer; and a mark measurement instrument having a third optical axis orthogonal to a position shifted from the first optical axis on the plane, a measuring step of measuring positions of the plurality of marks with the mark measuring device while controlling the driving mechanism based on outputs of the first interferometer and the second interferometer so that the stage is positioned at each of a plurality of positions corresponding to the plurality of marks, with a reference member having a plurality of marks aligned along the second direction being placed on the stage; a calculating step of calculating a shape of the plane mirror based on a measurement result of the positions of the plurality of marks by the mark measuring instrument and a distance between the first optical axis and the third optical axis on the plane; a control step of controlling exposure of the substrate while controlling the driving mechanism so that the stage is positioned based on outputs of the first interferometer and the second interferometer and the shape of the plane mirror; An exposure method comprising the steps of: (Item 15) In the calculation step, a shape of the plane mirror is obtained based on an arrangement interval of the plurality of marks in the second direction and a distance between the first optical axis and the second optical axis. 15. The exposure method according to item 14, (Item 16) the third optical axis is perpendicular to the first optical axis and the second optical axis at a position shifted from the first optical axis and the second optical axis on the plane; 16. The exposure method according to item 14 or 15, (Item 17) the third optical axis is perpendicular to a position shifted from the first optical axis, the second optical axis, and the third optical axis on the plane; 16. The exposure method according to item 14 or 15, (Item 18) In the control step, the stage is controlled based on the measurement results obtained by the first interferometer and the second interferometer and the shape of the plane mirror. 18. The exposure method according to any one of items 14 to 17. (Item 19) In the control step, measurement results obtained by the first interferometer and the second interferometer are corrected based on a shape of the plane mirror, and the stage is controlled based on the corrected measurement results. 18. The exposure method according to any one of items 14 to 17. (Item 20) An exposure step of exposing a substrate by the exposure method according to any one of items 14 to 19; a processing step for obtaining an article by processing the substrate exposed in the exposure step; A method for manufacturing an article, comprising:

[0121] The invention is not limited to the above-described embodiments, and various modifications and variations are possible without departing from the spirit and scope of the invention. Accordingly, the following claims are appended to apprise the public of the scope of the invention. [Explanation of symbols]

[0122] ST: stage device, 1: measuring instrument, 2X, 2Y: plane mirror, 3: substrate, 4X1, 4X2, 4Y: interferometer, 5: stage, 6: control unit, 6a: calculation unit, 6b: correction processing unit

Claims

1. Stage and a drive mechanism for driving the stage in first and second directions perpendicular to each other in a plane and in rotation about an axis perpendicular to the plane; a first interferometer having a first optical axis parallel to the first direction, the first interferometer being configured to measure a position of the stage in the first direction; a second interferometer having a second optical axis parallel to the first direction and spaced apart from the first optical axis, for measuring rotation of the stage; and a common plane mirror provided on the stage for measurement by the first interferometer and the second interferometer; a mark measuring device having a third optical axis perpendicular to the first optical axis at a position shifted from the first optical axis on the plane; a control unit that controls the drive mechanism so that the stage is positioned based on outputs of the first interferometer and the second interferometer, The control unit is a reference member having a plurality of marks aligned along the second direction is disposed on the stage, the mark measurement device measures positions of the plurality of marks while controlling the driving mechanism based on outputs of the first interferometer and the second interferometer so that the stage is positioned at each of a plurality of positions corresponding to the plurality of marks, respectively; obtaining a shape of the plane mirror based on a measurement result of the positions of the plurality of marks by the mark measuring device and a distance between the first optical axis and the third optical axis on the plane; A stage apparatus comprising:

2. the control unit further obtains a shape of the plane mirror based on an arrangement interval of the plurality of marks in the second direction and a distance between the first optical axis and the second optical axis.

2. The stage apparatus according to claim 1 .

3. the third optical axis is perpendicular to the first optical axis and the second optical axis at a position shifted from the first optical axis and the second optical axis on the plane; 2. The stage apparatus according to claim 1 .

4. the third optical axis is perpendicular to a position shifted from the first optical axis, the second optical axis, and the third optical axis on the plane; 2. The stage apparatus according to claim 1 .

5. the control unit controls the stage based on measurement results obtained by the first interferometer and the second interferometer and a shape of the plane mirror.

2. The stage apparatus according to claim 1 .

6. the control unit corrects measurement results obtained by the first interferometer and the second interferometer based on a shape of the plane mirror, and controls the stage based on the corrected measurement results.

2. The stage apparatus according to claim 1 .

7. An exposure apparatus for transferring a pattern of an original onto a substrate, comprising:

7. A stage apparatus according to claim 1 configured to position the substrate. An exposure apparatus comprising:

8. a drive mechanism for driving the stage in a first direction and a second direction that are orthogonal to each other in a plane and in rotation about an axis orthogonal to the plane; a first interferometer having a first optical axis parallel to the first direction and for measuring a position of the stage in the first direction; a second interferometer having a second optical axis that is parallel to the first direction and spaced from the first optical axis and for measuring a rotation of the stage; a common plane mirror provided on the stage for measurement by the first interferometer and the second interferometer; and a mark measurement instrument having a third optical axis orthogonal to a position shifted from the first optical axis on the plane, a measuring step of measuring positions of the plurality of marks with the mark measuring device while controlling the driving mechanism based on outputs of the first interferometer and the second interferometer so that the stage is positioned at each of a plurality of positions corresponding to the plurality of marks, with a reference member having a plurality of marks aligned along the second direction being placed on the stage; a calculation step of calculating a shape of the plane mirror based on a measurement result of the positions of the plurality of marks by the mark measuring instrument and a distance between the first optical axis and the third optical axis on the plane; a control step of controlling the driving mechanism so that the stage is positioned based on outputs of the first interferometer and the second interferometer and the shape of the plane mirror; A method for controlling a stage apparatus comprising the steps of:

9. the calculation step obtains a shape of the plane mirror based on an arrangement interval of the plurality of marks in the second direction and a distance between the first optical axis and the second optical axis; 9. The control method according to claim 8.

10. the third optical axis is perpendicular to the first optical axis and the second optical axis at a position shifted from the first optical axis and the second optical axis on the plane; 9. The control method according to claim 8.

11. the third optical axis is perpendicular to a position shifted from the first optical axis, the second optical axis, and the third optical axis on the plane; 9. The control method according to claim 8.

12. In the control step, the stage is controlled based on the measurement results obtained by the first interferometer and the second interferometer and the shape of the plane mirror.

9. The control method according to claim 8.

13. In the control step, measurement results obtained by the first interferometer and the second interferometer are corrected based on a shape of the plane mirror, and the stage is controlled based on the corrected measurement results.

12. The control method according to claim 8, wherein the control method is a control method for controlling a power supply.

14. an exposure method for exposing a substrate using an exposure apparatus comprising: a stage; a drive mechanism for driving the stage in first and second directions that are orthogonal to each other in a plane and in rotation about an axis orthogonal to the plane; a first interferometer having a first optical axis parallel to the first direction and for measuring a position of the stage in the first direction; a second interferometer having a second optical axis that is parallel to the first direction and spaced from the first optical axis and for measuring a rotation of the stage; a common plane mirror provided on the stage for measurement by the first interferometer and the second interferometer; and a mark measurement instrument having a third optical axis orthogonal to a position shifted from the first optical axis on the plane, a measuring step of measuring positions of the plurality of marks with the mark measuring device while controlling the driving mechanism based on outputs of the first interferometer and the second interferometer so that the stage is positioned at each of a plurality of positions corresponding to the plurality of marks, with a reference member having a plurality of marks aligned along the second direction being placed on the stage; a calculation step of calculating a shape of the plane mirror based on a measurement result of the positions of the plurality of marks by the mark measuring instrument and a distance between the first optical axis and the third optical axis on the plane; a control step of controlling exposure of the substrate while controlling the driving mechanism so that the stage is positioned based on outputs of the first interferometer and the second interferometer and the shape of the plane mirror; An exposure method comprising the steps of:

15. the calculation step obtains a shape of the plane mirror based on an arrangement interval of the plurality of marks in the second direction and a distance between the first optical axis and the second optical axis; 15. The exposure method according to claim 14.

16. the third optical axis is perpendicular to the first optical axis and the second optical axis at a position shifted from the first optical axis and the second optical axis on the plane; 15. The exposure method according to claim 14.

17. the third optical axis is perpendicular to a position shifted from the first optical axis, the second optical axis, and the third optical axis on the plane; 15. The exposure method according to claim 14.

18. In the control step, the stage is controlled based on the measurement results obtained by the first interferometer and the second interferometer and the shape of the plane mirror.

15. The exposure method according to claim 14.

19. In the control step, measurement results obtained by the first interferometer and the second interferometer are corrected based on a shape of the plane mirror, and the stage is controlled based on the corrected measurement results.

15. The exposure method according to claim 14.

20. an exposure step of exposing a substrate by the exposure method according to any one of claims 14 to 19; a processing step for obtaining an article by processing the substrate exposed in the exposure step; A method for manufacturing an article, comprising:

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