Abstract

We extend the method for the automatic computation of high-order optical aberration coefficients to include (1) a finite object distance and (2) an infinite entrance pupil position (telecentricity in object space). We present coefficients of the power series expansion of the transverse aberration vector with respect to the normalized aperture and field coordinates. Aberration coefficients of very high order (e.g., 21) can be computed easily and—as shown by comparisons with trigonometric ray tracing—reliably.

© 2008 Optical Society of America

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References

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  1. L. Seidel, “Zur Dioptrik. Über die Entwicklung der Glieder dritter Ordnung,” Astron. Nachr. 43, 289-322 (1856).
    [CrossRef]
  2. M. Berek, Grundlagen der Praktischen Optik (Walter de Gruyter, 1930).
  3. F. Wachendorf, “Bestimmung der Bildfehler fünfter Ordnung in zentrierten optischen Systemen,” Optik (Jena) 5, 80-122(1949).
  4. H. A. Buchdahl, Optical Aberration Coefficients (Dover, 1968).
  5. F. Bociort, “Computer algebra derivation of high-order optical aberration coefficients,” Technical Rep. 7 (Riaca, 1995), also available at http://www.optica.tn.tudelft.nl/users/bociort/riaca.pdf
  6. T. B. Andersen, “Automatic computation of optical aberration coefficients,” Appl. Opt. 19, 3800-3816 (1980).
    [CrossRef] [PubMed]
  7. T. B. Andersen, “Optical aberration coefficients: FORTRAN subroutines for symmetrical systems,” Appl. Opt. 20, 3263-3268 (1981).
    [CrossRef] [PubMed]
  8. W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).
  9. O. Marinescu and F. Bociort,” Optimization of extreme ultraviolet mirror systems comprising high-order aspheric surfaces,” Opt. Eng. 47, 033004 (2008).
    [CrossRef]
  10. D. Shafer, “Optical design and the relaxation response,” Proc. SPIE 0766, 2-9 (1987).
  11. T. Sasaya, K. Ushida, Y. Suenaga, and R. I. Mercado, “Projection optical system and projection exposure apparatus,” U.S. patent 5,805,344 (8 September 1998).
  12. J. B. Caldwell, “All-fused silica 248 nm lithographic projection lens,” Opt. Photon. News 9, 40-41 (1998).
    [CrossRef]
  13. D. C. Sinclair, “Optical design software,” in Handbook of Optics, Vol. 1 of Fundamentals, Techniques, and Design, M. Bass, ed. (McGraw-Hill, 1995), Chap. 34.
  14. F. Bociort, “Aberration balance in error functions calculated analytically,” Proc. SPIE 3482, 32-42 (1998).
    [CrossRef]

2008

O. Marinescu and F. Bociort,” Optimization of extreme ultraviolet mirror systems comprising high-order aspheric surfaces,” Opt. Eng. 47, 033004 (2008).
[CrossRef]

1998

J. B. Caldwell, “All-fused silica 248 nm lithographic projection lens,” Opt. Photon. News 9, 40-41 (1998).
[CrossRef]

F. Bociort, “Aberration balance in error functions calculated analytically,” Proc. SPIE 3482, 32-42 (1998).
[CrossRef]

1987

D. Shafer, “Optical design and the relaxation response,” Proc. SPIE 0766, 2-9 (1987).

1981

1980

1949

F. Wachendorf, “Bestimmung der Bildfehler fünfter Ordnung in zentrierten optischen Systemen,” Optik (Jena) 5, 80-122(1949).

1856

L. Seidel, “Zur Dioptrik. Über die Entwicklung der Glieder dritter Ordnung,” Astron. Nachr. 43, 289-322 (1856).
[CrossRef]

Andersen, T. B.

Berek, M.

M. Berek, Grundlagen der Praktischen Optik (Walter de Gruyter, 1930).

Bociort, F.

O. Marinescu and F. Bociort,” Optimization of extreme ultraviolet mirror systems comprising high-order aspheric surfaces,” Opt. Eng. 47, 033004 (2008).
[CrossRef]

F. Bociort, “Aberration balance in error functions calculated analytically,” Proc. SPIE 3482, 32-42 (1998).
[CrossRef]

F. Bociort, “Computer algebra derivation of high-order optical aberration coefficients,” Technical Rep. 7 (Riaca, 1995), also available at http://www.optica.tn.tudelft.nl/users/bociort/riaca.pdf

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, 1968).

Caldwell, J. B.

J. B. Caldwell, “All-fused silica 248 nm lithographic projection lens,” Opt. Photon. News 9, 40-41 (1998).
[CrossRef]

Marinescu, O.

O. Marinescu and F. Bociort,” Optimization of extreme ultraviolet mirror systems comprising high-order aspheric surfaces,” Opt. Eng. 47, 033004 (2008).
[CrossRef]

Mercado, R. I.

T. Sasaya, K. Ushida, Y. Suenaga, and R. I. Mercado, “Projection optical system and projection exposure apparatus,” U.S. patent 5,805,344 (8 September 1998).

Sasaya, T.

T. Sasaya, K. Ushida, Y. Suenaga, and R. I. Mercado, “Projection optical system and projection exposure apparatus,” U.S. patent 5,805,344 (8 September 1998).

Seidel, L.

L. Seidel, “Zur Dioptrik. Über die Entwicklung der Glieder dritter Ordnung,” Astron. Nachr. 43, 289-322 (1856).
[CrossRef]

Shafer, D.

D. Shafer, “Optical design and the relaxation response,” Proc. SPIE 0766, 2-9 (1987).

Sinclair, D. C.

D. C. Sinclair, “Optical design software,” in Handbook of Optics, Vol. 1 of Fundamentals, Techniques, and Design, M. Bass, ed. (McGraw-Hill, 1995), Chap. 34.

Suenaga, Y.

T. Sasaya, K. Ushida, Y. Suenaga, and R. I. Mercado, “Projection optical system and projection exposure apparatus,” U.S. patent 5,805,344 (8 September 1998).

Ushida, K.

T. Sasaya, K. Ushida, Y. Suenaga, and R. I. Mercado, “Projection optical system and projection exposure apparatus,” U.S. patent 5,805,344 (8 September 1998).

Wachendorf, F.

F. Wachendorf, “Bestimmung der Bildfehler fünfter Ordnung in zentrierten optischen Systemen,” Optik (Jena) 5, 80-122(1949).

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).

Appl. Opt.

Astron. Nachr.

L. Seidel, “Zur Dioptrik. Über die Entwicklung der Glieder dritter Ordnung,” Astron. Nachr. 43, 289-322 (1856).
[CrossRef]

Opt. Eng.

O. Marinescu and F. Bociort,” Optimization of extreme ultraviolet mirror systems comprising high-order aspheric surfaces,” Opt. Eng. 47, 033004 (2008).
[CrossRef]

Opt. Photon. News

J. B. Caldwell, “All-fused silica 248 nm lithographic projection lens,” Opt. Photon. News 9, 40-41 (1998).
[CrossRef]

Optik (Jena)

F. Wachendorf, “Bestimmung der Bildfehler fünfter Ordnung in zentrierten optischen Systemen,” Optik (Jena) 5, 80-122(1949).

Proc. SPIE

D. Shafer, “Optical design and the relaxation response,” Proc. SPIE 0766, 2-9 (1987).

F. Bociort, “Aberration balance in error functions calculated analytically,” Proc. SPIE 3482, 32-42 (1998).
[CrossRef]

Other

D. C. Sinclair, “Optical design software,” in Handbook of Optics, Vol. 1 of Fundamentals, Techniques, and Design, M. Bass, ed. (McGraw-Hill, 1995), Chap. 34.

T. Sasaya, K. Ushida, Y. Suenaga, and R. I. Mercado, “Projection optical system and projection exposure apparatus,” U.S. patent 5,805,344 (8 September 1998).

M. Berek, Grundlagen der Praktischen Optik (Walter de Gruyter, 1930).

H. A. Buchdahl, Optical Aberration Coefficients (Dover, 1968).

F. Bociort, “Computer algebra derivation of high-order optical aberration coefficients,” Technical Rep. 7 (Riaca, 1995), also available at http://www.optica.tn.tudelft.nl/users/bociort/riaca.pdf

W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986).

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Figures (7)

Fig. 1
Fig. 1

Three-lens system for 10 1 imaging of a mask for precision machining by an excimer laser ( 248 nm ). The application requires (1) a long backfocus to avoid lens contamination by debris and (2) color correction for 248 nm and 628 nm , the latter to allow alignment and focussing with a He–Ne laser. The positive lenses are made from Ca F 2 , the center negative lens from S i O 2 . Note the strong curvatures and short air spaces, which make the lens difficult to manufacture.

Fig. 2
Fig. 2

Aberration contributions up to order 11 for the system of Fig. 1. Spherical aberration terms (bS2, bS5, bS21, bS36) are dominant, the presence of high orders indicating a stressed design. Coma terms (bS4, bT2, bS7, bS9) are small and mutually compensating, third order field curvature (bS3, bT4) limits the range of application to a flat field of about 2 mm diameter.

Fig. 3
Fig. 3

Five-lens system for 4 1 imaging of a mask for precision machining by an excimer laser ( 248 nm ) with very large working distance and a flat field in excess of 8 mm diameter. All lens elements are made from S i O 2 , so there is no color correction.

Fig. 4
Fig. 4

Aberration contributions up to order 11 for the system of Fig. 3. Spherical aberration terms are small; the third order contribution is essentially compensated by fifth and a little seventh order, and coma terms (bT2, bS4, bS7) are all small. Field curvature (bS3, bT4) ultimately limits the applicability to a 8 mm flat field. There are virtually no aberration terms of order 7 or higher, indicating a relaxed design due to shallow curvatures, albeit at the expense of five lens elements. Note the higher scale factor compared to Fig. 2.

Fig. 5
Fig. 5

Lithographic lens for 4 1 imaging at 248 nm with a large field ( 23 mm diameter) and a high numerical aperture (0.56), taken from [12]. The system is telecentric in both object and image space.

Fig. 6
Fig. 6

Aberration contributions up to order 11 for the system of Fig. 5. Spherical aberration terms (bS2, bS5, bS21, bS36) are remarkably small with the exception of the term of seventh order (bS11), but oblique spherical terms, notably bS16, and to a lesser extent bT13 and bS41, account for a rapid increase of the marginal high oblique ray aberration at full field. Note the much higher scale factor compared to Figs. 2, 4.

Fig. 7
Fig. 7

Transverse aberration plot for the lithographic system shown in Fig. 5, showing the effect of oblique spherical aberration.

Tables (8)

Tables Icon

Table 1 Relationships Between the Third and the Fifth Order Aberration Coefficients Used in [13] and Those Used in the Present Paper a

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Table 2 Relationships Between the Third and Fifth Order Aberration Coefficients Used by Buchdahl [4] and Those Used in the Present Paper

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Table 3 Set of Four Indices n, i, j, and k Appearing in b s , n i j k and b t , n i j k a

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Table 4 Lens Data for the 10 1 Demagnifying System of Fig. 1 a

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Table 5 Lens Data for the 4 1 Demagnifying System of Fig. 3 a

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Table 6 Reconstruction for the System in Fig. 1 of the x and y Components of the Transverse Aberration of a Given Ray a

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Table 7 Reconstruction of Transverse Aberration for the System Shown in Fig. 3

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Table 8 Reconstruction of the Transverse Aberration for the System Shown in Fig. 5.

Equations (30)

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R ˜ 2 K + 1 ( R , D ) = k = 0 K j = 0 k n = 0 j ( S k , k j , j n , n R + T k , k j , j n , n D ) ( R 2 ) k j ( D 2 ) j n ( RD ) n .
R 2 = x 2 + y 2 , D 2 = D x 2 + D y 2 , RD = x D x + y D y
R = r s ,
D = u ¯ t ,
R ¯ = r ¯ t ,
D = u s + u ¯ t ,
R ˜ 2 K + 1 ( s , t ) = k = 0 K j = 0 k n = 0 j ( b s , k , k j , j n , n s + b t , k , k j , j n , n t ) ( s 2 ) k j ( t 2 ) j n ( st ) n ,
s 2 = s x 2 + s y 2 , t 2 = t x 2 + t y 2 , st = s x t x + s y t y .
b s , k , k j , j n , n = r 2 ( k j ) + n + 1 u ¯ 2 j n S k , k j , j n , n ,
b t , k , k j , j n , n = r 2 ( k j ) + n u ¯ 2 j n + 1 T k , k j , j n , n .
E k ( J ) = j = 0 k n = 0 j J k , k j , j n , n ( R 2 ) k j ( D 2 ) j n ( RD ) n ,
R ˜ 2 K + 1 ( R , D ) = k = 0 K ( E k ( S ) R + E k ( T ) D ) .
E k ( J ) = j = 0 k n = 0 j J k , k j , j n , n ( s 2 ) k j ( t 2 ) j n ( st ) n ,
b s , k , k j , j n , n = r S k , k j , j n , n + u T k , k j , j n , n ,
b t , k , k j , j n , n = u ¯ T k , k j , j n , n .
R 2 = r 2 s 2 , D 2 = u 2 s 2 + 2 u u ¯ st + u ¯ 2 t 2 , RD = r ( u s 2 + u ¯ st ) .
( R 2 ) k j ( D 2 ) j n ( RD ) n = p = 0 n p 1 = 0 j n p 2 = 0 j n p 1 C ( r , u , u ¯ ) ( s 2 ) e s 2 ( t 2 ) e t 2 ( st ) e s t ,
e s 2 = k j + p + p 1 , e st = n p + p 2 , e t 2 = j n p 1 p 2 ,
C ( r , u , u ¯ ) = 2 p 2 n ! p ! ( n p ) ! ( j n ) ! p 1 ! p 2 ! ( j n p 1 p 2 ) ! r e 1 u e 2 u ¯ e 3 ,
e 1 = 2 ( k j ) + n , e 2 = p + 2 p 1 + p 2 , e 3 = n p + p 2 + 2 ( j n p 1 p 2 ) .
p 1 = j j p , p 2 = n n + p .
E k ( J ) = j = 0 k n = 0 j p = 0 n j = n p j p n = n p j C ( r , u , u ¯ ) J k , k j , j n , n ( s 2 ) k j ( t 2 ) j n ( st ) n .
J k , k j , j n , n = u ¯ 2 j n j = j k n = 0 n + j j a k j n j n ( r , u ) J k , k j , j n , n ,
a k j n j n ( α , β ) = b j n j n α 2 k ( 2 j n ) β ( 2 j n ) ( 2 j n ) .
b j n j n = p = p min p max 2 p n + n n ! p ! ( n p ) ! ( j n ) ! ( j n ) ! ( j j p ) ! ( p n + n ) ! ,
p min = max ( n n , 0 ) , p max = min ( j j , n ) .
E ¯ k ( J ¯ ) = j = 0 k n = 0 j J ¯ k , k j , j n , n ( t 2 ) k j ( s 2 ) j n ( st ) n .
J ¯ k , k j , j n , n = u 2 j n j = j k n = 0 n + j j a k j n j n ( r ¯ , u ¯ ) J ¯ k , k j , j n , n ,
b t , k , k j , j n , n = r ¯ S ¯ k , j n , k j , n + u ¯ T ¯ k , j n , k j , n ,
b s , k , k j , j n , n = u T ¯ k , j n , k j , n .

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