Abstract

A position sensitive detector (PSD) has been used to determine the diameter of cylindrical pins based on the shift in a laser beam's centroid. The centroid of the light beam is defined here as the weighted average of position by the local intensity. A shift can be observed in the centroid of an otherwise axially symmetric light beam, which is partially obstructed. Additionally, the maximum shift in the centroid is a unique function of the obstructing cylinder diameter. Thus to determine the cylinder diameter, one only needs to detect this maximum shift as the cylinder is swept across the beam.

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References

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  1. J. Fraden, Handbook of Modern Sensors: Physics, Designs, and Applications, 3rd ed. (AIP, 2004).
  2. H. Wang, "Application of position-sensitive detectors to dynamic inspection of diameter of metal wires," Meas. Sci. Technol. 5, 942-946 (1994).
    [CrossRef]
  3. H. Wang and R. Valdivia-Hernandez, "Laser scanner and diffraction pattern detection: a novel concept for dynamic gauging of fine wires," Meas. Sci. Technol. 6, 452-457 (1995).
    [CrossRef]
  4. Cortex Technical R&D Ltd., H-1111 Budapest, Kende u. 13-17, Hungary, www.cortex.hu/.
  5. A. E. Seigman, Lasers (University Science Books, 1986), Chap. 17.
  6. Black oxide coated gauge pins manufactured by Vermont Gage in Swanton, Vermont, USA, www.vermontgage.com.
  7. Technical note, "Characteristics and use of PSD,"www.hamamatsu.com/.
  8. M. Glass, "Diffraction of a Gaussian beam around a strip mask," Appl. Opt. 37, 2550-2562 (1998).
    [CrossRef]
  9. R. G. Greenler, J. W. Hable, and P. O. Slane, "Diffraction around a fine wire: How good is the single slit approximation?"Am. J. Phys. 58, 330-331 (1990).
    [CrossRef]
  10. S. Ganci, "Fraunhofer diffraction by a thin wire and Babinet's principle," Am. J. Phys. 73, 83-84 (2005).
    [CrossRef]
  11. E. Zimmermann, R. Dandliker, N. Souli, and B. Krattiger, "Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach," J. Opt. Soc. Am. 12, 398-403 (1995).
    [CrossRef]
  12. H. T. Yura and T. S. Rose, "Gaussian beam transfer through hard-aperture optics," Appl. Opt. 34, 6826-6828 (1995).
    [CrossRef] [PubMed]
  13. E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002).
  14. M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley, 1986).
  15. MATHEMATICA software by Wolfram Research, Inc., www.wolfram.com/.

2005 (1)

S. Ganci, "Fraunhofer diffraction by a thin wire and Babinet's principle," Am. J. Phys. 73, 83-84 (2005).
[CrossRef]

1998 (1)

1995 (3)

H. T. Yura and T. S. Rose, "Gaussian beam transfer through hard-aperture optics," Appl. Opt. 34, 6826-6828 (1995).
[CrossRef] [PubMed]

E. Zimmermann, R. Dandliker, N. Souli, and B. Krattiger, "Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach," J. Opt. Soc. Am. 12, 398-403 (1995).
[CrossRef]

H. Wang and R. Valdivia-Hernandez, "Laser scanner and diffraction pattern detection: a novel concept for dynamic gauging of fine wires," Meas. Sci. Technol. 6, 452-457 (1995).
[CrossRef]

1994 (1)

H. Wang, "Application of position-sensitive detectors to dynamic inspection of diameter of metal wires," Meas. Sci. Technol. 5, 942-946 (1994).
[CrossRef]

1990 (1)

R. G. Greenler, J. W. Hable, and P. O. Slane, "Diffraction around a fine wire: How good is the single slit approximation?"Am. J. Phys. 58, 330-331 (1990).
[CrossRef]

Dandliker, R.

E. Zimmermann, R. Dandliker, N. Souli, and B. Krattiger, "Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach," J. Opt. Soc. Am. 12, 398-403 (1995).
[CrossRef]

Fraden, J.

J. Fraden, Handbook of Modern Sensors: Physics, Designs, and Applications, 3rd ed. (AIP, 2004).

Furtak, T. E.

M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley, 1986).

Ganci, S.

S. Ganci, "Fraunhofer diffraction by a thin wire and Babinet's principle," Am. J. Phys. 73, 83-84 (2005).
[CrossRef]

Glass, M.

Greenler, R. G.

R. G. Greenler, J. W. Hable, and P. O. Slane, "Diffraction around a fine wire: How good is the single slit approximation?"Am. J. Phys. 58, 330-331 (1990).
[CrossRef]

Hable, J. W.

R. G. Greenler, J. W. Hable, and P. O. Slane, "Diffraction around a fine wire: How good is the single slit approximation?"Am. J. Phys. 58, 330-331 (1990).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002).

Klein, M. V.

M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley, 1986).

Krattiger, B.

E. Zimmermann, R. Dandliker, N. Souli, and B. Krattiger, "Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach," J. Opt. Soc. Am. 12, 398-403 (1995).
[CrossRef]

Rose, T. S.

Seigman, A. E.

A. E. Seigman, Lasers (University Science Books, 1986), Chap. 17.

Slane, P. O.

R. G. Greenler, J. W. Hable, and P. O. Slane, "Diffraction around a fine wire: How good is the single slit approximation?"Am. J. Phys. 58, 330-331 (1990).
[CrossRef]

Souli, N.

E. Zimmermann, R. Dandliker, N. Souli, and B. Krattiger, "Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach," J. Opt. Soc. Am. 12, 398-403 (1995).
[CrossRef]

Valdivia-Hernandez, R.

H. Wang and R. Valdivia-Hernandez, "Laser scanner and diffraction pattern detection: a novel concept for dynamic gauging of fine wires," Meas. Sci. Technol. 6, 452-457 (1995).
[CrossRef]

Wang, H.

H. Wang and R. Valdivia-Hernandez, "Laser scanner and diffraction pattern detection: a novel concept for dynamic gauging of fine wires," Meas. Sci. Technol. 6, 452-457 (1995).
[CrossRef]

H. Wang, "Application of position-sensitive detectors to dynamic inspection of diameter of metal wires," Meas. Sci. Technol. 5, 942-946 (1994).
[CrossRef]

Yura, H. T.

Zimmermann, E.

E. Zimmermann, R. Dandliker, N. Souli, and B. Krattiger, "Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach," J. Opt. Soc. Am. 12, 398-403 (1995).
[CrossRef]

Am. J. Phys. (2)

R. G. Greenler, J. W. Hable, and P. O. Slane, "Diffraction around a fine wire: How good is the single slit approximation?"Am. J. Phys. 58, 330-331 (1990).
[CrossRef]

S. Ganci, "Fraunhofer diffraction by a thin wire and Babinet's principle," Am. J. Phys. 73, 83-84 (2005).
[CrossRef]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

E. Zimmermann, R. Dandliker, N. Souli, and B. Krattiger, "Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach," J. Opt. Soc. Am. 12, 398-403 (1995).
[CrossRef]

Meas. Sci. Technol. (2)

H. Wang, "Application of position-sensitive detectors to dynamic inspection of diameter of metal wires," Meas. Sci. Technol. 5, 942-946 (1994).
[CrossRef]

H. Wang and R. Valdivia-Hernandez, "Laser scanner and diffraction pattern detection: a novel concept for dynamic gauging of fine wires," Meas. Sci. Technol. 6, 452-457 (1995).
[CrossRef]

Other (8)

Cortex Technical R&D Ltd., H-1111 Budapest, Kende u. 13-17, Hungary, www.cortex.hu/.

A. E. Seigman, Lasers (University Science Books, 1986), Chap. 17.

Black oxide coated gauge pins manufactured by Vermont Gage in Swanton, Vermont, USA, www.vermontgage.com.

Technical note, "Characteristics and use of PSD,"www.hamamatsu.com/.

E. Hecht, Optics, 4th ed. (Addison-Wesley, 2002).

M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley, 1986).

MATHEMATICA software by Wolfram Research, Inc., www.wolfram.com/.

J. Fraden, Handbook of Modern Sensors: Physics, Designs, and Applications, 3rd ed. (AIP, 2004).

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Figures (6)

Fig. 1
Fig. 1

Experimental setup for measuring the shift in a laser beam's centroid as a gauge pin is swept across the beam.

Fig. 2
Fig. 2

(Color online) Shift in the beam centroid as a pin passes across the beam. The centroid position ( x c ) is plotted along the vertical axis and the pin position ( x p ) along the horizontal axis. d and d f i t are in units of millimeters. (a) d = 0.254 , d fit = 0.254 , Δ d / d = 0 % , (b) d = 0.406 , d fit = 0.407 , Δ d / d = 0.25 % , (c) d = 0.559 , d fit = 0.567 , Δ d / d = 1.4 % , (d) d = 0.711 , d fit = 0.718 , Δ d / d = 0.98 % , (e) d = 0.864 , d fit = 0.870 , Δ d / d = 0.69 % , (f) d = 1.016 , d fit = 1.027 , Δ d / d = 1.1 % , (g) d = 1.168 , d fit = 1.164 , Δ d / d = 0.34 % , (h) d = 1.321 , d fit = 1.315 , Δ d / d = 0.45 % .

Fig. 3
Fig. 3

Theoretical intensity profiles of a truncated Gaussian beam (dashed curves) and diffracted Gaussian beam (solid curves). Each plot is for the same pin size d = 0.711   mm [see Fig. 2(d)]. The four plots correspond to four different pin positions. The pin position and calculated centroid position (both in millimeters) are (a) x p = 0 , x c , w / diff = 0 , x c , w / o   diff = 0 , Δ x c / x c = 0 % , (b) x p = d / 3 , x c ,w / diff = 0.200 , x c ,w / o   diff = 0.200 , Δ x c / x c = 0 % , (c) x p = d / 2 , x c , w / diff = 0.243 , x c , w / o   diff = 0.231 , Δ x c / x c = 4.9 % , (d) x p = d , x c , w / diff = 0.160 , x c , w / o   diff = 0.156 , Δ x c / x c = 2.5 % .

Fig. 4
Fig. 4

(Color online) Theoretical shift in the beam centroid as a pin passes across the beam with and without diffraction considered: solid curves, without diffraction; dots, with diffraction. (a) d = 0.406   mm , (b) d = 0.711   mm , (c) d = 1.016   mm , (d) d = 1.321   mm .

Fig. 5
Fig. 5

(Color online) Known (nominal) pin diameter, d, and the theoretical predicted value, d fit , with and without diffraction as a function of the maximum shift in the centroid measured, max ( x c ,expt ) . The average percent difference between d and d fit without diffraction is 1.59 % and with diffraction is 1.73 % .

Fig. 6
Fig. 6

(Color online) d fit with and without diffraction plotted against the known nominal diameter, d. The 45° line aids the comparison.

Equations (9)

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x c , expt = L 2 ( i L i R i L + i R ) ,
I ( x ) = I o e 2 x 2 / w 2 .
x c ,thry = x I ( x ) d x I ( x ) d x .
x c ,thry = x p d / 2 x e 2 x 2 / w 2 d x + x p + d / 2 x e 2 x 2 / w 2 d x x p d / 2 e 2 x 2 / w 2 d x + x p + d / 2 e 2 x 2 / w 2 d x = w 2 π exp ( d 2 + 4 x p 2 w 2 ) [ exp ( ( d 2 x p ) 2 2 w 2 ) exp ( ( d + 2 x p ) 2 2 w 2 ) ] 2 Erf ( d 2 x p w 2 ) Erf ( d + 2 x p w 2 ) ,
E ( x , y , z ) = C E ( x , y , 0 ) e j k | r r | | r r | d x d y ,
| r r | z ( 1 x x x 2 2 z 2 y y y 2 2 z 2 ) ,
E ( x , 0 , z ) = C e j k z z [ e x 2 / w 2 exp [ j 2 π λ z ( x 2 2 x x ) ] d x x p d / 2 x p + d / 2 e x 2 / w 2 exp [ j 2 π λ z ( x 2 2 x x ) ] d x ] .
E norm ( 0 , 0 , z ) = C e j k z z [ e x 2 / w 2 exp [ j 2 π λ z ( x 2 2 ) ] d x ] .
I norm ( x , z ) = | E ( x , 0 , z ) E norm ( 0 , 0 , z ) | 2 .

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