Abstract

We consider four families of functions—the super-Gaussian, flattened Gaussian, Fermi–Dirac, and super-Lorentzian—that have been used to describe flattened irradiance profiles. We determine the shape and width parameters of the different distributions, when each flattened profile has the same radius and slope of the irradiance at its half-height point, and then we evaluate the implicit functional relationship between the shape and width parameters for matched profiles, which provides a quantitative way to compare profiles described by different families of functions. We conclude from an analysis of each profile with matched parameters using Kirchhoff–Fresnel diffraction theory and M2 analysis that the diffraction patterns as they propagate differ by small amounts, which may not be distinguished experimentally. Thus, beam shaping optics is designed to produce either of these four flattened output irradiance distributions with matched parameters will yield similar irradiance distributions as the beam propagates.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., Laser Beam Shaping Applications (CRC Press, 2005).
    [Crossref]
  2. J. Kreuzer, "Laser light redistribution in illuminating optical signal processing systems," in Optical and Electro-Optical Information Processing, J.T.Tippett, D.A.Berkowitz, L.C.Clapp, C.J.Koester, and A. Vanderburgh, Jr., eds. (Massachusetts Institute of Technology Press, 1965), pp. 365-369.
  3. B. R. Frieden, "Lossless conversion of a plane laser wave to a plane wave of uniform irradiance," Appl. Opt. 4, 1400-1403 (1965).
    [Crossref]
  4. J. L. Kreuzer, "Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface," U.S. patent 3,476,463 (4 November 1969).
  5. O. Bryngdahl, "Geometrical transforms in optics," J. Opt. Soc. Am. 64, 1092-1099 (1974).
    [Crossref]
  6. L. A. Romero and F. M. Dickey, "Lossless laser beam shaping," J. Opt. Soc. Am. A 13, 751-760 (1996).
    [Crossref]
  7. H. Aagedal, M. Schmid, S. Egner, J. Müller-Quade, J. Beth, and F. Wyrowski, "Analytical beam shaping with application to laser-diode arrays," J. Opt. Soc. Am. A 14, 1549-1553 (1997).
    [Crossref]
  8. F. M. Dickey and S. C. Holswade, eds., Laser Beam Shaping: Theory and Techniques (Marcel Dekker, 2000).
    [Crossref]
  9. S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH Verlag, 2003), pp. 271-299.
  10. H. Aagedal, F. Wyrowski, and M. Schmid, "Paraxial beam splitting and shaping," in Diffractive Optics for Industrial and Commercial Applications, J.Turunen and F.Wyrowski, eds. (Akademic-Verlag, 1997), Chap. 6.
  11. M. R. Feldman, A. D. Kathman, W. H. Welch, and T. Tkolste, "Integrated beam shaper and use thereof," U.S. patent 6,128,134 (3 October 2000).
  12. P. W. Scott and W. H. Southwell, "Reflective optics for irradiance redistribution of laser beams: design," Appl. Opt. 20, 1606-1610 (1981).
    [Crossref] [PubMed]
  13. P. W. Malyak, "Two-mirror unobscured optical system for reshaping the irradiance distribution of a laser beam," Appl. Opt. 31, 4377-4383 (1992).
    [Crossref] [PubMed]
  14. D. L. Shealy, "History of beam shaping," in Laser Beam Shaping Applications, F.M.Dickey, S.C.Holswade, and D.L.Shealy, eds. (CRC Press, 2005), pp. 307-347.
    [Crossref]
  15. W. Jiang, D. L. Shealy, and J. C. Martin, "Design and testing of a refractive reshaping system," in Current Developments in Optical Design and Optical Engineering III, R.E.Fischer and W.J.Smith, eds., Proc. SPIE 2000,64-75 (1993).
  16. W. Jiang and D. L. Shealy, "Development and testing of a refractive laser beam shaping system," in Laser Beam Shaping, F.M.Dickey and S.C.Holswade, eds., Proc. SPIE 4095,165-175 (2000).
  17. J. A. Hoffnagle and C. M. Jefferson, "Design and performance of a refractive optical system that converts a Gaussian to a flattop beam," Appl. Opt. 39, 5488-5499 (2000).
    [Crossref]
  18. J. A. Hoffnagle and C. M. Jefferson, "Refractive optical system that converts a laser beam to a collimated flat-top beam," U.S. patent 6,295,168 (25 September 2001).
  19. J. A. Hoffnagle and C. M. Jefferson, "Beam shaping with a plano-aspheric lens pair," Opt. Eng. 42, 3090-3099 (2003).
    [Crossref]
  20. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, and B. Majocchi, "Unstable laser resonators with super-Gaussian mirrors," Opt. Lett. 13, 201-203 (1988).
    [Crossref] [PubMed]
  21. A. Parent, M. Morin, and P. Lavigne, "Propagation of super-Gaussian field distributions," Opt. Quantum Electron. 24, S1071-S1079 (1992).
    [Crossref]
  22. F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
    [Crossref]
  23. R. Borghi, "Elegant Laguerre-Gauss beams as a new tool for describing axisymmetric flattened Gaussian beams," J. Opt. Soc. Am. A 18, 1627-1633 (2001).
    [Crossref]
  24. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambroshini, and G. Schirripa-Spagnolo, "Propagation of axially symmetric flattened Gaussian beams," J. Opt. Soc. Am. A 13, 1385-1394 (1996).
    [Crossref]
  25. B. Lü, B. Zhang, and S. Zuo, "Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams," Appl. Opt. 38, 4581-4585 (1999).
    [Crossref]
  26. Z. Mei and D. Zhao, "Approximate method for the generalizedM 2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams," Appl. Opt. 44,1381-1386 (2005).
    [PubMed]
  27. Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
    [Crossref]
  28. Y. Li, "Light beams with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002).
    [Crossref]
  29. F. M. Dickey and S. C. Holswade, "Gaussian laser beam profile shaping," Opt. Eng. 35, 3285-3295 (1996).
    [Crossref]
  30. Maple 9.50, Copyright held by Maplesoft, a division of Waterloo Maple, Inc.
  31. D. L. Shealy and J. A. Hoffnagle, "Beam shaping profiles and propagation," in Laser Beam Shaping VI, F.M.Dickey and D.L.Shealy, eds., Proc. SPIE 5876,99-109 (2005).
  32. M. Santarsiero and R. Borghi, "Correspondence between super-Gaussian and flattened Gaussian beams," J. Opt. Soc. Am. A 16, 188-190 (1999).
    [Crossref]
  33. J. K. Kasinski and R. L. Burnham, "Near-diffraction-limited laser beam shaping with diamond-turned aspheric optics," Opt. Lett. 22, 1062-1064 (1997).
    [Crossref] [PubMed]
  34. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).
  35. J. P. Campbell and L. G. DeShazer, "Near fields of truncated-Gaussian apertures," J. Opt. Soc. Am. 59, 1427-1429 (1969).
    [Crossref]
  36. S. Vicalvi, R. Borghi, M. Santarsiero, and F. Gori, "Shape-invariance error for axially symmetric light beams," IEEE J. Quantum Electron. QE-34, 2109-2116 (1998).
    [Crossref]
  37. F. Gori, S. Vicalvi, M. Santarsiero, and R. Borghi, "Shape-invariance range of a light beam," Opt. Lett. 21, 1205-1207 (1996).
    [Crossref] [PubMed]
  38. A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D.A.Holmes, ed., Proc. SPIE 1224,2-14 (1990).
  39. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series No. 55, 6th printing (U.S. Government Printing Office, 1967).
  40. H. J. Weaver, Applications of Discrete and Continuous Fourier Analysis (Wiley, 1983).
  41. G. B. Folland, Fourier Analysis and its Applications (Wadsworth & Brook, 1992), pp. 248-249.
  42. P. Rhodes, "Fermi-Dirac functions of integral order," Proc. R. Soc. 204, 396-405 (1950).
    [Crossref]

2003 (1)

J. A. Hoffnagle and C. M. Jefferson, "Beam shaping with a plano-aspheric lens pair," Opt. Eng. 42, 3090-3099 (2003).
[Crossref]

2002 (2)

Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
[Crossref]

Y. Li, "Light beams with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002).
[Crossref]

2001 (1)

2000 (2)

M. R. Feldman, A. D. Kathman, W. H. Welch, and T. Tkolste, "Integrated beam shaper and use thereof," U.S. patent 6,128,134 (3 October 2000).

J. A. Hoffnagle and C. M. Jefferson, "Design and performance of a refractive optical system that converts a Gaussian to a flattop beam," Appl. Opt. 39, 5488-5499 (2000).
[Crossref]

1999 (2)

1998 (1)

S. Vicalvi, R. Borghi, M. Santarsiero, and F. Gori, "Shape-invariance error for axially symmetric light beams," IEEE J. Quantum Electron. QE-34, 2109-2116 (1998).
[Crossref]

1997 (2)

1996 (4)

1994 (1)

F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
[Crossref]

1992 (2)

A. Parent, M. Morin, and P. Lavigne, "Propagation of super-Gaussian field distributions," Opt. Quantum Electron. 24, S1071-S1079 (1992).
[Crossref]

P. W. Malyak, "Two-mirror unobscured optical system for reshaping the irradiance distribution of a laser beam," Appl. Opt. 31, 4377-4383 (1992).
[Crossref] [PubMed]

1988 (1)

1981 (1)

1974 (1)

1969 (1)

1965 (1)

1950 (1)

P. Rhodes, "Fermi-Dirac functions of integral order," Proc. R. Soc. 204, 396-405 (1950).
[Crossref]

Aagedal, H.

H. Aagedal, M. Schmid, S. Egner, J. Müller-Quade, J. Beth, and F. Wyrowski, "Analytical beam shaping with application to laser-diode arrays," J. Opt. Soc. Am. A 14, 1549-1553 (1997).
[Crossref]

H. Aagedal, F. Wyrowski, and M. Schmid, "Paraxial beam splitting and shaping," in Diffractive Optics for Industrial and Commercial Applications, J.Turunen and F.Wyrowski, eds. (Akademic-Verlag, 1997), Chap. 6.

Abramowitz, M.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series No. 55, 6th printing (U.S. Government Printing Office, 1967).

Ambroshini, D.

Bagini, V.

Beth, J.

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Bryngdahl, O.

Burnham, R. L.

Campbell, J. P.

De Silvestri, S.

DeShazer, L. G.

Dickey, F. M.

F. M. Dickey and S. C. Holswade, "Gaussian laser beam profile shaping," Opt. Eng. 35, 3285-3295 (1996).
[Crossref]

L. A. Romero and F. M. Dickey, "Lossless laser beam shaping," J. Opt. Soc. Am. A 13, 751-760 (1996).
[Crossref]

F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., Laser Beam Shaping Applications (CRC Press, 2005).
[Crossref]

F. M. Dickey and S. C. Holswade, eds., Laser Beam Shaping: Theory and Techniques (Marcel Dekker, 2000).
[Crossref]

Egner, S.

Feldman, M. R.

M. R. Feldman, A. D. Kathman, W. H. Welch, and T. Tkolste, "Integrated beam shaper and use thereof," U.S. patent 6,128,134 (3 October 2000).

Folland, G. B.

G. B. Folland, Fourier Analysis and its Applications (Wadsworth & Brook, 1992), pp. 248-249.

Frieden, B. R.

Gori, F.

Hoffnagle, J. A.

J. A. Hoffnagle and C. M. Jefferson, "Beam shaping with a plano-aspheric lens pair," Opt. Eng. 42, 3090-3099 (2003).
[Crossref]

J. A. Hoffnagle and C. M. Jefferson, "Design and performance of a refractive optical system that converts a Gaussian to a flattop beam," Appl. Opt. 39, 5488-5499 (2000).
[Crossref]

J. A. Hoffnagle and C. M. Jefferson, "Refractive optical system that converts a laser beam to a collimated flat-top beam," U.S. patent 6,295,168 (25 September 2001).

D. L. Shealy and J. A. Hoffnagle, "Beam shaping profiles and propagation," in Laser Beam Shaping VI, F.M.Dickey and D.L.Shealy, eds., Proc. SPIE 5876,99-109 (2005).

Holswade, S. C.

F. M. Dickey and S. C. Holswade, "Gaussian laser beam profile shaping," Opt. Eng. 35, 3285-3295 (1996).
[Crossref]

F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., Laser Beam Shaping Applications (CRC Press, 2005).
[Crossref]

F. M. Dickey and S. C. Holswade, eds., Laser Beam Shaping: Theory and Techniques (Marcel Dekker, 2000).
[Crossref]

Jahns, J.

S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH Verlag, 2003), pp. 271-299.

Jefferson, C. M.

J. A. Hoffnagle and C. M. Jefferson, "Beam shaping with a plano-aspheric lens pair," Opt. Eng. 42, 3090-3099 (2003).
[Crossref]

J. A. Hoffnagle and C. M. Jefferson, "Design and performance of a refractive optical system that converts a Gaussian to a flattop beam," Appl. Opt. 39, 5488-5499 (2000).
[Crossref]

J. A. Hoffnagle and C. M. Jefferson, "Refractive optical system that converts a laser beam to a collimated flat-top beam," U.S. patent 6,295,168 (25 September 2001).

Jiang, W.

W. Jiang, D. L. Shealy, and J. C. Martin, "Design and testing of a refractive reshaping system," in Current Developments in Optical Design and Optical Engineering III, R.E.Fischer and W.J.Smith, eds., Proc. SPIE 2000,64-75 (1993).

W. Jiang and D. L. Shealy, "Development and testing of a refractive laser beam shaping system," in Laser Beam Shaping, F.M.Dickey and S.C.Holswade, eds., Proc. SPIE 4095,165-175 (2000).

Kasinski, J. K.

Kathman, A. D.

M. R. Feldman, A. D. Kathman, W. H. Welch, and T. Tkolste, "Integrated beam shaper and use thereof," U.S. patent 6,128,134 (3 October 2000).

Kreuzer, J.

J. Kreuzer, "Laser light redistribution in illuminating optical signal processing systems," in Optical and Electro-Optical Information Processing, J.T.Tippett, D.A.Berkowitz, L.C.Clapp, C.J.Koester, and A. Vanderburgh, Jr., eds. (Massachusetts Institute of Technology Press, 1965), pp. 365-369.

Kreuzer, J. L.

J. L. Kreuzer, "Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface," U.S. patent 3,476,463 (4 November 1969).

Laporta, P.

Lavigne, P.

A. Parent, M. Morin, and P. Lavigne, "Propagation of super-Gaussian field distributions," Opt. Quantum Electron. 24, S1071-S1079 (1992).
[Crossref]

Li, Y.

Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
[Crossref]

Y. Li, "Light beams with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002).
[Crossref]

Lü, B.

Magni, V.

Majocchi, B.

Malyak, P. W.

Martin, J. C.

W. Jiang, D. L. Shealy, and J. C. Martin, "Design and testing of a refractive reshaping system," in Current Developments in Optical Design and Optical Engineering III, R.E.Fischer and W.J.Smith, eds., Proc. SPIE 2000,64-75 (1993).

Mei, Z.

Z. Mei and D. Zhao, "Approximate method for the generalizedM 2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams," Appl. Opt. 44,1381-1386 (2005).
[PubMed]

Morin, M.

A. Parent, M. Morin, and P. Lavigne, "Propagation of super-Gaussian field distributions," Opt. Quantum Electron. 24, S1071-S1079 (1992).
[Crossref]

Müller-Quade, J.

Pacileo, A. M.

Parent, A.

A. Parent, M. Morin, and P. Lavigne, "Propagation of super-Gaussian field distributions," Opt. Quantum Electron. 24, S1071-S1079 (1992).
[Crossref]

Rhodes, P.

P. Rhodes, "Fermi-Dirac functions of integral order," Proc. R. Soc. 204, 396-405 (1950).
[Crossref]

Romero, L. A.

Santarsiero, M.

Schirripa-Spagnolo, G.

Schmid, M.

H. Aagedal, M. Schmid, S. Egner, J. Müller-Quade, J. Beth, and F. Wyrowski, "Analytical beam shaping with application to laser-diode arrays," J. Opt. Soc. Am. A 14, 1549-1553 (1997).
[Crossref]

H. Aagedal, F. Wyrowski, and M. Schmid, "Paraxial beam splitting and shaping," in Diffractive Optics for Industrial and Commercial Applications, J.Turunen and F.Wyrowski, eds. (Akademic-Verlag, 1997), Chap. 6.

Scott, P. W.

Shealy, D. L.

W. Jiang and D. L. Shealy, "Development and testing of a refractive laser beam shaping system," in Laser Beam Shaping, F.M.Dickey and S.C.Holswade, eds., Proc. SPIE 4095,165-175 (2000).

D. L. Shealy, "History of beam shaping," in Laser Beam Shaping Applications, F.M.Dickey, S.C.Holswade, and D.L.Shealy, eds. (CRC Press, 2005), pp. 307-347.
[Crossref]

W. Jiang, D. L. Shealy, and J. C. Martin, "Design and testing of a refractive reshaping system," in Current Developments in Optical Design and Optical Engineering III, R.E.Fischer and W.J.Smith, eds., Proc. SPIE 2000,64-75 (1993).

F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., Laser Beam Shaping Applications (CRC Press, 2005).
[Crossref]

D. L. Shealy and J. A. Hoffnagle, "Beam shaping profiles and propagation," in Laser Beam Shaping VI, F.M.Dickey and D.L.Shealy, eds., Proc. SPIE 5876,99-109 (2005).

Siegman, A. E.

A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D.A.Holmes, ed., Proc. SPIE 1224,2-14 (1990).

Sinzinger, S.

S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH Verlag, 2003), pp. 271-299.

Southwell, W. H.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series No. 55, 6th printing (U.S. Government Printing Office, 1967).

Svelto, O.

Tkolste, T.

M. R. Feldman, A. D. Kathman, W. H. Welch, and T. Tkolste, "Integrated beam shaper and use thereof," U.S. patent 6,128,134 (3 October 2000).

Vicalvi, S.

S. Vicalvi, R. Borghi, M. Santarsiero, and F. Gori, "Shape-invariance error for axially symmetric light beams," IEEE J. Quantum Electron. QE-34, 2109-2116 (1998).
[Crossref]

F. Gori, S. Vicalvi, M. Santarsiero, and R. Borghi, "Shape-invariance range of a light beam," Opt. Lett. 21, 1205-1207 (1996).
[Crossref] [PubMed]

Weaver, H. J.

H. J. Weaver, Applications of Discrete and Continuous Fourier Analysis (Wiley, 1983).

Welch, W. H.

M. R. Feldman, A. D. Kathman, W. H. Welch, and T. Tkolste, "Integrated beam shaper and use thereof," U.S. patent 6,128,134 (3 October 2000).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Wyrowski, F.

H. Aagedal, M. Schmid, S. Egner, J. Müller-Quade, J. Beth, and F. Wyrowski, "Analytical beam shaping with application to laser-diode arrays," J. Opt. Soc. Am. A 14, 1549-1553 (1997).
[Crossref]

H. Aagedal, F. Wyrowski, and M. Schmid, "Paraxial beam splitting and shaping," in Diffractive Optics for Industrial and Commercial Applications, J.Turunen and F.Wyrowski, eds. (Akademic-Verlag, 1997), Chap. 6.

Zhang, B.

Zhao, D.

Z. Mei and D. Zhao, "Approximate method for the generalizedM 2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams," Appl. Opt. 44,1381-1386 (2005).
[PubMed]

Zuo, S.

Appl. Opt. (5)

IEEE J. Quantum Electron. (1)

S. Vicalvi, R. Borghi, M. Santarsiero, and F. Gori, "Shape-invariance error for axially symmetric light beams," IEEE J. Quantum Electron. QE-34, 2109-2116 (1998).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Opt. Commun. (2)

F. Gori, "Flattened Gaussian beams," Opt. Commun. 107, 335-341 (1994).
[Crossref]

Y. Li, "New expressions for flat-topped light beams," Opt. Commun. 206, 225-234 (2002).
[Crossref]

Opt. Eng. (2)

F. M. Dickey and S. C. Holswade, "Gaussian laser beam profile shaping," Opt. Eng. 35, 3285-3295 (1996).
[Crossref]

J. A. Hoffnagle and C. M. Jefferson, "Beam shaping with a plano-aspheric lens pair," Opt. Eng. 42, 3090-3099 (2003).
[Crossref]

Opt. Lett. (4)

Opt. Quantum Electron. (1)

A. Parent, M. Morin, and P. Lavigne, "Propagation of super-Gaussian field distributions," Opt. Quantum Electron. 24, S1071-S1079 (1992).
[Crossref]

Proc. R. Soc. (1)

P. Rhodes, "Fermi-Dirac functions of integral order," Proc. R. Soc. 204, 396-405 (1950).
[Crossref]

U.S. patent 6,128,134 (1)

M. R. Feldman, A. D. Kathman, W. H. Welch, and T. Tkolste, "Integrated beam shaper and use thereof," U.S. patent 6,128,134 (3 October 2000).

Other (18)

Z. Mei and D. Zhao, "Approximate method for the generalizedM 2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams," Appl. Opt. 44,1381-1386 (2005).
[PubMed]

A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D.A.Holmes, ed., Proc. SPIE 1224,2-14 (1990).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series No. 55, 6th printing (U.S. Government Printing Office, 1967).

H. J. Weaver, Applications of Discrete and Continuous Fourier Analysis (Wiley, 1983).

G. B. Folland, Fourier Analysis and its Applications (Wadsworth & Brook, 1992), pp. 248-249.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Maple 9.50, Copyright held by Maplesoft, a division of Waterloo Maple, Inc.

D. L. Shealy and J. A. Hoffnagle, "Beam shaping profiles and propagation," in Laser Beam Shaping VI, F.M.Dickey and D.L.Shealy, eds., Proc. SPIE 5876,99-109 (2005).

J. A. Hoffnagle and C. M. Jefferson, "Refractive optical system that converts a laser beam to a collimated flat-top beam," U.S. patent 6,295,168 (25 September 2001).

D. L. Shealy, "History of beam shaping," in Laser Beam Shaping Applications, F.M.Dickey, S.C.Holswade, and D.L.Shealy, eds. (CRC Press, 2005), pp. 307-347.
[Crossref]

W. Jiang, D. L. Shealy, and J. C. Martin, "Design and testing of a refractive reshaping system," in Current Developments in Optical Design and Optical Engineering III, R.E.Fischer and W.J.Smith, eds., Proc. SPIE 2000,64-75 (1993).

W. Jiang and D. L. Shealy, "Development and testing of a refractive laser beam shaping system," in Laser Beam Shaping, F.M.Dickey and S.C.Holswade, eds., Proc. SPIE 4095,165-175 (2000).

F. M. Dickey and S. C. Holswade, eds., Laser Beam Shaping: Theory and Techniques (Marcel Dekker, 2000).
[Crossref]

S. Sinzinger and J. Jahns, Microoptics (Wiley-VCH Verlag, 2003), pp. 271-299.

H. Aagedal, F. Wyrowski, and M. Schmid, "Paraxial beam splitting and shaping," in Diffractive Optics for Industrial and Commercial Applications, J.Turunen and F.Wyrowski, eds. (Akademic-Verlag, 1997), Chap. 6.

J. L. Kreuzer, "Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface," U.S. patent 3,476,463 (4 November 1969).

F. M. Dickey, S. C. Holswade, and D. L. Shealy, eds., Laser Beam Shaping Applications (CRC Press, 2005).
[Crossref]

J. Kreuzer, "Laser light redistribution in illuminating optical signal processing systems," in Optical and Electro-Optical Information Processing, J.T.Tippett, D.A.Berkowitz, L.C.Clapp, C.J.Koester, and A. Vanderburgh, Jr., eds. (Massachusetts Institute of Technology Press, 1965), pp. 365-369.

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 (8)

Fig. 1
Fig. 1

Cross-sectional plot of the irradiance for (a) FG and (b) FD profiles for shape parameters N and β. All profiles are normalized such that the total energy contained within each irradiance distribution over all space is equal to unity. The FG profile for N = 0 reduces to the Gaussian profile.

Fig. 2
Fig. 2

Comparison of the irradiance distribution of the FG, SG, FD, and SL profiles with matched parameters: (a) N = 9 and R FG = 1 and (b) N = 49 and R FG = 1.

Fig. 3
Fig. 3

Comparison between shape parameters of the SG, FD, and SL shape parameters as a function of the FG shape parameters for profiles with matched parameters. The FD shape parameter β and the SL shaper parameter q have the same functional dependence on N for the scale of this figure and are represented by one curve.

Fig. 4
Fig. 4

Comparison between width parameters of the SG, FD, and SL width parameters as a function of the FG shape parameters for profiles with matched parameters.

Fig. 5
Fig. 5

Propagation of FD beams with matching parameters to a FG beam with R FG = 1. N = 18, 64, and 225; and β = 16.2003, 30.6677, and 57.8217 in (a)–(c), respectively. In each plot, the dashed curves show the FD functions before propagation and the solid curves show the irradiance distributions after propagation with a Fresnel number NF = 15.

Fig. 6
Fig. 6

Propagation of matched profiles (a) SL with q = 16.4 and R SL = 0.9295; (b) FD with β = 16.2 and R FD = 0.9295; (c) FG with N = 18 and R FG = 1; (d) SG with p = 11.14 and R SG = 1.0222. On each plot the solid curve shows the irradiance profile after propagation with Fresnel number NF = 15; the dashed curve is with NF = 10; and the dotted–dashed curve is with NF = 5.

Fig. 7
Fig. 7

Comparison of M 2 for the SG, FD, SL, and FG profiles as a function of their respective shape parameters ranging from 4 to 100. The M 2 values for the FD and SL distributions can be represented by one curve at the scale of this figure.

Fig. 8
Fig. 8

Comparison of M 2 for the SG, FG, FD, and SL beams with matching shape and beam width parameters for given FG shape parameter (N) and width (R FG = 1). Table 2 gives some shape and width parameters of the matched profiles.

Tables (3)

Tables Icon

Table 1 Irradiance Profile Normalization Constants I 0(σ, R 0) and Functions f σ , R R 0 for SG, FG, FD, and SL Distributions

Tables Icon

Table 2 Comparison of Beam Shape and Width Parameters of Matched Irradiance Profiles with R FG = 1

Tables Icon

Table 3 Asymptotic Expansions of σ r 2, σ p 2, and M 2 as the Respective Beam Profile Shape Parameter σ Approaches Infinity a

Equations (83)

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

2 π 0 I out ( R ) R d R = 1 ,
I out ( σ , R / R 0 ) = I 0 ( σ , R 0 ) f ( σ , R / R 0 ) .
[ d m f ( r ) d r m | r = 0 = 0 for m = 1 , 2 , 3 , .  .  .  ,
1. Equal   radius   when   the   irradiance   is   equal   to half   of   its   axial   value .
2. Equal   slope   of   the   irradiance   at   the   radius   of the   half-height   point .
N = nonnegative   integer ,
R FG = 1.
I out ( R ) = | u ( R ) | 2 .
u ( x ) = u 0 exp [ ( N + 1 ) x ] n = 0 N ( N + 1 ) n n ! x n ,
x = ( R / R FG ) 2 .
F ( x 1 / 2 ) = 0 ,
F ( x ) = 1 / 2 + exp [ ( N + 1 ) x ] n = 0 N ( N + 1 ) n n ! x n .
F ( x ) = exp [ ( N + 1 ) x ] ( N + 1 ) N + 1 N ! x N .
R 1 2 , FG = R FG x 1 / 2 .
m 1 2 , FG 2 2 I 0 , FG F ( x 1 / 2 ) R 1 2 , FG / R FG 2 .
R 1 2 , SG R SG = [ ln   2 2 ] 1 / p = R 1 2 , FG R FG ,
m 1 2 , FG + 4 1 / p p 2 [ ln 2 ] 1 + 1 p 2 π Γ ( 2 / p ) R 1 2 , FG 2 = 0 ,
R FD = R 1 2 , FG ,
R S L = R 1 2 , FG .
m 1 2 , FG + 3 β 3 4 π R 1 2 , FG 3 { 3 β 2 + 6   dilog [ 1 + exp ( β ) ] + π 2 } = 0.
m 1 2 , FG + q 2 sin ( 2 π / q ) 8 π 2 R 1 2 , FG 3 = 0.
u ( r , D ) = exp ( i φ ) k D 0 R max u 0 ( ρ ) J 0 ( k ρ r / D ) × exp ( i k ρ 2 / 2 D ) ρ d ρ ,
N F = R 0     2 / λ D ,
α = R max / R 0 ,
ξ = r / R 0 ,
τ = ρ / R 0 ,
v ( ξ , N F ) = u ( r , D ) ,
v 0 ( τ ) = u 0 ( ρ ) .
v ( ξ , N F ) = 2 π N F exp ( i φ ) 0 α v 0 ( τ ) J 0 ( 2 π N F ξ τ ) × exp ( i π N F τ 2 ) τ d τ .
M 2 = 2 π σ r 2 σ p 2 ,
σ r 2 = 2 π R 0 4 0 I ( r ) r 3 d r ,
σ p 2 = 2 π 0 I ^ ( k ) k 3 d k .
u ^ ( k ) = 2 π 0 u ( r ) J 0 ( 2 π r k ) r d r ,
I ^ ( k ) = | u ^ ( k ) | 2 .
σ p 2 = 1 2 π 0 | d u ( r ) d r | 2 r d r ,
M SG 2 = p Γ ( 4 / p ) 2 Γ ( 2 / p ) .
M FG 2 = 2 π I 0 , FG [ ( N + 2 ) Γ ( 2 N + 2 ) 2 2 N + 3 ( N + 1 ) ! N ! n = 0 N Γ ( 2 N + 2 ) Γ ( n + N + 3 ) F ( 1 , n + N + 3 ; N + 2 ; 1 2 ) 2 n + 3 N + 5 n ! ( N + 1 ) ! 3 ] 1 / 2 ,
M FD 2 = 1 2 F 1 ( β ) 1 2 F 3 ( β ) [ 1 1 + e β + ln ( 1 + e β ) ] ,
M SL 2 = 1 2 q sin ( 2 π / q ) [ 2 π sin ( 4 π / q ) ] 1 / 2 .
g ^ ( k x , k y ) = g ( x , y ) exp [ 2 π i ( x k x + y k y ) ] d x d y .
f ^ ( k ) = 2 π 0 f ( r ) J 0 ( 2 π r k ) r d r ,
f ( r ) = 2 π 0 f ^ ( k ) J 0 ( 2 π r k ) k d k ,
n f ( k ) = 2 π 0 f ( r ) J n ( 2 π r k ) r d r .
f ^ ( k ) = m f ( k ) ,
f ( r ) = 4 π 2 0 f ^ ( k ) J 0 ( 2 π r k ) k 2 d k .
J 0 ( z ) = J 1 ( z )
f ( r ) = 4 π 2 0 f ^ ( k ) J 1 ( 2 π r k ) k 2 d k ,
k f ^ ( k ) = 1 2 π 1 [ f ( r ) ] .
g ( x , y ) = | g ( x , y ) | 2 d x d y .
f ( r ) 2 = 2 π 0 | f ( r ) | 2 r d r .
0 | f ( r ) | 2 r d r = 0 | m f ( k ) | 2 k d k ,
σ p 2 = 2 π 0 | k u ^ ( k ) | 2 k d k ,
σ p 2 = 1 2 π 0 | 1 f ( k ) | 2 k d k .
σ p 2 = 1 2 π 0 | f ( r ) | 2 r d r ,
σ r 2 = 2 π I 0 , FG m , n = 0 N ( N + 1 ) ( m + n ) m ! n ! 0 r 2 ( m + n ) + 3 × exp [ ( N + 1 ) r 2 ] d r ,
= 2 π I 0 , FG m , n = 0 N Γ ( m + n + 2 ) 2 ( m + n + 3 ) m ! n ! ( N + 1 ) 2 ,
F ( a , b ; c ; z ) = Γ ( c ) Γ ( a ) Γ ( b ) m = 0 Γ ( a + m ) Γ ( b + m ) Γ ( c + m ) z m m ! .
S = 1 8 ( N + 1 ) 2 m = 0 N I ( m ) 2 m m ! ,
       I ( m ) = n = 0 N Γ ( m + n + 2 ) 2 n n ! ,
= n = 0 Γ ( m + n + 2 ) 2 n n ! n = N + 1 Γ ( m + n + 2 ) 2 n n ! ,
= Γ ( m + 2 ) F ( m + 2 , b ; b ; 1 2 ) k = 0 Γ ( k + N + 3 + m ) Γ ( k + 1 ) 2 n ( k + N + 1 ) ! k ! ,
= 2 m + 2 Γ ( m + 2 ) Γ ( m + N + 3 ) 2 N + 1 Γ ( N + 2 ) × F ( 1 , N + 3 + m ; N + 2 , 1 2 ) ,
σ r 2 = 2 π I 0 , FG [ ( N + 2 ) 4 ( N + 1 ) m = 0 N Γ ( m + N + 3 ) F ( 1 , m + N + 3 ; N + 2 ; 1 2 ) m ! ( N + 1 ) ! 2 ( m + N + 4 ) ( N + 1 ) 2 ] ,
d u ( r ) d r = - 2 I 0 , FG exp [ - ( N + 1 ) r 2 ] ( N + 1 ) N + 1 N ! r 2 N + 1 .
σ p 2 = I 0 , FG 2 π 4 ( N + 1 ) 2 ( N + 1 ) ( N ! ) 2 0 r 4 N + 3 × exp [ 2 ( N + 1 ) r 2 ] d r .
σ p 2 = I 0 , FG 2 π Γ [ 2 ( N + 1 ) ] 2 ( 2 N + 1 ) ( N ! ) 2 .
[ I 0 , FG ( N , R FG ) ] 1 = π R FG 2 2 ( N + 1 ) m , n = 0 N ( m + n ) ! 2 ( m + n ) m ! n ! ,
[ I 0 , FG ( N , R FG ) ] 1 = π R FG 2 π R FG 2 2 ( N + 2 ) ( N + 1 ) × n = 0 N Γ ( n + N + 2 ) 2 n n ! Γ ( N + 2 ) × F ( 1 , n + N + 2 ; N + 2 ; 1 2 ) ,
M 2 = 2 π I 0 , FG [ ( N + 2 ) Γ ( 2 N + 2 ) 2 2 N + 3 ( N + 1 ) ! N ! n = 0 N Γ ( 2 N + 2 ) Γ ( n + N + 3 ) F ( 1 , n + N + 3 ; N + 2 ; 1 2 ) 2 n + 3 N + 5 n ! ( N + 1 ) ! 3 ] 1 / 2 .
F n ( η ) 0 x n d x e x η + 1 ,
F 0 ( η ) = ln ( 1 + e η ) .
F n ( η ) = S n ( η ) + ( - ) n F n ( η ) ,
F n ( η ) = n ! m = 1 ( ) m + 1 e m η m n + 1 .
S 1 ( η ) = 1 2 η 2 + 1 6 π 2 ,
S 3 ( η ) = 1 4 η 4 + 1 2 π 2 η 2 + 7 60 π 4 .
σ r 2 = 2 π R FD 4 I 0 , FD 0 r 3 d r 1 + exp [ β ( r 1 ) ] ,
= R FD 2 F 3 ( β ) β 2 F 1 ( β ) ,
σ p 2 = I 0 , FD 2 π β 2 4 0 exp [ 2 β ( r 1 ) ] r d r { 1 + exp [ β ( r 1 ) ] } 3 .
σ p 2 = β 2 32 π 2 R FD 2 F 1 ( β ) [ 1 1 + e β + ln ( 1 + e β ) ] .
I FD ( r ) = I 0 , FD { 1 + exp [ β ( r 1 ) ] } 1 .
1 = 2 π R FD 2 I 0 , FD 0 r d r 1 + exp [ β ( r 1 ) ] .
I 0 , FD = β 2 2 π R FD 2 F 1 ( β ) ,
M 2 = 1 2 F 1 ( β ) 1 2 F 3 ( β ) [ 1 1 + e β + ln ( 1 + e β ) ] .

Metrics