Abstract

Extreme ultraviolet (EUV) lithography uses reflective ring-field projection systems. Geometrical obstruction limits the possible system configurations to small domains of the parameter space. We present an analysis, a search method, and a classification of these unobstructed domains. The exhaustive search method based on paraxial analysis provides an effective means for determining all possible design forms and for finding useful starting configurations for optimization. The approach is validated through comparison with finite ray tracing.

© 2003 Optical Society of America

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  1. U. Dinger, “Ringfeld-4-Spiegelsysteme mit konvexem Primarspiegel für die EUV-Lithography,” European patentEP 0 962 830 A1 (8December1999).
  2. J. Braat, “Mirror projection system for a scanning lithographic projection apparatus, and lithographic apparatus comprising such a system,” U.S. patent6,299,318 (9October2001).
  3. J. Braat, “Lithographic apparatus comprising a dedicated mirror projection system,” U.S. patent6,396,067 (28May2002).
  4. J. Braat, J. Verhoeven, “Method of imaging a mask pattern on a substrate by means of euv radiation, and apparatus and mask for performing the method,” U.S. patent6,280,906 (28August2001).
  5. J. Bruning, A. Phillips, D. Shafer, A. White, “X-ray projection lithography camera,” U.S. patent5,220,590 (15June1993).
  6. J. Bruning, A. Phillips, D. Shafer, A. White, “Lens system for X-ray projection lithography camera,” U.S. patent5,353,322 (4October1994).
  7. R. Hudyma, “High numerical aperture ring field projection system for extreme ultraviolet lithography,” U.S. patent6,033,079 (7March2000).
  8. R. Hudyma, “High numerical aperture ring field projection system for extreme ultraviolet lithography,” U.S. patent6,183,095 (6February2001).
  9. R. Hudyma, D. Shafer, “High numerical aperture ring field projection system for extreme ultraviolet lithography,” U.S. patent6,188,513 (2February2001).
  10. T. Jewell, J. Rodgers, “Apparatus for semiconductor lithography,” U.S. patent5,063,586 (5November1991).
  11. T. Jewell, K. Tompson, “X-ray ringfield lithography,” U.S. patent5,315,629 (24May1994).
  12. D. Shafer, “Reflective projection system comprising four spherical mirrors,” U.S. patent5,410,434 (25April1995).
  13. D. Shafer, “Projection lithography system and method using all-reflective optical elements,” U.S. patent5,686,728 (11November1997).
  14. M. Suzuki, N. Mochizuki, S. Minami, S. Ogura, Y. Fukuda, Y. Watanabe, Y. Kawai, T. Kariya, “X-ray reduction projection exposure system of reflection type,” U.S. patent5,153,898 (10October1992).
  15. V. Viswanathan, B. Newnam, “Reflective optical imaging system for extreme ultraviolet wavelengths,” U.S. patent5,212,588 (18May1993).
  16. D. Williamson, “High numerical aperture ring field optical reduction system,” U.S. patent5,815,310 (29September1998).
  17. D. Williamson, “Four mirror EUV projection optics,” U.S. patent5,956,192 (21September1999).
  18. S. A. Lerner, J. M. Sasian, M. R. Descour, “Design approach and comparison of projection cameras for EUV lithography,” Opt. Eng. 39, 792–802 (2000).
    [CrossRef]
  19. J. M. Howard, B. D. Stone, “Imaging a point with two spherical mirrors,” J. Opt. Soc. Am. A 15, 3045–3056 (1998).
    [CrossRef]
  20. J. M. Howard, B. D. Stone, “Imaging a point to a line with a single spherical mirror,” Appl. Opt. 37, 1826–1834 (1998).
    [CrossRef]
  21. J. M. Howard, B. D. Stone, “Imaging with three spherical mirrors,” Appl. Opt. 39, 3216–3231 (2000).
    [CrossRef]
  22. J. M. Howard, B. D. Stone, “Imaging with four spherical mirrors,” Appl. Opt. 39, 3232–3242 (2000).
    [CrossRef]
  23. F. Pedrotti, L. Pedrotti, Introduction to Optics (Prentice-Hall, Englewood Cliffs, N.J., 1993).
  24. code v, Optical Research Associates, Pasadena, Calif., 2001.
  25. oslo, Lambda Research Corporation, Littleton, Mass., 2001.
  26. M. Bal, F. Bociort, J. Braat, “Influence of multilayers on the optical performance of extreme-ultraviolet projection systems,” in International Optical Design Conference 2002, P. K. Manhart, J. M. Sasian, eds., Proc. SPIE4832, 149–157 (2002).
    [CrossRef]
  27. H.-J. Mann, W. Ulrich, G. Seitz, “8-mirrored microlithographic projector lens,” World Intellectual Property Organization patentWO 02/33467A1 (25April2002).
  28. M. Bal, F. Bociort, J. Braat, “Lithographic apparatus and device manufacturing method,” European patentEP 1 20 95 03 A2 (29May2002).

2000

1998

Bal, M.

M. Bal, F. Bociort, J. Braat, “Influence of multilayers on the optical performance of extreme-ultraviolet projection systems,” in International Optical Design Conference 2002, P. K. Manhart, J. M. Sasian, eds., Proc. SPIE4832, 149–157 (2002).
[CrossRef]

M. Bal, F. Bociort, J. Braat, “Lithographic apparatus and device manufacturing method,” European patentEP 1 20 95 03 A2 (29May2002).

Bociort, F.

M. Bal, F. Bociort, J. Braat, “Lithographic apparatus and device manufacturing method,” European patentEP 1 20 95 03 A2 (29May2002).

M. Bal, F. Bociort, J. Braat, “Influence of multilayers on the optical performance of extreme-ultraviolet projection systems,” in International Optical Design Conference 2002, P. K. Manhart, J. M. Sasian, eds., Proc. SPIE4832, 149–157 (2002).
[CrossRef]

Braat, J.

M. Bal, F. Bociort, J. Braat, “Influence of multilayers on the optical performance of extreme-ultraviolet projection systems,” in International Optical Design Conference 2002, P. K. Manhart, J. M. Sasian, eds., Proc. SPIE4832, 149–157 (2002).
[CrossRef]

M. Bal, F. Bociort, J. Braat, “Lithographic apparatus and device manufacturing method,” European patentEP 1 20 95 03 A2 (29May2002).

J. Braat, “Mirror projection system for a scanning lithographic projection apparatus, and lithographic apparatus comprising such a system,” U.S. patent6,299,318 (9October2001).

J. Braat, “Lithographic apparatus comprising a dedicated mirror projection system,” U.S. patent6,396,067 (28May2002).

J. Braat, J. Verhoeven, “Method of imaging a mask pattern on a substrate by means of euv radiation, and apparatus and mask for performing the method,” U.S. patent6,280,906 (28August2001).

Bruning, J.

J. Bruning, A. Phillips, D. Shafer, A. White, “X-ray projection lithography camera,” U.S. patent5,220,590 (15June1993).

J. Bruning, A. Phillips, D. Shafer, A. White, “Lens system for X-ray projection lithography camera,” U.S. patent5,353,322 (4October1994).

Descour, M. R.

S. A. Lerner, J. M. Sasian, M. R. Descour, “Design approach and comparison of projection cameras for EUV lithography,” Opt. Eng. 39, 792–802 (2000).
[CrossRef]

Dinger, U.

U. Dinger, “Ringfeld-4-Spiegelsysteme mit konvexem Primarspiegel für die EUV-Lithography,” European patentEP 0 962 830 A1 (8December1999).

Fukuda, Y.

M. Suzuki, N. Mochizuki, S. Minami, S. Ogura, Y. Fukuda, Y. Watanabe, Y. Kawai, T. Kariya, “X-ray reduction projection exposure system of reflection type,” U.S. patent5,153,898 (10October1992).

Howard, J. M.

Hudyma, R.

R. Hudyma, “High numerical aperture ring field projection system for extreme ultraviolet lithography,” U.S. patent6,033,079 (7March2000).

R. Hudyma, “High numerical aperture ring field projection system for extreme ultraviolet lithography,” U.S. patent6,183,095 (6February2001).

R. Hudyma, D. Shafer, “High numerical aperture ring field projection system for extreme ultraviolet lithography,” U.S. patent6,188,513 (2February2001).

Jewell, T.

T. Jewell, J. Rodgers, “Apparatus for semiconductor lithography,” U.S. patent5,063,586 (5November1991).

T. Jewell, K. Tompson, “X-ray ringfield lithography,” U.S. patent5,315,629 (24May1994).

Kariya, T.

M. Suzuki, N. Mochizuki, S. Minami, S. Ogura, Y. Fukuda, Y. Watanabe, Y. Kawai, T. Kariya, “X-ray reduction projection exposure system of reflection type,” U.S. patent5,153,898 (10October1992).

Kawai, Y.

M. Suzuki, N. Mochizuki, S. Minami, S. Ogura, Y. Fukuda, Y. Watanabe, Y. Kawai, T. Kariya, “X-ray reduction projection exposure system of reflection type,” U.S. patent5,153,898 (10October1992).

Lerner, S. A.

S. A. Lerner, J. M. Sasian, M. R. Descour, “Design approach and comparison of projection cameras for EUV lithography,” Opt. Eng. 39, 792–802 (2000).
[CrossRef]

Mann, H.-J.

H.-J. Mann, W. Ulrich, G. Seitz, “8-mirrored microlithographic projector lens,” World Intellectual Property Organization patentWO 02/33467A1 (25April2002).

Minami, S.

M. Suzuki, N. Mochizuki, S. Minami, S. Ogura, Y. Fukuda, Y. Watanabe, Y. Kawai, T. Kariya, “X-ray reduction projection exposure system of reflection type,” U.S. patent5,153,898 (10October1992).

Mochizuki, N.

M. Suzuki, N. Mochizuki, S. Minami, S. Ogura, Y. Fukuda, Y. Watanabe, Y. Kawai, T. Kariya, “X-ray reduction projection exposure system of reflection type,” U.S. patent5,153,898 (10October1992).

Newnam, B.

V. Viswanathan, B. Newnam, “Reflective optical imaging system for extreme ultraviolet wavelengths,” U.S. patent5,212,588 (18May1993).

Ogura, S.

M. Suzuki, N. Mochizuki, S. Minami, S. Ogura, Y. Fukuda, Y. Watanabe, Y. Kawai, T. Kariya, “X-ray reduction projection exposure system of reflection type,” U.S. patent5,153,898 (10October1992).

Pedrotti, F.

F. Pedrotti, L. Pedrotti, Introduction to Optics (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Pedrotti, L.

F. Pedrotti, L. Pedrotti, Introduction to Optics (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Phillips, A.

J. Bruning, A. Phillips, D. Shafer, A. White, “Lens system for X-ray projection lithography camera,” U.S. patent5,353,322 (4October1994).

J. Bruning, A. Phillips, D. Shafer, A. White, “X-ray projection lithography camera,” U.S. patent5,220,590 (15June1993).

Rodgers, J.

T. Jewell, J. Rodgers, “Apparatus for semiconductor lithography,” U.S. patent5,063,586 (5November1991).

Sasian, J. M.

S. A. Lerner, J. M. Sasian, M. R. Descour, “Design approach and comparison of projection cameras for EUV lithography,” Opt. Eng. 39, 792–802 (2000).
[CrossRef]

Seitz, G.

H.-J. Mann, W. Ulrich, G. Seitz, “8-mirrored microlithographic projector lens,” World Intellectual Property Organization patentWO 02/33467A1 (25April2002).

Shafer, D.

D. Shafer, “Reflective projection system comprising four spherical mirrors,” U.S. patent5,410,434 (25April1995).

D. Shafer, “Projection lithography system and method using all-reflective optical elements,” U.S. patent5,686,728 (11November1997).

J. Bruning, A. Phillips, D. Shafer, A. White, “X-ray projection lithography camera,” U.S. patent5,220,590 (15June1993).

J. Bruning, A. Phillips, D. Shafer, A. White, “Lens system for X-ray projection lithography camera,” U.S. patent5,353,322 (4October1994).

R. Hudyma, D. Shafer, “High numerical aperture ring field projection system for extreme ultraviolet lithography,” U.S. patent6,188,513 (2February2001).

Stone, B. D.

Suzuki, M.

M. Suzuki, N. Mochizuki, S. Minami, S. Ogura, Y. Fukuda, Y. Watanabe, Y. Kawai, T. Kariya, “X-ray reduction projection exposure system of reflection type,” U.S. patent5,153,898 (10October1992).

Tompson, K.

T. Jewell, K. Tompson, “X-ray ringfield lithography,” U.S. patent5,315,629 (24May1994).

Ulrich, W.

H.-J. Mann, W. Ulrich, G. Seitz, “8-mirrored microlithographic projector lens,” World Intellectual Property Organization patentWO 02/33467A1 (25April2002).

Verhoeven, J.

J. Braat, J. Verhoeven, “Method of imaging a mask pattern on a substrate by means of euv radiation, and apparatus and mask for performing the method,” U.S. patent6,280,906 (28August2001).

Viswanathan, V.

V. Viswanathan, B. Newnam, “Reflective optical imaging system for extreme ultraviolet wavelengths,” U.S. patent5,212,588 (18May1993).

Watanabe, Y.

M. Suzuki, N. Mochizuki, S. Minami, S. Ogura, Y. Fukuda, Y. Watanabe, Y. Kawai, T. Kariya, “X-ray reduction projection exposure system of reflection type,” U.S. patent5,153,898 (10October1992).

White, A.

J. Bruning, A. Phillips, D. Shafer, A. White, “X-ray projection lithography camera,” U.S. patent5,220,590 (15June1993).

J. Bruning, A. Phillips, D. Shafer, A. White, “Lens system for X-ray projection lithography camera,” U.S. patent5,353,322 (4October1994).

Williamson, D.

D. Williamson, “High numerical aperture ring field optical reduction system,” U.S. patent5,815,310 (29September1998).

D. Williamson, “Four mirror EUV projection optics,” U.S. patent5,956,192 (21September1999).

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Eng.

S. A. Lerner, J. M. Sasian, M. R. Descour, “Design approach and comparison of projection cameras for EUV lithography,” Opt. Eng. 39, 792–802 (2000).
[CrossRef]

Other

F. Pedrotti, L. Pedrotti, Introduction to Optics (Prentice-Hall, Englewood Cliffs, N.J., 1993).

code v, Optical Research Associates, Pasadena, Calif., 2001.

oslo, Lambda Research Corporation, Littleton, Mass., 2001.

M. Bal, F. Bociort, J. Braat, “Influence of multilayers on the optical performance of extreme-ultraviolet projection systems,” in International Optical Design Conference 2002, P. K. Manhart, J. M. Sasian, eds., Proc. SPIE4832, 149–157 (2002).
[CrossRef]

H.-J. Mann, W. Ulrich, G. Seitz, “8-mirrored microlithographic projector lens,” World Intellectual Property Organization patentWO 02/33467A1 (25April2002).

M. Bal, F. Bociort, J. Braat, “Lithographic apparatus and device manufacturing method,” European patentEP 1 20 95 03 A2 (29May2002).

U. Dinger, “Ringfeld-4-Spiegelsysteme mit konvexem Primarspiegel für die EUV-Lithography,” European patentEP 0 962 830 A1 (8December1999).

J. Braat, “Mirror projection system for a scanning lithographic projection apparatus, and lithographic apparatus comprising such a system,” U.S. patent6,299,318 (9October2001).

J. Braat, “Lithographic apparatus comprising a dedicated mirror projection system,” U.S. patent6,396,067 (28May2002).

J. Braat, J. Verhoeven, “Method of imaging a mask pattern on a substrate by means of euv radiation, and apparatus and mask for performing the method,” U.S. patent6,280,906 (28August2001).

J. Bruning, A. Phillips, D. Shafer, A. White, “X-ray projection lithography camera,” U.S. patent5,220,590 (15June1993).

J. Bruning, A. Phillips, D. Shafer, A. White, “Lens system for X-ray projection lithography camera,” U.S. patent5,353,322 (4October1994).

R. Hudyma, “High numerical aperture ring field projection system for extreme ultraviolet lithography,” U.S. patent6,033,079 (7March2000).

R. Hudyma, “High numerical aperture ring field projection system for extreme ultraviolet lithography,” U.S. patent6,183,095 (6February2001).

R. Hudyma, D. Shafer, “High numerical aperture ring field projection system for extreme ultraviolet lithography,” U.S. patent6,188,513 (2February2001).

T. Jewell, J. Rodgers, “Apparatus for semiconductor lithography,” U.S. patent5,063,586 (5November1991).

T. Jewell, K. Tompson, “X-ray ringfield lithography,” U.S. patent5,315,629 (24May1994).

D. Shafer, “Reflective projection system comprising four spherical mirrors,” U.S. patent5,410,434 (25April1995).

D. Shafer, “Projection lithography system and method using all-reflective optical elements,” U.S. patent5,686,728 (11November1997).

M. Suzuki, N. Mochizuki, S. Minami, S. Ogura, Y. Fukuda, Y. Watanabe, Y. Kawai, T. Kariya, “X-ray reduction projection exposure system of reflection type,” U.S. patent5,153,898 (10October1992).

V. Viswanathan, B. Newnam, “Reflective optical imaging system for extreme ultraviolet wavelengths,” U.S. patent5,212,588 (18May1993).

D. Williamson, “High numerical aperture ring field optical reduction system,” U.S. patent5,815,310 (29September1998).

D. Williamson, “Four mirror EUV projection optics,” U.S. patent5,956,192 (21September1999).

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Figures (22)

Fig. 1
Fig. 1

When mirrors unintentionally intersect and block rays, the system is obstructed. In the figure, points P i,0, P i,1, P i+1,0, P i+1,1, P j,0, and P j,1 are the intersections of the extreme meridional rays with the surfaces. The thick lines are the reflective surfaces, and, in the gray points, the beam propagates from mirror i to mirror i + 1.

Fig. 2
Fig. 2

A mirror can cause obstruction in different ways. On the left, the common situation is shown in which only a part of the mirror obstructs the beam. In the middle situation, the whole beam is obstructed. On the right, a part of the mirror is obstructed, and the intersection point P j,1′ is fictitious.

Fig. 3
Fig. 3

In an EUV arrangement a reflective projection system images the mask on the wafer. The requirements include a sufficiently large free working space, quasi-telecentricity on the mask side, perfect telecentricity on the wafer side, and a fixed magnification.

Fig. 4
Fig. 4

Sign of the incidence angle of the principal ray on a surface i gives the contribution of each surface to the class number. If that incidence angle θ i is positive, the contribution is a i = 0; otherwise, a i = 1. The class number C of an N-mirror system is the decimal form of the binary number a 1 · a 2 · … a N-1 · a N .

Fig. 5
Fig. 5

Three eight-mirror systems belonging to the class 153+. Systems a and b are in the same obstruction-free domain, whereas system c is a separate obstruction-free domain. Systems a and b cannot be changed into system c without having obstruction. In systems a and b the beams to and from the third mirror (encircled) cross the beams to and from the sixth mirror (within square).

Fig. 6
Fig. 6

Classes found in a paraxial search for four-mirror systems with positive or negative magnification. The height of the bars corresponds to the volume of an unobstructed domain. The total number of systems evaluated at a specific magnification is 1005. From the positive classes, only class 9+ leads to practically usable systems. Systems in classes 9+, 2-, 6-, and 10- were encountered in patent publications.

Fig. 7
Fig. 7

Classes found in a paraxial search for six-mirror systems with positive magnification. The total number of evaluated systems is 208.

Fig. 8
Fig. 8

Classes found in a paraxial search for six-mirror systems with negative magnification. The total number of evaluated systems is 308.

Fig. 9
Fig. 9

Paraxial results of a search for eight-mirror systems with negative magnification.

Fig. 10
Fig. 10

Paraxial results of a search for eight-mirror systems with positive magnification.

Fig. 11
Fig. 11

Analytical exploration of the paraxial obstruction borders for the class containing the systems shown in Figs. 12 and 14. A comparison of the paraxial and finite obstruction contours for the same domain is shown in Fig. 13. The lines in this figure are the obstruction boundaries, found with Eq. (2). Mirror number 4 obstructs the beam between mirror numbers 2 and 3 on lines a, b, and d; see point b in Fig. 14. The difference is in the location of the intermediate image: In a the intermediate image is after mirror number 4, in b the intermediate image is between mirror numbers 2 and 3, and in d the intermediate image is between mirror numbers 3 and 4. On line c, mirror number 2 borders the beam between the object and the first mirror; see point a in Fig. 14.

Fig. 12
Fig. 12

Paraxial rays agree well enough with the finite rays, for the purpose of a first evaluation of a system that comprises the examination of, e.g., the presence of obstruction, the workspace, and the telecentricity at the mask and wafer. This example shows a six-mirror system in class 37+.

Fig. 13
Fig. 13

Two-dimensional analysis of the solution space for six-mirror systems in class 37+. The distance between the first and the second reflective surfaces varies horizontally. On the vertical axis, the curvature of the fourth reflective surface changes. The other variables remain constant or are solved by constraints. The light gray points are paraxially unobstructed, and the dark gray points appear unobstructed with both paraxial and finite ray tracing. The black dot is the system shown in Fig. 12. We observe in this and other two-dimensional analyses that the paraxial unobstructed domains tend to include the smaller finite unobstructed domains.

Fig. 14
Fig. 14

Positive six-mirror system in class 37+. The object heights are between 108 and 120 mm, and the ray trajectories correspond to an numerical aperture of 0.3.

Fig. 15
Fig. 15

Positive six-mirror system in class 9+. Although the angles of incidence are large, the rms wave-front error can decrease to λ/2 at the chosen values of the numerical aperture (0.3) and the object heights (between 114 and 118 mm). However, the distortion is large.

Fig. 16
Fig. 16

Four-mirror system in class 6- with a rms wave-front error below 0.0266λ and distortion below 12 nm. This system has a wavelength of 13 nm, a numerical aperture of 0.15, and object heights between 114 and 119 mm.

Fig. 17
Fig. 17

This six-mirror system in class 26- resembles the four-mirror class 6- (see Fig. 16) with an additional pair of mirrors in the group on the image side. In the four-mirror design, the two mirrors nearest to the wafer almost cause obstruction. Here the additional pair of mirrors permits a larger numerical aperture.

Fig. 18
Fig. 18

This negative eight-mirror system is an example of systems belonging to class 150-.

Fig. 19
Fig. 19

This negative eight-mirror system is an example of systems belonging to class 182-.

Fig. 20
Fig. 20

This positive eight-mirror system is an example of systems belonging to class 169+.

Fig. 21
Fig. 21

This positive eight-mirror system is an example of systems belonging to class 173+.

Fig. 22
Fig. 22

This positive eight-mirror system is an example of systems belonging to class 181+.

Equations (26)

Equations on this page are rendered with MathJax. Learn more.

yi,k+yi+1,k-yi,kzi+1,k-zi,kzj,k-zi,k=yj,0+yj,1-yj,0zj,1-zj,0×zj,k-zj,0=yj,k,
k=01yj,kyj,0, yj,1=k=01zj,kzj,0, zj,1
k=01zj,kzi,k, zi+1,k=k=01yj,kyi,k, yi+1,k,
O=i=0Nj1jiji+1Nk=01yj,kyj,0, yj,1yj,kyi,k, yi+1,k,
H2i=1Ncini-1-nini-1ni=0
i=1N-1ici=0.
C=i=1Nai2N-i,
Oh0, NA, c1cN, d0dN=Of1h0, NA, c1f1cNf1, f1d0f1dN,
Oh0, NA, c1cN, d0dN=Of2h0, f2NA, c1cN, d0dN.
yj,k=yj,k,
Oh0, NA, c1cN, d0dN=Of1f2h0, NAf2, c1f1cNf1,f1d0f1dN.
f1=NAh0NAh0,
f2=NANA,
hN+1uN+1=M·h0u0+O3,
M=TN·RN·TN-1·B·A·T1·R1·T0.
A=a1a2a3a4=TS-1·RS-1·T2·R2,
B=b1b2b3b4=RN-1·TN-2·TS·RS.
Ti=1di01, Ri=10-2ci-1.
RN·TN-1·B·0uS=mh00.
dN-1=-b2b4-12cN.
A·T1·R1·T0·h0u0=0uS.
RN·TN-1·B·A·T1·R1·T0·h0u0=mh00.
d0=-12c1+h0-1-ma12-a2a3+a1a4c1b2+b4d5u0,
d1=mh0a1-2a2c1-u0a2a3-a1a4b2+b4dN-12mh0a1c1.
TN·RN·TN-1·B·A·T1·R1·T0·0mNA=0NA.
dN=a21+2c1d0b1+b3dN-1+a41+2c1d0b2+b4dN-1+d1+d0-1+2c1d1a1b1+b3dN-1+a3b2+b4dN-1/a21+2c1d0×b3+2b1cN+2b3cNdN-1+a41+2c1d0b4+2b2cN+2b4cN+2b4cNdN-1+d1+d0-1+2c1d1a3b4+2a3cNb2+b4dN-1+a1b3+2b1cN+2b3cNdN-1.

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