Abstract

A hybrid imaging system combines a modified optical imaging system and a digital postprocessing step. We describe a spatial-domain method for designing a pupil phase plate to extend the depth of field of an incoherent hybrid imaging system with a rectangular aperture. We use this method to obtain a pupil phase plate to extend the depth of field, which we refer to as a logarithmic phase plate. Introducing a logarithmic phase plate at the exit pupil of a simulated diffraction-limited system and digitally processing the detector’s output extend the depth of field by an order of magnitude more than the Hopkins defocus criterion. We also examine the effect of using a charge-coupled device optical detector, instead of an ideal optical detector, on the extension of the depth of field. Finally, we compare the performance of the logarithmic phase plate with that of a cubic phase plate in extending the depth of field of a hybrid imaging system with a rectangular aperture.

© 2004 Optical Society of America

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References

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2002 (1)

2001 (1)

1997 (1)

1995 (1)

1994 (1)

1990 (1)

1989 (1)

1988 (1)

1986 (1)

1984 (1)

1983 (1)

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

1972 (1)

G. Hausler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

1971 (1)

1964 (1)

E. N. Fowle, “The design of FM pulse compression signals,” IEEE Trans. Inf. Theory IT-10, 61–67 (1964).
[CrossRef]

1961 (1)

E. L. Key, E. N. Fowle, R. D. Haggarty, “A method for designing signals of large time-bandwidth product,” IRE Int. Conv. Rec. 4, 146–155 (1961).

1960 (1)

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Andres, P.

Berriel-Valdos, L. R.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1997).

Bradburn, S.

Brenner, K.

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

Cathey, W. T.

Chi, W.-L.

Copson, E. T.

E. T. Copson, Asymptotic Expansions (Cambridge U. Press, Cambridge, 1967).

Diaz, A.

Dowski, E. R.

Fowle, E. N.

E. N. Fowle, “The design of FM pulse compression signals,” IEEE Trans. Inf. Theory IT-10, 61–67 (1964).
[CrossRef]

E. L. Key, E. N. Fowle, R. D. Haggarty, “A method for designing signals of large time-bandwidth product,” IRE Int. Conv. Rec. 4, 146–155 (1961).

Franks, L. E.

L. E. Franks, Signal Theory, rev. ed. (Dowden Culver, Stroudsburg, Pa., 1981).

Frieden, B. R.

George, N.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 6, pp. 126–171.

Greivenkamp, J. E.

Haggarty, R. D.

E. L. Key, E. N. Fowle, R. D. Haggarty, “A method for designing signals of large time-bandwidth product,” IRE Int. Conv. Rec. 4, 146–155 (1961).

Hausler, G.

G. Hausler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Key, E. L.

E. L. Key, E. N. Fowle, R. D. Haggarty, “A method for designing signals of large time-bandwidth product,” IRE Int. Conv. Rec. 4, 146–155 (1961).

Lohmann, A.

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

Lowman, A. E.

Mino, M.

Noyola-Isgleas, A.

Ojeda-Castenada, J.

Okano, Y.

Ramos, R.

Rhodes, W. T.

Rushford, C. K.

Sherif, S. S.

Tepichin, E.

Welford, W. T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1997).

Appl. Opt. (8)

IEEE Trans. Inf. Theory (1)

E. N. Fowle, “The design of FM pulse compression signals,” IEEE Trans. Inf. Theory IT-10, 61–67 (1964).
[CrossRef]

IRE Int. Conv. Rec. (1)

E. L. Key, E. N. Fowle, R. D. Haggarty, “A method for designing signals of large time-bandwidth product,” IRE Int. Conv. Rec. 4, 146–155 (1961).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

G. Hausler, “A method to increase the depth of focus by two step image processing,” Opt. Commun. 6, 38–42 (1972).
[CrossRef]

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

Opt. Lett. (2)

Proc. R. Soc. London Ser. A (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Other (6)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Cambridge U. Press, Cambridge, 1997).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 6, pp. 126–171.

E. T. Copson, Asymptotic Expansions (Cambridge U. Press, Cambridge, 1967).

Focus Software Inc., “Zemax optical design program: user’s guide,” (Focus Software, Tucson, Ariz., 2000).

L. E. Franks, Signal Theory, rev. ed. (Dowden Culver, Stroudsburg, Pa., 1981).

A. K. Jain, Fundamentals of Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1989).

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Figures (15)

Fig. 1
Fig. 1

Hybrid imaging system with a phase plate at its exit pupil.

Fig. 2
Fig. 2

Profile of a logarithmic phase plate.

Fig. 3
Fig. 3

Defocused diffraction-limited PSF with a logarithmic phase plate.

Fig. 4
Fig. 4

Defocused diffraction-limited PSF with a clear rectangular aperture.

Fig. 5
Fig. 5

Spot diagrams through focus with a clear rectangular aperture.

Fig. 6
Fig. 6

Spot diagrams through focus with a logarithmic phase plate.

Fig. 7
Fig. 7

Diffraction-limited Hilbert space angles with a clear rectangular aperture and a logarithmic phase plate.

Fig. 8
Fig. 8

Magnitude of defocused diffraction-limited OTF with a logarithmic phase plate.

Fig. 9
Fig. 9

Phase angle of defocused diffraction-limited OTF with a logarithmic phase plate.

Fig. 10
Fig. 10

Defocused diffraction-limited OTF with a clear rectangular aperture.

Fig. 11
Fig. 11

Woodward function of a logarithmic phase plate.

Fig. 12
Fig. 12

Woodward function of a clear rectangular pupil function.

Fig. 13
Fig. 13

Simulated images of a spoke with using a clear rectangular aperture (left) and simulated optical and final images with a logarithmic phase plate (middle and right, respectively). (a)–(c) ψ = 0; (d)–(f) ψ = 5; (g)–(i) ψ = 15; (j)–(l) ψ = 30.

Fig. 14
Fig. 14

Diffraction-limited and CCD-limited Hilbert space angles with a logarithmic phase plate.

Fig. 15
Fig. 15

Diffraction-limited Hilbert space angles with a cubic phase plate and with a logarithmic phase plate.

Tables (4)

Tables Icon

Table 1 Imaging System Parameters

Tables Icon

Table 2 Logarithmic Phase Plate Optimum Parameters

Tables Icon

Table 3 Logarithmic Phase Plate Practical Parameters

Tables Icon

Table 4 Defocus Parameters Equivalent to Defocus Distances

Equations (48)

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

|hu, v, wx, wy|2=κ -ymaxymax-xmaxxmaxexpjkwxx2xmax2+wyy2ymax2-uxzi+vyzidxdy2,
|hu, v, w|2=κ -ymaxymax-xmaxxmaxexpjkwx2+wy2-uxzi+vyzidxdy2.
|hu, w|2=κ-xmaxxmaxexpjkwx2-uxzidx2.
|hu, w|2=-11expjkwxmax2x2-fx-uxmaxxzidx2.
|hu, w|2=|hu, -w|2.
|hu, w|2λ2wxmax2-fxs,
2xmax2xs+2wxmax2xsw-fxsxsw=0.
|hu, w|2λ2xmax2xs/wxs.
ddwλ2xmax2xs/wxs=0.
xsw, ud2dw2 xsw, u-ddw xsw, u2=0,
xsw, u=±1βexp±wα, α, β>0.
wxs=±α logβxs, xs>0,
wxs=±α log-βxs, xs<0.
wxs=12xmax2xsfxs+uxmaxzi.
±α logβxs=12xmax2xsfxs+uxmaxzi.
fxs=±αxmax2  xs logβxsdxs-uxmaxxszi+c1,
fxs=±αxmax2  xs logxs+xs logβdxs-uxmaxxszi+c1.
fxs=±αxmax2xs22logxs-12+β xs22-uxmaxxszi+c2, α>0,
fxs=±αxmax2xs2logxs+β-uxmaxxszi+c2, α>0,
fxs=±αxmax2xs2log-xs+β-uxmaxxszi+c2, α>0.
fx=sgnxαxmax2x2log|x|+β-uxmaxxzi.
fx, y=sgnxαxmax2x2log|x|+β-uxmaxxzi+sgnyαymax2y2log|y|+β-vymaxyzi.
cos θψ=|hu, 0|2, |hu, ψ|2|hu, 0|2 |hu, ψ|2,
|hu, 0|2, |hu, ψ|2=- |hu, 0|2|hu, ψ|2du,
|hu, 0|2=- |hu, 0|2|hu, 0|2du1/2,
|hu, ψ|2=- |hu, ψ|2|hu, ψ|2du1/2.
Ap, q=- Px+p2P*x-p2expj2πqxdx.
Hinvfx, fy=HCl-apfx, fyHLog-ph plfx, fy HLog-ph plfx, fy0 =0 HLog-ph plfx, fy=0,
|hu, w|CCD-lim2=|hu, w|Diff-lim2 * rectuacombuus,
fx, y=αx3+y3.
|hu, -w|2=-11expjk-wxmax2x2-fx-uxmaxxzidx2,
|hu, -w|2=-11expjkwxmax2x2+fx+uxmaxxzi*dx2.
|hu, -w|2=-11expjkwxmax2x2+fx+uxmaxxzidx2.
|hu, -w|2=-1-1expjkwxmax2x2+f-x-uxmaxxzidx2.
|hu, -w|2=-11expjkwxmax2x2+f-x-uxmaxxzidx2.
fx=-f-x.
-11expjkwxmax2x2-fx-uxmaxxzidx
ddxwxmax2x2-fx-uxmaxxzix=xs=0.
2wxmax2xs-fxs-uxmaxzi=0.
wxmax2x2-fx-uxmaxxziwxmax2xs2-fxs-uxmaxxszi+2wxmax2xs-fxs-uxmaxzix-xs+122wxmax2-fxs×x-xs2.
wxmax2x2-fx-uxmaxxziwxmax2xs2-fxs-uxmaxxszi+122wxmax2-fxsx-xs2.
hu, wexpjkwxmax2xs2-fxs-uxmaxxszi×xs-εxs+εexpjk2wxmax2-fxs2×x-xs2dx,
hu, wexpjkwxmax2xs2-fxs-uxmaxxszi-εεexp-Ψ22σ2dΨ,
σ2=-1jk2wxmax2-fxs.
hu, wexpjkwxmax2xs2-fxs-uxmaxxszi-exp-Ψ22σ2dΨ.
-exp-Ψ22σ2dΨ=2πσ;
hu, wexpjkwxmax2xs2-fxs-uxmaxxszi2πexp±jπ/4k|2wxmax2-fxs|1/2.
|hu, w|2λ2wxmax2-fxs.

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