Abstract

On the basis of generalized truncated second-order moments, a closed-form expression for the generalized M2 factor of hard-edge diffracted flattened Gaussian beams is derived that is determined by the beam order and the truncation parameter. Special cases are discussed. Moreover, it is shown that the M2 factor of truncated plane waves is equal to 43/3, independent of the aperture width.

© 2001 Optical Society of America

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References

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  1. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  2. A. E. Sigeman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.
  3. International Standards Organization (ISO) Document, (ISO, Geneva, Switzerland, 1995).
  4. R. Martı́nez-Herrero, P. M. Mejı́as, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18, 1669–1671 (1993).
    [CrossRef] [PubMed]
  5. R. Martı́nez-Herrero, P. M. Mejı́as, M. Arias, “Parametric characterization of coherent, lowest-order Gaussian beams propagating through hard-edged apertures,” Opt. Lett. 20, 124–126 (1995).
    [CrossRef] [PubMed]
  6. P.-A. Belanger, Y. Champagne, C. Pare, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
    [CrossRef]
  7. C. Pare, P.-A. Belanger, “Propagation law and quasi-invariance properties of the truncated second-order moment of a diffracted laser beam,” Opt. Commun. 123, 679–693 (1996).
    [CrossRef]
  8. M. Scholl, S. Muffer, O. Post, “Description of diffracted beams by weighted moments,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 112–122 (1996).
    [CrossRef]
  9. B. Lü, S. Luo, “Asymptotic approach to the truncated cosh-Gaussian beams,” Opt. Quantum Electron. (to be published).
  10. B. Lü, S. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
    [CrossRef]
  11. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  12. A. Erdelyi, Tables of Integral Transforms (McGraw–Hill, New York, 1954), Vol. 1, p. 387.
  13. B. Lü, B. Zhang, H. Ma, “Beam-propagation factor and mode-coherence coefficients of hyperbolic-cosine-Gaussian beams,” Opt. Lett. 24, 640–642 (1999).
    [CrossRef]
  14. B. Lü, B. Zhang, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999).
    [CrossRef]
  15. S. Luo, B. Lü, B. Zhang, “A comparison study on the propagation characteristics of flattened Gaussian beams and super-Gaussian beams,” Acta Phys. Sin. 48, 1446–1451 (1999) (in Chinese).
  16. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
    [CrossRef]
  17. S. Amarande, A. Giesen, H. Hügel, “Propagation analysis of self-convergent beam width and characterization of hard-edge diffracted beams,” Appl. Opt. 39, 3914–3924 (2000).
    [CrossRef]

2000 (2)

1999 (3)

1996 (2)

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

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

1995 (1)

1994 (2)

P.-A. Belanger, Y. Champagne, C. Pare, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[CrossRef]

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

1993 (1)

Amarande, S.

Ambrosini, D.

Arias, M.

Bagini, V.

Belanger, P.-A.

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

P.-A. Belanger, Y. Champagne, C. Pare, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[CrossRef]

Borghi, R.

Champagne, Y.

P.-A. Belanger, Y. Champagne, C. Pare, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[CrossRef]

Erdelyi, A.

A. Erdelyi, Tables of Integral Transforms (McGraw–Hill, New York, 1954), Vol. 1, p. 387.

Giesen, A.

Gori, F.

Hügel, H.

Lü, B.

B. Lü, S. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
[CrossRef]

B. Lü, B. Zhang, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999).
[CrossRef]

S. Luo, B. Lü, B. Zhang, “A comparison study on the propagation characteristics of flattened Gaussian beams and super-Gaussian beams,” Acta Phys. Sin. 48, 1446–1451 (1999) (in Chinese).

B. Lü, B. Zhang, H. Ma, “Beam-propagation factor and mode-coherence coefficients of hyperbolic-cosine-Gaussian beams,” Opt. Lett. 24, 640–642 (1999).
[CrossRef]

B. Lü, S. Luo, “Asymptotic approach to the truncated cosh-Gaussian beams,” Opt. Quantum Electron. (to be published).

Luo, S.

B. Lü, S. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
[CrossRef]

B. Lü, B. Zhang, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999).
[CrossRef]

S. Luo, B. Lü, B. Zhang, “A comparison study on the propagation characteristics of flattened Gaussian beams and super-Gaussian beams,” Acta Phys. Sin. 48, 1446–1451 (1999) (in Chinese).

B. Lü, S. Luo, “Asymptotic approach to the truncated cosh-Gaussian beams,” Opt. Quantum Electron. (to be published).

Ma, H.

Marti´nez-Herrero, R.

Meji´as, P. M.

Muffer, S.

M. Scholl, S. Muffer, O. Post, “Description of diffracted beams by weighted moments,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 112–122 (1996).
[CrossRef]

Pacileo, A. M.

Pare, C.

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

P.-A. Belanger, Y. Champagne, C. Pare, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[CrossRef]

Post, O.

M. Scholl, S. Muffer, O. Post, “Description of diffracted beams by weighted moments,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 112–122 (1996).
[CrossRef]

Santarsiero, M.

Schirripa Spagnolo, G.

Scholl, M.

M. Scholl, S. Muffer, O. Post, “Description of diffracted beams by weighted moments,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 112–122 (1996).
[CrossRef]

Siegman, A. E.

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

Sigeman, A. E.

A. E. Sigeman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.

Zhang, B.

Acta Phys. Sin. (1)

S. Luo, B. Lü, B. Zhang, “A comparison study on the propagation characteristics of flattened Gaussian beams and super-Gaussian beams,” Acta Phys. Sin. 48, 1446–1451 (1999) (in Chinese).

Appl. Opt. (2)

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

Opt. Commun. (4)

P.-A. Belanger, Y. Champagne, C. Pare, “Beam propagation factor of diffracted laser beams,” Opt. Commun. 105, 233–242 (1994).
[CrossRef]

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

B. Lü, S. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. 178, 275–281 (2000).
[CrossRef]

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

Opt. Lett. (3)

Other (6)

A. Erdelyi, Tables of Integral Transforms (McGraw–Hill, New York, 1954), Vol. 1, p. 387.

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

A. E. Sigeman, “How to (maybe) measure laser beam quality,” in DPSS Lasers: Application and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.

International Standards Organization (ISO) Document, (ISO, Geneva, Switzerland, 1995).

M. Scholl, S. Muffer, O. Post, “Description of diffracted beams by weighted moments,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 112–122 (1996).
[CrossRef]

B. Lü, S. Luo, “Asymptotic approach to the truncated cosh-Gaussian beams,” Opt. Quantum Electron. (to be published).

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Figures (2)

Fig. 1
Fig. 1

(a) Variation of the power fraction p with truncation parameter β. (b) Irradiance profiles of flattened Gaussian beams I(x, 0) at the z=0 plane.

Fig. 2
Fig. 2

Generalized M2 factor of truncated flattened Gaussian beams MG2 versus truncation parameter β.

Equations (31)

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E(x, 0)=exp-(N+1)x2w02n=0N1n!(N+1)x2w02n,
x2=1/I0-aax2|E|2dx,
u2=1k2I0-aa|E|2dx+4k2I0a [|E(a)|2+|E(-a)|2],
xu=12ikI0-aa{x[E(x)]*E(x)-xE(x)E*(x)}dx,
I0=-aa|E(x, 0)|2dx,
p=I0-|E(x, 0)|2dx.
γ(α, x)=0xexp(-t)tα-1dt,
I0=2-1/2w0(N+1)1/2 A,
p=An=0Nm=0N2-(n+m)n!m! Γn+m+12,
A=n=0Nm=0N2-(n+m)n!m! γn+m+12, 2(N+1)β2,
β=a/w0(truncationparameter).
x2=2-(3/2)w03(N+1)3/2BI0,
u2=(N+1)1/2w021/2k2I0C+42(N+1)1/2IDβ,
xu=0,
B=n=0Nm=0N2-(n+m)n!m! γn+m+32, 2(N+1)β2,
C=2-2N(N!)2 γ2N+32, 2(N+1)β2,
ID=|E(a, 0)|2=exp[-2(N+1)β2]
×n=0N1n! [(N+1)β2]n2.
x2 u2-xu=(MG2/2k)2
MG2=2{B[C+42ID/β(N+1)1/2]}1/2A.
MG2
=M2=21-NN!×Γ2N+32n=0Nm=0N2-(n+m)n!m! Γn+m+321/2n=0Nm=0N2-(n+m)n!m! Γn+m+12,
γ(α+1, x)=αγ(a, x)-xaexp(-x)
γ(1/2, x2)=πerf(x),
MG2=[1+4(2/π)1/2(2/β-β)erf-1(21/2β)exp(-2β2)
+(8/π)(β2-4)erf -2(21/2β)exp(-4β2)]1/2.
E(x, 0)=1xa0x>a
x2=a2/3,
u2=4k2a2,
xu=0,
MG2=43/3,

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