Abstract

Recently developed methods [ J. Opt. Soc. Am. A 11, 3292– 3307 ( 1994)] permit definitive studies of two-mirror systems for a variety of applications. As an illustration of such a study, unobstructed, telecentric, plane-symmetric systems of two conic mirrors that are intended for distortion-free projection are investigated here. The magnification, speed, and field size in the examples are arbitrarily chosen to correspond to values that are appropriate for full-field soft-x-ray projection lithography. Low-order imaging constraints are applied to eliminate most of the 13 parameters that specify the configuration of such a system. It is shown that there are just two three-parameter families of systems and a single four-parameter family. The associated three-and four-dimensional merit function spaces are mapped, and selected systems from various regions of the spaces are discussed.

© 1994 Optical Society of America

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References

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  1. B. D. Stone, G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9, 96–110 (1992).
    [CrossRef]
  2. B. D. Stone, G. W. Forbes, “Foundations of second-order layout for asymmetric systems,” J. Opt. Soc. Am. A 9, 2067–2082 (1992).
    [CrossRef]
  3. B. D. Stone, G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11, 3292–3307 (1994).
    [CrossRef]
  4. See, for example, the design requirements given by J. M. Rodgers, T. E. Jewell, “Design of reflective relay for soft x-ray lithography,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1354, 330–336 (1990).
    [CrossRef]
  5. An expression of s(y) appears in Eq. (6.1) of G. W. Forbes, B. D. Stone, “Hamilton’s angle characteristic in closed form for generally configured conic and toric interfaces,” J. Opt. Soc. Am. A 10, 1270–1278 (1993).
    [CrossRef]
  6. For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Dover, New York, 1993), Chap. 2.
  7. For a description of the imaging properties that correspond to the terms of degree two in the Taylor expansion of Hamilton’s point-angle mixed characteristic see, for example, B. D. Stone, G. W. Forbes, “Characterization of first-order optical properties for asymmetric systems,” J. Opt. Soc. Am. A 9, 478–489 (1992), and references therein.
    [CrossRef]
  8. The basal magnification is associated with planes in the object and image spaces that are normal to the associated base ray segments. Note that the basal magnification is generally not the same as the magnification associated with the (possibly) tilted object and image planes. When the principal basal magnifications are not the same, the basal magnification is said to be anisotropic, and otherwise it is isotropic.
  9. For a description of the Scheimpflug condition see, for example, R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), Chap. 5, Sec. VI.
  10. Here, M has the opposite sign of the magnification. This is an artifact of using the point-angle mixed characteristic in the work of Ref. 3. Also note that it was shown in Sec. 4.A.2 of Ref. 3 that in the nonanamorphic case the requirement of sharp imagery rules out the possibility that a system that left–right inverts images. That is, the principal magnifications cannot have opposite signs for that case.
  11. Equations (3.1a) and (3.1b) follow directly from the requirement that the right-hand sides of Eqs. (2.6d) and (2.6e) vanish. Equation (3.1c) follows from combining Eq. (2.3e) with the requirement that L˜221 [given by Eq. (2.6d)] vanish.
  12. This assumes, however, the constraint that the maximum separation to be considered remains 3 m. If the maximum separation between elements is allowed to increase without bounds, systems that are tens of meters long with merit function values of a few micrometers can result.
  13. T. E. Jewell, K. P. Thompson, J. M. Rodgers, “Reflective systems design study for soft x-ray projection lithography,”J. Vac. Sci. Technol. B 8, 1519–1523 (1990).
    [CrossRef]

1994

1993

1992

1990

T. E. Jewell, K. P. Thompson, J. M. Rodgers, “Reflective systems design study for soft x-ray projection lithography,”J. Vac. Sci. Technol. B 8, 1519–1523 (1990).
[CrossRef]

Buchdahl, H. A.

For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Dover, New York, 1993), Chap. 2.

Forbes, G. W.

Jewell, T. E.

T. E. Jewell, K. P. Thompson, J. M. Rodgers, “Reflective systems design study for soft x-ray projection lithography,”J. Vac. Sci. Technol. B 8, 1519–1523 (1990).
[CrossRef]

See, for example, the design requirements given by J. M. Rodgers, T. E. Jewell, “Design of reflective relay for soft x-ray lithography,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1354, 330–336 (1990).
[CrossRef]

Kingslake, R.

For a description of the Scheimpflug condition see, for example, R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), Chap. 5, Sec. VI.

Rodgers, J. M.

T. E. Jewell, K. P. Thompson, J. M. Rodgers, “Reflective systems design study for soft x-ray projection lithography,”J. Vac. Sci. Technol. B 8, 1519–1523 (1990).
[CrossRef]

See, for example, the design requirements given by J. M. Rodgers, T. E. Jewell, “Design of reflective relay for soft x-ray lithography,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1354, 330–336 (1990).
[CrossRef]

Stone, B. D.

Thompson, K. P.

T. E. Jewell, K. P. Thompson, J. M. Rodgers, “Reflective systems design study for soft x-ray projection lithography,”J. Vac. Sci. Technol. B 8, 1519–1523 (1990).
[CrossRef]

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. B

T. E. Jewell, K. P. Thompson, J. M. Rodgers, “Reflective systems design study for soft x-ray projection lithography,”J. Vac. Sci. Technol. B 8, 1519–1523 (1990).
[CrossRef]

Other

For a description of Hamilton’s characteristic functions see, for example, H. A. Buchdahl, Introduction to Hamiltonian Optics (Dover, New York, 1993), Chap. 2.

See, for example, the design requirements given by J. M. Rodgers, T. E. Jewell, “Design of reflective relay for soft x-ray lithography,” in 1990 International Lens Design Conference, G. N. Lawrence, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1354, 330–336 (1990).
[CrossRef]

The basal magnification is associated with planes in the object and image spaces that are normal to the associated base ray segments. Note that the basal magnification is generally not the same as the magnification associated with the (possibly) tilted object and image planes. When the principal basal magnifications are not the same, the basal magnification is said to be anisotropic, and otherwise it is isotropic.

For a description of the Scheimpflug condition see, for example, R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983), Chap. 5, Sec. VI.

Here, M has the opposite sign of the magnification. This is an artifact of using the point-angle mixed characteristic in the work of Ref. 3. Also note that it was shown in Sec. 4.A.2 of Ref. 3 that in the nonanamorphic case the requirement of sharp imagery rules out the possibility that a system that left–right inverts images. That is, the principal magnifications cannot have opposite signs for that case.

Equations (3.1a) and (3.1b) follow directly from the requirement that the right-hand sides of Eqs. (2.6d) and (2.6e) vanish. Equation (3.1c) follows from combining Eq. (2.3e) with the requirement that L˜221 [given by Eq. (2.6d)] vanish.

This assumes, however, the constraint that the maximum separation to be considered remains 3 m. If the maximum separation between elements is allowed to increase without bounds, systems that are tens of meters long with merit function values of a few micrometers can result.

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Figures (14)

Fig. 1
Fig. 1

Schematic representation of a plane-symmetric system that comprises two conic mirrors. The figure illustrates the cross section of the system in the plane of symmetry. The base ray through the system is shown, along with object and image planes. The axis associated with each mirror is represented by a thick dashed line. The points of intersection of the base ray with the axes are purposely drawn to be coincident.

Fig. 2
Fig. 2

Illustration of the size of the object and the 15 field points that are used to evaluate the systems investigated here. The size of the object is chosen to ensure that the image will be a square that is 2 cm on a side. Since the systems are all plane symmetric, rays need be traced only from field points on one side of the plane of symmetry.

Fig. 3
Fig. 3

The gray level in this plot represents the value of the merit function (maximum spot radius over the field) as a function of d0 and θ1. The object tilt for this plot is fixed at 9°. The black region corresponds to systems that are obstructed. Contours that represent systems with merit function values of 20, 50, 100, and 200 μm are also shown.

Fig. 4
Fig. 4

Cross section in the plane of symmetry for a system of two confocal parabolas that possesses no blur or distortion through second order. The merit function value and the values for the three parameters that specify the configuration of the system are listed.

Fig. 5
Fig. 5

The solid curve represents the maximum spot radius over the field as a function of the object tilt. The dashed curve is the tilt of the image plane. For both curves, d0 = 3 m and θ1 = 14.4°.

Fig. 6
Fig. 6

The dashed curve represents the values of θ1 that yield optimal systems plotted as a function of θobj. The merit function values for the optimal systems are represented by the solid curve. The value of d0 for all points is 3 m.

Fig. 7
Fig. 7

(a) and (b) Correspond to the points in Plate I labeled A and B, respectively. In both systems the first mirror is a hyperboloid and the second mirror is a paraboloid. The maximum spot radius over the field for this system is approximately 15.4 μm for system A and approximately 183 μm for system B. Unlike for the system illustrated in Fig. 4, the axes associated with the mirrors are not coincident in these systems.

Fig. 8
Fig. 8

Representation of the d0d1 plane in the configuration space for fixed values of θobj and d2. The line where the curvature of each mirror changes sign is shown, as is the line that represents the transition between prolate spheroids and hyperboloids for each mirror. When θobj is positive, the fine vertical hatching marks regions where θ1 is negative; the fine horizontal hatching marks regions where θ2 is positive.

Fig. 9
Fig. 9

Dependence of the merit function on the values of d0 and d1 for d2 = 0.2 m and θobj = 15°. The lines drawn in Fig. 8 are superimposed upon this plot. Seven points, labeled A–G, are also singled out.

Fig. 10
Fig. 10

(a)–(g) Correspond to the points labeled A–G in Fig. 9.

Fig. 11
Fig. 11

This system corresponds to the point labeled H in Plate II. It is the system with the smallest merit function value of all the systems that we evaluated to generate Plate II.

Fig. 12
Fig. 12

The system illustrated in this figure corresponds to point J in Plate II. This system has the lowest merit function value of all systems whose first mirror is convex that we evaluated to generate Plate II.

Plate I
Plate I

Each plot shows the merit function values as a function of d0 and d2 for the given value of θ1. Two points in the merit function space are singled out; they are labeled A and B. The line corresponding to d2 = Md0(1 + 2M) is also shown on each plot.

Plate II
Plate II

Each plot shows the merit function values as a function of d0 and d1 for fixed values of d2 and θobj. Each row contains plots for constant θobj, and each column contains plots of constant d2. Two points in the merit function space, labeled H and J, are singled out.

Tables (4)

Tables Icon

Table 1 System Parameters for the System Illustrated in Fig. 4

Tables Icon

Table 2 System Parameters for the Systems Illustrated in Fig. 7

Tables Icon

Table 3 System Parameters for the Systems Illustrated in Fig. 10

Tables Icon

Table 4 System Parameters for the Systems Illustrated in Figs. 11 and 12

Equations (62)

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s ( y ) = 1 2 ( S 11 y 2 + S 22 z 2 ) + y 6 ( S 111 y 2 + 3 S 122 z 2 ) + O ( 4 ) .
c = S 22 S 22 / S 11 ,
κ = S 122 2 + S 22 2 ( S 11 - S 22 ) 2 S 22 3 ( S 11 - S 22 ) ,
h = sgn ( S 122 ) ( S 11 - S 22 ) [ S 122 2 + S 22 2 ( S 11 - S 22 ) 2 ] 1 / 2 .
S 1 = [ d 2 - M ( d 0 + d 1 ) ] 2 M d 0 d 1 cos ( θ 1 ) [ cos 2 ( θ 1 ) 0 0 1 ] ,
S 2 = [ M d 0 - d 1 - d 2 ) ] 2 d 1 d 2 cos ( θ 2 ) [ cos 2 ( θ 2 ) 0 0 1 ] ,
S 122 I = [ M ( d 0 + d 1 ) - d 2 ] [ M ( d 0 - d 1 ) + d 2 ] tan ( θ 1 ) + 2 d 2 ( M d 0 - d 1 - d 2 ) tan ( θ 2 ) 4 M 2 d 0 2 d 1 2 ,
S 122 II = 2 M d 0 [ M ( d 0 + d 1 ) - d 2 ] tan ( θ 1 ) + ( M d 0 - d 1 - d 2 ) ( M d 0 - d 1 + d 2 ) tan ( θ 2 ) 4 d 1 2 d 2 2 ,
M tan ( θ obj ) + tan ( θ im ) = 2 d 1 { [ M ( d 0 + d 1 ) - d 2 ] tan ( θ 1 ) + ( M d 0 - d 1 - d 2 ) tan ( θ 2 ) } ,
S 1 : = [ S 11 I 0 0 S 22 I ] ,             S 2 : = [ S 11 II 0 0 S 22 II ] .
y im = - ( M ˜ 11 y obj + ½ L ˜ 111 y obj 2 + ½ L ˜ 221 z obj 2 ) + O ( 3 ) ,
z i m = - ( M ˜ 22 z obj ) + L ˜ 122 y obj z obj ) + O ( 3 ) ,
M ˜ 11 = cos ( θ obj ) cos ( θ im ) M ,
M ˜ 22 = M ,
L ˜ 111 = M cos 2 ( θ obj ) 2 d 0 d 2 cos ( θ im ) [ 4 ( d 1 + d 2 - M d 0 ) tan ( θ 2 ) + ( 3 M d 1 - 4 d 2 ) tan ( θ obj ) + ( 4 M d 0 - d 1 ) tan ( θ im ) ] ,
L ˜ 221 = M [ 4 ( d 1 + d 2 - M d 0 ) tan ( θ 2 ) + M d 1 tan ( θ obj ) + d 1 tan ( θ im ) ] 2 d 0 d 2 cos ( θ im ) ,
L ˜ 122 = M cos ( θ obj ) [ M d 1 - 2 d 2 ) tan ( θ obj ) + ( 2 M d 0 - d 1 ) tan ( θ im ) ] 2 d 0 d 2 .
( M d 1 - 2 d 2 ) tan ( θ obj ) = ( d 1 - 2 M d 0 ) tan ( θ im ) ,
M d 1 tan ( θ obj ) + d 1 tan ( θ im ) = 4 ( M d 0 - d 1 - d 2 ) tan ( θ 2 ) ,
[ M ( d 0 + d 1 ) - d 2 ] tan ( θ 1 ) = ( M d 0 - d 1 - d 2 ) tan ( θ 2 ) .
tan ( θ im ) = ( M d 1 - 2 d 2 ) ( d 1 - 2 M d 0 ) tan ( θ obj ) ,
tan ( θ 1 ) = d 1 [ d 2 - M ( d 1 - M d 0 ) ] 2 ( d 1 - 2 M d 0 ) [ d 2 - M ( d 0 + d 1 ) ] tan ( θ obj ) ,
tan ( θ 2 ) = d 1 [ d 2 - M ( d 1 - M d 0 ) ] 2 ( d 1 - 2 M d 0 ) [ d 1 + d 2 - M d 0 ) ] tan ( θ obj ) .
S 1 = [ d 2 - M ( d 0 + d 1 ) ] 2 M d 0 d 1 cos ( θ 1 ) [ cos 2 ( θ 1 ) 0 0 1 ] ,
S 2 = ( M d 0 - d 1 - d 2 ) ] 2 d 1 d 2 cos ( θ 2 ) [ cos 2 ( θ 2 ) 0 0 1 ] ,
S 122 I = [ 3 d 2 + M ( d 0 - d 1 ) ] [ d 2 - M ( d 1 - M d 0 ) ] 8 M 2 d 0 2 d 1 ( 2 M d 0 - d 1 ) × tan ( θ obj ) ,
S 122 II = ( 3 M d 0 - d 1 + d 2 ) [ d 2 - M ( d 1 - M d 0 ) ] 8 d 1 d 2 2 ( 2 M d 0 - d 1 ) tan ( θ obj ) .
M ˜ 11 = [ cos 2 ( θ obj ) + ( M d 1 - 2 d 2 ) 2 ( d 1 - 2 M d 0 ) 2 sin 2 ( θ obj ) ] 1 / 2 M .
c 1 = d 2 - M ( d 0 + d 1 ) 2 M d 0 d 1 cos 2 ( θ 1 ) ,
c 2 = M d 0 - d 1 - d 2 2 d 1 d 2 cos 2 ( θ 1 ) ,
κ 1 = 4 ( M d 0 + d 2 ) ( M d 1 - 2 d 2 ) [ d 2 - M ( d 0 + d 1 ) ] 2 cos 2 ( θ 1 ) - 1 ,
κ 2 = 4 ( M d 0 + d 2 ) ( d 1 - 2 M d 0 ) ( M d 0 - d 1 - d 0 ) 2 cos 2 ( θ 2 ) - 1 ,
d 1 = 2 M d 0 .
tan ( θ im ) = 2 [ M d 0 ( 1 + 2 M ) - d 2 ] M d 0 tan ( θ 1 ) ,
tan ( θ 2 ) = d 2 - M d 0 ( 1 + 2 M ) d 2 + M d 0 tan ( θ 1 ) .
S 1 = [ d 2 - M d 0 ( 1 + 2 M ) ] 4 M 2 d 0 2 cos ( θ 1 ) [ cos 2 ( θ 1 ) 0 0 1 ] ,
S 2 = - ( M d 0 + d 2 ) 4 M d 0 d 2 cos ( θ 2 ) [ cos 2 ( θ 2 ) 0 0 1 ] ,
S 122 I = [ M d 0 ( 1 + 2 M ) - d 2 ] [ 3 d 2 - M d 0 ( 1 - 2 M ) ] 16 M 4 d 0 4 × tan ( θ 1 ) ,
S 122 II = ( M d 0 + d 2 ) [ M d 0 ( 1 + 2 M ) - d 2 ] 16 M 2 d 0 2 d 2 2 tan ( θ 1 ) .
c 1 = d 2 - M d 0 ( 1 + 2 M ) 4 M 2 d 0 2 cos 2 ( θ 1 ) ,
c 2 = - ( M d 0 + d 2 ) 4 M d 0 d 2 cos 2 ( θ 2 ) ,
κ 1 = 8 ( M d 0 + d 2 ) ( M 2 d 0 - d 2 ) [ d 2 - M d 0 ( 1 + 2 M ) ] 2 cos 2 ( θ 1 ) - 1 ,
κ 2 = - 1.
M ˜ 11 = { 1 + 4 [ M d 0 ( 1 + 2 M ) - d 2 ] 2 ( M d 0 ) 2 tan 2 ( θ 1 ) } 1 / 2 M .
d 2 = M 2 d 0 .
2 M d 0 [ M tan ( θ obj ) + tan ( θ im ) + 2 ( 1 + M ) tan ( θ 2 ) ] = 0 ,
M d 0 ( 1 + M ) [ tan ( θ 1 ) + tan ( θ 2 ) ] = 0.
tan ( θ 2 ) = - tan ( θ 1 ) ,
tan ( θ im ) = 2 ( 1 + M ) tan ( θ 1 ) - M tan ( θ obj ) .
S 1 = - ( 1 + M ) 4 M d 0 cos ( θ 1 ) [ cos 2 ( θ 1 ) 0 0 1 ] ,
S 2 = 1 M S 1 ,
S 122 I = ( 1 + M ) 2 16 M 2 d 0 2 tan ( θ 1 ) ,
S 122 II = 1 M 2 S 122 I .
c 1 = - ( 1 + M ) 4 M d 0 cos 2 ( θ 1 ) ,
c 2 = c 1 / M ,
κ 1 = κ 2 = - 1.
M ˜ 11 = cos ( θ obj ) { 1 + [ 2 ( 1 + M ) tan ( θ 1 ) - M tan ( θ obj ) ] 2 } 1 / 2 M .
L y = cos ( θ im ) × 2 cm cos ( θ obj ) M ,
L z = 2 cm M .
L ˜ 111 = 2 M 11 cos 2 ( θ obj ) tan ( θ 2 ) d 0 d 2 ( M 11 + M 22 ) [ M 22 ( d 0 + d 1 ) - d 2 ] cos ( θ im ) × { M 11 d 0 d 2 ( 1 + M 22 ) 2 + M 11 M 22 d 1 ( d 0 + d 1 + d 2 ) + M 22 d 2 ( M 22 d 0 - d 1 - d 2 ) - M 11 2 d 0 [ M 22 ( d 0 + d 1 ) - d 2 ] } ,
L ˜ 221 = - ( M 11 + M 22 ) ( M 22 d 0 - d 1 - d 2 ) tan ( θ 2 ) d 0 d 2 cos ( θ im ) ,
L ˜ 122 = M 22 ( M 22 d 1 - d 2 - M 22 2 d 0 ) sin ( θ obj ) d 0 d 2 .

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