Abstract

The effect of diffraction on the transmission pattern of an infinite Ronchi grid and the moiré image of two equal Ronchi grids was studied. Transmission patterns and moiré images of two equal Ronchi grids with and without a phase object, at a wide range of conditions, are presented. Diffractional phenomena were found to affect the transmission pattern and the moiré image significantly. In order to minimize their effect, the distance between the grids must be exactly one of the Fourier-image planes of the grid. The moiré image of a given phase object, even under these conditions, has an inherent error that can be precisely calculated. Deflection mapping and the uncertainty (because of diffraction) of lenses were calculated.

© 1985 Optical Society of America

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References

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  1. F. J. Weinberg, Optics of Flames (Butterworth, London, 1963); A. K. Oppenheim, M. M. Kamel, Laser Cinematography of Explosions (Springer-Verlag, Berlin, 1971), Chap. 2; J. H. Burgoyne, F. J. Weinberg, Proc. R. Soc. London Ser. A 224, 286 (1954); F. J. Weinberg, Fuel (London) 34, S84 (1955); F. J. Weinberg, Proc. R. Soc. London Ser. A 235, 510 (1956); J. Reck, W. Sumi, F. J. Weinberg, Fuel (London) 35, 364 (1956); G. Dixon-Lewis, G. L. Isles, Eighth Symposium (International) on Combustion (Butterworth, London, 1962), p. 448.
    [CrossRef]
  2. Y. Nishijima, G. Oster, J. Opt. Soc. Am. 54, 1 (1964); G. Oster, M. Wasserman, C. Zwerling, J. Opt. Soc. Am. 54, 169 (1964).
    [CrossRef]
  3. S. Yokozeki, T. Suzuki, Appl. Opt. 10, 1575 (1971).
    [CrossRef] [PubMed]
  4. O. Kafri, Opt. Lett. 5, 555 (1980); O. Kafri, A. Livnat, Appl. Opt. 20, 3098 (1981); O. Kafri, A. Livnat, E. Keren, Appl. Opt. 21, 3884 (1982); Z. Karny, S. Lavi, O. Kafri, Opt. Lett. 8, 409 (1983).
    [CrossRef] [PubMed]
  5. J. Stricker, O. Kafri, AIAA J. 20, 820 (1982); J. Stricker, E. Keren, O. Kafri, “Axisymmetric density fields measurements by moire deflectometry,” AIAA J. 21, 1767 (1983).
    [CrossRef]
  6. E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, Appl. Opt. 20, 4263 (1981).
    [CrossRef] [PubMed]
  7. E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, Nineteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1982), p. 303.
    [CrossRef]
  8. E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, Appl. Opt. 22, 698 (1983).
    [CrossRef] [PubMed]
  9. E. Bar-Ziv, G. S. Rau, in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1983), p. 114; G. S. Rau, S.B. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1983); G. S. Rau, E. Bar-Ziv, “Deflection mapping of flames using the moiré effect,” Appl. Opt. 23, 2687 (1984).
    [CrossRef]
  10. E. Bar-Ziv, “Effect of diffraction on the moiré image for temperature mapping in flames,” Appl. Opt. 23, 4040 (1984).
    [CrossRef] [PubMed]
  11. J. M. Cowley, Diffraction Physics, 2nd ed. (North-Holland, Amsterdam, 1981).
  12. C. H. Palmer, B. Z. Hollmann, Appl. Opt. 11, 780 (1972).
    [CrossRef] [PubMed]
  13. A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).
  14. D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).
  15. E. Keren, Nuclear Research Center-Negev, P.O. Box 9001, Beer-Sheva 84190, Israel (personal communication).
  16. E. Bar-Ziv, S. Sgulim, D. Manor, J. Opt. Soc. Am. A 2, 380–384 (1985).
    [CrossRef]

1985 (1)

1984 (1)

1983 (1)

1982 (1)

J. Stricker, O. Kafri, AIAA J. 20, 820 (1982); J. Stricker, E. Keren, O. Kafri, “Axisymmetric density fields measurements by moire deflectometry,” AIAA J. 21, 1767 (1983).
[CrossRef]

1981 (1)

1980 (1)

1972 (1)

1971 (1)

1964 (1)

Bar-Ziv, E.

E. Bar-Ziv, S. Sgulim, D. Manor, J. Opt. Soc. Am. A 2, 380–384 (1985).
[CrossRef]

E. Bar-Ziv, “Effect of diffraction on the moiré image for temperature mapping in flames,” Appl. Opt. 23, 4040 (1984).
[CrossRef] [PubMed]

E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, Appl. Opt. 22, 698 (1983).
[CrossRef] [PubMed]

E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, Appl. Opt. 20, 4263 (1981).
[CrossRef] [PubMed]

E. Bar-Ziv, G. S. Rau, in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1983), p. 114; G. S. Rau, S.B. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1983); G. S. Rau, E. Bar-Ziv, “Deflection mapping of flames using the moiré effect,” Appl. Opt. 23, 2687 (1984).
[CrossRef]

E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, Nineteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1982), p. 303.
[CrossRef]

Cowley, J. M.

J. M. Cowley, Diffraction Physics, 2nd ed. (North-Holland, Amsterdam, 1981).

Durelli, A. J.

A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).

Glatt, I.

Hollmann, B. Z.

Kafri, O.

E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, Appl. Opt. 22, 698 (1983).
[CrossRef] [PubMed]

J. Stricker, O. Kafri, AIAA J. 20, 820 (1982); J. Stricker, E. Keren, O. Kafri, “Axisymmetric density fields measurements by moire deflectometry,” AIAA J. 21, 1767 (1983).
[CrossRef]

E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, Appl. Opt. 20, 4263 (1981).
[CrossRef] [PubMed]

O. Kafri, Opt. Lett. 5, 555 (1980); O. Kafri, A. Livnat, Appl. Opt. 20, 3098 (1981); O. Kafri, A. Livnat, E. Keren, Appl. Opt. 21, 3884 (1982); Z. Karny, S. Lavi, O. Kafri, Opt. Lett. 8, 409 (1983).
[CrossRef] [PubMed]

E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, Nineteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1982), p. 303.
[CrossRef]

Keren, E.

E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, Appl. Opt. 22, 698 (1983).
[CrossRef] [PubMed]

E. Keren, E. Bar-Ziv, I. Glatt, O. Kafri, Appl. Opt. 20, 4263 (1981).
[CrossRef] [PubMed]

E. Keren, Nuclear Research Center-Negev, P.O. Box 9001, Beer-Sheva 84190, Israel (personal communication).

E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, Nineteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1982), p. 303.
[CrossRef]

Manor, D.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

Nishijima, Y.

Oster, G.

Palmer, C. H.

Parks, V. J.

A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).

Rau, G. S.

E. Bar-Ziv, G. S. Rau, in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1983), p. 114; G. S. Rau, S.B. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1983); G. S. Rau, E. Bar-Ziv, “Deflection mapping of flames using the moiré effect,” Appl. Opt. 23, 2687 (1984).
[CrossRef]

Sgulim, S.

E. Bar-Ziv, S. Sgulim, D. Manor, J. Opt. Soc. Am. A 2, 380–384 (1985).
[CrossRef]

E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, Appl. Opt. 22, 698 (1983).
[CrossRef] [PubMed]

E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, Nineteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1982), p. 303.
[CrossRef]

Stricker, J.

J. Stricker, O. Kafri, AIAA J. 20, 820 (1982); J. Stricker, E. Keren, O. Kafri, “Axisymmetric density fields measurements by moire deflectometry,” AIAA J. 21, 1767 (1983).
[CrossRef]

Suzuki, T.

Weinberg, F. J.

F. J. Weinberg, Optics of Flames (Butterworth, London, 1963); A. K. Oppenheim, M. M. Kamel, Laser Cinematography of Explosions (Springer-Verlag, Berlin, 1971), Chap. 2; J. H. Burgoyne, F. J. Weinberg, Proc. R. Soc. London Ser. A 224, 286 (1954); F. J. Weinberg, Fuel (London) 34, S84 (1955); F. J. Weinberg, Proc. R. Soc. London Ser. A 235, 510 (1956); J. Reck, W. Sumi, F. J. Weinberg, Fuel (London) 35, 364 (1956); G. Dixon-Lewis, G. L. Isles, Eighth Symposium (International) on Combustion (Butterworth, London, 1962), p. 448.
[CrossRef]

Yokozeki, S.

AIAA J. (1)

J. Stricker, O. Kafri, AIAA J. 20, 820 (1982); J. Stricker, E. Keren, O. Kafri, “Axisymmetric density fields measurements by moire deflectometry,” AIAA J. 21, 1767 (1983).
[CrossRef]

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Lett. (1)

Other (7)

F. J. Weinberg, Optics of Flames (Butterworth, London, 1963); A. K. Oppenheim, M. M. Kamel, Laser Cinematography of Explosions (Springer-Verlag, Berlin, 1971), Chap. 2; J. H. Burgoyne, F. J. Weinberg, Proc. R. Soc. London Ser. A 224, 286 (1954); F. J. Weinberg, Fuel (London) 34, S84 (1955); F. J. Weinberg, Proc. R. Soc. London Ser. A 235, 510 (1956); J. Reck, W. Sumi, F. J. Weinberg, Fuel (London) 35, 364 (1956); G. Dixon-Lewis, G. L. Isles, Eighth Symposium (International) on Combustion (Butterworth, London, 1962), p. 448.
[CrossRef]

E. Bar-Ziv, G. S. Rau, in Digest of the Conference on Lasers and Electro-Optics (Optical Society of America, Washington, D.C., 1983), p. 114; G. S. Rau, S.B. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1983); G. S. Rau, E. Bar-Ziv, “Deflection mapping of flames using the moiré effect,” Appl. Opt. 23, 2687 (1984).
[CrossRef]

E. Bar-Ziv, S. Sgulim, O. Kafri, E. Keren, Nineteenth Symposium (International) on Combustion (Combustion Institute, Pittsburgh, Pa., 1982), p. 303.
[CrossRef]

J. M. Cowley, Diffraction Physics, 2nd ed. (North-Holland, Amsterdam, 1981).

A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).

D. Marcuse, Light Transmission Optics (Van Nostrand, New York, 1972).

E. Keren, Nuclear Research Center-Negev, P.O. Box 9001, Beer-Sheva 84190, Israel (personal communication).

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Figures (8)

Fig. 1
Fig. 1

A light ray traversing a phase object and two Ronchi grids.

Fig. 2
Fig. 2

A, The transmission pattern of a Ronchi grid. One period is shown, for δ = 0.0, 0.01, 0.02, 0.03, 0.04 for (a), (b), (c), (d), and (e), respectively. B, Calculated transmission patterns for various values of δ. δ = 0.05, 0.10, 0.15, 0.20, 0.25 for (a), (b), (c), (d), and (e), respectively. C, Calculated transmission patterns for various values of δ. δ = 0.30, 0.35, 0.40, 0.45, 0.50 for (a), (b), (c), (d), and (e), respectively.

Fig. 3
Fig. 3

A, Calculated moiré-intensity χ functions for various values of δ. δ = 0.0, 0.01, 0.02, 0.03, 0.04 for (a), (b), (c), (d), and (e), respectively. B, Calculated moiré-intensity χ functions for various values of δ. δ = 0.05, 0.10, 0.15, 0.20, 0.25 for (a), (b), (c), (d), and (e), respectively. C, Calculated moiré-intensity χ functions for various values of δ. δ = 0.30, 0.35, 0.40, 0.45, 0.50 for (a), (b), (c), (d), and (e), respectively.

Fig. 4
Fig. 4

A, Calculated moiré-image R* dependence without phase object. B, Results of A recalculated as C(R*) = [I(R*) − I(R* = 1/2)]/I(R* = 1/2).

Fig. 5
Fig. 5

A, Moiré-intensity x* profiles of a lens of R1* = 100, 50.8-mm diameter, at the first Fourier plane (m = 1) and for δ = 0, 0.01, 0.02, 0.03, 0.04 for (a), (b), (c), (d), and (e), respectively. B, The resultant relative error of the intensity profiles of A.

Fig. 6
Fig. 6

A, Delection-angle profiles calculated from the profiles of Fig. 5A. B, The resultant relative-error profile of the deflection-angle profiles of A.

Fig. 7
Fig. 7

The quantity (f* − R1*)/R1* and σ as a function of δ, for R1* = 100 and m = 1.

Fig. 8
Fig. 8

The error σ as a function of R1* for m = 1 and different values of δ.

Equations (40)

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ψ 1 ( X * ) = ( i / R * ) 1 / 2 exp ( - i 2 π R * / λ * 2 ) × A B q ( X * ) exp [ - i π ( x * - X * ) 2 / R * ] d X * ;
x * = x / p             X * = X / p .
λ * = λ / p .
R * = λ R / p 2 .
q ( X * ) = 1 / 2 + 2 / π n = 0 sin [ 2 π ( 2 n + 1 ) X * ] / ( 2 n + 1 ) .
ψ 1 ( x * ) = exp ( - i 2 π R * / λ * 2 ) × { 1 / 2 + 2 / π n = 0 sin [ 2 π ( 2 n + 1 ) x * ] × exp [ i π ( 2 n + 1 ) 2 R * ] / ( 2 n + 1 ) ] } .
G 1 , d ( x * , R * ) = ψ 1 ( x * ) ψ 1 * ( x * ) ,
G 1 , d ( x * , R * ) = 1 / 4 + 2 / π n = 0 A n ( x * , R * ) + 4 / π 2 { [ n = 0 A n ( x * , R * ) ] 2 + [ n = 0 B n ( x * , R * ) ] 2 } ,
A n ( x * , R * ) = sin [ 2 π ( 2 n + 1 ) x * ] × cos [ π ( 2 n + 1 ) 2 R * ] / ( 2 n + 1 ) , B n ( x * , R * ) = sin [ 2 π ( 2 n + 1 ) x * ] sin [ π ( 2 n + 1 ) 2 R * ] / ( 2 n + 1 ) .
I ( x * ) = x * x * + a G 1 ( x * ) G 2 ( x * ) d x * ,
G 2 ( x * , χ ) = 1 / 2 + 2 / π n = 0 sin [ 2 π ( 2 n + 1 ) × ( x * + χ ) ] / ( 2 n + 1 ) ,
I d ( χ , R * ) = 1 / 4 + 2 / π 2 n = 0 cos [ 2 π ( 2 n + 1 ) χ ] × cos [ π ( 2 n + 1 ) 2 R * ] / ( 2 n + 1 ) 2 .
ψ 1 ( X * ) = ( i / R * ) 1 / 2 exp ( - i 2 π R * / λ * 2 ) × - + exp [ - i π X * 2 / R 1 * ( X * ) ] q ( X * ) × exp [ - i π ( x * - X * ) 2 / R * ] d X * .
u ( x * ) = R 1 * ( x * ) / [ R 1 * ( x * ) + R * ]
R 1 * = λ R 1 / p 2 ,
ψ 1 ( x * ) = ( i / R * ) 1 / 2 exp ( - i 2 π R * / λ * 2 ) × ( ( 1 / 2 ) - + exp [ - i π u ( x * ) x * 2 / R 1 * ( x * ) ] × exp { - i π W 2 / [ u ( x * ) R * ] } d W + 2 / π n = 0 - + sin [ 2 π ( 2 n + 1 ) × [ x * u ( x * ) - W ] } exp { - i π W 2 / [ u ( x * ) R * ] } / × ( 2 n + 1 ) d W ) ,
W = x * u ( x * ) - X * .
R 1 * ( x ) R * , x * ;
ψ 1 ( x * ) = exp ( - i 2 π R * / λ * 2 ) u ( x * ) 1 / 2 × exp [ - i π u ( x * ) x * 2 / R 1 * ( x * ) ] × { 1 / 2 + 2 / π n = 0 sin [ 2 π ( 2 n + 1 ) x * u ( x * ) ] × exp [ i π ( 2 n + 1 ) 2 u ( x * ) R * ] / ( 2 n + 1 ) } .
G 1 , d ( x * , R * , u ) = u ( 1 / 4 + 2 / π n = 0 A n ( x * , R * , u ) + 4 / π 2 { [ n = 0 A n ( x * , R * , u ) ] 2 + [ n = 0 B n ( x * , R * ) ] 2 } ) ;
A n ( x * , R * , u ) = sin [ 2 π ( 2 n + 1 ) x * u ] cos [ π ( 2 n + 1 ) 2 × R * u ] / ( 2 n + 1 ) , B n ( x * , R * , u ) = sin [ 2 π ( 2 n + 1 ) x * u ] sin [ π ( 2 n + 1 ) 2 × R * u ] / ( 2 n + 1 ) .
I d ( x * , R * , R 1 * ) = x * x * + a G 1 ( x * , R * , R 1 * ) G 2 ( x * ) d x * .
I d ( x * , R * , R 1 * ) = u { 1 / 4 + 2 / π 2 n = 0 × cos [ 2 π ( 2 n + 1 ) x * u ( x * ) R * / R 1 * ] × cos [ π ( 2 n + 1 ) 2 u ( x * ) R * ] / ( 2 n + 1 ) 2 } .
φ ( x * ) = x * λ * / ( R 1 * + R * ) .
I d ( x * , R * ) = [ 1 - R * φ ( x * ) / x * λ * ] ( 1 / 4 + 2 / π 2 n = 0 cos [ 2 π ( 2 n + 1 ) R * φ ( x * ) / λ * ] × cos { π ( 2 n + 1 ) 2 R * [ 1 - R * φ ( x * ) / x * λ * ] } / ( 2 n + 1 ) 2 ) .
F 1 , m = m R 1 * / ( R 1 * - m ) ,             m = 1 , 2 , .
G 1 , d ( x * , m , R 1 * ) = ( m / R 1 * ) { 1 / 4 + 2 / π n = 0 A n ( x * , m , R 1 * ) + 4 / π 2 [ n = 0 A n ( x * , m , R 1 * ) ] 2 } ,
A n ( x * , m , R 1 * ) = sin [ 2 π ( 2 n + 1 ) x * ( 1 - m / R 1 * ) ] .
I d ( x * , m , R 1 * ) = 1 / 4 + 2 / π 2 × n = 0 cos [ 2 π ( 2 n + 1 ) x * m / R 1 * ] / ( 2 n + 1 ) 2 .
I ( x * , R * , R 1 * ) = u ( x * ) { 1 / 4 + 2 / π 2 × n = 0 cos [ 2 π ( 2 n + 1 ) x * u ( x * ) R * / R 1 * ] } .
Δ I ( x * ) / I ( x * ) = [ I d ( x * ) - I ( x * ) ] / I ( x * ) .
Δ φ ( x * ) / φ ( x * ) = [ φ d ( x * ) - φ ( x * ) ] / φ ( x * ) .
F m = m F 1 ,             m = 1 , 2 , ,
F 1 = p 2 / λ .
R m ( δ ) = F m ± δ F 1 ,
C ( R * ) = [ I d ( R * ) - I d ( R * = 1 / 2 ) ] / I d ( R * = 1 / 2 ) .
R m ( δ ) = F 1 , m ± δ F 1 ;
f * = R m [ ( 1 - A ) + ( 1 - 4 A ) 1 / 2 ] / 2 A ,
f = f * p 2 / λ .
σ = { Σ ( φ i - A x i * ) 2 / [ ( n - 1 ) Σ x i * 2 ] } 1 / 2 .

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