Abstract

A theoretical model of far-field interference from a sphere has been established, and its applications have been investigated. When two coherent parallel laser beams shine on a smooth sphere surface from opposite directions, the reflected lights form interference fringes at far field. The fringes have hyperbolic shapes and are not uniformly distributed. This paper derives a method for calculating the path-length difference between two parallel reflected lights, analyzes the interference field, and discusses reasons that cause the fringe density variations. A formula for calculating the highest orders of interference fringes is also provided. A method for using a spectrometer, CCD camera, and computer to measure the sphere diameter is demonstrated. The results agree with those from an Abbe comparator. The theory and methods are also suitable for measuring diameters of smooth cylinders.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. A. Akhmanov and S. Y. Nikitin, Physical Optics (Oxford University, 1997), pp. 183–293.
  2. M. Bass, Handbook of Optics, Vol. 1 (McGraw-Hill, 2010), pp. 1.1–3.38.
  3. K. J. Ggasvik, Optical Metrology (Wiley, 1987), pp. 20–33.
  4. L. M. Sanchez-Brea, “Diameter estimation of cylinders by the rigorous diffraction model,” J. Opt. Soc. Am. A 22, 1402–1407 (2005).
    [CrossRef]
  5. W. Tang, Y. Zhou, and J. Zhang, “Improvement on theoretical model for thin-wire and slot measurement by optical diffraction,” Meas. Sci. Technol. 10, N119–N123 (1999).
    [CrossRef]
  6. M. Glass, “Diffraction of a Gaussian beam around a strip mask,” Appl. Opt. 37, 2550–2562 (1998).
    [CrossRef]
  7. M. Glass, “Fresnel diffraction from curved fiber snippets with application to fiber diameter measurement,” Appl. Opt. 35, 1605–1616 (1996).
    [CrossRef]
  8. R. Ditteon, Modern Geometrical Optics (Wiley-Interscience, 1997), pp. 82–124.

2005 (1)

1999 (1)

W. Tang, Y. Zhou, and J. Zhang, “Improvement on theoretical model for thin-wire and slot measurement by optical diffraction,” Meas. Sci. Technol. 10, N119–N123 (1999).
[CrossRef]

1998 (1)

1996 (1)

Akhmanov, S. A.

S. A. Akhmanov and S. Y. Nikitin, Physical Optics (Oxford University, 1997), pp. 183–293.

Bass, M.

M. Bass, Handbook of Optics, Vol. 1 (McGraw-Hill, 2010), pp. 1.1–3.38.

Ditteon, R.

R. Ditteon, Modern Geometrical Optics (Wiley-Interscience, 1997), pp. 82–124.

Ggasvik, K. J.

K. J. Ggasvik, Optical Metrology (Wiley, 1987), pp. 20–33.

Glass, M.

Nikitin, S. Y.

S. A. Akhmanov and S. Y. Nikitin, Physical Optics (Oxford University, 1997), pp. 183–293.

Sanchez-Brea, L. M.

Tang, W.

W. Tang, Y. Zhou, and J. Zhang, “Improvement on theoretical model for thin-wire and slot measurement by optical diffraction,” Meas. Sci. Technol. 10, N119–N123 (1999).
[CrossRef]

Zhang, J.

W. Tang, Y. Zhou, and J. Zhang, “Improvement on theoretical model for thin-wire and slot measurement by optical diffraction,” Meas. Sci. Technol. 10, N119–N123 (1999).
[CrossRef]

Zhou, Y.

W. Tang, Y. Zhou, and J. Zhang, “Improvement on theoretical model for thin-wire and slot measurement by optical diffraction,” Meas. Sci. Technol. 10, N119–N123 (1999).
[CrossRef]

Appl. Opt. (2)

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

Meas. Sci. Technol. (1)

W. Tang, Y. Zhou, and J. Zhang, “Improvement on theoretical model for thin-wire and slot measurement by optical diffraction,” Meas. Sci. Technol. 10, N119–N123 (1999).
[CrossRef]

Other (4)

R. Ditteon, Modern Geometrical Optics (Wiley-Interscience, 1997), pp. 82–124.

S. A. Akhmanov and S. Y. Nikitin, Physical Optics (Oxford University, 1997), pp. 183–293.

M. Bass, Handbook of Optics, Vol. 1 (McGraw-Hill, 2010), pp. 1.1–3.38.

K. J. Ggasvik, Optical Metrology (Wiley, 1987), pp. 20–33.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

Sphere far-field interference and optical path difference.

Fig. 2.
Fig. 2.

Relation among all measurement angles.

Fig. 3.
Fig. 3.

Measurement device.

Fig. 4.
Fig. 4.

Interference pattern in θ0 direction. This is half of the taken interference patterns.

Equations (13)

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

ΔOAAΔOBB,1=2=3=4=45°,5=6=45°+α,7=8=45°α,BFAP,ABF=α.
δ=n(DA+EB+AF),in the air,n1.
δ=DA+EB+AF
δ=22rsinα=22rsinθ2.
Δδ=22rsinθ2222rsinθ12=Nλ.
d=2r=Nλ2(sinθ22sinθ12).
θ0=215°310195°1330=20.2917°.
βs=arctanSFf=arctan(1373593)×0.0034848.5=3.2033°,
βL=arctanFLf=arctan(18641373)×0.0034848.5=2.0177°,
θ1=θ0βS=17.0884°,
θ2=θ0+βL=22.3094°.
N=NSF+NFL=70.
d=2r=0.698mm.

Metrics