Abstract

An iterative model based on the LambertW function was developed for estimating the fundamental mode parameters of resonators with saturable gain guiding. The process of pulse buildup in passively Q-switched, end-pumped lasers was analyzed. The effective ABCD cavity matrix for consecutive round-trips was calculated, taking into account spatially variable saturated gain in an active medium and absorption bleaching in a saturable absorber. The twofold decrease in beam width, as compared with the fundamental mode of the bare cavity, was demonstrated. The application of such a model for resonators with other nonlinear elements is feasible.

© 2003 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  2. T. Y. Fan and R. L. Byer, “Diode laser-pumped solid state lasers,” IEEE J. Quantum Electron. 24, 895–912 (1988).
    [CrossRef]
  3. P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state lasers,” IEEE J. Quantum Electron. 27, 2319–2326 (1991).
    [CrossRef]
  4. R. Kapoor, P. K. Mukhopadhyay, and J. George, “A new approach to compute overlap efficiency in axially pumped solid state lasers,” Opt. Express 5, 125–133 (1999), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-5-6-125.
    [CrossRef] [PubMed]
  5. J. J. Zayhowski. “Thermal guiding in microchips,” in OSA Proceedings on Advanced Solid State Lasers, G. Dube and H. P. Jensen, eds. (Optical Society of America, Washington, D.C., 1990), pp. 9–13.
  6. M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave-end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
    [CrossRef]
  7. X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of passively Q-switched lasers,” J. Opt. Soc. Am B 17, 1166–1175 (2000).
    [CrossRef]
  8. F. Salin and J. Squier, “Gain guiding in solid state lasers,” Opt. Lett. 17, 2319–2326 (1992).
    [CrossRef]
  9. V. Magni, G. Cerullo, and S. De Silvestri, “Closed form gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. 101, 365–370 (1993).
    [CrossRef]
  10. A. J. Kemp, R. S. Conroy, G. J. Friel, and B. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
    [CrossRef]
  11. C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1341–1320 (1999).
    [CrossRef]
  12. O. Denchev, S. Kurtev, and P. Petrov, “Modes of unstable resonators with a saturable gain guide,” Appl. Opt. 40, 921–929 (2001).
    [CrossRef]
  13. E. J. Grace, G. H. New, and P. M. W. French, “Simple ABCD matrix treatment for transversely varying saturable gain,” Opt. Lett. 26, 1776–1778 (2001).
    [CrossRef]
  14. D. A. Barry, J. Y. Parlange, L. Li, H. Prommer, C. J. Cunningham, and F. Stagnitti, “Analytical approximations for real values of the LambertW-function,” Math Comput. Simulations 53, 95–103 (2000).
    [CrossRef]
  15. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, Proc. SPIE1224, 4–14 (1990).
  16. J. K. Jabczyński, J. Kwiatkowski, and W. Zendzian, “Laser beam propagation in gain media,” Appl. Opt. (to be published).
  17. J. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31, 1890–1901 (1995).
    [CrossRef]

2001 (2)

2000 (2)

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

X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of passively Q-switched lasers,” J. Opt. Soc. Am B 17, 1166–1175 (2000).
[CrossRef]

1999 (3)

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1341–1320 (1999).
[CrossRef]

R. Kapoor, P. K. Mukhopadhyay, and J. George, “A new approach to compute overlap efficiency in axially pumped solid state lasers,” Opt. Express 5, 125–133 (1999), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-5-6-125.
[CrossRef] [PubMed]

1995 (1)

J. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31, 1890–1901 (1995).
[CrossRef]

1993 (1)

V. Magni, G. Cerullo, and S. De Silvestri, “Closed form gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. 101, 365–370 (1993).
[CrossRef]

1992 (1)

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

1991 (1)

P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state lasers,” IEEE J. Quantum Electron. 27, 2319–2326 (1991).
[CrossRef]

1990 (1)

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave-end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

1988 (1)

T. Y. Fan and R. L. Byer, “Diode laser-pumped solid state lasers,” IEEE J. Quantum Electron. 24, 895–912 (1988).
[CrossRef]

Barry, D. A.

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

Brussard, M.

P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state lasers,” IEEE J. Quantum Electron. 27, 2319–2326 (1991).
[CrossRef]

Byer, R. L.

T. Y. Fan and R. L. Byer, “Diode laser-pumped solid state lasers,” IEEE J. Quantum Electron. 24, 895–912 (1988).
[CrossRef]

Cerullo, G.

V. Magni, G. Cerullo, and S. De Silvestri, “Closed form gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. 101, 365–370 (1993).
[CrossRef]

Conroy, R. S.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

Cunningham, C. J.

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

De Silvestri, S.

V. Magni, G. Cerullo, and S. De Silvestri, “Closed form gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. 101, 365–370 (1993).
[CrossRef]

Degnan, J. J.

J. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31, 1890–1901 (1995).
[CrossRef]

Denchev, O.

Fan, T. Y.

T. Y. Fan and R. L. Byer, “Diode laser-pumped solid state lasers,” IEEE J. Quantum Electron. 24, 895–912 (1988).
[CrossRef]

Fields, R. A.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave-end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Fincher, C. L.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave-end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

French, P. M. W.

Friel, G. J.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

George, J.

Grace, E. J.

Innocenzi, M. E.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave-end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Jabczynski, J. K.

J. K. Jabczyński, J. Kwiatkowski, and W. Zendzian, “Laser beam propagation in gain media,” Appl. Opt. (to be published).

Kapoor, R.

Kemp, A. J.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

Kurtev, S.

Kwiatkowski, J.

J. K. Jabczyński, J. Kwiatkowski, and W. Zendzian, “Laser beam propagation in gain media,” Appl. Opt. (to be published).

Laporta, P.

P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state lasers,” IEEE J. Quantum Electron. 27, 2319–2326 (1991).
[CrossRef]

Li, L.

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

Magni, V.

V. Magni, G. Cerullo, and S. De Silvestri, “Closed form gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. 101, 365–370 (1993).
[CrossRef]

Mukhopadhyay, P. K.

New, G. H.

Ozygus, B.

X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of passively Q-switched lasers,” J. Opt. Soc. Am B 17, 1166–1175 (2000).
[CrossRef]

Parlange, J. Y.

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

Petrov, P.

Prommer, H.

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

Salin, F.

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

Serrat, C.

C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1341–1320 (1999).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, Proc. SPIE1224, 4–14 (1990).

Sinclair, B.

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

Squier, J.

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

Stagnitti, F.

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

van Druten, N. J.

C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1341–1320 (1999).
[CrossRef]

van Exter, M. P.

C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1341–1320 (1999).
[CrossRef]

Wang, Q.

X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of passively Q-switched lasers,” J. Opt. Soc. Am B 17, 1166–1175 (2000).
[CrossRef]

Weber, H.

X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of passively Q-switched lasers,” J. Opt. Soc. Am B 17, 1166–1175 (2000).
[CrossRef]

Woerdman, J. P.

C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1341–1320 (1999).
[CrossRef]

Yura, H. T.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave-end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

Zayhowski, J. J.

J. J. Zayhowski. “Thermal guiding in microchips,” in OSA Proceedings on Advanced Solid State Lasers, G. Dube and H. P. Jensen, eds. (Optical Society of America, Washington, D.C., 1990), pp. 9–13.

Zendzian, W.

J. K. Jabczyński, J. Kwiatkowski, and W. Zendzian, “Laser beam propagation in gain media,” Appl. Opt. (to be published).

Zhang, X.

X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of passively Q-switched lasers,” J. Opt. Soc. Am B 17, 1166–1175 (2000).
[CrossRef]

Zhao, S.

X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of passively Q-switched lasers,” J. Opt. Soc. Am B 17, 1166–1175 (2000).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave-end-pumped solid-state lasers,” Appl. Phys. Lett. 56, 1831–1833 (1990).
[CrossRef]

IEEE J. Quantum Electron. (5)

T. Y. Fan and R. L. Byer, “Diode laser-pumped solid state lasers,” IEEE J. Quantum Electron. 24, 895–912 (1988).
[CrossRef]

P. Laporta and M. Brussard, “Design criteria for mode size optimization in diode-pumped solid-state lasers,” IEEE J. Quantum Electron. 27, 2319–2326 (1991).
[CrossRef]

A. J. Kemp, R. S. Conroy, G. J. Friel, and B. Sinclair, “Guiding effects in Nd:YVO4 microchip lasers operating well above threshold,” IEEE J. Quantum Electron. 35, 675–681 (1999).
[CrossRef]

C. Serrat, M. P. van Exter, N. J. van Druten, and J. P. Woerdman, “Transverse mode formation in microlasers by combined gain- and index-guiding,” IEEE J. Quantum Electron. 35, 1341–1320 (1999).
[CrossRef]

J. J. Degnan, “Optimization of passively Q-switched lasers,” IEEE J. Quantum Electron. 31, 1890–1901 (1995).
[CrossRef]

J. Opt. Soc. Am B (1)

X. Zhang, S. Zhao, Q. Wang, B. Ozygus, and H. Weber, “Modeling of passively Q-switched lasers,” J. Opt. Soc. Am B 17, 1166–1175 (2000).
[CrossRef]

Math Comput. Simulations (1)

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

Opt. Commun. (1)

V. Magni, G. Cerullo, and S. De Silvestri, “Closed form gaussian beam analysis of resonators containing a Kerr medium for femtosecond lasers,” Opt. Commun. 101, 365–370 (1993).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Other (4)

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, Proc. SPIE1224, 4–14 (1990).

J. K. Jabczyński, J. Kwiatkowski, and W. Zendzian, “Laser beam propagation in gain media,” Appl. Opt. (to be published).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

J. J. Zayhowski. “Thermal guiding in microchips,” in OSA Proceedings on Advanced Solid State Lasers, G. Dube and H. P. Jensen, eds. (Optical Society of America, Washington, D.C., 1990), pp. 9–13.

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Figures (7)

Fig. 1.
Fig. 1.

Scheme of an iterative procedure for a cavity including saturable gain medium.

Fig. 2.
Fig. 2.

Relative mode radius W act/W 00 with dependence on round-trip number, fundamental mode radius W 00=0.384 mm, gain profile radius wg =0.1 mm, stationary value of effective mode radius w last=0.359 mm, reabsorption loss L reab=0.15, transmission of output coupler T oc=0.15, small-signal gain g 0 l=10.

Fig. 3.
Fig. 3.

Variations of internal intensity with dependence on round-trip number.

Fig. 4.
Fig. 4.

Scheme of the iterative procedure for a cavity including a saturable gain medium and a saturable absorber.

Fig. 5.
Fig. 5.

Instantaneous intensity (continuous curve), relative radius of pulse (dotted curve), and width of passive Q-switch diaphragm (dashed curve) with dependence on round-trip number for a passively Q-switched laser with initial Q-switch transmission T ini=0.5, passive losses L pas=0.1, radius of the fundamental mode of the bare cavity W 00=0.546 mm, ratio of absorption to gain cross sections α=σ a e =5, and transmission of output coupler T oc=0.5.

Fig. 6.
Fig. 6.

Two-dimensional maps of instantaneous intensity during pulse formation for two passively Q-switched lasers having the same bare cavity matrix and different initial saturable losses: upper plot for passive Q switch of 10% initial transmission, lower plot for 50% initial transmission.

Fig. 7.
Fig. 7.

Dependence of fundamental mode width of the bare cavity (continuous curve), ratio of effective pulse beam radius Wm /W 00 to fundamental mode W 00 (dashed curve), and ratio of waist areas (W 00/Wm )2 of bare cavity and pulsed beam (dotted curve) on the trace of the ABCD round-trip matrix. Parameter values: ratio of absorption to gain cross sections α=σ a e =5, passive losses L pas=0.1, output coupler transmission T oc=0.25, initial transmission of Q switch T ini=0.5.

Equations (7)

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

1 I r d I r dz = g 1 + I r I r exp ( I r ) = G ( z ) = exp ( g z ) ,
W ( z ) exp [ W ( z ) ] = z .
I r , 1 exp ( I r , 1 ) = I r , 0 exp ( g l + I r , 0 ) .
I r , 1 = W [ I r , 0 exp ( I r , 0 ) G ( l ) ] .
M SG = [ 1 0 C SG 1 ] ; C SG = i λ π w SG 2 ,
w SG 2 = w out , sg 2 w inp 2 ,
w out , sg 2 = 2 0 x 3 I r , out ( x ) d x 0 x I r , out ( x ) d x .

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