Abstract

In this paper, we describe deflection mapping of phase objects using a Talbot interferometer. To examine the deflection of light by the phase objects, the moiré fringes are generated by superimposing the Fourier image of the first grating on the second one in the interferometer. The phase object is placed in front of the first grating. The light passing through the objects and impinging on the first grating produces the shifted Fourier image, and the resultant moiré fringes give the deflection mapping, which depends on the distribution of the refractive index of the phase object. The experiments show deflection mapping of a piece of plastic plate and a candle flame. This technique is used for measuring the focal length of a lens.

© 1984 Optical Society of America

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References

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  1. J. Guild, The Interference Systems of Crossed Diffraction Gratings (Oxford U.P., London, 1956).
  2. H. Takasaki, “Moire Topography,” Appl. Opt. 9, 1467 (1970).
    [CrossRef] [PubMed]
  3. Y. Nishijima, G. Oster, “Moire Patterns and Refractive-Index Measurement,” J. Opt. Soc. Am. 54, 1 (1964).
    [CrossRef]
  4. C. J. Van Oss, “The use of Gratings Producing Moiré Patterns for Measuring Refractive Index Gradients,” J. Sci. Instrum. 41, 227 (1964).
    [CrossRef]
  5. Z. Karny, O. Kafri, “Refractive-Index Measurements by Moire Deflectometry,” Appl. Opt. 21, 3326 (1982).
    [CrossRef] [PubMed]
  6. S. Yokozeki, T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” “Shearing Interferometer Using the Grating as the Beam Splitter. Part 2,” Appl. Opt. 10, 1575, 1690 (1971).
    [CrossRef] [PubMed]
  7. A. W. Lohmann, D. E. Silva, “An Interferometer Based on the Talbot Effect,” Opt. Commun. 2, 413 (1971).
    [CrossRef]
  8. A. W. Lohmann, D. E. Silva, “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1972).
    [CrossRef]
  9. D. E. Silva, “Talbot Interferometer for Radial and Lateral Derivatives,” Appl. Opt. 11, 2613 (1972).
    [CrossRef] [PubMed]
  10. F. Talbot, “Facts Relating to Optical Science. No. IV,” Philos. Mag. 9, 401 (1936).
  11. K. Murata, N. Baba, K. Kunugi, “Holographic Interferometry with a Wide Field Angle of View and Its Application to Reconstruction of Refractive Index Fields,” Optik 53, 285 (1979).

1982 (1)

1979 (1)

K. Murata, N. Baba, K. Kunugi, “Holographic Interferometry with a Wide Field Angle of View and Its Application to Reconstruction of Refractive Index Fields,” Optik 53, 285 (1979).

1972 (2)

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

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

1971 (2)

S. Yokozeki, T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” “Shearing Interferometer Using the Grating as the Beam Splitter. Part 2,” Appl. Opt. 10, 1575, 1690 (1971).
[CrossRef] [PubMed]

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

1970 (1)

1964 (2)

Y. Nishijima, G. Oster, “Moire Patterns and Refractive-Index Measurement,” J. Opt. Soc. Am. 54, 1 (1964).
[CrossRef]

C. J. Van Oss, “The use of Gratings Producing Moiré Patterns for Measuring Refractive Index Gradients,” J. Sci. Instrum. 41, 227 (1964).
[CrossRef]

1936 (1)

F. Talbot, “Facts Relating to Optical Science. No. IV,” Philos. Mag. 9, 401 (1936).

Baba, N.

K. Murata, N. Baba, K. Kunugi, “Holographic Interferometry with a Wide Field Angle of View and Its Application to Reconstruction of Refractive Index Fields,” Optik 53, 285 (1979).

Guild, J.

J. Guild, The Interference Systems of Crossed Diffraction Gratings (Oxford U.P., London, 1956).

Kafri, O.

Karny, Z.

Kunugi, K.

K. Murata, N. Baba, K. Kunugi, “Holographic Interferometry with a Wide Field Angle of View and Its Application to Reconstruction of Refractive Index Fields,” Optik 53, 285 (1979).

Lohmann, A. W.

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

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

Murata, K.

K. Murata, N. Baba, K. Kunugi, “Holographic Interferometry with a Wide Field Angle of View and Its Application to Reconstruction of Refractive Index Fields,” Optik 53, 285 (1979).

Nishijima, Y.

Oster, G.

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, “A Talbot Interferometer with Circular Gratings,” Opt. Commun. 4, 326 (1972).
[CrossRef]

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

Suzuki, T.

S. Yokozeki, T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” “Shearing Interferometer Using the Grating as the Beam Splitter. Part 2,” Appl. Opt. 10, 1575, 1690 (1971).
[CrossRef] [PubMed]

Takasaki, H.

Talbot, F.

F. Talbot, “Facts Relating to Optical Science. No. IV,” Philos. Mag. 9, 401 (1936).

Van Oss, C. J.

C. J. Van Oss, “The use of Gratings Producing Moiré Patterns for Measuring Refractive Index Gradients,” J. Sci. Instrum. 41, 227 (1964).
[CrossRef]

Yokozeki, S.

S. Yokozeki, T. Suzuki, “Shearing Interferometer Using the Grating as the Beam Splitter,” “Shearing Interferometer Using the Grating as the Beam Splitter. Part 2,” Appl. Opt. 10, 1575, 1690 (1971).
[CrossRef] [PubMed]

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

J. Sci. Instrum. (1)

C. J. Van Oss, “The use of Gratings Producing Moiré Patterns for Measuring Refractive Index Gradients,” J. Sci. Instrum. 41, 227 (1964).
[CrossRef]

Opt. Commun. (2)

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

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

Optik (1)

K. Murata, N. Baba, K. Kunugi, “Holographic Interferometry with a Wide Field Angle of View and Its Application to Reconstruction of Refractive Index Fields,” Optik 53, 285 (1979).

Philos. Mag. (1)

F. Talbot, “Facts Relating to Optical Science. No. IV,” Philos. Mag. 9, 401 (1936).

Other (1)

J. Guild, The Interference Systems of Crossed Diffraction Gratings (Oxford U.P., London, 1956).

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Figures (7)

Fig. 1
Fig. 1

Fourier image formation of a grating produced by a plane wave of monochromatic light tilted by a small angle Δϕ.

Fig. 2
Fig. 2

Moiré fringes produced by two Ronchi gratings.

Fig. 3
Fig. 3

Optical system for measuring focal length.

Fig. 4
Fig. 4

Rotated moiré fringes produced by a lens.

Fig. 5
Fig. 5

Experimental setup. The inverse telescope L1L2 produces a collimated light beam which passes through a phase object (P.O) and then Ronchi gratings G1 and G2. The moiré fringe may be viewed on screen D placed behind G2 by an observer.

Fig. 6
Fig. 6

Moiré deflection maps of phase objects: (a) without phase object, (b) with a candle flame, (c) with a perforated plate.

Fig. 7
Fig. 7

Deflection mapping obtained by a lens.

Equations (14)

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g ( x , y ) = m = - m = + A m exp [ 2 π i ( m / p ) x ] .
exp [ ( 2 π i / λ ) Δ ϕ · x ] .
U ( x , y ; z = 0 ) = m = - m = + A m exp { 2 π i [ ( m / p ) + ( Δ ϕ / λ ) ] x } .
U ( ξ , z ) - + U ( x , z = 0 ) exp [ i π λ z ( ξ - x ) 2 ] d x = m = - m = + A m - + exp { ( i π λ z ) × [ ( ξ - x ) 2 + 2 λ z ( m p + Δ ϕ λ ) x ] } d x = m = - m = + A m exp { ( i π λ z ) [ ξ 2 - ( ξ - m λ z p - Δ ϕ · z ) ] 2 } × - + exp { ( i π λ z ) [ x - ( ξ - m λ z p - Δ ϕ · z ) ] 2 } d x .
U ( ξ , z ) = m = - m = + A m exp [ ( i π λ ) ( 2 Δ ϕ · ξ - Δ ϕ 2 · z ) ] × exp [ i π ( m 2 λ z p 2 ) ] exp [ 2 π i m p ( ξ - Δ ϕ · z ) ] .
exp [ i π ( m 2 λ z k p ) ] = 1 ,
U ( ξ , z k ) = m = - m = + A m · B 1 · exp [ 2 π i m p ( ξ - Δ ϕ · z k ) ] ,
W = p 2 sin ( θ / 2. ) .
Δ η = Δ ξ 2 sin ( θ / 2 ) .
Δ η = 2 k p 2 λ · Δ ϕ 2 sin ( θ / 2 ) .
Δ ϕ 1 n s z o z s [ n ( x , y , z ) x ] y = const d z ,
tan Δ ϕ Δ ϕ = r x f ,
Δ η = ( r x - Δ ξ ) tan α ,
f = z k + z k 2 sin ( θ / 2 ) · 1 tan α ;

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