Abstract

A digital moiré fringe-scanning method for centering a circular fringe image is proposed. The image of a nondiffracting beam, whose cross section is a circular fringe, is first downloaded onto a computer. The image is then superposed with a digital circular grating, whose center is close to the center of the image, to generate circular moiré fringes. Changing the phase of a digital grating can cause moiré fringe scanning. The global center of the image can be calculated by use of sets of the scanned picture. Because all the image data are used for the calculation, the effect of random noise on centering is greatly reduced and the center position resolution can reach the order of a subelement of a CCD. The measurement of spatial straightness is discussed.

© 2004 Optical Society of America

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References

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  1. K. C. Fan, Y. Zhao, “A laser straightness measurement system using optical fiber and modulation techniques,” Int. J. Mach. Tools Manuf. 40, 2073–2081 (2000).
    [CrossRef]
  2. S.-T. Lin, “A laser interferometer for measuring straightness,” Opt. Laser Technol. 33, 195–199 (2001).
    [CrossRef]
  3. J. Durnin, J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  4. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  5. Z. Bin, L. Zhu, “Diffraction property of an axicon in oblique illumination,” Appl. Opt. 37, 2563–2568 (1998).
    [CrossRef]
  6. D. Malacara, Optical Shop Testing (Wiley, New York, 1992), Chap. 13, p. 409.

2001

S.-T. Lin, “A laser interferometer for measuring straightness,” Opt. Laser Technol. 33, 195–199 (2001).
[CrossRef]

2000

K. C. Fan, Y. Zhao, “A laser straightness measurement system using optical fiber and modulation techniques,” Int. J. Mach. Tools Manuf. 40, 2073–2081 (2000).
[CrossRef]

1998

1987

J. Durnin, J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

Bin, Z.

Durnin, J.

J. Durnin, J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Fan, K. C.

K. C. Fan, Y. Zhao, “A laser straightness measurement system using optical fiber and modulation techniques,” Int. J. Mach. Tools Manuf. 40, 2073–2081 (2000).
[CrossRef]

Lin, S.-T.

S.-T. Lin, “A laser interferometer for measuring straightness,” Opt. Laser Technol. 33, 195–199 (2001).
[CrossRef]

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, New York, 1992), Chap. 13, p. 409.

Miceli, J.

J. Durnin, J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Zhao, Y.

K. C. Fan, Y. Zhao, “A laser straightness measurement system using optical fiber and modulation techniques,” Int. J. Mach. Tools Manuf. 40, 2073–2081 (2000).
[CrossRef]

Zhu, L.

Appl. Opt.

Int. J. Mach. Tools Manuf.

K. C. Fan, Y. Zhao, “A laser straightness measurement system using optical fiber and modulation techniques,” Int. J. Mach. Tools Manuf. 40, 2073–2081 (2000).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Laser Technol.

S.-T. Lin, “A laser interferometer for measuring straightness,” Opt. Laser Technol. 33, 195–199 (2001).
[CrossRef]

Phys. Rev. Lett.

J. Durnin, J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Other

D. Malacara, Optical Shop Testing (Wiley, New York, 1992), Chap. 13, p. 409.

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Figures (7)

Fig. 1
Fig. 1

Experimental photograph of a nondiffracting beam. CS represents the central spot.

Fig. 2
Fig. 2

Nondiffracting beam. CS represents the central spot.

Fig. 3
Fig. 3

Illustration of the tracks at the center of each ring for the nondiffracting beam.

Fig. 4
Fig. 4

Calculation procedure of the common center of ring patterns by digital moiré fringe scanning: (a) original photograph of nondiffracting beam f(x, y), (b) f(x, y) times g(x, y), (c) M (x, y), (d) η0(x, y), (e) three-dimensional image of phase cone surface η(x, y), (f) common center, where the cross represents [x m , y m ].

Fig. 5
Fig. 5

Figure of moiré rings with center deviations of (a) Δx = 3 and (b) Δx = 6. The black cross represents the moiré fringe center [x m , y m ].

Fig. 6
Fig. 6

Experimental arrangement for the straightness measurement by use of a nondiffracting beam. BEP, beam expansion process.

Fig. 7
Fig. 7

Experimental results of the spatial straightness measurement: (a) *, x-direction data; o, y-direction data; and (b) space straightness curve of the slide.

Tables (1)

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Table 1 Numerical Simulation of Noise Effectsa

Equations (25)

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fx, y=Ax, ysinωfR+ξ+Nx, y,
R=x-x02+y-y021/2;
gx, y=sinωgR+φ,
f×g=12 Ax, ycosωg-ωfR+φ-ξ-12 Ax, ycosωg+ωfR+φ-ξ+Nx, ysinωgR+φ,
Mx, y=12 Ax, ycosωg-ωfR+φ-ξ,
η0x, y=ωg-ωfR-ξ=tan-102π M sin φdφ02π M cos φdφ,
ηx, y=η0x, y+2πk,
Gx, y=sinωgR+φ,
f×G=12 Ax, ycosωgR-ωfR+φ-ξ-12 Ax, ycosωgR+ωfR+φ-ξ+Nx, ysinωgR+φ,
ηx, y=ωgR-ωfR=ωgx-xg2+y-yg21/2-ωfx-x02+y-y021/2.
ηx, y=ωgx-Δx2+y21/2-ωfx2+y21/2,
ωg2-ωf2ρ2-2Δxωg2cos θ+Cωfρ+ωg2Δx2-C2=0,
ρ=Cωf+Δxωg2cos θ+Cωf+Δxωg2cos θ2+C2-ωg2Δx2ωg2-ωf21/2ωg2-ωf2.
xm-x0=ρθ=0-ρθ=π/2=ωgΔxωg-ωf,ym-y0=0,
x0=xm-xm-xgωgωf,y0=ym-ym-ygωgωf,
x0-xg2+y0-yg2=0,
gx, y=sinωgR+φ,
px, y=fx, ygx, y.
Mx, y=filterpx, y,
η0x, y=ωg-ωfR-ξ=tan-1j=1n Mjx, ysin2πj/nj=1n Mjx, ycos2πj/n.
ηx, y=unwrapη0x, y=η0x, y+2kπ.
cx, y=Ax-xc2+y-yc21/2+c0
εj=x,ycx, y-ηx, y2min,
xm=xc, ym=yc.
xg=x0=xm-xm-xgωgωf,yg=y0=ym-ym-ygωgωf.

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