Abstract

Diffraction of an obstructed focused Gaussian laser beam has been treated theoretically using the Huygens–Fresnel diffraction integral and was found to be in good agreement with experimental measurements. The obstruction is a vertically oriented opaque cylinder treated as a flat hard aperture. Measurements and calculations are compared for the diffracted irradiance profile and the beam centroid as a function of cylinder diameter and lateral and longitudinal placement along the optic axis. The cylinders used were gauge pins and/or wires with diameters from 0.5 to 100 mil.

© 2009 Optical Society of America

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References

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  1. W. D. St. John, “Cylinder gauge measurement using a position sensitive detector,” Appl. Opt. 46, 7469-7474 (2007).
    [CrossRef] [PubMed]
  2. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  3. J. C. Martinez-Anton, I. Serroukh, and E. Bernabeu, “On Babinet's principle and diffraction-interferometric technique to determine the diameter of cylindrical wires,” Metrologia 38, 125-134 (2001).
    [CrossRef]
  4. S. Ganci, “Fraunhofer diffraction by a thin wire and Babinet's principle,” Am. J. Phys. 73, 83-84 (2005).
    [CrossRef]
  5. N. M. Gagina and B. S. Rinkevicius, “Effects of Gaussian beam size on diffraction measurement errors,” Meas. Tech. (USSR) 40, 1069-1073 (1997).
    [CrossRef]
  6. M. Glass, “Diffraction of a Gaussian beam around a strip mask,” Appl. Opt. 37, 2550-2562 (1998).
    [CrossRef]
  7. H. T. Yura and T. S. Rose, “Gaussian beam transfer through hard-aperture optics,” Appl. Opt. 34, 6826-6828 (1995).
    [CrossRef] [PubMed]
  8. 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]
  9. 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. A 12, 398-403 (1995).
    [CrossRef]
  10. Vermont Gage, Swanton, Vermont, USA, www.vermontgage.com.
  11. Wire Tronic, Inc., Volcano, California, USA, www.wiretron.com.
  12. M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley,1986), Chap. 7.
  13. J. T. Verdeyen, Laser Electronics (Prentice-Hall,1981), Chap. 3.
  14. Mathematica Software, Wolfram Research, Inc., Champaign, Illinois, www.wolfram.com/.

2007 (1)

2005 (1)

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

2001 (1)

J. C. Martinez-Anton, I. Serroukh, and E. Bernabeu, “On Babinet's principle and diffraction-interferometric technique to determine the diameter of cylindrical wires,” Metrologia 38, 125-134 (2001).
[CrossRef]

1998 (1)

1997 (1)

N. M. Gagina and B. S. Rinkevicius, “Effects of Gaussian beam size on diffraction measurement errors,” Meas. Tech. (USSR) 40, 1069-1073 (1997).
[CrossRef]

1995 (2)

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]

Bernabeu, E.

J. C. Martinez-Anton, I. Serroukh, and E. Bernabeu, “On Babinet's principle and diffraction-interferometric technique to determine the diameter of cylindrical wires,” Metrologia 38, 125-134 (2001).
[CrossRef]

Dandliker, R.

Furtak, T. E.

M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley,1986), Chap. 7.

Gagina, N. M.

N. M. Gagina and B. S. Rinkevicius, “Effects of Gaussian beam size on diffraction measurement errors,” Meas. Tech. (USSR) 40, 1069-1073 (1997).
[CrossRef]

Ganci, S.

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

Glass, M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

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]

Klein, M. V.

M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley,1986), Chap. 7.

Krattiger, B.

Martinez-Anton, J. C.

J. C. Martinez-Anton, I. Serroukh, and E. Bernabeu, “On Babinet's principle and diffraction-interferometric technique to determine the diameter of cylindrical wires,” Metrologia 38, 125-134 (2001).
[CrossRef]

Rinkevicius, B. S.

N. M. Gagina and B. S. Rinkevicius, “Effects of Gaussian beam size on diffraction measurement errors,” Meas. Tech. (USSR) 40, 1069-1073 (1997).
[CrossRef]

Rose, T. S.

Serroukh, I.

J. C. Martinez-Anton, I. Serroukh, and E. Bernabeu, “On Babinet's principle and diffraction-interferometric technique to determine the diameter of cylindrical wires,” Metrologia 38, 125-134 (2001).
[CrossRef]

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.

St. John, W. D.

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics (Prentice-Hall,1981), Chap. 3.

Yura, H. T.

Zimmermann, E.

Am. J. Phys. (2)

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

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]

Appl. Opt. (3)

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

Meas. Tech. (USSR) (1)

N. M. Gagina and B. S. Rinkevicius, “Effects of Gaussian beam size on diffraction measurement errors,” Meas. Tech. (USSR) 40, 1069-1073 (1997).
[CrossRef]

Metrologia (1)

J. C. Martinez-Anton, I. Serroukh, and E. Bernabeu, “On Babinet's principle and diffraction-interferometric technique to determine the diameter of cylindrical wires,” Metrologia 38, 125-134 (2001).
[CrossRef]

Other (6)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Vermont Gage, Swanton, Vermont, USA, www.vermontgage.com.

Wire Tronic, Inc., Volcano, California, USA, www.wiretron.com.

M. V. Klein and T. E. Furtak, Optics, 2nd ed. (Wiley,1986), Chap. 7.

J. T. Verdeyen, Laser Electronics (Prentice-Hall,1981), Chap. 3.

Mathematica Software, Wolfram Research, Inc., Champaign, Illinois, www.wolfram.com/.

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Figures (4)

Fig. 1
Fig. 1

Experimental setup. The cylinders are represented by •. Motion in the X / x and Z / z direction is referred to as lateral and longitudinal, respectively. Not shown are the PSD amplifier and stepper motor controllers both interfaced to a common PC.

Fig. 2
Fig. 2

Shift in the beam centroid as a cylinder is swept laterally across the beam. Cases A, B, and C refer to different cylinder diameters at different z 1 planes; see Fig. 3. The horizontal axes are scaled by the local beam spot size.

Fig. 3
Fig. 3

Maximum shift in the beam centroid as a function of cylinder diameter and longitudinal position ( z 1 ). The special cases A, B, and C utilized in Figs. 2, 4 are circled.

Fig. 4
Fig. 4

Diffraction patterns for cases A, B, and C for the cylinder positioned at either the center of the beam ( x p = 0 , left column) or where the shift in the centroid is maximum ( x p = x p , max , right column). Solid curves, theory; dashed curves, experimental.

Equations (16)

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E ( x , y , z 2 ) = c o x y E ( x , y , z 1 ) e j k | r r | | r r | d x d y ,
| r r | Δ z ( x x + y y ) Δ z + x 2 + y 2 2 Δ z
E ( x , 0 , z 2 ) = c o e j k Δ z Δ z x y E ( x , y , z 1 ) exp [ j k 2 Δ z ( x 2 + y 2 2 x x ) ] d y d x .
E ( x , y , z ) = E o exp [ j ϕ ( z ) ] exp ( j k r 2 2 q ( z ) ) ,
1 q ( z ) = 1 R ( z ) j λ π w 2 ( z ) ,
w ( z ) = w o 1 + ( z / z o ) 2 ,
1 q 1 = C + D / q o A + B / q o ,
( A B C D ) = ( 1 z 1 0 1 ) ( 1 0 1 / f 1 ) ( 1 z lens 0 1 ) .
E ( x , 0 , z 2 ) = c ˜ 1 x y exp [ j k ( x 2 + y 2 ) 2 q 1 ] exp [ j k 2 Δ z ( x 2 + y 2 2 x x ) ] d y d x ,
E ( x , 0 , z 2 ) = c ˜ 2 x exp ( j k x 2 2 q 1 ) exp [ j k 2 Δ z ( x 2 2 x x ) ] d x ,
c ˜ 2 = c ˜ 1 2 π j k ( 1 q 1 1 Δ z ) .
E ( x , 0 , z 2 ) = c ˜ 2 [ x p d / 2 [ ] d x + x p + d / 2 [ ] d x ] ,
w ( z a ) = w o 1 + ( z a / z o ) 2 , w ( z b ) = w o 1 + ( z b / z o ) 2 , z b z a = s .
I ( x , z 2 ) = c 3 | E ( x , 0 , z 2 ) | 2 ,
I norm = I ( x , z 2 ) I peak ( x , z 2 ) | x p = x p , max ,
x c = x I ( x , z 2 ) d x I ( x , z 2 ) d x .

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