Abstract

We have developed an ABCD matrix that, for the first time to our knowledge, accurately describes the transformation of a Gaussian beam by a medium with transversely varying saturable gain. In contrast with the conventional ABCD matrix, the newly developed matrix is shown to be in excellent agreement with a full beam propagation code over a wide parameter range. Accurate treatment of transversely varying saturable gain in laser resonators is important for the optimization of end-pumped lasers, particularly for efficient diode-pumped solid-state and Kerr-lens mode-locked systems.

© 2001 Optical Society of America

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References

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  1. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4, 1562–1568 (1965).
    [Crossref]
  2. A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), pp. 790–792.
  3. T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19, 554–556 (1994).
    [Crossref] [PubMed]
  4. F. Salin and J. Squier, “Gain guiding in solid-state lasers,” Opt. Lett. 17, 2319–2326 (1992).
  5. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
    [Crossref]
  6. D. A. Barry, J. Y. Parlange, L. Li, H. Prommer, C. J. Cunningham, and E. Stagnitti, “Analytical approximations for real values of the Lambert W-function,” Math. Comput. Simulation 53, 95–103 (2000).
    [Crossref]

2000 (1)

D. A. Barry, J. Y. Parlange, L. Li, H. Prommer, C. J. Cunningham, and E. Stagnitti, “Analytical approximations for real values of the Lambert W-function,” Math. Comput. Simulation 53, 95–103 (2000).
[Crossref]

1996 (1)

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[Crossref]

1994 (1)

1992 (1)

F. Salin and J. Squier, “Gain guiding in solid-state lasers,” Opt. Lett. 17, 2319–2326 (1992).

1965 (1)

Barry, D. A.

D. A. Barry, J. Y. Parlange, L. Li, H. Prommer, C. J. Cunningham, and E. Stagnitti, “Analytical approximations for real values of the Lambert W-function,” Math. Comput. Simulation 53, 95–103 (2000).
[Crossref]

Corless, R. M.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[Crossref]

Cunningham, C. J.

D. A. Barry, J. Y. Parlange, L. Li, H. Prommer, C. J. Cunningham, and E. Stagnitti, “Analytical approximations for real values of the Lambert W-function,” Math. Comput. Simulation 53, 95–103 (2000).
[Crossref]

Fan, T. Y.

Gonnet, G. H.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[Crossref]

Hare, D. E. G.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[Crossref]

Jeffrey, D. J.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[Crossref]

Knuth, D. E.

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[Crossref]

Kogelnik, H.

Li, L.

D. A. Barry, J. Y. Parlange, L. Li, H. Prommer, C. J. Cunningham, and E. Stagnitti, “Analytical approximations for real values of the Lambert W-function,” Math. Comput. Simulation 53, 95–103 (2000).
[Crossref]

Parlange, J. Y.

D. A. Barry, J. Y. Parlange, L. Li, H. Prommer, C. J. Cunningham, and E. Stagnitti, “Analytical approximations for real values of the Lambert W-function,” Math. Comput. Simulation 53, 95–103 (2000).
[Crossref]

Prommer, H.

D. A. Barry, J. Y. Parlange, L. Li, H. Prommer, C. J. Cunningham, and E. Stagnitti, “Analytical approximations for real values of the Lambert W-function,” Math. Comput. Simulation 53, 95–103 (2000).
[Crossref]

Salin, F.

F. Salin and J. Squier, “Gain guiding in solid-state lasers,” Opt. Lett. 17, 2319–2326 (1992).

Siegman, A. E.

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), pp. 790–792.

Squier, J.

F. Salin and J. Squier, “Gain guiding in solid-state lasers,” Opt. Lett. 17, 2319–2326 (1992).

Stagnitti, E.

D. A. Barry, J. Y. Parlange, L. Li, H. Prommer, C. J. Cunningham, and E. Stagnitti, “Analytical approximations for real values of the Lambert W-function,” Math. Comput. Simulation 53, 95–103 (2000).
[Crossref]

Adv. Comput. Math. (1)

R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Adv. Comput. Math. 5, 329–359 (1996).
[Crossref]

Appl. Opt. (1)

Math. Comput. Simulation (1)

D. A. Barry, J. Y. Parlange, L. Li, H. Prommer, C. J. Cunningham, and E. Stagnitti, “Analytical approximations for real values of the Lambert W-function,” Math. Comput. Simulation 53, 95–103 (2000).
[Crossref]

Opt. Lett. (2)

T. Y. Fan, “Aperture guiding in quasi-three-level lasers,” Opt. Lett. 19, 554–556 (1994).
[Crossref] [PubMed]

F. Salin and J. Squier, “Gain guiding in solid-state lasers,” Opt. Lett. 17, 2319–2326 (1992).

Other (1)

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), pp. 790–792.

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Figures (2)

Fig. 1
Fig. 1

Calculated beam profiles (top row) and associated gain-factor profiles (bottom row). Note the dip in the middle of the saturated gain profile for the case of wg>ω and the excellent agreement of the new method (crosses) with the exact solution (solid curves).

Fig. 2
Fig. 2

Calculated intracavity powers for a simple three-element resonator as a function of normalized spot size for the labeled output couplings. Note the excellent agreement of the new method (crosses) with the BPC (solid curves), in marked contrast with the conventional matrix (dashed curves). The inset depicts the laser beam spot size at the gain medium end of the resonator for three values of the detuning parameter, y, with an output coupling of 20%.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

u˜x=u˜0exp-j1q˜πλ0x2,
Ix=I0exp-2x2/w2,
I0=2P/w2π,
w=21P-+x2Ixdx1/2,
P=-+Ixdx.
logIxIx+Ix-IxIs=logGx=gx,
IxIsexpIxIs=GxIxIsexpIxIs,
IxIs=WGxIxIsexpIxIs,
WzexpWz=z,
GxG0exp-2x2/wG2.
wG2=w2w2w2-w2,
Mgsat=[10-j2kwG21].
Msat=[102kwG2y-j1].

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