Abstract

The theoretical results of the effects of a small inclination misalignment, which is formed by rotation of the beam-splitter grating around the axis on the grating plane when the axis has an arbitrary angle with respect to the line direction of the grating, between the two grating planes on the moiré fringes in the Talbot interferometry are verified by experiment. The experimental results coincide well with theoretical ones. Consequently, the effect of the small arbitrary inclination on practical measurements based on the Talbot interferometry is further explained by an example that examines the beam collimation of a lens, and the advantages and limitations of the effect are also discussed.

© 2001 Optical Society of America

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References

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  1. Q. Liu, R. Ohba, “Effects of a small inclination misalignment in Talbot interferometry by use of gratings with arbitrary line orientation. I. Theoretical analysis,” Appl. Opt. 40, 3668–3676 (2001).
    [CrossRef]
  2. M. P. Kothiyal, K. V. Sriram, R. S. Sirohi, “Setting sensitivity in Talbot interferometry,” Opt. Laser Technol. 23, 361–365 (1991).
    [CrossRef]
  3. K. Patorski, Handbook of the Moiré Technique (Elsevier, Amsterdam, 1993), pp. 5, 15.
  4. K. V. Sriram, M. Kothiyal, R. S. Sirohi, “Direct determination of focal length by using Talbot interferometry,” Appl. Opt. 31, 5984–5987 (1992).
    [CrossRef] [PubMed]
  5. M. P. Kothiyal, R. S. Sirohi, K. J. Rosenbruch, “Improved techniques of collimating testing,” Opt. Laser Technol. 20, 139–144 (1988).
    [CrossRef]
  6. M. P. Kothiyal, K. V. Sriram, “Improved collimating testing using Talbot interferometry,” Appl. Opt. 26, 4056–4057 (1987).
    [CrossRef] [PubMed]
  7. D. E. Silva, “A simple interferometric method of beam collimation,” Appl. Opt. 10, 1980–1982 (1971).
    [CrossRef]
  8. S. Yokozeki, K. Patorski, K. Ohnishi, “Collimation method using Fourier imaging and moiré technique,” Opt. Commun. 14, 401–405 (1975).
    [CrossRef]
  9. K. Patorski, S. Yokozeki, “Collimation test by double grating shearing interferometer,” Appl. Opt. 15, 1234–1240 (1976).
    [CrossRef] [PubMed]
  10. J. C. Fouere, D. Malacare, “Focusing errors in a collimating lens or mirror: use of a moiré technique,” Appl. Opt. 13, 1322–1327 (1974).
    [CrossRef]
  11. C. W. Chang, D. C. Su, “An improved technique of measuring the focal length of a lens,” Opt. Commun. 73, 257–262 (1989).
    [CrossRef]
  12. 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]
  13. S. Yokozeki, K. Ohnishi, “Spherical aberration measurement with shearing interferometer using Fourier imaging and moiré method,” Appl. Opt. 14, 623–627 (1975).
    [CrossRef] [PubMed]
  14. E. Keren, O. Kafri, “Diffraction effects in moiré deflectometry,” J. Opt. Soc. Am. A 2, 111–120 (1985).
    [CrossRef]
  15. E. Keren, A. Livnat, I. Glatt, “Moire deflectometry with pure sinusoidal gratings,” Opt. Lett. 10, 167–169 (1985).
    [CrossRef] [PubMed]
  16. K. Patorski, “Conjugate lateral shear interferometry and its implementation,” J. Opt. Soc. Am. A 3, 1862–1870 (1986).
    [CrossRef]
  17. H. Canabal, J. A. Quiroga, E. Bernabeu, “Improved phase-shifting method for automatic processing deflectograms,” Appl. Opt. 37, 6227–6233 (1998).
    [CrossRef]
  18. J. Striker, “Performance of moiré deflectometry with deferred electronic heterodyne readout,” J. Opt. Soc. Am. A 4, 42–50 (1987).
  19. Y. Nakano, K. Murata, “Talbot interferometry for measuring the small tilt angle of an object surface,” Appl. Opt. 25, 2475–2477 (1986).
    [CrossRef] [PubMed]
  20. 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]

2001 (1)

1998 (1)

1992 (1)

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]

1988 (1)

M. P. Kothiyal, R. S. Sirohi, K. J. Rosenbruch, “Improved techniques of collimating testing,” Opt. Laser Technol. 20, 139–144 (1988).
[CrossRef]

1987 (3)

1986 (2)

1985 (2)

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 shearing interferometer using Fourier imaging and moiré method,” Appl. Opt. 14, 623–627 (1975).
[CrossRef] [PubMed]

1974 (1)

1971 (1)

Bernabeu, E.

Canabal, H.

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]

Fouere, J. C.

Glatt, I.

Kafri, O.

Keren, E.

Kothiyal, M.

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, K. J. Rosenbruch, “Improved techniques of collimating testing,” Opt. Laser Technol. 20, 139–144 (1988).
[CrossRef]

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

Liu, Q.

Livnat, A.

Malacare, D.

Murata, K.

Nakano, Y.

Ohba, R.

Ohnishi, K.

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

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

Patorski, K.

K. Patorski, “Conjugate lateral shear interferometry and its implementation,” J. Opt. Soc. Am. A 3, 1862–1870 (1986).
[CrossRef]

K. Patorski, S. Yokozeki, “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 Technique (Elsevier, Amsterdam, 1993), pp. 5, 15.

Quiroga, J. A.

Rosenbruch, K. J.

M. P. Kothiyal, R. S. Sirohi, K. J. Rosenbruch, “Improved techniques of collimating testing,” Opt. Laser Technol. 20, 139–144 (1988).
[CrossRef]

Silva, D. E.

Sirohi, R. S.

K. V. Sriram, M. Kothiyal, R. S. Sirohi, “Direct determination of focal length by using Talbot interferometry,” Appl. Opt. 31, 5984–5987 (1992).
[CrossRef] [PubMed]

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, K. J. Rosenbruch, “Improved techniques of collimating testing,” Opt. Laser Technol. 20, 139–144 (1988).
[CrossRef]

Sriram, K. V.

Striker, J.

J. Striker, “Performance of moiré deflectometry with deferred electronic heterodyne readout,” J. Opt. Soc. Am. A 4, 42–50 (1987).

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]

Yokozeki, S.

Appl. Opt. (10)

J. C. Fouere, D. Malacare, “Focusing errors in a collimating lens or mirror: use of a moiré technique,” Appl. Opt. 13, 1322–1327 (1974).
[CrossRef]

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

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

H. Canabal, J. A. Quiroga, E. Bernabeu, “Improved phase-shifting method for automatic processing deflectograms,” Appl. Opt. 37, 6227–6233 (1998).
[CrossRef]

Q. Liu, R. Ohba, “Effects of a small inclination misalignment in Talbot interferometry by use of gratings with arbitrary line orientation. I. Theoretical analysis,” Appl. Opt. 40, 3668–3676 (2001).
[CrossRef]

K. V. Sriram, M. Kothiyal, R. S. Sirohi, “Direct determination of focal length by using Talbot interferometry,” Appl. Opt. 31, 5984–5987 (1992).
[CrossRef] [PubMed]

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

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

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

J. Opt. Soc. Am. A (3)

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 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]

M. P. Kothiyal, R. S. Sirohi, K. J. Rosenbruch, “Improved techniques of collimating testing,” Opt. Laser Technol. 20, 139–144 (1988).
[CrossRef]

Opt. Lett. (1)

Other (1)

K. Patorski, Handbook of the Moiré Technique (Elsevier, Amsterdam, 1993), pp. 5, 15.

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Figures (6)

Fig. 1
Fig. 1

Schematic setup for experimental verification: LS, He–Ne laser; G1, beam-splitter grating; G2, detector grating; AMI, angle-measuring instrument.

Fig. 2
Fig. 2

Theoretical and experimental curves for |ϕLN| versus δ for θ = 5°. E and T refer to experimental and theoretical curves, respectively.

Fig. 3
Fig. 3

Theoretical and experimental curves for |ϕLN| versus δ for θ = 8°. E and T refer to experimental and theoretical curves, respectively.

Fig. 4
Fig. 4

Photographs of the moiré fringes used to detect |ΔϕLN| for θ = ±5°: (a) δ = 30°, γ = 1°, ϕLN = -2.598°; (b) δ = 30°, γ = 1°, ϕLN = 2.598°; (c) δ = 30°, γ = 3°, ϕLN = -3.145°; (d) δ = 30°, γ = 3°, ϕLN = 3.145°; (e) δ = 30°, γ = 5°, ϕLN = -4.345°; (f) δ = 30°, γ = 5°, ϕLN = 4.345°.

Fig. 5
Fig. 5

Schematic diagram of the twofold gratings used in Talbot interferometry: G1, beam-splitter grating; G2, detector grating; AB and BA, middle lines of G1 and G2, respectively.

Fig. 6
Fig. 6

Equivalent schematic diagram of top disposition on two gratings. G1 and G2, orientations of their grating lines; LN, perpendicular to G1; x axis, rotation axis of the beam-splitter grating.

Tables (2)

Tables Icon

Table 1 Measured |ϕLN| for |θ| = 5°

Tables Icon

Table 2 Measured |ϕLN| for |θ| = 8°

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

ϕLN=90°-δ+tan-1×sinθ+δsin2δ+cos2 δ cos2γ1/2-sinδcosθ+δsin2/δ+cos2 δ cos2 γ1/2cosδ.
ϕLN=90°-δ+tan-1sinθ+δ-sin δcosθ+δ-cos δ=90°-δ+ϕx=θ2,
ϕx=tan-1×sinθ+δsin2 δ+cos2 δ cos2 γ1/2-sin δcosθ+δsin2 δ+cos2 δ cos2 γ1/2-cos δ.

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