Abstract

Nodal aberration theory is used to calculate the third-order aberrations that result in image blur for an unobscured modified 4f relay (2f1 + 2f2) formed by two tilted spherical mirrors for objects at infinity (infinite conjugate) and near the front focal plane of the first mirror (finite conjugate). The field-averaged wavefront variance containing only non-rotationally symmetric aberration coefficients is then proposed as an optimization metric. Analytical and ray tracing optimization are demonstrated through sample designs. The particular cases of in-plane and orthogonal folding of the optical axis ray are discussed, followed by an analysis of a modified 2f1 + 2f2 relay in which the distance of the first mirror to the object or pupil is allowed to vary for aberration correction. The sensitivity of the infinite conjugate 2f1 + 2f2 relay to the input marginal ray angle is also examined. Finally, the optimization of multiple conjugate systems through a weighted combination of wavefront variances is proposed.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

2017 (1)

2014 (1)

2013 (2)

2012 (1)

2011 (3)

2010 (1)

2009 (2)

2005 (1)

2003 (2)

J. J. Green and S. B. Shaklan, “Optimizing coronagraph designs to minimize their contrast sensitivity to low-order optical aberrations,” Proc. SPIE 5170, 25–37 (2003).
[Crossref]

C. Iemmi, A. Moreno, J. Nicolás, and J. Campos, “Evaluation and correction of aberrations in an optical correlator by phase-shifting interferometry,” Opt. Lett. 28(13), 1117–1119 (2003).
[Crossref] [PubMed]

2002 (1)

J. M. Rodgers, “Unobscured mirror designs,” Proc. SPIE 4832, 33–60 (2002).
[Crossref]

2000 (4)

S. A. Lerner, J. M. Sasian, and M. R. Descour, “Design approach and comparison of projection cameras for EUV lithography,” Opt. Eng. 39(3), 792–802 (2000).
[Crossref]

D. Y.-H. Wang, D. M. Aikens, and R. E. English, “Design of optical systems with both near-field and far-field system requirements,” Opt. Eng. 39(7), 1788–1795 (2000).
[Crossref]

J. R. Rogers, “Techniques and tools for obtaining symmetrical performance from tilted-component systems,” Opt. Eng. 39(7), 1776–1787 (2000).
[Crossref]

J. M. Howard and B. D. Stone, “Imaging with three spherical mirrors,” Appl. Opt. 39(19), 3216–3231 (2000).
[Crossref] [PubMed]

1998 (2)

D. M. Williamson, “Evolution of ring-field systems in microlithography,” Proc. SPIE 3482, 369–376 (1998).
[Crossref]

J. M. Howard and B. D. Stone, “Imaging a point with two spherical mirrors,” J. Opt. Soc. Am. A 15(12), 3045–3056 (1998).
[Crossref]

1994 (3)

1992 (4)

1987 (1)

W. B. Wetherell, “All-reflecting afocal telescopes,” Proc. SPIE 0751, 126–134 (1987).
[Crossref]

1986 (2)

J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” Proc. SPIE 0554, 76–81 (1986).
[Crossref]

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE 0679, 21–24 (1986).
[Crossref]

1984 (1)

1981 (1)

W. E. Woehl, “An all-reflective zoom optical system for the infrared,” Opt. Eng. 20(3), 203450 (1981).
[Crossref]

1980 (1)

1978 (1)

1975 (1)

J. Prasad, G. Mitra, and P. K. Jain, “Aberrations of a system of arbitrarily inclined planar surfaces placed in non-collimated light beam,” Nouv. Rev. Opt. 6(6), 345–352 (1975).
[Crossref]

1974 (1)

R. Gelles, “Unobscured aperture stigmatic telescopes,” Opt. Eng. 13(6), 136534 (1974).
[Crossref]

1970 (1)

1955 (1)

B. Cuny, “Corrections des aberrations introduite par une lame incline en luminere convergent,” Rev. Opt. 34(9), 460–464 (1955).

1953 (1)

M. Montel, “Classification des aberrations geometriques des systems optiques sans symmetrie de revolution,” Rev. Opt. 32(11), 585–600 (1953).

1951 (2)

1950 (2)

J. Shearer, “Geometrical optics of concave mirrors and of combinations of mirrors,” Aust. J. Chem. 3(4), 532–540 (1950).
[Crossref]

W. Weinstein, “The computation of wave-front aberrations of oblique pencils in a symmetrical optical system,” Proc. Phys. Soc. B 63(9), 709–723 (1950).
[Crossref]

1949 (2)

W. Weinstein, “Wave-front aberrations of oblique pencils in a symmetrical optical system: refraction and transfer formulae,” Proc. Phys. Soc. B 62(11), 726–740 (1949).
[Crossref]

L. I. Epstein, “The aberrations of slightly decentered optical systems*,” J. Opt. Soc. Am. 39(10), 847–853 (1949).
[Crossref]

1946 (1)

H. H. Hopkins, “A transformation of known astigmatism formulae,” Proc. Phys. Soc. 58(6), 663–668 (1946).
[Crossref]

1919 (1)

A. E. Conrady, “Decentered lens systems,” Mon. Not. R. Astron. Soc. 79(5), 384–390 (1919).
[Crossref]

Aikens, D. M.

D. Y.-H. Wang, D. M. Aikens, and R. E. English, “Design of optical systems with both near-field and far-field system requirements,” Opt. Eng. 39(7), 1788–1795 (2000).
[Crossref]

Bahk, S.-W.

Bromage, J.

Buchroeder, R. A.

Cakmakci, O.

Campos, J.

Chang, S.

S. Chang, “Elimination of linear astigmatism in off-axis three-mirror telescope and its applications,” Proc. SPIE 8860, 88600U (2013).
[Crossref]

Conrady, A. E.

A. E. Conrady, “Decentered lens systems,” Mon. Not. R. Astron. Soc. 79(5), 384–390 (1919).
[Crossref]

Cuny, B.

B. Cuny, “Corrections des aberrations introduite par une lame incline en luminere convergent,” Rev. Opt. 34(9), 460–464 (1955).

Descour, M. R.

S. A. Lerner, J. M. Sasian, and M. R. Descour, “Design approach and comparison of projection cameras for EUV lithography,” Opt. Eng. 39(3), 792–802 (2000).
[Crossref]

Dubra, A.

English, R. E.

D. Y.-H. Wang, D. M. Aikens, and R. E. English, “Design of optical systems with both near-field and far-field system requirements,” Opt. Eng. 39(7), 1788–1795 (2000).
[Crossref]

Epstein, L. I.

Forbes, G. W.

Fuerschbach, K.

Gelles, R.

R. Gelles, “Unobscured aperture stigmatic telescopes,” Opt. Eng. 13(6), 136534 (1974).
[Crossref]

Gómez-Vieyra, A.

Green, J. J.

J. J. Green and S. B. Shaklan, “Optimizing coronagraph designs to minimize their contrast sensitivity to low-order optical aberrations,” Proc. SPIE 5170, 25–37 (2003).
[Crossref]

Gross, H.

Hopkins, H. H.

H. H. Hopkins, “A transformation of known astigmatism formulae,” Proc. Phys. Soc. 58(6), 663–668 (1946).
[Crossref]

Howard, J. M.

Iemmi, C.

Ih, C. S.

Jain, K.

Jain, P. K.

J. Prasad, G. Mitra, and P. K. Jain, “Aberrations of a system of arbitrarily inclined planar surfaces placed in non-collimated light beam,” Nouv. Rev. Opt. 6(6), 345–352 (1975).
[Crossref]

Kerth, R. T.

Kocaoglu, O. P.

Lee, K.-S.

Lerner, S. A.

S. A. Lerner, J. M. Sasian, and M. R. Descour, “Design approach and comparison of projection cameras for EUV lithography,” Opt. Eng. 39(3), 792–802 (2000).
[Crossref]

Liu, Z.

Macenka, S. A.

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE 0679, 21–24 (1986).
[Crossref]

Malacara-Hernández, D.

Miller, D. T.

Mitra, G.

J. Prasad, G. Mitra, and P. K. Jain, “Aberrations of a system of arbitrarily inclined planar surfaces placed in non-collimated light beam,” Nouv. Rev. Opt. 6(6), 345–352 (1975).
[Crossref]

Montel, M.

M. Montel, “Classification des aberrations geometriques des systems optiques sans symmetrie de revolution,” Rev. Opt. 32(11), 585–600 (1953).

Moreno, A.

Nicolás, J.

Norris, K. P.

Prasad, J.

J. Prasad, G. Mitra, and P. K. Jain, “Aberrations of a system of arbitrarily inclined planar surfaces placed in non-collimated light beam,” Nouv. Rev. Opt. 6(6), 345–352 (1975).
[Crossref]

Rodgers, J. M.

J. M. Rodgers, “Unobscured mirror designs,” Proc. SPIE 4832, 33–60 (2002).
[Crossref]

Rogers, J. R.

J. R. Rogers, “Techniques and tools for obtaining symmetrical performance from tilted-component systems,” Opt. Eng. 39(7), 1776–1787 (2000).
[Crossref]

J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” Proc. SPIE 0554, 76–81 (1986).
[Crossref]

Rolland, J. P.

Sasian, J. M.

S. A. Lerner, J. M. Sasian, and M. R. Descour, “Design approach and comparison of projection cameras for EUV lithography,” Opt. Eng. 39(3), 792–802 (2000).
[Crossref]

Schiesser, E. M.

Schmid, T.

Seeds, W. E.

Shafer, D. R.

Shaklan, S. B.

J. J. Green and S. B. Shaklan, “Optimizing coronagraph designs to minimize their contrast sensitivity to low-order optical aberrations,” Proc. SPIE 5170, 25–37 (2003).
[Crossref]

Shearer, J.

J. Shearer, “Geometrical optics of concave mirrors and of combinations of mirrors,” Aust. J. Chem. 3(4), 532–540 (1950).
[Crossref]

Stacy, J. E.

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE 0679, 21–24 (1986).
[Crossref]

Steel, W. H.

W. H. Steel, “The design of reflecting microscope objectives,” Aust. J. Chem. 4(1), 1–11 (1951).
[Crossref]

Stone, B. D.

Sulai, Y.

Sulai, Y. N.

Thompson, K.

Thompson, K. P.

Wang, D. Y.-H.

D. Y.-H. Wang, D. M. Aikens, and R. E. English, “Design of optical systems with both near-field and far-field system requirements,” Opt. Eng. 39(7), 1788–1795 (2000).
[Crossref]

Weinstein, W.

W. Weinstein, “The computation of wave-front aberrations of oblique pencils in a symmetrical optical system,” Proc. Phys. Soc. B 63(9), 709–723 (1950).
[Crossref]

W. Weinstein, “Wave-front aberrations of oblique pencils in a symmetrical optical system: refraction and transfer formulae,” Proc. Phys. Soc. B 62(11), 726–740 (1949).
[Crossref]

Wetherell, W. B.

W. B. Wetherell, “All-reflecting afocal telescopes,” Proc. SPIE 0751, 126–134 (1987).
[Crossref]

Wilkins, M. H. F.

Williams, D. R.

Williamson, D. M.

D. M. Williamson, “Evolution of ring-field systems in microlithography,” Proc. SPIE 3482, 369–376 (1998).
[Crossref]

Wilson, R. N.

R. N. Wilson, “Karl Schwarzschild and telescope optics,” Rev. Mod. Astron. 7, 1–30 (1994).

Woehl, W. E.

W. E. Woehl, “An all-reflective zoom optical system for the infrared,” Opt. Eng. 20(3), 203450 (1981).
[Crossref]

Xue, Q.

Yen, K.

Zhong, Y.

Zuegel, J. D.

Appl. Opt. (6)

Aust. J. Chem. (2)

J. Shearer, “Geometrical optics of concave mirrors and of combinations of mirrors,” Aust. J. Chem. 3(4), 532–540 (1950).
[Crossref]

W. H. Steel, “The design of reflecting microscope objectives,” Aust. J. Chem. 4(1), 1–11 (1951).
[Crossref]

Biomed. Opt. Express (2)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (9)

B. D. Stone and G. W. Forbes, “Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods,” J. Opt. Soc. Am. A 9(1), 96–109 (1992).
[Crossref]

B. D. Stone and G. W. Forbes, “Foundations of second-order layout for asymmetric systems,” J. Opt. Soc. Am. A 9(11), 2067–2082 (1992).
[Crossref]

B. D. Stone and G. W. Forbes, “First-order layout of asymmetric systems composed of three spherical mirrors,” J. Opt. Soc. Am. A 9(1), 110–120 (1992).
[Crossref]

B. D. Stone and G. W. Forbes, “First-order layout of asymmetric systems: sharp imagery of a single plane object,” J. Opt. Soc. Am. A 9(5), 832–843 (1992).
[Crossref]

B. D. Stone and G. W. Forbes, “Second-order design methods for definitive studies of plane-symmetric, two-mirror systems,” J. Opt. Soc. Am. A 11(12), 3292–3307 (1994).
[Crossref]

B. D. Stone and G. W. Forbes, “Illustration of second-order design methods: global merit function plots for a class of projection systems,” J. Opt. Soc. Am. A 11(12), 3308–3321 (1994).
[Crossref]

J. M. Howard and B. D. Stone, “Imaging a point with two spherical mirrors,” J. Opt. Soc. Am. A 15(12), 3045–3056 (1998).
[Crossref]

K. Thompson, “Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry,” J. Opt. Soc. Am. A 22(7), 1389–1401 (2005).
[Crossref] [PubMed]

K. P. Thompson, T. Schmid, O. Cakmakci, and J. P. Rolland, “Real-ray-based method for locating individual surface aberration field centers in imaging optical systems without rotational symmetry,” J. Opt. Soc. Am. A 26(6), 1503–1517 (2009).
[Crossref] [PubMed]

Mon. Not. R. Astron. Soc. (1)

A. E. Conrady, “Decentered lens systems,” Mon. Not. R. Astron. Soc. 79(5), 384–390 (1919).
[Crossref]

Nouv. Rev. Opt. (1)

J. Prasad, G. Mitra, and P. K. Jain, “Aberrations of a system of arbitrarily inclined planar surfaces placed in non-collimated light beam,” Nouv. Rev. Opt. 6(6), 345–352 (1975).
[Crossref]

Opt. Eng. (5)

R. Gelles, “Unobscured aperture stigmatic telescopes,” Opt. Eng. 13(6), 136534 (1974).
[Crossref]

S. A. Lerner, J. M. Sasian, and M. R. Descour, “Design approach and comparison of projection cameras for EUV lithography,” Opt. Eng. 39(3), 792–802 (2000).
[Crossref]

D. Y.-H. Wang, D. M. Aikens, and R. E. English, “Design of optical systems with both near-field and far-field system requirements,” Opt. Eng. 39(7), 1788–1795 (2000).
[Crossref]

W. E. Woehl, “An all-reflective zoom optical system for the infrared,” Opt. Eng. 20(3), 203450 (1981).
[Crossref]

J. R. Rogers, “Techniques and tools for obtaining symmetrical performance from tilted-component systems,” Opt. Eng. 39(7), 1776–1787 (2000).
[Crossref]

Opt. Express (5)

Opt. Lett. (3)

Proc. Phys. Soc. (1)

H. H. Hopkins, “A transformation of known astigmatism formulae,” Proc. Phys. Soc. 58(6), 663–668 (1946).
[Crossref]

Proc. Phys. Soc. B (2)

W. Weinstein, “Wave-front aberrations of oblique pencils in a symmetrical optical system: refraction and transfer formulae,” Proc. Phys. Soc. B 62(11), 726–740 (1949).
[Crossref]

W. Weinstein, “The computation of wave-front aberrations of oblique pencils in a symmetrical optical system,” Proc. Phys. Soc. B 63(9), 709–723 (1950).
[Crossref]

Proc. SPIE (7)

S. Chang, “Elimination of linear astigmatism in off-axis three-mirror telescope and its applications,” Proc. SPIE 8860, 88600U (2013).
[Crossref]

J. M. Rodgers, “Unobscured mirror designs,” Proc. SPIE 4832, 33–60 (2002).
[Crossref]

J. J. Green and S. B. Shaklan, “Optimizing coronagraph designs to minimize their contrast sensitivity to low-order optical aberrations,” Proc. SPIE 5170, 25–37 (2003).
[Crossref]

W. B. Wetherell, “All-reflecting afocal telescopes,” Proc. SPIE 0751, 126–134 (1987).
[Crossref]

J. R. Rogers, “Vector aberration theory and the design of off-axis systems,” Proc. SPIE 0554, 76–81 (1986).
[Crossref]

J. E. Stacy and S. A. Macenka, “Optimization of an unobscured optical system using vector aberration theory,” Proc. SPIE 0679, 21–24 (1986).
[Crossref]

D. M. Williamson, “Evolution of ring-field systems in microlithography,” Proc. SPIE 3482, 369–376 (1998).
[Crossref]

Rev. Mod. Astron. (1)

R. N. Wilson, “Karl Schwarzschild and telescope optics,” Rev. Mod. Astron. 7, 1–30 (1994).

Rev. Opt. (2)

M. Montel, “Classification des aberrations geometriques des systems optiques sans symmetrie de revolution,” Rev. Opt. 32(11), 585–600 (1953).

B. Cuny, “Corrections des aberrations introduite par une lame incline en luminere convergent,” Rev. Opt. 34(9), 460–464 (1955).

Other (11)

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

K. P. Thompson, “Aberration fields in tilted and decentered optical systems,” Ph.D. Dissertation (University of Arizona, 1980).

W. Wall, A History of Optical Telescopes in Astronomy (Springer Nature Switzerland AG, 2018).

H. King, The History of the Telescope (Charles Griffin Ltd., 1955).

C. R. Burch, “Reflecting microscopes,” Proc. Phys. Soc. 59(1), 41–46, 46-2 (1947).
[Crossref]

J. W. Hardy, Adaptive Optics for Astronomical Telescopes, Oxford Series in Optical and Imaging Sciences (Oxford University Press, 1998).

H. H. Hopkins, The Wave Theory of Aberrations (Oxford on Clarendon Press, 1950).

W. T. Welford, Aberrations of the Symmetrical Optical System (Academic Press Inc., 1974).

H. Gross, Aberration Theory and Correction of Optical Systems, Handbook of Optical Systems (WILEY-VCH, 2007).

J. R. Rogers, “Aberrations of Unobscured Reflected Optical Systems,” Ph.D. Dissertation (University of Arizona, 1983).

R. S. Kebo, “Four mirror afocal wide field of view optical system,” US4804258 (1989).

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Figures (16)

Fig. 1
Fig. 1 First-order layout of two identical consecutive infinite conjugate 4f relays, formed by either a single optical element (top) or two elements (bottom) with equal optical path length. The red and blue rays are the chief and marginal rays, respectively. The largest system element(s) are depicted in green.
Fig. 2
Fig. 2 First-order layout of a 2f1 + 2f2 relay (top) and a modified 2f1 + 2f2 (bottom) relay for type 1 (left) and type 2 (right). Note that the difference between the two forms is that distance between the object (type 1) or the pupil (type 2) to the first and last elements are different. The dashed black line represents the optical axis, the red lines the chief rays, and the blue lines the marginal rays.
Fig. 3
Fig. 3 Top view of unobscured modified 2f1 + 2f2 relays formed by two tilted spherical mirrors of type 1 (left) and type 2 (middle), along with the system of coordinates as it propagates through the system (right). Note that although in the figure the second mirror is only tilted around the x-axis, it could also be tilted in the y-axis, such that the optical axis leaves the plane of the page. The dashed black line represents the optical axis ray (OAR), the red lines the chief rays, and the blue lines the marginal rays. Note that the coordinate system follows the OAR, except the z-axis does not flip sign upon reflection.
Fig. 4
Fig. 4 Unobscured reflective 2f1 + 2f2 type 1 relay schematic showing mechanical clearances δ 1 and δ 2 .
Fig. 5
Fig. 5 Naming convention used to describe wavefront aberrations in terms of normalized coordinates in the pupil (ρ, φ) and field (H, θ) planes for rotationally symmetric optical systems.
Fig. 6
Fig. 6 Sample geometry used for illustrating the field displacement vector calculation for a tilted spherical mirror. OAR stands for optical axis ray, AS denotes the aperture stop, C.C. is the mirror center of curvature, N the object normal, R the incident ray, S the surface normal, and σ j is the field shift vector after the tilting and propagation of the coordinate system.
Fig. 7
Fig. 7 RMS wavefront error field maps as calculated by third-order nodal aberration theory (column 1), by real ray tracing (column 2), and relative error (see text for details) as compared to third-order nodal aberration theory prediction (column 3) for two unobscured reflective relay telescopes of type 1 and 2 (rows 1-4) over a normalized 5 × 5° and 5 × 5 mm field angle, respectively, with specification parameters listed in Table 2, and evaluated for 500 nm light.
Fig. 8
Fig. 8 Proposed optimization process for designing unobscured spherical reflective systems.
Fig. 9
Fig. 9 Contour plots of the NRS field-averaged wavefront variance as a function of mirror two tilt angle for type 1 (top) and type 2 (bottom) relays for three designs (rows 1-3) showing predicted (green cross) and actual angles (red circle) for minimum field-averaged wavefront variance. Each contour plot shows 20 levels with step sizes 0.006, 7x10−6, and 5x10−5 μm2 for type 1 designs 1-3, respectively, and 2x10−5, 0.02, and 0.5 μm2 for type 2 designs 1-3, respectively.
Fig. 10
Fig. 10 Infinite (left) and finite (right) conjugate field-averaged wavefront variance of design 1 as a function of mirror two tilt angle showing minimal required angle for a mechanical mounting clearance of 20 mm. The predicted (green cross) and actual angles (red circle) for minimum wavefront are also included.
Fig. 11
Fig. 11 Top view of an unobscured reflective modified 2f1 + 2f2 type 1 relay with the tilt of the second element such that it has an in-plane geometry (left) and an orthogonal geometry (right). The dashed black line represents the optical axis ray (OAR), the red lines the chief rays, and the blue lines the marginal rays.
Fig. 12
Fig. 12 Magnitude of non-rotationally symmetric coefficients from field-averaged wavefront variance for designs 1 and 2 (rows 1 and 2) both in-plane and orthogonal geometries, for type 1 and type 2 relays (columns 1 and 2). Note the difference in scales between designs 1 and 2.
Fig. 13
Fig. 13 Normalized absolute value of Seidel coefficients as a function of distance between pupil/object and the first optical element (with focal length f) for a modified 2f1 + 2f2 reflective relay.
Fig. 14
Fig. 14 RMS wavefront field maps of an example system of type 1 (top row) and type 2 (bottom row) at three different ɛ values: −250, 0, and 400 mm (columns 1-3).
Fig. 15
Fig. 15 Type 2 relay input marginal rays with deviation from collimation (i.e. vergence) represented by φv.
Fig. 16
Fig. 16 RMS wavefront field maps evaluated for 500 nm light for a type 2 relay with the parameters of design 2 in-plane (row 1) and orthogonal (row 2) geometries for 2D, 1D, 0D, −1D, and −2D of angle error (columns 1-5).

Tables (9)

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Table 1 Third-Order Aberrations (Excluding Distortion) in a Non-Rotationally Symmetric Optical System According to Nodal Aberration Theory a

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Table 2 Design Parameters for Two Sample Unobstructed 2f1 + 2f2 Type 1 and 2 Relays Formed by Two Spherical Mirrors with a Focal Length of the First Mirror f1 of 500 mm and Entrance Pupil Diameter of 5 mm a

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Table 3 Expressions for Seidel Coefficients for Unobscured Two-Mirror 2f1 + 2f2 Type 1 and 2 Relays a

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Table 4 Surface σ and Coefficients of Non-Rotationally Symmetric Field-Averaged Wavefront Variance for an Unobscured Two-Mirror Telescope

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Table 5 Design Parameters of Sample Relays a

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Table 6 Design Parameters for Two Infinite Conjugate 2f1 + 2f2 Relay Examples Formed by Two Spherical Mirrors (f1 = 500 mm, h0 = 4 mm and α0 = 1.0°) a

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Table 7 Seidel Coefficients, Their Respective Roots, and NAT Field Shifts ( σ ) for a Modified 2f1 + 2f2 Type 1 Relay

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Table 8 Seidel Coefficients, Their Respective Roots and NAT Field Shifts ( σ ) for a Modified 2f1 + 2f2 Type 2 Relay

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Table 9 Seidel Coefficients, Roots, and Field Shift Vectors σ for a Defocused 2f1 + 2f2 Type 2 Relay

Equations (29)

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δ 1 2 | θ y 1 ( f 1 + ε ) | | h 0 | | α 0 ( f 1 2 ε 2 ) f 1 | ,
δ 2 2 | m θ 2 ( f 1 + m ε ) | | m h 0 | | α 0 ( f 1 2 m 2 ε 2 ) f 1 | ,
W ( H , θ = 0 , ρ , φ ) = j p = 0 n = 0 m 0 ( W k l m ) j H k ρ l cos m φ ,
W ( H , ρ ) = j p = 0 n = 0 m 0 ( W k l m ) j ( H H ) p ( ρ ρ ) n ( H ρ ) m ,
W ( H , ρ ) = j p = 0 n = 0 m 0 ( W k l m ) j [ ( H σ j ) ( H σ j ) ] p ( ρ ρ ) n [ ( H σ j ) ρ ] m ,
σ j = i j * i j ,
i j * = N j × ( R j × S j ) ,
W var ( H ) = 1 π 0 2 π 0 1 [ W ( H , ρ ) ] 2 ρ d ρ d φ ( 1 π 0 2 π 0 1 W ( H , ρ ) ρ d ρ d φ ) 2 .
W var ¯ H = 1 π 0 2 π 0 1 W var ( H ) H d H d θ .
M F 2 = j W i ( T i V i ) 2 j W i ,
W ( H , ρ ) W 20 ( ρ ρ ) + W 11 ( H ρ ) + W 040 ( ρ ρ ) 2 + [ ( W 131 H A 131 ) ρ ] ( ρ ρ ) + [ W 220 M ( H H ) 2 ( H A 220 M ) + B 220 M ] ( ρ ρ ) + 1 2 [ W 222 H 2 2 H A 222 + B 222 2 ] ρ 2 ,
W var ¯ H = 1 720 [ 90 W 11 2 + 60 W 20 2 + 64 W 040 2 + 45 W 131 2 + 20 W 220 M 2 + 10 W 222 2 + 60 W 20 W 220 M + 120 W 11 W 131 + 60 W 040 ( 2 W 20 + W 220 M ) + 60 B 220 M ( 2 W 20 + 2 W 040 + W 220 M ) + 60 B 220 M 2 + 90 A 131 2 + 60 A 222 2 + 60 A 220 M 2 + 30 B 222 4 .
{ W var ¯ H } N R S = 1 24 [ 3 A 131 2 + 2 A 222 2 + 2 A 220 M 2 + B 222 4 ] .
{ W var ¯ H } T 1 N R S = α 0 4 384 m 4 { 4 m 2 h 0 2 [ ( m θ y 1 + θ y 2 ) 2 + θ x 2 2 ] + 3 f 1 2 α 0 2 [ ( m 2 θ y 1 + θ y 2 ) 2 + θ x 2 2 ] + 4 m 2 f 1 2 [ ( m θ y 1 2 θ y 2 2 + θ x 2 2 ) 2 + 4 θ x 2 2 θ y 2 2 ] }
{ W var ¯ H } T 2 N R S = h 0 4 384 f 1 4 { 4 f 1 2 α 0 2 [ ( θ y 1 + θ y 2 ) 2 + θ x 2 2 ] + 3 h 0 2 [ ( θ y 1 + m θ y 2 ) 2 + m 2 θ x 2 2 ] + 4 f 1 2 [ ( θ y 1 2 m θ y 2 2 + m θ x 2 2 ) 2 + 4 m 2 θ x 2 2 θ y 2 2 ] }
{ W var ¯ H } T o t a l N R S = i ω i { W var ¯ H } i N R S ,
m = h f h 0 = f 1 f 2 d 1 ( d 2 f 1 f 2 ) + f 1 ( f 2 d 2 ) ,
u a f = d 2 f 1 f 2 f 1 f 2 h 0 = 0
θ y 2 = 3 α 0 2 f 1 2 + 4 m h 0 2 16 m θ y 1 f 1 2 ,
θ x 2 = ± ( 3 α 0 2 f 1 2 + 4 m h 0 2 ) 2 32 f 1 2 θ y 1 2 ( 3 α 0 2 f 1 2 + 4 m 2 h 0 2 ) 256 m 3 f 1 4 θ y 1 4 16 m θ y 1 f 1 2 ,
θ y 2 = 48 ( 2 ) 1 / 3 [ 4 m 4 f 1 2 h 0 2 + f 1 4 ( 3 α 0 2 m 2 + 8 m 5 θ y 1 2 ) ] + 16 ( 3 ) 2 / 3 ( V ) 2 / 3 96 ( 2 ) 2 / 3 ( 3 ) 1 / 3 m 2 f 1 2 ( V ) 1 / 3 ,
θ x 2 = 0 ,
V = θ y 1 ( 54 m 6 f 1 6 α 0 2 + 72 m 7 f 1 4 h 0 2 ) + 6 m 6 f 1 6 { 54 m 2 f 1 2 θ y 1 2 [ 3 f 1 2 α 0 2 + 4 m h 0 2 ] 2 + [ 4 m 2 h 0 2 + f 1 2 ( 3 α 0 2 8 m 3 θ y 1 2 ) ] 3 } .
θ y 2 = 4 α 0 2 f 1 2 + 3 m h 0 2 16 m θ y 1 f 1 2 ,
θ x 2 = ± 9 m 2 h 0 4 24 m f 1 2 h 0 2 ( α 0 2 + 4 m θ y 1 2 ) 16 f 1 4 ( α 0 4 + 8 α 0 2 θ y 1 2 + 16 m θ y 1 4 ) 16 m θ y 1 f 1 2 ,
θ y 2 = 48 ( 2 ) 1 / 3 m 2 f 1 2 [ 3 m 2 h 0 2 + 4 f 1 2 ( α 0 2 2 m θ y 1 2 ) ] + 16 ( 3 ) 2 / 3 ( V ) 2 / 3 96 ( 2 ) 2 / 3 ( 3 ) 1 / 3 m 2 f 1 2 ( V ) 1 / 3 ,
θ x 2 = 0 ,
V = θ y 1 ( 72 m 4 f 1 6 α 0 2 + 54 m 5 f 1 4 h 0 2 ) + 6 m 6 f 1 6 { 54 m 2 f 1 2 θ y 1 2 [ 4 f 1 2 α 0 2 + 3 m h 0 2 ] 2 + [ 3 m 2 h 0 2 + 4 f 1 2 ( α 0 2 2 m θ y 1 2 ) ] 3 } .
ε = 2 ( 3 ) 1 / 3 m 3 f 1 2 + 2 1 / 3 m 8 / 3 f 1 2 [ 3 ( 27 m 2 + 58 m + 27 ) 9 ( m + 1 ) ] 2 / 3 6 2 / 3 m 10 / 3 f 1 [ 3 ( 27 m 2 + 58 m + 27 ) 9 ( m + 1 ) ] 1 / 3 .

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