Abstract

Incoherent imaging and spatial filtering systems may use illumination sources that are effectively periodic and discrete spatially (e.g., light-emitting-diode arrays or cathode-ray-tube rasters). We show that, despite the noncontinuous nature of such sources, linear-in-intensity imaging can, under certain conditions, be obtained. We model the source as a sampled continuous distribution and use Fourier optics methods to obtain a sampling condition that specifies the minimum permissible spacing between source points. This minimum spacing depends on the reciprocal of the smaller of the width of the object and the width of the imaging system’s coherent point-spread function. Relatively coarse sampling in the source plane is permitted in many cases of practical interest. We provide additional physical insight by examining the distributions (wave-amplitude distributions and transmittance function) in the pupil plane of the system.

© 1989 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. H. Bartelt, S. K. Case, R. Hauck, “Incoherent optical processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), pp. 509–510.
  3. N. Brousseau, H. H. Arsenault, “Les effets causes par les dimensions de la source dans les systemes optiques eclaires en lumiere spatialement incoherente,” Opt. Commun. 15, 389–391 (1975).
    [CrossRef]
  4. R. E. Swing, “Conditions for microdensitometer linearity,”J. Opt. Soc. Am. 62, 199–207 (1972).
    [CrossRef]
  5. P. Chavel, S. Lowenthal, “A method of incoherent optical-image processing using synthetic holograms,”J. Opt. Soc. Am. 66, 14–23 (1976).
    [CrossRef]
  6. D. Slepian, “On bandwidth,” Proc. IEEE 64, 292–300 (1976).
    [CrossRef]
  7. G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).
  8. B. E. A. Saleh, M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21, 2770–2777 (1982).
    [CrossRef] [PubMed]

1982 (1)

1976 (2)

1975 (1)

N. Brousseau, H. H. Arsenault, “Les effets causes par les dimensions de la source dans les systemes optiques eclaires en lumiere spatialement incoherente,” Opt. Commun. 15, 389–391 (1975).
[CrossRef]

1972 (1)

Arsenault, H. H.

N. Brousseau, H. H. Arsenault, “Les effets causes par les dimensions de la source dans les systemes optiques eclaires en lumiere spatialement incoherente,” Opt. Commun. 15, 389–391 (1975).
[CrossRef]

Bartelt, H.

H. Bartelt, S. K. Case, R. Hauck, “Incoherent optical processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), pp. 509–510.

Brousseau, N.

N. Brousseau, H. H. Arsenault, “Les effets causes par les dimensions de la source dans les systemes optiques eclaires en lumiere spatialement incoherente,” Opt. Commun. 15, 389–391 (1975).
[CrossRef]

Case, S. K.

H. Bartelt, S. K. Case, R. Hauck, “Incoherent optical processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), pp. 509–510.

Chavel, P.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hauck, R.

H. Bartelt, S. K. Case, R. Hauck, “Incoherent optical processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), pp. 509–510.

Lowenthal, S.

Rabbani, M.

Rogers, G. L.

G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).

Saleh, B. E. A.

Slepian, D.

D. Slepian, “On bandwidth,” Proc. IEEE 64, 292–300 (1976).
[CrossRef]

Swing, R. E.

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

N. Brousseau, H. H. Arsenault, “Les effets causes par les dimensions de la source dans les systemes optiques eclaires en lumiere spatialement incoherente,” Opt. Commun. 15, 389–391 (1975).
[CrossRef]

Proc. IEEE (1)

D. Slepian, “On bandwidth,” Proc. IEEE 64, 292–300 (1976).
[CrossRef]

Other (3)

G. L. Rogers, Noncoherent Optical Processing (Wiley, New York, 1977).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

H. Bartelt, S. K. Case, R. Hauck, “Incoherent optical processing,” in Applications of Optical Fourier Transforms, H. Stark, ed. (Academic, New York, 1982), pp. 509–510.

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Figures (2)

Fig. 1
Fig. 1

Diagram of 6-f imaging system assumed in the analysis.

Fig. 2
Fig. 2

Diagram used in analyzing the overlap of functions in the integrand of Eq. (9).

Equations (13)

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I ( x ; x s ) = K I s ( x s ) t 0 ( x ) exp ( - i 2 π x x s / λ f ) * h ( x ) 2 ,
I ( x ) = K - I s ( x s ) t 0 ( x ) exp ( - i 2 π x x s / λ f ) * h ( x ) 2 d x s .
I ( x ) = K - - t 0 ( ξ ) t 0 * ( η ) h ( x - ξ ) h * ( x - η ) × - I s ( x s ) exp [ - i 2 π x s ( ξ - η ) / λ f ] d x s d ξ d η .
I ( x ) = K - - t 0 ( ξ ) t 0 * ( η ) h ( x - ξ ) h * ( x - η ) I ˜ s ( ξ - η λ f ) d ξ d η .
I ( x ) = K t 0 ( x ) 2 * h ( x ) 2 .
I s ( x s ) = n = - δ ( x s - n D ) ,
I ˜ s ( ξ - η λ f ) = λ f D n = - δ ( ξ - η - n λ f D ) .
I ( x ) = K - - t 0 ( ξ ) t 0 * ( η ) h ( x - ξ ) h * ( x - η ) × n = - δ ( ξ - η - n λ f D ) d ξ d η ,
I ( x ) = K t 0 ( x ) 2 * h ( x ) 2 + K n 0 - t 0 ( η + n λ f D ) × t 0 * ( η ) h ( x - η - n λ f D ) h * ( x - η ) d η .
λ f D > Min ( w t , w h )
D < λ f Min ( w t , w h ) .
w h = 2 λ f F max .
D < 1 2 F max ,

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