Abstract

The signal processing method used in transaxial tomography, reconstruction from projections, is applied to the design of a spherical gradient index corrector for the concentric “noflare” lens. The method yields a spherical gradient index lens that corrects the entire field of the lens. Motivation for the method is provided by a comparison of the Schmidt and super-Schmidt camera designs.

© 2010 Optical Society of America

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References

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  1. R. M. Mercereau and A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE 62, 1319–1338 (1974).
    [CrossRef]
  2. A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, 1995), pp. 332 and 394.
  3. Walther’s reference to Hekker’s discussion of the “noflare” lens isF. Hekker, “On concentric optical systems,” Ph.D. thesis (Delft University of Technology, 1947).
  4. J. G. Baker, “Schmidt image former with spherical aberration corrector,” U.S. patent 2,458,132 (4 January 1949).
  5. F. L. Whipple, “The Harvard Photographic Meteor Program,” Sky Telesc. 8, 90 (1949).
  6. H. C. King, The History of the Telescope (Dover, 1979), pp. 365–369.
  7. H. Goldstein, Classical Mechanics (Addison-Wesley, 1950), pp. 81–82.
  8. The term “impact parameter” has been previously introduced in the optics literature with the refractive index as a multiplicative factor. See C. M. Vest, “Interferometry of strongly refracting axisymmetric phase objects,” Appl. Opt. 14, 1601–1606 (1975).
    [CrossRef] [PubMed]
  9. R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
    [CrossRef]
  10. M. Born and E. Wolf, Principles of Optics (Pergamon, 1969), p. 122.
  11. H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambridge U. Press, 1970), p. 7.
  12. M. Born and E. Wolf, Principles of Optics (Pergamon, 1969), pp. 147–149.
  13. J. F. Kordas, I. T. Lewis, B. A. Wilson, D. P. Nielsen, H-S. Park, R. E. Priest, R. Hills, M. J. Shannon, A. G. Ledebuhr, and L. D. Pleasance, “Star tracker stellar compass for the Clementine mission,” Proc. SPIE 2466, 70–83(1995).
    [CrossRef]
  14. V. D. Shargorodsky, V. P. Vasiliev, N. M. Soyuzova, V. B. Burmistrov, I. S. Gashkin, M. S. Belov, T. I. Khorosheva, and E. Nikolaev, “Experimental spherical retroreflector on board of the Meteor-3M satellite,” presented at the 12th International Workshop on Laser Ranging, Matera, Italy, 13–17 November 2000, http://cddis.gsfc.nasa.gov/lw12/index.html.
  15. J. P. Oakley, “Whole-angle spherical retroreflector using concentric layers of homogeneous optical material,” Appl. Opt. 46, 1026–1031 (2007).
    [CrossRef] [PubMed]
  16. B. E. Bernacki, N. C. Anheier, K. Krishnaswami, B. D. Cannon, and K. B. Binkley, “Design and fabrication of efficient miniature retroreflectors for the mid-infrared,” Proc. SPIE 6940, 69400X (2008).
    [CrossRef]
  17. K. Kikuchi, T. Morikawa, J. Shimada, and K. Sakurai, “Cladded radially inhomogeneous sphere lenses,” Appl. Opt. 20, 388–394 (1981).
    [CrossRef] [PubMed]
  18. K. Kikuchi, T. Morikawa, J. Shimada, and K. Sakurai, “Graded-index sphere lens with hemispherical rod cladding,” Appl. Opt. 21, 2734–2738 (1982).
    [CrossRef] [PubMed]
  19. Y. Koike, Y. Sumi, and Y. Ohtsuka, “Spherical gradient-index sphere lens,” Appl. Opt. 25, 3356–3363 (1986).
    [CrossRef] [PubMed]
  20. Y. Koike, A. Kanemitsu, Y. Shioda, E. Nihei, and Y. Ohtsuka, “Spherical gradient-index polymer lens with low spherical aberration,” Appl. Opt. 33, 3394–3400 (1994).
    [CrossRef] [PubMed]
  21. V. Handerek, H. McArdle, T. Willats, N. Psaila, and L. Laycock, “Experimental retroreflectors with very wide field of view,” Proc. SPIE 5986598611 (2005).
    [CrossRef]
  22. A. E. Conrady, Applied Optics and Optical Design, Part One (Dover, 1992), pp. 6–7.
  23. M. Kidger, Fundamental Optical Design (SPIE, 2002), p. 7.

2008 (1)

B. E. Bernacki, N. C. Anheier, K. Krishnaswami, B. D. Cannon, and K. B. Binkley, “Design and fabrication of efficient miniature retroreflectors for the mid-infrared,” Proc. SPIE 6940, 69400X (2008).
[CrossRef]

2007 (1)

2005 (1)

V. Handerek, H. McArdle, T. Willats, N. Psaila, and L. Laycock, “Experimental retroreflectors with very wide field of view,” Proc. SPIE 5986598611 (2005).
[CrossRef]

2002 (1)

M. Kidger, Fundamental Optical Design (SPIE, 2002), p. 7.

1995 (2)

J. F. Kordas, I. T. Lewis, B. A. Wilson, D. P. Nielsen, H-S. Park, R. E. Priest, R. Hills, M. J. Shannon, A. G. Ledebuhr, and L. D. Pleasance, “Star tracker stellar compass for the Clementine mission,” Proc. SPIE 2466, 70–83(1995).
[CrossRef]

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, 1995), pp. 332 and 394.

1994 (1)

1992 (1)

A. E. Conrady, Applied Optics and Optical Design, Part One (Dover, 1992), pp. 6–7.

1986 (1)

1982 (1)

1981 (1)

1979 (1)

H. C. King, The History of the Telescope (Dover, 1979), pp. 365–369.

1975 (1)

1974 (1)

R. M. Mercereau and A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE 62, 1319–1338 (1974).
[CrossRef]

1970 (1)

H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambridge U. Press, 1970), p. 7.

1969 (2)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1969), pp. 147–149.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1969), p. 122.

1956 (1)

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[CrossRef]

1950 (1)

H. Goldstein, Classical Mechanics (Addison-Wesley, 1950), pp. 81–82.

1949 (1)

F. L. Whipple, “The Harvard Photographic Meteor Program,” Sky Telesc. 8, 90 (1949).

1947 (1)

Walther’s reference to Hekker’s discussion of the “noflare” lens isF. Hekker, “On concentric optical systems,” Ph.D. thesis (Delft University of Technology, 1947).

Anheier, N. C.

B. E. Bernacki, N. C. Anheier, K. Krishnaswami, B. D. Cannon, and K. B. Binkley, “Design and fabrication of efficient miniature retroreflectors for the mid-infrared,” Proc. SPIE 6940, 69400X (2008).
[CrossRef]

Baker, J. G.

J. G. Baker, “Schmidt image former with spherical aberration corrector,” U.S. patent 2,458,132 (4 January 1949).

Belov, M. S.

V. D. Shargorodsky, V. P. Vasiliev, N. M. Soyuzova, V. B. Burmistrov, I. S. Gashkin, M. S. Belov, T. I. Khorosheva, and E. Nikolaev, “Experimental spherical retroreflector on board of the Meteor-3M satellite,” presented at the 12th International Workshop on Laser Ranging, Matera, Italy, 13–17 November 2000, http://cddis.gsfc.nasa.gov/lw12/index.html.

Bernacki, B. E.

B. E. Bernacki, N. C. Anheier, K. Krishnaswami, B. D. Cannon, and K. B. Binkley, “Design and fabrication of efficient miniature retroreflectors for the mid-infrared,” Proc. SPIE 6940, 69400X (2008).
[CrossRef]

Binkley, K. B.

B. E. Bernacki, N. C. Anheier, K. Krishnaswami, B. D. Cannon, and K. B. Binkley, “Design and fabrication of efficient miniature retroreflectors for the mid-infrared,” Proc. SPIE 6940, 69400X (2008).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1969), pp. 147–149.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1969), p. 122.

Bracewell, R. N.

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[CrossRef]

Buchdahl, H. A.

H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambridge U. Press, 1970), p. 7.

Burmistrov, V. B.

V. D. Shargorodsky, V. P. Vasiliev, N. M. Soyuzova, V. B. Burmistrov, I. S. Gashkin, M. S. Belov, T. I. Khorosheva, and E. Nikolaev, “Experimental spherical retroreflector on board of the Meteor-3M satellite,” presented at the 12th International Workshop on Laser Ranging, Matera, Italy, 13–17 November 2000, http://cddis.gsfc.nasa.gov/lw12/index.html.

Cannon, B. D.

B. E. Bernacki, N. C. Anheier, K. Krishnaswami, B. D. Cannon, and K. B. Binkley, “Design and fabrication of efficient miniature retroreflectors for the mid-infrared,” Proc. SPIE 6940, 69400X (2008).
[CrossRef]

Conrady, A. E.

A. E. Conrady, Applied Optics and Optical Design, Part One (Dover, 1992), pp. 6–7.

Gashkin, I. S.

V. D. Shargorodsky, V. P. Vasiliev, N. M. Soyuzova, V. B. Burmistrov, I. S. Gashkin, M. S. Belov, T. I. Khorosheva, and E. Nikolaev, “Experimental spherical retroreflector on board of the Meteor-3M satellite,” presented at the 12th International Workshop on Laser Ranging, Matera, Italy, 13–17 November 2000, http://cddis.gsfc.nasa.gov/lw12/index.html.

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, 1950), pp. 81–82.

Handerek, V.

V. Handerek, H. McArdle, T. Willats, N. Psaila, and L. Laycock, “Experimental retroreflectors with very wide field of view,” Proc. SPIE 5986598611 (2005).
[CrossRef]

Hekker, F.

Walther’s reference to Hekker’s discussion of the “noflare” lens isF. Hekker, “On concentric optical systems,” Ph.D. thesis (Delft University of Technology, 1947).

Hills, R.

J. F. Kordas, I. T. Lewis, B. A. Wilson, D. P. Nielsen, H-S. Park, R. E. Priest, R. Hills, M. J. Shannon, A. G. Ledebuhr, and L. D. Pleasance, “Star tracker stellar compass for the Clementine mission,” Proc. SPIE 2466, 70–83(1995).
[CrossRef]

Kanemitsu, A.

Khorosheva, T. I.

V. D. Shargorodsky, V. P. Vasiliev, N. M. Soyuzova, V. B. Burmistrov, I. S. Gashkin, M. S. Belov, T. I. Khorosheva, and E. Nikolaev, “Experimental spherical retroreflector on board of the Meteor-3M satellite,” presented at the 12th International Workshop on Laser Ranging, Matera, Italy, 13–17 November 2000, http://cddis.gsfc.nasa.gov/lw12/index.html.

Kidger, M.

M. Kidger, Fundamental Optical Design (SPIE, 2002), p. 7.

Kikuchi, K.

King, H. C.

H. C. King, The History of the Telescope (Dover, 1979), pp. 365–369.

Koike, Y.

Kordas, J. F.

J. F. Kordas, I. T. Lewis, B. A. Wilson, D. P. Nielsen, H-S. Park, R. E. Priest, R. Hills, M. J. Shannon, A. G. Ledebuhr, and L. D. Pleasance, “Star tracker stellar compass for the Clementine mission,” Proc. SPIE 2466, 70–83(1995).
[CrossRef]

Krishnaswami, K.

B. E. Bernacki, N. C. Anheier, K. Krishnaswami, B. D. Cannon, and K. B. Binkley, “Design and fabrication of efficient miniature retroreflectors for the mid-infrared,” Proc. SPIE 6940, 69400X (2008).
[CrossRef]

Laycock, L.

V. Handerek, H. McArdle, T. Willats, N. Psaila, and L. Laycock, “Experimental retroreflectors with very wide field of view,” Proc. SPIE 5986598611 (2005).
[CrossRef]

Ledebuhr, A. G.

J. F. Kordas, I. T. Lewis, B. A. Wilson, D. P. Nielsen, H-S. Park, R. E. Priest, R. Hills, M. J. Shannon, A. G. Ledebuhr, and L. D. Pleasance, “Star tracker stellar compass for the Clementine mission,” Proc. SPIE 2466, 70–83(1995).
[CrossRef]

Lewis, I. T.

J. F. Kordas, I. T. Lewis, B. A. Wilson, D. P. Nielsen, H-S. Park, R. E. Priest, R. Hills, M. J. Shannon, A. G. Ledebuhr, and L. D. Pleasance, “Star tracker stellar compass for the Clementine mission,” Proc. SPIE 2466, 70–83(1995).
[CrossRef]

McArdle, H.

V. Handerek, H. McArdle, T. Willats, N. Psaila, and L. Laycock, “Experimental retroreflectors with very wide field of view,” Proc. SPIE 5986598611 (2005).
[CrossRef]

Mercereau, R. M.

R. M. Mercereau and A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE 62, 1319–1338 (1974).
[CrossRef]

Morikawa, T.

Nielsen, D. P.

J. F. Kordas, I. T. Lewis, B. A. Wilson, D. P. Nielsen, H-S. Park, R. E. Priest, R. Hills, M. J. Shannon, A. G. Ledebuhr, and L. D. Pleasance, “Star tracker stellar compass for the Clementine mission,” Proc. SPIE 2466, 70–83(1995).
[CrossRef]

Nihei, E.

Nikolaev, E.

V. D. Shargorodsky, V. P. Vasiliev, N. M. Soyuzova, V. B. Burmistrov, I. S. Gashkin, M. S. Belov, T. I. Khorosheva, and E. Nikolaev, “Experimental spherical retroreflector on board of the Meteor-3M satellite,” presented at the 12th International Workshop on Laser Ranging, Matera, Italy, 13–17 November 2000, http://cddis.gsfc.nasa.gov/lw12/index.html.

Oakley, J. P.

Ohtsuka, Y.

Oppenheim, A. V.

R. M. Mercereau and A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE 62, 1319–1338 (1974).
[CrossRef]

Park, H-S.

J. F. Kordas, I. T. Lewis, B. A. Wilson, D. P. Nielsen, H-S. Park, R. E. Priest, R. Hills, M. J. Shannon, A. G. Ledebuhr, and L. D. Pleasance, “Star tracker stellar compass for the Clementine mission,” Proc. SPIE 2466, 70–83(1995).
[CrossRef]

Pleasance, L. D.

J. F. Kordas, I. T. Lewis, B. A. Wilson, D. P. Nielsen, H-S. Park, R. E. Priest, R. Hills, M. J. Shannon, A. G. Ledebuhr, and L. D. Pleasance, “Star tracker stellar compass for the Clementine mission,” Proc. SPIE 2466, 70–83(1995).
[CrossRef]

Priest, R. E.

J. F. Kordas, I. T. Lewis, B. A. Wilson, D. P. Nielsen, H-S. Park, R. E. Priest, R. Hills, M. J. Shannon, A. G. Ledebuhr, and L. D. Pleasance, “Star tracker stellar compass for the Clementine mission,” Proc. SPIE 2466, 70–83(1995).
[CrossRef]

Psaila, N.

V. Handerek, H. McArdle, T. Willats, N. Psaila, and L. Laycock, “Experimental retroreflectors with very wide field of view,” Proc. SPIE 5986598611 (2005).
[CrossRef]

Sakurai, K.

Shannon, M. J.

J. F. Kordas, I. T. Lewis, B. A. Wilson, D. P. Nielsen, H-S. Park, R. E. Priest, R. Hills, M. J. Shannon, A. G. Ledebuhr, and L. D. Pleasance, “Star tracker stellar compass for the Clementine mission,” Proc. SPIE 2466, 70–83(1995).
[CrossRef]

Shargorodsky, V. D.

V. D. Shargorodsky, V. P. Vasiliev, N. M. Soyuzova, V. B. Burmistrov, I. S. Gashkin, M. S. Belov, T. I. Khorosheva, and E. Nikolaev, “Experimental spherical retroreflector on board of the Meteor-3M satellite,” presented at the 12th International Workshop on Laser Ranging, Matera, Italy, 13–17 November 2000, http://cddis.gsfc.nasa.gov/lw12/index.html.

Shimada, J.

Shioda, Y.

Soyuzova, N. M.

V. D. Shargorodsky, V. P. Vasiliev, N. M. Soyuzova, V. B. Burmistrov, I. S. Gashkin, M. S. Belov, T. I. Khorosheva, and E. Nikolaev, “Experimental spherical retroreflector on board of the Meteor-3M satellite,” presented at the 12th International Workshop on Laser Ranging, Matera, Italy, 13–17 November 2000, http://cddis.gsfc.nasa.gov/lw12/index.html.

Sumi, Y.

Vasiliev, V. P.

V. D. Shargorodsky, V. P. Vasiliev, N. M. Soyuzova, V. B. Burmistrov, I. S. Gashkin, M. S. Belov, T. I. Khorosheva, and E. Nikolaev, “Experimental spherical retroreflector on board of the Meteor-3M satellite,” presented at the 12th International Workshop on Laser Ranging, Matera, Italy, 13–17 November 2000, http://cddis.gsfc.nasa.gov/lw12/index.html.

Vest, C. M.

Walther, A.

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, 1995), pp. 332 and 394.

Whipple, F. L.

F. L. Whipple, “The Harvard Photographic Meteor Program,” Sky Telesc. 8, 90 (1949).

Willats, T.

V. Handerek, H. McArdle, T. Willats, N. Psaila, and L. Laycock, “Experimental retroreflectors with very wide field of view,” Proc. SPIE 5986598611 (2005).
[CrossRef]

Wilson, B. A.

J. F. Kordas, I. T. Lewis, B. A. Wilson, D. P. Nielsen, H-S. Park, R. E. Priest, R. Hills, M. J. Shannon, A. G. Ledebuhr, and L. D. Pleasance, “Star tracker stellar compass for the Clementine mission,” Proc. SPIE 2466, 70–83(1995).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1969), p. 122.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1969), pp. 147–149.

Appl. Opt. (6)

Aust. J. Phys. (1)

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[CrossRef]

Proc. IEEE (1)

R. M. Mercereau and A. V. Oppenheim, “Digital reconstruction of multidimensional signals from their projections,” Proc. IEEE 62, 1319–1338 (1974).
[CrossRef]

Proc. SPIE (3)

J. F. Kordas, I. T. Lewis, B. A. Wilson, D. P. Nielsen, H-S. Park, R. E. Priest, R. Hills, M. J. Shannon, A. G. Ledebuhr, and L. D. Pleasance, “Star tracker stellar compass for the Clementine mission,” Proc. SPIE 2466, 70–83(1995).
[CrossRef]

B. E. Bernacki, N. C. Anheier, K. Krishnaswami, B. D. Cannon, and K. B. Binkley, “Design and fabrication of efficient miniature retroreflectors for the mid-infrared,” Proc. SPIE 6940, 69400X (2008).
[CrossRef]

V. Handerek, H. McArdle, T. Willats, N. Psaila, and L. Laycock, “Experimental retroreflectors with very wide field of view,” Proc. SPIE 5986598611 (2005).
[CrossRef]

Sky Telesc. (1)

F. L. Whipple, “The Harvard Photographic Meteor Program,” Sky Telesc. 8, 90 (1949).

Other (11)

H. C. King, The History of the Telescope (Dover, 1979), pp. 365–369.

H. Goldstein, Classical Mechanics (Addison-Wesley, 1950), pp. 81–82.

V. D. Shargorodsky, V. P. Vasiliev, N. M. Soyuzova, V. B. Burmistrov, I. S. Gashkin, M. S. Belov, T. I. Khorosheva, and E. Nikolaev, “Experimental spherical retroreflector on board of the Meteor-3M satellite,” presented at the 12th International Workshop on Laser Ranging, Matera, Italy, 13–17 November 2000, http://cddis.gsfc.nasa.gov/lw12/index.html.

A. E. Conrady, Applied Optics and Optical Design, Part One (Dover, 1992), pp. 6–7.

M. Kidger, Fundamental Optical Design (SPIE, 2002), p. 7.

A. Walther, The Ray and Wave Theory of Lenses (Cambridge U. Press, 1995), pp. 332 and 394.

Walther’s reference to Hekker’s discussion of the “noflare” lens isF. Hekker, “On concentric optical systems,” Ph.D. thesis (Delft University of Technology, 1947).

J. G. Baker, “Schmidt image former with spherical aberration corrector,” U.S. patent 2,458,132 (4 January 1949).

M. Born and E. Wolf, Principles of Optics (Pergamon, 1969), p. 122.

H. A. Buchdahl, An Introduction to Hamiltonian Optics(Cambridge U. Press, 1970), p. 7.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1969), pp. 147–149.

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Figures (12)

Fig. 1
Fig. 1

Diagram of a Schmidt camera: a concentric geometry is used but concentricity is broken with a planar corrector plate. The drawing exaggerates the curvature of corrector plate.

Fig. 2
Fig. 2

Early form of the super-Schmidt.

Fig. 3
Fig. 3

Illustrative noflare lens diagram showing the impact pa rameter s of a ray from the focal surface that passes through the core of the lens with focal surface radius R 0 and lens surface radii R 1 and R 2 . Rays from a focal point are fanned out as they cross the aperture stop. From the chief ray, the fan angle is β ( s ) . The chief ray from the focal point goes through the center of the aperture stop, which is itself centered on the center of the lens.

Fig. 4
Fig. 4

Illustrative ray diagram showing the ray paths through a concentric lens have mirror symmetry about the plane containing the line segment s, which is perpendicular to the ray as it passes the lens core. Corresponding angles of incidence and refraction on either side of the symmetry plane are equal.

Fig. 5
Fig. 5

This illustrative noflare lens diagram labels the angles used in ray tracing, calculation of the exit angle of a ray from the lens, and calculation of OPL ( s ) from the focal point to an external reference plane.

Fig. 6
Fig. 6

Curves of OPD ( s ) = [ OPL ( s ) OPL ( 0 ) ] for the noflare lens for the three values of R 0 provided in the text.

Fig. 7
Fig. 7

A smooth curve that tapers to zero at R 3 is spliced to the OPD curve at s = 3.73 mm , corresponding to the marginal ray. In this example, the OPD curve is for the “larger defocus” value of R 0 ( 28.46 mm ) and the OPD curve is shifted upward to join the taper curve.

Fig. 8
Fig. 8

Missing spherical gradient function obtained by taking the negative of the inverse Abel transformed, shifted, and tapered OPD curve. The extreme positive and negative slopes are 1.179 × 10 3 mm 1 and 1.703 × 10 3 mm 1 , respectively.

Fig. 9
Fig. 9

OPL ( s ) curves of the corrected and uncorrected lenses are plotted on the same scale. The peak-to-peak OPL variation of the uncorrected lens is 1.6 waves at 0.55 μm . This error is significantly reduced in the corrected lens.

Fig. 10
Fig. 10

OPL ( s ) curve of the corrected lens plotted with scale markers at steps of 1 × 10 6 mm ( 1 nm ). After correction, the peak- to-peak OPL variation of the “larger defocus” lens is 8.7 nm .

Fig. 11
Fig. 11

To fit the collimation error (deviation of collimated rays from the chief ray) of both the corrected and uncorrected lenses on one graph, the error of the uncorrected lens is divided by 32. With correction, the peak-to-peak collimation error is 3.7 × 10 5 rad .

Fig. 12
Fig. 12

To express the angular variables θ, α 0 , α 0 , α 1 , α 1 , β, γ 0 , and γ 1 as functions of s, start at the shaded triangle and work outward. Mirror symmetry about the line segment s simplifies the analysis.

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