Abstract

The effects of an arbitrary small inclination between the two crossed gratings on moiré fringes in Talbot interferometry are discussed. The small inclination is formed by the rotation by a small angle γ of the beam splitter’s grating about the axis that is on the plane of the grating and has an arbitrary angle δ with respect to the lines of the grating. The results indicate that the small inclination has a great influence on measurements for which Talbot interferometry is applied, such as beam collimation and measurement of the focal length of a lens. The theoretical analyses are proved by experimental results. Some methods for judging the size of a small inclination are also proposed.

© 2001 Optical Society of America

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References

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  1. Q. Liu, R. Ohba, “Effects of unparallel grating planes in Talbot interferometry,” Appl. Opt. 38, 4111–4116 (1999).
    [CrossRef]
  2. Q. Liu, R. Ohba, S. Kakuma, “Effects of unparallel grating planes in Talbot interferometry. II,” Appl. Opt. 39, 2084–2090 (2000).
    [CrossRef]
  3. Y. S. Chang, “Fringe formation with a cross-grating interferometer,” Appl. Opt. 25, 4185–4191 (1986).
    [CrossRef]
  4. P. Szwaykowski, “Producing binary diffraction gratings in the double-diffraction system,” Opt. Laser Technol. 17, 255–261 (1985).
    [CrossRef]
  5. H. O. Bartelt, Y. Li, “Lau interferometry with cross gratings,” Opt. Commun. 48, 1–6 (1983).
    [CrossRef]
  6. K. Patorski, “Talbot interferometry with increased shear,” Appl. Opt. 24, 4448–4453 (1985).
    [CrossRef] [PubMed]
  7. S. Yokozeki, T. Suzuki, “Shearing interferometer using the grating as the beam splitter,” Appl. Opt. 10, 1575–1580 (1971).
    [CrossRef] [PubMed]

2000 (1)

1999 (1)

1986 (1)

1985 (2)

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

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

1983 (1)

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

1971 (1)

Bartelt, H. O.

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

Chang, Y. S.

Kakuma, S.

Li, Y.

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

Liu, Q.

Ohba, R.

Patorski, K.

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.

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Figures (5)

Fig. 1
Fig. 1

Configuration of the Talbot interferometry under investigation: G1, crossed beam-splitter grating; G2, crossed detector grating; P, monochromatic plane wave with the x 0y 0 (xy) plane normal to the incident beam and the x 1y 1 plane not normal to the incident beam.

Fig. 2
Fig. 2

Configuration of G1 under investigation: R, rotation axis of G1; δ, arbitrary angle between rotation axis R and the lines of the grating.

Fig. 3
Fig. 3

Schematic of the tilt angles of the two sets of moiré fringes: M1 (M2) moiré fringes produced by the distribution only in the x (the y) direction of the two crossed gratings.

Fig. 4
Fig. 4

T, theoretical, and E, experimental, curves for ϕ versus γ for δ = 0°, 30°, with θ = ±5°: (a) ϕ x versus γ, (b) ϕ y versus γ.

Fig. 5
Fig. 5

Curves for ϕ versus δ for the values of γ shown and θ = ±5°: (a) ϕ x versus δ, (b) ϕ y versus δ.

Equations (9)

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Tx1, y1=A0+2A1 cos2πx1/d×B0+B1 cos2πy1/d,
Tx0, y0=A0+2A1 cos2πx0/dx0×B0+B1 cos2πy0/dy0,
dx0=dR cos γ2+dR21/2=dsin2 δ cos2 γ+sin2 δ1/2,
dy0=dR cos γ2+dR21/2=dcos2 δ cos2 γ+sin2 δ1/2,
Ix, y, zT=Ix, zTIy, zT=A0+2A1 cos2πx/dx02×B0+B1 cos2πy/dy02.
I=IxIy=A0+2A1 cos2π/dx cos θ-y sin θB0+2B1 cos2π/dx sin θ-y cos θ.
IxMx, y  A03+4A0A12 cos 2πcos θd-1d0x[-1d y sin θ.
tan ϕx=cot θ-1sin θsin2 δ cos2 γ+cos2 δ1/2.
tan ϕy=cot θ-1sin θcos2 γ cos2 δ+sin2 δ1/2,

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