Abstract

The properties of moiré fringes in Talbot interferometry are analyzed when the angle between the two grating planes is small. The results indicate that the tilt angle of the moiré fringes, observed just behind the test grating, is sensitive to the small angle. Based on this sensitivity, several features of parallelism of the two grating planes are presented, and the influence of the small angle when checking the beam collimation of a lens is also discussed. The validity of the theoretical analysis is illustrated by experiment.

© 1999 Optical Society of America

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References

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  1. D. E. Silva, “A simple interferometric method of beam collimation,” Appl. Opt. 10, 1980–1982 (1971).
    [CrossRef]
  2. J. C. Fouere, D. Malacara, “Focusing errors in a collimating lens or mirror: use of a moiré technique,” Appl. Opt. 13, 1322–1326 (1974).
    [CrossRef]
  3. S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation method using Fourier imaging and moiré technique,” Opt. Commun. 14, 401–405 (1975).
    [CrossRef]
  4. K. Patorski, S. Yokozeki, T. Suzuki, “Collimation test by double grating shearing interferometer,” Appl. Opt. 15, 1234–1240 (1976).
    [CrossRef] [PubMed]
  5. M. P. Kothiyal, “Improved collimating testing using the Talbot interferometer,” Appl. Opt. 26, 4056–4057 (1987).
    [CrossRef] [PubMed]
  6. S. Yokozeki, K. Ohnishi, “Spherical aberration measurement with a shearing interferometer using Fourier imaging and moiré method,” Appl. Opt. 14, 623–627 (1975).
    [CrossRef] [PubMed]
  7. M. P. Kothiyal, K. V. Sriram, R. S. Sirohi, “Setting sensitivity in Talbot interferometry,” Opt. Laser Technol. 23, 361–365 (1991).
    [CrossRef]
  8. C. W. Chang, D. C. Su, “An improved technique for measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
    [CrossRef]
  9. D. C. Su, C. W. Chang, “A new technique for measuring the effective focal length of a thick lens or a compound lens,” Opt. Commun. 78, 118–123 (1990).
    [CrossRef]
  10. K. Patorski, “Talbot interferometry with increased shear,” Appl. Opt. 24, 4448–4453 (1985).
    [CrossRef] [PubMed]
  11. K. Patorski, “Talbot interferometry with increased shear: further considerations,” Appl. Opt. 25, 1111–1116 (1986).
    [CrossRef] [PubMed]
  12. Y. Nakano, K. Murata, “Talbot interferometry for measuring the small tilt angle of an object surface,” Appl. Opt. 25, 2475–2477 (1986).
    [CrossRef] [PubMed]
  13. Y. Nakano, “Measurement of the small tilt-angle variation of an object surface using moiré interferometry and digital image processing,” Appl. Opt. 26, 3911–3914 (1987).
    [CrossRef] [PubMed]
  14. S. Yokozeki, T. Suzuki, “Shearing interferometer using a grating as a beam splitter,” Appl. Opt. 10, 1575–1580 (1971).
    [CrossRef] [PubMed]
  15. K. Patorski, “Fresnel diffraction field (self-imaging) of obliquely illuminated linear diffraction gratings,” Optik 69, 30–36 (1984).
  16. V. E. Lau, “Beugungserscheinungen an doppelrastern,” Ann. Phys. 6, 417–423 (1948).
    [CrossRef]
  17. Q. Liu, F. Y. Xu, “The relationship between the Lau effect and the moiré effect—a new explanation of the Lau effect,” in The Anthology of Theses of the Fourth Optical Academic Conference of Universities and Colleges of China (Chinese Publishing House of Geology, Peking, 1992), p. 238.
  18. K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), p. 15.

1991 (1)

M. P. Kothiyal, K. V. Sriram, R. S. Sirohi, “Setting sensitivity in Talbot interferometry,” Opt. Laser Technol. 23, 361–365 (1991).
[CrossRef]

1990 (1)

D. C. Su, C. W. Chang, “A new technique for measuring the effective focal length of a thick lens or a compound lens,” Opt. Commun. 78, 118–123 (1990).
[CrossRef]

1989 (1)

C. W. Chang, D. C. Su, “An improved technique for measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
[CrossRef]

1987 (2)

1986 (2)

1985 (1)

1984 (1)

K. Patorski, “Fresnel diffraction field (self-imaging) of obliquely illuminated linear diffraction gratings,” Optik 69, 30–36 (1984).

1976 (1)

1975 (2)

S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation method using Fourier imaging and moiré technique,” Opt. Commun. 14, 401–405 (1975).
[CrossRef]

S. Yokozeki, K. Ohnishi, “Spherical aberration measurement with a shearing interferometer using Fourier imaging and moiré method,” Appl. Opt. 14, 623–627 (1975).
[CrossRef] [PubMed]

1974 (1)

1971 (2)

1948 (1)

V. E. Lau, “Beugungserscheinungen an doppelrastern,” Ann. Phys. 6, 417–423 (1948).
[CrossRef]

Chang, C. W.

D. C. Su, C. W. Chang, “A new technique for measuring the effective focal length of a thick lens or a compound lens,” Opt. Commun. 78, 118–123 (1990).
[CrossRef]

C. W. Chang, D. C. Su, “An improved technique for measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
[CrossRef]

Fouere, J. C.

Kothiyal, M. P.

M. P. Kothiyal, K. V. Sriram, R. S. Sirohi, “Setting sensitivity in Talbot interferometry,” Opt. Laser Technol. 23, 361–365 (1991).
[CrossRef]

M. P. Kothiyal, “Improved collimating testing using the Talbot interferometer,” Appl. Opt. 26, 4056–4057 (1987).
[CrossRef] [PubMed]

Lau, V. E.

V. E. Lau, “Beugungserscheinungen an doppelrastern,” Ann. Phys. 6, 417–423 (1948).
[CrossRef]

Liu, Q.

Q. Liu, F. Y. Xu, “The relationship between the Lau effect and the moiré effect—a new explanation of the Lau effect,” in The Anthology of Theses of the Fourth Optical Academic Conference of Universities and Colleges of China (Chinese Publishing House of Geology, Peking, 1992), p. 238.

Malacara, D.

Murata, K.

Nakano, Y.

Ohnishi, K.

S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation method using Fourier imaging and moiré technique,” Opt. Commun. 14, 401–405 (1975).
[CrossRef]

S. Yokozeki, K. Ohnishi, “Spherical aberration measurement with a shearing interferometer using Fourier imaging and moiré method,” Appl. Opt. 14, 623–627 (1975).
[CrossRef] [PubMed]

Patorski, K.

K. Patorski, “Talbot interferometry with increased shear: further considerations,” Appl. Opt. 25, 1111–1116 (1986).
[CrossRef] [PubMed]

K. Patorski, “Talbot interferometry with increased shear,” Appl. Opt. 24, 4448–4453 (1985).
[CrossRef] [PubMed]

K. Patorski, “Fresnel diffraction field (self-imaging) of obliquely illuminated linear diffraction gratings,” Optik 69, 30–36 (1984).

K. Patorski, S. Yokozeki, T. Suzuki, “Collimation test by double grating shearing interferometer,” Appl. Opt. 15, 1234–1240 (1976).
[CrossRef] [PubMed]

S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation method using Fourier imaging and moiré technique,” Opt. Commun. 14, 401–405 (1975).
[CrossRef]

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), p. 15.

Silva, D. E.

Sirohi, R. S.

M. P. Kothiyal, K. V. Sriram, R. S. Sirohi, “Setting sensitivity in Talbot interferometry,” Opt. Laser Technol. 23, 361–365 (1991).
[CrossRef]

Sriram, K. V.

M. P. Kothiyal, K. V. Sriram, R. S. Sirohi, “Setting sensitivity in Talbot interferometry,” Opt. Laser Technol. 23, 361–365 (1991).
[CrossRef]

Su, D. C.

D. C. Su, C. W. Chang, “A new technique for measuring the effective focal length of a thick lens or a compound lens,” Opt. Commun. 78, 118–123 (1990).
[CrossRef]

C. W. Chang, D. C. Su, “An improved technique for measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
[CrossRef]

Suzuki, T.

Xu, F. Y.

Q. Liu, F. Y. Xu, “The relationship between the Lau effect and the moiré effect—a new explanation of the Lau effect,” in The Anthology of Theses of the Fourth Optical Academic Conference of Universities and Colleges of China (Chinese Publishing House of Geology, Peking, 1992), p. 238.

Yokozeki, S.

Ann. Phys. (1)

V. E. Lau, “Beugungserscheinungen an doppelrastern,” Ann. Phys. 6, 417–423 (1948).
[CrossRef]

Appl. Opt. (10)

K. Patorski, “Talbot interferometry with increased shear,” Appl. Opt. 24, 4448–4453 (1985).
[CrossRef] [PubMed]

K. Patorski, “Talbot interferometry with increased shear: further considerations,” Appl. Opt. 25, 1111–1116 (1986).
[CrossRef] [PubMed]

Y. Nakano, K. Murata, “Talbot interferometry for measuring the small tilt angle of an object surface,” Appl. Opt. 25, 2475–2477 (1986).
[CrossRef] [PubMed]

Y. Nakano, “Measurement of the small tilt-angle variation of an object surface using moiré interferometry and digital image processing,” Appl. Opt. 26, 3911–3914 (1987).
[CrossRef] [PubMed]

S. Yokozeki, T. Suzuki, “Shearing interferometer using a grating as a beam splitter,” Appl. Opt. 10, 1575–1580 (1971).
[CrossRef] [PubMed]

K. Patorski, S. Yokozeki, T. Suzuki, “Collimation test by double grating shearing interferometer,” Appl. Opt. 15, 1234–1240 (1976).
[CrossRef] [PubMed]

M. P. Kothiyal, “Improved collimating testing using the Talbot interferometer,” Appl. Opt. 26, 4056–4057 (1987).
[CrossRef] [PubMed]

S. Yokozeki, K. Ohnishi, “Spherical aberration measurement with a shearing interferometer using Fourier imaging and moiré method,” Appl. Opt. 14, 623–627 (1975).
[CrossRef] [PubMed]

D. E. Silva, “A simple interferometric method of beam collimation,” Appl. Opt. 10, 1980–1982 (1971).
[CrossRef]

J. C. Fouere, D. Malacara, “Focusing errors in a collimating lens or mirror: use of a moiré technique,” Appl. Opt. 13, 1322–1326 (1974).
[CrossRef]

Opt. Commun. (3)

S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation method using Fourier imaging and moiré technique,” Opt. Commun. 14, 401–405 (1975).
[CrossRef]

C. W. Chang, D. C. Su, “An improved technique for measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
[CrossRef]

D. C. Su, C. W. Chang, “A new technique for measuring the effective focal length of a thick lens or a compound lens,” Opt. Commun. 78, 118–123 (1990).
[CrossRef]

Opt. Laser Technol. (1)

M. P. Kothiyal, K. V. Sriram, R. S. Sirohi, “Setting sensitivity in Talbot interferometry,” Opt. Laser Technol. 23, 361–365 (1991).
[CrossRef]

Optik (1)

K. Patorski, “Fresnel diffraction field (self-imaging) of obliquely illuminated linear diffraction gratings,” Optik 69, 30–36 (1984).

Other (2)

Q. Liu, F. Y. Xu, “The relationship between the Lau effect and the moiré effect—a new explanation of the Lau effect,” in The Anthology of Theses of the Fourth Optical Academic Conference of Universities and Colleges of China (Chinese Publishing House of Geology, Peking, 1992), p. 238.

K. Patorski, Handbook of the Moiré Fringe Technique (Elsevier, Amsterdam, 1993), p. 15.

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Figures (4)

Fig. 1
Fig. 1

Schematic representation of Talbot interferometry: P, monochromatic plane wave; G1, beam-splitter grating; G2, detector grating.

Fig. 2
Fig. 2

Moiré patterns at α = 1°, 3°, and 5°: (a) α = 1°, θ = 5°, α = -2.66°; (b) α = 1°, θ = -5°, α = 2.66°; (c) α = 3°, θ = 5°, α = -3.55°; (d) α = 3°, θ = -5°, α = 3.55°; (e) α = 5°, θ = 5°, α = -4.98°, (f) α = 5°, θ = -5°, α = 4.98°.

Fig. 3
Fig. 3

Theoretical and experimental curves for tilt angle |φ| of the moiré fringe versus rotation angle α. T and E refer to the theoretical and the experimental curves, respectively.

Fig. 4
Fig. 4

Tilt angle |φ| of moiré fringes versus rotation angle |θ|.

Tables (1)

Tables Icon

Table 1 Measured |φ| by Experiment with a 5-line/mm Grating

Equations (14)

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Tx, y=A0+2A1 cos2πx/d,
sin αn-sin α=nλ/d,
pn=|αn-α|,
p+1p-1=p=λd cos α=λd1.
Ux, y, z=expikzA0+A1 expikpx-z2 p2+A1 exp-ikpx+z2 p2.
zt=n d cos α2λ=n d12λ,  n=1, 2, 3,
Ux, y, zt=expikztA0±2A1 cos kpx,
Tx, y=A0+2A1 cos2πdx cos θ±y sin θ=A0+2A1 cos2πd2x±y tan θ,
Ex, y, z=Tx, yUx, y, zt=A02+2A1A0 cos2πd2x±y tan θ±2A0A1 cos2πd1 x±4A12 cos2πd1 x cos2πd2x±y tan θ.
Ix, y, z=E2=A04+4A02A12cos2πd2x±y tan θ2±4A02A12cos2πd1 x2±16A14×cos2πd1 x cos2πd2x±y tan θ2±8A02A12 cos2πd1 x cos2πd2x±y tan θ+4A1A03 cos2πd2x±y tan θ±4A03A1 cos2πd1 x±8A02A12 cos2πd2×x±y tan θcos2πd1 x±16A13A0 cos2πd1 ×xcos2πd2x±y tan θ2±16A13A0×cos2πd1 x2 cos2πd2x±y tan θ.
I  A04±8A02A12 cos2πd2-2πd1x±2πd2 y tan θ.
2πd2-2πd1x±2πd2 y tan θ=2mπ,  m=0, ±1, ±2, ±3.
y=±m d2tan θ1d2-1d1d2tan θ x.
tan φ=1d2-1d1d2tan θ=±1sin θ cos α-cot θ,

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