Abstract

A radial shearing interferometer with two equispaced circular gratings is described for obtaining a constant radial displacement of the test wavefront. The procedure for the fringe analysis is shown by applying the Fourier transform method to the circular fringe pattern for precisely obtaining the differentiation of the test wavefront. A novel method is also shown for holographically making the equispaced circular gratings by using a cylindrical mirror and a collimating lens. It is shown that the deflection and misalignment of the gratings will not appear in the interferogram by using the holographic production method.

© 1989 Optical Society of America

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References

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  1. O. Bryngdahl, “Shearing Interferometry with Constant Radial Displacement,” J. Opt. Soc. Am. 61, 169–172 (1971).
    [CrossRef]
  2. A. W. Lohmann, D. E. Silva, “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326–328 (1972).
    [CrossRef]
  3. C. Colautti, L. M. Zerbino, E. E. Sicre, M. Garavaglia, “Lau Effect Using Circular Gratings,” Appl. Opt. 26, 2061–2062 (1987).
    [CrossRef] [PubMed]
  4. M. V. Perez, C. Gomez-Reino, J. M. Cuadrado, “Diffraction Patterns and Zone Plates Produced by Thin Linear Axicons,” Opt. Acta 33, 1161–1176 (1986).
    [CrossRef]
  5. M. Takeda, H. Ina, S. Kobayashi, “Fourier-Transform Method of Fringe Pattern Analysis for Computer-Based Topography and Interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  6. T. Kreis, “Digital Holographic Interference-Phase Measurement Using the Fourier-Transform Method,” J. Opt. Soc. Am. 3, 847–855 (1986).
    [CrossRef]
  7. M. Takeda, Q. S. Ru, “Computer-Based Highly Sensitive Electron-Wave Interferometry,” Appl. Opt. 24, 3068–3071 (1985).
    [CrossRef] [PubMed]

1987 (1)

1986 (2)

M. V. Perez, C. Gomez-Reino, J. M. Cuadrado, “Diffraction Patterns and Zone Plates Produced by Thin Linear Axicons,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

T. Kreis, “Digital Holographic Interference-Phase Measurement Using the Fourier-Transform Method,” J. Opt. Soc. Am. 3, 847–855 (1986).
[CrossRef]

1985 (1)

1982 (1)

1972 (1)

A. W. Lohmann, D. E. Silva, “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

1971 (1)

Bryngdahl, O.

Colautti, C.

Cuadrado, J. M.

M. V. Perez, C. Gomez-Reino, J. M. Cuadrado, “Diffraction Patterns and Zone Plates Produced by Thin Linear Axicons,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Garavaglia, M.

Gomez-Reino, C.

M. V. Perez, C. Gomez-Reino, J. M. Cuadrado, “Diffraction Patterns and Zone Plates Produced by Thin Linear Axicons,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Ina, H.

Kobayashi, S.

Kreis, T.

T. Kreis, “Digital Holographic Interference-Phase Measurement Using the Fourier-Transform Method,” J. Opt. Soc. Am. 3, 847–855 (1986).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, D. E. Silva, “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

Perez, M. V.

M. V. Perez, C. Gomez-Reino, J. M. Cuadrado, “Diffraction Patterns and Zone Plates Produced by Thin Linear Axicons,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Ru, Q. S.

Sicre, E. E.

Silva, D. E.

A. W. Lohmann, D. E. Silva, “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

Takeda, M.

Zerbino, L. M.

Appl. Opt. (2)

J. Opt. Soc. Am. (3)

Opt. Acta (1)

M. V. Perez, C. Gomez-Reino, J. M. Cuadrado, “Diffraction Patterns and Zone Plates Produced by Thin Linear Axicons,” Opt. Acta 33, 1161–1176 (1986).
[CrossRef]

Opt. Commun. (1)

A. W. Lohmann, D. E. Silva, “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326–328 (1972).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Constant radial shearing interferometer for testing a lens: S, coherent point source; TL, lens under test, G1 and G2, two identical equispaced circular gratings; L1 and L2, collimating lenses; SF, spatial filter for stopping the unwanted diffraction waves; P, image plane.

Fig. 2
Fig. 2

Interferogram obtained from the arrangement in Fig. 1. The fringe pattern shows the diffraction of the spherical aberration of the test lens.

Fig. 3
Fig. 3

Interferogram where a certain amount of radial spatial carrier frequency is introduced into the interferogram of Fig. 2 by slightly shifting the test lens along the optical axis.

Fig. 4
Fig. 4

Left: the power spectrum distribution of the fringe pattern shown in Fig 3; Right: the power spectrum after filtering.

Fig. 5
Fig. 5

Wrapped phase distribution obtained by using the 2-D Fourier transform fringe analysis method.

Fig. 6
Fig. 6

Resultant phase distribution obtained by unwrapping the discontinuous phase data of Fig. 5 and subtracting the radial spatial frequency component and the other asymmetrical components. The maximum value is 4.94 rad.

Fig. 7
Fig. 7

Arrangement for holographically producing equispaced circular gratings with a cylindrical mirror CM and a collimating lens L.

Equations (7)

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Δ = Z λ / ( p 2 λ 2 ) ,
I ( r , θ ) = a ( r , θ ) + b ( r , θ ) cos [ ω 0 r + ϕ ( r , θ ) ] ,
ϕ ( r ) = c r 3 + d r 5 + ,
A ( r ) = ( 1 / Δ ) ϕ ( r ) d r = ( 1 / Δ ) ( c r 4 / 4 + d r 6 / 6 + ) .
θ = tan 1 ( D / f ) .
I ( r ) = I 0 { 1 + cos [ ( 2 π sin θ / λ ) r ] } ,
p = λ ( 1 + f 2 / D 2 ) .

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