The controller 15 may output an optimum value of a proportional coefficient and the other analysis results determined by the analysis part 16 from the output part 18 to external equipment different from the measurement apparatus 1 via a network line NW.

The controller 15 may display the optimum value of the proportional coefficient and the other analysis results determined by the analysis part 16 on the display part 17.

Further, the measurement apparatus 1 or the exposure apparatus 2 may not include at least one of the output part 18 and the display part 17.

When measurement of the measurement positions MM or the displacement vectors DV of the plurality of measured-parts MP on the substrate WF is terminated, the controller 15 sends a signal S3 to the stage 30 on which the substrate WF is placed, and moves the stage 30 to a position facing the projection optical system 22. Then, the exposure apparatus 2 moves the position of the stage 30 based on the position of the conventional pattern on the substrate WF estimated by the analysis part 16, i.e., matches the position of the conventional pattern, and exposes the pattern formed on the mask 25 on the substrate WF.

The conventional pattern on the substrate WF may be referred to as a first pattern, and a pattern formed on the mask 25 and newly exposed on the substrate WF may be referred to as a second pattern.

The first pattern may be formed in plural at different positions on the substrate WF, and in this case, the plurality of second patterns may be exposed at different positions on the substrate WF to match the plurality of first patterns, respectively.

Further, in the measurement apparatus 1 shown in FIG. 1, while the controller 15 of the measurement apparatus 1 controls the exposure apparatus 2, the controller 15 performs only the control of the measurement apparatus 1 and a controller configured to control the exposure apparatus 2 may be provided separately from the controller 15.

(Analyzing Method of Embodiment)

The analyzing method of the embodiment is provided to analyze characteristics of the measured position information IMP by fitting a reference function, which will be described, to the measured position information IMP that is measured position information of the plurality of measured-parts MP formed on the substrate WF. As the reference function, the sum of at least one function obtained by multiplying a criterion function expressed using a first type Bessel function by a proportional coefficient is used.

An optimum value of the proportional coefficient determined by the fitting of the reference function to the measured position information IMP represents a deformation amount of the substrate WF corresponding to the predetermined deformation mode among the displacement amount of the plurality of measured-parts MP from the reference positions MD (the reference position information IRP).

Hereinafter, a specific example of the analyzing method of the embodiment will be described.

In the analyzing method of the embodiment, as a criterion function that constitutes a reference function, a function using a first type Bessel function J_m(μr) is used. Here, m is a subscript that represents an order of the first type Bessel function, and r is a distance from the origin O (center) of the substrate WF to the measured-part MP as shown in FIG. 2A. However, in order to facilitate mathematical handling, as described below, r is not treated as an actual distance, but as a value normalized to be 1 in the outer circumference of the circular substrate WF. As an example, if the measured-part MP is at the intermediate point between the origin O of the substrate WF and one arbitrary point on the outer circumference of the substrate WF, r of the measured-part MP is 0.5.

In addition, μ is a constant, which will be described below.

The criterion function is, for example, a function expressed by the following Equation (5) or Equation (6).

J_m(μ_mkr)×cos(mθ) (5)

J_m(μ_mkr)×sin(mθ) (6)

Here, as shown in FIG. 2A, θ is a deflection angle from the +X direction to the measured-part MP with reference to the origin O, and a value obtained by multiplying θ by an order m of the first type Bessel function J_mis an argument of a cos function contained in Equation (5) or a sin function contained in Equation (6). In other words, the criterion function is expressed using, for example, a product of the first type Bessel function and a trigonometric function based on a deflection angle θ from the reference position (the origin O) to the measured-part MP.

The above-mentioned constant μ is expressed as a constant μ_mkobtained by adding the order m of the first type Bessel function J_mand an arbitrary natural number k as subscripts in Equation (5) and Equation (6).

FIG. 3A and FIG. 3B are views showing an example of the first type Bessel function J_m(μ_mkr) contained in the criterion function. FIG. 3A is a numerical table showing some examples of values of the constant μ_mkdetermined by subscripts m and k. The value of the constant μ_mkshown in FIG. 3A is a value of r where J_m(r) takes a k^thextreme value (maximum value or minimum value) other than zero when r is increased from zero in a first type m^thorder of Bessel function J_m(r) using r as the argument. Accordingly, J_m(μ_mkr) contained in Equation (5) and Equation (6) is a function that takes an extreme value when r=1.

In the numerical table of FIG. 3A, the values of the constant μ_mkare shown within a range of 0 to 7 for m, and 1 to a predetermined value corresponding to a value of m for k. However, the value of m of the constant μ_mkmay be 8 or more, and the value of k may also be an arbitrary value of one or more. The value of the constant μ_mkcan be easily calculated from the first type m^thorder of Bessel function J_m(r) when m and k are determined.

Further, when m=0, an extreme value when k=1, i.e., a value of r that takes a first extreme value is zero, and it is not appropriate for use in Equation (5) and Equation (6). Here, the numerical table of FIG. 3A indicates that a horizontal line is drawn for the field with m=0 and k=1, and the numerical value of this field is inappropriate.

Further, in FIG. 3A, while the value of the constant μ_mkis displayed with six or more digits of significant figures, the value of the constant μ_mkdoes not necessarily have to match the value shown in FIG. 3A with an accuracy of six or more digits of significant figures. Depending on the usage situation, that is, the accuracy required for analysis, for example, it may be sufficient to match 2 digits of significant figures, or it may be sufficient to match 3 digits of significant figures.

FIG. 3B is a view in which some of the first type Bessel functions J_m(μ_mkr) determined by the subscripts m and k are expressed as graphs with r ranging from 0 to 1. A curve line C02 indicates a function J₀(μ₀₂r) of m=0 and k=2. Similarly, a curve line C11 indicates a function J₁(μ₁₁r), a curve line C12 indicates a function J₁(μ₁₂r), a curve line C21 indicates a function J₂(μ₂₁r), a curve line C22 indicates a function J₂(μ₂₂r), a curve line C31 indicates a function J₃(μ₃₁r), and a curve line C32 indicates a function J₃(μ₃₂r).

As described above, each function shown in FIG. 3B takes an extreme value (maximum value or minimum value) at r=1. In the curve lines C11, C21 and C31 equivalent to the function J_m(μ_m1r) corresponding to k=1, as r increases, it takes a first non-zero extreme value at r=1. In the curve lines C12, C22 and C32 equivalent to the function J_m(μ_m2r) corresponding to k=2, as r increases, it takes two non-zero extreme values at r=1.

FIG. 4 is a view showing an approximate value of a criterion function expressed by some of the subscripts in and k in the criterion functions expressed by Equation (5) or Equation (6). A criterion function C11c represents J₁(μ₁₁r)×cos(1θ) at m=1 and k=1, and a criterion function C11s represents J₁(μ₁₁r)×sin(1θ) at m=1 and k=1. A criterion function C21c represents J₂(μ₂₁r)×cos(2θ) at m=2 and k=1, and a criterion function C21s represents J₂(μ₂₁r)×cos(2θ) at m=2 and k=1.

A criterion function C31c represents J₃(μ₃₁r)×cos(3θ) at m=3 and k=1, and a criterion function C31s represents J₃(μ₃₁r)×sin(3θ) at m=3 and k=1. A criterion function C22c represents J₂(μ₂₂r)×cos(2θ) at m=2 and k=2, and a criterion function C22s represents J₂(μ₂₂r)×cos(2θ) at m=2 and k=2.

In FIG. 4, in the circle with the origin O as a center and a radius of r=1, regions where the value of each of the criterion functions (C11c to C22c, C11s to C22s) is greater than a predetermined positive value are indicated with “+,” and regions with a value smaller than a predetermined negative value are indicated with “−.” Regions with no “+” and “−” are regions in which a value of a criterion function is greater than the predetermined negative value or smaller than the predetermined positive value, i.e., close to zero.

As shown in FIG. 4, the criterion function given by Equation (5) and Equation (6) becomes a function whose value increases or decreases in various predetermined shapes within a circle of radius 1 by appropriately selecting the values of the subscripts m and k. Accordingly, for example, virtual deformation of the circular substrate WF in the XY in-plane direction can be expressed by one of these criterion functions (C11c to C22c, C11s to C22s, and the like) or combination of some of the criterion functions.

Hereinafter, a vector representing a virtual deformation amount of the measured-part MP of the substrate WF is expressed by the criterion function is referred to as a virtual displacement vector IV, and an X component and a Y component of the virtual displacement vector IV are referred to as Δx and Δy, respectively. The virtual displacement vector IVj of a J^th(j is an arbitrary natural number) measured-part MP is IVj=(Δxj, Δxj).

The X component Δxj and the Y component Δyj of the virtual displacement vector IVj are expressed by the following Equation (7) and Equation (8) using the criterion function.

Δ
⁢

x
j

=

∑
m

∑
k

{

α

1
⁢
mk

⁢

J
m

(

μ
mk

⁢

r
j

)

⁢

cos
⁡
(

m
⁢

θ
j

)

+

α

2
⁢
m
⁢
k

⁢

J
m

(

μ

m
⁢
k

⁢

r
j

)

⁢

sin
⁡
(

m
⁢

θ
j

)

}

(
7
)

Δ
⁢

y
j

=

∑
m

∑
k

{

α

3
⁢
m
⁢
k

⁢

J
m

(

μ
mk

⁢

r
j

)

⁢

cos
⁡
(

m
⁢

θ
j

)

+

α

4
⁢
m
⁢
k

⁢

J
m

(

μ

m
⁢
k

⁢

r
j

)

⁢

sin
⁡
(

m
⁢

θ
j

)

}

(
8
)

Here, α1_mk, α2_mk, α3_mk, and α4^mkare different proportional coefficients according to the values of the subscripts m and k, respectively. Hereinafter, the proportional coefficients α1_mk, α2_mk, α3_mk, and α4^mkare generally or individually simply referred to as a proportional coefficient α.

In addition, rj is a distance from the origin O to the j^thmeasured-part MPj, and θj is a deflection angle from the origin O to the j^thmeasured-part MPj.

As described above, in the specification, as shown by Equation (7) and Equation (8), a sum of at least one function obtained by multiplying the proportional coefficient α by the criterion function expressed using the first type Bessel function J_m(μ_mkr) is referred to as a reference function. Further, when the criterion function is one, the criterion function itself is the reference function.

FIG. 5 is a view showing, for example, a virtual deformation amount of the circular substrate WF in the XY in-plane direction, where a deformation amount in the X direction is expressed using the criterion function C21c and a deformation amount in a Y direction is expressed using the criterion function C21s. That is, in the example shown in FIG. 5, IVj=(Δxj, Δxj) is expressed by the following Equation (9) and Equation (10).

Δx_j=αJ₂(μ₂₁r_j)×cos(2θ_j) (9)

Δy_j=αj₂(μ₂₁r_j)×sin(2θ_j) (10)

Further, it is also possible to express Δxj and Δxj in a vector as the virtual displacement vector IVj (Δxj, Δxj), and combine Equation (9) and Equation (10) into one as in Equation (11).

I
⁢

V
j

=

(

Δ
⁢

x
j

Δ
⁢

y
j

)

=

α
⁢

J
2

(

μ
21

⁢

r
j

)

×

(

cos
⁡
(

2
⁢

θ
j

)

sin
⁡
(

2
⁢

θ
j

)

)

(
11
)

An actual deformation amount (the displacement vector DV) of the measured-part MP of the substrate WF can be expressed approximately using the reference function by setting the value of the proportional coefficient α to a proper value (optimum value). The value of the proportional coefficient α may be determined by, for example, a least squares method. That is, in the N measured-parts MPj of the substrate WF, the optimum value of the proportional coefficient α may be set such that values of Ex and Ey obtained by substituting Δxj and Δyj of Equation (9) and Equation (10) or Equation (11) into Equation (12) and Equation (13), respectively, are minimized.

Ex=Σ_j^N(ΔX_j−Δx_j)² (12)

Ey=Σ_j^N(ΔY_j−Δy_j)² (13)

Here, as described above, ΔXj and ΔYj are the X component and the Y component of the displacement vector DVj of the j^thmeasured-part MP.

Alternatively, the optimum value of the proportional coefficient α may be set using another known method.

The reference function shown by Equation (7) and Equation (8) can be fitted to the displacement vector DV (ΔX, ΔY) that is the measured position information IMP, i.e., position information measured by the plurality of measured-parts MP by setting the proper proportional coefficient α.

The value of the proportional coefficient α set to correspond to one criterion function is an index indicating how much a mode of deformation expressed by the criterion function is included in the measured position information IMP. Accordingly, amounts of various deformation modes included in the deformation of the substrate WF can be analyzed by calculating the proportional coefficient α corresponding to each of the criterion functions.

Further, the reference function expressed by Equation (7) and Equation (8) is expressed by a vector with Δxj and Δxj as the virtual displacement vector IVj (Δxj, Δxj) like the above-mentioned Equation (11), and can also be written as Equation (15).

I
⁢

V
j

=

(

Δ
⁢

x
j

Δ
⁢

y
j

)

=

∑
m

∑
k

[

J
m

(

μ
mk

⁢

r
j

)

×

{

β

1
⁢
m
⁢
k

 
(

cos
⁡
(

m
⁢

θ
j

)

sin
⁡
(

m
⁢

θ
j

)

)

+

β

2
⁢
m
⁢
k

 
(

sin
⁡
(

m
⁢

θ
j

)

-

cos
⁡
(

m
⁢

θ
j

)

)

+

β

3
⁢
m
⁢
k

 
(

cos
⁡
(

m
⁢

θ
j

)

-

sin
⁡
(

m
⁢

θ
j

)

)

+

β

4
⁢
m
⁢
k

 
(

sin
⁡
(

m
⁢

θ
j

)

cos
⁡
(

m
⁢

θ
j

)

)

}

]

(
15
)

In Equation (15), β1_mk, β2_mk, β3_mk, and β4_mkare proportional coefficients different according to values of the subscripts m and k, like α1_mk, α2_mk, α3_mk, and α4_mkof Equations (7) and (8). Hereinafter, the proportional coefficients β1_mk, β2_mk, β3_mk, and and β4_mkare generally or individually simply referred to as the proportional coefficient β.

Further, in Equation (15), when m=0, cos(mθ) is always 1, and sin(mθ) is always 0. For this reason, the proportional coefficient β1_0kand the proportional coefficient β3_0khave the same sign and overlap each other, and the proportional coefficient β2_0kand the proportional coefficient β4_0khave opposite signs and overlap each other. Accordingly, when m=0, it is appropriate to treat the proportional coefficient β2_0kand the proportional coefficient β3_0kas 0.

Further, even when disclosed as in Equation (15), the reference function is still the sum of at least one function obtained by multiplying the criterion function expressed using the first type Bessel function by the proportional coefficient.

The actual deformation amount (the displacement vector DV) of the measured-part MP of the substrate WF can be expressed approximately using the function including one or more reference function expressed by Equation (15) by setting the value of the proportional coefficient β of Equation (15) to a proper value. For example, a β value of the proportional coefficient may be determined by the above-mentioned least squares method, like the α value of the proportional coefficient.

Further, when the proportional coefficients α and β of the reference function expressed by Equation (7) and Equation (8) or Equation (15) are set, values of the proportional coefficients α and β corresponding to the subscripts m and k of the predetermined values may be determined as zero. In other words, when the proportional coefficients α and β are determined, it is not necessary to calculate the proportional coefficients α and β corresponding to the criterion functions corresponding to all combinations of the subscripts m and k. To take it to the extreme, as shown in Equation (9) to Equation (11), for only one reference function consisting of one combination of the subscripts m and k, the values of the proportional coefficients α and β corresponding thereto may be determined.

Hereinafter, in the criterion functions included in the reference function expressed by Equation (15), some criterion functions that are important when the actual deformation of the substrate WF is analyzed and the proportional coefficients corresponding thereto are exemplified. When the substrate WF is analyzed, for at least one of the criterion functions exemplified below, it is preferable to determine the value of the proportional coefficient β corresponding thereto.

In the first example, the subscripts are m=2 and k=1, and there are two criterion functions, i.e., a criterion function corresponding to the proportional coefficient β1_mkand a criterion function corresponding to the proportional coefficient β2_mkin Equation (15). In order to make it easier to determine the correspondence with Equation (15), the results obtained by multiplying the criterion functions by the proportional coefficient β1₂₁or the proportional coefficient β2₂₁are shown by Equation (16) and Equation (17), respectively.

β

1
⁢
2
⁢
1

⁢

J
2

(

μ

2
⁢
1

⁢

r
1

)

×

(

cos
⁡
(

2
⁢

θ
j

)

sin
⁡
(

2
⁢

θ
j

)

)

(
16
)

β

2
⁢
2
⁢
1

⁢

J
2

(

μ

2
⁢
1

⁢

r
j

)

×

(

sin
⁡
(

2
⁢

θ
j

)

-

cos
⁡
(

2
⁢

θ
j

)

)

(
17
)

Among these, the deformation of the substrate WF expressed by Equation (16) is the deformation shown in FIG. 5 described above.

In the second example, the subscripts are m=2 and k=2, and there are two criterion functions, i.e., a criterion function corresponding to the proportional coefficient β1₂₁and a criterion function corresponding to the proportional coefficient β2_mkin Equation (15). In order to make it easier to determine the correspondence with Equation (15), the results obtained by multiplying the criterion functions by the proportional coefficient β1₂₂or β2₂₂are shown in Equation (18) and Equation (19), respectively.

β

1
⁢
2
⁢
2

⁢

J
2

(

μ
22

⁢

r
j

)

×

(

cos
⁡
(

2
⁢

θ
j

)

sin
⁡
(

2
⁢

θ
j

)

)

(
18
)

β

2
⁢
2
⁢
2

⁢

J
2

(

μ

2
⁢
2

⁢

r
j

)

×

(

sin
⁡
(

2
⁢

θ
j

)

-

cos
⁡
(

2
⁢

θ
j

)

)

(
19
)

In the third example, there are two criterion functions, i.e., a criterion function corresponding to the proportional coefficient β1_mkin Equation (15) when the subscripts are m=1 and k=2, and, a criterion function corresponding to the proportional coefficient β1_mkin Equation (15) when the subscripts are m=1 and k=3. In order to make it easier to determine the correspondence with Equation (15), the results obtained by multiplying these criterion functions by the proportional coefficient β1₁₂or the proportional coefficient β1₁₃are shown in Equation (20) and Equation (21), respectively.

β

1
⁢
1
⁢
2

⁢

J
1

(

μ
12

⁢

r
j

)

×

(

cos
⁡
(

θ
j

)

sin
⁡
(

θ
j

)

)

(
20
)

β

1
⁢
1
⁢
3

⁢

J
1

(

μ

1
⁢
3

⁢

r
j

)

×

(

cos
⁡
(

θ
j

)

sin
⁡
(

θ
j

)

)

(
21
)

In the fourth example, the subscripts are m=2 and k=1, and there are two criterion functions, i.e., a criterion function corresponding to the proportional coefficient β3_mkand a criterion function corresponding to the proportional coefficient β4_mkin Equation (15). In order to make it easier to determine the correspondence with Equation (15), the results obtained by multiplying these criterion functions by the proportional coefficient β3₂₁and the proportional coefficient β4₂₁corresponding thereto are shown in Equation (22) and Equation (23), respectively.

β

3
⁢
2
⁢
1

⁢

J
2

(

μ
21

⁢

r
j

)

×

(

cos
⁡
(

2
⁢

θ
j

)

-

sin
⁡
(

2
⁢

θ
j

)

)

(
22
)

β

4
⁢
2
⁢
1

⁢

J
2

(

μ
21

⁢

r
j

)

×

(

sin
⁡
(

2
⁢

θ
j

)

cos
⁡
(

2
⁢

θ
j

)

)

(
23
)

Further, the plurality of criterion functions (C21c, C21s, and the like) that set the proportional coefficients α and β may be selected to become functions perpendicular to each other in a circle of r≤1 using the origin O as a center. For example, the criterion function C21c and the criterion function C21s are perpendicular to each other because the value obtained by integrating the product in the circle of r≤1 using the origin O as the center becomes zero.

In addition, in the plurality of first type Bessel functions J_m(μ_mkr) included in the plurality of criterion functions that set the proportional coefficients α and β, at least some or all of the subscripts m and k that differ may be perpendicular to each other within a circle of r≤1 using the origin O as the center. In other words, the reference function may include two or more criterion functions expressed by two or more first type Bessel functions perpendicular to each other in the surface (r≤1) of the substrate WF.

Further, a state in which the two first type Bessel functions J_m(μ_mkr) are perpendicular to each other is expressed by a state in which the value obtained by integrating the product of the two first type Bessel functions J_m(μ_mkr) within a circle of r≤1 using the origin O as the center becomes zero. In other words, a state in which the two first type Bessel functions J_m(μ_mkr) are perpendicular to each other is expressed by a state in which an inner product of the two first type Bessel functions J_m(μ_mkr) when integrated within a circle of r≤1 using the origin O as the center becomes zero.

Further, in determination of the optimum values of the proportional coefficients α and β, the remained position information that is the position information, from which a positional shift amount according to a translational shift, rotation, and uniform expansion and contraction of the substrate WF is removed from the measured position information (the displacement vector DV) of the plurality of measured-parts MP may be used as the measured position information IMP.

Here, the calculation of the positional shift amount according to the translational shift, the rotation and the uniform expansion and contraction of the substrate WF can be performed using general mathematical formulae shown in Equation (24) to Equation (26).

(

ε
⁢

x
j

ε
⁢

y
j

)

=

(

a

b

c

d

)

⁢

(

X
j

Y
j

)

+

(

e

f

)

-

(

X
j

Y
j

)

(
24
)

Lx
=

∑

j
N

⁢

(

Δ
⁢

X
j

-

ε
⁢

x
j

)

2

(
25
)

Ly
=

∑

j
N

⁢

(

Δ
⁢

Y
j

-

ε
⁢

y
j

)

2

(
26
)

Here, in Equation (24), a is a parameter expressing a uniform expansion and contraction of the substrate WF in the X direction, d is a parameter expressing a uniform expansion and contraction in the Y direction, b and c are parameters expressing rotations of the substrate WF, e is a parameter expressing a translational shift of the substrate WF in the X direction, and y is a parameter expressing a translational shift in the Y direction. In addition, Xj and Yj are an X position and a Y position of the reference position MD of the j^thmeasured-part MP, respectively, and N is the number of the measured-parts MP. Then, εxj and εyj express a shift amount of the X position and a shift amount of the Y position from the reference position MD of the j^thmeasured-part MPj according to the translational shift, the rotation, and the uniform expansion and contraction of the substrate WF determined by the parameters a to f.

The values of the parameters a to f are determined such that, for example, values of Lx and Ly are minimized using a least squares method by substituting εxj and εyj expressed by Equation (24) into Equation (25) and Equation (26), respectively. Then, εxj and εyj for each of the N measured-parts MP are calculated by substituting the values of the determined parameters a to f into Equation (22).

The remained position information that is the position information, from which the positional shift amounts (εxj, εyj) according to the translational shift, the rotation, and the uniform expansion and contraction of the substrate WF are removed, is calculated as (ΔXj-εxj, ΔYj-εyj) from the measured displacement vector DVj (ΔXj, ΔYj) of the jth measured-part MPj.

As described above, as the measured position information IMP that is the target of analysis, the analysis may be performed using the remained position information (ΔXj-εxj, ΔYj-εyj), instead of the measured position information (the displacement vector DV) of the plurality of measured-parts MP.

(Variant of Analyzing Method)

Hereinafter, a variant of an analyzing method will be described. However, the variant of the analyzing method has most of the same configurations as the analyzing method of the embodiment described above, so the differences will be described below. In the analyzing method of the embodiment, the first type Bessel functions that constitute the criterion function were all assumed to have extreme values at r=1. On the other hand, in the analyzing method of the variant, the first type Bessel functions that constitute the criterion function differ in that they all take 0 at r=1.

In the analyzing method of the variant, as the criterion function that constitutes the reference function, a function using the first type Bessel function J_m(λ_mkr) is used. Here, λ_mkis a constant to which the order m and the arbitrary natural number k of the first type Bessel function J_mare added as subscripts, like μ_mkin the analyzing method of the embodiment.

The criterion function used in the analyzing method of the variant is, for example, a function expressed by the next Equation (27) or Equation (28).

[Math. 27]

J_m(λ_mkr)×cos(mθ) (27)

[Math. 28]

J_m(λ_mkr)×sin(mθ) (28)

FIG. 6A and FIG. 6B are views showing examples of the first type Bessel function J_m(λ_mkr) included in the criterion function. FIG. 6A is a numerical table showing some examples of values of a constant λ_mkdetermined by the subscripts m and k. The value of the constant λ_mkshown in FIG. 6A is a value of r where J_m(r) takes zero at a k^thposition when r is increased from zero (r=0 is not included) in the first type m^thorder of Bessel function J_m(r) that uses r as an argument. Accordingly, J_m(λ_mkr) included in Equation (27) and Equation (28) becomes a function that takes zero at r=1.

In the numerical table of FIG. 6A, a value of the constant λ_mkwithin a range from 0 to 7 for m and from 1 to a predetermined value according to a value of m for k is shown. However, the value of m of the constant λ_mkis 8 or more, and the value of k may be an arbitrary value of one or more. The value of the constant λ_mkcan be easily calculated from the first type m^thorder of Bessel function J_m(r) when m and k are determined.

FIG. 6B is a view showing some of the first type Bessel functions J_m(λ_mkr) determined by the subscripts m and k as a graph within a range of r from 0 to 1. A curve line B01 shows a function J₀(λ₀₁r) at m=0 and k=1. Similarly, a curve line B11 shows a function J₁(λ₁₁r), a curve line B12 shows a function J₁(λ₁₂r), a curve line B21 shows a function J₂(λ₂₁r), a curve line B22 shows a function J₂(λ₂₂r), a curve line B31 shows a function J₃(λ₃₁r), and a curve line B32 shows a function J₃(λ₃₂r).

As described above, each function shown in FIG. 6B takes zero at r=1. In the curve lines B11, B21 and B31 equivalent to the functions J_m(λ_m1r) corresponding to k=1, zero is taken first at r=1 according to an increase of r (except r=0). In the curve lines B12, B22 and B32 equivalent to the functions J_m(λ_m2r) corresponding to k=2, zero is taken second at r=1 according to an increase of r (except r=0).

In the analyzing method of the variant, as the reference function, an expression in which the proportional coefficient μ_mkincluded in Equation (7) and Equation (8) or Equation (15) described above is replaced with the proportional coefficient λ_mkmay be used. Then, like the above-mentioned analyzing method of the embodiment, for example, optimum values of the proportional coefficients α and β may be determined using a least squares method.

In the analyzing method of the variant, the criterion function that constitutes the reference function is zero at r=1, that is, at the outer circumference of the substrate WF. In other words, the first type Bessel function included in the criterion function is a function that takes zero with respect to the reference position information IRP of the measured-part MP disposed on the circumferential edge of the substrate WF. For this reason, the values of the proportional coefficients α and β can be determined without much influence of deformation in the vicinity of the outer circumference of the substrate WF, i.e., influence of the displacement vector DV of the measured-part MP disposed in the vicinity of the outer circumference of the substrate WF.

Meanwhile, in the above-mentioned analyzing method of the embodiment, the extreme value is taken at a position where the first type Bessel function included in the criterion function that constitutes the reference function is r=1, i.e., the outer circumference of the substrate WF. In other words, the first type Bessel function included in the criterion function is a function that takes an extreme value with respect to the reference position information IRP of the measured-part MP disposed at the circumferential edge of the substrate WF. For this reason, at the outer circumference of the substrate WF, the change in the value of the criterion function with respect to the change in r is small. Accordingly, the values of the proportional coefficients α and β can be determined without much influence of a local change of the displacement vector DV of the measured-part MP disposed in the vicinity of the outer circumference of the substrate WF.

Further, the first type Bessel function J_m(κr) that constitutes the reference function does not necessarily have to be a function that takes an extreme value or 0 at r=1. That is, the proportional coefficient κ may be a value except the above-mentioned μ_mkor λ_mk.

In either the above-mentioned analyzing method of the embodiment or the analyzing method of the variant, the first type Bessel function alone may not be used as a reference function, but may be combined with another reference function. For example, it may be used in combination with a reference function using Zernike polynomial as a criterion function. In this case, fitting can be performed for low-order components using a reference function with a Zernike polynomial as a criterion function, and fitting can be performed for high-order components using a reference function with a first type Bessel function as a criterion function.

(Measuring Method of Embodiment)

Hereinafter, a measuring method of the embodiment will be described. The measuring method of the embodiment measures positions of the plurality of measured-parts MP formed on the substrate WF, and obtains the measured position information IMP of the measured-part MP, i.e., the displacement vector DV of the plurality of measured-parts MP. Then, optimum values of the proportional coefficients α and β included in the reference function expressed by Equation (7) and Equation (8) or Equation (15) are calculated by the above-mentioned analyzing method.

Since the method of measuring the position of the measured-part MP has been described in the description of the measurement apparatus and the exposure apparatus of the embodiment mentioned above, the description will not be repeated herein. In addition, since the method of calculating the optimum values of the reference function and the proportional coefficients α and β have also been described in the description of the analyzing method of the embodiment and variant, the description will not be repeated herein.

(Exposing Method of Embodiment)

Hereinafter, an exposing method of the embodiment will be described with reference to FIG. 7. FIG. 7 is a view showing the substrate WF on which a plurality of first patterns A1, outer edges of which are formed in a quadrangular shape by broken lines. A plurality of fine patterns (not shown) are formed inside the outer edge of the first pattern A1 shown by a broken line. The plurality of measured-parts MP are formed on the substrate WF at positions with predetermined positional relations with respect to the first pattern A1.

The exposing method of the embodiment measures positions of the plurality of measured-parts MP on the substrate WF using the above-mentioned measuring method of the embodiment, acquires measured position information IMP of the measured-part MP, and calculates optimum value of at least one of the proportional coefficients α and β of the above-mentioned reference functions. Then, the position of the first pattern A1 formed on the substrate WF with the predetermined positional relation with respect to the plurality of measured-parts MP is estimated based on the acquired measured position information IMP, and the optimum value of the at least one proportional coefficient.

The position of the first pattern A1 is estimated based on the reference function in which the optimum values of the proportional coefficients α and β are substituted, and the designed position data (AX, AY) of the first pattern A1. That is, the optimum values are substituted into the proportional coefficients α and β of the reference function shown by Equation (7) and Equation (8) or Equation (15). A position shift amount Δxj in the X direction and a position shift amount Δyj in the Y direction from the designed position are calculated (estimated) by inputting the value of the designed position (Arj, Aθj) in polar coordinates of the j^thfirst pattern A1j to θj and θj of the reference function.

Accordingly, the estimated position in the XY coordinates of the j^thfirst pattern A1j is an estimated position (AXj+Δxj, AYj+Δyj) where the above-mentioned Δxj and Δyj are added to the designed position (AXj, AYj) in the XY coordinates of the j^thfirst pattern A1j.

Then, a second pattern (not shown) is exposed to align with the first pattern A1 on the substrate WF based on the estimated position of the estimated first pattern A1.

Further, after removing the positional shift amount according to the translational shift, the rotation, and the uniform expansion and contraction of the substrate WF using Equation (24) or the like, when the optimum values of the proportional coefficients α and β are determined, correction is necessary for the estimated position (AXj+Δxj, AYj+Δyj) of the first pattern A1.

In the correction, εxj and εyj are calculated by substituting the designed position (AXj, AYj) in the XY coordinates of the j^thfirst pattern A1j into Equation (24) to which the values of the parameters a to f determined in this way are substituted. Then, the εxj and εyj may be added to the estimated position of the first pattern A1. That is, the estimated position of the first pattern A1 after correction is an estimated position (AXj+Δxj+εxj, AYj+Δyj+εyj).

The exposing method of the embodiment can be performed using the exposure apparatus 2 of the embodiment as an example. For example, the second pattern exposed to the substrate WF in the exposing method of the embodiment is to expose a pattern original plate formed on the mask 25 to the substrate WF via the projection optical system 22.

(Added Description of Measurement Apparatus and Exposure Apparatus of Embodiment)

Additional matters regarding the measurement apparatus 1 and the exposure apparatus 2 of the embodiment will be described.

The controller 15 may display the fitting result of the measured position information IMP as a vector diagram using a reference function to which the optimum values of the proportional coefficients α and β are applied (substituted). Here, as shown in FIG. 7, the vector diagram is a view that displays the deformation amount of the plurality of measured-parts MP on the substrate WF as a vector with the reference position MD of each of the measured-parts MP as the reference (starting point). Here, the scale representing the vector may be greatly expanded compared to the scale representing the position of the measured-part MP.

The vector diagram displayed on the display part 17 may be a vector diagram that indicates a virtual deformation amount of each of the measured-parts MP calculated by the fitted reference function. Alternatively, it may be a vector diagram that indicates a difference between the actual deformation amount (the displacement vector DV) of each of the measured-parts MP and the virtual deformation amount of each of the measured-parts MP calculated by the fitted reference function.

Further, the controller 15 may obtain information regarding the processing history of the substrate WF which has been analyzed via the network line NW, and analyze correlation between the information regarding the history and the analysis result performed by the analysis part 16. For example, correlation between a processing parameter of an external processing device (etching equipment, a deposition apparatus, chemical mechanical planarization (CMP), or the like), which is used before conveyance of the substrate WF to the measurement apparatus 1 and which is different from the measurement apparatus 1, and the optimum value of the proportional coefficient determined by the analysis part 16 may be taken, feed it back to the processing device, and the processing parameter may be changed and optimized. As the processing parameter, for example, the temperature added to the substrate WF during processing in a processing device, the polishing speed in CMP, or the like, may be changed and optimized. The analysis and feedback of such correlation need not necessarily be performed by the controller 15 or the analysis part 16, but may be performed by a host computer provided separately from the measurement apparatus 1 or an external processing device.

Further, the optimal value of the proportional coefficient determined by the analysis part 16 can be continuously monitored, and a warning can be issued if it exceeds a preset threshold. Abnormalities in external processing devices can be detected by the warning. For example, it is possible to estimate the deterioration (life span) of the electrode in etching equipment. In addition, a flag may be attached to the history of the substrate WF that falls outside the threshold.

(Added Description of Exposure Apparatus of Embodiment)

Additional matters of the exposure apparatus 2 of the embodiment will be described.

The exposure apparatus 2 of the embodiment determines an operation state of the substrate conveyance device 40 (the load guide 41, the load arm 42, and the substrate chuck part 31) based on the measurement results (the proportional coefficients α and β) obtained by the measuring method of the above-mentioned embodiment. The operation state of the substrate conveyance device 40 is, for example, a moving speed of the substrate WF with respect to the substrate chuck part 31 when the substrate WF is placed at the substrate chuck part 31 on the stage 30 or a relative inclination between the substrate WF and the substrate chuck part 31.

The operation state of the substrate conveyance device 40 may be timing of generation of a holding force for holding the substrate WF on the substrate chuck part 31. Change of the timing of generation of the holding force may be change of the timing when the holding force is generated almost all over the surface of the substrate chuck part 31 with respect to the tuning of placement of the substrate WF. Alternatively, the timing of generation of the holding force in the predetermined portion of the substrate chuck part 31 may be set earlier or later than the timing of generation of the holding force in other parts.

These operating states are changed by changing the value of the device constant stored in the controller 15 or the like of the exposure apparatus 2. Accordingly, it can be said that the exposure apparatus 2 determines a device constant that defines the operation of the substrate conveyance device 40 based on the measurement result (the proportional coefficients α and β) obtained by the measuring method of the embodiment.

By changing these device constants, the state in which the substrate WF is sucked to the substrate chuck part 31 can be changed, and the state in which the substrate WF is deformed by suction can be changed. As described above, the device constant includes any one of the device constants that determines a moving speed of the substrate WF when the substrate WF is placed on the stage 30, a relative inclination between the substrate WF and the stage 30, or timing of generation of a holding force for holding the substrate WF on the stage 30.

Further, the change of the device constant may be performed before the substrate WF is removed from the substrate chuck part 31 and the substrate WF is sucked to the substrate chuck part 31 again after the proportional coefficients α and β are determined by the measuring method of the embodiment in a state in which the substrate WF is sucked to the stage 30 (the substrate chuck part 31).

When exposing one lot of substrates consisting of a predetermined number of substrates WF, the device constant determined as above for the first substrate WF may be used to expose the second and subsequent substrates WF.