Abstract

Transmission functions are derived that are valid in the nonparaxial case for a class of lenses that will image a continuum of points along an optical axis to a single image point. This lens, which we call a logarithmic asphere, is then used in a digital camera. The resolution of the camera is limited by the pixel size of the CCD; i.e., it is not diffraction limited. Digital processing is used to recover the image, and image-plane processing is used for speed. We find a tenfold increase in the depth of field over that for the diffraction-limited case.

© 2001 Optical Society of America

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  1. J. Ojeda-Castaneda, L. R. Berriel-Valdos, and E. Montes, Opt. Lett. 8, 458 (1983).
    [CrossRef]
  2. T.-C. Poon and M. Motamedi, Appl. Opt. 26, 4612 (1987).
    [CrossRef] [PubMed]
  3. J. Ojeda-Castaneda and L. R. Berriel-Valdos, Appl. Opt. 29, 994 (1990).
    [CrossRef] [PubMed]
  4. E. R. Dowski and W. T. Cathey, Appl. Opt. 34, 1859 (1995).
    [CrossRef] [PubMed]
  5. J. van der Gracht, E. R. Dowski, W. T. Cathy, and J. P. Bowen, Proc. SPIE 2537, 279 (1995).
    [CrossRef]
  6. H. B. Wach, W. T. Cathey, and E. R. Dowski, Appl. Opt. 37, 5359 (1998).
    [CrossRef]
  7. S. C. Tucker, E. R. Dowski, and W. T. Cathey, Opt. Express 4, 467 (1999), http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  8. L. M. Soroko, in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), pp. 109–160, and references therein.
    [CrossRef]
  9. J. Sochacki, S. Bara, Z. Jaroszewicz, and A. Kolodziejczyk, Opt. Lett. 17, 7 (1992).
    [CrossRef] [PubMed]
  10. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bara, Appl. Opt. 31, 5326 (1992).
    [CrossRef] [PubMed]
  11. R. F. Steinert, C. T. Post, S. F. Brint, C. D. Fritch, D. L. Hall, L. W. Wilder, I. H. Fine, S. B. Lichtenstein, S. Masket, C. Casebeer, and H. Oksman, Opthalmology 99, 853 (1983).
    [CrossRef]
  12. R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964), p. 86.
  13. See, for instance, H. B. Dwight, Tables of Integrals and Other Mathematical Data (Macmillan, New York, 1947), Eq.  (380.001), p. 70.
  14. S. Wolfram, Mathematical Software 4.0 (Wolfram Reseach, Champaign, Ill., 2000), p. 859.
  15. This type of magneto-theological finisher is available commercially from QED Technologies, Rochester, New York.
  16. B. R. Hunt, IEEE Trans. Comput. C-22, 805 (1973).
    [CrossRef]
  17. R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), p. 218.

1999 (1)

1998 (1)

1995 (2)

E. R. Dowski and W. T. Cathey, Appl. Opt. 34, 1859 (1995).
[CrossRef] [PubMed]

J. van der Gracht, E. R. Dowski, W. T. Cathy, and J. P. Bowen, Proc. SPIE 2537, 279 (1995).
[CrossRef]

1992 (2)

1990 (1)

1987 (1)

1983 (2)

R. F. Steinert, C. T. Post, S. F. Brint, C. D. Fritch, D. L. Hall, L. W. Wilder, I. H. Fine, S. B. Lichtenstein, S. Masket, C. Casebeer, and H. Oksman, Opthalmology 99, 853 (1983).
[CrossRef]

J. Ojeda-Castaneda, L. R. Berriel-Valdos, and E. Montes, Opt. Lett. 8, 458 (1983).
[CrossRef]

1973 (1)

B. R. Hunt, IEEE Trans. Comput. C-22, 805 (1973).
[CrossRef]

Bara, S.

Berriel-Valdos, L. R.

Bowen, J. P.

J. van der Gracht, E. R. Dowski, W. T. Cathy, and J. P. Bowen, Proc. SPIE 2537, 279 (1995).
[CrossRef]

Brint, S. F.

R. F. Steinert, C. T. Post, S. F. Brint, C. D. Fritch, D. L. Hall, L. W. Wilder, I. H. Fine, S. B. Lichtenstein, S. Masket, C. Casebeer, and H. Oksman, Opthalmology 99, 853 (1983).
[CrossRef]

Casebeer, C.

R. F. Steinert, C. T. Post, S. F. Brint, C. D. Fritch, D. L. Hall, L. W. Wilder, I. H. Fine, S. B. Lichtenstein, S. Masket, C. Casebeer, and H. Oksman, Opthalmology 99, 853 (1983).
[CrossRef]

Cathey, W. T.

Cathy, W. T.

J. van der Gracht, E. R. Dowski, W. T. Cathy, and J. P. Bowen, Proc. SPIE 2537, 279 (1995).
[CrossRef]

Dowski, E. R.

Dwight, H. B.

See, for instance, H. B. Dwight, Tables of Integrals and Other Mathematical Data (Macmillan, New York, 1947), Eq.  (380.001), p. 70.

Fine, I. H.

R. F. Steinert, C. T. Post, S. F. Brint, C. D. Fritch, D. L. Hall, L. W. Wilder, I. H. Fine, S. B. Lichtenstein, S. Masket, C. Casebeer, and H. Oksman, Opthalmology 99, 853 (1983).
[CrossRef]

Fritch, C. D.

R. F. Steinert, C. T. Post, S. F. Brint, C. D. Fritch, D. L. Hall, L. W. Wilder, I. H. Fine, S. B. Lichtenstein, S. Masket, C. Casebeer, and H. Oksman, Opthalmology 99, 853 (1983).
[CrossRef]

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), p. 218.

Hall, D. L.

R. F. Steinert, C. T. Post, S. F. Brint, C. D. Fritch, D. L. Hall, L. W. Wilder, I. H. Fine, S. B. Lichtenstein, S. Masket, C. Casebeer, and H. Oksman, Opthalmology 99, 853 (1983).
[CrossRef]

Hunt, B. R.

B. R. Hunt, IEEE Trans. Comput. C-22, 805 (1973).
[CrossRef]

Jaroszewicz, Z.

Kolodziejczyk, A.

Lichtenstein, S. B.

R. F. Steinert, C. T. Post, S. F. Brint, C. D. Fritch, D. L. Hall, L. W. Wilder, I. H. Fine, S. B. Lichtenstein, S. Masket, C. Casebeer, and H. Oksman, Opthalmology 99, 853 (1983).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964), p. 86.

Masket, S.

R. F. Steinert, C. T. Post, S. F. Brint, C. D. Fritch, D. L. Hall, L. W. Wilder, I. H. Fine, S. B. Lichtenstein, S. Masket, C. Casebeer, and H. Oksman, Opthalmology 99, 853 (1983).
[CrossRef]

Montes, E.

Motamedi, M.

Ojeda-Castaneda, J.

Oksman, H.

R. F. Steinert, C. T. Post, S. F. Brint, C. D. Fritch, D. L. Hall, L. W. Wilder, I. H. Fine, S. B. Lichtenstein, S. Masket, C. Casebeer, and H. Oksman, Opthalmology 99, 853 (1983).
[CrossRef]

Poon, T.-C.

Post, C. T.

R. F. Steinert, C. T. Post, S. F. Brint, C. D. Fritch, D. L. Hall, L. W. Wilder, I. H. Fine, S. B. Lichtenstein, S. Masket, C. Casebeer, and H. Oksman, Opthalmology 99, 853 (1983).
[CrossRef]

Sochacki, J.

Soroko, L. M.

L. M. Soroko, in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), pp. 109–160, and references therein.
[CrossRef]

Steinert, R. F.

R. F. Steinert, C. T. Post, S. F. Brint, C. D. Fritch, D. L. Hall, L. W. Wilder, I. H. Fine, S. B. Lichtenstein, S. Masket, C. Casebeer, and H. Oksman, Opthalmology 99, 853 (1983).
[CrossRef]

Tucker, S. C.

van der Gracht, J.

J. van der Gracht, E. R. Dowski, W. T. Cathy, and J. P. Bowen, Proc. SPIE 2537, 279 (1995).
[CrossRef]

Wach, H. B.

Wilder, L. W.

R. F. Steinert, C. T. Post, S. F. Brint, C. D. Fritch, D. L. Hall, L. W. Wilder, I. H. Fine, S. B. Lichtenstein, S. Masket, C. Casebeer, and H. Oksman, Opthalmology 99, 853 (1983).
[CrossRef]

Wolfram, S.

S. Wolfram, Mathematical Software 4.0 (Wolfram Reseach, Champaign, Ill., 2000), p. 859.

Woods, R. E.

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), p. 218.

Appl. Opt. (5)

IEEE Trans. Comput. (1)

B. R. Hunt, IEEE Trans. Comput. C-22, 805 (1973).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Opthalmology (1)

R. F. Steinert, C. T. Post, S. F. Brint, C. D. Fritch, D. L. Hall, L. W. Wilder, I. H. Fine, S. B. Lichtenstein, S. Masket, C. Casebeer, and H. Oksman, Opthalmology 99, 853 (1983).
[CrossRef]

Proc. SPIE (1)

J. van der Gracht, E. R. Dowski, W. T. Cathy, and J. P. Bowen, Proc. SPIE 2537, 279 (1995).
[CrossRef]

Other (6)

L. M. Soroko, in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1989), pp. 109–160, and references therein.
[CrossRef]

R. C. Gonzalez and R. E. Woods, Digital Image Processing (Addison-Wesley, Reading, Mass., 1992), p. 218.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964), p. 86.

See, for instance, H. B. Dwight, Tables of Integrals and Other Mathematical Data (Macmillan, New York, 1947), Eq.  (380.001), p. 70.

S. Wolfram, Mathematical Software 4.0 (Wolfram Reseach, Champaign, Ill., 2000), p. 859.

This type of magneto-theological finisher is available commercially from QED Technologies, Rochester, New York.

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Figures (5)

Fig. 1
Fig. 1

Imaging setup: O, object; C, digital camera; DP, digital processor; D, display. See text for other definitions.

Fig. 2
Fig. 2

Notation for lens theory. See text for definitions.

Fig. 3
Fig. 3

Recovered image of a step chart from a 3-D object.

Fig. 4
Fig. 4

Segments of images obtained with (a) a Nikon conventional camera and (b) a logarithmic asphere camera.

Fig. 5
Fig. 5

Resolution versus object distance: dashed curve, Nikon camera; solid curve, logarithmic asphere; dotted line at 20 mm-1.

Equations (9)

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

rn=n/N1/2R,
xrn=s1+s2-s1n/N.
xr=s1+s2-s1r2/R2.
tr=exp-iϕr,
L=r2+x2+ϕrλ0/2π+r2+t2,
ϕr=-2πλ00r(rr2+t21/2+rr2+s1+s2-s1r2/R2212)dr.
ϕr=-{2πλ0r2+t212-t+πλ0R2s2-s1×[ln(2s2-s1R2r2+s1+s2-s1R2r2212+s1+s2-s1R2r2+1)-ln4s2-s1R2s1+1]}.
ϕr=-2πλ00.02r2-8×10-6r4+6.4×10-9r6-6.4×10-12r8+7×10-15r10+7.435×10-4r2+-7.563×10-5r4+5.1431×10-6r6+-1.803×10-7r8+2.4614×10-9r10.
[00-0.070000.51-0.850.510-0.07-0.855-0.85-0.0700.51-0.850.51000-0.0700],[0-10-16-10-10].

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