Abstract

A simple iterative model is introduced quantifying the interaction of saturable gain and nonlinear loss in a mode-locked laser cavity. The resulting geometrical description of the laser dynamics completely characterizes the generic multi-pulsing instability observed in experiments. The model further shows that the onset of multi-pulsing can be preceded by periodic and chaotic transitions as recently confirmed in theory and experiment. The results suggest ways to engineer the nonlinear losses in the cavity in order to achieve an enhanced performance.

© 2010 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2009 (2)

2008 (1)

2006 (1)

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

2004 (1)

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2004).
[CrossRef]

2000 (2)

1999 (2)

1998 (2)

B. Collings, K. Berman, and W. H. Knox, “Stable multigigahertz pulse train formation in a short cavity passively harmonic modelocked Er/Yb fiber laser,” Opt. Lett. 23, 123–125 (1998).
[CrossRef]

H. Kitano and S. Kinoshita, “Stable multipulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Commun. 157, 128–134 (1998).
[CrossRef]

1997 (3)

M. Lai, J. Nicholson, and W. Rudolph, “Multiple pulse operation of a femtosecond Ti:sapphire laser,” Opt. Commun. 142, 45–49 (1997).
[CrossRef]

C. Wang, W. Zhang, K. F. Lee, and K. M. Yoo, “Pulse splitting in a self-mode-locking Ti:sapphire laser,” Opt. Commun. 137, 89–92 (1997).
[CrossRef]

S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am B 14, 2099–2111 (1997).
[CrossRef]

1996 (1)

1994 (1)

1992 (1)

A. B. Grudinin, D. J. Richardson, and D. N. Payne, “Energy quantization in figure eight fibre laser,” Electron. Lett. 28, 67–68 (1992).
[CrossRef]

1991 (1)

R. P. Davey, N. Langford, and A. I. Ferguson, “Interacting solutions in erbium fibre laser,” Electron. Lett. 27, 1257–1259 (1991).
[CrossRef]

Akhmediev, N.

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2004).
[CrossRef]

N. Akhmediev and A. Ankiewicz, Dissipative Solitons, Lecture Notes in Physics (Springer-Verlag, 2005).
[CrossRef]

Akhmediev, N. N.

Ankiewicz, A.

N. Akhmediev and A. Ankiewicz, Dissipative Solitons, Lecture Notes in Physics (Springer-Verlag, 2005).
[CrossRef]

Bale, B. G.

Berman, K.

Boskovic, A.

Carruthers, T. F.

Chai, L.

Q. Xing, L. Chai, W. Zhang, and C. Wang, “Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser,” Opt. Commun. 162, 71–74 (1999).
[CrossRef]

Collings, B.

Davey, R. P.

R. P. Davey, N. Langford, and A. I. Ferguson, “Interacting solutions in erbium fibre laser,” Electron. Lett. 27, 1257–1259 (1991).
[CrossRef]

Devaney, R.

R. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed. (Addison-Wesley, 1989).

Ding, E.

Drazin, P. G.

P. G. Drazin, Nonlinear Systems (Cambridge University Press, 1992).

Duling, I. N.

Ferguson, A. I.

R. P. Davey, N. Langford, and A. I. Ferguson, “Interacting solutions in erbium fibre laser,” Electron. Lett. 27, 1257–1259 (1991).
[CrossRef]

Fermann, M. E.

Grapinet, M.

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2004).
[CrossRef]

Grelu, P.

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2004).
[CrossRef]

Grudinin, A. B.

A. B. Grudinin, D. J. Richardson, and D. N. Payne, “Energy quantization in figure eight fibre laser,” Electron. Lett. 28, 67–68 (1992).
[CrossRef]

Guy, M. J.

Haus, H. A.

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
[CrossRef]

S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am B 14, 2099–2111 (1997).
[CrossRef]

Horowitz, M.

Ippen, E. P.

S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am B 14, 2099–2111 (1997).
[CrossRef]

Jagadish, C.

Kieu, K.

Kinoshita, S.

H. Kitano and S. Kinoshita, “Stable multipulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Commun. 157, 128–134 (1998).
[CrossRef]

Kitano, H.

H. Kitano and S. Kinoshita, “Stable multipulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Commun. 157, 128–134 (1998).
[CrossRef]

Knox, W. H.

Kutz, J. N.

Lai, M.

M. Lai, J. Nicholson, and W. Rudolph, “Multiple pulse operation of a femtosecond Ti:sapphire laser,” Opt. Commun. 142, 45–49 (1997).
[CrossRef]

Langford, N.

R. P. Davey, N. Langford, and A. I. Ferguson, “Interacting solutions in erbium fibre laser,” Electron. Lett. 27, 1257–1259 (1991).
[CrossRef]

Lederer, M. J.

Lee, K. F.

C. Wang, W. Zhang, K. F. Lee, and K. M. Yoo, “Pulse splitting in a self-mode-locking Ti:sapphire laser,” Opt. Commun. 137, 89–92 (1997).
[CrossRef]

Luther-Davis, B.

Menyuk, C. R.

Minelly, J. D.

Namiki, S.

S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am B 14, 2099–2111 (1997).
[CrossRef]

Nicholson, J.

M. Lai, J. Nicholson, and W. Rudolph, “Multiple pulse operation of a femtosecond Ti:sapphire laser,” Opt. Commun. 142, 45–49 (1997).
[CrossRef]

Noske, D. U.

Payne, D. N.

A. B. Grudinin, D. J. Richardson, and D. N. Payne, “Energy quantization in figure eight fibre laser,” Electron. Lett. 28, 67–68 (1992).
[CrossRef]

Richardson, D. J.

A. B. Grudinin, D. J. Richardson, and D. N. Payne, “Energy quantization in figure eight fibre laser,” Electron. Lett. 28, 67–68 (1992).
[CrossRef]

Rudolph, W.

M. Lai, J. Nicholson, and W. Rudolph, “Multiple pulse operation of a femtosecond Ti:sapphire laser,” Opt. Commun. 142, 45–49 (1997).
[CrossRef]

Sandstede, B.

Soto-Crespo, J. M.

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2004).
[CrossRef]

M. J. Lederer, B. Luther-Davis, H. H. Tan, C. Jagadish, N. N. Akhmediev, and J. M. Soto-Crespo, “Multipulse operation of a Ti:Sapphire laser mode locked by an ion-implanted semiconductor saturable-absorber mirror,” J. Opt. Soc. Am. B 16, 895–904 (1999).
[CrossRef]

Tan, H. H.

Taylor, J. R.

Wang, C.

Q. Xing, L. Chai, W. Zhang, and C. Wang, “Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser,” Opt. Commun. 162, 71–74 (1999).
[CrossRef]

C. Wang, W. Zhang, K. F. Lee, and K. M. Yoo, “Pulse splitting in a self-mode-locking Ti:sapphire laser,” Opt. Commun. 137, 89–92 (1997).
[CrossRef]

Wise, F.

Xing, Q.

Q. Xing, L. Chai, W. Zhang, and C. Wang, “Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser,” Opt. Commun. 162, 71–74 (1999).
[CrossRef]

Yoo, K. M.

C. Wang, W. Zhang, K. F. Lee, and K. M. Yoo, “Pulse splitting in a self-mode-locking Ti:sapphire laser,” Opt. Commun. 137, 89–92 (1997).
[CrossRef]

Yu, C. X.

S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am B 14, 2099–2111 (1997).
[CrossRef]

Zhang, W.

Q. Xing, L. Chai, W. Zhang, and C. Wang, “Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser,” Opt. Commun. 162, 71–74 (1999).
[CrossRef]

C. Wang, W. Zhang, K. F. Lee, and K. M. Yoo, “Pulse splitting in a self-mode-locking Ti:sapphire laser,” Opt. Commun. 137, 89–92 (1997).
[CrossRef]

Electron. Lett. (2)

R. P. Davey, N. Langford, and A. I. Ferguson, “Interacting solutions in erbium fibre laser,” Electron. Lett. 27, 1257–1259 (1991).
[CrossRef]

A. B. Grudinin, D. J. Richardson, and D. N. Payne, “Energy quantization in figure eight fibre laser,” Electron. Lett. 28, 67–68 (1992).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am B (1)

S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am B 14, 2099–2111 (1997).
[CrossRef]

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

Opt. Commun. (4)

M. Lai, J. Nicholson, and W. Rudolph, “Multiple pulse operation of a femtosecond Ti:sapphire laser,” Opt. Commun. 142, 45–49 (1997).
[CrossRef]

C. Wang, W. Zhang, K. F. Lee, and K. M. Yoo, “Pulse splitting in a self-mode-locking Ti:sapphire laser,” Opt. Commun. 137, 89–92 (1997).
[CrossRef]

H. Kitano and S. Kinoshita, “Stable multipulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Commun. 157, 128–134 (1998).
[CrossRef]

Q. Xing, L. Chai, W. Zhang, and C. Wang, “Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser,” Opt. Commun. 162, 71–74 (1999).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. E (1)

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, “Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2004).
[CrossRef]

SIAM Rev. (1)

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

Other (3)

N. Akhmediev and A. Ankiewicz, Dissipative Solitons, Lecture Notes in Physics (Springer-Verlag, 2005).
[CrossRef]

R. Devaney, An Introduction to Chaotic Dynamical Systems, 2nd ed. (Addison-Wesley, 1989).

P. G. Drazin, Nonlinear Systems (Cambridge University Press, 1992).

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Figures (17)

Fig. 1
Fig. 1

Multi-pulsing bifurcation dynamics as a function of increasing gain. As the gain is increased, the mode-locked pulse peak power and spectral width (bold lines) are increased beyond the gain bandwidth (dashed line), leading to the formation of two mode-locked pulses whose spectra are within the gain bandwidth.

Fig. 2
Fig. 2

Simple cavity configuration involving a saturable gain element and a nonlinear loss element that gives the cavity saturable absorption. The remaining physical effects are assumed to balance each other in the formation of a mode-locked pulse.

Fig. 3
Fig. 3

Generic nonlinear loss (or saturable absorption or saturable fluency) curve (bold line) showing the standard effect of saturable absorption at high energies along with a fold-over due to higher-order nonlinear loss processes. The dashed line is the analytically computed threshold curve. Once the input energy is increased above the threshold point, any perturbation will cause the growth of an additional pulse, so the cavity jumps from N to N + 1 pulses. Note that the small and large signal transmission curves (dashed and solid lines, respectively) coalesce for low input energies.

Fig. 4
Fig. 4

Nonlinear loss and saturating gain curves for one-pulse ( N = 1 ) , two-pulse ( N = 2 ) , and three-pulse ( N = 3 ) per round trip configurations. The intersection of the gain and loss curves represents the mode-locked solution states of interest. As the gain parameter g 0 is increased, the gain curves shift to the right. The one-pulse solution first becomes periodic (b), and then chaotic (c) before ceasing to exist since it no longer intersects the loss curve. The solution then jumps to the next most energetically favorable configuration of two pulses per round trip (a). This qualitative picture describes the entire N to N + 1 pulse transition.

Fig. 5
Fig. 5

Iteration map dynamics for the nonlinear loss and saturating gain behavior. Possible iteration behaviors are (a) a steady-state solution, (b) a periodic solution, and (c) a chaotic dynamics. The interpretation of the periodic and chaotic dynamics in the mode-locking is given in Fig. 6. Note that the periodic and chaotic dynamics arise before the onset of multi-pulsing for the nonlinear loss curve chosen here.

Fig. 6
Fig. 6

Mode-locked cavity simulation where the saturable absorption is provided by waveguide arrays [4]. Shown is the intensity of the mode-locked field as a function of normalized propagation distance Z and time T. As the cavity gain is increased via g 0 , the stable one-pulse configuration first bifurcates to a periodic solution, and then bifurcates again to a chaotic solution, before finally going to the two-pulse configuration.

Fig. 7
Fig. 7

Nonlinear loss and saturating gain curves for one-pulse, two-pulse, and three-pulse per round trip configurations. The intersection of the gain and loss curves represents the mode-locked solution states of interest. As the cavity energy is increased, the gain curves shift to the right. For this case, the one-pulse solution ceases to exist beyond the threshold point indicated by the bold circle. Thus no periodic or chaotic behavior arises. The solution then jumps to the most energetically favorable configuration of two pulses per round trip.

Fig. 8
Fig. 8

Iteration map dynamics for the nonlinear loss and saturating gain behavior of Fig. 7. Shown is the total cavity energy E o u t (top panel) and the individual pulse energy E 1 (bottom panel) as functions of the cavity saturation energy E s a t . The transition dynamics between multi-pulse operations produces a discrete jump in the cavity energy. In this case, no periodic or chaotic dynamics is observed.

Fig. 9
Fig. 9

Nonlinear loss and saturating gain curves for one-pulse, two-pulse, and three-pulse per round trip configurations. The intersection of the gain and loss curves represents the mode-locked solution states of interest. As the cavity energy is increased, the gain curves shift to the right. Unlike Fig. 7, the one-pulse solution first experiences periodic and chaotic behavior before ceasing to exist beyond the threshold point indicated by the rightmost bold circle. The solution then jumps to the next most energetically favorable configuration of two pulses per round trip.

Fig. 10
Fig. 10

Iteration map dynamics for the nonlinear loss and saturating gain behavior of Fig. 9. Shown is the total cavity energy E o u t (top panel) and the individual pulse energy E 1 (bottom panel) as functions of the cavity saturation energy E s a t . The transition dynamics between multi-pulse operations produces a discrete jump in the cavity energy. In this case, both periodic and chaotic dynamics are observed preceding the multi-pulsing transition. This is consistent with recent theoretical and experimental findings [4, 5, 6].

Fig. 11
Fig. 11

Iteration map dynamics for the periodic nonlinear loss and saturating gain behavior of Fig. 9. Shown is the total cavity energy E o u t (top panel) and the individual pulse energy E 1 (bottom panel) as functions of the cavity saturation energy E s a t for two simulations. The transition dynamics shows that the chaotic behavior generates a more generic N to N + m pulses bifurcation.

Fig. 12
Fig. 12

Periodic nonlinear loss and saturating gain curves for one-pulse, two-pulse, and three-pulse per round trip configurations. The intersection of the gain and loss curves represents the mode-locked solution states of interest. As the cavity energy is increased, the gain curves shift to the right. The one-pulse solution first experiences periodic and chaotic behavior before ceasing to exist beyond the threshold point indicated by the rightmost bold circle. The solution then jumps to the next most energetically favorable configuration of two pulses per round trip. However, a high-energy one-pulse solution can also exist.

Fig. 13
Fig. 13

Iteration map dynamics for the periodic nonlinear loss and saturating gain behavior of Fig. 12. Shown is the total cavity energy E o u t (top panel) and the individual pulse energy E 1 (bottom panel) as functions of the cavity saturation energy E s a t . The transition dynamics between multi-pulse operations produces a discrete jump in the cavity energy. In this case, periodic dynamics is observed preceding the multi-pulsing transition to chaotic two-pulse solutions.

Fig. 14
Fig. 14

Periodic nonlinear loss and saturating gain curves for one-pulse, two-pulse, and three-pulse per round trip configurations. The intersection of the gain and loss curves represents the mode-locked solution states of interest. As the cavity energy is increased, the gain curves shift to the right. The low-energy one-pulse solution ceases to exist beyond the threshold point forcing the solution to jump to a high-energy one-pulse solution.

Fig. 15
Fig. 15

Iteration map dynamics for the periodic nonlinear loss and saturating gain behavior of Fig. 14. Shown is the total cavity energy in the one-pulse solution. Note the jump to the high-energy branch.

Fig. 16
Fig. 16

Periodic nonlinear loss and saturating gain curves for one-pulse, two-pulse, and three-pulse per round trip configurations. The intersection of the gain and loss curves represents the mode-locked solution states of interest. As the cavity energy is increased, the gain curves shift to the right. The low-energy one-pulse solution ceases to exist beyond the threshold point forcing the solution to jump to a high-energy one-pulse solution. The solution jumps to a chaotic state.

Fig. 17
Fig. 17

Iteration map dynamics for the periodic nonlinear loss and saturating gain behavior of Fig. 16. Shown is the total cavity energy in the one-pulse solution. Note the chaotic, then periodic, jump to the high-energy branch.

Equations (5)

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d E j d Z = g 0 1 + j = 1 N E j / E s a t E j ,
E o u t = T ( E i n ) E i n .
T ( E ) = 0.5 e α 1 ( E α 2 ) 8 + 0.1 e α 1 ( E α 2 ) 2 ,
T ( E ) = 0.1 + 0.2 [ 1 + cos ( 2 E 0.8 π ) ] ,
T ( E ) = 0.02 E + 0.1 + 0.2 [ 1 + cos ( ( 2 α 0 E ) E 0.8 π ) ] ,

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