Abstract

The diffraction orders and coherent bright and dark field images produced by isolated steps and grooves (notches and trenches) are examined. The isolated structures are approximated by means of a grating with a large period that contains small, widely separated, lamellar structures. The rigorous modal method for TE polarized incident light is used for the numerical calculations. Structures in two types of dielectric material, refractive index n = 1.5 and n = 4.0, are discussed. The images are examined for variations with several parameters, including width and thickness of the structure and focus position. Several approximate methods for calculating these images, based on first-order approximations, are suggested.

© 1993 Optical Society of America

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References

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    [CrossRef]
  4. J. W. Dockrey, “The application of coherence probe microscopy for submicron critical dimension linewidth measurement,” in Integrated Circuit Metrology, Inspection, and Process Control III, K. M. Monahan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1087, 120–137 (1989).
    [CrossRef]
  5. R. Petit, Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, New York, 1980).
    [CrossRef]
  6. D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXII, E. Wolf, ed. (Elsevier, Amsterdam, 1984), pp. 1–67.
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    [CrossRef]
  8. J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la determination numerique des efficacties de certains reseaux dielectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
    [CrossRef]
  9. Y. Nakata, M. Koshiba, “Boundary-element analysis of plane-wave diffraction from groove-type dielectric and metallic gratings,” J. Opt. Soc. Am. A 7, 1494–1502 (1990).
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    [CrossRef]
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  21. 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).
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    [CrossRef]
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    [CrossRef]
  29. R. Morf, “Rigorous diffraction theory,” Annu. Rep. 26 (Paul Scherrer Institute, Zurich, 1990).
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    [CrossRef]
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  34. C. J. R. Sheppard, H. J. Mathews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
    [CrossRef]
  35. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products4th ed. (Academic, New York, 1980),
  36. C. J. R. Sheppard, J. M. Heaton, “Images of surface steps in coherent illumination,” Optik 68, 267–280 (1984).
  37. C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Part I, Optik, 72, 131–133 (1986);Part II, Optik 74, 128–129 (1986).
  38. P. Beckmann, A. Spizzichino, The Scatter of Electromagnetic Waves from a Rough Surface (Artech, Northwood, Mass., 1987).
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    [CrossRef]

1991 (3)

1990 (4)

J. T. Sheridan, C. J. R. Sheppard, “An examination of the theories for the calculation of diffraction by square-wave gratings: 2. Angular variation,” Optik 85, 57–66 (1990).

J. T. Sheridan, C. J. R. Sheppard, “An examination of the theories for the calculation of diffraction by square-wave gratings: 3. Approximate theories,” Optik 85, 135–152 (1990).

Y. Nakata, M. Koshiba, “Boundary-element analysis of plane-wave diffraction from groove-type dielectric and metallic gratings,” J. Opt. Soc. Am. A 7, 1494–1502 (1990).
[CrossRef]

J. T. Sheridan, C. J. R. Sheppard, “An examination of the theories for the calculation of diffraction by square-wave gratings: 1. Thickness and period variations for normal incidence,” Optik 85, 25–32 (1990).

1989 (1)

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

1988 (3)

1987 (1)

1986 (1)

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Part I, Optik, 72, 131–133 (1986);Part II, Optik 74, 128–129 (1986).

1984 (2)

1983 (1)

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la determination numerique des efficacties de certains reseaux dielectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
[CrossRef]

1982 (2)

1978 (1)

1973 (1)

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

1966 (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]

Awada, K. A.

Beauchamp, K. G.

K. G. Beauchamp, A Guide to Signal Processing (Oxford Science, Oxford, 1987).

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scatter of Electromagnetic Waves from a Rough Surface (Artech, Northwood, Mass., 1987).

Bertoni, H. L.

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Born, M.

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

Bracewell, R. M.

R. M. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

Burckhardt, C. B.

Cadilhac, M.

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la determination numerique des efficacties de certains reseaux dielectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
[CrossRef]

Cheo, L. S.

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Chou, C-H.

G. S. Kino, C-H. Chou, G. Q. Xiao, ”Imaging theory for the scanning optical microscope,” in Scanning Imaging, T. Wilson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1028, 104–113 (1988).
[CrossRef]

DeSanto, J. A.

Dockrey, J. W.

J. W. Dockrey, “The application of coherence probe microscopy for submicron critical dimension linewidth measurement,” in Integrated Circuit Metrology, Inspection, and Process Control III, K. M. Monahan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1087, 120–137 (1989).
[CrossRef]

Forsyth, J. M.

Gaylord, T. K.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products4th ed. (Academic, New York, 1980),

Heaton, J. M.

C. J. R. Sheppard, J. M. Heaton, “Images of surface steps in coherent illumination,” Optik 68, 267–280 (1984).

Kaspar, F. G.

Kino, G. S.

G. S. Kino, C-H. Chou, G. Q. Xiao, ”Imaging theory for the scanning optical microscope,” in Scanning Imaging, T. Wilson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1028, 104–113 (1988).
[CrossRef]

Kirk, C. P.

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]

C. P. Kirk, D. Nyyssonen, “Modeling the optical microscope images of thick layers for the purpose of linewidth measurement,” in Optical Microlithography TV, H. L. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.538, 179–187 (1985).

Knop, K.

Korner, T. W.

T. W. Korner, Fourier Analysis (Cambridge U. Press, Cambridge, 1988).

Koshiba, M.

Lohmann, A.

A. Lohmann, “An array illuminator based on the Talbot-effect,” Optik 79, 41–45 (1988).

Mathews, H. J.

Maystre, D.

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXII, E. Wolf, ed. (Elsevier, Amsterdam, 1984), pp. 1–67.

Miller, J. M.

A. Vasara, E. Nopenen, J. Turunen, J. M. Miller, M. R. Taghizadeh, J. Tuovinen, “Rigorous diffraction theory of binary optical interconnects,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507, 224–238 (1991).
[CrossRef]

Moharam, M. G.

Morf, R.

R. Morf, “Rigorous diffraction theory,” Annu. Rep. 26 (Paul Scherrer Institute, Zurich, 1990).

R. MorfPaul Scherrer Institute, Zurich, “Diffraction theory—DOE efficiency,” contribution to the Workshop on Optical Information Technology, European Optical Computing Group (received as a personal communication, 1991).

Nakata, Y.

Nopenen, E.

A. Vasara, E. Nopenen, J. Turunen, J. M. Miller, M. R. Taghizadeh, J. Tuovinen, “Rigorous diffraction theory of binary optical interconnects,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507, 224–238 (1991).
[CrossRef]

Nyyssonen, D.

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]

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

C. P. Kirk, D. Nyyssonen, “Modeling the optical microscope images of thick layers for the purpose of linewidth measurement,” in Optical Microlithography TV, H. L. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.538, 179–187 (1985).

Pai, D. M.

Petit, R.

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la determination numerique des efficacties de certains reseaux dielectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
[CrossRef]

R. Petit, Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, New York, 1980).
[CrossRef]

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]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products4th ed. (Academic, New York, 1980),

Sheppard, C. J. R.

J. T. Sheridan, C. J. R. Sheppard, “An examination of the theories for the calculation of diffraction by square-wave gratings: 2. Angular variation,” Optik 85, 57–66 (1990).

J. T. Sheridan, C. J. R. Sheppard, “An examination of the theories for the calculation of diffraction by square-wave gratings: 3. Approximate theories,” Optik 85, 135–152 (1990).

J. T. Sheridan, C. J. R. Sheppard, “An examination of the theories for the calculation of diffraction by square-wave gratings: 1. Thickness and period variations for normal incidence,” Optik 85, 25–32 (1990).

C. J. R. Sheppard, H. J. Mathews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[CrossRef]

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Part I, Optik, 72, 131–133 (1986);Part II, Optik 74, 128–129 (1986).

C. J. R. Sheppard, J. M. Heaton, “Images of surface steps in coherent illumination,” Optik 68, 267–280 (1984).

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]

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

Sheridan, J. T.

J. T. Sheridan, C. J. R. Sheppard, “An examination of the theories for the calculation of diffraction by square-wave gratings: 1. Thickness and period variations for normal incidence,” Optik 85, 25–32 (1990).

J. T. Sheridan, C. J. R. Sheppard, “An examination of the theories for the calculation of diffraction by square-wave gratings: 2. Angular variation,” Optik 85, 57–66 (1990).

J. T. Sheridan, C. J. R. Sheppard, “An examination of the theories for the calculation of diffraction by square-wave gratings: 3. Approximate theories,” Optik 85, 135–152 (1990).

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]

Sincerbox, G. T.

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scatter of Electromagnetic Waves from a Rough Surface (Artech, Northwood, Mass., 1987).

Stagaman, G.

Strojwas, A.

Suratteau, J. Y.

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la determination numerique des efficacties de certains reseaux dielectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
[CrossRef]

Taghizadeh, M. R.

A. Vasara, E. Nopenen, J. Turunen, J. M. Miller, M. R. Taghizadeh, J. Tuovinen, “Rigorous diffraction theory of binary optical interconnects,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507, 224–238 (1991).
[CrossRef]

Tamir, T.

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Tuovinen, J.

A. Vasara, E. Nopenen, J. Turunen, J. M. Miller, M. R. Taghizadeh, J. Tuovinen, “Rigorous diffraction theory of binary optical interconnects,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507, 224–238 (1991).
[CrossRef]

Turunen, J.

A. Vasara, E. Nopenen, J. Turunen, J. M. Miller, M. R. Taghizadeh, J. Tuovinen, “Rigorous diffraction theory of binary optical interconnects,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507, 224–238 (1991).
[CrossRef]

Vasara, A.

A. Vasara, E. Nopenen, J. Turunen, J. M. Miller, M. R. Taghizadeh, J. Tuovinen, “Rigorous diffraction theory of binary optical interconnects,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507, 224–238 (1991).
[CrossRef]

Werlich, H.

Wilson, T.

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

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[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, 1980).

Wombell, D. J.

Xiao, G. Q.

G. S. Kino, C-H. Chou, G. Q. Xiao, ”Imaging theory for the scanning optical microscope,” in Scanning Imaging, T. Wilson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1028, 104–113 (1988).
[CrossRef]

Yuan, C.

Yung, B.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

J. Opt. (Paris) (1)

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la determination numerique des efficacties de certains reseaux dielectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (7)

Opt. Commun. (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Optik (6)

C. J. R. Sheppard, J. M. Heaton, “Images of surface steps in coherent illumination,” Optik 68, 267–280 (1984).

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Part I, Optik, 72, 131–133 (1986);Part II, Optik 74, 128–129 (1986).

A. Lohmann, “An array illuminator based on the Talbot-effect,” Optik 79, 41–45 (1988).

J. T. Sheridan, C. J. R. Sheppard, “An examination of the theories for the calculation of diffraction by square-wave gratings: 2. Angular variation,” Optik 85, 57–66 (1990).

J. T. Sheridan, C. J. R. Sheppard, “An examination of the theories for the calculation of diffraction by square-wave gratings: 3. Approximate theories,” Optik 85, 135–152 (1990).

J. T. Sheridan, C. J. R. Sheppard, “An examination of the theories for the calculation of diffraction by square-wave gratings: 1. Thickness and period variations for normal incidence,” Optik 85, 25–32 (1990).

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]

Other (16)

C. P. Kirk, D. Nyyssonen, “Modeling the optical microscope images of thick layers for the purpose of linewidth measurement,” in Optical Microlithography TV, H. L. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.538, 179–187 (1985).

R. Morf, “Rigorous diffraction theory,” Annu. Rep. 26 (Paul Scherrer Institute, Zurich, 1990).

R. MorfPaul Scherrer Institute, Zurich, “Diffraction theory—DOE efficiency,” contribution to the Workshop on Optical Information Technology, European Optical Computing Group (received as a personal communication, 1991).

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

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products4th ed. (Academic, New York, 1980),

P. Beckmann, A. Spizzichino, The Scatter of Electromagnetic Waves from a Rough Surface (Artech, Northwood, Mass., 1987).

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

G. S. Kino, C-H. Chou, G. Q. Xiao, ”Imaging theory for the scanning optical microscope,” in Scanning Imaging, T. Wilson, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1028, 104–113 (1988).
[CrossRef]

J. W. Dockrey, “The application of coherence probe microscopy for submicron critical dimension linewidth measurement,” in Integrated Circuit Metrology, Inspection, and Process Control III, K. M. Monahan, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1087, 120–137 (1989).
[CrossRef]

R. Petit, Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics (Springer-Verlag, New York, 1980).
[CrossRef]

D. Maystre, “Rigorous vector theories of diffraction gratings,” in Progress in Optics XXII, E. Wolf, ed. (Elsevier, Amsterdam, 1984), pp. 1–67.

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]

R. M. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

T. W. Korner, Fourier Analysis (Cambridge U. Press, Cambridge, 1988).

K. G. Beauchamp, A Guide to Signal Processing (Oxford Science, Oxford, 1987).

A. Vasara, E. Nopenen, J. Turunen, J. M. Miller, M. R. Taghizadeh, J. Tuovinen, “Rigorous diffraction theory of binary optical interconnects,” in Holographic Optics III: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1507, 224–238 (1991).
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

Two dielectric structures examined, G1 and G2, with TE polarized incident light. In all the cases examined, θ1 = 0, and the orders will be arranged symmetrically above the structure with the zero order traveling back up the input beam. For isolated steps WgWs and for isolated grooves WsWg.

Fig. 2
Fig. 2

(a) Variations of the zeroth through the third diffraction order intensities from an isolated G2, nstep = 1.49, groove as the thickness of the grating varies from 0 → 4λ are shown. Wg = 1.05λ, Λ = 6.1λ, and N = 11. (b) Again a G2 groove, Wg = 1.05λ, is examined. But now the period is Λ = 15.1λ. The intensities of the diffraction orders decrease, and their sizes remain similar over a large range of thickness. There are positions at which the diffraction orders contain relatively little power, i.e., T = 0.8λ. In this case N = 18. (c) Same as in (b) but N is increased to 19 to check the convergence of the method. Changes in the results are noticeable at larger thickness for the second and third orders.

Fig. 3
Fig. 3

The same procedure is now followed for G2 steps as was carried out for G2 grooves. Comparison of the corresponding diagrams from Figs. 2 and 3 demonstrates the effects in going from a step to a groove, (a) Ws =1.05λ, Λ = 6.1λ, and N = 11. (b) As in Fig. 2(c), the convergence of the modal method for a larger period is checked. For an isolated G2 step, nstep = 1.49, with Ws = 1.05λ, Λ = 6.1λ, and N = 19.

Fig. 4
Fig. 4

(a) For a G1 groove with Wg = 1.05λ, Λ = 13.1λ, and N = 17. The diffraction-order intensities, except the zeroth, are similar in size and shape. The intensities vary slowly with thickness, indicating that volume effects, not only boundary effects, dominate the response. (b) Calculation of the response of a wider G1 groove with Wg = 2.05λ, Λ = 13.1λ, and N = 17. The shapes and sizes of the diffraction orders have altered noticeably from the results in (a). Rapid Fresnel-like interference effects can be seen for the larger width. The orders differ in size decreasing in magnitude, (c) G1 groove, Wg = 1.05λ,Λ = 11.1λ, and N = 15. (c) Along with (a), illustrates the effects of reducing the isolation of the G1 grooves. A decrease in the period strengthens the volume effect. The period of the interference with depth decreases, and the sizes of the orders increase, as does the relative difference between them, (d) For a G1 step with Ws = 2.05λ,Λ = 13.1λ, and N = 17. Convergence has been achieved in this case, and Fresnel interference effects dominate the results. The orders differ significantly in size.

Fig. 5
Fig. 5

Coherent BF image of a G2, nstep = 1.49, groove structure with Wg = 1.05λ, centered halfway between two isolated grooves. Λ = 11.1λ, T = 1.0λ, N = 15, and the N.A. is 6/11. (The +6 to −6 diffraction orders are included in the image.) Although there are maxima above and below the structure, no image appears at x = 0, far from both grooves.

Fig. 6
Fig. 6

(a), (b) BF and DF images of a G2 groove Wg = 1.95λ, T = 0.75λ, Λ = 11.1λ, N = 15, and the N.A. is 6/11. The groove is centered at x = 5.55λ. (c), (d) BF and DF images of a G2 groove with Wg = 1.05λ. Again Λ = 11.1λ, N = 15, and the N.A. is 6/11. But in these cases, T = 1.75λ, i.e., the base of the groove is at −1.75λ on the z defocus axis

Fig 7
Fig 7

(a), (b) BF and DF images of a G2 isolated groove with Wg = 2.05λ, T = 1.0λ, Λ = 11.1λ,N = 17, and the N.A. = 8/11. (c), (d) BF and DF images of a G2 isolated groove with Wg = 4.05λ, T = 1.0λ, Λ = 15.1λ, N = 19 and the N.A. = 8/15. The edges of the groove are farther apart than in (a) and (b). The N.A., and thus resolution of the images, has decreased.

Fig. 8
Fig. 8

(a), (b) BF and DF images of a G2 step. Ws = 1.05λ, Λ = 11.1λ, N = 15, T = 1.05λ, and the N.A. = 6/11. The orders for this case are examined in Fig. 3(b). The BF image contains little accessible information. The DF image contains very little light, but the intensity maxima clearly indicate the position of the step, (c) DF image of a G2 step. Ws = 1.05λ, L = 11.1λ, N = 15, T = 1.5λ, and the N.A. = 6/11. There is only one maximum, and the position of the step is less clear.

Fig. 9
Fig. 9

(a) DF image of a G1 isolated groove, Wg = 0.5λ, L = 11.1λ, N = 15, T = 0.5λ, and the N.A. = 6/11. Such an object acts as a point source or line object, (b) DF image of the same structure as examined in (a); however, in this case the N.A. is increased to 8/11. There is a decrease in the slopes of the crossing ridges and a marked improvement in resolution.

Fig. 10
Fig. 10

(a), (b) BF and DF images of a G1 groove Wg = 1.05λ. Λ = 11.1λ, N = 15, T = 1.0λ, and N.A. = 6/11. In the BF image the ridges cross at z = 0, the top of the structure. (c), (d) BF and DF images for a deeper G1 groove Wg = 1.05λ. In this case Λ = 11.1λ, T = 1.5λ, and the N.A. = 6/11. The base of the groove physically at −1.5λ in the z defocus direction

Fig. 11
Fig. 11

Transmission geometry imaging; reflection geometry imaging is examined in the paper. The point scatterer scatters weakly and symmetrically in all directions. No skew waves are involved as a cylindrical imaging system is used.

Fig. 12
Fig. 12

(a) Diffracted light from a weakly scattering isolated step, made of the same material as the substrate, modeled by the light diffracted by a defocused point scatterer, z = ts, plus the large reflection from the substrate (Model S1.1). The point scatterer (ps) may not be positioned at the depth of the edge it represents, to produce optimum agreement, (b) In Model S1.2 the reflections from the surface of the substrate of the downward-traveling light from ps1, giving ps2, interfere with the upward traveling light from ps1 and the substrate. Allowance can be made for the matched condition at the bottom of the step if the width is larger. The defocus positions of the effective point scatters, ts1 and ts2, must be included, (c) Model S1.3 could be used for wider steps. The edges, left (l), right (r), top (t), and bottom (b), are each considered point scatterers, psl1, pst1, psr1, and psb. In the G1 case reflections from only three point scatterers are included plus the large reflection from the substrate surface. This model could be extended to allow for defocused reflections of the edge point scatterers from the substrate, psl2, psr2, and pst2. (d) Model G1.1 could be used for a G2 groove case. The field from a point scatterer is subtracted from a plane reflection from the substrate surface. In Model G1.2 a second point scatterer could be incorporated at the base. For a wider groove point scatterers could be introduced for the edges.

Fig. 13
Fig. 13

The step or the groove may be modeled by combinations of two-dimensional rectangular functions as shown.

Fig. 14
Fig. 14

Two-dimensional Ewald diagram for coherently imaging an object illuminated by a plane wave and collected with a cylindrical lens. All the spatial frequencies in the reflection image lie upon the arc of the shaded sector shown, the ends of which are defined by the N.A. of the lens.

Fig. 15
Fig. 15

(a) Model S2.1: For a weakly scattering isolated step, the step being made of the same material as the substrate, the response of the step can be approximated with the two-dimensional Fourier transform of a rectangular scatterer. This response, with the correct phase term, can then be added to the reflection from the substrate, (b) Model S2.2: When the step is made of different material from the substrate, reflections of the scattered field from the substrate must be included. For isolated grooves, the scatter field can be subtracted from the reflection from the substrate.

Equations (25)

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2 E ( x , z ) + k 2 ( x ) E ( x , z ) = 0 ,
( x ) = 0 + n = 1 ̂ n cos ( n K x ) ,
0 = 1 Λ ( α W g + s W s ) ,
̂ n = 2 n π ( α s ) sin ( n π W g / Λ ) ,
E g ( x , z ) = X ( x ) Z ( z ) .
U ( x , z ) = α + α s ( θ 1 , θ 2 ) P 2 ( θ 2 ) exp { + j k [ sin ( θ 1 ) + sin ( θ 2 ) ] x } × exp { + j k [ cos ( θ 1 ) + cos ( θ 2 ) ] z } d θ 2 ,
u = k z sin 2 ( α ) or u = 4 k z sin 2 ( α / 2 ) ; υ = k x sin ( α ) ,
X = sin θ 2 sin α ,
I ( u , υ ) = | 0 1 cos ( υ X ) exp ( 1 2 j u X 2 ) d X | 2 .
I s ( u , υ ) = | 0 1 cos ( υ X ) exp ( 1 2 j u 1 X 2 ) d X + 0 1 R ( θ 2 ) cos ( υ X ) exp ( 1 2 j u 2 X 2 ) d X | ,
s ( θ 1 , θ 2 ) = + + t ( x , z ) exp { j k [ ( sin θ 1 + sin θ 2 ) x + ( cos θ 1 + cos θ 2 ) z ] } d x d z ,
T ( m , r ) = + + t ( x , z ) exp [ 2 π j ( m x + r z ) ] d x d z ,
m = sin ( θ 2 ) λ ; r = 1 cos ( θ 2 ) λ ,
( λ m ) 2 + ( λ r ) 2 = 2 λ r .
sin ( α ) λ m sin ( α ) ; 1 cos ( α ) λ r 2 .
s ( θ 1 , θ 2 ) = [ ( λ m ) 2 + ( λ r ) 2 2 λ r ] T ( m , r ) .
s ( 0 , θ 2 ) = T ( m ) T ( r ) = T ( m , r ) .
2 U ( x , z ) + k 0 2 n 2 ( x , z ) U ( x , z ) = 0 ,
U ( x , z ) = U i ( x , z ) + U s ( x , z ) .
( 2 + k 0 2 ) U s ( x , z ) = F ( x , z ) U ( x , z ) = k 2 [ n 2 ( x , z ) 1 ] U ( x , z ) ,
F ( x , z ) = k 0 2 ( s a ) rect ( x a , z b ) ,
F ̂ ( u , w ) = k 0 2 ̂ a b sinc { a u , b w } ,
U s ( x , z ) = A ( ) ( p ; 0 ) exp [ j k 0 ( p x m z ) ] d m d p ,
A ( ) ( p ; 0 ) = j k 0 8 π 2 m F ( x , z ) exp { j k 0 ( p x ( m 1 ) z ] } d x d z .
I ( s ) [ x , z ( ) ] = | U ( s ) [ x , z ( ) ] | 2 .

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