## Intracavity pulse dynamics and stability for passively mode-locked lasers

Optics Express, Vol. 15, Issue 10, pp. 5919-5924 (2007)

http://dx.doi.org/10.1364/OE.15.005919

Acrobat PDF (169 KB)

### Abstract

We derive a general characterization of the intracavity pulse dynamics for passively mode-locked fiber lasers based on the use of the variational principle. As a first application this method is used for an efficient simulation of the laser dynamics of stretched pulse and similariton lasers and evaluation of its stability.

© 2007 Optical Society of America

## 1. Introduction

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

2. H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen “Stretched-Pulse Additive Pulse Mode-Locking in Fiber Abstruct- Stretched-pulse Ring Lasers: Theory and Experiment,” IEEE J. Quantum Electron. **31**, 591–598 (1995). [CrossRef]

3. V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers,” Opt. Lett. **25**, 1753–1755 (2000). [CrossRef]

4. 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**, 213902 (2004). [CrossRef] [PubMed]

5. J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B **15**, 87–96 (1998). [CrossRef]

6. C. Jirauscheck, U. Morgner, and F. X. Kärtner, “Variational analysis of spatio-temporal pulse dynamics in dispersive Kerr media,” J. Opt. Soc. Am. B **19**, 1716–1721 (2002). [CrossRef]

7. C. Jirauscheck, U. Morgner, and F. X. Kärtner, “Spatiotemporal Gaussian pulse dynamics in Kerr-lens mode-locked lasers,” J. Opt. Soc. Am. B **20**, 1356–1368 (2003). [CrossRef]

8. C. Jirauscheck and F. X. Kärtner, “Gaussian pulse dynamics in gain media with Kerr nonlinearity,” J. Opt. Soc. Am. B **23**, 1776–1784 (2006). [CrossRef]

9. J- G. Caputo, N. Flytzanis, and M. P. Sorensen, “Ring laser configuration studied by collective coordinates,” J. Opt. Soc. Am. B **12**, 139–145 (1995). [CrossRef]

10. S. Waiyapot and M. Matsumoto, “Jitter and time stability of an actively mode-locked dispersion-managed fiber laser,” Opt. Commun. **188**, 167–180 (2001). [CrossRef]

2. H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen “Stretched-Pulse Additive Pulse Mode-Locking in Fiber Abstruct- Stretched-pulse Ring Lasers: Theory and Experiment,” IEEE J. Quantum Electron. **31**, 591–598 (1995). [CrossRef]

4. 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**, 213902 (2004). [CrossRef] [PubMed]

## 2. Theory and system modeling

*u*=

*u*(

*z*,

*t*) is the complex pulse envelope and

*t*is the time in the co-moving frame. We denote by

*z*) the group velocity dispersion, γ(

*z*) is the self phase modulation coefficient and

*g*(

_{s}*z*) is the saturated gain coefficient given by

*g*(

_{s}*z*) =

*g*

_{0}(

*z*)/[1+

*E*(

*z*)/

*E*

_{sat}], where

*g*

_{0}(

*z*) is the small signal gain coefficient,

*E*(

*z*) is the pulse energy and

*E*

_{sat}is the gain saturation energy. We assume a parabolic spectral gain profile,

*g̃*(

*z*,

*ω*) ≃

*g*(

_{s}*z*) [1 - (

*ω*/Ω

_{g})

^{2}]

^{1}, although a similar analysis could be performed for a generic gain profile [8

8. C. Jirauscheck and F. X. Kärtner, “Gaussian pulse dynamics in gain media with Kerr nonlinearity,” J. Opt. Soc. Am. B **23**, 1776–1784 (2006). [CrossRef]

*z*in a step-wise fashion. The conservative part of the NLSE can be derived from the Lagrangian [5

5. J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B **15**, 87–96 (1998). [CrossRef]

^{+∞}

_{-∞}= ℒ(

*u*,

*u*;

_{z}*u*

^{*},

*u*

^{*}

_{z})d

*t*, with the lagrangian density ℒ(

*u*,

*u*;

_{z}*u*

^{*},

*u*

^{*}

_{z}) defined as follows:

*z*denotes differentiation with respect to

*z*. The experimental evidence of a linear chirp [2

2. H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen “Stretched-Pulse Additive Pulse Mode-Locking in Fiber Abstruct- Stretched-pulse Ring Lasers: Theory and Experiment,” IEEE J. Quantum Electron. **31**, 591–598 (1995). [CrossRef]

4. 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**, 213902 (2004). [CrossRef] [PubMed]

*P*

_{4}= ∫

^{+∞}

_{-∞}

*f*

^{4}(

*s*)d

*s*, it is straightforward algebra to calculate the Lagrangian for the ansatz (3):

*I*

_{2},

*J*

_{2}and

*P*

_{4}. The equations of motion for the pulse parameters can be derived by applying the Euler-Lagrange equations, appropriately modified to include the non-conservative terms [9

9. J- G. Caputo, N. Flytzanis, and M. P. Sorensen, “Ring laser configuration studied by collective coordinates,” J. Opt. Soc. Am. B **12**, 139–145 (1995). [CrossRef]

*u*

^{*}and integrating the real part [11

11. M. Horowitz and C. R. Menyuk, “Analysis of pulse dropout in harmonically mode-locked fiber lasers by use of the Lyapunov method,” Opt. Lett. **25**, 40–42 (2000). [CrossRef]

*S*+

*I*

_{2}

*J*

_{2}. In fact, for 1 -

*S*+

*I*

_{2}

*J*

_{2}< 0 the pulse width is reduced by gain filtering, whereas for 1 -

*S*+

*I*

_{2}

*J*

_{2}> 0 there is a critical value of the chirp, β

_{C}= √(1 -

*S*+

*I*

_{2}

*J*

_{2}) / (

*I*

_{4}-

*I*

^{2}

_{2}) such that for ∣β∣ < β

_{C}the gain filtering broadens the pulse. This value discriminates the two different ways the gain filtering influence the pulse width, namely increasing the pulse width when filtering for β < β

_{C}and reducing the pulse width by reducing the chirp for β > β

_{C}(for the Gaussian ansatz, for instance, β

_{C}= 1). From Eq. (11) one can see immediately that the frequency offset is forced to vanish by a permanent restoring force; in consequence of this the derivative of the time shift Eq. (12) also vanishes and

*t*

_{0}tends to a constant value that can be set to zero without loss of generality. In what follows we can safely set Ω = 0 and

*t*

_{0}= 0.

^{2}

## 3. Stretched pulse laser and similariton laser: intracavity pulse dynamics

3. V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers,” Opt. Lett. **25**, 1753–1755 (2000). [CrossRef]

13. K. Tamura, E. P. Ippen, H. A. Haus, and L. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. **18**, 1080–1082 (1993). [CrossRef] [PubMed]

**92**, 213902 (2004). [CrossRef] [PubMed]

*E*

_{max}, can also be obtained by neglecting the term accounting for gain filtering in the integration of Eq. (8). This is allowed by the fact that the energy dynamics is dominated by gain saturation [11

11. M. Horowitz and C. R. Menyuk, “Analysis of pulse dropout in harmonically mode-locked fiber lasers by use of the Lyapunov method,” Opt. Lett. **25**, 40–42 (2000). [CrossRef]

*E*

_{max}≃

*E*[2

_{L}*g*

_{0}

*L*+ log(Γ

_{g}_{SA}(1 -

*l*

_{0})

^{2})] /[1 - Γ

_{SA}(1 -

*l*

_{0})

^{2}]. For the systems of Figs. 1 and 2 this gives

*E*

_{max}≃ 0.36nJ and

*E*

_{max}≃ 34nJ respectively, which are very close to the real values.

## 4. Linear stability analysis

**x**is the perturbation column vector, Δ

**x**= (Δ

*E*, Δτ, Δρ)

^{T},

**A**(

*z*) is a 3 × 3 periodic matrix with period

*L*

_{t}and the matrix

**B**

_{L}accounts for the lumped contribution of the saturable absorber and also the grating pair in the similariton laser. The matrix

**Q**(

*z*) of the independent solutions of (15) can be written as

**Q**(

*z*) =

**U**(

*z*)exp(λ

*z*), being

**U**(

*z*) a periodic matrix with period

*L*

_{t}. The diagonal matrix λ contains the Floquet coefficients

*λ*,

_{i}*i*= 1,2,3: their real parts determine the decay rate of the perturbations, whereas their imaginary parts may account for relaxation oscillation type of behavior. They can be evaluated by using that

**Q**(

*L*) = exp(λ

_{t}*L*) for

_{t}**Q**(0) =

**1**. The Floquet coefficient with the smallest absolute real part determines the stability of the perturbations. The Floquet coefficient for the frequency shift, which plays a key role in the analysis of the timing jitter, can be evaluated explicitely linearizing Eq. (11): λ

_{Ω}= - 4 ∫

_{0}

^{Lt}

*g*(

_{s}*z*)ℬ

^{2}(

*z*)/Ω

_{g}

^{2}d

*z*/

*L*

_{t}. With regard to the stability against cw radiation, we assume as a stability criterion that the net (power) gain for the cw radiation must be less than unit for the laser to be stable (this criterion is equivalent to the one assumed in ref. [6]):

*g*(

_{s}*z*) is determined by the steady state pulse and

*L*is the length of the gain fiber. The gain for cw radiation is sensitive to the modulation depth of the saturable absorber and to the gain bandwidth. The results of the analysis of such dependence are reported in Fig. 3, where we have plotted the boundary of the stable region in the plane (

_{g}*l*

_{0},Ω

_{g}/2

*π*) for (a) the stretched pulse laser and for (b) the similariton laser. In the insets, for the values of

*l*

_{0}(and Ω

_{g}) corresponding to

*G*

_{cw}= 0.9, we have plotted the negative real part of the Floquet coefficients

*λ*(solid lines, only two because two coefficients are complex conjugate) and

_{i}*λ*

_{Ω}(dot-dashed lines), normalized to the ring length

*L*

_{t}. The plots show that, for the range of parameters under scrutiny, the two systems manifest the same robustness against perturbations (which is set by the negative real part of the Floquet coefficient closest to zero), despite the significantly higher energy of the similariton laser, which, interestingly, appears to be even more robust against frequency fluctuations.

## 5. Conclusion

## Acknowledgement

## Footnotes

1 | It is worth reminding that the quadratic term in the frequency domain is equivalent to second derivative in the time domain, ω
^{2} ↔ ∂
^{2}/∂t
^{2}. |

2 | The equation for the phase, inessential to this paper, is: |

## References and links

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

2. | H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen “Stretched-Pulse Additive Pulse Mode-Locking in Fiber Abstruct- Stretched-pulse Ring Lasers: Theory and Experiment,” IEEE J. Quantum Electron. |

3. | V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers,” Opt. Lett. |

4. | 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. |

5. | J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B |

6. | C. Jirauscheck, U. Morgner, and F. X. Kärtner, “Variational analysis of spatio-temporal pulse dynamics in dispersive Kerr media,” J. Opt. Soc. Am. B |

7. | C. Jirauscheck, U. Morgner, and F. X. Kärtner, “Spatiotemporal Gaussian pulse dynamics in Kerr-lens mode-locked lasers,” J. Opt. Soc. Am. B |

8. | C. Jirauscheck and F. X. Kärtner, “Gaussian pulse dynamics in gain media with Kerr nonlinearity,” J. Opt. Soc. Am. B |

9. | J- G. Caputo, N. Flytzanis, and M. P. Sorensen, “Ring laser configuration studied by collective coordinates,” J. Opt. Soc. Am. B |

10. | S. Waiyapot and M. Matsumoto, “Jitter and time stability of an actively mode-locked dispersion-managed fiber laser,” Opt. Commun. |

11. | M. Horowitz and C. R. Menyuk, “Analysis of pulse dropout in harmonically mode-locked fiber lasers by use of the Lyapunov method,” Opt. Lett. |

12. | C. Jirauschek and F. ö. Ilday, “Theory of the Self-Similar Laser Oscillator,” CLEO 2005, Paper JWB65 (2005). |

13. | K. Tamura, E. P. Ippen, H. A. Haus, and L. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. |

14. | Introduction to numerical analysis, ser. Texts in Applied Mathematics, vol. 12, Springer (2002). |

15. | Advanced Synergetics, ser. Springer Series, vol. 20, Springer-Verlag (1983). |

**OCIS Codes**

(140.3510) Lasers and laser optics : Lasers, fiber

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

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: February 23, 2007

Revised Manuscript: April 23, 2007

Manuscript Accepted: April 25, 2007

Published: April 30, 2007

**Citation**

Cristian Antonelli, Jeff Chen, and Franz X. Kartner, "Intracavity pulse dynamics and stability for passively mode-locked lasers," Opt. Express **15**, 5919-5924 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-5919

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### References

- H. A. Haus, "Mode-Locking of Lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000). [CrossRef]
- H. A. Haus, K. Tamura, L. E. Nelson and E. P. Ippen "Stretched-Pulse Additive Pulse Mode-Locking in Fiber Abstruct- Stretched-pulse Ring Lasers: Theory and Experiment," IEEE J. QuantumElectron. 31, 591-598 (1995). [CrossRef]
- V. I. Kruglov, A. C. Peacock, J. M. Dudley and J. D. Harvey, "Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers, "Opt. Lett. 25, 1753-1755 (2000). [CrossRef]
- F. Ö. Ilday, J. R. Buckley, W. G. Clark and F.W. Wise, "Self-Similar Evolution of Parabolic Pulses in a Laser," Phys. Rev. Lett. 92, 213902 (2004). [CrossRef] [PubMed]
- J. N. Kutz, P. Holmes, S. G. Evangelides, J. P. Gordon, " Hamiltonian dynamics of dispersion-managed breathers," J. Opt. Soc. Am. B 15, 87-96 (1998). [CrossRef]
- C. Jirauscheck, U. Morgner and F. X. Kärtner, "Variational analysis of spatio-temporal pulse dynamics in dispersive Kerr media," J. Opt. Soc. Am. B 19, 1716-1721 (2002). [CrossRef]
- C. Jirauscheck, U. Morgner and F. X. Kärtner, "Spatiotemporal Gaussian pulse dynamics in Kerr-lens modelocked lasers," J. Opt. Soc. Am. B 20, 1356-1368 (2003). [CrossRef]
- C. Jirauscheck and F. X. Kärtner, "Gaussian pulse dynamics in gain media with Kerr nonlinearity," J. Opt. Soc. Am. B 23, 1776-1784 (2006). [CrossRef]
- J- G. Caputo, N. Flytzanis, M. P. Sorensen, "Ring laser configuration studied by collective coordinates," J. Opt. Soc. Am. B 12, 139-145 (1995). [CrossRef]
- S. Waiyapot and M. Matsumoto, "Jitter and time stability of an actively mode-locked dispersion-managed fiber laser," Opt. Commun. 188, 167-180 (2001). [CrossRef]
- M. Horowitz and C. R. Menyuk, "Analysis of pulse dropout in harmonically mode-locked fiber lasers by use of the Lyapunov method," Opt. Lett. 25, 40-42 (2000). [CrossRef]
- C. Jirauschek and F. Ö. Ilday, "Theory of the Self-Similar Laser Oscillator," CLEO 2005, Paper JWB65 (2005).
- K. Tamura, E. P. Ippen, H. A. Haus and L. Nelson, "77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser," Opt. Lett. 18, 1080-1082 (1993). [CrossRef] [PubMed]
- Introduction to numerical analysis, ser. Texts in Applied Mathematics, vol. 12, Springer (2002).
- Advanced Synergetics, ser. Springer Series, vol. 20, Springer-Verlag (1983).

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