Abstract

We report experimental verification of an extended depth of focus (EDF) system with near-diffraction-limited performance capabilities. Dowski and Cathey [Appl. Opt. 34, 1859–1866 (1995)] described the theory of this system in detail. We can create an EDF system by modifying a standard incoherent optical system with a special cubic phase plate placed at the aperture stop. We briefly review the theory and present the first optical experimental verification of this EDF system. The phase plate codes the wave front, producing a modified optical transfer function. Once the image is transformed into digital form, a signal-processing step decodes the image and produces the final in-focus image. We have produced a number of images from various optical systems using the phase plate, thus demonstrating the success of this EDF system.

© 1997 Optical Society of America

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References

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  1. K. Brenner, A. Lohmann, J. Ojeda-Casteñeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
    [CrossRef]
  2. H. Bartelt, J. Ojeda-Casteñeda, E. S. Enrique, “Misfocus tolerance seen by simple inspection of the ambiguity function,” Appl. Opt. 23, 2693–2696 (1984).
    [CrossRef] [PubMed]
  3. E. R. Dowski, W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859–1866 (1995).
    [CrossRef] [PubMed]
  4. A. R. FitzGerrell, E. R. Dowski, W. T. Cathey, “Defocus transfer function for circularly symmetric pupils,” Appl. Opt. 36, 5796–5804 (1997).
    [CrossRef] [PubMed]
  5. J. van der Gracht, E. R. Dowski, M. G. Taylor, D. M. Deaver, “Broadband behavior of an optical-digital focus-invariant system,” Opt. Lett. 21, 919–921 (1996).
    [CrossRef] [PubMed]
  6. J. van der Gracht, W. T. Dowski, E. R. Cathey, J. P. Bowen, “Aspheric optical elements for extended depth of field imaging,” in Novel Optical Systems Design and Optimization, J. M. Sasian, ed., Proc. SPIE2537, 279–288 (1995).
  7. B. R. Frieden, “Image enhancement and restoration,” in Topics In Applied Physics Vol. 6 of Picture Processing and Digital Filtering, T. S. Huang, ed. (Springer-Verlag, New York, 1979), pp. 177–248.
  8. H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, New Jersey, 1977), Chap. 8, pp. 147–152.
  9. C. W. Helstrom, “Image restoration by the method of least squares,” J. Opt. Soc. Am. 57, 297–303 (1967).
    [CrossRef]
  10. Amnon Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University, Oxford, 1995), p. 431.

1997 (1)

1996 (1)

1995 (1)

1984 (1)

1983 (1)

K. Brenner, A. Lohmann, J. Ojeda-Casteñeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

1967 (1)

Andrews, H. C.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, New Jersey, 1977), Chap. 8, pp. 147–152.

Bartelt, H.

Bowen, J. P.

J. van der Gracht, W. T. Dowski, E. R. Cathey, J. P. Bowen, “Aspheric optical elements for extended depth of field imaging,” in Novel Optical Systems Design and Optimization, J. M. Sasian, ed., Proc. SPIE2537, 279–288 (1995).

Brenner, K.

K. Brenner, A. Lohmann, J. Ojeda-Casteñeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Cathey, E. R.

J. van der Gracht, W. T. Dowski, E. R. Cathey, J. P. Bowen, “Aspheric optical elements for extended depth of field imaging,” in Novel Optical Systems Design and Optimization, J. M. Sasian, ed., Proc. SPIE2537, 279–288 (1995).

Cathey, W. T.

Deaver, D. M.

Dowski, E. R.

Dowski, W. T.

J. van der Gracht, W. T. Dowski, E. R. Cathey, J. P. Bowen, “Aspheric optical elements for extended depth of field imaging,” in Novel Optical Systems Design and Optimization, J. M. Sasian, ed., Proc. SPIE2537, 279–288 (1995).

Enrique, E. S.

FitzGerrell, A. R.

Frieden, B. R.

B. R. Frieden, “Image enhancement and restoration,” in Topics In Applied Physics Vol. 6 of Picture Processing and Digital Filtering, T. S. Huang, ed. (Springer-Verlag, New York, 1979), pp. 177–248.

Helstrom, C. W.

Hunt, B. R.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, New Jersey, 1977), Chap. 8, pp. 147–152.

Lohmann, A.

K. Brenner, A. Lohmann, J. Ojeda-Casteñeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Ojeda-Casteñeda, J.

H. Bartelt, J. Ojeda-Casteñeda, E. S. Enrique, “Misfocus tolerance seen by simple inspection of the ambiguity function,” Appl. Opt. 23, 2693–2696 (1984).
[CrossRef] [PubMed]

K. Brenner, A. Lohmann, J. Ojeda-Casteñeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Taylor, M. G.

van der Gracht, J.

J. van der Gracht, E. R. Dowski, M. G. Taylor, D. M. Deaver, “Broadband behavior of an optical-digital focus-invariant system,” Opt. Lett. 21, 919–921 (1996).
[CrossRef] [PubMed]

J. van der Gracht, W. T. Dowski, E. R. Cathey, J. P. Bowen, “Aspheric optical elements for extended depth of field imaging,” in Novel Optical Systems Design and Optimization, J. M. Sasian, ed., Proc. SPIE2537, 279–288 (1995).

Yariv, Amnon

Amnon Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University, Oxford, 1995), p. 431.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

K. Brenner, A. Lohmann, J. Ojeda-Casteñeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323–326 (1983).
[CrossRef]

Opt. Lett. (1)

Other (4)

J. van der Gracht, W. T. Dowski, E. R. Cathey, J. P. Bowen, “Aspheric optical elements for extended depth of field imaging,” in Novel Optical Systems Design and Optimization, J. M. Sasian, ed., Proc. SPIE2537, 279–288 (1995).

B. R. Frieden, “Image enhancement and restoration,” in Topics In Applied Physics Vol. 6 of Picture Processing and Digital Filtering, T. S. Huang, ed. (Springer-Verlag, New York, 1979), pp. 177–248.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, New Jersey, 1977), Chap. 8, pp. 147–152.

Amnon Yariv, Optical Electronics in Modern Communications, 5th ed. (Oxford University, Oxford, 1995), p. 431.

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Figures (13)

Fig. 1
Fig. 1

DTF of a one-dimensional system with lines showing (a) in focus, (b) moderate defocus (π/2 waves), and (c) severe defocus (π waves). White denotes low values of the MTF, and black denotes high values of the MTF.

Fig. 2
Fig. 2

One-dimensional MTF’s for the three focus positions of Fig. 1 (0, π/2, and π waves of defocus).

Fig. 3
Fig. 3

Three-dimensional diagram of general phase mask.

Fig. 4
Fig. 4

DTF of a focus-invariant imaging system with a CPP.

Fig. 5
Fig. 5

One-dimensional MTF’s from a CPP focus-invariant system for the same focal planes used in Fig. 2.

Fig. 6
Fig. 6

Information systems view of CPP system.

Fig. 7
Fig. 7

Measured MTF’s of a standard system (a) in focus and (b) with 1.6 waves of defocus, and MTF’s of a focus-invariant system (c) in focus and (d) with 1.6 waves of defocus. The difference between the MTF’s of the focus-invariant systems at higher frequencies is reduced in the signal-processing step, as shown in Fig. 8.

Fig. 8
Fig. 8

Measured in-focus and defocused MTF’s of (a) and (b) a standard and (c) and (d) a focus-invariant system after signal processing. Note the attenuation at higher frequencies in the filtered focus-invariant MTF’s. This helps to eliminate noise magnification in the inverse filtering process.

Fig. 9
Fig. 9

Measured PSF’s of a standard imaging system (a) in focus and (b) with 1.6 waves of defocus (and increased exposure for visibility). Measured PSF’s of a focus-invariant imaging system (c) in focus and (d) with 1.6 waves of defocus.

Fig. 10
Fig. 10

General experimental setup of a focus-invariant optical–digital system. The CPP at the aperture stop of a two-lens system codes the wave front.

Fig. 11
Fig. 11

Square-wave imaging with (a) a standard and (c) an EDF system. The traces of image brightness for the two cases are shown in (b). The phase shift arising from misfocus in the standard image trace does not occur in the EDF image trace.

Fig. 12
Fig. 12

Images of small shells. (a) Standard imaging system with F/10.5. (b) Intermediate image from EDF imaging system with F/10.5. (c) Reduced-aperture standard imaging system with F/52.5. (d) EDF imaging system (F/10.5) image after signal processing.

Fig. 13
Fig. 13

Images of inclined text. (a) Standard imaging system with F/6.3. (b) Intermediate image from an EDF imaging system with F/6.3. (c) Reduced-aperture standard imaging system with F/37.5. (d) EDF imaging system image after signal processing.

Equations (10)

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Px, y=αx3+y3.
α=2πξ/λ,
SN=Psη2hνΔν,
SN=No. of electrons2.
e2=Hf-g2.
e2=Hf-g*Hf-g.
e2=Hf*-g*Hf-g=f*H*Hf-g*Hf-f*H*g+g*g.
de2f*=H*Hf+f*H*H*-g*H*-H*g.
H*Hf+H*Hf-H*g-H*g=0,2H*Hf-2H*g=0.
H*Hf=H*g,f=H*H-1H*g.

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