Abstract

The beam-quality factor M2 for the fundamental LP01 mode of a step-index fiber is calculated in the presence of gain, in a closed form, as a function of the complex generalized V number. It is shown that the M2 value of a single-mode gain-guided fiber laser can be arbitrarily large. The results are important for the interpretation of the beam-quality measurements in recent experiments on single-mode gain-guided fiber lasers.

© 2010 Optical Society of America

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References

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  1. A.-E. Siegman, J. Opt. Soc. Am. A 20, 1617 (2003).
    [CrossRef]
  2. A.-E. Siegman, Y. Chen, V. Sudesh, M.-C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, Appl. Phys. Lett. 89, 251101 (2006).
    [CrossRef]
  3. T. McComb, V. Sudesh, Y. Chen, M. Bass, M.-C. Richardson, J. Ballato, and A.-E. Siegman, in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2007), paper TuK1, pp. 242–243.
  4. Y. Chen, V. Sudesh, T. McComb, M.-C. Richardson, M. Bass, and J. Ballato, J. Opt. Soc. Am. B 24, 1683 (2007).
    [CrossRef]
  5. Y. Chen, T. McComb, V. Sudesh, M.-C. Richardson, and M. Bass, Opt. Lett. 32, 2505 (2007).
    [CrossRef] [PubMed]
  6. A.-E. Siegman, J. Opt. Soc. Am. B 24, 1677 (2007).
    [CrossRef]
  7. A. Mafi and J. V. Moloney, J. Lightwave Technol. 23, 2267 (2005).
    [CrossRef]
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    [CrossRef]
  9. T.-F. Johnston, Appl. Opt. 37, 4840 (1998).
    [CrossRef]

2007 (3)

2006 (2)

A.-E. Siegman, Y. Chen, V. Sudesh, M.-C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

H. Yoda, P. Polynkin, and M. Mansuripur, J. Lightwave Technol. 24, 1350 (2006).
[CrossRef]

2005 (1)

2003 (1)

1998 (1)

Ballato, J.

Y. Chen, V. Sudesh, T. McComb, M.-C. Richardson, M. Bass, and J. Ballato, J. Opt. Soc. Am. B 24, 1683 (2007).
[CrossRef]

A.-E. Siegman, Y. Chen, V. Sudesh, M.-C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

T. McComb, V. Sudesh, Y. Chen, M. Bass, M.-C. Richardson, J. Ballato, and A.-E. Siegman, in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2007), paper TuK1, pp. 242–243.

Bass, M.

Y. Chen, V. Sudesh, T. McComb, M.-C. Richardson, M. Bass, and J. Ballato, J. Opt. Soc. Am. B 24, 1683 (2007).
[CrossRef]

Y. Chen, T. McComb, V. Sudesh, M.-C. Richardson, and M. Bass, Opt. Lett. 32, 2505 (2007).
[CrossRef] [PubMed]

A.-E. Siegman, Y. Chen, V. Sudesh, M.-C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

T. McComb, V. Sudesh, Y. Chen, M. Bass, M.-C. Richardson, J. Ballato, and A.-E. Siegman, in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2007), paper TuK1, pp. 242–243.

Chen, Y.

Y. Chen, T. McComb, V. Sudesh, M.-C. Richardson, and M. Bass, Opt. Lett. 32, 2505 (2007).
[CrossRef] [PubMed]

Y. Chen, V. Sudesh, T. McComb, M.-C. Richardson, M. Bass, and J. Ballato, J. Opt. Soc. Am. B 24, 1683 (2007).
[CrossRef]

A.-E. Siegman, Y. Chen, V. Sudesh, M.-C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

T. McComb, V. Sudesh, Y. Chen, M. Bass, M.-C. Richardson, J. Ballato, and A.-E. Siegman, in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2007), paper TuK1, pp. 242–243.

Foy, P.

A.-E. Siegman, Y. Chen, V. Sudesh, M.-C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

Hawkins, W.

A.-E. Siegman, Y. Chen, V. Sudesh, M.-C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

Johnston, T.-F.

Mafi, A.

Mansuripur, M.

McComb, T.

Y. Chen, V. Sudesh, T. McComb, M.-C. Richardson, M. Bass, and J. Ballato, J. Opt. Soc. Am. B 24, 1683 (2007).
[CrossRef]

Y. Chen, T. McComb, V. Sudesh, M.-C. Richardson, and M. Bass, Opt. Lett. 32, 2505 (2007).
[CrossRef] [PubMed]

T. McComb, V. Sudesh, Y. Chen, M. Bass, M.-C. Richardson, J. Ballato, and A.-E. Siegman, in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2007), paper TuK1, pp. 242–243.

Moloney, J. V.

Polynkin, P.

Richardson, M.-C.

Y. Chen, V. Sudesh, T. McComb, M.-C. Richardson, M. Bass, and J. Ballato, J. Opt. Soc. Am. B 24, 1683 (2007).
[CrossRef]

Y. Chen, T. McComb, V. Sudesh, M.-C. Richardson, and M. Bass, Opt. Lett. 32, 2505 (2007).
[CrossRef] [PubMed]

A.-E. Siegman, Y. Chen, V. Sudesh, M.-C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

T. McComb, V. Sudesh, Y. Chen, M. Bass, M.-C. Richardson, J. Ballato, and A.-E. Siegman, in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2007), paper TuK1, pp. 242–243.

Siegman, A.-E.

A.-E. Siegman, J. Opt. Soc. Am. B 24, 1677 (2007).
[CrossRef]

A.-E. Siegman, Y. Chen, V. Sudesh, M.-C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

A.-E. Siegman, J. Opt. Soc. Am. A 20, 1617 (2003).
[CrossRef]

T. McComb, V. Sudesh, Y. Chen, M. Bass, M.-C. Richardson, J. Ballato, and A.-E. Siegman, in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2007), paper TuK1, pp. 242–243.

Sudesh, V.

Y. Chen, V. Sudesh, T. McComb, M.-C. Richardson, M. Bass, and J. Ballato, J. Opt. Soc. Am. B 24, 1683 (2007).
[CrossRef]

Y. Chen, T. McComb, V. Sudesh, M.-C. Richardson, and M. Bass, Opt. Lett. 32, 2505 (2007).
[CrossRef] [PubMed]

A.-E. Siegman, Y. Chen, V. Sudesh, M.-C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

T. McComb, V. Sudesh, Y. Chen, M. Bass, M.-C. Richardson, J. Ballato, and A.-E. Siegman, in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2007), paper TuK1, pp. 242–243.

Yoda, H.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A.-E. Siegman, Y. Chen, V. Sudesh, M.-C. Richardson, M. Bass, P. Foy, W. Hawkins, and J. Ballato, Appl. Phys. Lett. 89, 251101 (2006).
[CrossRef]

J. Lightwave Technol. (2)

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

J. Opt. Soc. Am. B (2)

Opt. Lett. (1)

Other (1)

T. McComb, V. Sudesh, Y. Chen, M. Bass, M.-C. Richardson, J. Ballato, and A.-E. Siegman, in 20th Annual Meeting of the IEEE Lasers and Electro-Optics Society (IEEE, 2007), paper TuK1, pp. 242–243.

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Figures (1)

Fig. 1
Fig. 1

Contour plot of M 2 as a function of Δ N and G in the (a) GG + IG region, (b) GG + IG region but zoomed in near the origin, and (c) GG + IAG region. The dashed lines represent the LP01 guiding threshold. (d) M 2 as a function of G for Δ N = 1000 (solid lines) and Δ N = 100 (dotted lines). The lines relate to the apodized beam and are marked by different values of α from the apodization function.

Equations (20)

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V ˜ 2 = Δ N + i G .
Δ N = ( 2 π a / λ ) 2 2 n 0 Δ n ,
G = ( 2 π a / λ ) 2 ( n 0 λ / 2 π ) g ,
E ( x , y , z 0 ) = { N ˜ J 0 ( u r / a ) / J 0 ( u ) , r a , N ˜ K 0 ( w r / a ) / K 0 ( w ) , r a .
u J 1 ( u ) / J 0 ( u ) = w K 1 ( w ) / K 0 ( w ) ,
u 2 + w 2 = V ˜ 2 .
d S | E ( x , y , z ) | 2 = 1 ,
x ( z ) = d S x | E ( x , y , z ) | 2 ,
σ x 2 ( z ) = d S ( x x ( z ) ) 2 | E ( x , y , z ) | 2 .
σ x 2 ( z ) = σ x 2 ( z 0 ) + A λ 2 π ( z z 0 ) + B λ 2 4 π 2 ( z z 0 ) 2 .
A = i d S ( x x ( z 0 ) ) [ E E x c . c . ] ,
B = d S | E x | 2 + 1 4 [ d S ( E E x c . c . ) ] 2 ,
z ˜ 0 x = z 0 π A / ( λ B ) ,
M x 2 = 4 B σ x 2 ( z 0 ) A 2 .
ζ n ; u m = ( u 2 + u 2 ) m / ( u 2 u 2 ) n ,
γ = u J 1 ( u ) / J 0 ( u ) ,
( 4 π a 2 N ˜ 2 ) 1 = i ( ζ 1 ; u 0 + ζ 1 ; w 0 ) Im ( γ ) ,
( π a 2 N ˜ 2 ) 1 A = 4 ( ζ 2 ; u 1 + ζ 2 ; w 1 ) Im ( γ ) i ( ζ 1 ; u 1 ζ 1 ; w 1 ) 2 i ( ζ 1 ; u 0 + ζ 1 ; w 0 ) | γ | 2
( 2 π N ˜ 2 ) 1 B = Re [ ( u 2 ζ 1 ; u 0 w 2 ζ 1 ; w 0 ) γ ] ,
( π a 4 N ˜ 2 ) 1 σ x 2 ( z 0 ) = 8 i ( ζ 3 ; u 1 ζ 3 ; w 1 ) Im ( γ ) + ( 2 π a 2 N ˜ 2 ) 1 + 2 ( ζ 2 ; u 1 + ζ 2 ; w 1 ) + 4 ( ζ 2 ; u 0 ζ 2 ; w 0 ) | γ | 2 .

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