## Self-starting of passive mode locking

Optics Express, Vol. 14, Issue 23, pp. 11142-11154 (2006)

http://dx.doi.org/10.1364/OE.14.011142

Acrobat PDF (227 KB)

### Abstract

It has been recently understood that mode locking of lasers has the signification of a thermodynamic phase transition in a system of many interacting light modes subject to noise. In the same framework, self starting of passive mode locking has the thermodynamic significance of a noise-activated escape process across an entropic barrier. Here we present the first dynamical study of the light mode system. While accordant with the predictions of some earlier theories, it is the first to give precise quantitative predictions for the distribution of self-start times, in closed form expressions, resolving the long standing self starting problem. Numerical simulations corroborate these results, which are also in good agreement with experiments.

© 2006 Optical Society of America

## 1. Introduction

*stochastic*theory of the onset of passive mode locking was presented only recently [15

15. A. Gordon and B. Fischer, “Phase Transition Theory of Many-Mode Ordering and Pulse Formation in Lasers”, Phys. Rev. Lett. **89**, 103901, (2002) [CrossRef] [PubMed]

16. A. Gordon and B. Fischer, Phase transition theory of pulse formation in passively mode-locked lasers with dispersion and Kerr nonlinearity” Opt. Commun. **223**, 151 (2003). [CrossRef]

17. A. Gordon and B. Fischer, “Inhibition of modulation instability in lasers by noise”, Opt. Lett **18**, 1326 (2003). [CrossRef]

18. O. Gat, A. Gordon, and B. Fischer, “Light-mode locking - A new class of solvable statistical physics systems”, New J. Phys. **7**, 151 (2005) [CrossRef]

19. R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, “Critical Behavior of Light in Mode-Locked Lasers,” Phys. Rev. Lett. **95**, 013903 (2005). [CrossRef] [PubMed]

*metastable*and the mode-locked pulsed state becomes stable, i. e., the ultimate stationary state of the laser is mode locked. However, such a metastable state can be very long lived, and the cw operation of the laser may therefore persist, even though the stable state is the mode-locked one, much as a supercooled liquid may stay unfrozen before an appropriate fluctuation drives it to the solid phase. The resulting hysteresis behavior is typical in first order phase transitions.

22. P. Hänggi, P. Talkner, and M. Borkovec, “Reaction-rate theory: fifty years after Kramers,” Rev. Mod. Phys. **62**, 251 (1990). [CrossRef]

*N*of modes. The number

*N*stands for the number of active modes in the

*initial*cw state, and not in the pulsed state. It is worth pointing out that the entropic barrier to mode locking exists only in such lasers, where the non mode-locked state is sometimes called “quasi-cw”. Section 2 introduces this model, reviews its steady-state properties, and identifies the self-start dimensionless parameter

*ε*, which measures the relative strength of the noise and the entropic barrier. In Sec. 3 we derive the self-start rate using the overdamped Kramers escape rate formula [21

21. H. A. Kramers, “Brownian motion in a field of fource and the diffusion model of chemical reactions”, Physica (Utrecht) **7**, 284 (1940). [CrossRef]

22. P. Hänggi, P. Talkner, and M. Borkovec, “Reaction-rate theory: fifty years after Kramers,” Rev. Mod. Phys. **62**, 251 (1990). [CrossRef]

## 2. The self-starting problem

*ψ*is constant over a time interval of the order of the coherence length. The physical reasoning leading to this model and its properties are discussed in Refs. [15

15. A. Gordon and B. Fischer, “Phase Transition Theory of Many-Mode Ordering and Pulse Formation in Lasers”, Phys. Rev. Lett. **89**, 103901, (2002) [CrossRef] [PubMed]

20. O. Gat, A. Gordon, and B. Fischer, “Solution of a statistical mechanics model for pulse formation in lasers”, Phys. Rev. E. **70**, 046108 (2004) [CrossRef]

24. M. Katz, A. Gordon, O. Gat, and B. Fischer, “Non-Gibbsian Stochastic Light-Mode Dynamics of Passive Mode Locking“, Phys. Rev. Lett. **97**, 113902 (2006) [CrossRef] [PubMed]

*N*of constant

*ψ*intervals is the number of active laser modes when it is operating in cw. When

*N*is large, cavity noise creates an entropic barrier which obstructs mode locking, and may make the cw configuration stable or metastable [15

15. A. Gordon and B. Fischer, “Phase Transition Theory of Many-Mode Ordering and Pulse Formation in Lasers”, Phys. Rev. Lett. **89**, 103901, (2002) [CrossRef] [PubMed]

*P*

_{0}. Here we consider the simplest form of gain saturation, in which the intracavity power is kept at a strictly fixed value

*P*

_{0}. In this case one can obtain an explicit expression for the net gain

### 2.1. The statistical steady state

*ρ*of the

*ψ*

_{m}variables under the dynamics defined by (1) with (4) is [20

20. O. Gat, A. Gordon, and B. Fischer, “Solution of a statistical mechanics model for pulse formation in lasers”, Phys. Rev. E. **70**, 046108 (2004) [CrossRef]

20. O. Gat, A. Gordon, and B. Fischer, “Solution of a statistical mechanics model for pulse formation in lasers”, Phys. Rev. E. **70**, 046108 (2004) [CrossRef]

*W*=

*NT*and taking the limit

*N*→∞ keeping

*T*,

*γ*

_{s}, and

*P*

_{0}fixed makes

*ρ*an equilibrium distribution of a nontrivial thermodynamic system where

*T*is the effective temperature, and the dimensionless parameter

*γ*increases the system exhibits a phase transition between the cw configuration and a pulsed configuration. In the cw state the intracavity power is roughly evenly divided and |

*ψ*

_{m}|

^{2}=

*O*(

*P*

_{0}) for all

*m*, while in a pulsed there is a single site with power of

*O*(

*NP*

_{0}) and the remaining power is divided between the other sites in a statistically homogeneous manner.

**70**, 046108 (2004) [CrossRef]

*F*of the form

*ξ*is the pulse power divided by

*p̄*=

*W*/(

*γ*

_{s}

*P*) (the motivation for this scaling is explained below in Sec. 2.2). In particular the steady state pulse power is equal to

*p̄*times the abscissa of the global minimum of

*F*.

*F*and their dependence on

*γ*. When

*γ*<

*γ*

_{c}=4,

*F*(

*ξ*) has a single minimum at

*ξ*

_{c}=0, corresponding to a stable cw state. When

*γ*>

*γ*

_{c}the minimum at zero persists, but another minimum appears at

*γ*

_{c}<

*γ*<

*γ*

_{e}≈4.91,

*F*(

*ξ*

_{p})>

*F*(

*ξ*

_{c}) and the pulsed state is metastable. The two states exchange stability at

*γ*

_{e}in the standard scenario of mean-field first-order phase transitions and the mode-locked state is the thermodynamically stable phase for larger values of

*γ*. These properties of the free energy are presented graphically in Fig. (2).

*γ*below

*γ*

_{e}, the pulsed state will persist unless a perturbation drives the system to the cw state, until the pulsed state becomes unstable when

*γ*=

*γ*

_{c}; this hysteresis scenario is born out in experiments [26

26. B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, and Baruch Fischer “Formation and Annihilation of Laser Pulse Quanta in a Thermodynamic-like Pathway”, Phys. Rev. Lett. **93**,153901 (2004). [CrossRef] [PubMed]

*always*(meta)stable; it follows that the transition to the pulsed state can only occur as a crossing of the entropic barrier by a fluctuation. Self starting means that the barrier crossing is activated by an internal noise, which is also the origin of the entropic barrier. This fact is the root of the self starting problem: The lifetime of metastable states grows exponentially with the height of the barrier, and spontaneous activation will only be observed when

*γ*is such that the strength of the barrier is not too large compared with the noise power.

### 2.2. The self-start parameter and the Arrhenius formula

21. H. A. Kramers, “Brownian motion in a field of fource and the diffusion model of chemical reactions”, Physica (Utrecht) **7**, 284 (1940). [CrossRef]

*E*required to escape the local minimum is much larger than the temperature, escape events are very rare and the metastable lifetime

*t*

_{esc}is much longer than the typical dynamical timescale

*t*

_{dyn}. In this case, the Arrhenius formula gives the lifetime as

*N*of degrees of freedom is large, while the dynamics of any particular degree of freedom is devoid of a potential barrier. The entropic barrier is exposed when one writes down the equation of motion for

*p*

_{m}=|

*ψ*

_{m}|

^{2}, the power in the

*m*-th interval, using the usual rules of stochastic calculus

*η*

_{m}(

*t*) is a real Gaussian white noise process derived from Γ

_{m}, which has

*p*

_{m}-dependent covariance

*η*

_{m}vanishes at the endpoints of the allowed interval 0≤

*p*

_{m}≤

*NP*

_{0}, while the deterministic part of Eq. (8) is positive at the lower endpoint and negative at the upper endpoint; together, these two properties guarantee that

*p*

_{m}indeed stays in this interval.

*p*

_{m}contains a restoring term proportional to the intensity

*W*of the noise. When

*p*

_{m}is small, the linear restoring term in Eq. (8) dominates, and

*p*

_{m}is trapped near zero, i. e. in the cw configuration. Assuming, as is argued below, that the term depending on

*p*

_{j},

*j*≠

*m*in Eq. (8) is subdominant, escape from cw is possible only with the help of the random term

*η*

_{m}, derived from the cavity noise. Hence in the self-starting problem, noise is responsible both for the

*existence*of the barrier, and the

*activation*which allows the system to overcome the barrier. It will be shown now, however, that the barrier height depends more strongly on the noise power than the activation, so that activation is harder and cw lifetime is longer for stronger noise intensity; this state of affairs is opposite to the standard Kramers-like scenario with a given potential, where stronger noise intensity implies a shorter lifetime.

*ξ*=

*p*

_{m}/

*p̄*, where

*p̄*=

*W*/(

*γ*

_{s}

*P*) is the power needed to accumulate in a single degree of freedom for the pulse buildup process to commence. This rescaling transforms Eq. (8) into

*γ*=

*γ*

_{s}

*T*as before,

*τ*=(

*W*/

*P*

_{0})

*t*is the rescaled time variable, and the prime designates derivative with respect to

*τ*.

*η*is a normalized white noise, with 〈

*η*(

*τ*)

*η*(

*τ*′)〉=

*δ*(

*τ*-

*τ*′).

*ε*, it is evident that there is an

*O*(1) negative drift for small

*ξ*, which becomes positive for larger

*ξ*if

*γ*is large enough, while the activating noise is of

*O*(√

*ε*). It follows that the parameter

*ε*, rather than

*γ*, determines the character of the self starting process. If

*ε*=

*O*(1) or larger, then the entropic barrier is too weak to inhibit the ordering interaction of the saturable absorber, and mode locking is achieved on a dynamical timescale 1/(

*γ*

_{s}

*P*

_{0}). On the other hand if

*ε*≪1, then

*ξ*remains trapped near zero by the drift force, and barrier crossing can occur only as a result of a rare event of a large fluctuation of

*η*. The Arrhenius formula indicates that self-start times are then slowed down by a factor of

*N*is large in passively mode locked lasers,

*γ*is significantly larger than the threshold value needed to maintain a mode locked pulse. This observation is the basic reason that self-staring of passive mode locking difficult to achieve in many practical systems, and external perturbations are needed instead to drive these systems to mode locked operation. The case

*ε*≪1 is the one considered in this paper. In this case the drift terms proportional to ε in Eq. (10) are small, and do not change the physical picture just described; nevertheless, because of the strong dependence of the self-start time on the drift, these small terms have an appreciable effect on the self-start process, and cannot be neglected. In the next section we carefully analyze the term proportional to ∑

_{j≠m}

## 3. The self-start time distribution

### 3.1. The mean field approximation

*ε*≪1. In this case self-starting occurs via noise activated barrier crossing, and the cw lifetime is exponentially larger than the other timescales in the system. Since the pulse buildup process occurs on a dynamical timescale, this fact implies that the probability of self starting occuring simultaneously in two different sites is exponentially small, and it can be safely assumed that the power at no more than a single site may simultaneously reach values which are much larger than the mean power

*P*

_{0}. Therefore the self-starting time is determined by the fastest of a system of

*N*-independent escape processes, each described by Eq. (10).

*ξ*, governed by Eq. (10). However, this is still not a single-variable escape problem, becuase it contains the term

*ξ*to the

*N*-1 other degrees of freedom in the system. Fortunately, in the case of interest where

*ε*≪1, we claim that the dynamics are such that the

*distribution*of

*Q*is determined to leading order by

*ξ*, enabling us to write a statistically equivalent equation, which involves

*ξ*alone.

*ψ*

_{j}with

*j*≠

*m*. Since self-starting occurs predominantly through the growth of a single degree of freedom, it can be assumed that

*ψ*

_{j}remains of

*O*(√

*P*

_{0}) throughout the process, and in the leading order (in

*ε*) the nonlinear terms in its dynamics, except those involving

*ψ*

_{m}can be neglected. It follows that the probability distribution of

*ψ*

_{j}, for a

*given*value of

*ψ*

_{m}, is gaussian.

*Q*is therefore the sum of the fourth powers of independent centered gaussian variables, the sum of whose variances is

*NP*

_{0}-

*p̄*

*ξ*; from this it follows that the expectation of

*Q*is [20

**70**, 046108 (2004) [CrossRef]

*Q*are smaller by a factor of

*O*(√

*N*) than its expectation, and will be neglected.

*Q*in (10) with its expectation value conditional on

*ξ*. The resulting equation in rescaled coordinates for the self-starting process at site

*m*is then

### 3.2. The mean lifetime of cw

*γ*>4, the

*O*(1) part of the drift force has three fixed points

*ξ*

_{c},

*ξ*

_{b}and

*ξ*

_{p}, whose values are precisely those of the three extrema of

*F*of Sec. 2.1. Since

*ε*≪1, there are exact fixed points of the drift near

*ξ*

_{c},

*ξ*

_{b}, and

*ξ*

_{p}, the first and last of which are (dynamically) stable, while the one near

*ξ*

_{b}is unstable. It follows that the drift force is derivable from a potential whose shape is well-represented by the steady-state free energy function

*F*shown in Fig. 2, with a potential barrier near

*ξ*

_{b}that inhibits self-starting. The escape time may be defined as the first time the variable

*ξ*with zero initial value reaches an arbitrary value

*strictly between*ξ ¯

*ξ*

_{b}and

*ξ*

_{p}. For typical escape times, which are much longer than the dynamical time scale, the distribtution of escape times is Poissonic

*τ*

_{esc}, and therefore the probability that it escapes is independent of history.

*τ*

_{esc}depends very little on the precise choice of the target pulse size

*, and is well-approximated by the mean first passage time to reach*ξ ¯

*. The self-start time*ξ ¯

*t*

_{cw}is obtained from

*τ*

_{esc}by converting back to the physical variable t and dividing the result by N to take into account the fact that self start can occur via

*N*independent escape processes. As the calculation of the mean first passage time for stochastic equations of the type of Eq. (12) is standard [25, 28], we merely cite the result, leaving details to the appendix

*F*(

*ξ*) is again the free energy function. Eq. (13) is the main quantitative result of this work. It contains an exponential dependence of the self-start time on

*ε*, which is equal to the one which can be deduced heuristically by applying the Arrhenius formula to the steady state free energy, and an algebraic prefactor, which has a

*ε*

^{5/2}dependence; it is verified by numerical simulationz in Sec. 3.4.

*ε*≪1 for any value of

*γ*>4; it is accompanied by a reverse process of activation from the pulsed state back to cw, which is not studied here. The steady state analysis [20

**70**, 046108 (2004) [CrossRef]

*γ*<

*γ*

_{e}the rate of activation of mode locking is slower than the rate of cw activation from the metastable mode-locked state, while the converse is true when

*γ*>

*γ*

_{e}, in which case the process indeed describes self-starting of mode locking.

*t*

_{cw}grows very fast as

*ε*decreases, systems where self-starting may be practically observed involve

*ε*values which are not too small; for example if

*F*(

*ξ*

_{b})/

*ε*is larger than 100, say, the probability of observing a self-starting event is less than 10

^{-35}per second. Since

*N*is large, this means that

*γ*may be assumed large in many cases. If

*γ*≫

*ε*

^{-1}, or equivalently,

*γ*≫√

*N*then Eq. (13) simplifies to

*N*are needed for this asymptotic expression for the mean lifetime to become precise, as can be seen in Fig. 3, where the exact and asymptotic expressions for the mean lifetime Eqs. (13) and (14) are displayed as a function of system parameters. For reference purposes we also include a version of Eq. (14) in the original unscaled variables,

### 3.3. Discussion and comparison with experiments

4. F. Krausz, T. Brabec, and Ch. Spielmann, “Self-starting passive mode locking”, Opt. Lett. **16**, 235 (1991) [CrossRef] [PubMed]

8. F. Krausz and T. Brabec, “Passive mode locking in standing-wave laser resonators”, Opt. Lett. **18**, 888 (1993) [CrossRef] [PubMed]

7. Ch. Spielman, F. Krausz, T. Brabec, E. Wintner, and A. J. Schmidt, “Experimental study of additive-pulse mode locking in an Nd:Glass laser”, IEEE J. Quantum Electron. **27**, 1207 (1991) [CrossRef]

*γ*

_{s}

*W*=1/2

*ε*, and the condition for self starting based on the decoherence argument is therefore

*ε*≳1. As shown above, this domain is where the noise is too weak for an entropic barrier to form, and self-starting occurs on a dynamical time scale. When

*ε*≪1 a significant entropic barrier forms, and the simple picture of decoherence versus saturable absorber-induced ordering is insufficient; rather, a full stochastic analysis is necessary leading ultimately to Eqs. (13) and (14).

23. B. Vodonos, A. Bekker, V. Smulakovsky, A. Gordon, O. Gat, N. K. Berger, and B. Fischer “Experimental study of the stochastic nature of the pulsation self-starting process in passive mode-locking,” Opt. Lett. **30**, 2787 (2005). [CrossRef] [PubMed]

*ε*≪1, and the experimental results agree well with our theoretical predictions. Firstly, as shown in the right panel of Fig. 4, the experiments have confirmed that the distribution of the self-start times has an exponential tail, as expected in this regime. Furthermore the approximately linear relation between log

*t*

_{cw}is 1/

### 3.4. Numerical analysis

*P*

_{0}in every time step. For a given value of

*N*the problem depends on

*γ*as a single dimensionless parameter. The initial values of the

*ψ*

_{j}’s were chosen as independent samples a random variable with a complex Gaussian distribution and normalized to the appropriate value of

*P*

_{0}. The time when the order parameter ∑

_{j}|

*ψ*

_{j}|

^{4}has reached half its maximal possible value was recorded as the self-starting time.

*γ*~

*ε*

^{-1}and Eq. (14) is not valid. Because of the

*ε*

^{5/2}preexponential factor, a considerably larger number of active modes

*N*is needed to reach the region of validity of the latter expression; on the other hand, such values of

*N*and larger ones are quite common in actual laser systems.

## 4. Conclusions

## A. Appendix: Calculation of the mean self-start time

*τ*

_{ξ¯}(

*ξ*) to reach the target point

*starting from the point*ξ ¯

*ξ*obeys the equation

*τ*(

_{ξ¯}*)=0, and*ξ ¯

*τ*′

*(0)=0 (reflecting boundary).*ξ ¯

*τ*

_{esc}=

*τ*(0),

_{ξ¯}*F*is the steady state free energy defined in Sec. 2.1.

*ε*≪1, the main contribution to the integral arises from the vicinity of the maximum of

*F*(

*ξ*)-

*F*(

*ξ*′) in the region 0≤

*ξ*′≤

*ξ*≤

*, which is reached when*ξ ¯

*ξ*=

*ξ*

_{b}, the abcissa of the potential barrier of

*F*, and

*ξ*′=0. The leading term in the asymptotic expansion of

*τ*

_{esc}in powers of

*ε*is obtained by approximating the integrand as a function quadratic in

*ξ*and linear in

*ξ*′ near the maximum (Laplace’s method), giving

*N*to take into account the

*N*possible self-starting paths.

## References and links

1. | E. P. Ippen, L. Y. Liu, and H. A. Haus, “Self-starting condition for additive-pulse mode-locked lasers”, Opt. Lett. |

2. | C. J. Chen, P. K. A. Wai, and C. R. Menyuk, “Self-starting of passively mode-locked lasers with fast saturable absorbers”, Opt. Lett. |

3. | H. A. Haus and E. P. Ippen, “Self-starting of passively mode-locked lasers”, Opt. Lett. |

4. | F. Krausz, T. Brabec, and Ch. Spielmann, “Self-starting passive mode locking”, Opt. Lett. |

5. | K. Tamura, J. Jacobson, E. P. Ippen, H. A. Haus, and J. G. Fujimoto, “Unidirectional ring resonators for self-starting passively mode-locked lasers”, Opt. Lett. |

6. | Y.-F. Chou, J. Wang, H.-H. Liu, and N.-P. Kuo, “Measurements of the self-starting threshold of Kerr-lens modelocking lasers”, Opt. Lett. |

7. | Ch. Spielman, F. Krausz, T. Brabec, E. Wintner, and A. J. Schmidt, “Experimental study of additive-pulse mode locking in an Nd:Glass laser”, IEEE J. Quantum Electron. |

8. | F. Krausz and T. Brabec, “Passive mode locking in standing-wave laser resonators”, Opt. Lett. |

9. | Y.-F. Chou, J. Wang, H.-H. Liu, and N.-P. Kuo, “Measurements of the self-starting threshold of Kerr-lens mode-locking lasers”, Opt. Lett. |

10. | J. Hermann, “Starting dynamic, self-starting condition and mode-locking threshold in passive, coupled-cavity or Kerr-lens mode locked solid-state lasers”, Opt. Comm. |

11. | A. K. Komarov, K. P. Komarov, and F. M. Mitschke, “Phase-modulation bistability and threshold self-start of laser passive mode locking”, Phys. Rev. A. |

12. | J. M. Soto-Crespo, N. Akhmediev, and G. Town, “Continuous-wave versus pulse regime in a passively mode-locked laser with a fast saturable absorber”, J. Opt. Soc. Am. B |

13. | H. A. Haus, “Theory of mode locking with a fast saturable absorber”, J. Appl. Phys. , |

14. | H. A. Haus, “Mode-Locking of Lasers”, IEEE J. Sel. Top. Quantum Electron. |

15. | A. Gordon and B. Fischer, “Phase Transition Theory of Many-Mode Ordering and Pulse Formation in Lasers”, Phys. Rev. Lett. |

16. | A. Gordon and B. Fischer, Phase transition theory of pulse formation in passively mode-locked lasers with dispersion and Kerr nonlinearity” Opt. Commun. |

17. | A. Gordon and B. Fischer, “Inhibition of modulation instability in lasers by noise”, Opt. Lett |

18. | O. Gat, A. Gordon, and B. Fischer, “Light-mode locking - A new class of solvable statistical physics systems”, New J. Phys. |

19. | R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, “Critical Behavior of Light in Mode-Locked Lasers,” Phys. Rev. Lett. |

20. | O. Gat, A. Gordon, and B. Fischer, “Solution of a statistical mechanics model for pulse formation in lasers”, Phys. Rev. E. |

21. | H. A. Kramers, “Brownian motion in a field of fource and the diffusion model of chemical reactions”, Physica (Utrecht) |

22. | P. Hänggi, P. Talkner, and M. Borkovec, “Reaction-rate theory: fifty years after Kramers,” Rev. Mod. Phys. |

23. | B. Vodonos, A. Bekker, V. Smulakovsky, A. Gordon, O. Gat, N. K. Berger, and B. Fischer “Experimental study of the stochastic nature of the pulsation self-starting process in passive mode-locking,” Opt. Lett. |

24. | M. Katz, A. Gordon, O. Gat, and B. Fischer, “Non-Gibbsian Stochastic Light-Mode Dynamics of Passive Mode Locking“, Phys. Rev. Lett. |

25. | R. L. Stratonovich, “Some Markov methods in the theory of stochastic processes in nonlinear dynamical systems”, in “ |

26. | B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat, and Baruch Fischer “Formation and Annihilation of Laser Pulse Quanta in a Thermodynamic-like Pathway”, Phys. Rev. Lett. |

27. | H. H. Risken, “The Fokker-Planck Equation”, Second edition, Springler-Verlag (1989, 1996). |

28. | C. W. Gardiner |

29. | P. E. Kloeden and E. Platen, |

30. | W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, |

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(140.4050) Lasers and laser optics : Mode-locked lasers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: April 11, 2006

Revised Manuscript: September 20, 2006

Manuscript Accepted: September 21, 2006

Published: November 13, 2006

**Citation**

Ariel Gordon, Omri Gat, Baruch Fischer, and Franz X. Kärtner, "Self-starting of passive mode locking," Opt. Express **14**, 11142-11154 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-23-11142

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### References

- E. P. Ippen, L. Y. Liu, and H. A. Haus, "Self-starting condition for additive-pulse mode-locked lasers", Opt. Lett. 15, 183 (1990). [CrossRef] [PubMed]
- C. J. Chen, P. K. A. Wai and C. R. Menyuk, "Self-starting of passively mode-locked lasers with fast saturable absorbers", Opt. Lett. 20, 350 (1995) [CrossRef] [PubMed]
- H. A. Haus and E. P. Ippen, "Self-starting of passively mode-locked lasers", Opt. Lett. 16, 1331 (1991) [CrossRef] [PubMed]
- F. Krausz, T. Brabec and Ch. Spielmann, "Self-starting passive mode locking", Opt. Lett. 16, 235 (1991) [CrossRef] [PubMed]
- K. Tamura, J. Jacobson, E. P. Ippen, H. A. Haus, and J. G. Fujimoto, "Unidirectional ring resonators for selfstarting passively mode-locked lasers", Opt. Lett. 18, 220 (1993) [CrossRef] [PubMed]
- Y.-F. Chou, J. Wang, H.-H. Liu, and N.-P. Kuo, "Measurements of the self-starting threshold of Kerr-lens modelocking lasers", Opt. Lett. 19, 566 (1994) [CrossRef] [PubMed]
- Ch. Spielman, F. Krausz, T. Brabec, E. Wintner and A. J. Schmidt, "Experimental study of additive-pulse mode locking in an Nd:Glass laser", IEEE J. Quantum Electron. 27, 1207 (1991) [CrossRef]
- F. Krausz and T. Brabec, "Passive mode locking in standing-wave laser resonators", Opt. Lett. 18, 888 (1993) [CrossRef] [PubMed]
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