Abstract

We point out that there is no oddness in the expression of the cross-spectral density of a partially coherent flat-topped beam given by Ge et al. [Appl. Opt. 43, 4732 (2004)]. The criticism of the comment by Wu et al. [Appl. Opt. 45, 366 (2006)] is not appropriate because no one has proved that the M2 factor as defined by them must be greater than or equal to 1. We propose a new definition of the M2 factor that really confines the propagation of partially coherent beams. The new definition leads to M2>1 for the partially coherent beam given by Ge et al.

© 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. G. Wu, H. Guo, and D. Deng, "Comment on 'Partially coherent flat-topped beam and its propagation,"' Appl. Opt. 45, 366-368 (2006).
    [CrossRef] [PubMed]
  3. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 1999).
  4. A. E. Siegman, "New developments in laser resonators," in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2-14 (1990).
    [CrossRef]
  5. M. A. Porras, "Non-paraxial vectorial moment theory of light beam propagation," Opt. Commun. 127, 79-95 (1996).
    [CrossRef]
  6. M. A. Porras, "Finiteness and propagation law of the power density second-order moment for diffracted scalar light beams," Optik (Stuttgart) 110, 417-420 (1999).
  7. F. Gori and M. Santarsiero, "The change of width for a partially coherent beam on paraxial propagation," Opt. Commun. 82, 197-203 (1991).
    [CrossRef]
  8. 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]
  9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

2006 (1)

2004 (1)

1999 (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]

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

M. A. Porras, "Finiteness and propagation law of the power density second-order moment for diffracted scalar light beams," Optik (Stuttgart) 110, 417-420 (1999).

1996 (1)

M. A. Porras, "Non-paraxial vectorial moment theory of light beam propagation," Opt. Commun. 127, 79-95 (1996).
[CrossRef]

1995 (1)

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

1991 (1)

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

1990 (1)

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

Born, M.

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

Cai, Y.

Deng, D.

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]

Guo, H.

Lin, Q.

Mandel, L.

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

Porras, M. A.

M. A. Porras, "Finiteness and propagation law of the power density second-order moment for diffracted scalar light beams," Optik (Stuttgart) 110, 417-420 (1999).

M. A. Porras, "Non-paraxial vectorial moment theory of light beam propagation," Opt. Commun. 127, 79-95 (1996).
[CrossRef]

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]

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.

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

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

Wu, G.

Appl. Opt. (2)

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

Opt. Commun. (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]

M. A. Porras, "Non-paraxial vectorial moment theory of light beam propagation," Opt. Commun. 127, 79-95 (1996).
[CrossRef]

Optik (1)

M. A. Porras, "Finiteness and propagation law of the power density second-order moment for diffracted scalar light beams," Optik (Stuttgart) 110, 417-420 (1999).

Proc. SPIE (1)

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

Other (2)

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

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

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Figures (1)

Fig. 1
Fig. 1

M W 2 factor of a PCFB versus 1∕η for several values of N.

Equations (26)

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2 E ( r ) + k 2 E ( r ) = 0 ,
2 E ( r ) x 2 + 2 E ( r ) y 2 2 i k E ( r ) z = 0.
M 2 = 4 π Δ x Δ p ,
Δ x = [ 1 I ( x x ¯ ) 2 | E ( x ) 2 | d x ] 1 / 2 ,
Δ p = [ 1 I ( p p ¯ ) 2 | E ˜ ( p ) | 2 d p ] 1 / 2 ,
E ˜ ( p ) = E ( x ) exp ( 2 π i p x ) d x .
x ¯ = 1 I x | E ( x ) | 2 d x ,
p ¯ = 1 I p | E ˜ ( p ) | 2 d p ,
I = | E ( x ) | 2 d x = | E ˜ ( p ) | 2 d p .
M e 2 = 4 π Δ x Δ p ,
Δ x = [ 1 N ( x x ¯ ) 2 W ( x , x ) d x ] 1 / 2 ,
Δ p = [ 1 N ( p p ¯ ) 2 W ˜ ( p , p ) d p ] 1 / 2 ,
N = W ( x , x ) d x = W ˜ ( p , p ) d p ,
M e 2 = 2 [ W ( x , x ) d x ] 1 { x 2 W ( x , x ) d x × [ 2 W ( x 1 , x 2 ) x 1 x 2 ] x , x d x } 1 / 2 .
1 2 W ( r 1 , r 2 ) + k 2 W ( r 1 , r 2 ) = 0 ,
2 2 W ( r 1 , r 2 ) + k 2 W ( r 1 , r 2 ) = 0.
2 W ( r 1 , r 2 ) x i 2 + 2 W ( r 1 , r 2 ) y i 2 2 i k W ( r 1 , r 2 ) z i = 0 ( i = 1 , 2 ) .
M W 2 = 4 π Δ x i Δ p i ( i = 1 , 2 ) ,
Δ x i = [ 1 N ( x i - x ¯ i ) 2 | W ( x 1 , x 2 ) | 2 d x 1 d x 2 ] 1 / 2 ,
Δ p i = [ 1 N ( p i p ¯ i ) 2 | W ˜ ( p 1 , p 2 ) | 2 d p 1 d p 2 ] 1 / 2 ,
W ˜ ( p 1 , p 2 ) = W ( x 1 , x 2 ) exp [ 2 π i ( p 1 x 1 + p 2 x 2 ) ] d x 1 d x 2 ,
N = | W ( x 1 , x 2 ) | 2 d x 1 d x 2 = - | W ˜ ( p 1 , p 2 ) | 2 d p 1 d p 2 ,
x ¯ i = 1 N - - x i | W ( x 1 , x 2 ) | 2 d x 1 d x 2 ,
p ¯ i = 1 N p i | W ˜ ( p 1 , p 2 ) | 2 d p 1 d p 2 .
W ( x 1 , x 2 ) = n = 1 N ( 1 ) n 1 N ( N n ) exp { n [ x 1 2 + x 2 2 4 w 0 2 + ( x 1 x 2 ) 2 2 σ 0 2 ] } .
M W 2 = 2 ( 1 + 1 η 2 + η 4 / 4 ) 1 / 2 [ n = 1 N m = 1 N ( 1 ) n + m 2 N 2 ( n + m ) 2 ( N n ) × ( N m ) n = 1 N m = 1 N ( 1 ) n + m 2 n m N 2 ( n + m ) 2 ( 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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