Abstract

Spatially incoherent imaging systems are characterized by a linear-in-intensity relationship between the object and the image distributions. While strict spatial incoherence is theoretically not achievable, a particular imaging system may be made effectively linear in intensity by a choice of the appropriate illumination source location and size. The requirement for source size for effectively incoherent illumination of a two-dimensional object is well known. I extend the arguments for choosing the source size in a two-dimensional imaging system to develop necessary conditions for the source size for effectively spatially incoherent illumination of a three-dimensional object. While the conditions are necessary, they are not sufficient, since coherence in the direction of the optical axis is not addressed.

© 2001 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

1990 (1)

1989 (1)

1984 (1)

N. Streibl, “Depth transfer by an imaging system,” Opt. Acta 31, 1233–1241 (1984).
[CrossRef]

1983 (1)

1980 (1)

1979 (1)

B. E. A. Saleh, “Optical bilinear transforms,” Opt. Acta 26, 777–799 (1979).
[CrossRef]

1976 (1)

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

1975 (1)

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

1972 (1)

1953 (1)

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London Ser. A 217408–432 (1953).
[CrossRef]

Arseneault, H. H.

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

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1991).

Brousseau, N.

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

Goeke, W. C.

Goodman, D. S.

D. S. Goodman, “Stationary optical projectors,” Ph.D dissertation (University of Arizona, Tucson, Ariz., 1979).

D. S. Goodman, A. E. Rosenbluth, “Condenser aberrations in Köhler illumination,” in Optical/Laser Microlithography, B. J. Lin, ed., Proc. SPIE922, 108–134 (1988).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985).

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London Ser. A 217408–432 (1953).
[CrossRef]

Mao, X. Q.

Rabanni, M.

Rhodes, W. T.

Rosenbluth, A. E.

D. S. Goodman, A. E. Rosenbluth, “Condenser aberrations in Köhler illumination,” in Optical/Laser Microlithography, B. J. Lin, ed., Proc. SPIE922, 108–134 (1988).
[CrossRef]

Saleh, B. E. A.

Sheppard, C. J. R.

Sitter, D. N.

D. N. Sitter, W. T. Rhodes, “Three-dimensional imaging: a space invariant model for space variant systems,” Appl. Opt. 29, 3789–3794 (1990).
[CrossRef] [PubMed]

D. N. Sitter, “Space invariant modeling in three-dimensional image formation,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Ga., 1991).

Slepian, D.

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

Streibl, N.

N. Streibl, “Depth transfer by an imaging system,” Opt. Acta 31, 1233–1241 (1984).
[CrossRef]

Swing, R. E.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1991).

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

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

Opt. Acta (2)

N. Streibl, “Depth transfer by an imaging system,” Opt. Acta 31, 1233–1241 (1984).
[CrossRef]

B. E. A. Saleh, “Optical bilinear transforms,” Opt. Acta 26, 777–799 (1979).
[CrossRef]

Opt. Commun. (1)

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

Proc. IEEE (1)

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

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

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London Ser. A 217408–432 (1953).
[CrossRef]

Other (5)

D. S. Goodman, A. E. Rosenbluth, “Condenser aberrations in Köhler illumination,” in Optical/Laser Microlithography, B. J. Lin, ed., Proc. SPIE922, 108–134 (1988).
[CrossRef]

J. W. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985).

D. S. Goodman, “Stationary optical projectors,” Ph.D dissertation (University of Arizona, Tucson, Ariz., 1979).

D. N. Sitter, “Space invariant modeling in three-dimensional image formation,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Ga., 1991).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1991).

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Figures (6)

Fig. 1
Fig. 1

Köhler illumination imaging system comprised of a doubly telecentric condenser coupled to a doubly telecentric imaging system. The source is indicated by the wavy line and is considered spatially incoherent and quasi-monochromatic. Lenses L1,L2, and L3 have focal lengths F1,F2, and F3, respectively.

Fig. 2
Fig. 2

Condenser portion of the imaging system emphasizing the light originating from the end points of a finite extent source and ending as projections of the object Fraunhofer pattern at the pupil plane. The extent of the pupil opening is WP, the effective Fraunhofer pattern width is WF, and the source width is WS.

Fig. 3
Fig. 3

Condenser portion of the imaging system emphasizing the transmission of light from an off-axis source point P2 (same location as the point P2 in Fig. 2) impinging on a planar object located away from best focus. The shift of the object from best focus does not change the extent and location of the projected Fraunhofer pattern.

Fig. 4
Fig. 4

Condenser portion of the imaging system with a field stop maintained at the best-focus position and a planar object shown displaced from best focus. The light shown originating at the end points of the source does not uniformly illuminate the displaced object. Only the shaded region of object space receives light from all points on the finite-extent source.

Fig. 5
Fig. 5

Condenser portion of the imaging system showing the path from various points on a large-extent source traveling through a particular point in object space. The resulting effective source at the pupil plane is large but does not overfill the pupil. The extent of the effective source can be determined by ray tracing as shown in Fig. 6.

Fig. 6
Fig. 6

A detailed view of rays traveling from the source plane through a particular point in object space to conjugate points at the pupil plane: (a) Light rays restricted by the lower edge of a field stop of width WC that limits XSa, the extent of the effective source above the optical axis. (b) Light rays restricted by the upper edge of the field stop and limiting XSb, the extent of the effective source below the optical axis.

Equations (8)

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

I(x)=|o(ξ)|2|h(x-ξ)|2dξ,
I(x)=o(ξ)o*(η)h(x-ξ)h*(x-η)J0(ξ; η)dξdη,
J0(ξ-η)=S(x)exp(jkx¯(ξ-η)/F)dx,
WS>WP+WF,
XSa>(WP+WF)/2,
XSb>(WP+WF)/2,
XSa=F2(Wc/2+Δx)/Δz
XSb=F2(Wc/2-Δx)/Δz.

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