JP2005012190A - Estimation method and adjusting method of imaging optical system, exposure apparatus and method - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide an estimation method of aberration in an imaging optical system by which the aberration can be estimated analytically by expressing the distribution of the aberration in the pupil and in an image surface at the same time. <P>SOLUTION: This estimation method of aberration in an imaging optical system using an aberration polynomial comprises processes of: (S11), setting an aberration polynominal by using only an orthonormal function series, wherein the aberration polynominal expresses aberration in the imaging optical system as functions of an image surface coordinate and a pupil coordinate of the imaging optical system; (S12), obtaining wavefront aberrations at a plurality of points on the imaging surface in the imaging optical system; (S13), approximating wavefront aberration obtained in the aberration obtaining process (S12) by a predetermined polynominal as a function of the pupil coordinate; and (S14), determining coefficients of respective terms in the aberration polynomial based on a coefficient of each term in the predetermined polynomial obtained in the approximating process. <P>COPYRIGHT: (C)2005,JPO&NCIPI

Description

  The present invention relates to an imaging optical system evaluation method, an imaging optical system adjustment method, an exposure apparatus, and an exposure method, and more particularly to an aberration evaluation of a projection optical system mounted on the exposure apparatus.

  For example, in a lithography process for forming a circuit pattern in the manufacture of an LSI, an exposure apparatus incorporating a projection optical system for transferring a mask pattern onto a resist on a wafer is used. In the current lithography, as the integration degree of LSI increases and the k1 factor (line width = kl × λ / NA: λ is the exposure wavelength, NA is the numerical aperture of the projection optical system), the aberration of the projection optical system Is required to be reduced to the limit.

  For this reason, in recent years, measurement and analysis of wavefront aberrations using various wavefront aberration measuring instruments have been performed in the optical adjustment process of the projection optical system. In the wavefront aberration analysis process, the measured wavefront aberration is often approximated (fitted) using a Fringe Zernike polynomial as a function of pupil coordinates. Here, the Zernike polynomial is a function expressing the distribution of wavefront aberration in the pupil.

  As described above, the Zernike polynomial is a function suitable for expressing the wavefront aberration in the pupil. However, in this case, based on the coefficient of each term of the Zernike polynomial, it is necessary to perform optical adjustment by determining the optical adjustment method and the optical adjustment amount by trial and error by optimization calculation using a computer.

  On the other hand, as a function expressing the distribution of wavefront aberration in the image plane, a function derived from an aberration theory assuming a rotationally symmetric optical system, or an aberration of an optical system including a decentration error up to the third order (ray aberration) Functions derived from theory are known. However, the conventional function for expressing the distribution of wavefront aberration in the image plane is insufficient to express the aberration state before and during the optical adjustment of the projection optical system having a very large numerical aperture and field.

  The present invention has been made in view of the foregoing problems, and analytically evaluates aberrations of the imaging optical system by simultaneously expressing the aberration distribution of the imaging optical system in the pupil and the distribution in the image plane. It aims at providing the evaluation method which can do. Another object of the present invention is to provide an adjustment method that can satisfactorily optically adjust the imaging optical system based on the analytical evaluation of the aberration obtained by the evaluation method of the present invention. It is another object of the present invention to provide an exposure apparatus and an exposure method capable of performing good projection exposure using an imaging optical system that is optically adjusted by the adjustment method of the present invention.

In order to solve the above problem, in a first embodiment of the present invention, in a method for evaluating aberration of an imaging optical system using an aberration polynomial,
Setting the aberration polynomial using only orthonormal function series as a function of image plane coordinates and pupil coordinates of the imaging optical system;
An aberration acquisition step of obtaining wavefront aberration of the imaging optical system for a plurality of points on the image plane of the imaging optical system;
An approximation step of approximating the wavefront aberration obtained in the aberration acquisition step with a predetermined polynomial as a function of pupil coordinates;
And determining a coefficient of each term of the aberration polynomial based on a coefficient of each term in the predetermined polynomial obtained in the approximating step.

  According to a preferred aspect of the first aspect, the setting step includes a step of expressing the orthonormal function sequence by a Zernike function in pupil coordinates and a Zernike function in image plane coordinates. The predetermined polynomial preferably includes a Zernike polynomial. The aberration polynomial includes at least one aberration component of a rotationally symmetric aberration component, an eccentric aberration component, an as (Toric) aberration component, and a trefoil aberration component with respect to the optical axis of the imaging optical system. It is preferable to set as follows. In this case, it is preferable that the aberration polynomial is set so as to include at least one aberration component of the decentering aberration component, the astolic aberration component, and the trefoil aberration component.

  According to a preferred aspect of the first aspect, the rotationally symmetric aberration component is expressed as an invariant power series with respect to rotation in the image plane coordinates and the pupil coordinates. In this case, the decentration aberration component is preferably expressed as a product of a first-order dependent component of the coordinates in the image plane coordinates and the pupil coordinates and a power series of invariants with respect to the rotation. Alternatively, the astigmatic aberration component is a quadratic dependent component of the coordinates in the image plane coordinates and the pupil coordinates and a periodic function of 180 degrees with respect to the rotation of the coordinates, and an invariant of the rotation. It is preferable to include a product with a power series. Alternatively, the trefoil aberration component is a third-order dependent component of the coordinates in the image plane coordinates and the pupil coordinates and a periodic function of 120 degrees with respect to the rotation of the coordinates, and an invariant power series with respect to the rotation. It is preferable to include the product.

  Further, according to a preferred aspect of the first aspect, the determining step includes a predetermined polynomial as a function of the image plane coordinates, and the distribution in the image plane of the coefficient of each term in the predetermined polynomial obtained in the approximating step. A second approximation step that approximates The aberration acquisition step preferably includes a step of measuring a wavefront aberration of the imaging optical system. The aberration acquisition step preferably includes a step of calculating a wavefront aberration of the imaging optical system by ray tracing.

According to a second aspect of the present invention, in the method for evaluating the aberration of the imaging optical system using an aberration polynomial, based on the wavefront aberration obtained for a plurality of points on the image plane of the imaging optical system,
Setting the aberration polynomial using only orthonormal function series as a function of image plane coordinates and pupil coordinates of the imaging optical system;
An approximation step of approximating the obtained wavefront aberration with a predetermined polynomial as a function of pupil coordinates;
And determining a coefficient of each term of the aberration polynomial based on a coefficient of each term in the predetermined polynomial obtained in the approximating step.

  According to a preferred aspect of the second aspect, the setting step includes a step of expressing the orthonormal function sequence by a Zernike function in pupil coordinates and a Zernike function in image plane coordinates. The aberration polynomial is preferably set so as to include at least one aberration component of an eccentric aberration component, an as (toric) aberration component, and a trefoil aberration component. The determining step includes a second approximating step of approximating the in-image distribution of the coefficient of each term in the predetermined polynomial obtained in the approximating step with a predetermined polynomial as a function of the image plane coordinates. It is preferable.

  According to a third aspect of the present invention, there is provided an adjustment method characterized in that the imaging optical system is optically adjusted based on aberration information of the imaging optical system obtained by the evaluation method of the first or second aspect. provide.

  According to a fourth aspect of the present invention, an imaging optical system optically adjusted by the adjustment method of the third aspect is provided as a projection optical system for projecting and exposing a mask pattern onto a photosensitive substrate. An exposure apparatus is provided.

  According to a fifth aspect of the present invention, the image of the pattern formed on the mask is projected and exposed onto a photosensitive substrate using the imaging optical system optically adjusted by the adjustment method of the third aspect. Provide a method. According to a sixth aspect of the present invention, there is provided an imaging optical system that is optically adjusted by the adjustment method of the third aspect.

  According to a seventh aspect of the present invention, there is provided a recording medium in which a program for executing the evaluation method of the first aspect or the second aspect is recorded. According to an eighth aspect of the present invention, there is provided a computer-receivable carrier wave that is equipped with a signal including a program that executes the evaluation method according to the first or second aspect.

  In the evaluation method of the present invention, the aberration of the imaging optical system can be analytically evaluated by simultaneously expressing the distribution in the pupil and the distribution in the image plane of the aberration of the imaging optical system. Therefore, it is possible to satisfactorily optically adjust the imaging optical system based on the analytical evaluation of the aberration obtained by the evaluation method of the present invention. In addition, it is possible to perform good projection exposure by using the imaging optical system that is optically adjusted by the adjustment method of the present invention, and thus it is possible to manufacture a good microdevice.

Embodiments of the present invention will be described with reference to the accompanying drawings.
FIG. 1 is a view schematically showing a configuration of an exposure apparatus provided with a projection optical system to which an imaging optical system evaluation method according to an embodiment of the present invention is applied. 1, the Z-axis is parallel to the optical axis AX of the projection optical system PL, the Y-axis is parallel to the paper surface of FIG. 1 in a plane perpendicular to the optical axis AX, and the plane is perpendicular to the optical axis AX. The X axis is set perpendicular to the paper surface.

The exposure apparatus shown in FIG. 1 includes, for example, an F ₂ laser light source (wavelength 157 nm) as a light source LS for supplying illumination light. The light emitted from the light source LS illuminates the reticle (mask) R on which a predetermined pattern is formed via the illumination optical system IL. The optical path between the light source LS and the illumination optical system IL is sealed with a casing (not shown), and the space from the light source LS to the optical member on the most reticle side in the illumination optical system IL absorbs exposure light. It is replaced with an inert gas such as helium gas or nitrogen, which is a low-rate gas, or is kept in a substantially vacuum state.

  The reticle R is held parallel to the XY plane on the reticle stage RS via the reticle holder RH. A pattern to be transferred is formed on the reticle R. For example, a rectangular pattern region having a long side along the X direction and a short side along the Y direction in the entire pattern region is illuminated. Reticle stage RS can be moved two-dimensionally along the reticle plane (ie, XY plane) by the action of a drive system (not shown), and its position coordinates are measured by interferometer RIF using reticle moving mirror RM. And the position is controlled.

  Light from the pattern formed on the reticle R forms a reticle pattern image on the wafer W, which is a photosensitive substrate, via the projection optical system PL. The wafer W is held parallel to the XY plane on the wafer stage WS via a wafer table (wafer holder) WT. A rectangular exposure area having a long side along the X direction and a short side along the Y direction on the wafer W so as to optically correspond to the rectangular illumination area on the reticle R. A pattern image is formed. The wafer stage WS can be moved two-dimensionally along the wafer surface (that is, the XY plane) by the action of a drive system (not shown), and its position coordinates are measured by an interferometer WIF using a wafer moving mirror WM. In addition, the position is controlled.

  In the illustrated exposure apparatus, the interior of the projection optical system PL is hermetically sealed between the optical member disposed on the most reticle side and the optical member disposed on the most wafer side among the optical members constituting the projection optical system PL. The gas inside the projection optical system PL is replaced with an inert gas such as helium gas or nitrogen, or is almost kept in a vacuum state.

  Further, a reticle R and a reticle stage RS are arranged in a narrow optical path between the illumination optical system IL and the projection optical system PL, but a casing (not shown) that hermetically surrounds the reticle R and the reticle stage RS. Is filled with an inert gas such as nitrogen or helium gas, or is kept in a vacuum state.

  A narrow optical path between the projection optical system PL and the wafer W includes a wafer W and a wafer stage WS. The inside of a casing (not shown) that hermetically encloses the wafer W and the wafer stage WS. Is filled with an inert gas such as nitrogen or helium gas, or is kept in a vacuum state. Thus, an atmosphere in which the exposure light is hardly absorbed is formed over the entire optical path from the light source LS to the wafer W.

  As described above, the illumination area on the reticle R and the exposure area on the wafer W (that is, the effective exposure area) defined by the projection optical system PL are rectangular shapes having short sides along the Y direction. Accordingly, the reticle stage RS is controlled along the short side direction of the rectangular exposure area and the illumination area, that is, the Y direction, while controlling the positions of the reticle R and the wafer W using a drive system and an interferometer (RIF, WIF). By moving (scanning) the wafer stage WS and thus the reticle R and the wafer W synchronously, the wafer W has a width equal to the long side of the exposure area and the scanning amount (movement amount) of the wafer W. The reticle pattern is scanned and exposed to an area having a length corresponding to).

  In the present embodiment, the evaluation method and the adjustment method of the present invention are applied to the projection optical system PL as the imaging optical system. Prior to this description, the aberration of the projection optical system PL is represented by the image plane coordinates and the pupil coordinates. An aberration polynomial (aberration function) generally expressed as a function of is newly derived (set). FIG. 2 is a diagram for explaining image plane coordinates and pupil coordinates of the projection optical system PL. When attention is paid to the light ray passing through the image plane orthogonal coordinate (y, z) and the pupil orthogonal coordinate (ξ, η) in FIG. 2, the wavefront aberration W of this light ray is expanded to a power series of y, z, ξ, η. It should be.

  Therefore, first of all, the rotationally symmetric aberration component Wr related to the optical axis AX of the projection optical system PL among the components of the wavefront aberration W will be considered. The invariant with respect to the rotation of the coordinates is expressed by the following equations (1) to (3). The rotationally symmetric aberration component Wr is expressed by an invariant of Expression (1), an invariant of Expression (2), and a power series of the invariant of Expression (3). In other words, the rotationally symmetric aberration component Wr is expressed by {a power series from (1) to (3)}.

y ² + z ² (1)
ξ ² + η ² (2)
y · ξ + z · η (3)

  Next, among the components of the wavefront aberration W, the decentration aberration component Ws related to the optical axis AX of the projection optical system PL will be considered. The dependency of aberration components newly generated by decentration on coordinates (image plane coordinates or pupil coordinates) is only primary. Therefore, the aberration including the decentering component, that is, the decentering aberration component Ws, is any one of the primary coordinate-dependent components represented by the following formulas (4) to (7), and the formulas (1) to (7). It is expressed as the product of the invariant power series for the rotation represented by (3). In other words, the decentration aberration component Ws is represented by {a power series from (1) to (3)} × {any one of (4) to (7)}.

y (4)
z (5)
ξ (6)
η (7)

  Next, among the components of the wavefront aberration W, an astigmatic aberration component Wa will be considered. The dependence of the aberration component newly generated in the as (Toric) component on the coordinates (image plane coordinates or pupil coordinates) is only second order, and is a periodic function of 180 degrees with respect to the rotation of the coordinates. For this reason, an aberration including an as (toric) component, that is, an as (toric) aberration component Wa is a second-order dependent component of the coordinates represented by the following equations (8) to (13), and with respect to the rotation of the coordinates. It is expressed as a product of a component that is a periodic function of 180 degrees and a power series of invariants with respect to the rotation represented by equations (1) to (3). In other words, the astoric aberration component Wa is represented by {a power series from (1) to (3)} × {any one of (8) to (13)}.

y ² −z ² (8)
2y · z (9)
ξ ² −η ² (10)
2ξ · η (11)
y · ξ−z · η (12)
y · η + z · ξ (13)

  Finally, the trefoil aberration component Wt among the components of the wavefront aberration W will be considered. The dependency of the aberration component newly generated in the Trefoil component on the coordinates (image plane coordinates or pupil coordinates) is only the third order, and is a periodic function of 120 degrees with respect to the rotation of the coordinates. For this reason, the aberration including the trefoil component, that is, the trefoil aberration component Wt is a third-order dependent component of coordinates expressed by the following equations (14) to (21) and is a periodic function of 120 degrees with respect to the rotation of the coordinates. It is expressed as the product of the component and the invariant power series with respect to the rotation represented by the equations (1) to (3). In other words, the trefoil aberration component Wt is represented by {a power series from (1) to (3)} × {any one of (14) to (21)}.

^{y (y 2 -3z 2) (} 14)
z (3y ² −z ² ) (15)
ξ (ξ ² −3η ² ) (16)
η (3ξ ² −η ² ) (17)
(Y ² −z ² ) ξ−2yzη (18)
2yzξ + (y ² −z ² ) η (19)
y (ξ ² −η ² ) −2zξη (20)
z (ξ ² −η ² ) + 2yξη (21)

  Thus, the wavefront aberration W including the rotationally symmetric aberration component Wr, the decentering aberration component Ws, the as (toric) aberration component Wa, and the trefoil aberration component Wt is converted into {the power series of (1) to (3)}, or { (1) to (3) power series} × {any one of (4) to (21)}. On the other hand, referring to FIG. 2, between the image plane orthogonal coordinates (y, z) and the pupil orthogonal coordinates (ξ, η) and the image plane polar coordinates (h, α) and the pupil polar coordinates (ρ, θ), The relationships shown in equations (a) to (d) are established. Here, h and ρ are standardized half mysteries, and α and θ are radial angles of polar coordinates.

y = hcosα (a)
z = hsinα (b)
ξ = ρcosθ (c)
η = ρsinθ (d)

  Therefore, based on the relationships shown in the expressions (a) to (d), the above expressions (1) to (21) can be transformed into the following expressions (A) to (U), respectively.

y ² + z ² (1) is h ² (A)
ξ ² + η ² (2) is ρ ² (B)
y · ξ + z · η (3) is ρhcos (θ−α) (C)
y (4) is hcosα (D)
z (5) is hsinα (E)
ξ (6) is ρcosθ (F)
η (7) is ρsinθ (G)
y ² −z ² (8) is h ² cos2α (H)
2y · z (9) is h ² sin2α (I)
ξ ² −η ² (10) is ρ ² cos 2θ (J)
2ξ · η (11) is ρ ² sin2θ (K)
y · ξ−z · η (12) is hρcos (θ + α) (L)
y · η + z · ξ (13) is hρsin (θ + α) (M)
y (y ² −3z ² ) (14) is h ³ cos3α (N)
z (3y ² −z ² ) (15) is h ³ sin3α (O)
ξ (ξ ² −3η ² ) (16) is ρ ³ cos 3θ (P)
η (3ξ ² −η ² ) (17) is ρ ³ sin3θ (Q)
(Y ² −z ² ) ξ−2yzη (18) is h ² ρcos (θ + 2α) (R)
2yzξ + (y ² −z ² ) η (19) is h ² ρsin (θ + 2α) (S)
y (ξ ² −η ² ) −2zξη (20) is hρ ² cos (2θ + α) (T)
z (ξ ² −η ² ) + 2yξη (21) is hρ ² sin (2θ + α) (U)

  Accordingly, the wavefront aberration W including the rotationally symmetric aberration component Wr, the decentering aberration component Ws, the as (toric) aberration component Wa, and the trefoil aberration component Wt is expressed by the following aberration polynomial (e).

W = Σ (Mi × FMi)
However,
^{FMi = (A j1 · B j2} · C j3) × {(D k1 · E k2 · F k3 · G k4)
^{× (H k5 · I k6 ·} J k7 · K k8 · L k9 · M k10)
^{× (N k11 · O k12 ·} P k13 · Q k14 · R k15
· S ^k16 · T ^k17 · U ^k18 )} (e)

  Here, Σ is a summation symbol for a positive integer i (i = 1, 2, 3,...), And Mi and FMi are coefficients and functions of terms in the aberration polynomial Σ (Mi · FMi). . J1 to j3 are non-negative integers (0, 1, 2,...). Furthermore, k1 to k18 are 0 or 1, and satisfy Σki ≦ 1. In other words, k1 to k18 are all 0, or only one of them is 1 and the others are 0. Specifically, when k1 to k18 are all 0, the term represents the rotationally symmetric aberration component Wr. On the other hand, when only one of k1 to k18 is 1, the term represents the decentering aberration component Ws, the astolic aberration component Wa, or the trefoil aberration component Wt.

  In the following tables (1) and (2), the value of i corresponding to the aberration function FMi of each term in the aberration polynomial Σ (Mi · FMi), ω dependency, aberration order, and j1 to j3 and k1 to k18 Indicates a combination of orders (blanks are 0 in each table). Here, the combinations of the orders of j1 to j3 and k1 to k18 are defined such that the aberration function FMi of each term other than the constant term FM1 includes at least ρ. As for ω dependency, there is no rotation dependency when ω = 0, 360 ° rotation dependency (one rotation dependency) when ω = 1, and ω = 2. Indicates that there is a 180 ° rotation dependency (twice rotation dependency), and that when ω = 3, there is a 120 ° rotation dependency (three rotation rotation dependency).

  Further, if any one of j1 to j3 is 1, the aberration order is increased by 2, if any one of k1 to k4 is 1, the aberration order is increased by 1, and any one of k5 to k10 is 1. If there is, the aberration order is increased by 2, and if any one of k11 to k18 is 1, the aberration order is increased by 3. In Tables (1) and (2), the aberration function FMi after the 47th term is not shown.

Table (1)

Table (2)

  Next, according to Tables (1) and (2), the aberration function FMi of each term in the aberration polynomial Σ (Mi · FMi) is expressed by image plane orthogonal coordinates (z, y) and pupil polar coordinates (ρ, θ). It is shown in the following table (3). In the aberration classification of Table (3), Focus indicates focus, Dist indicates distortion, Toric indicates toric, Coma indicates coma, Trefoil indicates trefoil, and as indicates as (toric) aberration. In Table (3), the order represents the aberration order. In the aberration function FMi in Table (3), “+...” Represents a portion that can be expressed by the above-described aberration function. In Table (3), the aberration function FMi after the 47th term is not shown.

Table (3)

  Here, basic matters regarding the Zernike polynomial representing the distribution of the wavefront aberration in the pupil will be described. In the Zernike polynomial expression, the above-mentioned pupil polar coordinates (ρ, θ) are used as the coordinate system, and the Zernike cylindrical function is used as the orthogonal function system. That is, the wavefront aberration W (ρ, θ) is developed as shown in the following equation (f) using Zernike's cylindrical function Zi (ρ, θ).

W (ρ, θ) = ΣCi · Zi (ρ, θ)
= C1 · Z1 (ρ, θ) + C2 · Z2 (ρ, θ)
・ ・ ・ ・ + Cn ・ Zn (ρ, θ) (f)

  Here, Ci is a coefficient of each term of the Zernike polynomial. Hereinafter, among the function system Zi (ρ, θ) of each term of the Zernike polynomial, functions Z1 to Z36 relating to the first term to the 36th term are shown in the following table (4).

Table (4)
Z1: 1
Z2: ρcosθ
Z3: ρsinθ
Z4: 2ρ ² −1
Z5: ρ ² cos2θ
Z6: ρ ² sin2θ
Z7: (3ρ ² -2) ρcosθ
Z8: (3ρ ² -2) ρsinθ
Z9: 6ρ ⁴ -6ρ ² +1
Z10: ρ ³ cos3θ
Z11: ρ ³ sin3θ
Z12: (4ρ ² -3) ρ ² cos2θ
Z13: (4ρ ² -3) ρ ² sin2θ
Z14: (10ρ ⁴ −12ρ ² +3) ρcosθ
Z15: (10ρ ⁴ −12ρ ² +3) ρsinθ
Z16: 20ρ ⁶ −30ρ ⁴ + 12ρ ² −1
Z17: ρ ⁴ cos4θ
Z18: ρ ⁴ sin4θ
Z19: (5ρ ² -4) ρ ³ cos3θ
Z20: (5ρ ² -4) ρ ³ sin3θ
Z21: (15ρ ⁴ −20ρ ² +6) ρ ² cos 2θ
Z22: (15ρ ⁴ −20ρ ² +6) ρ ² sin2θ
Z23: (35ρ ⁶ -60ρ ⁴ + 30ρ ² -4) ρcosθ
Z24: (35ρ ⁶ -60ρ ⁴ + 30ρ ² -4) ρsinθ
Z25: 70ρ ⁸ −140ρ ⁶ + 90ρ ⁴ −20ρ ² +1
Z26: ρ ⁵ cos 5θ
Z27: ρ ⁵ sin5θ
Z28: (6ρ ² -5) ρ ⁴ cos4θ
Z29: (6ρ ² -5) ρ ⁴ sin4θ
Z30: (21ρ ⁴ −30ρ ² +10) ρ ³ cos3θ
Z31: (21ρ ⁴ −30ρ ² +10) ρ ³ sin3θ
Z32: (56ρ ⁶ −104ρ ⁴ + 60ρ ² −10) ρ ² cos 2θ
Z33: (56ρ ⁶ −104ρ ⁴ + 60ρ ² −10) ρ ² sin2θ
Z34: (126ρ ⁸ -280ρ ⁶ + 210ρ ⁴ -60ρ ² +5) ρcosθ
Z35: (126ρ ⁸ −280ρ ⁶ + 210ρ ⁴ −60ρ ² +5) ρsinθ
Z36: 252ρ ¹⁰ −630ρ ⁸ + 560ρ ⁶ −210ρ ⁴ + 30ρ ² −1

  Next, the aberration function FMi of each term shown in the table (3) of the aberration polynomial Σ (Mi · FMi) is transformed into a form of linear combination of the Zernike function Zi of each term shown in the table (4) of the Zernike polynomial. This is shown in the following table (5) as a new aberration function for the aberration polynomial, that is, the aberration function FNi expressed in the Zernike function. When the aberration function FMi of each term is modified, a Zernike function Zi having the same order of ρ and the same order of θ and the same type of sin and cos is introduced.

  In addition, when the aberration function FMi of each term is modified as a result of the deformation of the aberration function FMi of each term, the aberration function FNj is changed from the aberration function FNi to avoid redundancy. The part corresponding to is omitted. Specifically, referring to FM2 and Z4 as the simplest example, the aberration function FN2 of the second term is initially (Z4-Z1), but the aberration function FN1 of the other term (first term). = Z1 is included, the portion Z1 corresponding to the aberration function FN1 is omitted from the aberration function FN2, and the aberration function FN2 = Z4.

  Furthermore, even when the aberration function FNi of a certain term matches the aberration function FNj of another term as a result of modifying the aberration function FMi of each term, the adoption of the aberration function FNj is omitted in order to avoid redundancy. . Specifically, referring to Table (5), since the aberration functions FN39 and FN40 of the 39th term and the 40th term coincide with the aberration functions FN25 and FN26 of the 25th term and the 26th term, respectively, the aberration polynomial In Σ (Mi · FNi), the aberration functions FN39 and FN40 in the 39th and 40th terms are not used.

  In Table (5), the aberration functions FN2 to FN8 correspond to the rotationally symmetric aberration component Wr, the aberration functions FN9 to FN20 correspond to the decentration aberration component Ws, and the aberration functions FN21 to FN40 represent the astigmatic aberration component. Corresponding to Wa, the aberration functions FN41 to FN46 correspond to the trefoil aberration component Wt. In Table (5), the aberration function FN1 = Z1 of the first term and the aberration function FNi after the 47th term are omitted.

Table (5)

  In this embodiment, in order to suppress a fitting error when fitting (approximate) wavefront aberration with an aberration polynomial, the aberration polynomial is set as a function of image plane coordinates and pupil coordinates using only an orthonormal function sequence. For this purpose, in this embodiment, a Zernike function Fi represented by image plane polar coordinates (h, α) is introduced in correspondence with the Zernike function Zi represented by pupil polar coordinates (ρ, θ). Zernike functions F1 to F36 in image plane coordinates according to the first term to the 36th term are shown in the following table (6).

Table (6)
F1: 1
F2: hcosα
F3: hsinα
F4: 2h ² -1
F5: h ² cos2α
F6: h ² sin2α
F7: (3h ² -2) hcosα
F8: (3h ² -2) hsinα
F9: 6h ⁴ -6h ² +1
F10: h ³ cos3α
F11: h ³ sin3α
F12: (4h ² -3) h ² cos2α
F13: (4h ² -3) h ² sin2α
F14: (10h ⁴ -12h ² +3) hcosα
F15: (10h ⁴ -12h ² +3) hsinα
F16: 20h ⁶ -30h ⁴ + 12h ² -1
F17: h ⁴ cos4α
F18: h ⁴ sin4α
F19: (5h ² -4) h ³ cos3α
F20: (5h ² -4) h ³ sin3α
F21: (15h ⁴ -20h ² +6) h ² cos2α
F22: (15h ⁴ -20h ² +6) h ² sin2α
F23: (35h ⁶ -60h ⁴ + 30h ² -4) hcosα
F24: (35h ⁶ -60h ⁴ + 30h ² -4) hsinα
F25: 70h ⁸ -140h ⁶ + 90h ⁴ -20h ² +1
F26: h ⁵ cos5α
F27: h ⁵ sin5α
F28: (6h ² -5) h ⁴ cos4α
F29: (6h ² -5) h ⁴ sin4α
F30: (21h ⁴ -30h ² +10) h ³ cos3α
F31: (21h ⁴ -30h ² +10) h ³ sin3α
F32: (56h ⁶ −104h ⁴ + 60h ² −10) h ² cos2α
F33: (56h ⁶ −104h ⁴ + 60h ² −10) h ² sin2α
F34: (126h ⁸ -280h ⁶ + 210h ⁴ -60h ² +5) hcosα
F35: (126h ⁸ -280h ⁶ + 210h ⁴ -60h ² +5) hsinα
F36: 252h ¹⁰ -630h ⁸ + 560h ⁶ -210h ⁴ + 30h ² -1

  When expressing the wavefront aberration of an actual projection optical system as a function of image plane coordinates and pupil coordinates, the wavefront aberration obtained by measurement (or ray tracing calculation) at a plurality of points in the exposure area is expressed as a Zernike function. And each aberration component is calculated by approximating the aberration polynomial of this embodiment. At this time, orthogonalization of the aberration function FNi shown in Table (5) becomes a problem in order to suppress the fitting error when fitting the wavefront aberration with the aberration polynomial.

  In the present embodiment, on the condition that the image surface is circular and the maximum image height is normalized to 1, the aberration function is orthogonalized by, for example, the Gram-Schmidt orthogonalization method, and finally the normal orthogonalization is performed. An aberration function series is derived. That is, the Zernike function Zi in pupil coordinates and the Zernike function Fi in image plane coordinates represent orthonormal function sequences TAi, TBi, TCi, and TDi, and only these orthonormal function sequences TAi, TBi, TCi, and TDi. Is used to set an aberration polynomial as shown in the following equation (g) as a function of image plane coordinates and pupil coordinates.

W = Σ (MAi × TAi + MBi × TBi + MCi × TCi
+ MDi × TDi) (g)

  In the aberration polynomial shown in the equation (g), MAi and TAi are the coefficient of each term and the orthogonalized aberration function regarding the rotationally symmetric aberration component Wr. MBi and TBi are the coefficient of each term and the orthogonalized aberration function regarding the decentration aberration component Ws. MCi and TCi are the coefficient of each term and the orthogonalized aberration function regarding the astolic aberration component Wa. MDi and TDi are the coefficient of each term and the orthogonalized aberration function regarding the trefoil aberration component Wt.

  The following tables (7) and (8) show orthogonalized aberration functions TAi for each term related to the rotationally symmetric aberration component Wr. The following tables (9) to (12) show orthogonalized aberration functions TBi of the terms related to the decentration aberration component Ws. The following table (13) shows the orthogonalized aberration function TCi of each term related to the as (toric) aberration component Wa. Table (14) below shows the orthogonalized aberration function TDi of each term related to the trefoil aberration component Wt. In the orthogonalized aberration function TAi, the function after the 51st term is displayed. In the orthogonalized aberration function TBi, the function after the 131st term is displayed. In the orthogonalized aberration function TCi, the function after the 19th term is displayed. In the orthogonalized aberration function TDi, the display of functions after the seventh term is omitted.

Table (7)

Table (8)

Table (9)

Table (10)

Table (11)

Table (12)

Table (13)

Table (14)

  Thus, in the present embodiment, an aberration polynomial (g) using only an orthonormal function sequence is finally set as an aberration polynomial expressing the aberration of the projection optical system PL as a function of image plane coordinates and pupil coordinates. In this embodiment, as an aberration function of each term in the aberration polynomial (g), a function expressing a rotationally symmetric aberration component up to the ninth order (ray aberration), and a decentration aberration component up to the eighth order (ray aberration) are expressed. A function that expresses an as (toric) aberration component up to the third order (ray aberration), and a function that expresses a trefoil aberration component up to the second order (ray aberration). Thus, it is also possible to calculate an aberration function expressing a higher-order aberration distribution.

Furthermore, in order to express the focus component more accurately, each evaluation is performed by replacing Z4 in Tables (7) to (13) with a defocus aberration D (shown by the following equation (h)) as follows. The fitting accuracy of the focus component or spherical aberration component at a point can be improved. This is particularly effective when evaluating an imaging optical system having a high numerical aperture.
D = (ρ ² −1) ^1/2 −1 (h)

  FIG. 3 is a flowchart showing steps of the evaluation method and adjustment method of the projection optical system PL in the present embodiment. Referring to FIG. 3, in the present embodiment, using the above-described method, an aberration polynomial (g) representing the aberration of the projection optical system PL as a function of image plane coordinates and pupil coordinates is used only by using an orthonormal function sequence. Set (S11). Next, the wavefront aberration is measured for a plurality of points on the image plane of the projection optical system PL (S12). When measuring the wavefront aberration of the projection optical system PL, for example, a Fizeau interferometer disclosed in US Pat. No. 5,898,501 (corresponding to JP-A-10-38757 and JP-A-10-38758) is used. Can be used.

  Further, PDI (Point Deflection Interferometer) disclosed in Japanese Patent Laid-Open No. 2000-97617, phase recovery method disclosed in Japanese Patent Laid-Open No. 10-284368 and US Pat. No. 4,309,602, WO99 / 60361 No., WO 00/55890, and Japanese Patent Application No. 2000-258085, the S / H (Shack-Hartmann) method, and US Pat. No. 5,828,455 and US Pat. No. 5,978,085. Litel Instruments Inc. Company methods can also be used.

  Further, a method using a halftone phase shift mask disclosed in JP-A No. 2000-146757, JP-A No. 10-170399, Jena Review 1991/1, pp8-12 “Wavefront analysis of photolithographic lenses” Wolfgang Freitag et al. , Applied Optics Vol. 31, No. 13, May 1, 1992, pp2284-2290. "Aberration analysis in aerial images formed by lithographic lenses", Wolfgang Freitag et al., And Japanese Patent Application Laid-Open No. 2002-22609. As described above, a method using a light beam passing through a part of the pupil can be used. In the above description, the wavefront aberration of the projection optical system PL is measured using an interferometer or the like. However, for example, the wavefront aberration of the projection optical system PL can be calculated by ray tracing.

  Next, in the present embodiment, the wavefront aberration obtained in the aberration acquisition step (S12) is approximated by a Zernike polynomial as a function of pupil coordinates (S13). Specifically, the wavefront aberration obtained for a plurality of points on the image plane is fitted with a Zernike polynomial, and the Zernike coefficient Ci of each term is calculated for each image point. Next, the coefficients MAi, MBi, MCi, and MDi of each term in the aberration polynomial (g) of the present embodiment are determined based on the Zernike coefficient Ci of each term in the Zernike polynomial obtained in the approximation step (S13). (S14).

  Specifically, for example, focusing on the Zernike function Zi of the specific term, the coefficient of the specific term in the aberration polynomial (g) based on the in-plane distribution of the corresponding Zernike coefficient Ci (the distribution of the coefficient Ci at each image point) MAi, MBi, MCi, and MDi are determined using, for example, the least square method. Further, paying attention to the Zernike function Zi of another specific term, based on the in-image distribution of the corresponding Zernike coefficient Ci, the coefficients MAi, MBi, MCi, and MDi of other terms in the aberration polynomial (g) are, for example, Sequentially determined using the least squares method.

  Thus, in the present embodiment, the orthogonalization functions TAi, TBi, TCi, and TDi defined in Tables (7) to (14), and the coefficients MAi, MBi, MCi, and MDi determined in the determination step (S14). Based on the above, an aberration polynomial that simultaneously represents the in-pupil distribution and the in-plane distribution of aberration of the projection optical system PL is finally obtained. Finally, the projection optical system PL is optically adjusted based on the aberration information (ie, finally obtained aberration polynomial) of the projection optical system PL obtained by the evaluation method (S11 to S14) of the present embodiment (S15). ).

  In this embodiment, the aberration component of the projection optical system PL can be analyzed analytically by using the aberration polynomial (g) that simultaneously expresses the aberration distribution of the projection optical system PL in the pupil and the distribution in the image plane. Thus, it is possible to calculate a corrected solution, that is, an optical adjustment method and an optical adjustment amount quickly and accurately as compared with the conventional method of performing numerical optimization by trial and error using a computer. That is, since it becomes easy to grasp the characteristics of the aberration state of the projection optical system PL by the aberration polynomial (g), it can be expected that the prospect of optical adjustment can be easily made.

  Furthermore, in various error analysis at the design stage, conventionally, a technique using automatic correction has been widely used. However, by using the aberration polynomial (g) of the present embodiment, the aberration state of the projection optical system PL is unambiguous. Therefore, simple and accurate analysis can be expected. Further, by using the aberration polynomial (g) of the present embodiment set using only the orthonormal function series TAi, TBi, TCi, and TDi, fitting error when fitting wavefront aberration with the aberration polynomial (g) Can be kept small.

  In the embodiment described above, the order is limited in the derivation of the aberration polynomial (g) so that the aberration of the projection optical system PL can be sufficiently expressed while avoiding the complexity of the calculation. In the derivation method, the order can be further increased as necessary. In the above-described embodiment, when the aberration polynomial (g) is derived, the rotationally symmetric aberration component Wr, the decentration aberration component Ws, the astolic aberration component Wa, and the trefoil aberration component Wt are considered. However, other appropriate aberration components can be taken into consideration as necessary.

  In the exposure apparatus of the above-described embodiment, the reticle (mask) is illuminated by the illumination device (illumination process), and the transfer pattern formed on the mask is exposed to the photosensitive substrate using the projection optical system (exposure process). Thus, a micro device (semiconductor element, imaging element, liquid crystal display element, thin film magnetic head, etc.) can be manufactured. Hereinafter, referring to the flowchart of FIG. 4 for an example of a method for obtaining a semiconductor device as a micro device by forming a predetermined circuit pattern on a wafer or the like as a photosensitive substrate using the exposure apparatus of this embodiment. I will explain.

  First, in step 301 of FIG. 4, a metal film is deposited on one lot of wafers. In the next step 302, a photoresist is applied on the metal film on the one lot of wafers. Thereafter, in step 303, using the exposure apparatus of the present embodiment, the image of the pattern on the mask is sequentially exposed and transferred to each shot area on the wafer of one lot via the projection optical system. Thereafter, in step 304, the photoresist on the one lot of wafers is developed, and in step 305, the resist pattern is etched on the one lot of wafers to form a pattern on the mask. Corresponding circuit patterns are formed in each shot area on each wafer.

  Thereafter, a device pattern such as a semiconductor element is manufactured by forming a circuit pattern of an upper layer. According to the semiconductor device manufacturing method described above, a semiconductor device having an extremely fine circuit pattern can be obtained with high throughput. In steps 301 to 305, a metal is deposited on the wafer, a resist is applied on the metal film, and exposure, development, and etching processes are performed. Prior to these processes, on the wafer. It is needless to say that after forming a silicon oxide film, a resist may be applied on the silicon oxide film, and steps such as exposure, development, and etching may be performed.

  In the exposure apparatus of this embodiment, a liquid crystal display element as a micro device can be obtained by forming a predetermined pattern (circuit pattern, electrode pattern, etc.) on a plate (glass substrate). Hereinafter, an example of the technique at this time will be described with reference to the flowchart of FIG. In FIG. 5, in a pattern forming process 401, a so-called photolithography process is performed in which a mask pattern is transferred and exposed to a photosensitive substrate (such as a glass substrate coated with a resist) using the exposure apparatus of the present embodiment. By this photolithography process, a predetermined pattern including a large number of electrodes and the like is formed on the photosensitive substrate. Thereafter, the exposed substrate undergoes steps such as a developing step, an etching step, and a resist stripping step, whereby a predetermined pattern is formed on the substrate, and the process proceeds to the next color filter forming step 402.

  Next, in the color filter forming step 402, a large number of sets of three dots corresponding to R (Red), G (Green), and B (Blue) are arranged in a matrix or three of R, G, and B A color filter is formed by arranging a plurality of stripe filter sets in the horizontal scanning line direction. Then, after the color filter forming step 402, a cell assembly step 403 is executed. In the cell assembly step 403, a liquid crystal panel (liquid crystal cell) is assembled using the substrate having the predetermined pattern obtained in the pattern formation step 401, the color filter obtained in the color filter formation step 402, and the like. In the cell assembly step 403, for example, liquid crystal is injected between the substrate having the predetermined pattern obtained in the pattern formation step 401 and the color filter obtained in the color filter formation step 402, and a liquid crystal panel (liquid crystal cell) is obtained. ).

  Thereafter, in a module assembling step 404, components such as an electric circuit and a backlight for performing a display operation of the assembled liquid crystal panel (liquid crystal cell) are attached to complete a liquid crystal display element. According to the above-described method for manufacturing a liquid crystal display element, a liquid crystal display element having an extremely fine circuit pattern can be obtained with high throughput.

  In the above-described embodiment, the present invention is applied to the projection optical system mounted on the exposure apparatus. However, the present invention is not limited to this, and the present invention is applied to other general imaging optical systems. The invention can also be applied. In the above-described embodiment, the present invention is applied to the projection optical system mounted on the so-called scan exposure type exposure apparatus. However, the present invention is not limited to this, and is mounted on the collective exposure type exposure apparatus. The present invention can also be applied to the projected optical system.

Furthermore, in the above-described embodiment, an F ₂ laser light source that supplies light having a wavelength of 157 nm is used. However, the present invention is not limited to this. For example, a KrF excimer laser light source that supplies light having a wavelength of 248 nm, Deep ultraviolet light source such as ArF excimer laser light source that supplies light, Kr ₂ laser light source that supplies light with a wavelength of 146 nm, vacuum ultraviolet light source such as Ar ₂ laser light source that supplies light with a wavelength of 126 nm, g-line (436 nm), A mercury lamp or the like that supplies i-line (365 nm) can also be used.

It is a figure which shows schematically the structure of the exposure apparatus provided with the projection optical system which applies the evaluation method of the imaging optical system concerning embodiment of this invention. It is a figure explaining the image plane coordinate and pupil coordinate of projection optical system PL. It is a flowchart which shows the process of the evaluation method and adjustment method of projection optical system PL in this embodiment. It is a flowchart of the method at the time of obtaining the semiconductor device as a microdevice. It is a flowchart of the method at the time of obtaining the liquid crystal display element as a microdevice.

Explanation of symbols

LS Light source IL Illumination optical system R Reticle RS Reticle stage PL Projection optical system W Wafer WS Wafer stage

Claims (16)

In a method for evaluating aberration of an imaging optical system using an aberration polynomial,
Setting the aberration polynomial using only orthonormal function series as a function of image plane coordinates and pupil coordinates of the imaging optical system;
An aberration acquisition step of obtaining wavefront aberration of the imaging optical system for a plurality of points on the image plane of the imaging optical system;
An approximation step of approximating the wavefront aberration obtained in the aberration acquisition step with a predetermined polynomial as a function of pupil coordinates;
And a determining step of determining a coefficient of each term of the aberration polynomial based on a coefficient of each term in the predetermined polynomial obtained in the approximating step. 2. The evaluation method according to claim 1, wherein the setting step includes a step of expressing the orthonormal function sequence by a Zernike function in pupil coordinates and a Zernike function in image plane coordinates. The evaluation method according to claim 1, wherein the predetermined polynomial includes a Zernike polynomial. The aberration polynomial is set so as to include at least one aberration component of a rotationally symmetric aberration component, an eccentric aberration component, an as (toric) aberration component, and a trefoil aberration component with respect to the optical axis of the imaging optical system. The evaluation method according to any one of claims 1 to 3, wherein: 5. The evaluation method according to claim 4, wherein the aberration polynomial is set so as to include at least one aberration component among the decentering aberration component, the as (toric) aberration component, and the trefoil aberration component. . 6. The evaluation method according to claim 4, wherein the rotationally symmetric aberration component is expressed as an invariant power series with respect to rotation in the image plane coordinates and the pupil coordinates. The evaluation according to claim 6, wherein the decentration aberration component is expressed as a product of a first-order dependent component of the coordinates in the image plane coordinates and the pupil coordinates and a power series of invariants with respect to the rotation. Method. The astigmatic aberration component is a quadratic dependent component of the coordinates in the image plane coordinates and the pupil coordinates and a periodic function of 180 degrees with respect to the rotation of the coordinates, and an invariable power series with respect to the rotation The evaluation method according to claim 6, further comprising: The trefoil aberration component is a product of a third-order dependent component of the coordinates in the image plane coordinates and the pupil coordinates and a periodic function of 120 degrees with respect to the rotation of the coordinates and a power series of invariants with respect to the rotation. The evaluation method according to claim 6, further comprising: The determining step includes a second approximating step of approximating the in-image distribution of the coefficient of each term in the predetermined polynomial obtained in the approximating step with a predetermined polynomial as a function of the image plane coordinates. The evaluation method according to any one of claims 1 to 9, wherein the evaluation method is characterized. The evaluation method according to claim 1, wherein the aberration acquisition step includes a step of measuring a wavefront aberration of the imaging optical system. The evaluation method according to claim 1, wherein the aberration acquisition step includes a step of calculating a wavefront aberration of the imaging optical system by ray tracing. In a method for evaluating the aberration of the imaging optical system using an aberration polynomial, based on the wavefront aberration obtained for a plurality of points on the image plane of the imaging optical system,
Setting the aberration polynomial using only orthonormal function series as a function of image plane coordinates and pupil coordinates of the imaging optical system;
An approximation step of approximating the obtained wavefront aberration with a predetermined polynomial as a function of pupil coordinates;
And a determining step of determining a coefficient of each term of the aberration polynomial based on a coefficient of each term in the predetermined polynomial obtained in the approximating step. An adjustment method comprising: optically adjusting the imaging optical system based on aberration information of the imaging optical system obtained by the evaluation method according to claim 1. 15. An exposure apparatus comprising an imaging optical system optically adjusted by the adjustment method according to claim 14 as a projection optical system for projecting and exposing a mask pattern onto a photosensitive substrate. An exposure method comprising: projecting and exposing a pattern image formed on a mask onto a photosensitive substrate using an imaging optical system optically adjusted by the adjustment method according to claim 14.

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