Abstract

We revisited the well known Khosrofian and Garetz inversion algorithm [Appl. Opt. 22, 3406–3410 (1983)APOPAI0003-6935] that was developed to analyze data obtained by the application of the traveling knife-edge technique. We have analyzed the approximated fitting function that was used for adjusting their experimental data and have found that it is not optimized to work with a full range of the experimentally-measured data. We have numerically calculated a new set of coefficients, which makes the approximated function suitable for a full experimental range, considerably improving the accuracy of the measurement of a radius of a focused Gaussian laser beam.

© 2009 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  3. M. L. Baesso, J. Shen, and R. D. Snook, ”Mode-mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths, J. Appl. Phys. 75, 3732-3737 (1994).
    [CrossRef]
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    [CrossRef] [PubMed]
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  7. J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. de la Clavière, E. A. Franke, and J. M. Franke, “Technique for fast measurement of Gaussian laser beam parameters,” Appl. Opt. 10, 2775-2776 (1971).
    [PubMed]
  8. D. R. Skinner and R. E. Whitcher, “Measurement of the radius of a high-power laser beam near the focus of a lens,” J. Phys. E 5, 237-238 (1972).
    [CrossRef]
  9. J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data,” Appl. Opt. 22, 3406-3410 (1983).
    [CrossRef] [PubMed]
  10. D. Wright, P. Greve, J. Fleischer, and L. Austin, “Laser beam width, divergence and beam propagation factor: an international standardization approach,” Opt. Quantum Electron. 24, S993-S1000 (1992).
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    [CrossRef] [PubMed]
  13. Z. A. Talib and W. M. M. Yunus, “Measuring Gaussian laser beam diameter using piezoelectric detection,” Meas. Sci. Technol. 4, 22-25 (1993).
    [CrossRef]
  14. L. Bachmann, D. M. Zezell, and E. P. Maldonado, “Determination of beam width and quality for pulsed lasers using the knife-edge method,” Instrum. Sci. Technol. 31, 47-52 (2003).
    [CrossRef]
  15. P. Van Halen, “Accurate analytical approximations for error function and its integral,” Electron. Lett. 25, 561-563(1989).
    [CrossRef]

2003 (1)

L. Bachmann, D. M. Zezell, and E. P. Maldonado, “Determination of beam width and quality for pulsed lasers using the knife-edge method,” Instrum. Sci. Technol. 31, 47-52 (2003).
[CrossRef]

1994 (2)

M. L. Baesso, J. Shen, and R. D. Snook, ”Mode-mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths, J. Appl. Phys. 75, 3732-3737 (1994).
[CrossRef]

P. B. Chapple, “Beam waist and M2 measurement using a finite slit,” Opt. Eng. 332461-2466 (1994).
[CrossRef]

1993 (1)

Z. A. Talib and W. M. M. Yunus, “Measuring Gaussian laser beam diameter using piezoelectric detection,” Meas. Sci. Technol. 4, 22-25 (1993).
[CrossRef]

1992 (1)

D. Wright, P. Greve, J. Fleischer, and L. Austin, “Laser beam width, divergence and beam propagation factor: an international standardization approach,” Opt. Quantum Electron. 24, S993-S1000 (1992).
[CrossRef]

1990 (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

1989 (1)

P. Van Halen, “Accurate analytical approximations for error function and its integral,” Electron. Lett. 25, 561-563(1989).
[CrossRef]

1987 (1)

1986 (1)

1984 (1)

1983 (1)

1979 (1)

1978 (1)

1972 (1)

D. R. Skinner and R. E. Whitcher, “Measurement of the radius of a high-power laser beam near the focus of a lens,” J. Phys. E 5, 237-238 (1972).
[CrossRef]

1971 (1)

Arnaud, J. A.

Austin, L.

D. Wright, P. Greve, J. Fleischer, and L. Austin, “Laser beam width, divergence and beam propagation factor: an international standardization approach,” Opt. Quantum Electron. 24, S993-S1000 (1992).
[CrossRef]

Bachmann, L.

L. Bachmann, D. M. Zezell, and E. P. Maldonado, “Determination of beam width and quality for pulsed lasers using the knife-edge method,” Instrum. Sci. Technol. 31, 47-52 (2003).
[CrossRef]

Baesso, M. L.

M. L. Baesso, J. Shen, and R. D. Snook, ”Mode-mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths, J. Appl. Phys. 75, 3732-3737 (1994).
[CrossRef]

Chapple, P. B.

P. B. Chapple, “Beam waist and M2 measurement using a finite slit,” Opt. Eng. 332461-2466 (1994).
[CrossRef]

de la Clavière, B.

Fleischer, J.

D. Wright, P. Greve, J. Fleischer, and L. Austin, “Laser beam width, divergence and beam propagation factor: an international standardization approach,” Opt. Quantum Electron. 24, S993-S1000 (1992).
[CrossRef]

Franke, E. A.

Franke, J. M.

Garetz, B. A.

Greve, P.

D. Wright, P. Greve, J. Fleischer, and L. Austin, “Laser beam width, divergence and beam propagation factor: an international standardization approach,” Opt. Quantum Electron. 24, S993-S1000 (1992).
[CrossRef]

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Hubbard, W. M.

Khosrofian, J. M.

Maldonado, E. P.

L. Bachmann, D. M. Zezell, and E. P. Maldonado, “Determination of beam width and quality for pulsed lasers using the knife-edge method,” Instrum. Sci. Technol. 31, 47-52 (2003).
[CrossRef]

Mandeville, G. D.

Mauck, M.

McCally, R. L.

Nemoto, S.

O'Connell, R. M.

Said, A. A.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Shayler, P. J.

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Shen, J.

M. L. Baesso, J. Shen, and R. D. Snook, ”Mode-mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths, J. Appl. Phys. 75, 3732-3737 (1994).
[CrossRef]

Skinner, D. R.

D. R. Skinner and R. E. Whitcher, “Measurement of the radius of a high-power laser beam near the focus of a lens,” J. Phys. E 5, 237-238 (1972).
[CrossRef]

Snook, R. D.

M. L. Baesso, J. Shen, and R. D. Snook, ”Mode-mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths, J. Appl. Phys. 75, 3732-3737 (1994).
[CrossRef]

Talib, Z. A.

Z. A. Talib and W. M. M. Yunus, “Measuring Gaussian laser beam diameter using piezoelectric detection,” Meas. Sci. Technol. 4, 22-25 (1993).
[CrossRef]

Van Halen, P.

P. Van Halen, “Accurate analytical approximations for error function and its integral,” Electron. Lett. 25, 561-563(1989).
[CrossRef]

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Vogel, R. A.

Wei, T.-H.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Whitcher, R. E.

D. R. Skinner and R. E. Whitcher, “Measurement of the radius of a high-power laser beam near the focus of a lens,” J. Phys. E 5, 237-238 (1972).
[CrossRef]

Wright, D.

D. Wright, P. Greve, J. Fleischer, and L. Austin, “Laser beam width, divergence and beam propagation factor: an international standardization approach,” Opt. Quantum Electron. 24, S993-S1000 (1992).
[CrossRef]

Yunus, W. M. M.

Z. A. Talib and W. M. M. Yunus, “Measuring Gaussian laser beam diameter using piezoelectric detection,” Meas. Sci. Technol. 4, 22-25 (1993).
[CrossRef]

Zezell, D. M.

L. Bachmann, D. M. Zezell, and E. P. Maldonado, “Determination of beam width and quality for pulsed lasers using the knife-edge method,” Instrum. Sci. Technol. 31, 47-52 (2003).
[CrossRef]

Appl. Opt. (7)

Electron. Lett. (1)

P. Van Halen, “Accurate analytical approximations for error function and its integral,” Electron. Lett. 25, 561-563(1989).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

Instrum. Sci. Technol. (1)

L. Bachmann, D. M. Zezell, and E. P. Maldonado, “Determination of beam width and quality for pulsed lasers using the knife-edge method,” Instrum. Sci. Technol. 31, 47-52 (2003).
[CrossRef]

J. Appl. Phys. (1)

M. L. Baesso, J. Shen, and R. D. Snook, ”Mode-mismatched thermal lens determination of temperature coefficient of optical path length in soda lime glass at different wavelengths, J. Appl. Phys. 75, 3732-3737 (1994).
[CrossRef]

J. Phys. E (1)

D. R. Skinner and R. E. Whitcher, “Measurement of the radius of a high-power laser beam near the focus of a lens,” J. Phys. E 5, 237-238 (1972).
[CrossRef]

Meas. Sci. Technol. (1)

Z. A. Talib and W. M. M. Yunus, “Measuring Gaussian laser beam diameter using piezoelectric detection,” Meas. Sci. Technol. 4, 22-25 (1993).
[CrossRef]

Opt. Eng. (1)

P. B. Chapple, “Beam waist and M2 measurement using a finite slit,” Opt. Eng. 332461-2466 (1994).
[CrossRef]

Opt. Quantum Electron. (1)

D. Wright, P. Greve, J. Fleischer, and L. Austin, “Laser beam width, divergence and beam propagation factor: an international standardization approach,” Opt. Quantum Electron. 24, S993-S1000 (1992).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Simplified scheme for the measurement of laser beam radius using the knife-edge technique. The gray color area represents the shadow caused by the knife edge.

Fig. 2
Fig. 2

Comparison of the data obtained from Eq. (3) with f ( s ) defined by Eq. (8).

Fig. 3
Fig. 3

Fitting the data obtained from Eq. (3) with f ( s ) defined by Eq. (9).

Fig. 4
Fig. 4

Differences between f ( s ) and P N ( x ) . (a)  f ( s ) is given by Eq. (8) with the parameters w = 1.0 and x 0 = 0.0 (solid line) and w = 0.9612 and x 0 = 0.0132 (dashed line). (b)  f ( s ) is given by Eq. (9) when only the coefficients a 1 and a 3 are considered (solid line), and when the new set of coefficients that includes a 5 is considered (dashed line).

Fig. 5
Fig. 5

Fitting of the experimental data using Eq. (9). A similar curve is obtained by using Eq. (8), but with the adjusted laser beam radius 3.8 % lower.

Equations (15)

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I ( x , y ) = I 0 exp [ - ( x - x 0 ) 2 + ( y - y 0 ) 2 w 2 ] ,
P N = - x - I ( x , y ) d y d x - - I ( x , y ) d y d x ,
P N ( x ) = 1 2 [ 1 + erf ( x - x 0 w ) ] ,
d P N ( x ) d x = 1 π w exp [ - ( x - x 0 ) 2 w 2 ] .
f ( s ) = 1 1 + exp [ p ( s ) ] ,
p ( s ) = i = 0 m a i s i ,
s = 2 ( x - x 0 ) w .
f ( s ) = 1 1 + exp ( a 0 + a 1 s + a 2 s 2 + a 3 s 3 ) .
a 0 = - 6.71387 × 10 - 3 , a 1 = - 1.55115 ,
a 2 = - 5.13306 × 10 - 2 ,
a 3 = - 5.49164 × 10 - 2 .
a 1 = - 1.597106847 , a 3 = - 7.0924013 × 10 - 2 .
f ( s ) = 1 1 + exp ( a 1 s + a 3 s 3 ) .
a 1 = - 1.5954086 , a 3 = - 7.3638857 × 10 - 2 ,
a 5 = + 6.4121343 × 10 - 4 .

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