## Semi-analytic technique for analyzing mode-locked lasers

Optics Express, Vol. 13, Issue 6, pp. 2075-2081 (2005)

http://dx.doi.org/10.1364/OPEX.13.002075

Acrobat PDF (115 KB)

### Abstract

A semi-analytic tool is developed for investigating pulse dynamics in mode-locked lasers. It provides a set of rate equations for pulse energy, width, and chirp, whose solutions predict how these pulse parameters evolve from one round trip to the next and how they approach their final steady-state values. An actively mode-locked laser is investigated using this technique and the results are in excellent agreement with numerical simulations and previous analytical studies.

© 2005 Optical Society of America

## 1. Introduction

1. D. J. Kuizenga and A. E. Siegman, “FM and AM Mode Locking of the Homogeneous Laser-Part I: Theory,” IEEE J. Quantum Electron. **QE-6**, 694–708 (1970). [CrossRef]

2. S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Radiophys. Quantum Electron. **14**, 1062–1070 (1971). [CrossRef]

3. C. J. McKinstrie, “Effects of filtering on Gordon-Haus timing jitter in dispersion-managed systems,” J. Opt. Soc. Am. B **19**, 1275–1285 (2002). [CrossRef]

4. J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. **222**, 413–420 (2003). [CrossRef]

## 2. The master equation of mode-locking

5. H. A. Haus and Y. Silberberg, “Laser Mode Locking with Addition of Nonliner Index,” IEEE J. Quantum Electron. **QE-22**, 325–331 (1986). [CrossRef]

*L*

_{R}. This equation takes the form [5

5. H. A. Haus and Y. Silberberg, “Laser Mode Locking with Addition of Nonliner Index,” IEEE J. Quantum Electron. **QE-22**, 325–331 (1986). [CrossRef]

*T=z/v*

_{g}

*, v*

_{g}is the group velocity, and

*A*(

*T, t*) is the slowly varying envelope of the electric field. As in Refs. [5

5. H. A. Haus and Y. Silberberg, “Laser Mode Locking with Addition of Nonliner Index,” IEEE J. Quantum Electron. **QE-22**, 325–331 (1986). [CrossRef]

7. F. X. Kärtner, D. Kopf, and U. Keller, “Solitary-pulse stabilization and shortening in actively mode-locked lasers,” J. Opt. Soc. Am. B **12**, 486–496 (1995). [CrossRef]

*t*measured in the frame of the moving pulse and the propagation time

*T*, often called the coarse-grained time [7

7. F. X. Kärtner, D. Kopf, and U. Keller, “Solitary-pulse stabilization and shortening in actively mode-locked lasers,” J. Opt. Soc. Am. B **12**, 486–496 (1995). [CrossRef]

*T*is measured in terms of the round-trip time

*TR=L*

_{R}

*/v*

_{g}. It is assumed that the time scale associated with the pulse is sufficiently smaller than

*T*

_{R}so the two times are essentially decoupled. This treatment is valid for most lasers for which

*T*

_{R}exceeds 1 ns and pulse widths are typically less than 100 ps.

*ḡ*=

*ḡ*

_{0}(1+

*P*

_{ave}/

*P*

_{sat})

^{-1}where

*P*

_{ave}represents the average power defined as

**QE-22**, 325–331 (1986). [CrossRef]

*T*

_{m}

*m*

^{-1}

*T*

_{R}, where

*m*is an integer representing the harmonic the laser is mode-locked at and Frep is the repetition rate of the pulses. The overbar in Eq. (1) denotes the value of the corresponding parameter averaged over a round trip. More specifically,

*β̄*

_{2}represents the averaged second-order dispersion of the cavity elements, while

*γ̄*takes into account the averaged nonlinear parameter and

_{w}=2/

*T*

_{2}.

*a, τ, q*, and

*φ*(

*T*) are determined uniquely by the parameters appearing in Eq. (1). In the absence of a mode-locker, a stable pulse will neither form nor survive multiple round trips in the cavity. However, the active fiber will try to impose the autosoliton shape on any pulse circulating in such a laser. This fact was previously exploited by Haus and Silberberg in their investigation of pulse shortening in AM mode-locked lasers in the presence of dispersive and nonlinear elements [5

**QE-22**, 325–331 (1986). [CrossRef]

## 3. The moment method

2. S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Radiophys. Quantum Electron. **14**, 1062–1070 (1971). [CrossRef]

4. J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. **222**, 413–420 (2003). [CrossRef]

## 4. Mode-locking rate equations

*M*(

*A, t*)=-Δ

_{AM}[1-cos (

*ω*

_{m}

*t*)]

*A*in Eq. (1), where Δ

_{AM}is the modulation depth experienced by a pulse during a single round trip,

*ω*

_{m}

*=2π/T*

_{m}is the modulation frequency (assumed to be identical to the repetition rate of the mode-locked pulse train), and the average loss of modulator has been incorporated into

*T*

_{m}, we approximate the effect of the AM mode-locker as

*M*(

*A, t*)≈-Δ

_{AM}

*t*

^{2}/2

*A*. Although this approximation is not required, it simplifies the appearance of the equations in the rest of this paper and is applicable in most cases of practical interest.

*C*

_{n}(

*n*=0 to 6) are introduced such that they all equal 1 for a Gaussian pulse. In the case of an autosoliton,

*C*

_{0}=2/3,

*C*

_{1}=

*π*

^{2}/6,

*C*

_{2}=4/

*π*

^{2},

*C*

_{3}=2,

*C*

_{4}=4

*π*

^{2}/15,

*C*

_{5}=1/3, and

*E, τ*, and

*q*change from one round trip to the next. For example, Eq. (11) shows that the energy is enhanced by the gain (first term) but reduced by both the gain filtering (second term) and the AM mode-locker (third term). Similarly, Eq. (12) shows that the modulator shortens optical pulses as they pass through it (last term).

*β̄*

_{2}=±0.014 ps

^{2}/m,

*γ̄*=0.012 W-1/m,

*ḡ*

_{0}=0.55 m

^{-1},

*ᾱ*=0.17 m

^{-1},

*T*

_{2}=47 fs/rad,

*P*

_{sat}=12.5 mW,

*F*

_{rep}=10 GHz,

*L*

_{R}=4 m,

*T*

_{R}=40 ns, and Δ

_{AM}=0.3. Figure 1 shows the approach to steady state in both the normal-and anomalous-dispersion regimes. It reveals that the pulse converges quickly in the normal-dispersion region but takes >1000 round trips before converging in the anomalous-dispersion region. Although the rate of convergence depends on the initial conditions used (

*E*=1 fJ,

*q*=0, and τ=0.5 ps), this type of behavior is expected since the nonlinear effects are weaker in the normal dispersion region. In the normal-dispersion regime the nonlinear effects add to the effect of dispersion and broaden the pulse to τ=3.73 ps, thus reducing its peak power and the role played by nonlinearity. In the anomalous-dispersion regime the interplay between dispersion and nonlinearity prolongs the convergence. We also point out the robustness of our approach since the initial energy used is more than 2500 times smaller than the steady state value obtained.

_{AM}.

*q*≃2), whereas the chirp is nearly zero (

*q*≈0) in the anomalous-dispersion regime. This behavior is a consequence of the interplay between dispersion and nonlinearity. In the anomalous regime, the two effects produce chirps with opposite signs, which partially cancel one another, whereas, the chirps add in the normal-dispersion regime [6].

## 5. Steady-state

*d*is defined as

*d*→∞ and

*q*

_{ss}=0 is the only physically valid solution. Of course, we should use the Gaussian pulse shape under such conditions. Setting

*C*

_{n}=1 in Eq. (15), we find that the steady state pulse width in the absence of dispersive and nonlinear effects is given by

1. D. J. Kuizenga and A. E. Siegman, “FM and AM Mode Locking of the Homogeneous Laser-Part I: Theory,” IEEE J. Quantum Electron. **QE-6**, 694–708 (1970). [CrossRef]

*β̄*

_{2}and

*β̄*

_{2}>0); they also reveal a discrepancy between our theory and the full model [i.e. the solution to Eq. (1)]. This error, which is <15%, is again due to the pulse shape which deviates from the assumed Gaussian shape. Plots (c) and (d), both made assuming anomalous dispersion, exhibit excellent agreement between our theory and the full model even over the large parameter space explored. Comparing Figs. 2(a) and (c) it is found that as the magnitude of dispersion is decreased, pulse width is reduced but chirp increases. Qualitatively, this behavior is the same in both regions. Comparing Figs. 2(b) and (d), however, we observe an interesting feature. Increasing the nonlinearity in the normal-dispersion region results in a large increase in both pulse width and chirp. However, in the anomalous-dispersion regime, an increase in nonlinearity reduces the pulse width while the chirp increases only slightly. For example, pulse width

*τ*

_{ss}, is reduced below 1 ps for

*γ̄*>0.012W

^{-1}/m while chirp is nearly zero. If the nonlinearity is increased to

*γ̄*=0.028 W

^{-1}/m the theory predicts pulses with

*τ*

_{ss}=0.36 ps indicating a pulse 3 times smaller than those predicted by the Kuizenga-Siegman limit [1

1. D. J. Kuizenga and A. E. Siegman, “FM and AM Mode Locking of the Homogeneous Laser-Part I: Theory,” IEEE J. Quantum Electron. **QE-6**, 694–708 (1970). [CrossRef]

*γ̄*into the cavity. Photonic-crystal fibers or tapered fibers can be used for this purpose. Since chirp can be nearly eliminated in the anomalous-dispersion regime, optimized actively mode-locked fiber lasers should produce near transform-limited femtosecond pulses.

## 6. Conclusion

**QE-6**, 694–708 (1970). [CrossRef]

## Acknowledgments

## References and links

1. | D. J. Kuizenga and A. E. Siegman, “FM and AM Mode Locking of the Homogeneous Laser-Part I: Theory,” IEEE J. Quantum Electron. |

2. | S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Radiophys. Quantum Electron. |

3. | C. J. McKinstrie, “Effects of filtering on Gordon-Haus timing jitter in dispersion-managed systems,” J. Opt. Soc. Am. B |

4. | J. Santhanam and G. P. Agrawal, “Raman-induced spectral shifts in optical fibers: general theory based on the moment method,” Opt. Commun. |

5. | H. A. Haus and Y. Silberberg, “Laser Mode Locking with Addition of Nonliner Index,” IEEE J. Quantum Electron. |

6. | G. P. Agrawal, |

7. | F. X. Kärtner, D. Kopf, and U. Keller, “Solitary-pulse stabilization and shortening in actively mode-locked lasers,” J. Opt. Soc. Am. B |

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(140.3510) Lasers and laser optics : Lasers, fiber

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

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 26, 2005

Revised Manuscript: March 9, 2005

Published: March 21, 2005

**Citation**

Nicholas Usechak and Govind Agrawal, "Semi-analytic technique for analyzing mode-locked lasers," Opt. Express **13**, 2075-2081 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-6-2075

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### References

- D. J. Kuizenga and A. E. Siegman, �??FM and AM Mode Locking of the Homogeneous Laser�??Part I: Theory,�?? IEEE J. Quantum Electron. QE-6, 694-708 (1970). [CrossRef]
- S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov �??Averaged description of wave beams in linear and nonlinear media (the method of moments),�?? Radiophys. Quantum Electron. 14, 1062-1070 (1971). [CrossRef]
- C. J. McKinstrie, �??Effects of filtering on Gordon-Haus timing jitter in dispersion-managed systems,�?? J. Opt. Soc. Am. B 19, 1275-1285 (2002). [CrossRef]
- J. Santhanam and G. P. Agrawal, �??Raman-induced spectral shifts in optical fibers: general theory based on the moment method,�?? Opt. Commun. 222, 413-420 (2003). [CrossRef]
- H. A. Haus and Y. Silberberg, �??Laser Mode Locking with Addition of Nonliner Index,�?? IEEE J. Quantum Electron. QE-22, 325-331 (1986). [CrossRef]
- G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic Press, New York, 2001).
- F. X. Kärtner, D. Kopf, and U. Keller, �??Solitary-pulse stabilization and shortening in actively mode-locked lasers,�?? J. Opt. Soc. Am. B 12, 486-496 (1995). [CrossRef]

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