Abstract

The resolution of optical systems with annular pupil objectives is discussed, particularly with respect to the imaging properties of soft-x-ray imaging optical systems. Images of two dark bars in a bright field and knife-edge responses, which are calculated with partially coherent transfer functions, were used to estimate the resolution. Resolution for two bright points in a dark field, which is conventionally used in Rayleigh’s analysis, is also discussed for comparison. It was found that annular systems with obstruction ratios of more than 0.6 show resolution properties different from those of systems with low-obstruction-ratios. The high-obstruction-ratio systems show lower resolution for the two-dark-bar objects and higher resolution for the two-bright-point objects, whereas the low-obstruction-ratio systems show almost the same resolution for both objects.

© 1994 Optical Society of America

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References

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  1. J. Kirz, H. Rarback, “Soft x-ray microscopes,” Rev. Sci. Instrum. 56, 1–13 (1985).
    [CrossRef]
  2. M. Howells, J. Kirz, D. Sayre, G. Schmahl, “Soft-x-ray microscopes,” Phys. Today 38 (8), 22–32 (1985).
    [CrossRef]
  3. H. Wolter, “Spiegelsysteme streifenden Einfalls als ab-bibldende Optiken für Rontgenstrahlen,” Ann. Phys. 10, 94–113 (1952).
    [CrossRef]
  4. I. Lovas, W. Santy, E. Spiller, R. Tibbetts, J. Wilczynski, “Design and assembly of a high resolution Schwarzschild microscope for soft x rays,” in Very High Speed Integrated Circuit Technology for Electro-Optic Applications,W. S. Chan, J. T. Hall, eds., Proc. Soc. Photo-Opt. Instrum. Eng.319, 90–97 (1981).
  5. G. Schmahl, D. Rudolph, B. Niemann, “Imaging and scanning soft x-ray microscope,” in Scanned Image Microscopy,E. A. Ash, ed. (Academic, New York, 1980), pp. 393–412.
  6. Y. Horikawa, “Consideration on the condenser for an imaging type Schwarzschild soft x-ray microscope,” J. Mod. Opt. 40, 289–298 (1993).
    [CrossRef]
  7. E. H. Linfoot, E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. London B 66, 145–149 (1953).
    [CrossRef]
  8. W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt. Soc Am. 50, 749–753 (1960).
    [CrossRef]
  9. T. S. McKechnie, “The effect of condenser obstruction on the two-point resolution of a microscope,” Opt. Acta 19, 729–737 (1972).
    [CrossRef]
  10. C. J. R. Sheppard, T. Wilson, “Imaging properties of annular lenses,” Appl. Opt. 18, 3764–3769 (1979).
    [PubMed]
  11. Y. Horikawa, “Imaging properties of an annular pupil optical system; first stage for understanding of soft x-ray imaging optics,” Optik 95, 115–124 (1993).
  12. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 414–424.
  13. Y. Vladimirsky, D. Kern, T. H. P. Chang, D. Attwood, H. Ade, J. Kirz, I. McNulty, H. Rarback, D. Shu, “High-resolution Fresnel zone plates for soft x rays,” J. Vac. Sci Technol. B 6, 311–315 (1988).
    [CrossRef]
  14. J. A. Trail, R. L. Byer, “Compact scanning soft x-ray microscope using a laser-produced plasma source and normal-incidence multilayer mirrors,” Opt. Lett. 14, 539–541 (1989).
    [CrossRef] [PubMed]
  15. H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London A 217, 408–432 (1953).
    [CrossRef]
  16. K. Yamamoto, J. Tokumitu, K. Matsuoka, Y. Ichioka, T. Suzuki, “The resolution of small double-bar targets in a bright background,” Ouyobutsuri 46, 139–144 (1977) (a publication of the Japan Society of Applied Physics, in Japanese).
  17. Y. Horikawa, “Imaging properties of the Schwarzschild soft x-ray microscopes,” J. Mod. Opt. 40, 2025–2038 (1993).
    [CrossRef]
  18. H. H. Hopkins, P. M. Barham, “The influence of the condenser on microscopic resolution,” Proc Phys. Soc. 63, 737–744 (1950).
    [CrossRef]
  19. Y. Horikawa, S. Mochimaru, Y. Iketaki, H. Nagai, K. Okawa, S. Iura, “Design and fabrication of the Schwarzschild objective for soft x-ray microscopes,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation,J. Tsujiuchi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1720, 217–225 (1992).
    [CrossRef]
  20. W. Werner, “Imaging properties of Wolter I type x-ray telescopes,” Appl. Opt. 16, 764–773 (1977).
    [CrossRef] [PubMed]

1993

Y. Horikawa, “Consideration on the condenser for an imaging type Schwarzschild soft x-ray microscope,” J. Mod. Opt. 40, 289–298 (1993).
[CrossRef]

Y. Horikawa, “Imaging properties of the Schwarzschild soft x-ray microscopes,” J. Mod. Opt. 40, 2025–2038 (1993).
[CrossRef]

Y. Horikawa, “Imaging properties of an annular pupil optical system; first stage for understanding of soft x-ray imaging optics,” Optik 95, 115–124 (1993).

1989

1988

Y. Vladimirsky, D. Kern, T. H. P. Chang, D. Attwood, H. Ade, J. Kirz, I. McNulty, H. Rarback, D. Shu, “High-resolution Fresnel zone plates for soft x rays,” J. Vac. Sci Technol. B 6, 311–315 (1988).
[CrossRef]

1985

J. Kirz, H. Rarback, “Soft x-ray microscopes,” Rev. Sci. Instrum. 56, 1–13 (1985).
[CrossRef]

M. Howells, J. Kirz, D. Sayre, G. Schmahl, “Soft-x-ray microscopes,” Phys. Today 38 (8), 22–32 (1985).
[CrossRef]

1979

1977

W. Werner, “Imaging properties of Wolter I type x-ray telescopes,” Appl. Opt. 16, 764–773 (1977).
[CrossRef] [PubMed]

K. Yamamoto, J. Tokumitu, K. Matsuoka, Y. Ichioka, T. Suzuki, “The resolution of small double-bar targets in a bright background,” Ouyobutsuri 46, 139–144 (1977) (a publication of the Japan Society of Applied Physics, in Japanese).

1972

T. S. McKechnie, “The effect of condenser obstruction on the two-point resolution of a microscope,” Opt. Acta 19, 729–737 (1972).
[CrossRef]

1960

W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt. Soc Am. 50, 749–753 (1960).
[CrossRef]

1953

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London A 217, 408–432 (1953).
[CrossRef]

E. H. Linfoot, E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. London B 66, 145–149 (1953).
[CrossRef]

1952

H. Wolter, “Spiegelsysteme streifenden Einfalls als ab-bibldende Optiken für Rontgenstrahlen,” Ann. Phys. 10, 94–113 (1952).
[CrossRef]

1950

H. H. Hopkins, P. M. Barham, “The influence of the condenser on microscopic resolution,” Proc Phys. Soc. 63, 737–744 (1950).
[CrossRef]

Ade, H.

Y. Vladimirsky, D. Kern, T. H. P. Chang, D. Attwood, H. Ade, J. Kirz, I. McNulty, H. Rarback, D. Shu, “High-resolution Fresnel zone plates for soft x rays,” J. Vac. Sci Technol. B 6, 311–315 (1988).
[CrossRef]

Attwood, D.

Y. Vladimirsky, D. Kern, T. H. P. Chang, D. Attwood, H. Ade, J. Kirz, I. McNulty, H. Rarback, D. Shu, “High-resolution Fresnel zone plates for soft x rays,” J. Vac. Sci Technol. B 6, 311–315 (1988).
[CrossRef]

Barham, P. M.

H. H. Hopkins, P. M. Barham, “The influence of the condenser on microscopic resolution,” Proc Phys. Soc. 63, 737–744 (1950).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 414–424.

Byer, R. L.

Chang, T. H. P.

Y. Vladimirsky, D. Kern, T. H. P. Chang, D. Attwood, H. Ade, J. Kirz, I. McNulty, H. Rarback, D. Shu, “High-resolution Fresnel zone plates for soft x rays,” J. Vac. Sci Technol. B 6, 311–315 (1988).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London A 217, 408–432 (1953).
[CrossRef]

H. H. Hopkins, P. M. Barham, “The influence of the condenser on microscopic resolution,” Proc Phys. Soc. 63, 737–744 (1950).
[CrossRef]

Horikawa, Y.

Y. Horikawa, “Consideration on the condenser for an imaging type Schwarzschild soft x-ray microscope,” J. Mod. Opt. 40, 289–298 (1993).
[CrossRef]

Y. Horikawa, “Imaging properties of the Schwarzschild soft x-ray microscopes,” J. Mod. Opt. 40, 2025–2038 (1993).
[CrossRef]

Y. Horikawa, “Imaging properties of an annular pupil optical system; first stage for understanding of soft x-ray imaging optics,” Optik 95, 115–124 (1993).

Y. Horikawa, S. Mochimaru, Y. Iketaki, H. Nagai, K. Okawa, S. Iura, “Design and fabrication of the Schwarzschild objective for soft x-ray microscopes,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation,J. Tsujiuchi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1720, 217–225 (1992).
[CrossRef]

Howells, M.

M. Howells, J. Kirz, D. Sayre, G. Schmahl, “Soft-x-ray microscopes,” Phys. Today 38 (8), 22–32 (1985).
[CrossRef]

Ichioka, Y.

K. Yamamoto, J. Tokumitu, K. Matsuoka, Y. Ichioka, T. Suzuki, “The resolution of small double-bar targets in a bright background,” Ouyobutsuri 46, 139–144 (1977) (a publication of the Japan Society of Applied Physics, in Japanese).

Iketaki, Y.

Y. Horikawa, S. Mochimaru, Y. Iketaki, H. Nagai, K. Okawa, S. Iura, “Design and fabrication of the Schwarzschild objective for soft x-ray microscopes,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation,J. Tsujiuchi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1720, 217–225 (1992).
[CrossRef]

Iura, S.

Y. Horikawa, S. Mochimaru, Y. Iketaki, H. Nagai, K. Okawa, S. Iura, “Design and fabrication of the Schwarzschild objective for soft x-ray microscopes,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation,J. Tsujiuchi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1720, 217–225 (1992).
[CrossRef]

Kern, D.

Y. Vladimirsky, D. Kern, T. H. P. Chang, D. Attwood, H. Ade, J. Kirz, I. McNulty, H. Rarback, D. Shu, “High-resolution Fresnel zone plates for soft x rays,” J. Vac. Sci Technol. B 6, 311–315 (1988).
[CrossRef]

Kirz, J.

Y. Vladimirsky, D. Kern, T. H. P. Chang, D. Attwood, H. Ade, J. Kirz, I. McNulty, H. Rarback, D. Shu, “High-resolution Fresnel zone plates for soft x rays,” J. Vac. Sci Technol. B 6, 311–315 (1988).
[CrossRef]

M. Howells, J. Kirz, D. Sayre, G. Schmahl, “Soft-x-ray microscopes,” Phys. Today 38 (8), 22–32 (1985).
[CrossRef]

J. Kirz, H. Rarback, “Soft x-ray microscopes,” Rev. Sci. Instrum. 56, 1–13 (1985).
[CrossRef]

Linfoot, E. H.

E. H. Linfoot, E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. London B 66, 145–149 (1953).
[CrossRef]

Lovas, I.

I. Lovas, W. Santy, E. Spiller, R. Tibbetts, J. Wilczynski, “Design and assembly of a high resolution Schwarzschild microscope for soft x rays,” in Very High Speed Integrated Circuit Technology for Electro-Optic Applications,W. S. Chan, J. T. Hall, eds., Proc. Soc. Photo-Opt. Instrum. Eng.319, 90–97 (1981).

Matsuoka, K.

K. Yamamoto, J. Tokumitu, K. Matsuoka, Y. Ichioka, T. Suzuki, “The resolution of small double-bar targets in a bright background,” Ouyobutsuri 46, 139–144 (1977) (a publication of the Japan Society of Applied Physics, in Japanese).

McKechnie, T. S.

T. S. McKechnie, “The effect of condenser obstruction on the two-point resolution of a microscope,” Opt. Acta 19, 729–737 (1972).
[CrossRef]

McNulty, I.

Y. Vladimirsky, D. Kern, T. H. P. Chang, D. Attwood, H. Ade, J. Kirz, I. McNulty, H. Rarback, D. Shu, “High-resolution Fresnel zone plates for soft x rays,” J. Vac. Sci Technol. B 6, 311–315 (1988).
[CrossRef]

Mochimaru, S.

Y. Horikawa, S. Mochimaru, Y. Iketaki, H. Nagai, K. Okawa, S. Iura, “Design and fabrication of the Schwarzschild objective for soft x-ray microscopes,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation,J. Tsujiuchi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1720, 217–225 (1992).
[CrossRef]

Nagai, H.

Y. Horikawa, S. Mochimaru, Y. Iketaki, H. Nagai, K. Okawa, S. Iura, “Design and fabrication of the Schwarzschild objective for soft x-ray microscopes,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation,J. Tsujiuchi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1720, 217–225 (1992).
[CrossRef]

Niemann, B.

G. Schmahl, D. Rudolph, B. Niemann, “Imaging and scanning soft x-ray microscope,” in Scanned Image Microscopy,E. A. Ash, ed. (Academic, New York, 1980), pp. 393–412.

Okawa, K.

Y. Horikawa, S. Mochimaru, Y. Iketaki, H. Nagai, K. Okawa, S. Iura, “Design and fabrication of the Schwarzschild objective for soft x-ray microscopes,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation,J. Tsujiuchi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1720, 217–225 (1992).
[CrossRef]

Rarback, H.

Y. Vladimirsky, D. Kern, T. H. P. Chang, D. Attwood, H. Ade, J. Kirz, I. McNulty, H. Rarback, D. Shu, “High-resolution Fresnel zone plates for soft x rays,” J. Vac. Sci Technol. B 6, 311–315 (1988).
[CrossRef]

J. Kirz, H. Rarback, “Soft x-ray microscopes,” Rev. Sci. Instrum. 56, 1–13 (1985).
[CrossRef]

Rudolph, D.

G. Schmahl, D. Rudolph, B. Niemann, “Imaging and scanning soft x-ray microscope,” in Scanned Image Microscopy,E. A. Ash, ed. (Academic, New York, 1980), pp. 393–412.

Santy, W.

I. Lovas, W. Santy, E. Spiller, R. Tibbetts, J. Wilczynski, “Design and assembly of a high resolution Schwarzschild microscope for soft x rays,” in Very High Speed Integrated Circuit Technology for Electro-Optic Applications,W. S. Chan, J. T. Hall, eds., Proc. Soc. Photo-Opt. Instrum. Eng.319, 90–97 (1981).

Sayre, D.

M. Howells, J. Kirz, D. Sayre, G. Schmahl, “Soft-x-ray microscopes,” Phys. Today 38 (8), 22–32 (1985).
[CrossRef]

Schmahl, G.

M. Howells, J. Kirz, D. Sayre, G. Schmahl, “Soft-x-ray microscopes,” Phys. Today 38 (8), 22–32 (1985).
[CrossRef]

G. Schmahl, D. Rudolph, B. Niemann, “Imaging and scanning soft x-ray microscope,” in Scanned Image Microscopy,E. A. Ash, ed. (Academic, New York, 1980), pp. 393–412.

Sheppard, C. J. R.

Shu, D.

Y. Vladimirsky, D. Kern, T. H. P. Chang, D. Attwood, H. Ade, J. Kirz, I. McNulty, H. Rarback, D. Shu, “High-resolution Fresnel zone plates for soft x rays,” J. Vac. Sci Technol. B 6, 311–315 (1988).
[CrossRef]

Spiller, E.

I. Lovas, W. Santy, E. Spiller, R. Tibbetts, J. Wilczynski, “Design and assembly of a high resolution Schwarzschild microscope for soft x rays,” in Very High Speed Integrated Circuit Technology for Electro-Optic Applications,W. S. Chan, J. T. Hall, eds., Proc. Soc. Photo-Opt. Instrum. Eng.319, 90–97 (1981).

Suzuki, T.

K. Yamamoto, J. Tokumitu, K. Matsuoka, Y. Ichioka, T. Suzuki, “The resolution of small double-bar targets in a bright background,” Ouyobutsuri 46, 139–144 (1977) (a publication of the Japan Society of Applied Physics, in Japanese).

Tibbetts, R.

I. Lovas, W. Santy, E. Spiller, R. Tibbetts, J. Wilczynski, “Design and assembly of a high resolution Schwarzschild microscope for soft x rays,” in Very High Speed Integrated Circuit Technology for Electro-Optic Applications,W. S. Chan, J. T. Hall, eds., Proc. Soc. Photo-Opt. Instrum. Eng.319, 90–97 (1981).

Tokumitu, J.

K. Yamamoto, J. Tokumitu, K. Matsuoka, Y. Ichioka, T. Suzuki, “The resolution of small double-bar targets in a bright background,” Ouyobutsuri 46, 139–144 (1977) (a publication of the Japan Society of Applied Physics, in Japanese).

Trail, J. A.

Vladimirsky, Y.

Y. Vladimirsky, D. Kern, T. H. P. Chang, D. Attwood, H. Ade, J. Kirz, I. McNulty, H. Rarback, D. Shu, “High-resolution Fresnel zone plates for soft x rays,” J. Vac. Sci Technol. B 6, 311–315 (1988).
[CrossRef]

Welford, W. T.

W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt. Soc Am. 50, 749–753 (1960).
[CrossRef]

Werner, W.

Wilczynski, J.

I. Lovas, W. Santy, E. Spiller, R. Tibbetts, J. Wilczynski, “Design and assembly of a high resolution Schwarzschild microscope for soft x rays,” in Very High Speed Integrated Circuit Technology for Electro-Optic Applications,W. S. Chan, J. T. Hall, eds., Proc. Soc. Photo-Opt. Instrum. Eng.319, 90–97 (1981).

Wilson, T.

Wolf, E.

E. H. Linfoot, E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. London B 66, 145–149 (1953).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 414–424.

Wolter, H.

H. Wolter, “Spiegelsysteme streifenden Einfalls als ab-bibldende Optiken für Rontgenstrahlen,” Ann. Phys. 10, 94–113 (1952).
[CrossRef]

Yamamoto, K.

K. Yamamoto, J. Tokumitu, K. Matsuoka, Y. Ichioka, T. Suzuki, “The resolution of small double-bar targets in a bright background,” Ouyobutsuri 46, 139–144 (1977) (a publication of the Japan Society of Applied Physics, in Japanese).

Ann. Phys.

H. Wolter, “Spiegelsysteme streifenden Einfalls als ab-bibldende Optiken für Rontgenstrahlen,” Ann. Phys. 10, 94–113 (1952).
[CrossRef]

Appl. Opt.

J. Mod. Opt.

Y. Horikawa, “Consideration on the condenser for an imaging type Schwarzschild soft x-ray microscope,” J. Mod. Opt. 40, 289–298 (1993).
[CrossRef]

Y. Horikawa, “Imaging properties of the Schwarzschild soft x-ray microscopes,” J. Mod. Opt. 40, 2025–2038 (1993).
[CrossRef]

J. Opt. Soc Am.

W. T. Welford, “Use of annular apertures to increase focal depth,” J. Opt. Soc Am. 50, 749–753 (1960).
[CrossRef]

J. Vac. Sci Technol. B

Y. Vladimirsky, D. Kern, T. H. P. Chang, D. Attwood, H. Ade, J. Kirz, I. McNulty, H. Rarback, D. Shu, “High-resolution Fresnel zone plates for soft x rays,” J. Vac. Sci Technol. B 6, 311–315 (1988).
[CrossRef]

Opt. Acta

T. S. McKechnie, “The effect of condenser obstruction on the two-point resolution of a microscope,” Opt. Acta 19, 729–737 (1972).
[CrossRef]

Opt. Lett.

Optik

Y. Horikawa, “Imaging properties of an annular pupil optical system; first stage for understanding of soft x-ray imaging optics,” Optik 95, 115–124 (1993).

Ouyobutsuri

K. Yamamoto, J. Tokumitu, K. Matsuoka, Y. Ichioka, T. Suzuki, “The resolution of small double-bar targets in a bright background,” Ouyobutsuri 46, 139–144 (1977) (a publication of the Japan Society of Applied Physics, in Japanese).

Phys. Today

M. Howells, J. Kirz, D. Sayre, G. Schmahl, “Soft-x-ray microscopes,” Phys. Today 38 (8), 22–32 (1985).
[CrossRef]

Proc Phys. Soc.

H. H. Hopkins, P. M. Barham, “The influence of the condenser on microscopic resolution,” Proc Phys. Soc. 63, 737–744 (1950).
[CrossRef]

Proc. Phys. Soc. London B

E. H. Linfoot, E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. London B 66, 145–149 (1953).
[CrossRef]

Proc. R. Soc. London A

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London A 217, 408–432 (1953).
[CrossRef]

Rev. Sci. Instrum.

J. Kirz, H. Rarback, “Soft x-ray microscopes,” Rev. Sci. Instrum. 56, 1–13 (1985).
[CrossRef]

Other

I. Lovas, W. Santy, E. Spiller, R. Tibbetts, J. Wilczynski, “Design and assembly of a high resolution Schwarzschild microscope for soft x rays,” in Very High Speed Integrated Circuit Technology for Electro-Optic Applications,W. S. Chan, J. T. Hall, eds., Proc. Soc. Photo-Opt. Instrum. Eng.319, 90–97 (1981).

G. Schmahl, D. Rudolph, B. Niemann, “Imaging and scanning soft x-ray microscope,” in Scanned Image Microscopy,E. A. Ash, ed. (Academic, New York, 1980), pp. 393–412.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 414–424.

Y. Horikawa, S. Mochimaru, Y. Iketaki, H. Nagai, K. Okawa, S. Iura, “Design and fabrication of the Schwarzschild objective for soft x-ray microscopes,” in International Symposium on Optical Fabrication, Testing, and Surface Evaluation,J. Tsujiuchi, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1720, 217–225 (1992).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Soft-x-ray imaging optical elements: (a) zone plate, (b) Schwarzschild optical system consisting of a concave multilayer mirror and a convex multilayer mirror, (c) Wolter mirror system consisting of a hyperboloidal mirror and an ellipsoidal mirror.

Fig. 2
Fig. 2

Amplitude transmittance of a two-dark-bar object.

Fig. 3
Fig. 3

Definition of the limit of resolution for the two dark bars in a bright field by analogy with the Rayleigh resolution for a circular system in incoherent illumination.

Fig. 4
Fig. 4

Image profiles of the two-dark-bar object at the limit of resolution of each system.

Fig. 5
Fig. 5

Image profiles of the two-dark-bar object with (a) a circular system, (b) a Schwarzschild system, and (c) a Wolter system at the resolution limit of each system; (d) image profiles with a Wolter system near the resolution limits of the other systems.

Fig. 6
Fig. 6

Image profiles of the two-dark-bar object with the Wolter system and use of a circular-pupil condenser. The limit of resolution is improved with the circular condenser.

Fig. 7
Fig. 7

Image profiles of the two-bright-point object with each system at (a) the Rayleigh limit of resolution for a circular system and (b) the resolution limit of each system, ρ, obstruction ratio; δ, resolution.

Fig. 8
Fig. 8

Knife-edge responses for (a) partially coherent systems with various obstruction ratios and (b) incoherent systems. Each response is improved slightly compared with responses with partially coherent systems, ρ, obstruction ratio; δ, resolution.

Fig. 9
Fig. 9

(a) Knife-edge responses for circular systems with condenser apertures smaller than those of the objective; R is the ratio of the numerical aperture of the condenser to that of the objective. (b) Knife-edge responses for annular systems with condenser annuli thinner than those of the objective. i and o are the inner and outer radii, respectively, of the condenser pupil.

Fig. 10
Fig. 10

Image profiles of the three-dark-bar object with (a) a circular system, (b) a Schwarzschild system, and (c) a Wolter system near the resolution limit of each system; (d) image profiles with a Wolter system at the resolution limits of the other systems.

Fig. 11
Fig. 11

Fidelity of the images of the three-dark-bar object with each system at (a) s = 2.0 and (b) s = 5.0.

Tables (1)

Tables Icon

Table 1 Resolution Limit with Use of Various Criteria for the Circular, Schwarzschild, and Wolter Systemsa

Equations (17)

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

I ( x , y ) = const . C ( m , n ; p , q ) T ( m , n ) T * ( p , q ) × exp { 2 π j [ ( m p ) x + ( n q ) y ] } d m d n d p d q ,
C ( m , n ; p , q ) = | P 2 ( u , υ ) | 2 P 1 ( m + u , ñ + υ ) × P 1 * ( p + u , q + υ ) d u d υ ,
t ( x , y ) = 2 f α + N = 1 2 N π sin ( 2 N π f α ) cos ( 2 N π f x ) ,
t ( x , y ) = 1 f s 4 π N = 1 1 N sin ( N π f s 2 ) × cos ( N π f s ) cos ( 2 N π f x ) .
T ( u , υ ) = ( 1 f s ) δ ( u , υ ) 2 π N = 1 1 N sin ( N π f s 2 ) × cos ( N π f s ) [ δ ( N f + u , υ ) + δ ( N f + u , υ ) ] .
I ( x , y ) = ( 1 f s ) 2 ( 1 f s ) 8 π N = 1 1 N Re { C ( N f ; 0 ) } × sin ( N π f s 2 ) cos ( N π f s ) cos ( 2 N π f x ) + 8 π 2 N = 1 M = 1 1 N M sin ( N π f s 2 ) sin ( M π f s 2 ) × cos ( N π f s ) cos ( M π f s ) { C ( N f ; M f ) cos [ 2 π ( N M ) f x ] + C ( N f ; M f ) cos [ 2 π ( N + M ) f x ] } ,
C ( p ; m ) = C * ( m ; p ) ,
C ( m ; p ) = C ( m ; p ) ,
t ( x , y ) = 1 2 + 2 π N = 1 ( 1 ) N + 1 2 N 1 cos ( 2 N 1 ) 2 π f x .
T ( u , υ ) = 1 2 δ ( u , υ ) + 1 π N = 1 ( 1 ) N + 1 2 N 1 { δ [ ( 2 N 1 ) f + u , υ ] + δ [ ( 2 N 1 ) f + u , υ ] } .
I ( x , y ) = 1 4 + 2 π N = 1 ( 1 ) N + 1 2 N 1 × Re { C [ 2 N 1 ) f ; 0 ] } cos [ 2 π ( 2 N 1 ) f x ] + 2 π 2 M = 1 N = 1 ( 1 ) M + N ( 2 M 1 ) ( 2 N 1 ) × { C [ ( 2 M 1 ) f ; ( 2 N 1 ) f ] cos [ 4 π ( M N ) f x ] + C [ ( 2 M 1 ) f ; ( 2 N 1 ) f ] × cos [ 4 π ( M + N 1 ) f x ] } .
I ( x , y ) = const . | P 2 ( u , υ ) t ( x x , y y ) × h 1 ( x , y ) exp [ 2 π j ( x u + y υ ) ] d x d y | 2 d u d υ ,
t ( x ) = δ ( x d / 2 ) + δ ( x + d / 2 ) ,
I ( x ) = [ h 1 ( x d / 2 ) ] 2 + [ h 1 ( x + d / 2 ) ] 2 + 2 μ ( d ) h 1 ( x d / 2 ) h 1 ( x + d / 2 ) ,
t ( x , y ) = 1 3 2 f s 2 π N = 1 1 N sin ( N π f s 2 ) × [ 2 cos ( 2 N π f s ) + 1 ] cos ( 2 N π f x ) .
T ( u , υ ) = ( 1 3 2 f s ) δ ( u , υ ) 1 π N = 1 1 N sin ( N π f s 2 ) × [ 2 cos ( 2 N π f s ) + 1 ] × [ δ ( n f + u , υ ) + δ ( n f + u , υ ) ] ,
I ( x , y ) = ( 1 3 2 f s ) 2 ( 1 3 2 f s ) 4 π N = 1 1 N × Re { C ( N f ; 0 ) } sin ( N π f s 2 ) × [ 2 cos ( 2 N π f s ) + 1 ] cos ( 2 N π f x ) + 2 π 2 N = 1 M = 1 1 N M sin ( N π f s 2 ) sin ( M π f s 2 ) × [ 2 cos ( 2 N π f s ) + 1 ] [ 2 cos ( 2 M π f s ) + 1 ] × { C ( N f ; M f ) cos [ 2 π ( N M ) f x ] + C ( N f ; M f ) cos [ 2 π ( N + M ) f x ] } .

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