Abstract

A rigorous numerical simulation of imaging in an optical scanning microscope is described. The sample is assumed to be a perfectly conducting one-dimensional surface-relief object. The approach is capable of taking into account multiple interactions between the illuminating field and the surface and is also valid for optically thick objects. The results are compared with those obtained by use of the Kirchhoff approximation and the thin-phase-screen model. We find that for the structures considered, the Kirchhoff approximation is adequate for describing the process of image formation, except in those cases in which the surface leads to significant amounts of multiple scattering.

© 1994 Optical Society of America

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References

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  1. W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
    [CrossRef]
  2. P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  3. M. Nieto-Vesperinas, Sattering and Diffraction in Physical Optics (Wiley, New York, 1991).
  4. J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).
  5. D. Nyyssonen, “Theory of optical edge detection and imaging of thick layers,”J. Opt. Soc. Am. 72, 1425–1436 (1982).
    [CrossRef]
  6. D. Nyyssonen, C. P. Kirk, “Optical microscope imaging of lines patterned in thick layers with variable edge geometry: theory,” J. Opt. Soc. Am. A 5, 1270–1280 (1988).
    [CrossRef]
  7. D. M. Gale, M. I. Pether, F. C. Reavell, “Interference microscopy of surface relief structures,” in Optical Microlithographic Technology for Integrated Circuit Fabrication and Inspection, H. L. Stover, S. Wittekoek, eds., Proc. Soc. Photo-Opt. Instrum. Eng.811, 40–47 (1987).
    [CrossRef]
  8. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980).
    [CrossRef]
  9. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  10. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,”J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  11. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,”J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [CrossRef]
  12. F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectric constant,”J. Opt. Soc. Am. 63, 37–45 (1973).
    [CrossRef]
  13. See, for example, A. Wirgin, “A new theoretical approach to scattering from a periodic interface,” Opt. Commun. 27, 189–194 (1978); D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
    [CrossRef]
  14. E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 83, 78–92 (1988).
    [CrossRef]
  15. J. M. Soto-Crespo, M. Nieto-Vesperinas, “Electromagnetic scattering from very rough random surfaces and deep ref lection gratings,” J. Opt. Soc. Am. A 6, 367–384 (1989).
    [CrossRef]
  16. A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
    [CrossRef]
  17. B. J. Kachoyan, C. Macaskill, “Acoustic scattering from an arbitrary rough surface,”J. Acoust. Soc. Am. 82, 1720–1726 (1987).
    [CrossRef]
  18. C. J. R. Sheppard, J. T. Sheridan, “Micrometrology of thick structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1139, 32–39 (1989).
    [CrossRef]
  19. C. M. Yuan, A. J. Strojwas, “Modeling optical microscope images of integrated-circuit structures,” J. Opt. Soc. Am. A 8, 778–790 (1991).
    [CrossRef]
  20. Some preliminary results were presented in J. F. Aguilar, E. R. Méndez, “Numerical simulations of images of perfect conducting rough surfaces in scanning microscopy,” in New Methods in Microscopy and Low Light Imaging, J. E. Wampler, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1161, 286–296 (1989).
    [CrossRef]
  21. W. T. Welford, “Length measurement at the optical resolution limit by scanning microscopy,” in Optics in Metrology, P. Mollet, ed. (Pergamon, London, 1960), pp. 85–91.
  22. M. E. Barnett, “The reciprocity theorem and the equivalence of conventional and scanning transmission microscopes,” Optik 38, 585–588 (1974).
  23. D. Kermisch, “Principle of equivalence between scanning and conventional optical imaging systems,”J. Opt. Soc. Am. 67, 1357–1360 (1977).
    [CrossRef]
  24. C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1978).
    [CrossRef]
  25. C. J. R. Sheppard, T. Wilson, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).
  26. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1986), Chap. 11, p. 560.
  27. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 7, p. 300.
  28. See, e.g., Ref. 26, Chap. 8, p. 441.
  29. Ref. 27, Chap. 9, p. 364.
  30. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 2, p. 31.
  31. W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,”J. Rat. Mech. Anal. 5, 323–333 (1956).
  32. E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary—extension of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
    [CrossRef]
  33. See, e.g., Ref. 26, Chap. 8, p. 436, and J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 13, p. 378.

1991

1990

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

1989

1988

D. Nyyssonen, C. P. Kirk, “Optical microscope imaging of lines patterned in thick layers with variable edge geometry: theory,” J. Opt. Soc. Am. A 5, 1270–1280 (1988).
[CrossRef]

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

1987

B. J. Kachoyan, C. Macaskill, “Acoustic scattering from an arbitrary rough surface,”J. Acoust. Soc. Am. 82, 1720–1726 (1987).
[CrossRef]

1985

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1982

E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary—extension of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
[CrossRef]

D. Nyyssonen, “Theory of optical edge detection and imaging of thick layers,”J. Opt. Soc. Am. 72, 1425–1436 (1982).
[CrossRef]

1981

1978

See, for example, A. Wirgin, “A new theoretical approach to scattering from a periodic interface,” Opt. Commun. 27, 189–194 (1978); D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
[CrossRef]

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1978).
[CrossRef]

1977

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

D. Kermisch, “Principle of equivalence between scanning and conventional optical imaging systems,”J. Opt. Soc. Am. 67, 1357–1360 (1977).
[CrossRef]

1974

M. E. Barnett, “The reciprocity theorem and the equivalence of conventional and scanning transmission microscopes,” Optik 38, 585–588 (1974).

1973

1966

1956

W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,”J. Rat. Mech. Anal. 5, 323–333 (1956).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 7, p. 300.

Aguilar, J. F.

Some preliminary results were presented in J. F. Aguilar, E. R. Méndez, “Numerical simulations of images of perfect conducting rough surfaces in scanning microscopy,” in New Methods in Microscopy and Low Light Imaging, J. E. Wampler, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1161, 286–296 (1989).
[CrossRef]

Barnett, M. E.

M. E. Barnett, “The reciprocity theorem and the equivalence of conventional and scanning transmission microscopes,” Optik 38, 585–588 (1974).

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1986), Chap. 11, p. 560.

Burckhardt, C. B.

Choudhury, A.

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1978).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 2, p. 31.

Gale, D. M.

D. M. Gale, M. I. Pether, F. C. Reavell, “Interference microscopy of surface relief structures,” in Optical Microlithographic Technology for Integrated Circuit Fabrication and Inspection, H. L. Stover, S. Wittekoek, eds., Proc. Soc. Photo-Opt. Instrum. Eng.811, 40–47 (1987).
[CrossRef]

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,”J. Opt. Soc. Am. 71, 811–818 (1981).
[CrossRef]

Kachoyan, B. J.

B. J. Kachoyan, C. Macaskill, “Acoustic scattering from an arbitrary rough surface,”J. Acoust. Soc. Am. 82, 1720–1726 (1987).
[CrossRef]

Kaspar, F. G.

Kermisch, D.

Kirk, C. P.

Liszka, E. G.

E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary—extension of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
[CrossRef]

Macaskill, C.

B. J. Kachoyan, C. Macaskill, “Acoustic scattering from an arbitrary rough surface,”J. Acoust. Soc. Am. 82, 1720–1726 (1987).
[CrossRef]

Maradudin, A. A.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

McCoy, J. J.

E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary—extension of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
[CrossRef]

McGurn, A. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Meecham, W. C.

W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,”J. Rat. Mech. Anal. 5, 323–333 (1956).

Méndez, E. R.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Some preliminary results were presented in J. F. Aguilar, E. R. Méndez, “Numerical simulations of images of perfect conducting rough surfaces in scanning microscopy,” in New Methods in Microscopy and Low Light Imaging, J. E. Wampler, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1161, 286–296 (1989).
[CrossRef]

Michel, T.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,”J. Opt. Soc. Am. 71, 811–818 (1981).
[CrossRef]

Nieto-Vesperinas, M.

Nyyssonen, D.

Ogilvy, J. A.

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).

Pether, M. I.

D. M. Gale, M. I. Pether, F. C. Reavell, “Interference microscopy of surface relief structures,” in Optical Microlithographic Technology for Integrated Circuit Fabrication and Inspection, H. L. Stover, S. Wittekoek, eds., Proc. Soc. Photo-Opt. Instrum. Eng.811, 40–47 (1987).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 2, p. 31.

Reavell, F. C.

D. M. Gale, M. I. Pether, F. C. Reavell, “Interference microscopy of surface relief structures,” in Optical Microlithographic Technology for Integrated Circuit Fabrication and Inspection, H. L. Stover, S. Wittekoek, eds., Proc. Soc. Photo-Opt. Instrum. Eng.811, 40–47 (1987).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1978).
[CrossRef]

C. J. R. Sheppard, J. T. Sheridan, “Micrometrology of thick structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1139, 32–39 (1989).
[CrossRef]

C. J. R. Sheppard, T. Wilson, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

Sheridan, J. T.

C. J. R. Sheppard, J. T. Sheridan, “Micrometrology of thick structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1139, 32–39 (1989).
[CrossRef]

Soto-Crespo, J. M.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 7, p. 300.

Strojwas, A. J.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 2, p. 31.

Thorsos, E. I.

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 2, p. 31.

Welford, W. T.

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

W. T. Welford, “Length measurement at the optical resolution limit by scanning microscopy,” in Optics in Metrology, P. Mollet, ed. (Pergamon, London, 1960), pp. 85–91.

Wilson, T.

C. J. R. Sheppard, T. Wilson, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

Wirgin, A.

See, for example, A. Wirgin, “A new theoretical approach to scattering from a periodic interface,” Opt. Commun. 27, 189–194 (1978); D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1986), Chap. 11, p. 560.

Yuan, C. M.

Ann. Phys.

A. A. Maradudin, T. Michel, A. R. McGurn, E. R. Méndez, “Enhanced backscattering of light from a random grating,” Ann. Phys. 203, 255–307 (1990).
[CrossRef]

J. Acoust. Soc. Am.

B. J. Kachoyan, C. Macaskill, “Acoustic scattering from an arbitrary rough surface,”J. Acoust. Soc. Am. 82, 1720–1726 (1987).
[CrossRef]

E. G. Liszka, J. J. McCoy, “Scattering at a rough boundary—extension of the Kirchhoff approximation,”J. Acoust. Soc. Am. 71, 1093–1100 (1982).
[CrossRef]

E. I. Thorsos, “The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum,”J. Acoust. Soc. Am. 83, 78–92 (1988).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Rat. Mech. Anal.

W. C. Meecham, “On the use of the Kirchhoff approximation for the solution of reflection problems,”J. Rat. Mech. Anal. 5, 323–333 (1956).

Opt. Acta

C. J. R. Sheppard, A. Choudhury, “Image formation in the scanning microscope,” Opt. Acta 24, 1051–1073 (1978).
[CrossRef]

Opt. Commun.

See, for example, A. Wirgin, “A new theoretical approach to scattering from a periodic interface,” Opt. Commun. 27, 189–194 (1978); D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
[CrossRef]

Opt. Quantum Electron.

W. T. Welford, “Optical estimation of statistics of surface roughness from light scattering measurements,” Opt. Quantum Electron. 9, 269–287 (1977).
[CrossRef]

Optik

M. E. Barnett, “The reciprocity theorem and the equivalence of conventional and scanning transmission microscopes,” Optik 38, 585–588 (1974).

Proc. IEEE

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Other

C. J. R. Sheppard, J. T. Sheridan, “Micrometrology of thick structures,” in Optical Storage and Scanning Technology, T. Wilson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1139, 32–39 (1989).
[CrossRef]

P. Beckmann, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).

M. Nieto-Vesperinas, Sattering and Diffraction in Physical Optics (Wiley, New York, 1991).

J. A. Ogilvy, Theory of Wave Scattering from Random Rough Surfaces (Hilger, Bristol, UK, 1991).

D. M. Gale, M. I. Pether, F. C. Reavell, “Interference microscopy of surface relief structures,” in Optical Microlithographic Technology for Integrated Circuit Fabrication and Inspection, H. L. Stover, S. Wittekoek, eds., Proc. Soc. Photo-Opt. Instrum. Eng.811, 40–47 (1987).
[CrossRef]

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980).
[CrossRef]

C. J. R. Sheppard, T. Wilson, Theory and Practice of Scanning Optical Microscopy (Academic, London, 1984).

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1986), Chap. 11, p. 560.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 7, p. 300.

See, e.g., Ref. 26, Chap. 8, p. 441.

Ref. 27, Chap. 9, p. 364.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 2, p. 31.

See, e.g., Ref. 26, Chap. 8, p. 436, and J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986), Chap. 13, p. 378.

Some preliminary results were presented in J. F. Aguilar, E. R. Méndez, “Numerical simulations of images of perfect conducting rough surfaces in scanning microscopy,” in New Methods in Microscopy and Low Light Imaging, J. E. Wampler, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1161, 286–296 (1989).
[CrossRef]

W. T. Welford, “Length measurement at the optical resolution limit by scanning microscopy,” in Optics in Metrology, P. Mollet, ed. (Pergamon, London, 1960), pp. 85–91.

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Figures (15)

Fig. 1
Fig. 1

Schematic diagram of the simulated microscope.

Fig. 2
Fig. 2

Modes of image formation: (a) bright-field coherent, (b) dark-field coherent, (c) partially coherent, (d) confocal.

Fig. 3
Fig. 3

Schematic diagram showing the scattering geometry and the notation. The surface profile ζ(x) divides two regions: the region z > ζ(x) is vacuum, and the region z < ζ(x) is the perfect conductor. The position vector of the point of observation is denoted by r, and that of a point on the surface by r′. The surface has length L and is centered with the origin of the x axis.

Fig. 4
Fig. 4

Incident field: (a) the intensity as a function of (v, u); (b) cross section of the intensity along the line v = 0, (c) cross section of the intensity along the line u = 0.

Fig. 5
Fig. 5

Surface profile used in the calculations: (a), (b) steplike surface profiles; (c) a rectangular groove, (d) a rectangular ridge, (e) a Gaussian groove, (f) a Gaussian ridge.

Fig. 6
Fig. 6

s-Polarized, bright-field, coherent images of a perfectly conducting surface defined by the surface profile shown in Fig. 5(a). (a) Comparison of the rigorous results (solid curves) and those based on the Kirchhoff approximation (circles), (b) comparison of the results based on the Kirchhoff approximation (circles) and those based on the thin-phase-screen model (solid curve).

Fig. 7
Fig. 7

s-Polarized, bright-field, coherent images of a perfectly conducting surface defined by the surface profile shown in Fig. 5(b). (a) Comparison of the rigorous results (solid curve) and those based on the Kirchhoff approximation (circles), (b) comparison of the results based on the Kirchhoff approximation (circles) and those based on the thin-phase-screen model (solid curve).

Fig. 8
Fig. 8

s-Polarized, bright-field, coherent images of perfectly conducting surfaces defined by the rectangular surface profiles shown in Fig. 5. (a) Comparison of the rigorous results (solid curves) and those based on the Kirchhoff approximation (circles) for the groove shown in Fig. 5(c), (b) comparison of the rigorous results (solid curve) and those based on the Kirchhoff approximation (circles) for the ridge shown in Fig. 5(d).

Fig. 9
Fig. 9

s-Polarized, bright-field, coherent images of perfectly conducting surfaces defined by the Gaussian surface profiles shown in Fig. 5. (a) Comparison of the rigorous results (solid curve) and those based on the Kirchhoff approximation (circles) for the groove shown in Fig. 5(e), (b) comparison of the rigorous results (solid curve) and those based on the Kirchhoff approximation (circles) for the ridge shown in Fig. 5(f).

Fig. 10
Fig. 10

s-Polarized, bright-field, coherent images of perfectly conducting surfaces defined by the groovelike surface profiles shown in Fig. 5. (a) Images of the groove shown in Fig. 5(c); (b) images of the groove shown in Fig. 5(e). We compare the rigorous results (solid curves) with those based on the first iterative solution of Eq. (31) below (circles), which includes single- and double-scattering processes.

Fig. 11
Fig. 11

p-Polarized, bright-field, coherent images of a perfectly conducting surface defined by the surface profile shown in Fig. 5(c). (a) Comparison of the rigorous results (solid curve) and those based on the Kirchhoff approximation (circles), (b) comparison of the rigorous results for p polarization (circles) and for s polarization (solid curve).

Fig. 12
Fig. 12

p-Polarized, bright-field, coherent images of a perfectly conducting surface defined by the surface profile shown in Fig. 5(d). (a) Comparison of the rigorous results (solid curve) and those based on the Kirchhoff approximation (circles), (b) comparison of the rigorous results for p polarization (circles) and for s polarization (solid curve).

Fig. 13
Fig. 13

s-Polarized, bright-field, partially coherent images of the perfectly conducting surface defined by the surface profile shown in Fig. 5(c). (a) The numerical aperture of the condenser is 0.2, and that of the objective is 0.4; (b) both numerical apertures are set at 0.4. We compare the results obtained with the rigorous technique (solid curve) with those based on the Kirchhoff approximation (circles).

Fig. 14
Fig. 14

Perfectly conducting surface and its s-polarized confocal images: (a) surface profile, (b) microscope focused on the plane z = 1.0 μm, (c) microscope focused on the plane z = −1.0 μm, (d) microscope focused on the plane z = 0.0 μm. We compare the results obtained with the rigorous technique (solid curves) with those based on the Kirchhoff approximation (circles).

Fig. 15
Fig. 15

Perfectly conducting surface and its s-polarized confocal images: (a) surface profile, (b) microscope focused on the plane z = 1.0 μm, (c) microscope focused on the plane z = −1.0 μm, (d) microscope focused on the plane z = 0.0 μm. We compare the results obtained with the rigorous technique (solid curves) with those based on the Kirchhoff approximation (circles).

Equations (49)

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

E i ( x , 0 ) = sinc ( 2 α m x λ ) ,
E i ( v , u ) = exp [ - i ( u α m 2 + v 2 2 u ) ] π 4 u [ f ( u π + v u π ) + f ( u π - v u π ) ] ,
f ( x ) = 0 x exp ( i π 2 ξ 2 ) d ξ ,
u = k α m 2 z ,
v = k α m x .
E x = - i k z B y ,
E z = - i k x B y ,
E ( x , z ) = E i ( x , z ) - 1 4 π S [ G ( x , z ; x , z ) × E ( x , z ) n - E ( x , z ) n G ( x , z ; x , z ) ] d s ,
n = 1 { 1 + [ ζ ( x ) ] 2 } 1 / 2 [ - ζ ( x ) x + z ]
G ( x , z ; x , z ) = i π H 0 ( 1 ) { k [ ( x - x ) 2 + ( z - z ) 2 ] 1 / 2 } ,
E ( x , z ) = E i ( x , z ) + 1 4 i - L / 2 L / 2 H 0 ( 1 ) { k [ ( x - x ) 2 + ( z - z ) 2 ] 1 / 2 } F ( x ) d x ,
F ( x ) = [ - ζ ( x ) x + z ] E ( x , z ) .
E i ( x 0 ) = - L / 2 L / 2 L 0 ( x 0 , x ) F ( x ) d x ,
L 0 ( x 0 , x ) = - 1 4 i lim 0 H 0 ( 1 ) ( k { ( x 0 - x ) 2 + [ ζ ( x 0 ) + ɛ - ζ ( x ) ] 2 } 1 / 2 )
E s ( r , θ ) = Φ r - L / 2 L / 2 - F ( x ) exp { - i k [ x sin θ + ζ ( x ) cos θ ] } × d x ,
Φ r = ( 1 4 ) 2 π k exp ( i π / 4 ) exp ( i k r ) r
B ( x , z ) n = 0.
B ( x , z ) = B i ( x , z ) + i 4 - L / 2 L / 2 B ( x ) ( [ - ζ ( x ) x + z ] H 0 ( 1 ) × { k [ ( x - x ) 2 + ( z - z ) 2 ] 1 / 2 } ) d x ,
B ( x 0 ) = B i ( x 0 ) + - L / 2 L / 2 B ( x ) M 0 ( x 0 , x ) d x ,
M 0 ( x 0 , x ) = i 4 lim 0 ( - ζ ( x ) x + z ) × H 0 ( 1 ) ( k { ( x 0 - x ) 2 + [ ζ ( x 0 ) + ɛ - ζ ( x ) ] 2 } 1 / 2 )
B s ( r , θ ) = Φ r - L / 2 L / 2 ( i k ) [ ζ ( x ) sin θ - cos θ ] B ( x ) × exp { - i k [ x sin θ + ζ ( x ) cos θ ] } d x ,
x j = - L 2 + ( j - 1 2 ) Δ x ,
E i ( x m ) = n = 1 N x n - Δ x / 2 x n + Δ x / 2 L 0 ( x m , x ) F ( x ) d x .
L m n = x n - Δ x / 2 x n + Δ x / 2 L 0 ( x m , x ) d x ,
E i m = n = 1 N F n L m n ,
L m n = { - Δ x 4 i H 0 ( 1 ) ( k { ( x m - x n ) 2 + [ ζ ( x m ) - ζ ( x n ) ] 2 } 1 / 2 ) for m n - Δ x 4 i H 0 ( 1 ) ( k Δ x 2 e { 1 + [ ζ ( x m ) ] 2 } 1 / 2 ) for m = n .
E s ( r , θ ) = Φ r n = 1 N F n exp { - i k [ x n sin θ + ζ ( x n ) cos θ ] } Δ x .
B m = B i m + n = 1 N B n M m n ,
M m n = x n - Δ x / 2 x n + Δ x / 2 M 0 ( x m , x ) d x .
B i m = n = 1 N B n M m n ,
M m n = { k 2 Δ x 4 i { [ ζ ( x m ) - ζ ( x n ) ] - ζ ( x n ) ( x m - x n ) } H 1 ( 1 ) ( k { ( x m - x n ) 2 + [ ζ ( x m ) - ζ ( x n ) ] 2 } 1 / 2 ) k { ( x m - x n ) 2 + [ ζ ( x m ) - ζ ( x n ) ] 2 } 1 / 2 for m n 1 2 - Δ x ζ ( x m ) 4 π { 1 + [ ζ ( x m ) ] 2 } for m = n .
B s ( r , θ ) = Φ r ( i k ) n = 1 N [ ζ ( x n ) sin θ - cos θ ] B n × exp { - i k [ x n sin θ + ζ ( x n ) cos θ ] } Δ x .
F ( x 0 ) = 2 F i ( x 0 ) + 1 2 i - L / 2 L / 2 ( [ - ζ ( x 0 ) x + z × H 0 ( 1 ) { k [ ( x - x ) 2 + ( z - z ) 2 ] 1 / 2 } ] ) x = x 0 x = ζ ( x 0 ) F ( x ) d x .
Ψ i 0 ( x , 0 ) = - A 0 ( α λ ) exp ( i k α x ) d ( α λ ) ,
A 0 ( α λ ) = λ 2 α m rect ( α 2 α m ) ,
rect ( x ) = { 1 if x 1 2 0 if x > 1 2 .
A ( α λ ; z ) = A 0 ( α λ ) exp ( - i k z 1 - α 2 ) ,
Ψ i ( x , z ) = 1 2 α m - α m α m exp ( - i k z 1 - α 2 ) exp ( i k α x ) d α .
1 - α 2 1 - α 2 2 .
Ψ i ( x , z ) = exp ( - i k z ) 2 - 1 1 exp ( i k α m 2 t 2 2 z ) exp ( i k α m t x ) d t .
Ψ i ( v , u ) = exp ( - i u α m 2 ) 1 2 - 1 1 exp ( i u 2 t 2 ) exp ( i v t ) d t ,
Ψ i ( v , u ) v = i u { exp [ - i ( u α m 2 - u 2 ) ] sin ( v ) - v Ψ i ( v , u ) } ,
Ψ i ( v , u ) v = exp [ - i ( u α m 2 - u 2 ) ] v 2 2 u 2 [ u cos ( v ) v 2 - i sin ( v ) v ] + [ i ( v 2 2 u 2 - 1 α m 2 ) - 1 2 u ] Ψ i ( v , u ) .
Ψ i ( v , u ) v | v 0 = 0 ,
Ψ i ( v , u ) v | u = 0 = 1 v [ cos ( v ) - sin ( v ) v ] ,
Ψ i ( v , u ) v | v 0 u 0 = 0 ;
Ψ i ( v , u ) u | v 0 = exp [ - i ( u / α m 2 - u / 2 ) ] 2 u - ( 1 2 u + i α m 2 ) exp ( - i u α m 2 ) π u f ( u π ) ,
Ψ i ( v , u ) u | u 0 = i v 2 [ cos ( v ) + sin ( v ) v ( v 2 2 - v 2 α m 2 - 1 ) ] ,
Ψ i ( v , u ) u | v 0 u 0 = i 6 - i α m 2 .

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