Abstract

We discuss the effects of a general small inclination misalignment, which is formed by rotation of the beam-splitter grating around an axis that is laid on the grating plane and that has an arbitrary angle with respect to the line direction of the grating, between the two grating planes on the moiré fringes in Talbot interferometry. It is shown that the small inclination angle has a significant influence on measurement results based on Talbot interferometry because both the period and the slope of the moiré fringes are sensitive to the angle, especially when the rotation axis is nearly parallel to the lines of the grating. Simple and practical detection methods for the small inclination angle are proposed, and the effects of the inclination angle on the contrast in the moiré fringes are also briefly discussed.

© 2001 Optical Society of America

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References

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  1. K. Patorski, “Talbot interferometry with increased shear,” Appl. Opt. 24, 4448–4453 (1985).
    [CrossRef] [PubMed]
  2. K. Patorski, “Talbot interferometry with increased shear: further considerations,” Appl. Opt. 25, 1111–1116 (1986).
    [CrossRef] [PubMed]
  3. Q. Liu, R. Ohba, “Effects of unparallel grating planes in Talbot interferometry,” Appl. Opt. 38, 4111–4116 (1999).
    [CrossRef]
  4. Q. Liu, R. Ohba, “Effects of unparallel grating planes in Talbot interferometry. II,” Appl. Opt. 39, 2084–2090 (2000).
    [CrossRef]
  5. M. P. Kothiyal, K. V. Sriram, “Improved collimating testing using Talbot interferometer,” Appl. Opt. 26, 4056–4057 (1987).
    [CrossRef] [PubMed]
  6. M. P. Kothiyal, K. V. Sriram, R. S. Sirohi, “Setting sensitivity in Talbot interferometry,” Opt. Laser Technol. 23, 361–365 (1991).
    [CrossRef]
  7. 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]
  8. M. P. Kothiyal, R. S. Sirohi, K. J. Rosenbruch, “Improved techniques of collimating testing,” Opt. Laser Technol. 20, 139–144 (1991).
    [CrossRef]
  9. Y. Nakano, K. Murata, “Talbot interferometry for measuring the small tilt angle of an object surface,” Appl. Opt. 25, 2475–2477 (1986).
    [CrossRef] [PubMed]
  10. 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]
  11. H. Canabal, J. A. Quiroga, E. Bernabeu, “Improved phase-shifting method for automatic processing deflectograms,” Appl. Opt. 37, 6227–6233 (1998).
    [CrossRef]
  12. J. Striker, “Performance of moiré deflectometry with deferred electronic heterodyne readout,” J. Opt. Soc. Am. A 4, 42–50 (1987).
  13. S. Yokozeki, T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt. 10, 1575–1580 (1971).
    [CrossRef] [PubMed]
  14. K. Patorski, “Fresnel diffraction field (self-imaging) of obliquely illuminated liner diffraction gratings,” Optik 69, 30–36 (1984).
  15. Q. Liu, R. Ohba, “Effects of a small inclination alignment in Talbot interferometry by use of gratings with arbitrary line orientation. II. Experimental verification,” Appl. Opt. (to be published).

2000 (1)

1999 (1)

1998 (1)

1992 (1)

1991 (2)

M. P. Kothiyal, R. S. Sirohi, K. J. Rosenbruch, “Improved techniques of collimating testing,” Opt. Laser Technol. 20, 139–144 (1991).
[CrossRef]

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

1987 (3)

1986 (2)

1985 (1)

1984 (1)

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

1971 (1)

Bernabeu, E.

Canabal, H.

Kothiyal, M.

Kothiyal, M. P.

M. P. Kothiyal, R. S. Sirohi, K. J. Rosenbruch, “Improved techniques of collimating testing,” Opt. Laser Technol. 20, 139–144 (1991).
[CrossRef]

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

Liu, Q.

Q. Liu, R. Ohba, “Effects of unparallel grating planes in Talbot interferometry. II,” Appl. Opt. 39, 2084–2090 (2000).
[CrossRef]

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 a small inclination alignment in Talbot interferometry by use of gratings with arbitrary line orientation. II. Experimental verification,” Appl. Opt. (to be published).

Murata, K.

Nakano, Y.

Ohba, R.

Q. Liu, R. Ohba, “Effects of unparallel grating planes in Talbot interferometry. II,” Appl. Opt. 39, 2084–2090 (2000).
[CrossRef]

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 a small inclination alignment in Talbot interferometry by use of gratings with arbitrary line orientation. II. Experimental verification,” Appl. Opt. (to be published).

Patorski, K.

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 (1991).
[CrossRef]

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, R. S. Sirohi, K. J. Rosenbruch, “Improved techniques of collimating testing,” Opt. Laser Technol. 20, 139–144 (1991).
[CrossRef]

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.

Striker, J.

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

Suzuki, T.

Yokozeki, S.

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]

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 unparallel grating planes in Talbot interferometry. II,” Appl. Opt. 39, 2084–2090 (2000).
[CrossRef]

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

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

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

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

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

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 (1991).
[CrossRef]

Optik (1)

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

Other (1)

Q. Liu, R. Ohba, “Effects of a small inclination alignment in Talbot interferometry by use of gratings with arbitrary line orientation. II. Experimental verification,” Appl. Opt. (to be published).

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Figures (7)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Relationship between the line orientation of G1 and rotation axis: L, line direction of G1; x 1, rotation axis; δ, arbitrary angle between the line direction of the grating and the x 1 axis.

Fig. 3
Fig. 3

Period difference ΔT of moiré fringes between γ = i(i = 1°, 2°, 3°, 4°, 5°) and γ = 0° versus δ for θ = 2°, 6°, 10°, and 12°.

Fig. 4
Fig. 4

Period T of the moiré fringes versus δ at γ = 0°, 1°, 2°, 3°, 4°, and 5° for θ = 2°, 6°, 10°, and 14°.

Fig. 5
Fig. 5

Geometry relationship between angles, ϕLN and ϕ x : L, line direction of G1; LN, normal direction of the lines of G1; LM, orientation of the moiré fringes.

Fig. 6
Fig. 6

Tilt-angle difference ΔϕLN(i) between γ = i(i = 1°, 2°, 3°, 4°, 5°) and γ = 0° versus angle δ for θ = 2°, 4°, 6°, and 8°.

Fig. 7
Fig. 7

ϕLN for γ = 0° versus δ for θ = 2°, 4°, 6°, and 8°.

Equations (19)

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Tx1, y1=- An expi 2πndx1 sin δ-y1 cos δ=- An expi 2πndx1-y1 cot δ,
Tx0, y0=- An expi 2πnd0x0 sin δ-y0 cos δ=- An expi 2πnd0x0-y0 cot δ,
d0=d0x2+d0y21/2=dx12+dy12 cos2 γ1/2=dsin2 δ+cos2 δ cos2 γ1/2.
Ux, y, z=expikziλz  Tx0, y0, 0×expik x-x02+y-y022zdx0dy0=expikziλzexp Tx0, y0, 0qx0, y0:1λz×exp-i 2πλzxx0+yy0dx0dy0,
Ux, y, z=iϕx, y- An expi 2πnd0x-y cot θ×exp-i πλzn2d021+cot2 δ,
zT=md02λ1+cot2 δ=md02λ1+cot2 δsin2 δ=md2λsin2 δ+cos2 δ cos2 γ,
Ux, y, z=expiϕx, y-±1nAn×expi 2πnd0x-y cot δ.
|Ux, y, zT|2=-±1nAn expi 2πnd0x-y cot δ2.
I1x, y, zT=A0±2A1 cos2πd0x-y cot δ2 =A02±4A0A1 cos2πd0x-y cot δ±4A12cos2πd0x-y cot δ2.
I2x, y, zT=A0+2A1cos2πd2x-y cot ω,
y=l+1d2-1d0xcot ωd2-cot δd0, l=0, ±1, ±2,
p=11d2-1d02+cot ωd2-cot δd021/2=d1+1sin2 δ+cos2 δ cos2 γ-2 sin δ sin ωsin2 δ+cos2 δ cos2 γ1/2-2 cos δ cos ωsin2 δ+cos2 δ cos2 γ1/2-1/2,
tan ϕx=1d2-1d0Xcot ωd2-cot δd0=sin ωsin2 δ+cos2 δ cos2 γ1/2-sin δcos ωsin2 δ+cos2 δ cos2 γ1/2-cos δ,
Δp=d1+1sin2 δ+cos2 δ cos2 γ-2 sin δ sin ωsin2 δ+cos2 δ cos2 γ1/2-2 cos δ cos ωsin2 δ+cos2 δ cos2 γ1/2-1/2-d1+1sin2 δ+cos2 δ-2 sin δ sin ωsin2 δ+cos2 δ1/2-2 cos δ cos ωsin2 δ+cos2 δ1/2-1/2=dΔT,
T=p/d=1+1sin2 δ+cos2 δ cos2 γ-2 sin δ sin ωsin2 δ+cos2 δ cos2 γ1/2-2 cos δ cos ωsin2 δ+cos2 δ cos2 γ1/2-1/2.
ϕLN=90°-δ+tan-1×sin ωsin2 δ+cos2 δ cos2 γ1/2-sin δcos ωsin2 δ+cos2 δ cos2 γ1/2-cos δ,
ϕLN=90°-δ+tan-1sin ω-sin δcos ω-cos δ.
tan ϕLN=tanπ2+tan-1sin θ cos γcos θ cos γ-1=cot tan-1sin θ cos γcos θ cos γ-1=1sin θ cos γ-cot θ.
ϕLN=θ/2.

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