Abstract

We point out that the expression of the cross-spectral density (CSD) of a partially coherent flat-topped beam (PCFB) given by Ge et al. [Appl. Opt. 43, 4732 (2004)] is incorrect. The results show that the M2 factor derived based on Ge's expression leads to M2<1 in some cases, which is physically unacceptable. A new expression of the CSD for a PCFB is given.

© 2006 Optical Society of America

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References

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  1. D. Ge, Y. Cai, and Q. Lin, "Partially coherent flat-topped beam and its propagation," Appl. Opt. 43, 4732-4738 (2004).
    [CrossRef] [PubMed]
  2. F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
    [CrossRef]
  3. M. Santarsiero and F. Gori, "Spreading properties of beams radiated by partially coherent Schell-model sources," J. Opt. Soc. Am. A 16, 106-112 (1999).
    [CrossRef]
  4. J. Serna, R. Martínez-Herrero, and P. M. Mejias, "Parametric characterization of general partially coherent beams propagating through ABCD optical systems," J. Opt. Soc. Am. A 8, 1094-1098 (1991).
    [CrossRef]
  5. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  6. A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2-14 (1990).
    [CrossRef]
  7. Y. Li, "Light beams with flat-topped profiles," Opt. Lett. 27, 1007-1009 (2002).
    [CrossRef]

2004

2002

1999

1995

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

1991

1990

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

Cai, Y.

Ge, D.

Gori, F.

M. Santarsiero and F. Gori, "Spreading properties of beams radiated by partially coherent Schell-model sources," J. Opt. Soc. Am. A 16, 106-112 (1999).
[CrossRef]

F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

Li, Y.

Lin, Q.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Martínez-Herrero, R.

Mejias, P. M.

Santarsiero, M.

M. Santarsiero and F. Gori, "Spreading properties of beams radiated by partially coherent Schell-model sources," J. Opt. Soc. Am. A 16, 106-112 (1999).
[CrossRef]

F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

Serna, J.

Siegman, A. E.

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

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
[CrossRef]

Opt. Lett.

Proc. SPIE

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

Other

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

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Figures (2)

Fig. 1
Fig. 1

Beam quality factor M 2 as a function of ratio η and beam order N based on Eq. (1).

Fig. 2
Fig. 2

Beam quality factor M 2 as a function of ratio η and beam order N based on Eq. (11).

Equations (12)

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W ( x 1 , x 2 ) = n = 1 N ( 1 ) n 1 N ( N n ) exp { n [ x 1 2 + x 2 2 4 ω 0 2 + ( x 1 x 2 ) 2 2 σ 0 2 ] } ,
x 2 = 1 I x 2 W ( x , x ) d x ,
u 2 = 1 k 2 I 2 W ( x 1 , x 2 ) x 1 x 2 | x 1 = x 2 = x d x ,
x u = 1 2 i k I [ x 1 W ( x 1 , x 2 ) x 2 x 2 W ( x 1 , x 2 ) x 1 ] x 1 = x 2 = x d x ,
I = W ( x , x ) d x .
M 2 = 2 k [ x 2 u 2 x u 2 ] 1 / 2 .
M 2 = [ n = 1 N m = 1 N ( 1 ) n + m 2 m N 2 n 3 ( N n ) ( N m ) ( 1 + 4 η 2 ) ] 1 / 2 / n = 1 N [ ( 1 ) n 1 / N n ]( N n ) ,
g ( x ) = n = 1 N exp [ ( n x 2 / 2 σ 0 2 ) ] N .
E N ( x , 0 ) = n = 1 N ( 1 ) n 1 N ( N n ) exp ( n x 2 ω 0 2 ) ,
W ( x 1 , x 2 ) = E N ( x 1 , 0 ) E N ( x 2 , 0 ) g ( x 1 x 2 ) .
W ( x 1 , x 2 ) = n = 1 N m = 1 N ( 1 ) n + m 2 N 4 ( N n ) ( N m ) × exp { [ n x 1 2 + m x 2 2 4 ω 0 2 + ( n + m ) ( x 1 - x 2 ) 2 4 σ 0 2 ] } .
M 2 = 2 { n = 1 N m = 1 N ( 1 ) n + m 2 N 2 n + m 3 ( N n ) ( N m ) [ n m + ( n + m ) 2 η 2 ] n = 1 N m = 1 N ( 1 ) n + m 2 N 2 n + m 3 × ( N n ) ( N m ) } 1 / 2 / n = 1 N m = 1 N ( 1 ) n + m 2 N 2 n + m ( N n ) ( N m ) .

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