Theory of energy evolution in laser resonators with saturated gain and non-saturated loss

S. K. Turitsyn

doi:10.1364/OE.17.011898

Abstract

Theory of the energy evolution in laser resonators with saturated gain and non-saturated loss is revisited. An explicit analytical expression for the output energy/average power in terms of the gain saturation energy, cavity loss and small signal gain parameters is derived for a ring cavity configuration.

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References

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Year
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Author
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Publication

A. E. Siegman, Lasers (University Science Books, 1986).
L. F. Mollenauer, and J. P. Gordon, Solitons in Optical Fibers: Fundamentals and Applications (Academic Press (2006).
A. C. Newell, and J. V. Moloney, Nonlinear Optics (Addison-Wesley Publishing Company, 1992).
A. Hasegawa, and Y. Kodama, Solitons in Optical Communications (Clarendon, 1995).
V. E. Zakharov, and E. S. Wabnitz, Optical Solitons: Theoretical Challenges and Industrial Perspectives (Springer-Verlag, 1998).
H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11(9), 736–746 (1975).
[Crossref]
H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992)Sh. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14(8), 2099 (1997).
[Crossref]
A. M. Dunlop, W. J. Firth, D. R. Heatley, and E. M. Wright, “Generalized mean-field or master equation for nonlinear cavities with transverse effects,” Opt. Lett. 21(11), 770–772 (1996).
[Crossref] [PubMed]
J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48(4), 629–678 (2006).
[Crossref]
N. Akhmediev, and A. Ankiewicz, eds., Dissipative Solitons, Lecture Notes in Physics, (Springer, 2005) Vol. 661.
N. Akhmediev, and A. Ankiewicz, eds., Dissipative Solitons: From optics to biology and medicine, Lecture Notes in Physics, (Springer, Berlin-Heidelberg, 2008). Vol. 751.
J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(4), 4783–4796 (1997).
[Crossref]
A. I. Chernykh and S. K. Turitsyn, “Soliton and collapse regimes of pulse generation in passively mode-locking laser systems,” Opt. Lett. 20(4), 398 (1995).
[Crossref] [PubMed]
L. Kramer, E. A. Kuznetsov, S. Popp, and S. K. Turitsyn, “Optical pulse collapse in defocusing active medium,” JETP Lett. 61, 904 (1995).
I. S. Aranson and L. Kramer, “The world of the complex Ginzburg Landau equation,” Rev. Mod. Phys. 74(1), 99–143 (2002).
[Crossref]
N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2002).
W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36(8), 2487–2490 (1965).
[Crossref]
F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

2006 (1)

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48(4), 629–678 (2006).
[Crossref]

2004 (1)

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

2002 (1)

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg Landau equation,” Rev. Mod. Phys. 74(1), 99–143 (2002).
[Crossref]

1997 (2)

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(4), 4783–4796 (1997).
[Crossref]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992)Sh. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14(8), 2099 (1997).
[Crossref]

1996 (1)

A. M. Dunlop, W. J. Firth, D. R. Heatley, and E. M. Wright, “Generalized mean-field or master equation for nonlinear cavities with transverse effects,” Opt. Lett. 21(11), 770–772 (1996).
[Crossref] [PubMed]

1995 (2)

A. I. Chernykh and S. K. Turitsyn, “Soliton and collapse regimes of pulse generation in passively mode-locking laser systems,” Opt. Lett. 20(4), 398 (1995).
[Crossref] [PubMed]

L. Kramer, E. A. Kuznetsov, S. Popp, and S. K. Turitsyn, “Optical pulse collapse in defocusing active medium,” JETP Lett. 61, 904 (1995).

1975 (1)

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11(9), 736–746 (1975).
[Crossref]

1965 (1)

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36(8), 2487–2490 (1965).
[Crossref]

Afanasjev, V. V.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(4), 4783–4796 (1997).
[Crossref]

Akhmediev, N. N.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(4), 4783–4796 (1997).
[Crossref]

Aranson, I. S.

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg Landau equation,” Rev. Mod. Phys. 74(1), 99–143 (2002).
[Crossref]

Buckley, J. R.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Chernykh, A. I.

A. I. Chernykh and S. K. Turitsyn, “Soliton and collapse regimes of pulse generation in passively mode-locking laser systems,” Opt. Lett. 20(4), 398 (1995).
[Crossref] [PubMed]

Clark, W. G.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11(9), 736–746 (1975).
[Crossref]

Heatley, D. R.

A. M. Dunlop, W. J. Firth, D. R. Heatley, and E. M. Wright, “Generalized mean-field or master equation for nonlinear cavities with transverse effects,” Opt. Lett. 21(11), 770–772 (1996).
[Crossref] [PubMed]

Ilday, F. O.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Ippen, E. P.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992)Sh. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14(8), 2099 (1997).
[Crossref]

Kramer, L.

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg Landau equation,” Rev. Mod. Phys. 74(1), 99–143 (2002).
[Crossref]

L. Kramer, E. A. Kuznetsov, S. Popp, and S. K. Turitsyn, “Optical pulse collapse in defocusing active medium,” JETP Lett. 61, 904 (1995).

Kutz, J. N.

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48(4), 629–678 (2006).
[Crossref]

Kuznetsov, E. A.

L. Kramer, E. A. Kuznetsov, S. Popp, and S. K. Turitsyn, “Optical pulse collapse in defocusing active medium,” JETP Lett. 61, 904 (1995).

Namiki, Sh.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992)Sh. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14(8), 2099 (1997).
[Crossref]

Popp, S.

L. Kramer, E. A. Kuznetsov, S. Popp, and S. K. Turitsyn, “Optical pulse collapse in defocusing active medium,” JETP Lett. 61, 904 (1995).

Rigrod, W. W.

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36(8), 2487–2490 (1965).
[Crossref]

Soto-Crespo, J. M.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(4), 4783–4796 (1997).
[Crossref]

Turitsyn, S. K.

A. I. Chernykh and S. K. Turitsyn, “Soliton and collapse regimes of pulse generation in passively mode-locking laser systems,” Opt. Lett. 20(4), 398 (1995).
[Crossref] [PubMed]

L. Kramer, E. A. Kuznetsov, S. Popp, and S. K. Turitsyn, “Optical pulse collapse in defocusing active medium,” JETP Lett. 61, 904 (1995).

Wabnitz, S.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(4), 4783–4796 (1997).
[Crossref]

Wise, F. W.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Wright, E. M.

A. M. Dunlop, W. J. Firth, D. R. Heatley, and E. M. Wright, “Generalized mean-field or master equation for nonlinear cavities with transverse effects,” Opt. Lett. 21(11), 770–772 (1996).
[Crossref] [PubMed]

Yu, C. X.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992)Sh. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14(8), 2099 (1997).
[Crossref]

IEEE J. Quantum Electron. (1)

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11(9), 736–746 (1975).
[Crossref]

J. Appl. Phys. (1)

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36(8), 2487–2490 (1965).
[Crossref]

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

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. 28(10), 2086–2096 (1992)Sh. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14(8), 2099 (1997).
[Crossref]

JETP Lett. (1)

L. Kramer, E. A. Kuznetsov, S. Popp, and S. K. Turitsyn, “Optical pulse collapse in defocusing active medium,” JETP Lett. 61, 904 (1995).

Opt. Lett. (2)

A. I. Chernykh and S. K. Turitsyn, “Soliton and collapse regimes of pulse generation in passively mode-locking laser systems,” Opt. Lett. 20(4), 398 (1995).
[Crossref] [PubMed]

A. M. Dunlop, W. J. Firth, D. R. Heatley, and E. M. Wright, “Generalized mean-field or master equation for nonlinear cavities with transverse effects,” Opt. Lett. 21(11), 770–772 (1996).
[Crossref] [PubMed]

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(4), 4783–4796 (1997).
[Crossref]

Phys. Rev. Lett. (1)

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92(21), 213902 (2004).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

I. S. Aranson and L. Kramer, “The world of the complex Ginzburg Landau equation,” Rev. Mod. Phys. 74(1), 99–143 (2002).
[Crossref]

SIAM Rev. (1)

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48(4), 629–678 (2006).
[Crossref]

Other (8)

N. Akhmediev, and A. Ankiewicz, eds., Dissipative Solitons, Lecture Notes in Physics, (Springer, 2005) Vol. 661.

N. Akhmediev, and A. Ankiewicz, eds., Dissipative Solitons: From optics to biology and medicine, Lecture Notes in Physics, (Springer, Berlin-Heidelberg, 2008). Vol. 751.

A. E. Siegman, Lasers (University Science Books, 1986).

L. F. Mollenauer, and J. P. Gordon, Solitons in Optical Fibers: Fundamentals and Applications (Academic Press (2006).

A. C. Newell, and J. V. Moloney, Nonlinear Optics (Addison-Wesley Publishing Company, 1992).

A. Hasegawa, and Y. Kodama, Solitons in Optical Communications (Clarendon, 1995).

V. E. Zakharov, and E. S. Wabnitz, Optical Solitons: Theoretical Challenges and Industrial Perspectives (Springer-Verlag, 1998).

N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2002).

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Fig. 1 The inverse function

x = f_{s}^{- 1} (y)

for different values of the parameter s.

Download Full Size | PPT Slide | PDF

Fig. 2 Contour plot of the normalized energy

E / E_{s a t}

in the plane (gain, R); here s = 0.03.

Download Full Size | PPT Slide | PDF

Fig. 4 Normalized energy

E / E_{s a t}

vs. the reflectivity R for different gains and s.

Download Full Size | PPT Slide | PDF

Fig. 3 Normalized energy

E / E_{s a t}

as a function of an effective gain for different R and s.

Download Full Size | PPT Slide | PDF

Equations (10)

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

i \frac{\partial A}{\partial z} + \sum_{n = 2}^{\infty} \frac{{(i)}^{n} β_{n}}{n!} \frac{\partial^{n} A}{\partial t^{n}} + γ {| A |}^{2} A = i (\frac{g (A)}{2} - \frac{α}{2}) A

\begin{array}{l} g (A) = \frac{g_{0}}{1 + P_{a v} / P_{s a t}} = \frac{g_{0}}{1 + E / E_{s a t}}, \\ P_{a v} = \frac{1}{T_{R}} \int {| A (t) |}^{2} d t, E = P_{a v} T_{R}, E_{s a t} = P_{s a t} T_{R} . \end{array}

\frac{d E}{d z} = \frac{g_{0} E}{1 + E / E_{s a t}} - α E

\frac{E (z)}{E_{s a t}} {[1 - s (1 + \frac{E (z)}{E_{s a t}})]}^{- \frac{1}{s}} = \exp [(g_{0} - α) (z - z_{0})], s = \frac{α}{g_{0}} .

\frac{E (z)}{E_{s a t}} {[1 - s (1 + \frac{E (z)}{E_{s a t}})]}^{- \frac{1}{s}} = \exp [(g_{0} - α) (z - z_{0})] = G (z) \times \frac{E (0)}{E_{s a t}} {[1 - s (1 + \frac{E (0)}{E_{s a t}})]}^{- \frac{1}{s}} .

E (z) = E_{s a t} \times f_{s}^{- 1} [G (z) f_{s} (\frac{E (0)}{E_{s a t}})] .

\frac{E (0)}{E_{s a t}} = R_{2} \times f_{s}^{- 1} {G f_{s} [R_{1} f_{s}^{- 1} (G f_{s} (\frac{E (0)}{E_{s a t}}))]}

E_{o u t} = E_{s a t} \times \frac{g_{0} - α}{α} \times \frac{1 - R}{\sqrt{R}} \times \frac{\sinh {\frac{α [L (g_{0} - α) + \ln R]}{2 g_{0}}}}{\sinh {\frac{α L (g_{0} - α) + (α - g_{0}) \ln R]}{2 g_{0}}}}

E_{o u t} = (1 - R) \times E_{s a t} \times \frac{1 - s}{s} \times \frac{1 - R^{- s} \times \exp [- s Δ G (L)]}{1 - R^{1 - s} \times \exp [- s Δ G (L)]} = E_{o u t} (R, L, s)

E_{o u t} / E_{s a t} \to (1 - R) \times \frac{1 - s}{s},

Abstract

References

2006 (1)

2004 (1)

2002 (1)

1997 (2)

1996 (1)

1995 (2)

1975 (1)

1965 (1)

Afanasjev, V. V.

Akhmediev, N. N.

Aranson, I. S.

Buckley, J. R.

Chernykh, A. I.

Clark, W. G.

Dunlop, A. M.

Firth, W. J.

Haus, H. A.

Heatley, D. R.

Ilday, F. O.

Ippen, E. P.

Kramer, L.

Kutz, J. N.

Kuznetsov, E. A.

Namiki, Sh.

Popp, S.

Rigrod, W. W.

Soto-Crespo, J. M.

Turitsyn, S. K.

Wabnitz, S.

Wise, F. W.

Wright, E. M.

Yu, C. X.

IEEE J. Quantum Electron. (1)

J. Appl. Phys. (1)

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

JETP Lett. (1)

Opt. Lett. (2)

Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

Phys. Rev. Lett. (1)

Rev. Mod. Phys. (1)

SIAM Rev. (1)

Other (8)

Cited By

Figures (4)

Equations (10)

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