Abstract

We introduce new definitions of spot size, mean curvature radius, divergence angle, and quality of laser beams that are based on Shannon’s information-entropy formula and study their properties.

© 1995 Optical Society of America

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References

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  1. A. E. Siegman, “Defining and measuring laser beam parameters,” in Proceedings of the First Workshop on Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Urena, eds. (Sociedad Espanola de Óptica, Madrid, 1993), pp. 1–21.
  2. G. Herziger, M. Scholl, P. Loosen, “Beam characterization for materials processing,” in Proceedings of the Second Workshop on Laser Beam Characterization, H. Weber, N. Reng, J. Lüdtke, P. M. Mejías, eds. (Festkörper Laser Institut, Berlin, 1994), pp. 304–319.
  3. A. E. Siegman, M. W. Sasnett, T. F. Johnston, “Choice of clip levels for beam width measurements using knife-edge techniques,” IEEE J. Quantum Electron. 27, 1098–1104 (1991).
    [CrossRef]
  4. D. Wright, “Beamwidths of a diffracted laser using four proposed methods,” Opt. Quantum Electron. 24, 1129–1135 (1992).
    [CrossRef]
  5. P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
    [CrossRef] [PubMed]
  6. 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]
  7. M. A. Porras, “Experimental investigation on aperture-diffracted laser beam characterization,” Opt. Commun. 109, 5–9 (1994).
    [CrossRef]
  8. N. Reng, B. Eppich, “Definition and measurements of high-power laser beam parameters,” Opt. Quantum Electron. 24, 973–992 (1992).
    [CrossRef]
  9. C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. of Illinois Press, Urbana, Ill., 1949).
  10. I. Bialynicki-Birula, J. Mycielski, “Uncertainty relations for information entropy in wave mechanics,” Commun. Math. Phys. 44, 129–132 (1975).
    [CrossRef]
  11. H. Maassen, J. B. M. Uffink, “Generalized entropic uncertainty relations,” Phys. Rev. Lett. 60, 1103–1106 (1988).
    [CrossRef] [PubMed]
  12. By introducing a factor (8/eΠ)1/2 in the definition of DS, we could calibrate DS so as to yield 2w for the Gaussian profile. In this case we would have DS = 1.936a for the slit. It seems, however, more natural to assign 2a to the slit spot size than 2w to the Gaussian one.
  13. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]

1994

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

1992

N. Reng, B. Eppich, “Definition and measurements of high-power laser beam parameters,” Opt. Quantum Electron. 24, 973–992 (1992).
[CrossRef]

D. Wright, “Beamwidths of a diffracted laser using four proposed methods,” Opt. Quantum Electron. 24, 1129–1135 (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

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

A. E. Siegman, M. W. Sasnett, T. F. Johnston, “Choice of clip levels for beam width measurements using knife-edge techniques,” IEEE J. Quantum Electron. 27, 1098–1104 (1991).
[CrossRef]

1988

H. Maassen, J. B. M. Uffink, “Generalized entropic uncertainty relations,” Phys. Rev. Lett. 60, 1103–1106 (1988).
[CrossRef] [PubMed]

1975

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

I. Bialynicki-Birula, J. Mycielski, “Uncertainty relations for information entropy in wave mechanics,” Commun. Math. Phys. 44, 129–132 (1975).
[CrossRef]

Alda, J.

Bélanger, P. A.

Bernabeu, E.

Bialynicki-Birula, I.

I. Bialynicki-Birula, J. Mycielski, “Uncertainty relations for information entropy in wave mechanics,” Commun. Math. Phys. 44, 129–132 (1975).
[CrossRef]

Eppich, B.

N. Reng, B. Eppich, “Definition and measurements of high-power laser beam parameters,” Opt. Quantum Electron. 24, 973–992 (1992).
[CrossRef]

Herziger, G.

G. Herziger, M. Scholl, P. Loosen, “Beam characterization for materials processing,” in Proceedings of the Second Workshop on Laser Beam Characterization, H. Weber, N. Reng, J. Lüdtke, P. M. Mejías, eds. (Festkörper Laser Institut, Berlin, 1994), pp. 304–319.

Johnston, T. F.

A. E. Siegman, M. W. Sasnett, T. F. Johnston, “Choice of clip levels for beam width measurements using knife-edge techniques,” IEEE J. Quantum Electron. 27, 1098–1104 (1991).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Loosen, P.

G. Herziger, M. Scholl, P. Loosen, “Beam characterization for materials processing,” in Proceedings of the Second Workshop on Laser Beam Characterization, H. Weber, N. Reng, J. Lüdtke, P. M. Mejías, eds. (Festkörper Laser Institut, Berlin, 1994), pp. 304–319.

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Maassen, H.

H. Maassen, J. B. M. Uffink, “Generalized entropic uncertainty relations,” Phys. Rev. Lett. 60, 1103–1106 (1988).
[CrossRef] [PubMed]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mycielski, J.

I. Bialynicki-Birula, J. Mycielski, “Uncertainty relations for information entropy in wave mechanics,” Commun. Math. Phys. 44, 129–132 (1975).
[CrossRef]

Porras, M. A.

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

Reng, N.

N. Reng, B. Eppich, “Definition and measurements of high-power laser beam parameters,” Opt. Quantum Electron. 24, 973–992 (1992).
[CrossRef]

Sasnett, M. W.

A. E. Siegman, M. W. Sasnett, T. F. Johnston, “Choice of clip levels for beam width measurements using knife-edge techniques,” IEEE J. Quantum Electron. 27, 1098–1104 (1991).
[CrossRef]

Scholl, M.

G. Herziger, M. Scholl, P. Loosen, “Beam characterization for materials processing,” in Proceedings of the Second Workshop on Laser Beam Characterization, H. Weber, N. Reng, J. Lüdtke, P. M. Mejías, eds. (Festkörper Laser Institut, Berlin, 1994), pp. 304–319.

Shannon, C. E.

C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. of Illinois Press, Urbana, Ill., 1949).

Siegman, A. E.

A. E. Siegman, M. W. Sasnett, T. F. Johnston, “Choice of clip levels for beam width measurements using knife-edge techniques,” IEEE J. Quantum Electron. 27, 1098–1104 (1991).
[CrossRef]

A. E. Siegman, “Defining and measuring laser beam parameters,” in Proceedings of the First Workshop on Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Urena, eds. (Sociedad Espanola de Óptica, Madrid, 1993), pp. 1–21.

Uffink, J. B. M.

H. Maassen, J. B. M. Uffink, “Generalized entropic uncertainty relations,” Phys. Rev. Lett. 60, 1103–1106 (1988).
[CrossRef] [PubMed]

Weaver, W.

C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. of Illinois Press, Urbana, Ill., 1949).

Wright, D.

D. Wright, “Beamwidths of a diffracted laser using four proposed methods,” Opt. Quantum Electron. 24, 1129–1135 (1992).
[CrossRef]

Appl. Opt.

Commun. Math. Phys.

I. Bialynicki-Birula, J. Mycielski, “Uncertainty relations for information entropy in wave mechanics,” Commun. Math. Phys. 44, 129–132 (1975).
[CrossRef]

IEEE J. Quantum Electron.

A. E. Siegman, M. W. Sasnett, T. F. Johnston, “Choice of clip levels for beam width measurements using knife-edge techniques,” IEEE J. Quantum Electron. 27, 1098–1104 (1991).
[CrossRef]

Opt. Commun.

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

Opt. Lett.

Opt. Quantum Electron.

N. Reng, B. Eppich, “Definition and measurements of high-power laser beam parameters,” Opt. Quantum Electron. 24, 973–992 (1992).
[CrossRef]

D. Wright, “Beamwidths of a diffracted laser using four proposed methods,” Opt. Quantum Electron. 24, 1129–1135 (1992).
[CrossRef]

Phys. Rev. A

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Phys. Rev. Lett.

H. Maassen, J. B. M. Uffink, “Generalized entropic uncertainty relations,” Phys. Rev. Lett. 60, 1103–1106 (1988).
[CrossRef] [PubMed]

Other

By introducing a factor (8/eΠ)1/2 in the definition of DS, we could calibrate DS so as to yield 2w for the Gaussian profile. In this case we would have DS = 1.936a for the slit. It seems, however, more natural to assign 2a to the slit spot size than 2w to the Gaussian one.

C. E. Shannon, W. Weaver, The Mathematical Theory of Communication (U. of Illinois Press, Urbana, Ill., 1949).

A. E. Siegman, “Defining and measuring laser beam parameters,” in Proceedings of the First Workshop on Laser Beam Characterization, P. M. Mejías, H. Weber, R. Martínez-Herrero, A. González-Urena, eds. (Sociedad Espanola de Óptica, Madrid, 1993), pp. 1–21.

G. Herziger, M. Scholl, P. Loosen, “Beam characterization for materials processing,” in Proceedings of the Second Workshop on Laser Beam Characterization, H. Weber, N. Reng, J. Lüdtke, P. M. Mejías, eds. (Festkörper Laser Institut, Berlin, 1994), pp. 304–319.

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Figures (4)

Fig. 1
Fig. 1

a, SG profiles; b, Hermite–Gauss mode of order 3; c, pedestal; d, noisy Gaussian profile. The lengths of the horizontal arrows are equal to the corresponding entropic spot sizes. The vertical scale is arbitrary.

Fig. 2
Fig. 2

a, D S and D sm for the SG profies Ψ p ∝ exp(−|x/w| p ). b, D S and D sm for the Hermite–Gauss modes Ψ H m ( 2 x / w ) exp ( - x 2 / w 2 ) . c, D S (solid curves) and D sm (dashed curves) for pedestal beams as a function of the pedestal width and for several amounts of power in the pedestal. The parameter w is the Gaussian width of the central Gaussian lobe. d, D S (solid curves) and D sm (dashed curves) for noisy Gaussian profiles as a function of the integration area for several noise-to-signal ratios. Both spot sizes are normalized to the noise-free Gaussian spot sizes.

Fig. 3
Fig. 3

a, D S of a diffracted (solid curve) and a nondiffracted (dashed curve) Gaussian beam of initial Gaussian width 2w = 2 mm, at several planes behind the diffracting aperture of width 2a = 3 mm. The wavelength is 633 nm. b, Propagation of D S with distance. The dashed curve corresponds to the initial distribution (at z = 0) Ψ = exp(−x 4/w 4)exp(−iΠx 2R), with R = −1000 mm, w = 1 mm, λ = 633 nm. The solid curve corresponds to the same initial distribution truncated at x = 1.1 mm. c, Entropic spot size and divergence of truncated Gaussian beams as functions of the truncation parameter a/w. d, Entropic quality factors of truncated Gaussian beams as a function of the truncation parameter a/w.

Fig. 4
Fig. 4

a, Intensity change on free propagation of the initial profile Ψ(x) = exp(−x 2/w 2)exp[−iΠ(x 2R)]exp[−2Πia(x 4/w 4)] (w = 1 mm, R = −500 mm, λ = 633 nm, a = 1). b, D S (z) (solid curve) and D sm(z) (dashed curve) for the same beam. c, Intensity profiles at the entropic waist (z ≃ 700 mm) (solid curve) and the second-moment waist (z ≃ 41 mm) (dashed curve).

Equations (8)

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D sm = 2 [ 4 P - d x ( x - x 0 ) 2 Ψ ( x ) 2 ] 1 / 2 ,
P = - d x Ψ ( x ) 2 ,             x 0 = 1 P - d x x Ψ ( x ) 2 ,
D S = exp S ,             S = - - d x Ψ ( x ) 2 P ln Ψ ( x ) 2 P .
D S 2 p [ - 1 + ( 1 / s p ) ] exp ( 1 / p ) Γ ( 1 / p ) x - x 0 p Ψ 1 / p ,
D S Γ ( 1 / p ) w 2 [ - 1 + ( 1 / p ) ] p exp ( 2 w p x - a p Ψ ) ,
A 2 z + 1 k φ x A 2 x = - 1 k A 2 2 φ x 2 ,
d S d z = 1 P - Ψ 2 ( 1 k 2 φ x 2 ) d x 1 R S ,
d S d z = - 1 P - d Ψ 2 d τ d x ,

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