Abstract

The properties of the moiré fringes in Talbot interferometry are analyzed for a small inclined angle β between the two grating planes, which is produced by rotation of the beam splitter grating about the axis perpendicular to the lines of the grating. Theoretical analyses indicate that the tilt angle of the resultant moiré fringes is less sensitive to β than when the small inclined angle is formed by rotation of the beam splitter grating about the axis parallel to the lines direction of the grating as described earlier [Appl. Opt. 38, 4111 (1999)] and that contrast of the moiré fringes decreases with an increase in β or in the spatial frequency of the grating and may result in impaired measurement accuracy. The validity of the theoretical analyses is illustrated by experiments.

© 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, 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, R. S. Sirohi, “Improved collimating testing using Talbot interferometry,” Appl. Opt. 26, 4056–4057 (1987).
    [CrossRef] [PubMed]
  6. S. Yokozeki, K. Ohnishi, “Spherical aberration measurement with 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 of 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. Q. Liu, R. Ohba, “Effects of unparallel grating planes in Talbot interferometry,” Appl. Opt. 38, 4111–4116 (1999).
    [CrossRef]
  11. K. Patorski, “Talbot interferometry with increased shear,” Appl. Opt. 24, 4448–4453 (1985).
    [CrossRef] [PubMed]
  12. S. Yokozeki, T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt. 10, 1575–1580 (1971).
    [CrossRef] [PubMed]
  13. K. Patorski, “Fresnel diffraction field of obliquely illuminated linear diffraction gratings,” Optik (Stuttgart) 69, 30–36 (1984).
  14. E. Keren, O. Kafri, “Diffraction effects in moiré deflectometry,” J. Opt. Soc. Am. A 2, 111–120 (1985).
    [CrossRef]

1999 (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]

1987 (1)

1985 (2)

1984 (1)

K. Patorski, “Fresnel diffraction field of obliquely illuminated linear diffraction gratings,” Optik (Stuttgart) 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 shearing interferometer using Fourier imaging and moiré method,” Appl. Opt. 14, 623–627 (1975).
[CrossRef] [PubMed]

1974 (1)

1971 (2)

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.

Kafri, O.

Keren, E.

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 Talbot interferometry,” Appl. Opt. 26, 4056–4057 (1987).
[CrossRef] [PubMed]

Liu, Q.

Malacara, D.

Ohba, R.

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 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,” Appl. Opt. 24, 4448–4453 (1985).
[CrossRef] [PubMed]

K. Patorski, “Fresnel diffraction field of obliquely illuminated linear diffraction gratings,” Optik (Stuttgart) 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]

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]

M. P. Kothiyal, R. S. Sirohi, “Improved collimating testing using Talbot interferometry,” 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.

Yokozeki, S.

Appl. Opt. (8)

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

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. (1)

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

Optik (Stuttgart) (1)

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

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Figures (4)

Fig. 1
Fig. 1

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

Fig. 2
Fig. 2

Tilt difference Δφ of the two moiré patterns versus rotation angle θ for β = 1°, 3°, 5°.

Fig. 3
Fig. 3

Period difference of the two constituent patterns, Δp, versus rotation angle θ for β = 1°, 3°, 5°.

Fig. 4
Fig. 4

Contrast of moiré fringe versus small angle β for θ = 2°: (a) β = 0°, (b) β = 5°.

Equations (26)

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Tx, y=A0+2A1 cos2π/dx,
Ux, y, z=A0 expikz+A1 exp(ikpx+qy+z1-p2+q21/2)+A1exp(ik-px+qy+z1-p2+q21/2)=expikzA0+A1 expikpx+qy-z p2+q22+A1 expik-px+qy-z p2+q22,
p=λ/d,
q=sin β cos β1-1-p/cos β21/2.
q=sin β cos β1-1-p22 cos2 β+p22tan β.
I1x, y, z=A02+2A12+4A0A1 cos kqy-zp2+q2/2cos kpx+2A12 cos 2kpx.
I1=A02+2A12+4A0A1 cos kpxcos kqy cos zk p2+q22+sin kqy sin zk p2+q22+2A12 cos 2kpx.
z=mλp2+q2,  m=0, 1, 2, 3,
z=m+12λp2+q2,  m=0, 1, 2, 3.
I1=A02+2A12±4A0A1 cos kqy cos kpx+2A12 cos 2kpx,
I1=A02+2A12±4A0A1 sin kqy cos kpx+2A12 cos 2kpx=A02+2A12±4A0A1 cosπ/2-kqycos kpx+2A12 cos 2kpx.
I2=A0+2A1 cos2π/Dx-y tan θ,
I=I1I2=A02+2A12±2A0A1 cos kpx+qy±2A0A1 cos kpx-qy+2A12 cos 2kpx×A0+2A1 cos2π/Dx-y tan θ,=A0A02+2A12+2A1A02+2A12cos2π/Dx-y tan θ±2A02A1 cos kpx+qy±4A0A12 cos kpx+qycos2π/Dx-y tan θ±2A02A1 cos kpx-qy±4A0A12 cos kpx-qycos2π/Dx-y tan θ+2A0A12 cos 2kpx+4A13 cos2π/Dx-y tan θcos 2kpx.
I  A0A02+2A12±2A0A12cos 2π1d-1Dx+1Dtan θ+qλy+cos 2π1d-1Dx+1Dtan θ-qλy.
y1=n1Dtan θ+qλ-1d-1Dx1Dtan θ+qλ,
y2=n1Dtan θ-qλ-1d-1Dx1Dtan θ-qλ,
2π1d-1Dx+1Dtan θ+qλy=2πn,
2π1d-1Dx+1Dtan θ-qλy=2πn,
y1=n+141Dtan θ+qλ-1d-1Dx1Dtan θ+qλ,
y2=n-141Dtan θ-qλ-1d-1Dx1Dtan θ-qλ.
φ1=tan-1cos θ-1sin θ+λ/2f tan β,
φ2=tan-1cos θ-1sin θ-λ/2f tan β,
φ0=tan-1cos θ-1sin θ=-θ2.
p1=1fsin θ+λ/2f tan β2+cos θ-121/2,
p2=1fsin θ-λ/2f tan β2+cos θ-121/2.
I  A0A02+2A12±4A0A12cos 2π1d-1Dx+1Dtan θycos 2π qλ y=A0A02+2A12±4A0A12cos2πd1-cos θx+sin θycosπλd2 y tan β.

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