## Pulse dynamics in mode-locked lasers: relaxation oscillations and frequency pulling

Optics Express, Vol. 15, Issue 11, pp. 6677-6689 (2007)

http://dx.doi.org/10.1364/OE.15.006677

Acrobat PDF (210 KB)

### Abstract

A theoretical description of the pulse dynamics in a mode-locked laser including gain dynamics is developed. Relaxation oscillations and frequency pulling are predicted that influence the pulse parameters. Experimental observations of the response of a mode-locked Ti:sapphire laser to an abrupt change in the pump power confirm that the predicted behavior occurs. These results provide a framework for understanding the effects of noise on the spectrum of the laser.

© 2007 Optical Society of America

## 1. Introduction

1. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science **288**, 635–639 (2000). [CrossRef] [PubMed]

2. Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature (London) **416**, 233–237 (2002). [CrossRef]

3. S. A. Diddams, A. Bartels, T. M. Ramond, C. W. Oates, E. A. Curtis, and J. C. Bergquist, “Design and control of femtosecond lasers for optical clocks and synthesis of low-noise optical and microwave signals,” J. Sel. Top. Quantum Electron. **9**, 1072–1080 (2003). [CrossRef]

4. M. M. Boyd, T. Zelevinsky, A. D. Ludlow, S. M. Foreman, S. Blatt, T. Ido, and J. Ye, “Optical atomic coherence at the one second time scale,” Science **314**, 1430–1433 (2006). [CrossRef] [PubMed]

5. E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science **305**, 1267–1269 (2004). [CrossRef] [PubMed]

6. L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T.W. Hänsch,“Route to phase control of ultrashort light pulses,” Opt. Lett. **21**, 2008–2010 (1996). [CrossRef] [PubMed]

7. A. Poppe, R. Holzwarth, A. Apolonski, G. Tempea, Ch. Spielmann, T.W. Hänsch, and F. Krausz, “Few-cycle optical waveform synthesis,” Appl. Phys. B **72**, 373–376 (2001). [CrossRef]

8. K. W. Holman, R. J. Jones, A. Marian, S. T. Cundiff, and J. Ye, “Detailed studies and control of intensityrelated dynamics of femtosecond frequency combs from mode-Locked Ti:sapphire lasers,” J. Sel. Top. Quantum Electron. **9**, 1018–1024 (2003). [CrossRef]

6. L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T.W. Hänsch,“Route to phase control of ultrashort light pulses,” Opt. Lett. **21**, 2008–2010 (1996). [CrossRef] [PubMed]

8. K. W. Holman, R. J. Jones, A. Marian, S. T. Cundiff, and J. Ye, “Detailed studies and control of intensityrelated dynamics of femtosecond frequency combs from mode-Locked Ti:sapphire lasers,” J. Sel. Top. Quantum Electron. **9**, 1018–1024 (2003). [CrossRef]

9. H. A. Haus and E. P. Ippen, “Group velocity of solitons,” Opt. Lett. **26**, 1654–1656 (2001). [CrossRef]

10. M.J. Ablowitz, B. Ilan, and S.T. Cundiff, “Carrier-envelope phase slip of ultrashort dispersion-managed solitons,” Opt. Lett. **29**, 1808–1810 (2004). [CrossRef] [PubMed]

8. K. W. Holman, R. J. Jones, A. Marian, S. T. Cundiff, and J. Ye, “Detailed studies and control of intensityrelated dynamics of femtosecond frequency combs from mode-Locked Ti:sapphire lasers,” J. Sel. Top. Quantum Electron. **9**, 1018–1024 (2003). [CrossRef]

11. D. von der Linde, “Characterization of the noise in continuously operating mode-locked lasers,” Appl. Phys. B **39**, 201–217 (1986). [CrossRef]

12. A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. **112**, 1940–1949 (1958). [CrossRef]

## 2. Background

13. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. **29**, 983–995 (1993). [CrossRef]

13. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. **29**, 983–995 (1993). [CrossRef]

*T*is the laser cavity round trip time,

_{R}*u*is the complex field envelope, normalized so that |

*u*|

^{2}equals the instantaneous power,

*T*is a slow time corresponding to

*z/v¯*, the folded distance along the laser divided by the average group velocity, and

_{g}*t*is fast (retarded) time. We have explicitly added a phase slip

*θ*

_{sl}to the Haus-Mecozzi equations, so that the phase slip of

*u*per round trip corresponds to the actual value at low power, where nonlinear effects can be neglected. Haus and Mecozzi assume that the dispersion

*D*= −

*β″ L*is constant. We use

*β″*to designate the usual chromatic dispersion and

*L*to designate the roundtrip length of the laser. Finally,

*g*and

*l*are linear gain and loss per round trip at the central frequency of the laser pulse, Ω

_{g}is the gain bandwidth,

*γ*is the Kerr coefficient, and

*δ*is the fast saturable loss (or gain) parameter, which in a Ti: sapphire laser arises from the Kerr lens effect. We note that we have changed the definitions of

*D*,

*γ*, and

*δ*from those of Haus and Mecozzi to bring the notation into closer alignment with the majority of the literature on optical solitons. However, like Haus and Mecozzi, we use

*T/T*and not propagation length

_{R}*z*as an independent variable. Consequently, the coefficient

*γ*has units of inverse power, in contrast to the usual case in optical fiber solitons, where it has units of inverse power × inverse length.

*ω*

_{0}, usually referred to as the angular carrier frequency, is removed from

*u*(

*T,t*), so that its spectrum is shifted in the frequency domain towards zero frequency by this amount and is located in the neighborhood of zero frequency. A complete derivation of Eq. (1) in the context of optical fibers can be found in Refs. [14

14. C. R. Menyuk and B. S. Marks, “Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems,” J. Lightwave Technol. **24**, 2806–2826 (2006). [CrossRef]

15. C. R. Menyuk, “Application of multiple-scale-length methods to the study of optical fiber transmission,” J. Eng. Math. **36**, 113–136 (1999). [CrossRef]

*t*as a function of

*T*, we must choose

*ω*

_{0}equal to this frequency. Otherwise, a group velocity term proportional to

*∂u/∂t*with a real coefficient must be added to Eq. (1), or the pulse’s central time changes at a constant rate as a function of

*T*. There is some freedom in our choice of

*ϖ*

_{eq}, the equilibrium pulse central frequency. In the laser system that we are considering, it is convenient to choose

*ϖ*

_{eq}=

*ω*

_{0}.

*g*to the pulse energy

*w*, for which Haus and Mecozzi take

*g*

_{0}is the unsaturated gain and

*P*is the saturation power. The soliton limit corresponds to

_{s}*g,l*≪ 1,

*δ*≪

*γ*, and

*g*/Ω

^{2}

_{g}≪ |

*D*|. In this limit, Eq. (1) reduces at lowest order to the nonlinear Schrödinger equation, and both the linear and nonlinear gain and loss contributions appear as perturbations.

*w*=

*w*

_{eq}+ Δ

*w*, the central frequency

*ϖ*=

*ϖ*

_{eq}+ Δ

*ϖ*, the central pulse time

*τ*=

*τ*

_{eq}+ Δ

*τ*, and the phase

*θ*=

*θ*

_{eq}+ Δ

*θ*, where

*w*

_{eq},

*ϖ*

_{eq},

*τ*

_{eq}, and

*θ*

_{eq}are the equilibrium values of these quantities, while Δ

*w*, Δ

*ϖ*, Δ

*τ*, and Δ

*θ*are their changes. Since the system is invariant under time and phase translations, we may without loss of generality choose

*τ*

_{eq}= 0 and

*θ*

_{eq}= -

*θ*

_{ceo}(

*ω*

_{0})

*T/T*, where

_{R}*θ*

_{ceo}is the carrier envelope offset phase shift per round trip in the laser at

*ω*=

*ω*

_{0}. We note that

*θ*

_{ceo}≠

*θ*

_{sl}in general because of the nonlinear phase shift. This choice of phase, which is standard [13

13. H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. **29**, 983–995 (1993). [CrossRef]

16. H. A. Haus, “Quantum noise in a solitonlike repeater,” J. Opt. Soc. Am. B **8**, 1122–1126 (1991). [CrossRef]

17. T. Georges, “Perturbation theory for the assessment of soliton transmission control,” J. Opt. Fiber Technol. **1**, 97–116 (1995). [CrossRef]

*p*corresponds to -Δ

*ϖ*.

**v**= (Δ

*w*,Δ

*ϖ*,Δ

*τ*,Δ

*θ*)

^{t}is the vector of the changes in the four pulse parameters (superscript

*t*denotes the transpose). The quantity A is the 4×4 matrix of the constant coefficients that govern the linear response of each parameter to changes in either itself or the other parameters, while

**S**is the vector of noise sources [13

**29**, 983–995 (1993). [CrossRef]

**29**, 983–995 (1993). [CrossRef]

18. Y. Takushima, H. Sotobayashi, M. E. Grein, E. P. Ippen, and H. A. Haus, “Linewidth of mode combs of passively and actively mode-locked semiconductor laser diodes,” Proc. SPIE **5595**, 213–227 (2004). [CrossRef]

*D*, but closer to Gaussian when the system is dispersion-managed), are known at best qualitatively. This difficulty is particularly acute for

*A*, which depends on the unsaturated gain, the saturation power, and the nonlinear Kerr coefficient — none of which are easily measurable. Indeed, in Ti:sapphire lasers with soft Kerr lens modelocking, the variation of the fast saturable gain with power is not linear as implied by Eq. (1) [20

_{ww}20. D.-G. Juang, Y.-C. Chen, S.-H. Hsu, K.-H. Lin, and W.-F. Hsieh, “Differential gain and buildup dynamics of selfstarting Kerr lens mode-locked Ti:sapphire laser without an internal aperture,” J. Opt. Soc. Am. B **14**, 2116–2121 (1997). [CrossRef]

*A*(where

_{ϖx}*x*is

*g*or

*w*) are zero, which the experiments to be described shortly show is not the case. Thus, an accurate calculation of the line shape based on this approach is not possible.

**S**but are difficult to directly measure. These measurements yield important insights into the laser behavior that point the way toward more complete and accurate underlying physical models than Eq. (1).

## 3. Experiment

*ϖ*, the laser intensity, spectrally resolved with a monochromator, was measured using a photodiode. We measure time traces for a range of wavelengths, covering the entire laser spectrum. The central frequency is approximated as the centroid of the frequency spectrum.

21. S.T. Cundiff, J.M. Soto-Crespo, and N. Akhmediev, “Experimental evidence for soliton explosions,” Phys. Rev. Lett. **88**, 073903 (2002). [CrossRef] [PubMed]

*τ*is the fluorescence lifetime of the medium [22], and we are assuming that the Ti:sapphire crystal may be treated as an ideal four-level system, so that

_{f}*g*∝

*N*

_{2}, the population of the upper lasing level. As a consequence, Eq. (3) as a 4-dimensional system is incomplete. It must be replaced by a similar 5-dimensional system in which the vector

**v**becomes (Δ

*w*, Δ

*ϖ*, Δ

*τ*, Δ

*θ*, Δ

*g*)

^{t}, where Δ

*g*is the change in the gain, and A becomes a 5 × 5 matrix. Second, there is significant frequency pulling. Frequency pulling is not included in Eq. (1), which must be appropriately modified.

*A*,

_{wx}*A*, and

_{gx}*A*, where

_{ϖx}*x*=

*w*,

*ϖ*,

*τ*,

*θ*, or

*g*. We can also infer

*N*

_{2}, the number of atoms in the upper lasing level. This parameter is important in determining

**S**. Finally, we may infer minimal modifications to (1) that incorporate frequency pulling, although these modifications are not unique. In principle, if the gain and loss as a function of frequency were known, we could directly calculate the frequency-pulling coefficients. However, the gain and loss as a function of frequency are difficult to measure and not well known.

## 4. Gain and intensity dynamics

*A*= (d

_{ww}*τ*

_{ph}

^{-1}/d

*w*)

*w*

_{eq},

*A*= -2

_{wg}*w*

_{eq}/

*T*,

_{R}*A*= (

_{gw}*g*

_{eq}/

*τ*)(1/

_{f}*P*), and

_{s}T_{R}*A*= (1 +

_{gg}*w*

_{eq}/

*P*)(1/

_{s}T_{R}*τ*). All other

_{f}*A*and

_{wx}*A*equal zero.

_{gx}*g*

_{0}, leading to the damped oscillations shown in Fig. 1. The damping rate is given by

*α*= (

*A*+

_{ww}*A*)/2 and the oscillation frequency is given by

_{gg}*ω*

^{2}

_{osc}= -

*A*+

_{wg}A_{gw}*A*-

_{ww}A_{gg}*α*

^{2}. We show measured

*ω*

_{osc}and

*α*in Fig. 2 as a function of the pump power for both modelocked and continuous wave (cw) operation. These values are obtained by explicitly solving Eq. (6) and fitting

*ω*

_{osc}and

*α*to the analytical form, shown in Appendix A, using the method of least squares. As the pump power varies between 4.7 and 5.4 W, we find that that

*ω*

_{osc}varies from 2.6×10

^{6}rad/s to 2.8×10

^{6}rad/s and

*α*varies from 0.2×10

^{6}s

^{-1}to 1.0×10

^{6}s

^{-1}for the modelocked operation. The variation of

*ω*

_{osc}in the cw case is qualitatively similar to the variation in the modelocked case, but the variation of

*α*differs significantly, being far more gradual.

*N*

_{ph}=

*w*/

*h¯ϖ*and the number of atoms in the upper lasing level

*N*

_{2}= (

*V*/

*σ*)

_{lg}*g*, where

*h¯*is Planck’s constant,

*V*is the effective gain volume,

*l*is the effective gain length, and

_{g}*σ*is the gain cross section. We then find where Δ

*N*

_{20}indicates the abrupt change in

*N*

_{2}corresponding to Δ

*g*

_{0}, and

*N*

_{ph,eq}and

*N*

_{2,eq}indicate the equilibrium photon number and upper state population, respectively. Ignoring the damping contributions, we find that -

*A*= (1/

_{wg}A_{gw}*τ*

_{ph})

^{2}(

*N*

_{ph,eq}/

*N*

_{2,eq}), from which we obtain the important result, where we used the experimental value for the cavity lifetime

*τ*

_{ph}= 0.1

*μ*s, and we may infer (-

*A*)

_{wg}A_{gw}^{1/2}≃ 2.8 × 10

^{6}rad/s by using this value and showing that it produces the variation of

*ω*

_{osc}shown in Fig. 2. Using

*w*

_{eq}= 55 nJ and

*ϖ*

_{eq}= 2.3 × 10

^{15}rad/s, we obtain

*N*

_{ph,eq}= 2.3 × 10

^{11}and

*N*

_{2,eq}= 2.9 × 10

^{12}. Using the expression

*A*= (1/

_{gg}*τ*

_{ph}) + (1/

*τ*

_{ph})(

*N*

_{ph,eq}/

*N*

_{2,eq}) and the measured value

*τ*= 2.5

_{f}*μ*s, we find

*A*= 1.2 × 10

_{gg}^{6}s

^{-1}, and we also find that

*A*= 2α-

_{ww}*A*varies from -0.8 × 10

_{gg}^{6}s

^{-1}at a pump power of 4.7 W to 0.8 × 10

^{6}s

^{-1}at a pump power of 5.4 W. We note that the laser remains stable when

*A*becomes negative, although the relaxation oscillations become long-lived. This behavior is very different from the cw behavior shown in Fig. 2 and indicates that relaxation oscillations may be important in modelocked lasers even when they are not important in the same laser generating cw light. Using (-

_{ww}*A*)

_{wg}A_{gw}^{1/2}= 2.8 × 10

^{6}rad/s and the relationship

*ω*

^{2}

_{osc}= -

*A*+

_{wg}A_{gw}*A*-

_{ww}A_{gg}*α*

^{2}implies that

*ω*

_{osc}varies from 2.6 × 10

^{6}rad/s to 2.8 × 10

^{6}rad/s, consistent with Fig. 2. Using the relation

*A*= -2

_{wg}*w*

_{eq}/

*T*and the measured value

_{R}*w*

_{eq}= 55 nJ, we infer

*A*= -11 Js

_{wg}^{-1}and

*A*= 7.1 × 10

_{gw}^{11}J

^{-1}s

^{-1}.

## 5. Frequency pulling

*A*=

_{ϖτ}*A*= 0. We determine the

_{ϖθ}*A*from the experimental data by using the method of least squares to fit the analytical form of the solution to Eq. (9), shown in Appendix A. We find

_{ϖx}*A*= 1.2 × 10

_{ϖϖ}^{6}s

^{-1},

*A*= -3.0 × 10

_{ϖg}^{8}THz/s, and

*A*= 1.2 × 10

_{ϖw}^{5}THz/(nJs) at 5.1 W pump power.

**29**, 983–995 (1993). [CrossRef]

*g̃*(

*ω*) and the loss

*l̃*(

*ω*) become

*ω*indicates the change in frequency with respect to the carrier frequency

*ω*

_{0}, so that the actual frequency is given by

*ω*

_{0}+

*ω*. The coefficients

*g*

^{(m)}and

*l*

^{(m)}indicate the

*m*

^{th}derivatives of the gain and loss with respect to frequency. With this expansion, the operator -

*l*+

*g*[1 + (1/Ω

^{2}

_{g})

*∂*

^{2}/

*∂t*

^{2}] is replaced by [

*g*

^{(0)}-

*l*

^{(0)}] +

*i*[

*g*

^{(1)}-

*l*

^{(1)}]

*∂*/

*∂t*- (1/2)[

*g*

^{(2)}-

*l*

^{(2)}]

*∂*

^{2}/

*∂t*

^{2}-

*i*(1/6)[

*g*

^{(3)}-

*l*

^{(3)}]

*∂*

^{3}/

*∂t*

^{3}. Changes in the gain will produce an additional contribution of

*i*Δ

*g*

^{(1)}

*∂/∂t*at lowest non-trivial order. Identifying (1/2)[

*g*

^{(2)}-

*l*

^{(2)}] ≡ Ω

^{2}

_{g}, we see that we are adding an additional perturbation to Eq. (1) of the form

*P*[

*u*] =

*i*[

*g*

^{(1)}-

*l*

^{(1)}]

*∂u/∂t*-

*i*[

*g*

^{(3)}-

*l*

^{(3)}]

*∂*

^{3}

*u*/

*∂t*

^{3}+

*i*Δ

*g*

^{(1)}

*∂u/∂t*. Since our system is dispersion-managed, the pulses are nearly Gaussian in shape. So, it is appropriate to use a perturbation expansion based on Gaussian-shaped pulses. We show in Appendix B that

*t*

_{FWHM}is the full width half maximum pulse duration of 15 fs. Assuming with Haus and Mecozzi [13

**29**, 983–995 (1993). [CrossRef]

_{g}= 1.6 × 10

^{15}rad/s and using the experimentally determined value of

*A*, we obtain

_{ϖg}*g*

^{(1)}Ω

_{g}/g^{(0)}= 0.42, which is consistent with the loss bandwidth peak lying below the gain bandwidth peak. Using the experimental value of

*A*, we also find [

_{ϖw}*g*

^{(3)}-

*l*

^{(3)}]Ω

^{3}

_{g}= -4.5, indicating a substantial asymmetry over the nominal gain bandwidth. We note that these inferences are not unique, since they assume that higher orders of the Taylor expansion of the gain and loss curves do not contribute. Nonetheless, these results show that it is possible to infer corrections to Eq. (1) from the measurements of A.

## 6. Conclusion

*A*,

_{wx}*A*, and

_{gx}*A*. One may use these results to infer

_{ϖx}*N*

_{2}, which is needed to determine the strengths of the quantum noise sources. The next step, measurement of

*A*and

_{τx}*A*, is underway. This information can be used with a full perturbation theory for this system with realistic pulse shapes that take into account dispersion management to yield a complete calculation of the entire line shape.

_{θx}## Appendix A.

## Solutions to Eqs. (6) and (9)

*g*

_{0}changes instantaneously at

*t*= 0 to its final value. The solution is:

*α¯*≡

*α*-

*Aϖϖ*, we integrate this equation to obtain

## Appendix B.

## Generalized Perturbation Theory

27. M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, “Quantum-limited timing jitter in actively mode-locked lasers,” IEEE J. Quantum Electron. **40**, 1458–1470 (2004). [CrossRef]

9. H. A. Haus and E. P. Ippen, “Group velocity of solitons,” Opt. Lett. **26**, 1654–1656 (2001). [CrossRef]

10. M.J. Ablowitz, B. Ilan, and S.T. Cundiff, “Carrier-envelope phase slip of ultrashort dispersion-managed solitons,” Opt. Lett. **29**, 1808–1810 (2004). [CrossRef] [PubMed]

*et al*. [28

28. H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: Theory and experiment,” IEEE J. Quantum Electron. **31**, 591–598 (1995). [CrossRef]

*et al*. [29

29. N. J. Smith, N. J. Doran, W. Forysiak, and F. M. Knox, “Soliton transmission using periodic dispersion compensation,” J. Lightwave Technol. **15**, 1808–1822 (1997). [CrossRef]

30. M. J. Ablowitz and G. Biondini, “Multiscale pulse dynamics in communication systems with strong dispersion management,” Opt. Lett. **23**, 1668–1670 (1998). [CrossRef]

31. I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. **63**, 861–866 (1996) [Pis’ma Zh. Eksp. Teor. Fiz. **63**, 814–819 (1996)]. [CrossRef]

*i. e.*, the variation of the dispersion is large compared to the average. Typically, the map strengths in laser systems are moderate, and the ratio of the variation to the average is close to 1. Moreover, one would like to have an approach that can be applied to computational studies like those of Paschotta [25

25. R. Paschotta, “Noise of mode-locked lasers (Part I): Numerical model,” Appl. Phys. B **79**, 153–162 (2004). [CrossRef]

26. R. Paschotta, “Noise of mode-locked lasers (Part II): Timing jitter and other fluctuations,” Appl. Phys. B **79**, 163–173 (2004). [CrossRef]

*T*,

*i.e*.,

*T*(

_{R}∂u*T,t*)/

*∂T*=

*F*[

*u*(

*T,t*)]. While Eq. (1) is an example of just such a model, the computational model of Paschotta [25

25. R. Paschotta, “Noise of mode-locked lasers (Part I): Numerical model,” Appl. Phys. B **79**, 153–162 (2004). [CrossRef]

26. R. Paschotta, “Noise of mode-locked lasers (Part II): Timing jitter and other fluctuations,” Appl. Phys. B **79**, 163–173 (2004). [CrossRef]

*u*

_{0}(

*T,t*) that depends only on the four parameters

*w*,

*ϖ*, τ, and

*θ*. Gain and loss are needed to set the equilibrium values of

*w*and

*ϖ*, and loss is needed to damp the continuum radiation, but in the dispersion-managed soliton regime in which virtually all of today’s short-pulse lasers operate, it is appropriate to treat these terms perturbatively. In addition, we will assume that the underlying equations are time- and phase-invariant at zero order, so that

*u*

_{0}(

*T,t*;

*w*,

*ϖ*,

*τ*,

*θ*) =

*u*

_{0}(

*T,t*-

*τ*;

*w*,

*ϖ*)exp(

*iθ*). While it is not necessary for our development, we will also assume that at zero order the underlying equations are frequency-invariant at zero order. It is not difficult to find model systems with no gain or loss that violate this assumption. If higher-order dispersion for example is included in the zero-order system, then this assumption is not valid. However, this assumption holds at zero order in the dispersion-managed systems that are of interest to us here, and we find

*u*

_{0}(

*T,t*) =

*u*

_{0}(

*T,t*-

*τ*;

*w*)exp[-

*i*Δ

*ϖ*(

*i*-

*τ*) +

*iθ*], where we have used the assumption

*ω*

_{0}=

*ϖ*

_{eq}.

*u*(

*T,t*) =

*u*

_{0}(

*T,t*) + Δ

*u*(

*T,t*) and we linearize

*F*[

*u*] about the equilibrium (periodically stationary) solution

*u*

_{0}(

*T, t*), we obtain a linear Bloch-Floquet equation with periodically varying coefficients. Starting at any location in the laser, we may integrate this equation over one round trip and divide by

*T*to obtain an averaged equation that governs the slow evolution and that may be written in the form

_{R}*M*and

*N*are operators that may be non-local in time. In the case of the nonlinear Schrödinger equation with constant dispersion,

*M*= (

*D*/2)

*∂*

^{2}/

*∂t*

^{2}+ 2γ|

*u*

_{0}|

^{2}and

*N*=

*γu*

^{2}

_{0}. In keeping with our assumption that the zero-order system has no gain or loss, we assume that

*M*is a Hermitian operator and

*N*is symmetric, by which we mean that given any

*u*(

*t*) and

*v*(

*t*),

*v*|

*u*〉 = (1/2) ∫

^{∞}

_{-∞}d

*t*(

*v**

*u*+

*vu**) and note that if

*u*satisfies Eq. (B.1), then

*v*=

*iu*satisfies the dual equation

*u*=

*u*Δ

_{w}*w*+

*u*Δ

_{ϖ}*ϖ*+

*u*Δτ +

_{τ}*uθ*Δ

*θ*+ Δ

*u*, where

_{c}*u*=

_{x}*∂u*

_{0}/

*∂x*,

*x*=

*w*,

*ϖ*,

*τ*, or θ, and Δ

*u*is a dispersive wave continuum. We first have the important result that since the

_{c}*u*satisfy Eq. (B.1),

_{x}*v*≡

_{x}*iu*must satisfy the dual equations. Second, we may show 〈

_{x}*v*|Δ

_{x}*u*〉 = 0, using an approach that is analogous to the approach used in traditional soliton perturbation theory [32

_{c}32. H. A. Haus and M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. **21**, 1172–1188 (1985). See sec. 9. [CrossRef]

*v*| on any perturbed equation, we may find the effect of the perturbation on the four soliton parameters, just as in traditional soliton perturbation theory.

_{x}*u*

_{0}, the 〈

*v*|

_{x}*u*〉 are non-zero in general for all combinations of

_{y}*x*and

*y*. However, in the special case in which the pulses are symmetric about

*t*-

*τ*and are entirely in one phase — as is the case for both standard and dispersion-managed solitons at their point of minimum compression — then we find 〈

*v*|

_{x}*u*〈 = 0 when

_{y}*x*≠

*y*. In practice, the point of minimum compression is arranged to be at the exit mirror of the laser, so that is the point at which the pulses are observed.

*u*

_{0}(

*t*) =

*A*exp [-(

*t*-

*τ*)

^{2}/2

*t*

_{p}^{2}exp[-

*i*Δ

*ϖ*(

*t*-

*τ*) +

*iθ*]. We must now relate

*A*and

*t*to the pulse energy

_{p}*w*. Here, we may appeal to the computationally-determined relations [33

33. N. J. Smith, N. J. Doran, F. M. Knox, and W. Forysiak, “Energy-scaling characteristics of solitons in strongly dispersion-managed fibers,” Opt. Lett. **21**, 1981–1983 (1996). [CrossRef] [PubMed]

34. T. Yu, E. A. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Dispersion-managed soliton interactions in optical fibers,” Opt. Lett. **22**, 793–795 (1997). [CrossRef] [PubMed]

*L*

_{1}and

*L*

_{2}with dispersions

*w*= (

*r*/

*t*)(1 +

_{p}*s*/

*t*

_{p}^{4}), where

33. N. J. Smith, N. J. Doran, F. M. Knox, and W. Forysiak, “Energy-scaling characteristics of solitons in strongly dispersion-managed fibers,” Opt. Lett. **21**, 1981–1983 (1996). [CrossRef] [PubMed]

34. T. Yu, E. A. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Dispersion-managed soliton interactions in optical fibers,” Opt. Lett. **22**, 793–795 (1997). [CrossRef] [PubMed]

*A*to

*w*. We now find,

*v*|

_{x}*u*〉 = 1. Operating on

_{x}*P*[

*u*

_{0}] with 〈

*v*|, we obtain

_{ϖ}*ω*= 0. In order to enforce the condition that dΔ

*ϖ*/d

*T*= 0 at equilibrium, we must set [

*g*

^{(1)}-

*l*

^{(1)}] = -[

*g*

^{(3)}-

*l*

^{(3)}]/4

*t*

^{2}

_{p,eq}, from which we find

*t*= (d

_{p}*t*/d

_{p}*w*)Δ

*w*= -[(

*t*/

_{p}*w*)(1 +

*s*/

*t*

_{p}^{4})/(1 + 5

*s*/

*t*

_{p}^{4})] and Δ

*g*

^{(1)}= (

*g*

^{(1)}/

*g*

^{(0)})Δ

*g*, we conclude

33. N. J. Smith, N. J. Doran, F. M. Knox, and W. Forysiak, “Energy-scaling characteristics of solitons in strongly dispersion-managed fibers,” Opt. Lett. **21**, 1981–1983 (1996). [CrossRef] [PubMed]

*s*/

*t*

_{p}^{4}= 0.7|(

*β*

_{1}

35. Q. Quraishi, S. T. Cundiff, B. Ilan, and M. J. Ablowitz, “Dynamics of nonlinear and dispersion managed solitons,” Phys. Rev. Lett. **94**, 243904 (2005). [CrossRef]

*t*

_{FWHM}= 15 fs, we have

*s*/

*t*

_{p}^{4}= 0.05 and (1+

*s*/

*t*

_{p}^{4})/(1 + 5

*s*/

*t*

_{p}^{4}) = 0.84.

*A*becomes 3.00 and the coefficient 2.77 in the expression for

_{ϖw}*A*becomes 2.07. So, as is often the case, the traditional theory agrees well with the generalized approach described here. However, there is no compelling reason to use the traditional approach. The generalized approach is no more difficult to apply than the traditional approach and makes use of the true zero-order pulse shapes with whatever exactitude the problem at hand requires.

_{ϖg}## References and links

1. | D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science |

2. | Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature (London) |

3. | S. A. Diddams, A. Bartels, T. M. Ramond, C. W. Oates, E. A. Curtis, and J. C. Bergquist, “Design and control of femtosecond lasers for optical clocks and synthesis of low-noise optical and microwave signals,” J. Sel. Top. Quantum Electron. |

4. | M. M. Boyd, T. Zelevinsky, A. D. Ludlow, S. M. Foreman, S. Blatt, T. Ido, and J. Ye, “Optical atomic coherence at the one second time scale,” Science |

5. | E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, “Direct measurement of light waves,” Science |

6. | L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T.W. Hänsch,“Route to phase control of ultrashort light pulses,” Opt. Lett. |

7. | A. Poppe, R. Holzwarth, A. Apolonski, G. Tempea, Ch. Spielmann, T.W. Hänsch, and F. Krausz, “Few-cycle optical waveform synthesis,” Appl. Phys. B |

8. | K. W. Holman, R. J. Jones, A. Marian, S. T. Cundiff, and J. Ye, “Detailed studies and control of intensityrelated dynamics of femtosecond frequency combs from mode-Locked Ti:sapphire lasers,” J. Sel. Top. Quantum Electron. |

9. | H. A. Haus and E. P. Ippen, “Group velocity of solitons,” Opt. Lett. |

10. | M.J. Ablowitz, B. Ilan, and S.T. Cundiff, “Carrier-envelope phase slip of ultrashort dispersion-managed solitons,” Opt. Lett. |

11. | D. von der Linde, “Characterization of the noise in continuously operating mode-locked lasers,” Appl. Phys. B |

12. | A. L. Schawlow and C. H. Townes, “Infrared and optical masers,” Phys. Rev. |

13. | H. A. Haus and A. Mecozzi, “Noise of mode-locked lasers,” IEEE J. Quantum Electron. |

14. | C. R. Menyuk and B. S. Marks, “Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems,” J. Lightwave Technol. |

15. | C. R. Menyuk, “Application of multiple-scale-length methods to the study of optical fiber transmission,” J. Eng. Math. |

16. | H. A. Haus, “Quantum noise in a solitonlike repeater,” J. Opt. Soc. Am. B |

17. | T. Georges, “Perturbation theory for the assessment of soliton transmission control,” J. Opt. Fiber Technol. |

18. | Y. Takushima, H. Sotobayashi, M. E. Grein, E. P. Ippen, and H. A. Haus, “Linewidth of mode combs of passively and actively mode-locked semiconductor laser diodes,” Proc. SPIE |

19. | F. X. Kärtner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Few-cycle pulses directly from a laser,” in |

20. | D.-G. Juang, Y.-C. Chen, S.-H. Hsu, K.-H. Lin, and W.-F. Hsieh, “Differential gain and buildup dynamics of selfstarting Kerr lens mode-locked Ti:sapphire laser without an internal aperture,” J. Opt. Soc. Am. B |

21. | S.T. Cundiff, J.M. Soto-Crespo, and N. Akhmediev, “Experimental evidence for soliton explosions,” Phys. Rev. Lett. |

22. | W. Koechner and M. Bass, |

23. | L. A. Jiang, M. E. Grein, H. A. Haus, and E. P. Ippen, “Noise of mode-locked semiconductor lasers,” IEEE J. Sel. Top. Quantum Electron. |

24. | L. Matos, O. D. Mücke, J. Chen, and F. X. Kärtner, “Carrier-envelope phase dynamics and noise analysis in octave-spanning Ti:sapphire lasers,” Opt. Express |

25. | R. Paschotta, “Noise of mode-locked lasers (Part I): Numerical model,” Appl. Phys. B |

26. | R. Paschotta, “Noise of mode-locked lasers (Part II): Timing jitter and other fluctuations,” Appl. Phys. B |

27. | M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, “Quantum-limited timing jitter in actively mode-locked lasers,” IEEE J. Quantum Electron. |

28. | H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber ring lasers: Theory and experiment,” IEEE J. Quantum Electron. |

29. | N. J. Smith, N. J. Doran, W. Forysiak, and F. M. Knox, “Soliton transmission using periodic dispersion compensation,” J. Lightwave Technol. |

30. | M. J. Ablowitz and G. Biondini, “Multiscale pulse dynamics in communication systems with strong dispersion management,” Opt. Lett. |

31. | I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” JETP Lett. |

32. | H. A. Haus and M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. |

33. | N. J. Smith, N. J. Doran, F. M. Knox, and W. Forysiak, “Energy-scaling characteristics of solitons in strongly dispersion-managed fibers,” Opt. Lett. |

34. | T. Yu, E. A. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, “Dispersion-managed soliton interactions in optical fibers,” Opt. Lett. |

35. | Q. Quraishi, S. T. Cundiff, B. Ilan, and M. J. Ablowitz, “Dynamics of nonlinear and dispersion managed solitons,” Phys. Rev. Lett. |

**OCIS Codes**

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

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: January 23, 2007

Revised Manuscript: May 4, 2007

Manuscript Accepted: May 10, 2007

Published: May 16, 2007

**Citation**

Curtis R. Menyuk, Jared K. Wahlstrand, John Willits, Ryan P. Smith, Thomas R. Schibli, and Steven T. Cundiff, "Pulse dynamics in mode-locked lasers: relaxation oscillations and frequency pulling," Opt. Express **15**, 6677-6689 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-11-6677

Sort: Year | Journal | Reset

### References

- D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288, 635-639 (2000). [CrossRef] [PubMed]
- Th. Udem, R. Holzwarth, and T. W. Hänsch, "Optical frequency metrology," Nature (London) 416, 233-237 (2002). [CrossRef]
- S. A. Diddams, A. Bartels, T. M. Ramond, C. W. Oates, E. A. Curtis, and J. C. Bergquist, "Design and control of femtosecond lasers for optical clocks and synthesis of low-noise optical and microwave signals," J. Sel. Top. Quantum Electron. 9, 1072-1080 (2003). [CrossRef]
- M. M. Boyd, T. Zelevinsky, A. D. Ludlow, S. M. Foreman, S. Blatt, T. Ido, and J. Ye, "Optical atomic coherence at the one second time scale," Science 314, 1430-1433 (2006). [CrossRef] [PubMed]
- E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Baltuska, V. Yakovlev, A. Scrinzi, Th. Westerwalbesloh, U. Kleineberg, U. Heinzmann, M. Drescher, and F. Krausz, "Direct measurement of light waves," Science 305, 1267-1269 (2004). [CrossRef] [PubMed]
- L. Xu, Ch. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T.W. Hänsch,"Route to phase control of ultrashort light pulses," Opt. Lett. 21, 2008-2010 (1996). [CrossRef] [PubMed]
- A. Poppe, R. Holzwarth, A. Apolonski, G. Tempea, Ch. Spielmann, T.W. H¨ansch, and F. Krausz, "Few-cycle optical waveform synthesis," Appl. Phys. B 72, 373-376 (2001). [CrossRef]
- K. W. Holman, R. J. Jones, A. Marian, S. T. Cundiff, and J. Ye, "Detailed studies and control of intensityrelated dynamics of femtosecond frequency combs from mode-Locked Ti:sapphire lasers," J. Sel. Top. Quantum Electron. 9, 1018-1024 (2003). [CrossRef]
- H. A. Haus and E. P. Ippen, "Group velocity of solitons," Opt. Lett. 26, 1654-1656 (2001). [CrossRef]
- M. J. Ablowitz, B. Ilan, and S. T. Cundiff, "Carrier-envelope phase slip of ultrashort dispersion-managed solitons," Opt. Lett. 29, 1808-1810 (2004). [CrossRef] [PubMed]
- D. von der Linde, "Characterization of the noise in continuously operating mode-locked lasers," Appl. Phys. B 39, 201-217 (1986). [CrossRef]
- A. L. Schawlow and C. H. Townes, "Infrared and optical masers," Phys. Rev. 112, 1940-1949 (1958). [CrossRef]
- H. A. Haus and A. Mecozzi, "Noise of mode-locked lasers," IEEE J. Quantum Electron. 29, 983-995 (1993). [CrossRef]
- C. R. Menyuk and B. S. Marks, "Interaction of polarization mode dispersion and nonlinearity in optical fiber transmission systems," J. Lightwave Technol. 24, 2806-2826 (2006). [CrossRef]
- C. R. Menyuk, "Application of multiple-scale-length methods to the study of optical fiber transmission," J. Eng. Math. 36, 113-136 (1999). [CrossRef]
- H. A. Haus, "Quantum noise in a solitonlike repeater," J. Opt. Soc. Am. B 8, 1122-1126 (1991). [CrossRef]
- T. Georges, "Perturbation theory for the assessment of soliton transmission control," J. Opt. Fiber Technol. 1, 97-116 (1995). [CrossRef]
- Y. Takushima, H. Sotobayashi, M. E. Grein, E. P. Ippen, and H. A. Haus, "Linewidth of mode combs of passively and actively mode-locked semiconductor laser diodes," Proc. SPIE 5595, 213-227 (2004). [CrossRef]
- F. X. K¨artner, U. Morgner, T. Schibli, R. Ell, H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Few-cycle pulses directly from a laser," in Few-cycle Laser Pulse Generation and its Applications, F. X. K¨artner, ed., Topics in Applied Physics, vol. 95 (Springer, Berlin, Germany, 2004), pp. 73-135.
- D.-G. Juang, Y.-C. Chen, S.-H. Hsu, K.-H. Lin, and W.-F. Hsieh, "Differential gain and buildup dynamics of selfstarting Kerr lens mode-locked Ti:sapphire laser without an internal aperture," J. Opt. Soc. Am. B 14, 2116-2121 (1997). [CrossRef]
- S. T. Cundiff, J. M. Soto-Crespo, and N. Akhmediev, "Experimental evidence for soliton explosions," Phys. Rev. Lett. 88, 073903 (2002). [CrossRef] [PubMed]
- W. Koechner and M. Bass, Solid-state lasers (Springer, New York, NY, 2003), pp. 72-75.
- L. A. Jiang, M. E. Grein, H. A. Haus, and E. P. Ippen, "Noise of mode-locked semiconductor lasers," IEEE J. Sel. Top. Quantum Electron. 7, 159-167 (2001). [CrossRef]
- L. Matos, O. D. Mucke, J. Chen, and F. X. Kartner, "Carrier-envelope phase dynamics and noise analysis in octave-spanning Ti:sapphire lasers," Opt. Express 14, 2497-2511 (2006). [CrossRef] [PubMed]
- R. Paschotta, "Noise of mode-locked lasers (Part I): Numerical model," Appl. Phys. B 79, 153-162 (2004). [CrossRef]
- R. Paschotta, "Noise of mode-locked lasers (Part II): Timing jitter and other fluctuations," Appl. Phys. B 79, 163-173 (2004). [CrossRef]
- M. E. Grein, H. A. Haus, Y. Chen, and E. P. Ippen, "Quantum-limited timing jitter in actively mode-locked lasers," IEEE J. Quantum Electron. 40, 1458-1470 (2004). [CrossRef]
- H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, "Stretched-pulse additive pulse mode-locking in fiber ring lasers: Theory and experiment," IEEE J. Quantum Electron. 31, 591-598 (1995). [CrossRef]
- N. J. Smith, N. J. Doran, W. Forysiak, and F. M. Knox, "Soliton transmission using periodic dispersion compensation," J. Lightwave Technol. 15, 1808-1822 (1997). [CrossRef]
- M. J. Ablowitz and G. Biondini, "Multiscale pulse dynamics in communication systems with strong dispersion management," Opt. Lett. 23, 1668-1670 (1998). [CrossRef]
- I. Gabitov and S. K. Turitsyn, "Breathing solitons in optical fiber links," JETP Lett. 63, 861-866 (1996) [Pis’ma Zh. Eksp. Teor. Fiz. 63, 814-819 (1996)]. [CrossRef]
- H. A. Haus and M. N. Islam, "Theory of the soliton laser," IEEE J. Quantum Electron. 21, 1172-1188 (1985). [CrossRef]
- N. J. Smith, N. J. Doran, F. M. Knox, and W. Forysiak, "Energy-scaling characteristics of solitons in strongly dispersion-managed fibers," Opt. Lett. 21, 1981-1983 (1996). [CrossRef] [PubMed]
- T. Yu, E. A. Golovchenko, A. N. Pilipetskii, and C. R. Menyuk, "Dispersion-managed soliton interactions in optical fibers," Opt. Lett. 22, 793-795 (1997). [CrossRef] [PubMed]
- Q. Quraishi, S. T. Cundiff, B. Ilan, andM. J. Ablowitz, "Dynamics of nonlinear and dispersion managed solitons," Phys. Rev. Lett. 94, 243904 (2005). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.