Abstract

The fundamental limit to resolution of a perfect lens for off-axis object points is considered. It has been shown previously that resolution decreases with illumination. Here it is shown that the relative decrease in resolution with illumination angle is reduced for lenses of higher numerical aperture. These results are of importance for optical systems that combine high aperture with large field of view, such as lithographic lenses.

© 1998 Optical Society of America

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References

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  1. M. V. R. K. Murty, “On the theoretical limit of resolution,” J. Opt. Soc. Am. 47, 667–668 (1957).
    [CrossRef]
  2. V. A. Zverev, “Illumination distribution in the diffraction image of an off-axis point,” Sov. J. Opt. Technol. 53, 451–454 (1986).
  3. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986).
  4. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1983).
  5. H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
    [CrossRef]
  6. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
    [CrossRef]
  7. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [CrossRef]
  8. W. Hsu, R. Barakat, “Stratton–Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems,” J. Opt. Soc. Am. A 11, 623–629 (1994).
    [CrossRef]
  9. C. J. R. Sheppard, “Imaging in optical systems of finite Fresnel number,” J. Opt. Soc. Am. A 3, 1428–1432 (1986).
    [CrossRef]
  10. S. F. Gibson, F. Lanni, “Diffraction by a circular aperture as a model for three-dimensional optical microscopy,” J. Opt. Soc. Am. A 6, 1357–1367 (1989).
    [CrossRef] [PubMed]
  11. C. J. R. Sheppard, P. P. Roberts, M. Gu, “Fresnel approximation for off-axis illumination of a circular aperture,” J. Opt. Soc. Am. A 10, 984–986 (1993).
    [CrossRef]
  12. C. J. R. Sheppard, M. Hrynevych, “Diffraction by a circular aperture: a generalization of Fresnel diffraction theory,” J. Opt. Soc. Am. A 9, 274–281 (1992).
    [CrossRef]
  13. R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983).

1994 (1)

1993 (1)

1992 (1)

1989 (1)

1986 (2)

C. J. R. Sheppard, “Imaging in optical systems of finite Fresnel number,” J. Opt. Soc. Am. A 3, 1428–1432 (1986).
[CrossRef]

V. A. Zverev, “Illumination distribution in the diffraction image of an off-axis point,” Sov. J. Opt. Technol. 53, 451–454 (1986).

1984 (1)

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

1957 (1)

1943 (1)

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
[CrossRef]

Barakat, R.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1983).

Gibson, S. F.

Gu, M.

Hopkins, H. H.

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
[CrossRef]

Hrynevych, M.

Hsu, W.

Kingslake, R.

R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983).

Lanni, F.

Li, Y.

Murty, M. V. R. K.

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Roberts, P. P.

Sheppard, C. J. R.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986).

Wolf, E.

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1983).

Zverev, V. A.

V. A. Zverev, “Illumination distribution in the diffraction image of an off-axis point,” Sov. J. Opt. Technol. 53, 451–454 (1986).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Proc. Phys. Soc. London (1)

H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943).
[CrossRef]

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

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959).
[CrossRef]

Sov. J. Opt. Technol. (1)

V. A. Zverev, “Illumination distribution in the diffraction image of an off-axis point,” Sov. J. Opt. Technol. 53, 451–454 (1986).

Other (3)

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1983).

R. Kingslake, Optical System Design (Academic, Orlando, Fla., 1983).

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Figures (4)

Fig. 1
Fig. 1

Geometry of the system.

Fig. 2
Fig. 2

Edge of the pupil function for illumination at an angle θ. The scale is normalized so that for on-axis illumination the radius is independent of aperture.

Fig. 3
Fig. 3

Normalized width of the pupil function along (a) the x axis and (b) the y axis.

Fig. 4
Fig. 4

System with aperture stop in the front focal plane, so that it is telecentric in image space and behaves as a system with infinite value for the Fresnel number.

Equations (7)

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

d=a cot α.
p=(a cos ϕ-d tan θ)i+a sin ϕj-dk,
|p|=dcos α cos θ(1-sin2 α sin2 θ-2 sin α cos α sin θ cos θ cos ϕ)1/2.
m=cos θ cos ϕ-sin θ cot α(1-sin2 α sin2 θ-2 sin α cos α sin θ cos θ cos ϕ)1/2,
n=cos θ sin ϕ(1-sin2 α sin2 θ-2 sin α cos α sin θ cos θ cos ϕ)1/2.
m0=cos θ1-sin2 θ1-12sin2 α,
n0=cos θ1+12sin2 θ sin2 α.

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