Abstract

We present an analysis of the relaxation oscillations in a laser with a Gaussian mirror by taking into account the three-dimensional spatial field distribution of the laser modes and the spatial hole burning effect. In particular, we discuss the influence of the Gaussian mirror peak reflectivity and a Gaussian parameter on the damping rate and frequency of the relaxation oscillation for two different laser structures, i.e., with a classically unstable resonator and a classically stable resonator.

© 2002 Optical Society of America

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References

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  1. G. Vakimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron Phys. 10, 1439–1446 (1965).
  2. H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
    [CrossRef]
  3. Y. A. Anan’ev, V. E. Sherstobitov, “Influence of the edge effects of the properties of unstable resonators,” Sov. J. Quantum Electron. 1, 263–267 (1971).
    [CrossRef]
  4. A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
    [CrossRef]
  5. L. W. Casperson, S. D. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
    [CrossRef] [PubMed]
  6. U. Ganiel, Y. Silberberg, “Stability of optical laser resonators with mirrors of Gaussian reflectivity profiles which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
    [CrossRef]
  7. U. Ganiel, A. Hardy, “Eigenmodes of optical resonators with mirrors having Gaussian reflectivity profiles,” Appl. Opt. 9, 2145–2149 (1976).
    [CrossRef]
  8. N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: misalignment sensitivity,” Appl. Opt. 22, 2704–2708 (1983).
    [CrossRef] [PubMed]
  9. N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: output beam characteristics,” Appl. Opt. 23, 3845–3850 (1984).
    [CrossRef] [PubMed]
  10. D. M. Walsh, L. V. Knight, “Transverse modes of a laser resonator with Gaussian mirrors,” Appl. Opt. 25, 2947–2954 (1986).
    [CrossRef] [PubMed]
  11. A. Parent, N. McCarthy, P. Lavigne, “Effects of hard apertures on mode properties of resonators with Gaussian reflectivity mirrors,” IEEE J. Quantum Electron. QE-23, 222–228 (1987).
    [CrossRef]
  12. J. P. Taché, “Derivation of ABCD law for Laguerre-Gaussian beams,” Appl. Opt. 26, 2698–2700 (1987).
    [CrossRef] [PubMed]
  13. P. Lavigne, N. McCarthy, A. P. Parent, K. J. Snell, “Laser mode control with variable reflectivity mirrors,” Can. J. Phys. 66, 888–894 (1988).
    [CrossRef]
  14. P. Lavigne, N. McCarthy, J. G. Demers, “Design and characterization of complementary Gaussian reflectivity mirrors,” Appl. Opt. 24, 2581–2586 (1985).
    [CrossRef] [PubMed]
  15. E. Armandillo, G. Giuliani, “Achievement of large-sized TEM00 mode from an excimer laser by means of a novel apoditic filter,” Opt. Lett. 10, 445–447 (1985).
    [CrossRef] [PubMed]
  16. N. McCarthy, P. Lavigne, “Large-size Gaussian mode in unstable resonators using Gaussian mirrors,” Opt. Lett. 10, 553–555 (1985).
    [CrossRef] [PubMed]
  17. D. J. Harter, J. C. Walling, “Low magnification unstable resonators used with ruby and alexandrite lasers,” Opt. Lett. 11, 706–708 (1986).
    [CrossRef] [PubMed]
  18. S. De Silvestri, P. Laporta, V. Magni, “Laser output coupler based on a radially variable interferometer,” J. Opt. Soc. Am. A 4, 1413–1418 (1987).
    [CrossRef]
  19. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Radially variable reflectivity output coupler of novel design for unstable resonators,” Opt. Lett. 12, 84–86 (1987).
    [CrossRef] [PubMed]
  20. K. J. Snell, N. McCarthy, M. Piché, “Single mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
    [CrossRef]
  21. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, B. Majocchi, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (1988).
    [CrossRef] [PubMed]
  22. A. Piegari, G. Emiliani, “Laser mirrors with variable reflected intensity and uniform phase shift: design process,” Appl. Opt. 32, 5454–5461 (1993).
    [CrossRef] [PubMed]
  23. D. V. Willetts, M. R. Harris, “Output characteristics of a compact 1 J carbon dioxide laser with Gaussian reflectivity resonator,” IEEE J. Quantum Electron. QE-24, 849–855 (1988).
    [CrossRef]
  24. S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
    [CrossRef]
  25. P. Witoński, P. Szczepański, “Output power optimization of a slab-waveguide laser with Gaussian output mirror,” Appl. Phys. B 71, 831–839 (1995).
    [CrossRef]
  26. S. R. Chinn, “Relaxation oscillations in distributed feedback lasers,” Opt. Commun. 19, 208–211 (1976).
    [CrossRef]
  27. P. Szczepański, “Relaxation oscillations in distributed feedback gas lasers,” IEEE J. Quantum Electron. 27, 886–890 (1991).
    [CrossRef]
  28. P. Szczepański, A. Mossakowska, D. Dejnarowicz, “Relaxation oscillations in waveguide distributed feedback lasers,” J. Lightwave Technol. 2, 220–226 (1992).
    [CrossRef]
  29. W. Koechner, Solid-State Laser Engineering, 2nd ed. (Springer-Verlag, New York, 1988), Chap. 2.3, pp. 48–53.
  30. P. Witoński, P. Szczepański, A. Kujawski, “Model of the nonlinear operation of a laser with a Gaussian mirror,” J. Mod. Opt. 45, 1957–1974 (1998).
    [CrossRef]
  31. A. Kujawski, P. Szczepański, “Model of gain saturation in a two-mirror laser: single mode operation,” J. Mod. Opt. 39, 2519–2529 (1992).
    [CrossRef]
  32. P. Szczepański, P. Witoński, “Optimization of output power in hollow-waveguide lasers,” Appl. Opt. 34, 6099–6107 (1995).
    [CrossRef] [PubMed]
  33. A. Tyszka-Zawadzka, P. Szczepański, “Influence of mode nonorthogonality on the correlation function of the amplitude and of the intensity fluctuation of a distributed-feedback laser,” J. Opt. Soc. Am. B 13, 300–305 (1996).
    [CrossRef]
  34. See, for example, M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Chaps. 8 and 9, pp. 96–143.

1998

P. Witoński, P. Szczepański, A. Kujawski, “Model of the nonlinear operation of a laser with a Gaussian mirror,” J. Mod. Opt. 45, 1957–1974 (1998).
[CrossRef]

1996

1995

P. Szczepański, P. Witoński, “Optimization of output power in hollow-waveguide lasers,” Appl. Opt. 34, 6099–6107 (1995).
[CrossRef] [PubMed]

P. Witoński, P. Szczepański, “Output power optimization of a slab-waveguide laser with Gaussian output mirror,” Appl. Phys. B 71, 831–839 (1995).
[CrossRef]

1993

1992

P. Szczepański, A. Mossakowska, D. Dejnarowicz, “Relaxation oscillations in waveguide distributed feedback lasers,” J. Lightwave Technol. 2, 220–226 (1992).
[CrossRef]

A. Kujawski, P. Szczepański, “Model of gain saturation in a two-mirror laser: single mode operation,” J. Mod. Opt. 39, 2519–2529 (1992).
[CrossRef]

1991

P. Szczepański, “Relaxation oscillations in distributed feedback gas lasers,” IEEE J. Quantum Electron. 27, 886–890 (1991).
[CrossRef]

1990

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

1988

K. J. Snell, N. McCarthy, M. Piché, “Single mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[CrossRef]

P. Lavigne, N. McCarthy, A. P. Parent, K. J. Snell, “Laser mode control with variable reflectivity mirrors,” Can. J. Phys. 66, 888–894 (1988).
[CrossRef]

D. V. Willetts, M. R. Harris, “Output characteristics of a compact 1 J carbon dioxide laser with Gaussian reflectivity resonator,” IEEE J. Quantum Electron. QE-24, 849–855 (1988).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, B. Majocchi, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (1988).
[CrossRef] [PubMed]

1987

1986

1985

1984

1983

1976

S. R. Chinn, “Relaxation oscillations in distributed feedback lasers,” Opt. Commun. 19, 208–211 (1976).
[CrossRef]

U. Ganiel, A. Hardy, “Eigenmodes of optical resonators with mirrors having Gaussian reflectivity profiles,” Appl. Opt. 9, 2145–2149 (1976).
[CrossRef]

1975

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

U. Ganiel, Y. Silberberg, “Stability of optical laser resonators with mirrors of Gaussian reflectivity profiles which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

L. W. Casperson, S. D. Lunnam, “Gaussian modes in high loss laser resonators,” Appl. Opt. 14, 1193–1199 (1975).
[CrossRef] [PubMed]

1971

Y. A. Anan’ev, V. E. Sherstobitov, “Influence of the edge effects of the properties of unstable resonators,” Sov. J. Quantum Electron. 1, 263–267 (1971).
[CrossRef]

1970

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
[CrossRef]

1965

G. Vakimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron Phys. 10, 1439–1446 (1965).

Anan’ev, Y. A.

Y. A. Anan’ev, V. E. Sherstobitov, “Influence of the edge effects of the properties of unstable resonators,” Sov. J. Quantum Electron. 1, 263–267 (1971).
[CrossRef]

Armandillo, E.

Casperson, L. W.

Chinn, S. R.

S. R. Chinn, “Relaxation oscillations in distributed feedback lasers,” Opt. Commun. 19, 208–211 (1976).
[CrossRef]

De Silvestri, S.

Dejnarowicz, D.

P. Szczepański, A. Mossakowska, D. Dejnarowicz, “Relaxation oscillations in waveguide distributed feedback lasers,” J. Lightwave Technol. 2, 220–226 (1992).
[CrossRef]

Demers, J. G.

Emiliani, G.

Ganiel, U.

U. Ganiel, A. Hardy, “Eigenmodes of optical resonators with mirrors having Gaussian reflectivity profiles,” Appl. Opt. 9, 2145–2149 (1976).
[CrossRef]

U. Ganiel, Y. Silberberg, “Stability of optical laser resonators with mirrors of Gaussian reflectivity profiles which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

Giuliani, G.

Hardy, A.

U. Ganiel, A. Hardy, “Eigenmodes of optical resonators with mirrors having Gaussian reflectivity profiles,” Appl. Opt. 9, 2145–2149 (1976).
[CrossRef]

Harris, M. R.

D. V. Willetts, M. R. Harris, “Output characteristics of a compact 1 J carbon dioxide laser with Gaussian reflectivity resonator,” IEEE J. Quantum Electron. QE-24, 849–855 (1988).
[CrossRef]

Harter, D. J.

Knight, L. V.

Koechner, W.

W. Koechner, Solid-State Laser Engineering, 2nd ed. (Springer-Verlag, New York, 1988), Chap. 2.3, pp. 48–53.

Kujawski, A.

P. Witoński, P. Szczepański, A. Kujawski, “Model of the nonlinear operation of a laser with a Gaussian mirror,” J. Mod. Opt. 45, 1957–1974 (1998).
[CrossRef]

A. Kujawski, P. Szczepański, “Model of gain saturation in a two-mirror laser: single mode operation,” J. Mod. Opt. 39, 2519–2529 (1992).
[CrossRef]

Lamb, W. E.

See, for example, M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Chaps. 8 and 9, pp. 96–143.

Laporta, P.

Lavigne, P.

Lunnam, S. D.

Magni, V.

Majocchi, B.

McCarthy, N.

K. J. Snell, N. McCarthy, M. Piché, “Single mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[CrossRef]

P. Lavigne, N. McCarthy, A. P. Parent, K. J. Snell, “Laser mode control with variable reflectivity mirrors,” Can. J. Phys. 66, 888–894 (1988).
[CrossRef]

A. Parent, N. McCarthy, P. Lavigne, “Effects of hard apertures on mode properties of resonators with Gaussian reflectivity mirrors,” IEEE J. Quantum Electron. QE-23, 222–228 (1987).
[CrossRef]

N. McCarthy, P. Lavigne, “Large-size Gaussian mode in unstable resonators using Gaussian mirrors,” Opt. Lett. 10, 553–555 (1985).
[CrossRef] [PubMed]

P. Lavigne, N. McCarthy, J. G. Demers, “Design and characterization of complementary Gaussian reflectivity mirrors,” Appl. Opt. 24, 2581–2586 (1985).
[CrossRef] [PubMed]

N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: output beam characteristics,” Appl. Opt. 23, 3845–3850 (1984).
[CrossRef] [PubMed]

N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: misalignment sensitivity,” Appl. Opt. 22, 2704–2708 (1983).
[CrossRef] [PubMed]

Mossakowska, A.

P. Szczepański, A. Mossakowska, D. Dejnarowicz, “Relaxation oscillations in waveguide distributed feedback lasers,” J. Lightwave Technol. 2, 220–226 (1992).
[CrossRef]

Parent, A.

A. Parent, N. McCarthy, P. Lavigne, “Effects of hard apertures on mode properties of resonators with Gaussian reflectivity mirrors,” IEEE J. Quantum Electron. QE-23, 222–228 (1987).
[CrossRef]

Parent, A. P.

P. Lavigne, N. McCarthy, A. P. Parent, K. J. Snell, “Laser mode control with variable reflectivity mirrors,” Can. J. Phys. 66, 888–894 (1988).
[CrossRef]

Piché, M.

K. J. Snell, N. McCarthy, M. Piché, “Single mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[CrossRef]

Piegari, A.

Sargent, M.

See, for example, M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Chaps. 8 and 9, pp. 96–143.

Scully, M. O.

See, for example, M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Chaps. 8 and 9, pp. 96–143.

Sherstobitov, V. E.

Y. A. Anan’ev, V. E. Sherstobitov, “Influence of the edge effects of the properties of unstable resonators,” Sov. J. Quantum Electron. 1, 263–267 (1971).
[CrossRef]

Silberberg, Y.

U. Ganiel, Y. Silberberg, “Stability of optical laser resonators with mirrors of Gaussian reflectivity profiles which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

Snell, K. J.

K. J. Snell, N. McCarthy, M. Piché, “Single mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[CrossRef]

P. Lavigne, N. McCarthy, A. P. Parent, K. J. Snell, “Laser mode control with variable reflectivity mirrors,” Can. J. Phys. 66, 888–894 (1988).
[CrossRef]

Svelto, O.

Szczepanski, P.

P. Witoński, P. Szczepański, A. Kujawski, “Model of the nonlinear operation of a laser with a Gaussian mirror,” J. Mod. Opt. 45, 1957–1974 (1998).
[CrossRef]

A. Tyszka-Zawadzka, P. Szczepański, “Influence of mode nonorthogonality on the correlation function of the amplitude and of the intensity fluctuation of a distributed-feedback laser,” J. Opt. Soc. Am. B 13, 300–305 (1996).
[CrossRef]

P. Szczepański, P. Witoński, “Optimization of output power in hollow-waveguide lasers,” Appl. Opt. 34, 6099–6107 (1995).
[CrossRef] [PubMed]

P. Witoński, P. Szczepański, “Output power optimization of a slab-waveguide laser with Gaussian output mirror,” Appl. Phys. B 71, 831–839 (1995).
[CrossRef]

A. Kujawski, P. Szczepański, “Model of gain saturation in a two-mirror laser: single mode operation,” J. Mod. Opt. 39, 2519–2529 (1992).
[CrossRef]

P. Szczepański, A. Mossakowska, D. Dejnarowicz, “Relaxation oscillations in waveguide distributed feedback lasers,” J. Lightwave Technol. 2, 220–226 (1992).
[CrossRef]

P. Szczepański, “Relaxation oscillations in distributed feedback gas lasers,” IEEE J. Quantum Electron. 27, 886–890 (1991).
[CrossRef]

Taché, J. P.

Tyszka-Zawadzka, A.

Vakimov, G.

G. Vakimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron Phys. 10, 1439–1446 (1965).

Valentini, G.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

Walling, J. C.

Walsh, D. M.

Willetts, D. V.

D. V. Willetts, M. R. Harris, “Output characteristics of a compact 1 J carbon dioxide laser with Gaussian reflectivity resonator,” IEEE J. Quantum Electron. QE-24, 849–855 (1988).
[CrossRef]

Witonski, P.

P. Witoński, P. Szczepański, A. Kujawski, “Model of the nonlinear operation of a laser with a Gaussian mirror,” J. Mod. Opt. 45, 1957–1974 (1998).
[CrossRef]

P. Szczepański, P. Witoński, “Optimization of output power in hollow-waveguide lasers,” Appl. Opt. 34, 6099–6107 (1995).
[CrossRef] [PubMed]

P. Witoński, P. Szczepański, “Output power optimization of a slab-waveguide laser with Gaussian output mirror,” Appl. Phys. B 71, 831–839 (1995).
[CrossRef]

Yariv, A.

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

Yeh, P.

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

Zucker, H.

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
[CrossRef]

Appl. Opt.

Appl. Phys. B

P. Witoński, P. Szczepański, “Output power optimization of a slab-waveguide laser with Gaussian output mirror,” Appl. Phys. B 71, 831–839 (1995).
[CrossRef]

Bell Syst. Tech. J.

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
[CrossRef]

Can. J. Phys.

P. Lavigne, N. McCarthy, A. P. Parent, K. J. Snell, “Laser mode control with variable reflectivity mirrors,” Can. J. Phys. 66, 888–894 (1988).
[CrossRef]

IEEE J. Quantum Electron.

D. V. Willetts, M. R. Harris, “Output characteristics of a compact 1 J carbon dioxide laser with Gaussian reflectivity resonator,” IEEE J. Quantum Electron. QE-24, 849–855 (1988).
[CrossRef]

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

A. Parent, N. McCarthy, P. Lavigne, “Effects of hard apertures on mode properties of resonators with Gaussian reflectivity mirrors,” IEEE J. Quantum Electron. QE-23, 222–228 (1987).
[CrossRef]

P. Szczepański, “Relaxation oscillations in distributed feedback gas lasers,” IEEE J. Quantum Electron. 27, 886–890 (1991).
[CrossRef]

J. Lightwave Technol.

P. Szczepański, A. Mossakowska, D. Dejnarowicz, “Relaxation oscillations in waveguide distributed feedback lasers,” J. Lightwave Technol. 2, 220–226 (1992).
[CrossRef]

J. Mod. Opt.

P. Witoński, P. Szczepański, A. Kujawski, “Model of the nonlinear operation of a laser with a Gaussian mirror,” J. Mod. Opt. 45, 1957–1974 (1998).
[CrossRef]

A. Kujawski, P. Szczepański, “Model of gain saturation in a two-mirror laser: single mode operation,” J. Mod. Opt. 39, 2519–2529 (1992).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

K. J. Snell, N. McCarthy, M. Piché, “Single mode oscillation from an unstable resonator Nd:YAG laser using a variable reflectivity mirror,” Opt. Commun. 65, 377–382 (1988).
[CrossRef]

S. R. Chinn, “Relaxation oscillations in distributed feedback lasers,” Opt. Commun. 19, 208–211 (1976).
[CrossRef]

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

U. Ganiel, Y. Silberberg, “Stability of optical laser resonators with mirrors of Gaussian reflectivity profiles which contain an active medium,” Opt. Commun. 14, 290–293 (1975).
[CrossRef]

Opt. Lett.

Radio Eng. Electron Phys.

G. Vakimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron Phys. 10, 1439–1446 (1965).

Sov. J. Quantum Electron.

Y. A. Anan’ev, V. E. Sherstobitov, “Influence of the edge effects of the properties of unstable resonators,” Sov. J. Quantum Electron. 1, 263–267 (1971).
[CrossRef]

Other

W. Koechner, Solid-State Laser Engineering, 2nd ed. (Springer-Verlag, New York, 1988), Chap. 2.3, pp. 48–53.

See, for example, M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974), Chaps. 8 and 9, pp. 96–143.

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Figures (12)

Fig. 1
Fig. 1

Configuration of a laser with a Gaussian mirror.

Fig. 2
Fig. 2

Frequency Ω of the relaxation oscillations plotted versus the Gaussian parameter w g for two levels of distributed losses and Gaussian mirror peak reflectivity ρ0 as parameters. Both mirrors are convex.

Fig. 3
Fig. 3

Frequency Ω of the relaxation oscillations plotted versus the Gaussian parameter w g for two levels of distributed losses and output power levels P out/P s as parameters. Both mirrors are convex.

Fig. 4
Fig. 4

Damping rate coefficient Λ of the relaxation oscillations plotted versus the Gaussian parameter w g for two levels of distributed losses and Gaussian mirror peak reflectivity ρ0 as parameters. Both mirrors are convex.

Fig. 5
Fig. 5

Damping rate coefficient Λ of the relaxation oscillations plotted versus the Gaussian parameter w g for two levels of the distributed losses and output power levels P out/P s as parameters. Both mirrors are convex.

Fig. 6
Fig. 6

Frequency Ω of the relaxation oscillations plotted versus the Gaussian parameter w g for the spatial hole burning effect and for output power levels P out/P s as parameters. Both mirrors are convex.

Fig. 7
Fig. 7

Damping rate coefficient Λ of the relaxation oscillations plotted versus the Gaussian parameter w g for the spatial hole burning effect and for output power levels P out/P s as parameters. Both mirrors are convex.

Fig. 8
Fig. 8

Frequency Ω of the relaxation oscillations plotted versus the Gaussian parameter w g for two levels of the distributed losses and Gaussian mirror peak reflectivity ρ0 as parameters. Both mirrors are concave.

Fig. 9
Fig. 9

Frequency Ω of the relaxation oscillations plotted versus the Gaussian parameter w g for two levels of the distributed losses and output power levels P out/P s as parameters. Both mirrors are concave.

Fig. 10
Fig. 10

Frequency Ω of the relaxation oscillations plotted versus the Gaussian parameter w g for the spatial hole burning effect and output power levels P out/P s as parameters. Both mirrors are concave.

Fig. 11
Fig. 11

Damping rate coefficient Λ of the relaxation oscillations plotted versus the Gaussian parameter w g for two levels of the distributed losses and Gaussian mirror peak reflectivity ρ0 as parameters. Both mirrors are concave.

Fig. 12
Fig. 12

Damping rate coefficient Λ of the relaxation oscillations plotted versus the Gaussian parameter w g for the spatial hole burning effect and output power levels P out/P s as parameters. Both mirrors are concave.

Equations (24)

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

dNdt=-ImnqrT, z, tIsNrT, z, tτ-NrT, z, tτ+pr,
dQdt= dνImnqrT, z, tIsNrT, z, tτ-QτQ,
ErT, z, t=12m,n,q EmnqrT, z, t+c.c. =12m,n,qRqmnz, tAmnRrT, zexpikmnqz+Sqmnz, tAmnSrT, zexp-ikmnqz×exp-iωmnqt+c.c.,
 dνEmnqrT, z, tEmnq*rT, z, tδmnq.
|Rqz|=|Mqmn|expβqz, |Sqz|=1ρc|Mqmn|exp-βqz,
βq=12Lln1ρcρgeff.
ρgeff=ρ0Ω  dΩexp-rT2wg2exp-rT2wR2L2Ω  dΩexp-rT2wR2L21/2=ρ0Ω  dΩexp-rT2wS2L2Ω  dΩexp-rT2wR2L21/2=ρ0wSLwRL,
|Rq0AmnRrT, 0|=ρc|Sq0AmnSrT, 0|, 1-ρc2|Sq0|2rT  drT|AmnSrT, 0|2=PoutS,
|SqLAmnSrT, L|=ρgeff|RqLAmnRrT, L|, 1-ρgeff2|RqL|2rT  drT|AmnRrT, L|2=PoutR,
|Mqmn|2=PoutρcNmnS0ρc1-ρc2+NmnRLρgeff1-ρgeff2,
Pout=PoutR+PoutS, NmnRz=rT  |AmnR|2drT, NmnSz=rT  |AmnS|2drT
ImnqrT, z, tIs=PoutPs B|RqmnAmnR+SqmnAmnS|2=PoutPs B|RqmnAmnR|2+|SqmnAmnS|2+ηRqmnAmnR*SqmnAmnS+SqmnAmnS*RqmnAmnR,
B=ρcNmnS0ρc1-ρc2+NmnRLρgeff1-ρgeff2.
dNˆdt=-Nˆτ1+PoutPs B|RqmnAmnR+SqmnAmnS|2C-N0τQˆQ0PoutPs B|RqmnAmnR+SqmnAmnS|2C,
dQˆdt=C  dνPoutPs B|RqmnAmnR+SqmnAmnS|2Nˆτ.
dNˆdt=-brT, z, tNˆ-arT, z, tQˆ,
brT, z, t=1τ1+PoutPs B|RqmnAmnR+SqmnAmnS|2C,
arT, z, t=1τN0Q0PoutPs B|RqmnAmnR+SqmnAmnS|2C.
Nˆ=-arT, z, texp-brT, z, tt0texpbξQˆξdξ,
Qˆ=Qˆ0 exp-ΛtcosΩt,
Λ=1τ dν|RqmnAmnR+SqmnAmnS|4 dν|RqmnAmnR+SqmnAmnS|41+PoutPs B|RqmnAmnR+SqmnAmnS|2C,
Ω2=cñ2α0τ  dνPoutPs B|RqmnAmnR+SqmnAmnS|4C1+PoutPs B|RqmnAmnR+SqmnAmnS|2C,
α=α01+|RqmnAmnR+SqmnAmnS|2Ps.
2α0=1ρc1ρgeff+1ρc-ρgeff+ρcρc1+αLβq dz1NmnRrT  drTfrT, zexp2βqz|A00R|21+PoutPsBζrT, z+1NmnSrT  drTfrT, zexp-2βqz|A00S|21+PoutPs BζrT, z, ζrT, z=exp2βqz|A00RrT, z|2+1ρc2exp-2βqz|A00SrT, z|2+η 1ρcA00RrT, zA00S*rT, z+A00R*rT, zA00SrT, z,

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