Abstract

Characteristics of the diffraction field of two spatially separated linear diffraction gratings under incoherent illumination are studied. In contrast to the previous studies concerning the so-called Lau effect the grating separation distance is assumed to be infinitely large. The analytical model is based on the superimposition theory of mutually incoherent Talbot effects. It gives a simple explanation of the basic parameters of the diffraction images for prediction and discussion of the performance of grating shearing interferometers using a periodic, spatially extended light source. The experimental verification of theoretical analyses is given.

© 1986 Optical Society of America

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References

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  1. E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417 (1948).
    [CrossRef]
  2. M. L. Roblin, “Réalisation d’un systeme de Franges Achromatiques a Pas Variable,” Opt. Acta 7, 539 (1971).
    [CrossRef]
  3. M. L. Roblin, “Modulation Spatiale d’Intensite Produite par l’Association de Deux Reseaux, Paralleles,” Doctoral Thesis, U. Paris VI (1973).
  4. J. Ebbeni, “Nouveaux Aspects du Phénomène de Moiré. I. Formation des Franges de Moire en Lumiere Incoherente,” Nouv. Rev. Opt. Appl. 1, 333 (1970).
    [CrossRef]
  5. J. Jahns, A. W. Lohmann, “The Lau Effect: A Diffraction Experiment with Incoherent Illumination,” Opt. Commun. 28, 263 (1979).
    [CrossRef]
  6. F. Gori, “Lau Effect and Coherence Theory,” Opt. Commun. 31, 4 (1979).
    [CrossRef]
  7. R. Sudol, B. J. Thompson, “An Explanation of the Lau Effect Based on the Coherence Theory,” Opt. Commun. 31, 105 (1979).
    [CrossRef]
  8. R. J. Sudol, B. J. Thompson, “Lau Effect: Theory and Experiment,” Appl. Opt. 20, 1107 (1981).
    [CrossRef] [PubMed]
  9. G. J. Swanson, E. N. Leith, “Lau Effect and Grating Imaging,” J. Opt. Soc. Am. 72, 552 (1982).
    [CrossRef]
  10. G. J. Swanson, “Partially Coherent Imaging and Interferometry Based on Diffraction Gratings,” Ph.D. Dissertation, U. Michigan, Ann Arbor (1983).
  11. G. J. Swanson, E. N. Leith, “Analysis of the Lau Effect and Generalized Grating Imaging,” J. Opt. Soc. Am. A 2, 789 (1985).
    [CrossRef]
  12. K. Patorski, “Incoherent Superposition of Multiple Self-Imaging. Lau Effect and Moire Fringe Explanation,” Opt. Acta 30, 745 (1983).
    [CrossRef]
  13. K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau Effect: OTF Theory,” Opt. Commun. 46, 14 (1983).
    [CrossRef]
  14. J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, a Parageometrical Approach,” Opt. Acta 31, 313 (1984).
    [CrossRef]
  15. K. Patorski, “Periodic Source Ronchi-Talbot Shearing Interferometer,” Optik 62, 207 (1982).
  16. K. Patorski, “Heuristic Explanation of Grating Shearing Interferometry Using Incoherent Illumination,” Opt. Acta 31, 33 (1984).
    [CrossRef]
  17. H. O. Bartelt, J. Jahns, “Interferometry Based on the Lau Effect,” Opt. Commun. 30, 268 (1979).
    [CrossRef]
  18. S. Yokozeki, T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” Appl. Opt. 10, 1575 (1971).
    [CrossRef] [PubMed]
  19. A. W. Lohmann, D. E. Silva, “An Interferometer Based on the Talbot Effect,” Opt. Commun. 2, 413 (1971).
    [CrossRef]
  20. D. E. Silva, “Talbot Interferometer for Radial and Lateral Derivatives,” Appl. Opt. 11, 2613 (1972).
    [CrossRef] [PubMed]
  21. S. Mallick, “Interferometry with an Achromatic Fringe System,” Opt. Acta 19, 739 (1972).
    [CrossRef]
  22. K. Patorski, “Fresnel Diffraction Field (Self-Imaging) of Obliquely Illuminated Linear Diffraction Gratings,” Optik 69, 30 (1984).
  23. P. Bialobrzski, K. Patorski, “Self-Imaging Phenomenon of Tilted Linear Periodic Objects,” Opt. Appl. 15, 295 (1985).
  24. J. T. Winthrop, C. R. Worthington, “Theory of Fresnel Images. I. Plane Periodic Objects in Monochromatic Light,” J. Opt. Soc. Am. 55, 373 (1965).
    [CrossRef]
  25. K. Patorski, “Self-Imaging Phenomenon. Lateral Shift of Fresnel Images,” Opt. Acta 30, 1255 (1983).
    [CrossRef]
  26. P. Szwaykowski, “The Lateral Shift of Fresnel Images of Periodic Objects Under Coherent Plane Wave Illumination,” Opt. Acta 31, 563 (1984).
    [CrossRef]
  27. K. Patorski, “Production of Binary Amplitude Gratings with Arbitrary Opening Ratio and Variable Period,” Opt. Laser Technol. 12, 267 (1980).
    [CrossRef]
  28. R. Sudol, “Fresnel Images, Coherence Theory and the Lau Effect,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 155 (1980).
  29. K. Patorski, S. Yokozeki, T. Suzuki, “Moire Profile Prediction Using Fourier Series Formalism,” Jpn. J. Appl. Phys. 15, 443 (1976).
    [CrossRef]
  30. H. Fujiwara, “Effects of Spatial Coherence on Fourier Imaging of a Periodic Object,” Opt. Acta 21, 861 (1969).
    [CrossRef]
  31. Ch. Deckers, “Etude de l’Influence de la Coherence Partielle sur le Phenomene de Moire,” Nouv. Rev. Opt. 6, 197 (1975).
    [CrossRef]
  32. K. Patorski, G. Parfjanowicz, “Self-Imaging Phenomenon of a Sinusoidal Complex Object,” Opt. Acta 28, 357 (1981).
    [CrossRef]

1985

G. J. Swanson, E. N. Leith, “Analysis of the Lau Effect and Generalized Grating Imaging,” J. Opt. Soc. Am. A 2, 789 (1985).
[CrossRef]

P. Bialobrzski, K. Patorski, “Self-Imaging Phenomenon of Tilted Linear Periodic Objects,” Opt. Appl. 15, 295 (1985).

1984

K. Patorski, “Fresnel Diffraction Field (Self-Imaging) of Obliquely Illuminated Linear Diffraction Gratings,” Optik 69, 30 (1984).

P. Szwaykowski, “The Lateral Shift of Fresnel Images of Periodic Objects Under Coherent Plane Wave Illumination,” Opt. Acta 31, 563 (1984).
[CrossRef]

K. Patorski, “Heuristic Explanation of Grating Shearing Interferometry Using Incoherent Illumination,” Opt. Acta 31, 33 (1984).
[CrossRef]

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, a Parageometrical Approach,” Opt. Acta 31, 313 (1984).
[CrossRef]

1983

K. Patorski, “Incoherent Superposition of Multiple Self-Imaging. Lau Effect and Moire Fringe Explanation,” Opt. Acta 30, 745 (1983).
[CrossRef]

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau Effect: OTF Theory,” Opt. Commun. 46, 14 (1983).
[CrossRef]

K. Patorski, “Self-Imaging Phenomenon. Lateral Shift of Fresnel Images,” Opt. Acta 30, 1255 (1983).
[CrossRef]

1982

G. J. Swanson, E. N. Leith, “Lau Effect and Grating Imaging,” J. Opt. Soc. Am. 72, 552 (1982).
[CrossRef]

K. Patorski, “Periodic Source Ronchi-Talbot Shearing Interferometer,” Optik 62, 207 (1982).

1981

R. J. Sudol, B. J. Thompson, “Lau Effect: Theory and Experiment,” Appl. Opt. 20, 1107 (1981).
[CrossRef] [PubMed]

K. Patorski, G. Parfjanowicz, “Self-Imaging Phenomenon of a Sinusoidal Complex Object,” Opt. Acta 28, 357 (1981).
[CrossRef]

1980

K. Patorski, “Production of Binary Amplitude Gratings with Arbitrary Opening Ratio and Variable Period,” Opt. Laser Technol. 12, 267 (1980).
[CrossRef]

R. Sudol, “Fresnel Images, Coherence Theory and the Lau Effect,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 155 (1980).

1979

J. Jahns, A. W. Lohmann, “The Lau Effect: A Diffraction Experiment with Incoherent Illumination,” Opt. Commun. 28, 263 (1979).
[CrossRef]

F. Gori, “Lau Effect and Coherence Theory,” Opt. Commun. 31, 4 (1979).
[CrossRef]

R. Sudol, B. J. Thompson, “An Explanation of the Lau Effect Based on the Coherence Theory,” Opt. Commun. 31, 105 (1979).
[CrossRef]

H. O. Bartelt, J. Jahns, “Interferometry Based on the Lau Effect,” Opt. Commun. 30, 268 (1979).
[CrossRef]

1976

K. Patorski, S. Yokozeki, T. Suzuki, “Moire Profile Prediction Using Fourier Series Formalism,” Jpn. J. Appl. Phys. 15, 443 (1976).
[CrossRef]

1975

Ch. Deckers, “Etude de l’Influence de la Coherence Partielle sur le Phenomene de Moire,” Nouv. Rev. Opt. 6, 197 (1975).
[CrossRef]

1972

D. E. Silva, “Talbot Interferometer for Radial and Lateral Derivatives,” Appl. Opt. 11, 2613 (1972).
[CrossRef] [PubMed]

S. Mallick, “Interferometry with an Achromatic Fringe System,” Opt. Acta 19, 739 (1972).
[CrossRef]

1971

S. Yokozeki, T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” Appl. Opt. 10, 1575 (1971).
[CrossRef] [PubMed]

A. W. Lohmann, D. E. Silva, “An Interferometer Based on the Talbot Effect,” Opt. Commun. 2, 413 (1971).
[CrossRef]

M. L. Roblin, “Réalisation d’un systeme de Franges Achromatiques a Pas Variable,” Opt. Acta 7, 539 (1971).
[CrossRef]

1970

J. Ebbeni, “Nouveaux Aspects du Phénomène de Moiré. I. Formation des Franges de Moire en Lumiere Incoherente,” Nouv. Rev. Opt. Appl. 1, 333 (1970).
[CrossRef]

1969

H. Fujiwara, “Effects of Spatial Coherence on Fourier Imaging of a Periodic Object,” Opt. Acta 21, 861 (1969).
[CrossRef]

1965

1948

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417 (1948).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, J. Jahns, “Interferometry Based on the Lau Effect,” Opt. Commun. 30, 268 (1979).
[CrossRef]

Bialobrzski, P.

P. Bialobrzski, K. Patorski, “Self-Imaging Phenomenon of Tilted Linear Periodic Objects,” Opt. Appl. 15, 295 (1985).

Brenner, K. H.

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau Effect: OTF Theory,” Opt. Commun. 46, 14 (1983).
[CrossRef]

Deckers, Ch.

Ch. Deckers, “Etude de l’Influence de la Coherence Partielle sur le Phenomene de Moire,” Nouv. Rev. Opt. 6, 197 (1975).
[CrossRef]

Ebbeni, J.

J. Ebbeni, “Nouveaux Aspects du Phénomène de Moiré. I. Formation des Franges de Moire en Lumiere Incoherente,” Nouv. Rev. Opt. Appl. 1, 333 (1970).
[CrossRef]

Fujiwara, H.

H. Fujiwara, “Effects of Spatial Coherence on Fourier Imaging of a Periodic Object,” Opt. Acta 21, 861 (1969).
[CrossRef]

Gori, F.

F. Gori, “Lau Effect and Coherence Theory,” Opt. Commun. 31, 4 (1979).
[CrossRef]

Jahns, J.

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, a Parageometrical Approach,” Opt. Acta 31, 313 (1984).
[CrossRef]

H. O. Bartelt, J. Jahns, “Interferometry Based on the Lau Effect,” Opt. Commun. 30, 268 (1979).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau Effect: A Diffraction Experiment with Incoherent Illumination,” Opt. Commun. 28, 263 (1979).
[CrossRef]

Lau, E.

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417 (1948).
[CrossRef]

Leith, E. N.

Lohmann, A. W.

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, a Parageometrical Approach,” Opt. Acta 31, 313 (1984).
[CrossRef]

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau Effect: OTF Theory,” Opt. Commun. 46, 14 (1983).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau Effect: A Diffraction Experiment with Incoherent Illumination,” Opt. Commun. 28, 263 (1979).
[CrossRef]

A. W. Lohmann, D. E. Silva, “An Interferometer Based on the Talbot Effect,” Opt. Commun. 2, 413 (1971).
[CrossRef]

Mallick, S.

S. Mallick, “Interferometry with an Achromatic Fringe System,” Opt. Acta 19, 739 (1972).
[CrossRef]

Ojeda-Castaneda, J.

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, a Parageometrical Approach,” Opt. Acta 31, 313 (1984).
[CrossRef]

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau Effect: OTF Theory,” Opt. Commun. 46, 14 (1983).
[CrossRef]

Parfjanowicz, G.

K. Patorski, G. Parfjanowicz, “Self-Imaging Phenomenon of a Sinusoidal Complex Object,” Opt. Acta 28, 357 (1981).
[CrossRef]

Patorski, K.

P. Bialobrzski, K. Patorski, “Self-Imaging Phenomenon of Tilted Linear Periodic Objects,” Opt. Appl. 15, 295 (1985).

K. Patorski, “Fresnel Diffraction Field (Self-Imaging) of Obliquely Illuminated Linear Diffraction Gratings,” Optik 69, 30 (1984).

K. Patorski, “Heuristic Explanation of Grating Shearing Interferometry Using Incoherent Illumination,” Opt. Acta 31, 33 (1984).
[CrossRef]

K. Patorski, “Incoherent Superposition of Multiple Self-Imaging. Lau Effect and Moire Fringe Explanation,” Opt. Acta 30, 745 (1983).
[CrossRef]

K. Patorski, “Self-Imaging Phenomenon. Lateral Shift of Fresnel Images,” Opt. Acta 30, 1255 (1983).
[CrossRef]

K. Patorski, “Periodic Source Ronchi-Talbot Shearing Interferometer,” Optik 62, 207 (1982).

K. Patorski, G. Parfjanowicz, “Self-Imaging Phenomenon of a Sinusoidal Complex Object,” Opt. Acta 28, 357 (1981).
[CrossRef]

K. Patorski, “Production of Binary Amplitude Gratings with Arbitrary Opening Ratio and Variable Period,” Opt. Laser Technol. 12, 267 (1980).
[CrossRef]

K. Patorski, S. Yokozeki, T. Suzuki, “Moire Profile Prediction Using Fourier Series Formalism,” Jpn. J. Appl. Phys. 15, 443 (1976).
[CrossRef]

Roblin, M. L.

M. L. Roblin, “Réalisation d’un systeme de Franges Achromatiques a Pas Variable,” Opt. Acta 7, 539 (1971).
[CrossRef]

M. L. Roblin, “Modulation Spatiale d’Intensite Produite par l’Association de Deux Reseaux, Paralleles,” Doctoral Thesis, U. Paris VI (1973).

Silva, D. E.

D. E. Silva, “Talbot Interferometer for Radial and Lateral Derivatives,” Appl. Opt. 11, 2613 (1972).
[CrossRef] [PubMed]

A. W. Lohmann, D. E. Silva, “An Interferometer Based on the Talbot Effect,” Opt. Commun. 2, 413 (1971).
[CrossRef]

Sudol, R.

R. Sudol, “Fresnel Images, Coherence Theory and the Lau Effect,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 155 (1980).

R. Sudol, B. J. Thompson, “An Explanation of the Lau Effect Based on the Coherence Theory,” Opt. Commun. 31, 105 (1979).
[CrossRef]

Sudol, R. J.

Suzuki, T.

K. Patorski, S. Yokozeki, T. Suzuki, “Moire Profile Prediction Using Fourier Series Formalism,” Jpn. J. Appl. Phys. 15, 443 (1976).
[CrossRef]

S. Yokozeki, T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” Appl. Opt. 10, 1575 (1971).
[CrossRef] [PubMed]

Swanson, G. J.

Szwaykowski, P.

P. Szwaykowski, “The Lateral Shift of Fresnel Images of Periodic Objects Under Coherent Plane Wave Illumination,” Opt. Acta 31, 563 (1984).
[CrossRef]

Thompson, B. J.

R. J. Sudol, B. J. Thompson, “Lau Effect: Theory and Experiment,” Appl. Opt. 20, 1107 (1981).
[CrossRef] [PubMed]

R. Sudol, B. J. Thompson, “An Explanation of the Lau Effect Based on the Coherence Theory,” Opt. Commun. 31, 105 (1979).
[CrossRef]

Winthrop, J. T.

Worthington, C. R.

Yokozeki, S.

K. Patorski, S. Yokozeki, T. Suzuki, “Moire Profile Prediction Using Fourier Series Formalism,” Jpn. J. Appl. Phys. 15, 443 (1976).
[CrossRef]

S. Yokozeki, T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” Appl. Opt. 10, 1575 (1971).
[CrossRef] [PubMed]

Ann. Phys.

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. 6, 417 (1948).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

K. Patorski, S. Yokozeki, T. Suzuki, “Moire Profile Prediction Using Fourier Series Formalism,” Jpn. J. Appl. Phys. 15, 443 (1976).
[CrossRef]

Nouv. Rev. Opt.

Ch. Deckers, “Etude de l’Influence de la Coherence Partielle sur le Phenomene de Moire,” Nouv. Rev. Opt. 6, 197 (1975).
[CrossRef]

Nouv. Rev. Opt. Appl.

J. Ebbeni, “Nouveaux Aspects du Phénomène de Moiré. I. Formation des Franges de Moire en Lumiere Incoherente,” Nouv. Rev. Opt. Appl. 1, 333 (1970).
[CrossRef]

Opt. Acta

M. L. Roblin, “Réalisation d’un systeme de Franges Achromatiques a Pas Variable,” Opt. Acta 7, 539 (1971).
[CrossRef]

K. Patorski, “Incoherent Superposition of Multiple Self-Imaging. Lau Effect and Moire Fringe Explanation,” Opt. Acta 30, 745 (1983).
[CrossRef]

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, a Parageometrical Approach,” Opt. Acta 31, 313 (1984).
[CrossRef]

K. Patorski, “Heuristic Explanation of Grating Shearing Interferometry Using Incoherent Illumination,” Opt. Acta 31, 33 (1984).
[CrossRef]

K. Patorski, G. Parfjanowicz, “Self-Imaging Phenomenon of a Sinusoidal Complex Object,” Opt. Acta 28, 357 (1981).
[CrossRef]

H. Fujiwara, “Effects of Spatial Coherence on Fourier Imaging of a Periodic Object,” Opt. Acta 21, 861 (1969).
[CrossRef]

S. Mallick, “Interferometry with an Achromatic Fringe System,” Opt. Acta 19, 739 (1972).
[CrossRef]

K. Patorski, “Self-Imaging Phenomenon. Lateral Shift of Fresnel Images,” Opt. Acta 30, 1255 (1983).
[CrossRef]

P. Szwaykowski, “The Lateral Shift of Fresnel Images of Periodic Objects Under Coherent Plane Wave Illumination,” Opt. Acta 31, 563 (1984).
[CrossRef]

Opt. Appl.

P. Bialobrzski, K. Patorski, “Self-Imaging Phenomenon of Tilted Linear Periodic Objects,” Opt. Appl. 15, 295 (1985).

Opt. Commun.

H. O. Bartelt, J. Jahns, “Interferometry Based on the Lau Effect,” Opt. Commun. 30, 268 (1979).
[CrossRef]

A. W. Lohmann, D. E. Silva, “An Interferometer Based on the Talbot Effect,” Opt. Commun. 2, 413 (1971).
[CrossRef]

K. H. Brenner, A. W. Lohmann, J. Ojeda-Castaneda, “Lau Effect: OTF Theory,” Opt. Commun. 46, 14 (1983).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau Effect: A Diffraction Experiment with Incoherent Illumination,” Opt. Commun. 28, 263 (1979).
[CrossRef]

F. Gori, “Lau Effect and Coherence Theory,” Opt. Commun. 31, 4 (1979).
[CrossRef]

R. Sudol, B. J. Thompson, “An Explanation of the Lau Effect Based on the Coherence Theory,” Opt. Commun. 31, 105 (1979).
[CrossRef]

Opt. Laser Technol.

K. Patorski, “Production of Binary Amplitude Gratings with Arbitrary Opening Ratio and Variable Period,” Opt. Laser Technol. 12, 267 (1980).
[CrossRef]

Optik

K. Patorski, “Fresnel Diffraction Field (Self-Imaging) of Obliquely Illuminated Linear Diffraction Gratings,” Optik 69, 30 (1984).

K. Patorski, “Periodic Source Ronchi-Talbot Shearing Interferometer,” Optik 62, 207 (1982).

Proc. Soc. Photo-Opt. Instrum. Eng.

R. Sudol, “Fresnel Images, Coherence Theory and the Lau Effect,” Proc. Soc. Photo-Opt. Instrum. Eng. 240, 155 (1980).

Other

M. L. Roblin, “Modulation Spatiale d’Intensite Produite par l’Association de Deux Reseaux, Paralleles,” Doctoral Thesis, U. Paris VI (1973).

G. J. Swanson, “Partially Coherent Imaging and Interferometry Based on Diffraction Gratings,” Ph.D. Dissertation, U. Michigan, Ann Arbor (1983).

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Figures (6)

Fig. 1
Fig. 1

Geometry of the system under study: G1, incoherently illuminated amplitude diffraction grating placed at the front focal plane of the collimator lens LC; G2, second diffraction grating with lines mutually parallel to the lines of G1. Observation plane OP is located at a distance z in the Fresnel diffraction region of G2.

Fig. 2
Fig. 2

Lau-phase interferometer17 using spatially incoherent light source S coupled with a narrow slit SF in the frequency plane: LC, collimator lens; G1 and G2, diffraction gratings; O, phase object under study; L1 and L2, imaging optics; SF, spatial filter; OP, observation plane.

Fig. 3
Fig. 3

Modified Talbot (or Lau-phase) interferometer using a periodic incoherent light source: G1 and LC, illumination system providing multiple plane beam illumination of the self-imaging grating G2; O phase object; G3, detection grating.

Fig. 4
Fig. 4

Another version of the modified Talbot interferometer using a periodic incoherent light source. Grating G3 detects one of the virtual self-images of G2; the phase object is placed in the plane of this self-image; IL, imaging optics, other symbols are the same as in Fig. 3.

Fig. 5
Fig. 5

Grating shearing interferometer with continuous variation of the shear amount21: S axial slit or point source; GS and LC, multiple beam illumination system; L1 and L2, imaging optics; O, phase object; OP, observation plane containing the detection grating.

Fig. 6
Fig. 6

Optical system for the formation of the source grating G1: S, point source; G2, diffraction grating; LC, collimator objective; P, focal plane of LC where one of the self-images of G2 is recorded.

Equations (7)

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

tan α = d 1 / f ,
z = 2 m d 2 2 / λ ,
Δ x = z tan α .
Δ x = 2 m n d 2 2 tan α / λ .
Δ x = 2 m n d 2 .
Δ x = 2 m r s d 2 .
d 1 = ( d 2 / z 0 ) f = ( λ / N d 2 ) f ,

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