Optimization and matching of optical systems by use of orientation Zernike polynomials
View Patent ↗The present disclosure relates to specification, optimization and matching of optical systems by use of orientation Zernike polynomials. In some embodiments, a method for assessing the suitability of an optical system of a microlithographic projection exposure apparatus is provided. The method can include determining a Jones pupil of the optical system, at least approximately describing the Jones pupil using an expansion into orientation Zernike polynomials, and assessing the suitability of the optical system on the basis of the expansion coefficient of at least one of the orientation Zernike polynomials in the expansion.
1. A method, comprising:
at least approximately describing a Jones pupil of a microlithographic projection exposure apparatus using an expansion into orientation Zernike polynomials;
assessing the suitability of the microlithographic projection exposure apparatus on the basis of an expansion coefficient of at least one of the orientation Zernike polynomials in the expansion; and
modifying the microlithographic projection exposure apparatus based on the assessment,
wherein the orientation Zernike polynomials are defined as
{right arrow over (W)} nmε =R n |m| Φ mε ;
wherein R n m are radial polynomials given by
R
n
m
(
r
)
=
∑
s
=
0
(
n
-
m
)
/
2
(
-
1
)
s
(
n
-
s
)
!
s
!
(
1
2
(
n
+
m
)
-
s
)
!
(
1
2
(
n
-
m
)
-
s
)
!
r
n
-
2
s
with m, n, s being integers, m=−n, . . . , n and ε=0 or 1; and
wherein Φ mε is given by
Φ
→
m
0
=
(
cos
m
φ
2
-
sin
m
φ
2
)
,
Φ
→
m
1
=
(
sin
m
φ
2
cos
m
φ
2
)
or a linear combination thereof.
2. The method according to claim 1 , wherein the optical system is considered within tolerance if the expansion coefficient of the at least one of the orientation Zernike polynomials is less than a threshold value, and the optical system is considered not within tolerance if the expansion coefficient of the at least one of the orientation Zernike polynomials is not less than the threshold value.
3. The method according to claim 1 , wherein the optical system is rated as being within tolerance only if the expansion coefficient of the at least one of the orientation Zernike polynomials between a center and an edge of an illumination field is less than a threshold value.
4. The method according to claim 1 , further comprising:
determining a sensitivity function, which describes, for at least for some of the orientation Zernike polynomials, an impact of the respective orientation Zernike polynomial on a lithography parameter; and
assessing the suitability of the optical system using the sensitivity function.
5. The method according to claim 4 , wherein the lithography parameter comprises a parameter selected from the group consisting of CD deviation, image placement errors and deviation between horizontal and vertical structures.
6. The method according to claim 4 , wherein the sensitivity function is multiplied with the expansion.
7. The method according to claim 1 , wherein the microlithographic projection exposure apparatus comprises at least one layer selected from the group consisting of anti-reflective layers and high-reflective layers, and a thickness and/or material of the at least one layer is modified in dependence of the assessment.
8. The method according to claim 1 , wherein, during assessing, the expansion coefficients of only a subgroup of orientation Zernike polynomials are considered, and the number of orientation Zernike polynomials in the subgroup does not exceed 25.
9. The method according to claim 1 , wherein, during assessing, only a subgroup of orientation Zernike polynomials are considered, and the order of orientation Zerknike polynomials in the subgroup does not exceed 20.
10. The method according to claim 1 , wherein the optical system is a projection lens of the microlithographic projection exposure apparatus.
11. The method according to claim 1 , wherein the optical system is a single optical element of the microlithographic projection exposure apparatus, or a group of elements of a projection lens of the microlithographic projection exposure apparatus.
12. A method, comprising:
at least approximately describing, for a microlithographic projection exposure apparatus comprising a plurality of optical systems, respective Jones pupils of at least two of the optical systems using an expansion into orientation Zernike polynomials; and
modifying at least one of the at least two optical systems such that a difference between an expansion coefficient of at least one of the orientation Zernike polynomials in the expansions for the at least two optical systems is reduced,
wherein the orientation Zernike polynomials are defined as
{right arrow over (W)} nmε =R n |m| Φ mε ;
wherein R n m are radial polynomials given by
R
n
m
(
r
)
=
∑
s
=
0
(
n
-
m
)
/
2
(
-
1
)
s
(
n
-
s
)
!
s
!
(
1
2
(
n
+
m
)
-
s
)
!
(
1
2
(
n
-
m
)
-
s
)
!
r
n
-
2
s
with m, n, s being integers, m=−n, . . . , n and ε=0 or 1; and
wherein Φ mε is given by
Φ
→
m
0
=
(
cos
m
φ
2
-
sin
m
φ
2
)
,
Φ
→
m
1
=
(
sin
m
φ
2
cos
m
φ
2
)
or a linear combination thereof.
13. A method, comprising:
at least approximately describing a Jones pupil for an initial design of an optical system of a microlithographic projection exposure apparatus using an expansion into orientation Zernike polynomials;
establishing a quality function that describes a variation in a lithography parameter, and which incorporates an expansion coefficient of at least one of the orientation Zernike polynomials in the expansion; and
modifying the initial design of the optical system such that the quality function is improved for the modified design with respect to the initial design,
wherein the orientation Zernike polynomials are defined as
{right arrow over (W)} nmε =R n |m| Φ mε ;
wherein R n m are radial polynomials given by
R
n
m
(
r
)
=
∑
s
=
0
(
n
-
m
)
/
2
(
-
1
)
s
(
n
-
s
)
!
s
!
(
1
2
(
n
+
m
)
-
s
)
!
(
1
2
(
n
-
m
)
-
s
)
!
r
n
-
2
s
with m, n, s being integers, m=−n, . . . , n and ε=0 or 1; and
wherein Φ mε is given by
Φ
→
m
0
=
(
cos
m
φ
2
-
sin
m
φ
2
)
,
Φ
→
m
1
=
(
sin
m
φ
2
cos
m
φ
2
)
or a linear combination thereof.
14. The method according to claim 13 , wherein the optical system comprises at least one layer selected from the group consisting of an anti-reflective and a high-reflective layer, and designing comprises varying the thickness and/or the material of at least one of the layers.
15. The method according to claim 12 , wherein a number of orientation Zernike polynomials in the expansion does not exceed 25.
16. The method according to claim 12 , wherein each of the at least two optical systems is a single optical element of the microlithographic projection exposure apparatus, or a group of elements of a projection lens of the microlithographic projection exposure apparatus.
17. The method according to claim 13 , wherein the lithography parameter comprises a parameter selected from the group consisting of CD deviation, image placement errors and deviation between horizontal and vertical structures.
18. The method according to claim 13 , wherein the optical system is a projection lens of the microlithographic projection exposure apparatus.
19. The method according to claim 13 , wherein the optical system is a single optical element of the microlithographic projection exposure apparatus, or a group of elements of a projection lens of the microlithographic projection exposure apparatus.
20. The method according to claim 13 , wherein a number of orientation Zernike polynomials in the expansion does not exceed 25.