Abstract

Both the Talbot and the Lau effects of Ronchi and binary phase gratings on an incident wavefront of arbitrary wavefront are discussed. General and analytic solutions are obtained in terms of the Fresnel diffraction theory. Numerical simulations and supporting verification for Gaussian beams are demonstrated. It is found that the distortion of the self-image is closely linked to that of the far field distribution in a two-grating system, and both depend on the Talbot distance and the beam wavefront. Some interesting phenomena are explored such as light modulation and phase compensation. Possible applications are accordingly suggested.

© 1989 Optical Society of America

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References

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  1. F. Talbot, “Facts Relating to Optical Science No. IV,” Philos. Mag. 9, 401–407 (1836).
  2. E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–423 (1948).
  3. W. D. Montgomery, “Self-Imaging Objects of Infinite Aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  4. J. T. Winthrop, C. R. Worthington, “Theory of Fresnel Images. I. Plane Periodic Objects in Monochromatic Light,” J. Opt. Soc. Am. 55, 373–381 (1965).
    [CrossRef]
  5. J. Jahns, A. W. Lohmann, “The Lau Effect (a Diffaction Experiment with Incoherent Illumination),” Opt. Commun. 28, 263–267 (1979).
    [CrossRef]
  6. F. Gori, “Lau Effect and Coherence Theory,” Opt. Commun. 31, 4–8 (1979).
    [CrossRef]
  7. R. Sudal, R. J. Thompson, “Lau Effect, Theory and Experiment,” Appl. Opt. 20, 1107–1116 (1981).
    [CrossRef]
  8. G. J. Swanson, E. N. Leith, “Analysis of the Lau Effect and Generalized Grating Imaging,” J. Opt. Soc. Am. A 2, 789–793 (1985).
    [CrossRef]
  9. K. Patorski, “Incoherent Superposition of Multiple Self-Imaging Lau Effect and Moire Fringe Explanation,” Opt. Acta 30, 754–758 (1983).
    [CrossRef]
  10. J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, a Parageometrical Approach,” Opt. Acta 31, 313–324 (1984).
    [CrossRef]
  11. J. Ojeda-Castaneda, E. E. Sicre, “Quasi Ray-Optical Approach to Longitudinal Periodicities of Free and Bounded Wavefields,” Opt. Acta 32, 17–26 (1985).
    [CrossRef]
  12. L. Liu, “Ambiguity Function and General Talbot-Lau Effects,” Acta Opt. Sin. 7, 501–510 (1987).
  13. A. W. Lohamnn, J. Ojeda-Castaneda, “Spatial Periodicities in Partially Coherent Fields,” Opt. Acta 30, 475–479 (1983).
    [CrossRef]
  14. G. Indebetouw, “Propagation of Spatially Periodic Wavefields,” Opt. Acta 31, 531–539 (1984).
    [CrossRef]
  15. K. Hane, S. Hattori, C. P. Grover, “Lau Effect in Reflection,” J. Mod. Opt. 34, 1481–1490 (1987).
    [CrossRef]
  16. L. Liu, “Partially Coherent Diffraction Effect Between Lau and Talbot Effects,” J. Opt. Soc. Am. A 5, 1709–1716 (1988).
    [CrossRef]
  17. W. Veldkamp, “Binary Optics: An Emerging Diffractive Optics Technology,” Phys. TodayS49–S50 (Jan.1987).
  18. W. Veldkamp, C. Kastner, “Beam Profile Shaping for Laser Radars That use Detector Arrays,” Appl. Opt. 21, 345–356 (1982).
    [CrossRef] [PubMed]
  19. G. J. Swanson, W. B. Veldkamp, “Binary Lenses for Use at 10.6 Micrometers,” Opt. Eng. 24, 791–795 (1985).
    [CrossRef]
  20. W. Veldkamp, “Development in Laser-Beam Control with Holographic Diffraction Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 255, 136–144 (1980).
  21. U. Killat, G. Rabe, W. Rave, “Binary Phase Gratings for Star Couplers with High Splitting Ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
    [CrossRef]
  22. H. Dammann, K. Gortler, “High-Efficiency In-Line Multiple Imaging by Means of Multiple Phase Holograms,” Opt. Commun. 3, 312–315 (1971).
    [CrossRef]
  23. W. Veldkamp, J. Leger, G. Swanson, “Coherent Summation of Laser Beams Using Binary Phase Gratings,” Opt. Lett. 11, 303–305 (1986).
    [CrossRef] [PubMed]
  24. J. P. Guigay, “On Fresnel Diffraction by One-Dimensional Periodic Objects, with Application to Structure Determination of Phase Objects,” Opt. Acta 18, 677–682 (1971).
    [CrossRef]
  25. L. Liu, “Theory for Lau Effect of Plane Objects,” Acta Opt. Sin. 6, 807–814 (1986).
  26. A. W. Lohmann, “An Array Illuminator Based on the Talbot-Effect,” Optik (Stuttgart) 79, 41–45 (1988).
  27. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics, (Wiley, New York, 1978). Chap. 10.
  28. L. Liu, “A New Grating Diffraction-Interference Effect: Theory on Phenomenon of Lau Setup Illuminated by Partially Coherent Quasihomogeneous Source,” Science in China A 32, 570–584 (1989), in English.
  29. L. Liu, “Optical Modulation with Binary Phase Gratings,” Chin. J. Lasers, in press.
  30. G. Swanson, J. Leger, M. Holz, “Aperture Filling of Phase-Locked Laser Arrays,” Opt. Lett. 12, 245–247 (1987).
    [CrossRef] [PubMed]
  31. A. A. Golubentsev, V. V. Likhanskii, A. P. Napartovich, “Theory of Phase Locking of an Array of Lasers,” Sov. Phys. JETP 66, 676–682 (1987).
  32. L. Liu, L. Zhao, “Aperture Filling of Phase Locked Laser Arrays by Phase Correlation of Self-Imaging,” Chin. J. Lasers 16, 37–40 (1989).

1989 (2)

L. Liu, “A New Grating Diffraction-Interference Effect: Theory on Phenomenon of Lau Setup Illuminated by Partially Coherent Quasihomogeneous Source,” Science in China A 32, 570–584 (1989), in English.

L. Liu, L. Zhao, “Aperture Filling of Phase Locked Laser Arrays by Phase Correlation of Self-Imaging,” Chin. J. Lasers 16, 37–40 (1989).

1988 (2)

A. W. Lohmann, “An Array Illuminator Based on the Talbot-Effect,” Optik (Stuttgart) 79, 41–45 (1988).

L. Liu, “Partially Coherent Diffraction Effect Between Lau and Talbot Effects,” J. Opt. Soc. Am. A 5, 1709–1716 (1988).
[CrossRef]

1987 (5)

W. Veldkamp, “Binary Optics: An Emerging Diffractive Optics Technology,” Phys. TodayS49–S50 (Jan.1987).

L. Liu, “Ambiguity Function and General Talbot-Lau Effects,” Acta Opt. Sin. 7, 501–510 (1987).

G. Swanson, J. Leger, M. Holz, “Aperture Filling of Phase-Locked Laser Arrays,” Opt. Lett. 12, 245–247 (1987).
[CrossRef] [PubMed]

A. A. Golubentsev, V. V. Likhanskii, A. P. Napartovich, “Theory of Phase Locking of an Array of Lasers,” Sov. Phys. JETP 66, 676–682 (1987).

K. Hane, S. Hattori, C. P. Grover, “Lau Effect in Reflection,” J. Mod. Opt. 34, 1481–1490 (1987).
[CrossRef]

1986 (2)

1985 (3)

G. J. Swanson, W. B. Veldkamp, “Binary Lenses for Use at 10.6 Micrometers,” Opt. Eng. 24, 791–795 (1985).
[CrossRef]

J. Ojeda-Castaneda, E. E. Sicre, “Quasi Ray-Optical Approach to Longitudinal Periodicities of Free and Bounded Wavefields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

G. J. Swanson, E. N. Leith, “Analysis of the Lau Effect and Generalized Grating Imaging,” J. Opt. Soc. Am. A 2, 789–793 (1985).
[CrossRef]

1984 (2)

G. Indebetouw, “Propagation of Spatially Periodic Wavefields,” Opt. Acta 31, 531–539 (1984).
[CrossRef]

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, a Parageometrical Approach,” Opt. Acta 31, 313–324 (1984).
[CrossRef]

1983 (2)

A. W. Lohamnn, J. Ojeda-Castaneda, “Spatial Periodicities in Partially Coherent Fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

K. Patorski, “Incoherent Superposition of Multiple Self-Imaging Lau Effect and Moire Fringe Explanation,” Opt. Acta 30, 754–758 (1983).
[CrossRef]

1982 (2)

W. Veldkamp, C. Kastner, “Beam Profile Shaping for Laser Radars That use Detector Arrays,” Appl. Opt. 21, 345–356 (1982).
[CrossRef] [PubMed]

U. Killat, G. Rabe, W. Rave, “Binary Phase Gratings for Star Couplers with High Splitting Ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
[CrossRef]

1981 (1)

1980 (1)

W. Veldkamp, “Development in Laser-Beam Control with Holographic Diffraction Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 255, 136–144 (1980).

1979 (2)

J. Jahns, A. W. Lohmann, “The Lau Effect (a Diffaction Experiment with Incoherent Illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau Effect and Coherence Theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

1971 (2)

H. Dammann, K. Gortler, “High-Efficiency In-Line Multiple Imaging by Means of Multiple Phase Holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

J. P. Guigay, “On Fresnel Diffraction by One-Dimensional Periodic Objects, with Application to Structure Determination of Phase Objects,” Opt. Acta 18, 677–682 (1971).
[CrossRef]

1967 (1)

1965 (1)

1948 (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–423 (1948).

1836 (1)

F. Talbot, “Facts Relating to Optical Science No. IV,” Philos. Mag. 9, 401–407 (1836).

Dammann, H.

H. Dammann, K. Gortler, “High-Efficiency In-Line Multiple Imaging by Means of Multiple Phase Holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics, (Wiley, New York, 1978). Chap. 10.

Golubentsev, A. A.

A. A. Golubentsev, V. V. Likhanskii, A. P. Napartovich, “Theory of Phase Locking of an Array of Lasers,” Sov. Phys. JETP 66, 676–682 (1987).

Gori, F.

F. Gori, “Lau Effect and Coherence Theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

Gortler, K.

H. Dammann, K. Gortler, “High-Efficiency In-Line Multiple Imaging by Means of Multiple Phase Holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Grover, C. P.

K. Hane, S. Hattori, C. P. Grover, “Lau Effect in Reflection,” J. Mod. Opt. 34, 1481–1490 (1987).
[CrossRef]

Guigay, J. P.

J. P. Guigay, “On Fresnel Diffraction by One-Dimensional Periodic Objects, with Application to Structure Determination of Phase Objects,” Opt. Acta 18, 677–682 (1971).
[CrossRef]

Hane, K.

K. Hane, S. Hattori, C. P. Grover, “Lau Effect in Reflection,” J. Mod. Opt. 34, 1481–1490 (1987).
[CrossRef]

Hattori, S.

K. Hane, S. Hattori, C. P. Grover, “Lau Effect in Reflection,” J. Mod. Opt. 34, 1481–1490 (1987).
[CrossRef]

Holz, M.

Indebetouw, G.

G. Indebetouw, “Propagation of Spatially Periodic Wavefields,” Opt. Acta 31, 531–539 (1984).
[CrossRef]

Jahns, J.

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, a Parageometrical Approach,” Opt. Acta 31, 313–324 (1984).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau Effect (a Diffaction Experiment with Incoherent Illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Kastner, C.

Killat, U.

U. Killat, G. Rabe, W. Rave, “Binary Phase Gratings for Star Couplers with High Splitting Ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
[CrossRef]

Lau, E.

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–423 (1948).

Leger, J.

Leith, E. N.

Likhanskii, V. V.

A. A. Golubentsev, V. V. Likhanskii, A. P. Napartovich, “Theory of Phase Locking of an Array of Lasers,” Sov. Phys. JETP 66, 676–682 (1987).

Liu, L.

L. Liu, L. Zhao, “Aperture Filling of Phase Locked Laser Arrays by Phase Correlation of Self-Imaging,” Chin. J. Lasers 16, 37–40 (1989).

L. Liu, “A New Grating Diffraction-Interference Effect: Theory on Phenomenon of Lau Setup Illuminated by Partially Coherent Quasihomogeneous Source,” Science in China A 32, 570–584 (1989), in English.

L. Liu, “Partially Coherent Diffraction Effect Between Lau and Talbot Effects,” J. Opt. Soc. Am. A 5, 1709–1716 (1988).
[CrossRef]

L. Liu, “Ambiguity Function and General Talbot-Lau Effects,” Acta Opt. Sin. 7, 501–510 (1987).

L. Liu, “Theory for Lau Effect of Plane Objects,” Acta Opt. Sin. 6, 807–814 (1986).

L. Liu, “Optical Modulation with Binary Phase Gratings,” Chin. J. Lasers, in press.

Lohamnn, A. W.

A. W. Lohamnn, J. Ojeda-Castaneda, “Spatial Periodicities in Partially Coherent Fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, “An Array Illuminator Based on the Talbot-Effect,” Optik (Stuttgart) 79, 41–45 (1988).

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, a Parageometrical Approach,” Opt. Acta 31, 313–324 (1984).
[CrossRef]

J. Jahns, A. W. Lohmann, “The Lau Effect (a Diffaction Experiment with Incoherent Illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Montgomery, W. D.

Napartovich, A. P.

A. A. Golubentsev, V. V. Likhanskii, A. P. Napartovich, “Theory of Phase Locking of an Array of Lasers,” Sov. Phys. JETP 66, 676–682 (1987).

Ojeda-Castaneda, J.

J. Ojeda-Castaneda, E. E. Sicre, “Quasi Ray-Optical Approach to Longitudinal Periodicities of Free and Bounded Wavefields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, a Parageometrical Approach,” Opt. Acta 31, 313–324 (1984).
[CrossRef]

A. W. Lohamnn, J. Ojeda-Castaneda, “Spatial Periodicities in Partially Coherent Fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

Patorski, K.

K. Patorski, “Incoherent Superposition of Multiple Self-Imaging Lau Effect and Moire Fringe Explanation,” Opt. Acta 30, 754–758 (1983).
[CrossRef]

Rabe, G.

U. Killat, G. Rabe, W. Rave, “Binary Phase Gratings for Star Couplers with High Splitting Ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
[CrossRef]

Rave, W.

U. Killat, G. Rabe, W. Rave, “Binary Phase Gratings for Star Couplers with High Splitting Ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
[CrossRef]

Sicre, E. E.

J. Ojeda-Castaneda, E. E. Sicre, “Quasi Ray-Optical Approach to Longitudinal Periodicities of Free and Bounded Wavefields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

Sudal, R.

Swanson, G.

Swanson, G. J.

G. J. Swanson, E. N. Leith, “Analysis of the Lau Effect and Generalized Grating Imaging,” J. Opt. Soc. Am. A 2, 789–793 (1985).
[CrossRef]

G. J. Swanson, W. B. Veldkamp, “Binary Lenses for Use at 10.6 Micrometers,” Opt. Eng. 24, 791–795 (1985).
[CrossRef]

Talbot, F.

F. Talbot, “Facts Relating to Optical Science No. IV,” Philos. Mag. 9, 401–407 (1836).

Thompson, R. J.

Veldkamp, W.

W. Veldkamp, “Binary Optics: An Emerging Diffractive Optics Technology,” Phys. TodayS49–S50 (Jan.1987).

W. Veldkamp, J. Leger, G. Swanson, “Coherent Summation of Laser Beams Using Binary Phase Gratings,” Opt. Lett. 11, 303–305 (1986).
[CrossRef] [PubMed]

W. Veldkamp, C. Kastner, “Beam Profile Shaping for Laser Radars That use Detector Arrays,” Appl. Opt. 21, 345–356 (1982).
[CrossRef] [PubMed]

W. Veldkamp, “Development in Laser-Beam Control with Holographic Diffraction Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 255, 136–144 (1980).

Veldkamp, W. B.

G. J. Swanson, W. B. Veldkamp, “Binary Lenses for Use at 10.6 Micrometers,” Opt. Eng. 24, 791–795 (1985).
[CrossRef]

Winthrop, J. T.

Worthington, C. R.

Zhao, L.

L. Liu, L. Zhao, “Aperture Filling of Phase Locked Laser Arrays by Phase Correlation of Self-Imaging,” Chin. J. Lasers 16, 37–40 (1989).

Acta Opt. Sin. (2)

L. Liu, “Ambiguity Function and General Talbot-Lau Effects,” Acta Opt. Sin. 7, 501–510 (1987).

L. Liu, “Theory for Lau Effect of Plane Objects,” Acta Opt. Sin. 6, 807–814 (1986).

Ann. Phys. (Leipzig) (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 6, 417–423 (1948).

Appl. Opt. (2)

Chin. J. Lasers (1)

L. Liu, L. Zhao, “Aperture Filling of Phase Locked Laser Arrays by Phase Correlation of Self-Imaging,” Chin. J. Lasers 16, 37–40 (1989).

Fiber Integr. Opt. (1)

U. Killat, G. Rabe, W. Rave, “Binary Phase Gratings for Star Couplers with High Splitting Ratio,” Fiber Integr. Opt. 4, 159–167 (1982).
[CrossRef]

J. Mod. Opt. (1)

K. Hane, S. Hattori, C. P. Grover, “Lau Effect in Reflection,” J. Mod. Opt. 34, 1481–1490 (1987).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Acta (6)

A. W. Lohamnn, J. Ojeda-Castaneda, “Spatial Periodicities in Partially Coherent Fields,” Opt. Acta 30, 475–479 (1983).
[CrossRef]

G. Indebetouw, “Propagation of Spatially Periodic Wavefields,” Opt. Acta 31, 531–539 (1984).
[CrossRef]

K. Patorski, “Incoherent Superposition of Multiple Self-Imaging Lau Effect and Moire Fringe Explanation,” Opt. Acta 30, 754–758 (1983).
[CrossRef]

J. Jahns, A. W. Lohmann, J. Ojeda-Castaneda, “Talbot and Lau Effects, a Parageometrical Approach,” Opt. Acta 31, 313–324 (1984).
[CrossRef]

J. Ojeda-Castaneda, E. E. Sicre, “Quasi Ray-Optical Approach to Longitudinal Periodicities of Free and Bounded Wavefields,” Opt. Acta 32, 17–26 (1985).
[CrossRef]

J. P. Guigay, “On Fresnel Diffraction by One-Dimensional Periodic Objects, with Application to Structure Determination of Phase Objects,” Opt. Acta 18, 677–682 (1971).
[CrossRef]

Opt. Commun. (3)

J. Jahns, A. W. Lohmann, “The Lau Effect (a Diffaction Experiment with Incoherent Illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

F. Gori, “Lau Effect and Coherence Theory,” Opt. Commun. 31, 4–8 (1979).
[CrossRef]

H. Dammann, K. Gortler, “High-Efficiency In-Line Multiple Imaging by Means of Multiple Phase Holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Opt. Eng. (1)

G. J. Swanson, W. B. Veldkamp, “Binary Lenses for Use at 10.6 Micrometers,” Opt. Eng. 24, 791–795 (1985).
[CrossRef]

Opt. Lett. (2)

Optik (Stuttgart) (1)

A. W. Lohmann, “An Array Illuminator Based on the Talbot-Effect,” Optik (Stuttgart) 79, 41–45 (1988).

Philos. Mag. (1)

F. Talbot, “Facts Relating to Optical Science No. IV,” Philos. Mag. 9, 401–407 (1836).

Phys. Today (1)

W. Veldkamp, “Binary Optics: An Emerging Diffractive Optics Technology,” Phys. TodayS49–S50 (Jan.1987).

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

W. Veldkamp, “Development in Laser-Beam Control with Holographic Diffraction Gratings,” Proc. Soc. Photo-Opt. Instrum. Eng. 255, 136–144 (1980).

Science in China A (1)

L. Liu, “A New Grating Diffraction-Interference Effect: Theory on Phenomenon of Lau Setup Illuminated by Partially Coherent Quasihomogeneous Source,” Science in China A 32, 570–584 (1989), in English.

Sov. Phys. JETP (1)

A. A. Golubentsev, V. V. Likhanskii, A. P. Napartovich, “Theory of Phase Locking of an Array of Lasers,” Sov. Phys. JETP 66, 676–682 (1987).

Other (2)

L. Liu, “Optical Modulation with Binary Phase Gratings,” Chin. J. Lasers, in press.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics, (Wiley, New York, 1978). Chap. 10.

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Figures (14)

Fig. 1
Fig. 1

Schematic diagram of the optical arrangement.

Fig. 2
Fig. 2

Numerical simulation of the self-image of the Ronchi grating (h/T = 0.5) illuminated by the Gaussian beam with b = 10T and at the distance p = 0.5: (a) amplitude profile and (b) phase profile.

Fig. 3
Fig. 3

Same as in Fig. 2 but with p = 1: (a) amplitude profile and (b) phase profile.

Fig. 4
Fig. 4

Numerical stimulation of the self-image of the −π/4 −π/4 binary phase grating illuminated by a Gaussian beam with b = 10T and at the distance p = 0.5: (a) amplitude profile and (b) phase profile.

Fig. 5
Fig. 5

Same as in Fig. 4 but with p = 1: (a) amplitude profile and (b) phase profile.

Fig. 6
Fig. 6

Profiles of the zero-order far field diffraction in the two-phase grating system illuminated by a Gaussian beam with b = 10T, h/T = 0.5 and p = 1 for (a) d = 0, (b) d = T/4, and (c) d = T/2.

Fig. 7
Fig. 7

Same as in Fig. 6 but with p = 2 for (a) d = 0, (b) d = T/4, and (c) d = T/2.

Fig. 8
Fig. 8

Same as in Fig. 6 but with p = 3 for (a) d = 0, (b) d = T/4, and (c) d = T/2.

Fig. 9
Fig. 9

Same as in Fig. 6 but with b = 5Tand p = 1 for(a) d = 0, (b)d = T/4, and (c) d = T/2.

Fig. 10
Fig. 10

(a) Photograph of the Ronchi grating (T = 0.1 mm and h = 0.05 mm) illuminated by a He–Ne laser beam, (b) Photograph of the self-image of the illuminated grating at the distance 7.9 mm (p =0.5).

Fig. 11
Fig. 11

Photographs of the far field diffraction of the He–Ne laser beam in the Lau setup with two identical Ronchi gratings (T = 0.1 mm and h = 0.05 mm) and the interplanar distance 15.8 mm (p = 1): (a) for the center shift of the second grating d = 0.05 and (b) for d = 0 mm.

Fig. 12
Fig. 12

Same as in Fig. 11 but with the interplanar distance 31.6 mm (p = 2) for (a) d = 0 and (b) d = 0.05 mm.

Fig. 13
Fig. 13

Same as in Fig. 11 but with the interplanar distance 47.4 mm (p = 3) for (a) d = 0.05 and (b) d = 0 mm.

Fig. 14
Fig. 14

Ideal modulation curve, where I0 signifies the intensity of the zero-order diffraction and I1 the first-order diffraction.

Equations (50)

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g r ( x , y ) = n rect ( x n T h ) ,
g p ( x , y ) = n [ exp ( j π 4 ) rect ( x n T T 2 ) + exp ( j π 4 ) rect ( x n T T 2 T 2 ) ] .
Z = p T 2 λ ,
e 2 ( x , y ) = exp ( jkZ ) j λ Z [ e 0 ( x , y ) g 1 ( x , y ) ] * * q ( x , y ; 1 λ Z ) ,
e 2 ( x , y ) = exp ( jkZ ) j λ Z { q * ( x λ Z , y λ Z ; Z ) [ e 0 ( x , y ) * * q ( x , y : 1 λ Z ) ] } * * ( λ Z ) 2 G ( x λ Z ) δ ( x λ Z ) .
e 0 ( x , y ) = exp ( jkZ ) j λ Z e 0 ( x , y ) * * q ( x , y ; 1 λ Z ) .
G ( f x ) = n C n δ ( f x n T ) ,
C n = h T sinc ( n h T ) ( for Ronchi gratings ) , C n = 2 2 sinc ( n 2 ) exp ( π i n 2 2 ) ( for binary π 4 π 4 phase grating ) .
e 2 ( x , y ) = n C n exp ( π i p n 2 ) e 0 ( x npT , y ) exp ( 2 π i n x T ) .
e 0 ( x , y ) = Gauss ( x 2 + y 2 b 1 ) q ( x , y ; 1 λ R 1 ) ,
e 0 ( x , y ) = A 2 Gauss ( x 2 + y 2 b 2 ) q ( x , y ; 1 λ Z ) ,
b 2 = λ Z b 1 1 + ( b 1 2 λ ) 2 ( 1 Z + 1 R 1 ) 2 , A 2 = ( b 1 b 2 ) exp [ j ( k Z + ϕ ) ] , R 2 = Z [ 1 + ( b 1 2 λ ) 2 ( 1 Z + 1 R 1 ) 2 1 + ( b 1 2 λ ) 2 ( 1 R 1 ) ( 1 Z + 1 R 1 ) ] , ϕ = tan 1 [ ( λ b 2 ) 1 Z + 1 R 1 ] .
e 2 ( x , y ) = n C n exp [ π i p n 2 ) exp ( 2 π i n T x ) .
e 2 ( x , y ) = g r ( x p 2 T ) ( when p is an integer ) , e 2 ( x , y ) = 2 2 g p [ x ( p 2 + 1 4 ) T ] ( when p is an odd multiple of 1 2 and h T = 0.5 ) .
e 2 ( x , y ) = g p ( x p 2 T ) ( when p is an integer ) , e 2 ( x , y ) = 2 g r [ x ( p 2 + 1 4 ) T ] ( when p is a multiple of 1 2 and h T = 0.5 ) .
e 3 ( x , y ) = e 2 ( x , y ) g 2 ( x d , y ) ,
E 3 ( f x , f y ) = exp ( jkZ ) n m C n 1 C m 2 exp ( 2 π i m T d ) × exp ( π i p m 2 ) q * ( f x , f y ; λ Z ) × E 0 [ f x ( m T + n T ) , f y ] exp ( 2 π i m T λ Z f x ) ,
e 4 ( x , y ) = K E 3 ( x λ f , y λ f ) exp ( jkZ ) q * ( x λ f , y λ f ; λ Z ) ,
K = exp [ π i ( f x 2 + f y 2 ) ( 1 1 f ) ] j λ Z q * ( x λ f , y λ f ; λ Z ) exp ( jkZ ) ,
e 4 ( x , y ) = K k E 0 ( x λ f k T , y λ f ) M ( x ; k ) ,
M ( x ; k ) = m C k m 1 C m 2 exp ( 2 π i m T d ) exp ( π i p m 2 ) × exp ( 2 π i m T Z f x ) .
M ( x ; k ) = g 21 ( Z f x d ) [ g 1 ( Z f x ) exp ( 2 π i k f Z f x ) ] ,
g 21 ( Z f x ) = m C m 2 exp ( π i p m 2 ) exp ( 2 π i m T Z f x ) ,
I 4 ( x , y ) = | K | 2 k | E 0 ( x f k T , y f ) M ( x ; k ) | 2 + | K | 2 k m k E 0 ( x λ f k T , y λ f ) E 0 * ( x λ f k T m T , y λ f ) × M ( x ; k ) M * ( x ; k + m ) .
I 4 ( x , y ) = | K | 2 k | E 0 ( x f k T , y f ) M ( x ; k ) | 2 .
M ( x ; k ) = 1 T C T + C g r ( α + x ) g r ( α ) exp ( 2 π i k T α ) d α ,
x = Z T x p 2 T d ,
h ( x ) = T 2 n Λ ( x n T T 2 ) ,
c ( x ) = n 1 2 F ( x n T T 2 ) .
F ( x T 2 ) = { x for T 2 < x < T 2 , 0 for | x | T 2 .
M ( x ; k ) = 1 2 π k [ sin ( 2 π k T α ) i cos ( 2 π k T α ) ] | c ( x ) + h ( x ) 2 , c ( x ) h ( x ) 2 .
M ( x ; 0 ) = 1 2 n Λ ( x n T T 2 ) .
M ( x ; k 0 ) = 1 π k k sin [ π k 2 Λ ( x n T T 2 ) ] × exp [ π k T i F ( x n T T 2 ) ] .
M ( x ; 0 ) = n Λ ( x n T T 2 T 2 ) ,
M ( x ; k 0 ) = exp ( j π 2 ) π k n sin [ π k 2 Λ ( x n T T 2 ) ] × { 1 + exp [ j π ( 1 k ) ] } exp [ π k T i F ( x n T T 2 ) ] + 2 π k cos ( π k 2 ) sin [ π k 2 Λ ( x n T T 2 ) ] × exp [ π k T i F ( x n T T 2 ) ] .
M ( x ; 0 ) = 2 2 n Λ ( x n T T 4 T 2 ) ,
M ( x ; k 0 ) = 2 π k n sin [ π k 2 Λ ( x n T T 4 T 2 ) ] × exp [ π k T i F ( x n T T 4 T 2 ) ] .
T d = λ f T ,
T 1 = f Z T = T d p .
E 0 ( f x , f y ) = b 2 λ R b 4 + ( λ R ) 2 ( λ R + i b 2 ) exp { π [ ( λ R ) 2 b 2 ( λ R ) 2 + b 4 + i b 4 λ R ( λ R ) 2 + b 4 ] ( f x 2 + f y 2 ) } .
e 4 ( x , y ) = K k δ ( x λ f k T ) M ( x ; k ) .
I 4 ( x , y ) | K | 2 = k | M ( x ; k ) | 2 = 1 4 n Λ 2 ( Z f x n T T 2 p d T 2 ) + k = 1 , 3 2 k 2 π 2 cos 2 [ π k T ( Z f x p 2 T d ) ] + k = 2 , 4 2 k 2 π 2 sin 2 [ π k T ( Z f x p 2 T d ) ] = 1 2 n Λ ( Z f x n T p 2 T d T 2 ) .
n Λ ( x n T T 2 ) = 1 2 + n = 1 , 3 4 n 2 π 2 cos ( 2 π n T x ) , n Λ 2 ( x n T T 2 ) = 1 3 + n 0 4 n 2 π 2 cos ( 2 π n T x ) , n = 1 , 2 1 n 2 = π 2 6 .
I 4 ( x , y ) | K | 2 = n Λ 2 ( Z f x n T p + 1 2 T d T 2 ) + k = 1.3 8 k 2 π 2 cos 2 [ ( π k T ( Z f x p 2 T d ) ] + k = 2.4 8 k 2 π 2 sin 2 [ ( π k T ( Z f x p 2 T d ) ] = 1 .
E 0 ( x , y ) = b 1 2 exp ( π b 1 2 x 2 + y 2 λ 2 f 2 ) .
E e = exp [ π ( 2 b λ f ) 2 ( x 2 + y 2 ) ] dxdy = b 2 2 .
E max = 2 0 0 n Λ 2 ( Z f x n T T 2 ) exp [ π ( 2 b λ f ) 2 ( x 2 + y 2 ) ] dxdy = E e ( 1 2 p T π b ) .
E min = 2 0 0 n Λ 2 ( Z f x n T T 2 T 2 ) × exp [ π ( 2 b λ f ) 2 ( x 2 + y 2 ] dxdy = E e p 2 T 2 2 π b .
C 1 = E max E min = 2 π b 2 p 2 T 2 ,
C 2 = E e E max E e = 2 p T π b .

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