Abstract

The modification of Talbot interferometer configuration for obtaining moire fringe shearing interferograms with an increased shear amount is proposed. The problem of decrease of interferogram quality that occurs when using higher frequency gratings is significantly eliminated by rotating the beam splitter grating about the axis perpendicular to the grating lines. The theory of the interferometer and its experimental verification are presented.

© 1985 Optical Society of America

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References

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  1. S. Yokozeki, T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” Appl. Opt. 10, 1575 (1971).
    [CrossRef] [PubMed]
  2. A. W. Lohmann, D. E. Silva, “An Interferometer Based on the Talbot Effect,” Opt. Commun. 2, 413 (1971).
    [CrossRef]
  3. D. E. Silva, “Talbot Interferometer for Radial and Lateral Derivatives,” Appl. Opt. 11, 2613 (1972).
    [CrossRef] [PubMed]
  4. K. Patorski, D. Szwaykowski, “Optical Differentiation of Quasi-periodic Patterns using Talbot Interferometry,” Opt. Acta 31, 23 (1984).
    [CrossRef]
  5. J. Schwider, “Single Sideband Ronchi Test,” Appl. Opt. 20, 2635 (1981).
    [CrossRef] [PubMed]
  6. K. Patorski, S. Yokozeki, T. Suzuki, “Collimation Test by Double Grating Shearing Interferometer,” Appl. Opt. 15, 1234 (1976).
    [CrossRef] [PubMed]
  7. B. R. Hunt, “Matrix Formulation of the Reconstruction of Phase Values from Phase Differences,” J. Opt. Soc. Am. 69, 393 (1979).
    [CrossRef]
  8. S. Ganci, “Fourier Diffraction through a Tilted Slit,” Eur. J. Phys. 2, 158 (1981).
    [CrossRef]
  9. K. Patorski, “Fraunhofer Diffraction Patterns of Tilted Planar Objects,” Opt. Acta 30, 673 (1983); “Errata,” Opt. Acta 31, 147 (1984).
    [CrossRef]
  10. K. Patorski, “Fresnel Diffraction Field (self-imaging) of Obliquely Illuminated Linear Diffraction Gratings,” Optik 69, 30 (1984).
  11. P. Bialobrzeski, K. Patorski, “Self-imaging Phenomenon of Tilted Linear Periodic Objects,” Opt. Appl.15, 000 (1985), in press.
  12. K. Patorski, P. Szwaykowski, “Producing and Testing Binary Amplitude Gratings using a Self-imaging and Double Exposure Technique,” Opt. Laser Technol. 15, 316 (1983).
    [CrossRef]

1984 (2)

K. Patorski, D. Szwaykowski, “Optical Differentiation of Quasi-periodic Patterns using Talbot Interferometry,” Opt. Acta 31, 23 (1984).
[CrossRef]

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

1983 (2)

K. Patorski, P. Szwaykowski, “Producing and Testing Binary Amplitude Gratings using a Self-imaging and Double Exposure Technique,” Opt. Laser Technol. 15, 316 (1983).
[CrossRef]

K. Patorski, “Fraunhofer Diffraction Patterns of Tilted Planar Objects,” Opt. Acta 30, 673 (1983); “Errata,” Opt. Acta 31, 147 (1984).
[CrossRef]

1981 (2)

S. Ganci, “Fourier Diffraction through a Tilted Slit,” Eur. J. Phys. 2, 158 (1981).
[CrossRef]

J. Schwider, “Single Sideband Ronchi Test,” Appl. Opt. 20, 2635 (1981).
[CrossRef] [PubMed]

1979 (1)

1976 (1)

1972 (1)

1971 (2)

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]

Bialobrzeski, P.

P. Bialobrzeski, K. Patorski, “Self-imaging Phenomenon of Tilted Linear Periodic Objects,” Opt. Appl.15, 000 (1985), in press.

Ganci, S.

S. Ganci, “Fourier Diffraction through a Tilted Slit,” Eur. J. Phys. 2, 158 (1981).
[CrossRef]

Hunt, B. R.

Lohmann, A. W.

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

Patorski, K.

K. Patorski, D. Szwaykowski, “Optical Differentiation of Quasi-periodic Patterns using Talbot Interferometry,” Opt. Acta 31, 23 (1984).
[CrossRef]

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

K. Patorski, P. Szwaykowski, “Producing and Testing Binary Amplitude Gratings using a Self-imaging and Double Exposure Technique,” Opt. Laser Technol. 15, 316 (1983).
[CrossRef]

K. Patorski, “Fraunhofer Diffraction Patterns of Tilted Planar Objects,” Opt. Acta 30, 673 (1983); “Errata,” Opt. Acta 31, 147 (1984).
[CrossRef]

K. Patorski, S. Yokozeki, T. Suzuki, “Collimation Test by Double Grating Shearing Interferometer,” Appl. Opt. 15, 1234 (1976).
[CrossRef] [PubMed]

P. Bialobrzeski, K. Patorski, “Self-imaging Phenomenon of Tilted Linear Periodic Objects,” Opt. Appl.15, 000 (1985), in press.

Schwider, J.

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]

Suzuki, T.

Szwaykowski, D.

K. Patorski, D. Szwaykowski, “Optical Differentiation of Quasi-periodic Patterns using Talbot Interferometry,” Opt. Acta 31, 23 (1984).
[CrossRef]

Szwaykowski, P.

K. Patorski, P. Szwaykowski, “Producing and Testing Binary Amplitude Gratings using a Self-imaging and Double Exposure Technique,” Opt. Laser Technol. 15, 316 (1983).
[CrossRef]

Yokozeki, S.

Appl. Opt. (4)

Eur. J. Phys. (1)

S. Ganci, “Fourier Diffraction through a Tilted Slit,” Eur. J. Phys. 2, 158 (1981).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (2)

K. Patorski, “Fraunhofer Diffraction Patterns of Tilted Planar Objects,” Opt. Acta 30, 673 (1983); “Errata,” Opt. Acta 31, 147 (1984).
[CrossRef]

K. Patorski, D. Szwaykowski, “Optical Differentiation of Quasi-periodic Patterns using Talbot Interferometry,” Opt. Acta 31, 23 (1984).
[CrossRef]

Opt. Commun. (1)

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

Opt. Laser Technol. (1)

K. Patorski, P. Szwaykowski, “Producing and Testing Binary Amplitude Gratings using a Self-imaging and Double Exposure Technique,” Opt. Laser Technol. 15, 316 (1983).
[CrossRef]

Optik (1)

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

Other (1)

P. Bialobrzeski, K. Patorski, “Self-imaging Phenomenon of Tilted Linear Periodic Objects,” Opt. Appl.15, 000 (1985), in press.

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of the Talbot interferometer under investigation: S, point source; L, lens under test; G1 and G2, linear amplitude-type diffraction gratings axially separated by a distance z0. In the usual configuration G1 lies in the x-y plane, in the configuration proposed it lies in the x-η plane.

Fig. 2
Fig. 2

Intensity distributions observed in the plane of detecting grating G2 when testing lens L with spherical aberration. The usual Talbot interferometer setup with gratings of 40-lines/mm spatial frequency: (a) z0 = 58.3 mm, (b) z0 = 180 mm, λ = 0.6328 μm.

Fig. 3
Fig. 3

Intensity distributions observed in the plane of detecting grating G2 for beam splitter grating G1 rotated about the x axis: (a) β = 45°, (b) β = 65°.

Fig. 4
Fig. 4

Finite fringe detection mode with grating G2 slightly rotated about the z axis: (a) β = 45°, (b) β = 65°.

Fig. 5
Fig. 5

Fourier spectrum of the double grating system G1–G2 (with G1 rotated about the axis perpendicular to the grating lines): (a) overall view, (b) enlarged part of the first order of double diffraction.

Equations (9)

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T ( x , y ) = A 0 + 2 A 1 cos ( 2 π x / d ) ,
E ( x , y , z ) = A 0 exp [ i k g ( x , y ) ] + A 1 exp { i k [ p x + q y + g ( x Δ , y δ ) z ( p 2 + q 2 ) / 2 ] } + A 1 exp { i k [ p x q y g ( x + Δ , y δ ) + z ( p 2 + q 2 ) / 2 ] } ,
p = λ / d , q = sin β cos β [ 1 1 ( p / cos β ) 2 ] ,
I ( x , y , z ) = A 0 2 + 2 A 1 2 + 2 A 0 A 1 cos k [ p x + q y g ( x , y ) x Δ g ( x , y ) y δ z ( p 2 + q 2 ) / 2 ] + 2 A 0 A 1 cos k [ p x q y g ( x , y ) x Δ + g ( x , y ) y δ + z ( p 2 + q 2 ) / 2 ] + 2 A 1 2 cos k [ 2 p x 2 Δ g ( x , y δ ) x ] = A 0 2 + 2 A 1 2 + 4 A 0 A 1 cos k [ q y g ( x , y ) y δ z ( p 2 + q 2 ) / 2 ] · cos k [ p x g ( x , y ) x Δ ] + 2 A 1 2 cos k [ 2 p x 2 Δ g ( x , y δ ) x ] .
g ( x , y ) g ( x Δ , y δ ) = Δ g ( x , y ) x + δ g ( x , y ) y ,
I G 2 ( x , y ) = A 0 + 2 A 1 cos ( 2 π / d ) ( x cos φ ± y sin φ ) ,
I G 2 ( x , y ) A 0 + 2 A 1 cos k ( p x ± q y ) ,
I M ( x , y , z ) ( A 0 2 + 2 A 1 2 ) A 0 + 2 A 0 A 1 2 { cos k [ g ( x , y ) x Δ + g ( x , y ) x δ + z ( p 2 + q 2 ) / 2 ] + cos k [ 2 q y + g ( x , y ) x Δ g ( x , y ) y δ z ( p 2 + q 2 ) / 2 ] } .
I M ( x , y , z ) ( A 0 2 + 2 A 1 2 ) A 0 + 2 A 0 A 1 2 { cos k [ g ( x , y ) x Δ g ( x , y ) y δ z ( p 2 + q 2 ) / 2 ] + cos k [ 2 q y g ( x , y ) x Δ g ( x , y ) y δ z ( p 2 + q 2 ) / 2 ] } .

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