Abstract

The distribution model of wavefront aberrations, which takes on a significant role in the designs and alignments of imaging optical systems without vignetting, is newly presented. This model decomposes the complicated distributions into the characteristic components, which clarifies the alignment criteria. For the actual alignments, only small displacements (decentering, tilt, and surface distance) of rotationally symmetric surfaces in the system are assumed. Then, the model, which regards the aberration distributions of the system as the sum of the contributions of each surface, is extended for the system with surface displacements. As a result of the derivation, it is concluded that the aberration distributions in the rotationally nonsymmetric systems can be expressed as the sum of several folds of rotationally symmetric components. In addition, it is presented that, based on this model, suitable distribution models, even of the arbitrary higher order, can be constructed for any aberration coefficients in various optical systems.

© 2011 Optical Society of America

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References

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  1. I. W. Kwee and J. J. M. Braat, “Double Zernike expansion of the optical aberration function,” Pure Appl. Opt. 2, 21-32(1993).
    [CrossRef]
  2. I. Agurok, “Double expansion of wavefront deformation in Zernike polynomials over the pupil and the field-of-view of optical systems: lens design, testing, and alignment,” Proc. SPIE 3430, 80-87 (1998).
    [CrossRef]
  3. I. Agurok, “Aberrations of perturbed and unobscured optical systems,” Proc. SPIE 3779, 166-177 (1999).
    [CrossRef]
  4. I. Agurok, “The optimum position and minimum number of field-of-view points for optical system wavefront comprehension,” Proc. SPIE 4092, 38-47 (2000).
    [CrossRef]
  5. T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev. 11, 199-207 (2004).
    [CrossRef]
  6. K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22, 1389-1401 (2005).
    [CrossRef]
  7. K. P. Thompson, “Multinodal fifth-order optical aberrations of optical systems without rotational symmetry: spherical aberration,” J. Opt. Soc. Am. A 26, 1090-1100 (2009).
    [CrossRef]
  8. H. H. Hopkins, Wave Theory of Aberrations (Clarendon, 1950).
  9. R. A. Buchroeder, “Tilted component optical systems,” Ph.D dissertation (University of Arizona, 1976).
  10. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).
  11. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995).
  12. A. M. Manuel and J. H. Burge, “Alignment aberrations of the new solar telescope,” Proc. SPIE 7433, 74330A (2009).
    [CrossRef]
  13. M. A. van den Brink, C. G. M. de Mol, and R. A. George, “Matching performance for multiple wafer steppers using an advanced metrology procedure,” Proc. SPIE 921, 180-197 (1988).
  14. D. C. Brown, “Decentering distortion of lenses,” Photogramm. Eng. 32, 444-462 (1966).
  15. D. C. Brown, “Close-range camera calibration,” Photogramm. Eng. 37, 855-866 (1971).

2009 (2)

2005 (1)

2004 (1)

T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev. 11, 199-207 (2004).
[CrossRef]

2000 (1)

I. Agurok, “The optimum position and minimum number of field-of-view points for optical system wavefront comprehension,” Proc. SPIE 4092, 38-47 (2000).
[CrossRef]

1999 (1)

I. Agurok, “Aberrations of perturbed and unobscured optical systems,” Proc. SPIE 3779, 166-177 (1999).
[CrossRef]

1998 (1)

I. Agurok, “Double expansion of wavefront deformation in Zernike polynomials over the pupil and the field-of-view of optical systems: lens design, testing, and alignment,” Proc. SPIE 3430, 80-87 (1998).
[CrossRef]

1993 (1)

I. W. Kwee and J. J. M. Braat, “Double Zernike expansion of the optical aberration function,” Pure Appl. Opt. 2, 21-32(1993).
[CrossRef]

1988 (1)

M. A. van den Brink, C. G. M. de Mol, and R. A. George, “Matching performance for multiple wafer steppers using an advanced metrology procedure,” Proc. SPIE 921, 180-197 (1988).

1971 (1)

D. C. Brown, “Close-range camera calibration,” Photogramm. Eng. 37, 855-866 (1971).

1966 (1)

D. C. Brown, “Decentering distortion of lenses,” Photogramm. Eng. 32, 444-462 (1966).

Agurok, I.

I. Agurok, “The optimum position and minimum number of field-of-view points for optical system wavefront comprehension,” Proc. SPIE 4092, 38-47 (2000).
[CrossRef]

I. Agurok, “Aberrations of perturbed and unobscured optical systems,” Proc. SPIE 3779, 166-177 (1999).
[CrossRef]

I. Agurok, “Double expansion of wavefront deformation in Zernike polynomials over the pupil and the field-of-view of optical systems: lens design, testing, and alignment,” Proc. SPIE 3430, 80-87 (1998).
[CrossRef]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995).

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

Braat, J. J. M.

I. W. Kwee and J. J. M. Braat, “Double Zernike expansion of the optical aberration function,” Pure Appl. Opt. 2, 21-32(1993).
[CrossRef]

Brown, D. C.

D. C. Brown, “Close-range camera calibration,” Photogramm. Eng. 37, 855-866 (1971).

D. C. Brown, “Decentering distortion of lenses,” Photogramm. Eng. 32, 444-462 (1966).

Buchroeder, R. A.

R. A. Buchroeder, “Tilted component optical systems,” Ph.D dissertation (University of Arizona, 1976).

Burge, J. H.

A. M. Manuel and J. H. Burge, “Alignment aberrations of the new solar telescope,” Proc. SPIE 7433, 74330A (2009).
[CrossRef]

de Mol, C. G. M.

M. A. van den Brink, C. G. M. de Mol, and R. A. George, “Matching performance for multiple wafer steppers using an advanced metrology procedure,” Proc. SPIE 921, 180-197 (1988).

George, R. A.

M. A. van den Brink, C. G. M. de Mol, and R. A. George, “Matching performance for multiple wafer steppers using an advanced metrology procedure,” Proc. SPIE 921, 180-197 (1988).

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, 1950).

Kwee, I. W.

I. W. Kwee and J. J. M. Braat, “Double Zernike expansion of the optical aberration function,” Pure Appl. Opt. 2, 21-32(1993).
[CrossRef]

Manuel, A. M.

A. M. Manuel and J. H. Burge, “Alignment aberrations of the new solar telescope,” Proc. SPIE 7433, 74330A (2009).
[CrossRef]

Matsuyama, T.

T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev. 11, 199-207 (2004).
[CrossRef]

Thompson, K.

Thompson, K. P.

Ujike, T.

T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev. 11, 199-207 (2004).
[CrossRef]

van den Brink, M. A.

M. A. van den Brink, C. G. M. de Mol, and R. A. George, “Matching performance for multiple wafer steppers using an advanced metrology procedure,” Proc. SPIE 921, 180-197 (1988).

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

J. Opt. Soc. Am. A (2)

Opt. Rev. (1)

T. Matsuyama and T. Ujike, “Orthogonal aberration functions for microlithographic optics,” Opt. Rev. 11, 199-207 (2004).
[CrossRef]

Photogramm. Eng. (2)

D. C. Brown, “Decentering distortion of lenses,” Photogramm. Eng. 32, 444-462 (1966).

D. C. Brown, “Close-range camera calibration,” Photogramm. Eng. 37, 855-866 (1971).

Proc. SPIE (5)

A. M. Manuel and J. H. Burge, “Alignment aberrations of the new solar telescope,” Proc. SPIE 7433, 74330A (2009).
[CrossRef]

M. A. van den Brink, C. G. M. de Mol, and R. A. George, “Matching performance for multiple wafer steppers using an advanced metrology procedure,” Proc. SPIE 921, 180-197 (1988).

I. Agurok, “Double expansion of wavefront deformation in Zernike polynomials over the pupil and the field-of-view of optical systems: lens design, testing, and alignment,” Proc. SPIE 3430, 80-87 (1998).
[CrossRef]

I. Agurok, “Aberrations of perturbed and unobscured optical systems,” Proc. SPIE 3779, 166-177 (1999).
[CrossRef]

I. Agurok, “The optimum position and minimum number of field-of-view points for optical system wavefront comprehension,” Proc. SPIE 4092, 38-47 (2000).
[CrossRef]

Pure Appl. Opt. (1)

I. W. Kwee and J. J. M. Braat, “Double Zernike expansion of the optical aberration function,” Pure Appl. Opt. 2, 21-32(1993).
[CrossRef]

Other (4)

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, 1950).

R. A. Buchroeder, “Tilted component optical systems,” Ph.D dissertation (University of Arizona, 1976).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980).

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. (Academic, 1995).

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Figures (4)

Fig. 1
Fig. 1

Arrangement where the center axis (b) of the kth surface has the decentering with respect to the optical axis (a) under the rotationally symmetric state. The principal ray of the on-axis object point is shown by the gray dashed line, which coincides with the origins of the object and pupil planes of each surface.

Fig. 2
Fig. 2

Arrangement where the center axis (b) has the tilt with respect to the optical axis (a). We assume the virtual object plane (c) is perpendicular to the center axis (b). The paraxial image plane (d) of the kth surface also tilts.

Fig. 3
Fig. 3

Terms of the Taylor expansion of w can be arranged in a check pattern. This is an example of m = 2 and max = 2 .

Fig. 4
Fig. 4

Field terms are arranged like this pattern. This is an example of m = 2 and max = 2 .

Tables (9)

Tables Icon

Table 1 Field Terms Multiplied by Pupil Terms and Decentering Terms Appear as Six Combinations a

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Table 2 Field Terms of the Spherical Aberration and Defocus ( m = 0 )

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Table 3 Field Term Vectors of Coma and Tilt ( m = 1 )

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Table 4 Field Term Vectors of Astigmatism ( m = 2 )

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Table 5 Relation between the Index s of Field Terms and the Deformation of Image Field Distribution

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Table 6 Correspondence between Curvature Terms and the Field Terms in Table 2 a

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Table 7 Correspondence between Distortion Terms and the Field Terms in Table 3 a

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Table 8 List of Curvature Terms

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Table 9 List of Distortion Terms

Equations (57)

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r 2 = x 2 + y 2 , ρ 2 = ξ 2 + η 2 , κ 2 = x ξ + y η ,
Φ ( ξ , η ; x , y ) = = 0 n = 0 m = 0 a n m ( x 2 + y 2 ) ( ξ 2 + η 2 ) n ( x ξ + y η ) m ,
Φ ( ρ , θ ; r , φ ) = = 0 n = 0 m = 0 a n m r m + 2 ρ m + 2 n cos m ( θ φ ) ,
Φ ( ρ , θ ; r , φ ) = = 0 m = 0 n = 0 A n m r m + 2 R m + 2 n m ( ρ ) cos [ m ( θ φ ) ] ,
Φ ( ρ , θ ; r , φ ) = m = 0 n = 0 F m ( r , φ ) · P n m ( ρ , θ ) ,
F m ( r , φ ) = 0 A n m r m + 2 ( cos m φ sin m φ ) , P n m ( ρ , θ ) R m + 2 n m ( ρ ) ( cos m θ sin m θ ) .
Φ ( ρ , θ ; r , φ ) = = 0 n = 0 m = 0 A n m R m + 2 m ( r ) R m + 2 n m ( ρ ) cos [ m ( θ φ ) ] .
Φ k ( ξ k Δ ξ k , η k Δ η k ; x k Δ x k , y k Δ y k ) .
Φ k ( ξ k , η k ; x k Δ x k , y k Δ y k ) .
Φ ( ξ , η ; x , y ) = k Φ k ( ξ , η ; x Δ x k , y Δ y k ) ,
( X ( x , y ) Y ( x , y ) ) = ( X s Y s ) + ( β + β x x + β y y ) ( x y ) ,
Z ( x , y ) = Z s + t x x + t y y ,
A n m r m + 2 R m + 2 n m ( ρ ) cos m ( θ φ ) .
{ u r m + 2 R m + 2 n m ( ρ ) cos m ( θ φ ) v r m + 2 R m + 2 n m ( ρ ) sin m ( θ φ ) .
w u + i v = r m + 2 R m + 2 n m ( ρ ) exp [ i m ( θ φ ) ] .
w r m + 2 exp ( i m φ ) ,
w ( x + Δ x , y + Δ y ) = k = 0 m + 2 j = 0 k ( k j ) ( + m j ) r m + 2 k ( Δ r ) k exp { i [ ( m + k 2 j ) φ ( k 2 j ) Δ φ ] } ,
{ r m + 2 max k ( Δ r ) k [ O exp ( i m φ ) ] ( s = 0 ) r m + 2 max k ( Δ r ) k ( P exp { i [ ( m + s ) φ s Δ φ ] } + Q exp { i [ ( m s ) φ + s Δ φ ] } ) ( s 0 ) ,
O ( max k / 2 ) ( max + m k / 2 ) , P ( max ( k + s ) / 2 ) ( max + m ( k s ) / 2 ) , Q ( max ( k s ) / 2 ) ( max + m ( k + s ) / 2 ) .
F m ( r , φ ) = = 0 max { A O r m + 2 ( cos m φ sin m φ ) + s = 1 m + [ B s r m + 2 s ( P cos ( m + s ) φ + Q cos ( m s ) φ P sin ( m + s ) φ + Q sin ( m s ) φ ) + C s r m + 2 s ( P sin ( m + s ) φ Q sin ( m s ) φ P cos ( m + s ) φ + Q cos ( m s ) φ ) ] } .
Q P = ( + m ) ! ! ( s ) ! ( + m s ) ! = ( + m m ) / ( + m s m ) .
P { ( + m s m ) ( s ) 0 ( s > ) , Q ( + m m ) .
| F ( r , φ ) | = { O r m + 2 ( coefficient A ) r m + 2 s [ P 2 + Q 2 + 2 P Q cos ( 2 s φ ) ] 1 / 2 ( coefficient B ) r m + 2 s [ P 2 + Q 2 2 P Q cos ( 2 s φ ) ] 1 / 2 ( coefficient C ) .
F m ( r , φ ) = = 0 { A r m + 2 ( cos m φ sin m φ ) + s = 1 m + [ B s r m + 2 s P + Q ( P cos ( m + s ) φ + Q cos ( m s ) φ P sin ( m + s ) φ + Q sin ( m s ) φ ) + C s r m + 2 s P + Q ( P sin ( m + s ) φ Q sin ( m s ) φ P cos ( m + s ) φ + Q cos ( m s ) φ ) ] } .
F 0 ( r , φ ) = = 0 { A r 2 + s = 1 [ B s r 2 s cos ( s φ ) + C s r 2 s sin ( s φ ) ] } .
Φ ( ρ , θ ; r , φ ) = m = 0 n = 0 = 0 R m + 2 n m ( ρ ) [ A r m + 2 cos m ( θ φ ) + s = 1 m + r m + 2 s P + Q ( B s { P cos [ m ( θ φ ) s φ ] + Q cos [ m ( θ φ ) + s φ ] } + C s { P sin [ m ( θ φ ) s φ ] + Q sin [ m ( θ φ ) + s φ ] } ) ] .
Φ ( ρ , θ ψ ; r , φ ψ ) = Φ ( ρ , θ ; r , φ ) .
ψ = 2 π s n ,
r 2 1 F α ( r , φ ) · F β ( r , φ ) r d r d φ = A α δ α β ,
r 2 ( cos 2 φ sin 2 φ ) , r 4 5 ( cos 4 φ + 4 cos 2 φ sin 4 φ 4 sin 2 φ ) , r 6 7 ( 2 cos 4 φ + 5 cos 2 φ 2 sin 4 φ 5 sin 2 φ ) .
R 2 2 ( r ) ( cos 2 φ sin 2 φ ) , 1 2 [ R 4 4 ( r ) ( cos 4 φ sin 4 φ ) + R 4 2 ( r ) ( cos 2 φ sin 2 φ ) ] , 1 2 [ R 6 4 ( r ) ( cos 4 φ sin 4 φ ) + R 6 2 ( r ) ( cos 2 φ sin 2 φ ) ] .
F m ( r , φ ) = = 0 ( A R m + 2 m ( r ) ( cos m φ sin m φ ) + s = 1 m + { B s 2 [ P ( r ) ( cos ( m + s ) φ sin ( m + s ) φ ) + Q ( r ) ( cos ( m s ) φ sin ( m s ) φ ) ] + C s 2 [ P ( r ) ( sin ( m + s ) φ cos ( m + s ) φ ) + Q ( r ) ( sin ( m s ) φ cos ( m s ) φ ) ] } ) ,
P ( r ) { R m + 2 s m + s ( r ) ( s ) 0 ( s > ) , Q ( r ) { R m + 2 s m s ( r ) ( s ) 2 R m + 2 s m s ( r ) ( s > ) .
{ π m + 2 + 1 ( coefficient A ) π 2 ( m + 2 s + 1 ) ( coefficient B , C ) ( s ) π m + 2 s + 1 ( coefficient B , C ) ( s > ) .
R n ± m ( r ) = k = 0 ( n m ) / 2 ( 1 ) k ( n k ) ! k ! [ ( n + m ) / 2 k ] ! [ ( n m ) / 2 k ] ! r n 2 k ,
q k ( r ) p k ( r ) = ( k + m ) ! ( k s ) ! ( k + m s ) ! ( k ) ! ( k = 0 , 1 , , s ) ,
Φ ( ρ , θ ; r , φ ) = m = 0 n = 0 = 0 R m + 2 n m ( ρ ) [ A R m + 2 m ( r ) cos m ( θ φ ) + s = 1 m + ( ( B s / 2 ) { P ( r ) cos [ m ( θ φ ) s φ ] + Q ( r ) cos [ m ( θ φ ) + s φ ] } + ( C s / 2 ) { P ( r ) sin [ m ( θ φ ) s φ ] + Q ( r ) sin [ m ( θ φ ) + s φ ] } ) ] .
Δ Z ( x , y ) = = 0 { A r 2 + s = 1 [ B s r 2 s cos ( s φ ) + C s r 2 s sin ( s φ ) ] } .
Z s A 0 , t x x B 11 x , t y y C 11 y .
( Δ X ( x , y ) Δ Y ( x , y ) ) = = 0 { A r 2 + 1 ( cos φ sin φ ) + s = 1 + 1 [ B s r 2 s + 1 P + Q ( P cos ( s + 1 ) φ + Q cos ( s 1 ) φ P sin ( s + 1 ) φ Q sin ( s 1 ) φ ) + C s r 2 s + 1 P + Q ( P sin ( s + 1 ) φ + Q sin ( s 1 ) φ P cos ( s + 1 ) φ + Q cos ( s 1 ) φ ) ] } .
( X s Y s ) ( B 01 C 01 ) , β ( x y ) A 1 ( x y ) .
β x ( x 2 x y ) , β y ( x y y 2 ) ,
Z ( x , y ) = i = 1 C i c i ( x r max , y r max ) ,
1 r max [ 1 β 0 ( X ( x , y ) Y ( x , y ) ) ( x y ) ] = i = 1 D i d i ( x r max , y r max ) ,
( X ( x , y ) Y ( x , y ) ) = β 0 [ ( x y ) + r max i = 1 D i d i ( x r max , y r max ) ] .
d 4 ( r , φ ) = 1 2 [ R 2 2 ( r ) ( cos 2 φ sin 2 φ ) R 2 0 ( r ) ( 1 0 ) ] , d 5 ( r , φ ) = 1 2 [ R 2 2 ( r ) ( sin 2 φ cos 2 φ ) R 2 0 ( r ) ( 0 1 ) ] ,
( Δ X Δ Y ) = ( d X d Y ) + M s ( X Y ) + Φ s ( Y X ) T x ( X 2 X Y ) T y ( X Y Y 2 ) + M a ( X Y ) + Φ a ( Y X ) + W x ( Y 2 X Y ) + W y ( X Y X 2 ) + D 3 ( X 2 + Y 2 ) ( X Y ) + D 5 ( X 2 + Y 2 ) 2 ( X Y ) .
( x x y y ) = ( K 1 r ¯ 2 + K 2 r ¯ 4 + K 3 r ¯ 6 + ) ( x ¯ y ¯ ) + [ P 1 ( r ¯ 2 + 2 x ¯ 2 2 x ¯ y ¯ ) + P 2 ( 2 x ¯ y ¯ r ¯ 2 + 2 y ¯ 2 ) ] ( 1 + P 3 r ¯ 2 + ) ,
w m ( r , φ ) r m + 2 exp ( i m φ ) .
w m ( x + Δ x , y + Δ y ) = k = 0 1 k ! ( Δ r · ) k w m ( x , y ) , where Δ r ( Δ x , Δ y ) T , ( x , y ) T ,
{ x w m = w 1 m + 1 + ( + m ) w m 1 y w m = i [ w 1 m + 1 ( + m ) w m 1 ] .
( Δ r · ) w m = ( Δ x + i Δ y ) w 1 m + 1 + ( Δ x i Δ y ) ( + m ) w m 1 .
( Δ r · ) w m = ε w 1 m + 1 + ε * ( + m ) w m 1 ,
1 2 ! ( Δ r · ) 2 w m = 1 2 ! { ε [ ( Δ r · ) w 1 m + 1 ] + ε * ( + m ) [ ( Δ r · ) w m 1 ] } .
1 2 ! ( Δ r · ) 2 w m = 1 2 ! [ ε 2 ( 1 ) w 2 m + 2 + 2 ε ε * ( + m ) w 1 m + ε * 2 ( + m ) ( + m 1 ) w m 2 ] .
1 k ! ( Δ r · ) k w m = 1 k ! j = 0 k ( k j ) ε k j ε * j ! ( k + j ) ! ( + m ) ! ( + m j ) ! w k + j m + k 2 j .
1 k ! ( Δ r · ) k w m = j = 0 k ( k j ) ( + m j ) r m + 2 k ( Δ r ) k exp { i [ ( m + k 2 j ) φ ( k 2 j ) Δ φ ] } ,

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