Abstract

The intensity profile of a radially symmetric laser beam is shown to be invertible from knife-edge data via the inverse Abel transform of the first derivative of the data. Two numerical algorithms for inverting pulsed-beam data are applied to a knife-edge scan of the focused output of a confocal unstable resonator laser. The resulting beam profiles are shown to be in good agreement with each other but significantly different from the profile obtained by assuming the beam to be approximately Gaussian. The results are shown to predict that the laser damage threshold of polymethylmethacrylate is ~4 times as large as previously reported.

© 1987 Optical Society of America

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References

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  1. J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. de la Claviere, E. A. Franke, J. M. Franke, “Technique for Fast Measurement of Gaussian Laser Beam Parameters,” Appl. Opt. 10, 2775 (1971).
    [CrossRef] [PubMed]
  2. D. R. Skinner, R. E. Whitcher, “Measurement of the Radius of a High-Power Laser Beam Near the Focus of a Lens,” J. Phys. E 5, 237 (1972).
    [CrossRef]
  3. Y. Suzaki, A. Tachibana, “Measurement of the am Sized Radius of Gaussian Laser Beam Using the Scanning Knife-Edge,” Appl. Opt. 14, 2809 (1975).
    [CrossRef] [PubMed]
  4. J. M. Khosrofian, B. A. Garetz, “Measurement of a Gaussian Laser Beam Diameter Through the Direct Inversion of Knife-Edge Data,” Appl. Opt. 22, 3406 (1983).
    [CrossRef] [PubMed]
  5. A. H. Firester, M. E. Heller, P. Sheng, “Knife-Edge Scanning Measurements of Subwavelength Focused Light Beam,” Appl. Opt. 16, 1971 (1977).
    [CrossRef] [PubMed]
  6. M. Mauck, “Knife-Edge Profiling of Q-Switched Nd:YAG Laser Beam and Waist,” Appl. Opt. 18, 599 (1979).
    [CrossRef] [PubMed]
  7. G. Brost, P. D. Horn, A. Abtahi, “Convenient Spatial Profiling of Pulsed Laser Beam,” Appl. Opt. 24, 38 (1985).
    [CrossRef] [PubMed]
  8. R. M. O’Connell, T. F. Deaton, T. T. Saito, “Single- and Multiple-Shot Laser-Damaged Properties of Commercial Grade PMMA,” Appl. Opt. 23, 682 (1984).
    [CrossRef]
  9. J. Ebert, E. Kiesel, “Measurement of Laser-Induced Damage with an Unstable Resonator-Type Laser,” Appl. Opt. 23, 3759 (1984).
    [CrossRef] [PubMed]
  10. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 339–345.
  11. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), pp. 262–266.
  12. R. M. O’Connell, A. B. Romberger, A. A. Shaffer, T. T. Saito, T. F. Deaton, K. E. Siegenthaler, “Improved Laser-Damage-Resistant Polymethyl Methacrylate,” J. Opt. Soc. Am. B 1, 853 (1984).
    [CrossRef]
  13. W. F. Krupke, W. R. Sooy, “Properties of an Unstable Confocal Resonator CO2 Laser System,” IEEE J. Quantum Electron. QE-5, 575 (1969).
    [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), pp. 416–417.
  15. O. H. Nestor, H. N. Olsen, “Numerical Methods for Reducing Line and Surface Probe Data,” SIAM Rev. 2, 200 (1960).
    [CrossRef]
  16. C. J. Cremers, R. C. Birkebak, “Application of the Abel Integral Equation to Spectrographic Data,” Appl. Opt. 5, 1057 (1966).
    [CrossRef] [PubMed]
  17. S. R. Foltyn, B. E. Newnam, “Ultraviolet Damage Resistance of Dielectric Reflectors Under Multiple-Shot Irradiation,” IEEE J. Quantum Electron. QE-17, 2092 (1981).
    [CrossRef]
  18. J. O. Porteus, S. C. Seitel, “Absolute Onset of Optical Surface Damage Using Distributed Defect Ensembles,” Appl. Opt. 23, 3796 (1984).
    [CrossRef] [PubMed]
  19. E. W. Hansen, P. L. Law, “Recursive Methods for Computing the Abel Transform and its Inverse,” J. Opt. Soc. Am. A 2, 510 (1985).
    [CrossRef]

1985 (2)

1984 (4)

1983 (1)

1981 (1)

S. R. Foltyn, B. E. Newnam, “Ultraviolet Damage Resistance of Dielectric Reflectors Under Multiple-Shot Irradiation,” IEEE J. Quantum Electron. QE-17, 2092 (1981).
[CrossRef]

1979 (1)

1977 (1)

1975 (1)

1972 (1)

D. R. Skinner, R. E. Whitcher, “Measurement of the Radius of a High-Power Laser Beam Near the Focus of a Lens,” J. Phys. E 5, 237 (1972).
[CrossRef]

1971 (1)

1969 (1)

W. F. Krupke, W. R. Sooy, “Properties of an Unstable Confocal Resonator CO2 Laser System,” IEEE J. Quantum Electron. QE-5, 575 (1969).
[CrossRef]

1966 (1)

1960 (1)

O. H. Nestor, H. N. Olsen, “Numerical Methods for Reducing Line and Surface Probe Data,” SIAM Rev. 2, 200 (1960).
[CrossRef]

Abtahi, A.

Arnaud, J. A.

Birkebak, R. C.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), pp. 416–417.

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), pp. 262–266.

Brost, G.

Cremers, C. J.

de la Claviere, B.

Deaton, T. F.

Ebert, J.

Firester, A. H.

Foltyn, S. R.

S. R. Foltyn, B. E. Newnam, “Ultraviolet Damage Resistance of Dielectric Reflectors Under Multiple-Shot Irradiation,” IEEE J. Quantum Electron. QE-17, 2092 (1981).
[CrossRef]

Franke, E. A.

Franke, J. M.

Garetz, B. A.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 339–345.

Hansen, E. W.

Heller, M. E.

Horn, P. D.

Hubbard, W. M.

Khosrofian, J. M.

Kiesel, E.

Krupke, W. F.

W. F. Krupke, W. R. Sooy, “Properties of an Unstable Confocal Resonator CO2 Laser System,” IEEE J. Quantum Electron. QE-5, 575 (1969).
[CrossRef]

Law, P. L.

Mandeville, G. D.

Mauck, M.

Nestor, O. H.

O. H. Nestor, H. N. Olsen, “Numerical Methods for Reducing Line and Surface Probe Data,” SIAM Rev. 2, 200 (1960).
[CrossRef]

Newnam, B. E.

S. R. Foltyn, B. E. Newnam, “Ultraviolet Damage Resistance of Dielectric Reflectors Under Multiple-Shot Irradiation,” IEEE J. Quantum Electron. QE-17, 2092 (1981).
[CrossRef]

O’Connell, R. M.

Olsen, H. N.

O. H. Nestor, H. N. Olsen, “Numerical Methods for Reducing Line and Surface Probe Data,” SIAM Rev. 2, 200 (1960).
[CrossRef]

Porteus, J. O.

Romberger, A. B.

Saito, T. T.

Seitel, S. C.

Shaffer, A. A.

Sheng, P.

Siegenthaler, K. E.

Skinner, D. R.

D. R. Skinner, R. E. Whitcher, “Measurement of the Radius of a High-Power Laser Beam Near the Focus of a Lens,” J. Phys. E 5, 237 (1972).
[CrossRef]

Sooy, W. R.

W. F. Krupke, W. R. Sooy, “Properties of an Unstable Confocal Resonator CO2 Laser System,” IEEE J. Quantum Electron. QE-5, 575 (1969).
[CrossRef]

Suzaki, Y.

Tachibana, A.

Whitcher, R. E.

D. R. Skinner, R. E. Whitcher, “Measurement of the Radius of a High-Power Laser Beam Near the Focus of a Lens,” J. Phys. E 5, 237 (1972).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), pp. 416–417.

Appl. Opt. (10)

C. J. Cremers, R. C. Birkebak, “Application of the Abel Integral Equation to Spectrographic Data,” Appl. Opt. 5, 1057 (1966).
[CrossRef] [PubMed]

A. H. Firester, M. E. Heller, P. Sheng, “Knife-Edge Scanning Measurements of Subwavelength Focused Light Beam,” Appl. Opt. 16, 1971 (1977).
[CrossRef] [PubMed]

J. M. Khosrofian, B. A. Garetz, “Measurement of a Gaussian Laser Beam Diameter Through the Direct Inversion of Knife-Edge Data,” Appl. Opt. 22, 3406 (1983).
[CrossRef] [PubMed]

R. M. O’Connell, T. F. Deaton, T. T. Saito, “Single- and Multiple-Shot Laser-Damaged Properties of Commercial Grade PMMA,” Appl. Opt. 23, 682 (1984).
[CrossRef]

J. Ebert, E. Kiesel, “Measurement of Laser-Induced Damage with an Unstable Resonator-Type Laser,” Appl. Opt. 23, 3759 (1984).
[CrossRef] [PubMed]

J. O. Porteus, S. C. Seitel, “Absolute Onset of Optical Surface Damage Using Distributed Defect Ensembles,” Appl. Opt. 23, 3796 (1984).
[CrossRef] [PubMed]

G. Brost, P. D. Horn, A. Abtahi, “Convenient Spatial Profiling of Pulsed Laser Beam,” Appl. Opt. 24, 38 (1985).
[CrossRef] [PubMed]

J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. de la Claviere, E. A. Franke, J. M. Franke, “Technique for Fast Measurement of Gaussian Laser Beam Parameters,” Appl. Opt. 10, 2775 (1971).
[CrossRef] [PubMed]

Y. Suzaki, A. Tachibana, “Measurement of the am Sized Radius of Gaussian Laser Beam Using the Scanning Knife-Edge,” Appl. Opt. 14, 2809 (1975).
[CrossRef] [PubMed]

M. Mauck, “Knife-Edge Profiling of Q-Switched Nd:YAG Laser Beam and Waist,” Appl. Opt. 18, 599 (1979).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (2)

W. F. Krupke, W. R. Sooy, “Properties of an Unstable Confocal Resonator CO2 Laser System,” IEEE J. Quantum Electron. QE-5, 575 (1969).
[CrossRef]

S. R. Foltyn, B. E. Newnam, “Ultraviolet Damage Resistance of Dielectric Reflectors Under Multiple-Shot Irradiation,” IEEE J. Quantum Electron. QE-17, 2092 (1981).
[CrossRef]

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

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

J. Phys. E (1)

D. R. Skinner, R. E. Whitcher, “Measurement of the Radius of a High-Power Laser Beam Near the Focus of a Lens,” J. Phys. E 5, 237 (1972).
[CrossRef]

SIAM Rev. (1)

O. H. Nestor, H. N. Olsen, “Numerical Methods for Reducing Line and Surface Probe Data,” SIAM Rev. 2, 200 (1960).
[CrossRef]

Other (3)

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), pp. 339–345.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), pp. 262–266.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1970), pp. 416–417.

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Figures (4)

Fig. 1
Fig. 1

Relation between the line response function L(x) and an arbitrary beam profile function F(x,y).

Fig. 2
Fig. 2

Geometry used to relate a radially symmetric beam profile function F(r) to the line response function.

Fig. 3
Fig. 3

Measured knife-edge data from a focused Q-Switched Nd:YAGlaser. The knife-edge position is relative to beam center.

Fig. 4
Fig. 4

Normalized beam profiles obtained by inverting the knife-edge data of Fig. 3 using the Nestor/Olsen algorithm (curve A), the Cremers/Birkebak algorithm (curve B), and differentiation (curve C).

Equations (12)

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

L ( x ) = - E ( x ) .
L ( x ) = - F ( x , y ) d y .
F ( x , y ) = f ( x ) g ( y ) .
L ( x ) = C F ( x ) ,
L ( x ) = 2 0 ( R 2 - x 2 ) 1 / 2 F ( r ) d y .
x 2 + y 2 = r 2 ,
d y = r d r ( r 2 - x 2 ) 1 / 2 ,
L ( x ) = 2 x R F ( r ) r d r ( r 2 - x 2 ) 1 / 2 ,
F ( r ) = - 1 π r R L ( x ) ( x 2 - r 2 ) 1 / 2 d x ,
F ( r ) = 1 π r R E ( x ) ( x 2 - r 2 ) 1 / 2 d x ,
A = E R F R = 2 π 0 R F ( r ) r d r F R cm 2 .
F P = E T A .

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