Abstract

The effects of an arbitrary small inclination between two cross gratings on the moiré fringes in Talbot interferometry are discussed when the frequencies of the grating differ in two perpendicular directions. We show that the small angles, α and β, by which the beam-splitter cross grating is rotated around the two axes parallel to the two perpendicular line directions of the cross grating, have a greater influence on the moiré fringes with cross gratings than that with one-dimensional gratings. A simple and practical detection method for the angles between the two unparallel grating planes in Talbot interferometry is also proposed. The theoretical analyses are proved by experimental results.

© 2000 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, 1981–1982 (1971).
    [CrossRef]
  2. S. Yokozeki, K. Patorski, K. Ohnshi, “Collimation method using Fourier imaging and the moiré technique,” Opt. Commun. 14, 401–405 (1975).
    [CrossRef]
  3. K. Patorski, S. Yokozeki, “Collimation test by a double grating shearing interferometer,” Appl. Opt. 15, 1234–1240 (1976).
    [CrossRef] [PubMed]
  4. M. P. Kothiyal, R. S. Sirohi, “Improved collimating testing using the Talbot interferometer,” Appl. Opt. 26, 4056–4057 (1987).
    [CrossRef] [PubMed]
  5. J. C. Fouere, D. Malacare, “Focusing errors in a collimating lens or mirror: use of a moiré technique,” Appl. Opt. 13, 1322–1326 (1974).
    [CrossRef]
  6. C. W. Chang, D. C. Su, “An improved technique of measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
    [CrossRef]
  7. 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]
  8. S. Yokozeki, K. Ohnishi, “Spherical aberration measurement with a shearing interferometer using Fourier imaging and a moiré method,” Appl. Opt. 14, 623–627 (1975).
    [CrossRef] [PubMed]
  9. M. P. Kothiyal, K. V. Sriram, R. S. Sirohi, “Setting sensitivity in Talbot interferometry,” Opt. Laser Technol. 23, 361–365 (1991).
    [CrossRef]
  10. Q. Liu, R. Ohba, “Effects of unparallel grating planes in Talbot interferometry,” Appl. Opt. 38, 4111–4116 (1999).
    [CrossRef]
  11. Q. Liu, R. Ohba, “Effects of arbitrary discrepancy between the unparallel gratings in Talbot interferometry,” Opt. Commun. 170, 193–199 (1999).
    [CrossRef]
  12. H. O. Bartelt, Y. Li, “Lau interferometry with cross gratings,” Opt. Commun. 48, 1–6 (1983).
    [CrossRef]
  13. Y. S. Chang, “Fringe formation with a cross-grating interferometer,” Appl. Opt. 25, 4185–4191 (1986).
    [CrossRef]
  14. P. Szwaykowski, “Producing binary diffraction gratings in the double-diffraction system,” Opt. Laser Technol. 17, 255–261 (1985).
    [CrossRef]
  15. K. Patorski, “Talbot interferometry with increased shear,” Appl. Opt. 24, 4448–4453 (1985).
    [CrossRef] [PubMed]
  16. S. Yokozeki, T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt. 10, 1575–1580 (1971).
    [CrossRef] [PubMed]

1999 (2)

Q. Liu, R. Ohba, “Effects of arbitrary discrepancy between the unparallel gratings in Talbot interferometry,” Opt. Commun. 170, 193–199 (1999).
[CrossRef]

Q. Liu, R. Ohba, “Effects of unparallel grating planes in Talbot interferometry,” Appl. Opt. 38, 4111–4116 (1999).
[CrossRef]

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 of measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
[CrossRef]

1987 (1)

1986 (1)

1985 (2)

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

P. Szwaykowski, “Producing binary diffraction gratings in the double-diffraction system,” Opt. Laser Technol. 17, 255–261 (1985).
[CrossRef]

1983 (1)

H. O. Bartelt, Y. Li, “Lau interferometry with cross gratings,” Opt. Commun. 48, 1–6 (1983).
[CrossRef]

1976 (1)

1975 (2)

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

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

1974 (1)

1971 (2)

D. E. Silva, “A simple interferometric method of beam collimation,” Appl. Opt. 10, 1981–1982 (1971).
[CrossRef]

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

Bartelt, H. O.

H. O. Bartelt, Y. Li, “Lau interferometry with cross gratings,” Opt. Commun. 48, 1–6 (1983).
[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 of measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
[CrossRef]

Chang, Y. S.

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

Li, Y.

H. O. Bartelt, Y. Li, “Lau interferometry with cross gratings,” Opt. Commun. 48, 1–6 (1983).
[CrossRef]

Liu, Q.

Q. Liu, R. Ohba, “Effects of unparallel grating planes in Talbot interferometry,” Appl. Opt. 38, 4111–4116 (1999).
[CrossRef]

Q. Liu, R. Ohba, “Effects of arbitrary discrepancy between the unparallel gratings in Talbot interferometry,” Opt. Commun. 170, 193–199 (1999).
[CrossRef]

Malacare, D.

Ohba, R.

Q. Liu, R. Ohba, “Effects of arbitrary discrepancy between the unparallel gratings in Talbot interferometry,” Opt. Commun. 170, 193–199 (1999).
[CrossRef]

Q. Liu, R. Ohba, “Effects of unparallel grating planes in Talbot interferometry,” Appl. Opt. 38, 4111–4116 (1999).
[CrossRef]

Ohnishi, K.

Ohnshi, K.

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

Patorski, K.

Silva, D. E.

D. E. Silva, “A simple interferometric method of beam collimation,” Appl. Opt. 10, 1981–1982 (1971).
[CrossRef]

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]

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

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 of measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
[CrossRef]

Suzuki, T.

Szwaykowski, P.

P. Szwaykowski, “Producing binary diffraction gratings in the double-diffraction system,” Opt. Laser Technol. 17, 255–261 (1985).
[CrossRef]

Yokozeki, S.

Appl. Opt. (9)

Opt. Commun. (5)

Q. Liu, R. Ohba, “Effects of arbitrary discrepancy between the unparallel gratings in Talbot interferometry,” Opt. Commun. 170, 193–199 (1999).
[CrossRef]

H. O. Bartelt, Y. Li, “Lau interferometry with cross gratings,” Opt. Commun. 48, 1–6 (1983).
[CrossRef]

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

C. W. Chang, D. C. Su, “An improved technique of 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. (2)

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

P. Szwaykowski, “Producing binary diffraction gratings in the double-diffraction system,” Opt. Laser Technol. 17, 255–261 (1985).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Moiré fringes at |θ| = 5°: (a) α = 5°, β = 3°; |Δφ x | = 10.120°, |Δφ y | = 6.680°; (b) α = 5°, β = 5°; |Δφ x | = 10.120°, |Δφ y | = 9.792°; (c) fringes satisfying only Eq. (5); (d) fringes satisfying only Eq. (4).

Tables (1)

Tables Icon

Table 1 Measured and Theoretical Values of |φ y | at |θ| = 5°

Equations (16)

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Tx, y=TxTy=A0+2A1 cos2πx/dxB0+B1 cos2πy/dy,
Tx, y=A0+2A1 cos2πx/dxB0+B1 cos2πy/dy,
UX, Y, z=expikziλz Tx, y, 0×expik X-x2+Y-y22zdxdy=expikφx, yA0+2A1 cos2πdx X×exp-iπλz 1dx2×B0+2B1 cos2πdy Y×exp-iπλz 1dy2,
φx, y=kz+k2zX2+Y2-πλzX2+Y2.
λzxdx2=kx,
λzydy2=ky,
kxdx2=kydy2.
I=|UX, Y, z|2=A0+2A1 cos2πdx XB0+2B1 cos2πdy Y2.
I=A0+2A1 cos2πdxX cos θ-Y sin θ×B0+2B1 cos2πdyY cos θ-X sin θ.
Y=m dxsin θ+1cos α sin θ-cot θX,
X=n dysin θ+1cos β sin θ-cot θY,
Px=dx1+1-cos θcos α sin θ21/2,
Py=dy1+1-cos θcos β sin θ21/2
tan φx=1sin θ cos α-cot θ,
tan φy=1sin θ cos β-cot θ,
kxky=dydx2

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