Abstract

For a laser beam diffracted by a hard-edge aperture, propagation of the beam width, defined by the second-order moment of its irradiance distribution truncated according to the self-convergent-width criterion, obeys the familiar hyperbolic law. It is demonstrated numerically that, with the self-convergent-width approach, the beam-propagation parameters for three beam types (Gaussian, Hermite–Gaussian, and flattened Gaussian) diffracted by hard-edge apertures can be determined with the second-moment-based procedure that is recommended by the present draft standard only for unapertured laser beams.

© 2000 Optical Society of America

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References

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  1. L. W. Austin, A. Giesen, eds., Beam Control, Diagnostic, Standards and Propagation, Proc. SPIE2375 (1995).
  2. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  3. P.-A. Bélanger, “Beam propagation and ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
    [CrossRef]
  4. A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
    [CrossRef]
  5. M. A. Porras, J. Alda, E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams,” Appl. Opt. 31, 6389–6402 (1992).
    [CrossRef] [PubMed]
  6. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  7. M. W. Sasnett, “Propagation of multimode laser beams—the M2 factor,” in The Physics and Technology of Laser Resonators, D. R. Hall, P. E. Jackson, eds. (Adam Hilger, New York, 1989), pp. 132–142.
  8. M. Forbes, “Laser beam spatial measurement: comparing instrument,” Lasers Optron. 9, 51–55 (1990).
  9. M. W. Sasnett, T. F. Johnston, “Beam characterization of propagation attributes,” in Laser Beam Diagnostics, R. N. Hindy, Y. Kohanzadeh, eds., Proc. SPIE1414, 1–32 (1991).
  10. International Standards Organization, “Test methods for laser beam parameters: beam widths, divergence angle, and beam propagation factor,” (Deutsches Institut für Normung, Pforzheim, Germany, 1994).
  11. M. A. Porras, “Experimental investigation on aperture-diffracted laser beam characterization,” Opt. Commun. 109, 5–9 (1994).
    [CrossRef]
  12. G. Gbur, P. S. Carney, “Convergence criteria and optimization techniques for beam moments,” Pure Appl. Opt. 7, 1221–1230 (1998).
    [CrossRef]
  13. G. N. Lawrence, “Proposed international standard for laser-beam quality falls short,” Laser Focus World 30(7), 109–114 (1994).
  14. D. Wright, “Can we ignore the weak spatial range of diffracted beams?” in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 207–213.
  15. R. Martínez-Herrero, P. M. Mejías, “Parametric characterization of hard-edge diffracted beams,” in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 197–206.
  16. P.-A. Bélanger, Y. Champagne, C. Paré, “The beam quality factor MQ2 of diffracted beams,” in Laser Beam Characterization, P. M. Mejías, H. Weber, Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 173–183.
  17. M. A. Porras, “Entropy-based definition of laser beam spot size,” Appl. Opt. 34, 8247–8251 (1995).
    [CrossRef] [PubMed]
  18. C. Paré, P.-A. Bélanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted beam,” Opt. Commun. 123, 679–693 (1996).
    [CrossRef]
  19. C. Paré, P.-A. Bélanger, “Propagation analysis of the truncated second-order moment,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 104–111 (1996).
    [CrossRef]
  20. R. Mästle, A. Giesen, “Characterization of hard edge diffracted beams,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 123–132 (1996).
    [CrossRef]
  21. F. Varnik, R. Mästle, A. Giesen, “Measurement of moments for diffracted beams: comparison with theory,” in Fourth International Workshop on Laser Beam and Optics Characterization, A. Giesen, M. Morin, eds. (Institut für Strahlwerkzeuge, Stuttgart, 1997), pp. 488–506.
  22. Y. Champagne, P.-A. Bélanger, “Method for measurement of realistic second-moment propagation parameters for non-ideal laser beams,” Opt. Quantum Electron. 27, 813–824 (1995).
    [CrossRef]
  23. T. F. Johnston, “Beam propagation factor (M2) measurement made as easy as it gets: the four-cuts method,” Appl. Opt. 37, 4840–4850 (1998).
    [CrossRef]
  24. S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, 2nd ed. (Addison-Wesley, Redwood City, Calif., 1991).
  25. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  26. S.-A. Amarande, “Beam propagation factor and kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).
  27. K.-M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
    [CrossRef]
  28. H. Weber, “Propagation of higher-order intensity moments in quadratic index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
    [CrossRef]

1998 (2)

G. Gbur, P. S. Carney, “Convergence criteria and optimization techniques for beam moments,” Pure Appl. Opt. 7, 1221–1230 (1998).
[CrossRef]

T. F. Johnston, “Beam propagation factor (M2) measurement made as easy as it gets: the four-cuts method,” Appl. Opt. 37, 4840–4850 (1998).
[CrossRef]

1996 (2)

C. Paré, P.-A. Bélanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted beam,” Opt. Commun. 123, 679–693 (1996).
[CrossRef]

S.-A. Amarande, “Beam propagation factor and kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).

1995 (2)

Y. Champagne, P.-A. Bélanger, “Method for measurement of realistic second-moment propagation parameters for non-ideal laser beams,” Opt. Quantum Electron. 27, 813–824 (1995).
[CrossRef]

M. A. Porras, “Entropy-based definition of laser beam spot size,” Appl. Opt. 34, 8247–8251 (1995).
[CrossRef] [PubMed]

1994 (3)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

G. N. Lawrence, “Proposed international standard for laser-beam quality falls short,” Laser Focus World 30(7), 109–114 (1994).

M. A. Porras, “Experimental investigation on aperture-diffracted laser beam characterization,” Opt. Commun. 109, 5–9 (1994).
[CrossRef]

1992 (3)

K.-M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

H. Weber, “Propagation of higher-order intensity moments in quadratic index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[CrossRef]

M. A. Porras, J. Alda, E. Bernabeu, “Complex beam parameter and ABCD law for non-Gaussian and nonspherical light beams,” Appl. Opt. 31, 6389–6402 (1992).
[CrossRef] [PubMed]

1991 (2)

P.-A. Bélanger, “Beam propagation and ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
[CrossRef]

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

1990 (1)

M. Forbes, “Laser beam spatial measurement: comparing instrument,” Lasers Optron. 9, 51–55 (1990).

Alda, J.

Amarande, S.-A.

S.-A. Amarande, “Beam propagation factor and kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).

Bélanger, P.-A.

C. Paré, P.-A. Bélanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted beam,” Opt. Commun. 123, 679–693 (1996).
[CrossRef]

Y. Champagne, P.-A. Bélanger, “Method for measurement of realistic second-moment propagation parameters for non-ideal laser beams,” Opt. Quantum Electron. 27, 813–824 (1995).
[CrossRef]

P.-A. Bélanger, “Beam propagation and ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
[CrossRef]

C. Paré, P.-A. Bélanger, “Propagation analysis of the truncated second-order moment,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 104–111 (1996).
[CrossRef]

P.-A. Bélanger, Y. Champagne, C. Paré, “The beam quality factor MQ2 of diffracted beams,” in Laser Beam Characterization, P. M. Mejías, H. Weber, Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 173–183.

Bernabeu, E.

Carney, P. S.

G. Gbur, P. S. Carney, “Convergence criteria and optimization techniques for beam moments,” Pure Appl. Opt. 7, 1221–1230 (1998).
[CrossRef]

Champagne, Y.

Y. Champagne, P.-A. Bélanger, “Method for measurement of realistic second-moment propagation parameters for non-ideal laser beams,” Opt. Quantum Electron. 27, 813–824 (1995).
[CrossRef]

P.-A. Bélanger, Y. Champagne, C. Paré, “The beam quality factor MQ2 of diffracted beams,” in Laser Beam Characterization, P. M. Mejías, H. Weber, Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 173–183.

Du, K.-M.

K.-M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

Forbes, M.

M. Forbes, “Laser beam spatial measurement: comparing instrument,” Lasers Optron. 9, 51–55 (1990).

Gbur, G.

G. Gbur, P. S. Carney, “Convergence criteria and optimization techniques for beam moments,” Pure Appl. Opt. 7, 1221–1230 (1998).
[CrossRef]

Giesen, A.

R. Mästle, A. Giesen, “Characterization of hard edge diffracted beams,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 123–132 (1996).
[CrossRef]

F. Varnik, R. Mästle, A. Giesen, “Measurement of moments for diffracted beams: comparison with theory,” in Fourth International Workshop on Laser Beam and Optics Characterization, A. Giesen, M. Morin, eds. (Institut für Strahlwerkzeuge, Stuttgart, 1997), pp. 488–506.

Gori, F.

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

Herziger, G.

K.-M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

Johnston, T. F.

T. F. Johnston, “Beam propagation factor (M2) measurement made as easy as it gets: the four-cuts method,” Appl. Opt. 37, 4840–4850 (1998).
[CrossRef]

M. W. Sasnett, T. F. Johnston, “Beam characterization of propagation attributes,” in Laser Beam Diagnostics, R. N. Hindy, Y. Kohanzadeh, eds., Proc. SPIE1414, 1–32 (1991).

Lawrence, G. N.

G. N. Lawrence, “Proposed international standard for laser-beam quality falls short,” Laser Focus World 30(7), 109–114 (1994).

Loosen, P.

K.-M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

Martínez-Herrero, R.

R. Martínez-Herrero, P. M. Mejías, “Parametric characterization of hard-edge diffracted beams,” in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 197–206.

Mästle, R.

R. Mästle, A. Giesen, “Characterization of hard edge diffracted beams,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 123–132 (1996).
[CrossRef]

F. Varnik, R. Mästle, A. Giesen, “Measurement of moments for diffracted beams: comparison with theory,” in Fourth International Workshop on Laser Beam and Optics Characterization, A. Giesen, M. Morin, eds. (Institut für Strahlwerkzeuge, Stuttgart, 1997), pp. 488–506.

Mejías, P. M.

R. Martínez-Herrero, P. M. Mejías, “Parametric characterization of hard-edge diffracted beams,” in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 197–206.

Paré, C.

C. Paré, P.-A. Bélanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted beam,” Opt. Commun. 123, 679–693 (1996).
[CrossRef]

P.-A. Bélanger, Y. Champagne, C. Paré, “The beam quality factor MQ2 of diffracted beams,” in Laser Beam Characterization, P. M. Mejías, H. Weber, Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 173–183.

C. Paré, P.-A. Bélanger, “Propagation analysis of the truncated second-order moment,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 104–111 (1996).
[CrossRef]

Porras, M. A.

Rühl, F.

K.-M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

Sasnett, M. W.

M. W. Sasnett, T. F. Johnston, “Beam characterization of propagation attributes,” in Laser Beam Diagnostics, R. N. Hindy, Y. Kohanzadeh, eds., Proc. SPIE1414, 1–32 (1991).

M. W. Sasnett, “Propagation of multimode laser beams—the M2 factor,” in The Physics and Technology of Laser Resonators, D. R. Hall, P. E. Jackson, eds. (Adam Hilger, New York, 1989), pp. 132–142.

Siegman, A. E.

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Varnik, F.

F. Varnik, R. Mästle, A. Giesen, “Measurement of moments for diffracted beams: comparison with theory,” in Fourth International Workshop on Laser Beam and Optics Characterization, A. Giesen, M. Morin, eds. (Institut für Strahlwerkzeuge, Stuttgart, 1997), pp. 488–506.

Weber, H.

H. Weber, “Propagation of higher-order intensity moments in quadratic index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[CrossRef]

Wolfram, S.

S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, 2nd ed. (Addison-Wesley, Redwood City, Calif., 1991).

Wright, D.

D. Wright, “Can we ignore the weak spatial range of diffracted beams?” in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 207–213.

Appl. Opt. (3)

IEEE J. Quantum Electron. (1)

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[CrossRef]

Laser Focus World (1)

G. N. Lawrence, “Proposed international standard for laser-beam quality falls short,” Laser Focus World 30(7), 109–114 (1994).

Lasers Optron. (1)

M. Forbes, “Laser beam spatial measurement: comparing instrument,” Lasers Optron. 9, 51–55 (1990).

Opt. Commun. (4)

C. Paré, P.-A. Bélanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted beam,” Opt. Commun. 123, 679–693 (1996).
[CrossRef]

M. A. Porras, “Experimental investigation on aperture-diffracted laser beam characterization,” Opt. Commun. 109, 5–9 (1994).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

S.-A. Amarande, “Beam propagation factor and kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).

Opt. Lett. (1)

Opt. Quantum Electron. (3)

K.-M. Du, G. Herziger, P. Loosen, F. Rühl, “Coherence and intensity moments of laser light,” Opt. Quantum Electron. 24, 1081–1093 (1992).
[CrossRef]

H. Weber, “Propagation of higher-order intensity moments in quadratic index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[CrossRef]

Y. Champagne, P.-A. Bélanger, “Method for measurement of realistic second-moment propagation parameters for non-ideal laser beams,” Opt. Quantum Electron. 27, 813–824 (1995).
[CrossRef]

Pure Appl. Opt. (1)

G. Gbur, P. S. Carney, “Convergence criteria and optimization techniques for beam moments,” Pure Appl. Opt. 7, 1221–1230 (1998).
[CrossRef]

Other (13)

S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, 2nd ed. (Addison-Wesley, Redwood City, Calif., 1991).

C. Paré, P.-A. Bélanger, “Propagation analysis of the truncated second-order moment,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 104–111 (1996).
[CrossRef]

R. Mästle, A. Giesen, “Characterization of hard edge diffracted beams,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 123–132 (1996).
[CrossRef]

F. Varnik, R. Mästle, A. Giesen, “Measurement of moments for diffracted beams: comparison with theory,” in Fourth International Workshop on Laser Beam and Optics Characterization, A. Giesen, M. Morin, eds. (Institut für Strahlwerkzeuge, Stuttgart, 1997), pp. 488–506.

M. W. Sasnett, T. F. Johnston, “Beam characterization of propagation attributes,” in Laser Beam Diagnostics, R. N. Hindy, Y. Kohanzadeh, eds., Proc. SPIE1414, 1–32 (1991).

International Standards Organization, “Test methods for laser beam parameters: beam widths, divergence angle, and beam propagation factor,” (Deutsches Institut für Normung, Pforzheim, Germany, 1994).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

M. W. Sasnett, “Propagation of multimode laser beams—the M2 factor,” in The Physics and Technology of Laser Resonators, D. R. Hall, P. E. Jackson, eds. (Adam Hilger, New York, 1989), pp. 132–142.

D. Wright, “Can we ignore the weak spatial range of diffracted beams?” in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 207–213.

R. Martínez-Herrero, P. M. Mejías, “Parametric characterization of hard-edge diffracted beams,” in Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 197–206.

P.-A. Bélanger, Y. Champagne, C. Paré, “The beam quality factor MQ2 of diffracted beams,” in Laser Beam Characterization, P. M. Mejías, H. Weber, Martínez-Herrero, A. González-Ureña, eds. (Sociedad Espanola de Optica, Madrid, 1993), pp. 173–183.

L. W. Austin, A. Giesen, eds., Beam Control, Diagnostic, Standards and Propagation, Proc. SPIE2375 (1995).

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Optical setup for characterization of laser beams.

Fig. 2
Fig. 2

Solid curves, propagation of the Gaussian beam diffracted by the hard-edge aperture with a/w 0 = 0.94 calculated for 10 transverse planes inside the measurement regions. Dashed curves, propagation of the same beam unaffected by the hard-edge aperture.

Fig. 3
Fig. 3

Propagation of the self-convergent width calculated with several values of the self-convergent-width factor F s (squares, F s = 2.19; circles, F s = 3.46; diamonds, F s = 4.73; asterisks, F s = 6) for the Gaussian beam: (a) unaffected by the aperture, (b), (c) diffracted by the aperture.

Fig. 4
Fig. 4

Beam-propagation parameters of the Gaussian beam versus the self-convergent-width factor F s (dashed curves, without a hard-edge aperture; thinner solid curves, aperture with a/w = 1.5; thicker solid curves, aperture with a/w = 0.94).

Fig. 5
Fig. 5

Solid curves, propagation of the Hermite–Gaussian beam of order n = 3 diffracted by a hard-edge aperture with a/w 0 = 1.73 calculated for 10 transverse planes inside the measurement region (dashed curves, the propagation of the same beam unaffected by the hard-edge aperture).

Fig. 6
Fig. 6

Propagation of the self-convergent width calculated with several values of the self-convergent-width factor (squares, F s = 2; circles, F s = 3; diamonds, F s = 4, asterisks, F s = 5) for the Hermite–Gaussian beam: (a) unaffected by aperture, (b), (c) diffracted by the aperture.

Fig. 7
Fig. 7

Beam-propagation parameters of the Hermite–Gaussian beam of order n = 3 versus the self-convergent-width factor F s (dashed curves, without a hard-edge aperture; thinner solid curves, aperture with a/w 0 = 1.90; thicker solid curves, aperture with a/w 0 = 1.73).

Fig. 8
Fig. 8

Solid curves, propagation of the multimode beam diffracted by a hard-edge aperture with a/w 0 = 1.56 calculated for 10 transverse planes inside the measurement region (dashed curves, the same beam unaffected by a hard-edge aperture).

Fig. 9
Fig. 9

Propagation of the self-convergent width calculated with several values of the self-convergent-width factor (squares, F s = 2.5; circles, F s = 3.3; diamonds; F s = 4.1; asterisks, F s = 4.9) for the Hermite–Gaussian beam: (a) unaffected by aperture, (b), (c) diffracted by the aperture.

Fig. 10
Fig. 10

Beam-propagation parameters of the flattened Gaussian beam of order N = 3 versus the self-convergent-width factor F s (dashed curves, without a hard-edge aperture, thinner solid curves, aperture with a/w 0 = 1.90; thicker solid curves, aperture with a/w 0 = 1.56).

Tables (1)

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Table 1 Values of Self-Convergent-Width Factor Fs

Equations (12)

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

W=2σ,
σ2=1P-xlimxlim Ix, zx2dx
P-xlimxlim Ix, zdx,
xlim=FsWFs,
Wz2=W02+λM2πW02z-z02,
ux=2πw021/4 exp-xw02.
ux=2πw021/42nn!-1/2Hn2x/w0exp-xw02,
M2=2n+1.
ux=n=0N cnu2nx,  N=0, 1, ,
cn=Aπ21/42n2n!1/2k=nN123·kk!2k!k-n!,
A=π2-1/4n=0N22n2n!k=nN123·kk!2k!k-n!-2-1/2,  N=0, 1, .
M2=n=0N2n+1|cn|22-4n=0Nn+1n+21/2Recncn+2*21/2.

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