Abstract

Interferometric methods are widely used in surface metrology. A question that arises is how much information about the surface can be extracted from a given interferogram. For examination of the resolution limit of interferometry with coherent monochromatic light, interferograms of several surface relief gratings calculated with the use of approximate and rigorous theories are presented. The limits of the usefulness of scalar theory based on the use of the Fourier transform are indicated. Interferograms of dielectric and metallic structures are examined, including simple lamellar gratings and gratings made up of trapezoidal steps with varying slopes and depths. In all cases TE illumination is assumed. The effects of changing numerical aperture and defocus on the interferograms are also examined.

© 1995 Optical Society of America

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References

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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]
  29. 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).
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    [CrossRef]
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    [CrossRef]

1994

J. T. Sheridan, C. J. R. Sheppard, “Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory,” Opt. Commun. 105, 367–378 (1994).
[CrossRef]

T. O. Körner, J. T. Sheridan, “Near fields of periodic gratings calculated using rigorous electromagnetic theory,” J. Scanning 16, 343–352 (1994).

M. Heissmeier, J. T. Sheridan, J. Schwider, N. Streibl, “Detection of errors in microlithographic grating fabrication using a simple methodology,” Optik 95, 161–167 (1994).

1993

L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
[CrossRef]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[CrossRef]

J. T. Sheridan, C. J. R. Sheppard, “Coherent imaging of periodic thick fine isolated structures,” J. Opt. Soc. Am. A 10, 1–19 (1993).
[CrossRef]

S.-E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: an exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

1991

1990

J. Schwider, “Advanced evaluation techniques in interferometry,” Prog. Opt. 28, 271–359 (1990).
[CrossRef]

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).

1988

1985

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

1984

D. Maystre, “Rigorous vector theories of diffraction gratings,” Prog. Opt. 21, 1–67 (1984).
[CrossRef]

G. Tayeb, R. Petit, “On the numerical study of deep dielectric lamellar gratings,” Opt. Acta 31, 1361–1365 (1984).
[CrossRef]

1983

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains résaux diélectriques profonds,” J. Opt.(Paris) 14, 273–288 (1983).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

1982

1981

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

1978

1959

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]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 7.

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Cadilhac, M.

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains résaux diélectriques profonds,” J. Opt.(Paris) 14, 273–288 (1983).
[CrossRef]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[CrossRef]

Ehrhardt, R.

J. T. Sheridan, R. Ehrhardt, T. O. Körner, “Optimization techniques for the design of resonance domain diffractive optical elements,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 12–15.

Françon, M.

M. Françon, Optical Interferometry (Academic, New York, 1966).

Gale, M. T.

H. P. Herzig, M. T. Gale, H. W. Lehmann, R. Morf, “Diffractive components: computer-generated elements,” in Perspectives for Parallel Optical Interconnects, Ph. Lalanne, P. Chavel, eds., ESPRIT Basic Research Ser.71–108 (Springer-VerlagBerlin, 1993), Chap. 5.
[CrossRef]

Gaylord, T. K.

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

Haggans, C. W.

Heissmeier, M.

M. Heissmeier, J. T. Sheridan, J. Schwider, N. Streibl, “Detection of errors in microlithographic grating fabrication using a simple methodology,” Optik 95, 161–167 (1994).

Herzig, H. P.

H. P. Herzig, M. T. Gale, H. W. Lehmann, R. Morf, “Diffractive components: computer-generated elements,” in Perspectives for Parallel Optical Interconnects, Ph. Lalanne, P. Chavel, eds., ESPRIT Basic Research Ser.71–108 (Springer-VerlagBerlin, 1993), Chap. 5.
[CrossRef]

Kirk, C. P.

Knop, K.

Körner, T. O.

T. O. Körner, J. T. Sheridan, “Near fields of periodic gratings calculated using rigorous electromagnetic theory,” J. Scanning 16, 343–352 (1994).

J. T. Sheridan, R. Ehrhardt, T. O. Körner, “Optimization techniques for the design of resonance domain diffractive optical elements,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 12–15.

J. T. Sheridan, T. O. Körner, “Imaging periodic surface relief structures,” J. Microsc. (to be published).

Lehmann, H. W.

H. P. Herzig, M. T. Gale, H. W. Lehmann, R. Morf, “Diffractive components: computer-generated elements,” in Perspectives for Parallel Optical Interconnects, Ph. Lalanne, P. Chavel, eds., ESPRIT Basic Research Ser.71–108 (Springer-VerlagBerlin, 1993), Chap. 5.
[CrossRef]

Li, L.

Maystre, D.

D. Maystre, “Rigorous vector theories of diffraction gratings,” Prog. Opt. 21, 1–67 (1984).
[CrossRef]

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Moharam, M. G.

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

Morf, R.

H. P. Herzig, M. T. Gale, H. W. Lehmann, R. Morf, “Diffractive components: computer-generated elements,” in Perspectives for Parallel Optical Interconnects, Ph. Lalanne, P. Chavel, eds., ESPRIT Basic Research Ser.71–108 (Springer-VerlagBerlin, 1993), Chap. 5.
[CrossRef]

Nyyssonen, D.

Petit, R.

S.-E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: an exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

G. Tayeb, R. Petit, “On the numerical study of deep dielectric lamellar gratings,” Opt. Acta 31, 1361–1365 (1984).
[CrossRef]

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains résaux diélectriques profonds,” J. Opt.(Paris) 14, 273–288 (1983).
[CrossRef]

Reid, G. T.

D. W. Robinson, G. T. Reid, Interferogram Analysis (Institute of Physics, Bristol, UK, 1993).

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]

Robinson, D. W.

D. W. Robinson, G. T. Reid, Interferogram Analysis (Institute of Physics, Bristol, UK, 1993).

Sandström, S.-E.

S.-E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: an exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

Schwider, J.

M. Heissmeier, J. T. Sheridan, J. Schwider, N. Streibl, “Detection of errors in microlithographic grating fabrication using a simple methodology,” Optik 95, 161–167 (1994).

J. Schwider, “Advanced evaluation techniques in interferometry,” Prog. Opt. 28, 271–359 (1990).
[CrossRef]

Sheppard, C. J. R.

J. T. Sheridan, C. J. R. Sheppard, “Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory,” Opt. Commun. 105, 367–378 (1994).
[CrossRef]

J. T. Sheridan, C. J. R. Sheppard, “Coherent imaging of periodic thick fine isolated structures,” J. Opt. Soc. Am. A 10, 1–19 (1993).
[CrossRef]

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

Sheridan, J. T.

M. Heissmeier, J. T. Sheridan, J. Schwider, N. Streibl, “Detection of errors in microlithographic grating fabrication using a simple methodology,” Optik 95, 161–167 (1994).

T. O. Körner, J. T. Sheridan, “Near fields of periodic gratings calculated using rigorous electromagnetic theory,” J. Scanning 16, 343–352 (1994).

J. T. Sheridan, C. J. R. Sheppard, “Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory,” Opt. Commun. 105, 367–378 (1994).
[CrossRef]

J. T. Sheridan, C. J. R. Sheppard, “Coherent imaging of periodic thick fine isolated structures,” J. Opt. Soc. Am. A 10, 1–19 (1993).
[CrossRef]

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

J. T. Sheridan, R. Ehrhardt, T. O. Körner, “Optimization techniques for the design of resonance domain diffractive optical elements,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 12–15.

J. T. Sheridan, T. O. Körner, “Imaging periodic surface relief structures,” J. Microsc. (to be published).

Streibl, N.

M. Heissmeier, J. T. Sheridan, J. Schwider, N. Streibl, “Detection of errors in microlithographic grating fabrication using a simple methodology,” Optik 95, 161–167 (1994).

Strojwas, A. S.

Suratteau, J. Y.

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains résaux diélectriques profonds,” J. Opt.(Paris) 14, 273–288 (1983).
[CrossRef]

Tayeb, G.

S.-E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: an exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

G. Tayeb, R. Petit, “On the numerical study of deep dielectric lamellar gratings,” Opt. Acta 31, 1361–1365 (1984).
[CrossRef]

Tolansky, S.

S. Tolansky, An Introduction to Interferometry (Longmans, Green, London, 1955).

Wolf, E.

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), Chap. 7.

Yuan, C. M.

Comput. Phys. Commun.

L. C. Botten, M. S. Craig, R. C. McPhedran, “Complex zeros of analytic functions,” Comput. Phys. Commun. 29, 245–259 (1983).
[CrossRef]

J. Electromagn. Waves Appl.

S.-E. Sandström, G. Tayeb, R. Petit, “Lossy multistep lamellar gratings in conical diffraction mountings: an exact eigenfunction solution,” J. Electromagn. Waves Appl. 7, 631–649 (1993).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt.(Paris)

J. Y. Suratteau, M. Cadilhac, R. Petit, “Sur la détermination numérique des efficacités de certains résaux diélectriques profonds,” J. Opt.(Paris) 14, 273–288 (1983).
[CrossRef]

J. Scanning

T. O. Körner, J. T. Sheridan, “Near fields of periodic gratings calculated using rigorous electromagnetic theory,” J. Scanning 16, 343–352 (1994).

Opt. Acta

G. Tayeb, R. Petit, “On the numerical study of deep dielectric lamellar gratings,” Opt. Acta 31, 1361–1365 (1984).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Opt. Commun.

J. T. Sheridan, C. J. R. Sheppard, “Modelling of images of square-wave gratings and isolated edges using rigorous diffraction theory,” Opt. Commun. 105, 367–378 (1994).
[CrossRef]

Optik

M. Heissmeier, J. T. Sheridan, J. Schwider, N. Streibl, “Detection of errors in microlithographic grating fabrication using a simple methodology,” Optik 95, 161–167 (1994).

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).

Proc. IEEE

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

Proc. R. Soc. London Ser. A

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]

Prog. Opt.

D. Maystre, “Rigorous vector theories of diffraction gratings,” Prog. Opt. 21, 1–67 (1984).
[CrossRef]

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[CrossRef]

J. Schwider, “Advanced evaluation techniques in interferometry,” Prog. Opt. 28, 271–359 (1990).
[CrossRef]

Other

D. W. Robinson, G. T. Reid, Interferogram Analysis (Institute of Physics, Bristol, UK, 1993).

H. P. Herzig, M. T. Gale, H. W. Lehmann, R. Morf, “Diffractive components: computer-generated elements,” in Perspectives for Parallel Optical Interconnects, Ph. Lalanne, P. Chavel, eds., ESPRIT Basic Research Ser.71–108 (Springer-VerlagBerlin, 1993), Chap. 5.
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. 7.

M. Françon, Optical Interferometry (Academic, New York, 1966).

S. Tolansky, An Introduction to Interferometry (Longmans, Green, London, 1955).

D. Malacara, ed., Optical Shop Testing (Wiley, New York, 1978).

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

J. T. Sheridan, R. Ehrhardt, T. O. Körner, “Optimization techniques for the design of resonance domain diffractive optical elements,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 12–15.

J. T. Sheridan, T. O. Körner, “Imaging periodic surface relief structures,” J. Microsc. (to be published).

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

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Figures (23)

Fig. 1
Fig. 1

Surface profile f(x) on a dielectric material.

Fig. 2
Fig. 2

Field distribution behind an edge in a dielectric grating according to Eq. (3). In the two homogeneous regions of constant thickness, Φ(x) takes on the constant values Φ1 and Φ2.

Fig. 3
Fig. 3

Interferogram for a reference wave rotated about the x axis for a field distribution behind an edge as in Fig. 2.

Fig. 4
Fig. 4

Mach–Zehnder interferogram of a binary grating etched into fused silica (n = 1.46), Λ = 9 μm, h = 0.656 μm illuminated at a wavelength λ = 632.8 nm.

Fig. 5
Fig. 5

Geometry and notation for a trapezoidal grating.

Fig. 6
Fig. 6

Diffraction efficiencies for a lamellar grating with β = 0°, S = 1, Λ = 10.1λ, h = 1.0λ, St = 2.25, 0 = Gr = 2 = 1.0, and t = 0.5, computed with Fourier-based approximate theory. Diffraction orders: solid curve, T3; dotted–dashed curve, T5.

Fig. 7
Fig. 7

As in Fig. 6 but computed by the Legendre method with number of eigenfunctions N = 31, maximum polynomial order in groove m0 = 24, and step m1 = 36. Diffraction orders: solid curve, T2; dotted–dashed curve, T3; dashed curve, T4; dotted curve, T5.

Fig. 8
Fig. 8

Interferogram according to AF for a grating with Λ = 10.1λ, 1 = Gr = S+2 = 1.0, St = 2.25, t = 0.5, and h = 0.4λ; Imax = 4.86 (with intensity incident onto the grating set to 1.0).

Fig. 9
Fig. 9

As in Fig. 8 but computed according to rigorous LM with N = 31, m0 = 24, and m1 = 36; Imax = 5.51.

Fig. 10
Fig. 10

Interferogram according to AF for a grating with Λ = 10.1λ, 1 = Gr = S+2 = 10, St = 2.25, t = 0.5, and h = 1.0λ; Imax = 5.5.

Fig. 11
Fig. 11

As in Fig. 10 but computed according to rigorous LM with N = 31, m0 = 20, and m1 = 30; Imax = 5.51.

Fig. 12
Fig. 12

Interferogram according to LM with N = 17, m0 = 10, and m1 = 15 for a grating with Λ = 5.1λ, 1 = Gr = S+2 = 1.0, St = 2.25, t = 0.5, and h = 1.0λ; Imax = 7.06.

Fig. 13
Fig. 13

Interferogram according to LM with N = 47, m0 = 30, and m1 = 45 for a grating with Λ = 15.1λ, 1 = Gr = S+2 = 1.0, St = 2.25, t = 0.5, and h = 1.0 λ; Imax = 7.68.

Fig. 14
Fig. 14

Convergence of diffraction efficiency as a function of angle of incidence with varying number of layers S ∈{4, 6, 10}for the structure from Fig. 6 with N = 37. (a) Minus-tenth transmitted order, (b) minus-first diffracted order. Long-dashed curves, S = 4; solid curves, S = 6; dotted curves, S = 10.

Fig. 15
Fig. 15

Interferogram according to LM with N = 31, m0 = 24, and m1 = 36 (at half-thickness) for a trapezoidal grating with Λ = 10.1λ, β = 45°, 1 = Gr = S+2 = 1.0, St = 2.25, t = 0.5, and h = 1.0λ; S = 10, Imax = 6.45.

Fig. 16
Fig. 16

Interferogram according to LM with N = 31, m0 = 24, and m1 = 36 (at half-thickness) for a trapezoidal grating with Λ = 10.1λ, β = 30°, 1 = Gr = S+2 = 10, St = 2.25, t = 0.5, and h = 1.0λ; S = 6 and Imax = 7.52.

Fig. 17
Fig. 17

Interferogram according to LM with N = 31, m0 = 24, and m1 = 36 (at half-thickness) for a trapezoidal grating with Λ = 10.1λ, β = 60°, 1 = Gr = S+2 = 1.0, St = 2.25, t = 0.5, and h = 1.0λ; S = 10 and Imax = 7.52.

Fig. 18
Fig. 18

Interferogram according to LM with N = 47, m0 = 30, and m1 = 45 (at half-thickness) for a trapezoidal grating with Λ = 15.1λ, β = 30°, 1 = Gr = S+2 = 1.0 St = 2.25, t = 0.5, and h = 1.0 λ; S = 6, Imax = 6.25.

Fig. 19
Fig. 19

Interferogram according to LM with N = 37, m0 = 20, and m1 = 30 for a binary metallic grating with Λ = 10.1λ, 1 = Gr = 1.0, St = S+2 = 26.22 +i3.19, t = 0.5, and h = 0.25λ; Imax = 5.97.

Fig. 20
Fig. 20

Interferogram according to LM with N = 37, m0 = 20, and m1 = 30 (at half-thickness) for a trapezoidal metallic grating with Λ = 10.1λ, β = 45°, 1 = Gr = 1.0, St = S+2 = 26.22 +i3.19, t = 0.5, and h = 0.25λ; S = 4, Imax = 5.94.

Fig. 21
Fig. 21

Interferogram according to LM with N = 37, m0 = 20, and m1 = 30 (at half-thickness) for a trapezoidal metallic grating with Λ = 10.1 λ, β = 75°, 1 = Gr = 1.0, St = S+2 = 26.22 +i3.19, t = 0.5, and h = 0.25λ; Imax = 5.11.

Fig. 22
Fig. 22

Interferogram according to LM for a trapezoidal metallic grating as in Fig. 19, imaged with N.A. = 0.7, focused at the top (y = 0.25λ) of the grating, Imax = 4.6.

Fig. 23
Fig. 23

Interferogram according to LM for a trapezoidal metallic grating as in Fig. 19, imaged with N.A. = 1.0, focused at the bottom of the grating at y = 0, Imax = 4.88.

Equations (5)

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t ( x ) = | t ( x ) | exp [ i Φ ( x ) ] with x : Φ ( x ) ,
E ( x ) y = y max = U 0 t ( x ) for x [ x 1 , x 2 ] .
Φ ( x ) = 2 π λ f ( x ) ( n 2 n 1 ) .
E ref ( y , z ) = E 0 ref exp [ i ( y k 0 cos ϕ + z k 0 sin ϕ ) ] .
I ( x b , z b ) = | E ref ( y , z ) + E d ( x , y ) | 2 , x = x b , z = z b ,

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