## Pulse shapes and stability in Kerr and Active Mode-Locking (KAML)

Optics Express, Vol. 2, Issue 5, pp. 204-211 (1998)

http://dx.doi.org/10.1364/OE.2.000204

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### Abstract

We present numerical simulations of laser mode-locking using a spatio-temporal master equation. We look at active mode-locking using an amplitude modulator and compare the results with those found using a phase modulator. We find gaussian pulses and stability conditions consistent with the Kuizenga-Siegman theory of mode-locking. We then add a Kerr medium to the cavity and examine the effect this has on the mode-locking process, the stability, and the shape of the final pulses. We find that the pulses are significantly compressed in both space and time, and the profiles become more sech-like.

© Optical Society of America

## 1. Introduction

^{11. T. Deutsch, Appl. Phys. Lett. 7, 80 (1965). [CrossRef] ,22. D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 709 (1970). [CrossRef] }and theoretically

^{33. D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 694 (1970). [CrossRef] ,44. H. A. Haus, IEEE J. Quantum Electron. QE-11, 323 (1975). [CrossRef] }. Theoretical (and numerical) modelling of the evolution of the optical field in laser cavity can be a formidable task, necessitating the description of both transverse and longitudinal field variations as well as dynamical evolution, perhaps over a huge range of time scales. Previous analysis has tended to concentrate on either spatial or temporal analysis. The former generally assumes that the response time of the nonlinear medium is instantaneous so that the steady-state analysis can be applied to the pulsed regime

^{5–85. T. Brabec, C. Spielmann, P. F. Curley, and F. Krausz, Opt. Lett. 17, 1292 (1992). [CrossRef] [PubMed] }, while the latter does not include spatial effects explicitly, but includes an effective saturable absorption to take the self-focusing due to the nonlinear medium into account

^{9–119. H. A. Haus and Y. Silberberg, IEEE J. Quantum Electron. QE-22, 325 (1986). [CrossRef] }. One example of this is the well-known master equation approach of Haus

*et al*.

^{1010. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, J. Opt. Soc. Am. B 8, 2068 (1991). [CrossRef] }in which a partial differential equation is used to describe evolution on fast and slow time scales, considered as independent variables.

^{1212. A. M. Dunlop, W. J. Firth, E. M. Wright, and D. R. Heatley, Opt. Lett. 21, 770 (1996). [CrossRef] [PubMed] }. Our ME can also handle nonlinearity, although our present requirement of a single nonlinear element limits our model to a ring cavity with one nonlinear element, or a Fabry-Perot cavity with the nonlinear element at an end mirror. In short-pulse (mode-locked) lasers, our partial differential equation (pde) approach treats pulse evolution (dispersion) and transverse effects (diffraction) in mathematically and computationally very similar ways.

## 2. Master Equation

_{x}and

_{t}respectively, and may also include gain and nonlinear terms:

^{1212. A. M. Dunlop, W. J. Firth, E. M. Wright, and D. R. Heatley, Opt. Lett. 21, 770 (1996). [CrossRef] [PubMed] }:

*ψ*=

*S*≡ (

*A*+

*D*)/2 is independent of the cavity reference plane,

*T*

_{R}is the round-trip time, based on the group velocity.

*g*, which we assume to be uniform, linear, and unsaturated, and a nonlinear function

*N*(

*E*) which depends only on the local field

*E*at the reference plane. For a Kerr lens

*N*(

*E*) =

*ik*|

*E*|

^{2}

*E*. The reference plane must be set at the nonlinear element in this ME.

## 3. Intracavity modulation

^{1010. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, J. Opt. Soc. Am. B 8, 2068 (1991). [CrossRef] }:

*β*is a complex parameter whose real and imaginary parts describe group velocity dispersion and gain bandwidth respectively. Re (

*β*) > 0, as we have chosen in these simulations, corresponds to the case of anomalous dispersion. Other fast-time effects (such as intracavity modulation) can also be incorporated. In this paper we are interested in the effect of an intracavity amplitude or phase modulator. It is well-known

^{3}that pulses form at an extremum of the modulation cycle, and are usually much shorter than the modulation period. We can therefore set

*t*= 0 at an extremum, and replace the modulation cycle by a term in the ME which is quadratic in

*t*, i.e. by

*αt*

^{2}, where

*α*is an arbitrary complex parameter, real in the case of amplitude modulation and imaginary for phase modulation.

## 4. Properties of the linear Master Equation

^{1212. A. M. Dunlop, W. J. Firth, E. M. Wright, and D. R. Heatley, Opt. Lett. 21, 770 (1996). [CrossRef] [PubMed] }that this has gaussian solutions in

*x*of the form

*R*is the phase front curvature of the field and

*w*is its width. We expect a similar solution in

*t*, and so we consider a field of the form

*q*

_{t}describes the temporal properties of the field in a similar manner to

*q*

_{x}.

*x*and

*t*we obtain the following three equations:

*dq*

_{x}/

*dT*=

*dq*

_{t}/

*dT*= 0.

*Im*(

*Q*

_{x}) > 0 and

*Im*(

*Q*

_{t}) > 0. This determines the choice of the signs in (8) and (9).

*Q*

_{x}=

*Q*

_{xm}+ Δ

_{x};

*Q*

_{t}=

*Q*

_{tm}+ Δ

_{t}, and neglect

*Re*(∓2

*iψ*) < 0 and

*Re*(∓4√-

*iαβ*) < 0.

*arg*(

*α*) = ±π/2) is stable for either choice of sign, but amplitude modulation (

*arg*(

*α*) = 0 or

*π*) is only stable if

*arg*(

*α*) =

*π*. Thus our ME reproduces the known properties of linear field evolution in optical cavities containing apertures and/or modulators.

## 5. Numerical analysis

^{1515. J. V. Roey, J. van der Donk, and P. Lagasse, J. Opt. Soc. Am. 71, 803 (1981). [CrossRef] }, with periodic boundary conditions. At first sight the form of the ME seems to make this method unsuitable, since the “(A - D)” term belongs in both real space and

*k*-space.

*E*(

*x*,

*t*,

*T*) as follows

^{1414. A. M. Dunlop, W. J. Firth, and E. M. Wright, Opt. Commun. 138, 211 (1997). [CrossRef] }:

*U*(

*x*), and need only be done at the start and end of the simulation, the computational penalty is very slight.

*Ẽ*(

*x*,

*t*,

*T*)

## 6. Active mode-locking

^{1414. A. M. Dunlop, W. J. Firth, and E. M. Wright, Opt. Commun. 138, 211 (1997). [CrossRef] }. We also include an internal modulator, and a gain bandwidth (to limit pulse shortening). At this stage we do not include a Kerr medium, and so we set

*κ*= 0 in (13), which is then linear. The total cavity length is 2

*m*and the strength of the thin lenses is chosen such that the beam focuses close to the slit and to the Kerr lens. The linear power loss in the cavity is ≈ 8%. We include a simple, unsaturable linear gain, which we assume can be large enough to overcome the linear and/or nonlinear losses. The initial field was gaussian in

*x*and flat in

*t*, with an amplitude corresponding to a nonlinear phase shift of 0.1 radians when the Kerr medium is included. We use a 128 × 128 grid, and scale

*x*and

*t*to the width of the gaussian mode.

*E dx dt*), but by rescaling we can investigate the evolution of the beam profile. We find the stable field profiles correspond to the mode profiles calculated from (6) with 1/

*q*

_{x}=

*Q*

_{xm}and 1/

*q*

_{t}=

*Q*

_{tm}. Note that although we have considered anomalous dispersion here pulse formation is also seen with normal dispersion. However, in order to get pulse

*compression*it would be necessary to use a self-defocusing lens and to change the location of the Kerr medium.

*α*|. As expected

^{33. D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 694 (1970). [CrossRef] }, we find that we can get stable solutions with

*arg*(

*α*) = ±

*π*/2 or

*π*, but not with

*arg*(

*α*) = 0. As figure 1 shows, the phase modulated pulses have different temporal widths depending on the sign of

*arg*(

*α*). Note that we use arbitrary units throughout, since our main interest is the

*relative*width, and change in width, in each case. All pulse parameters agree with the analytic predictions (8), (9).

## 7. Active mode-locking plus a Kerr medium

*iκ*|

*U*(

*x*)

^{-1}

*Ẽ*|

^{2}

*E*(i.e.

*κ*≠ 0 in (13)). Note that although the function

*U*(

*x*) appears explicitly in the nonlinear term there is no problem with the split-step method as nonlinear terms are handled in real space. We expect that the effect of the Kerr medium will result in some soliton-like pulse shaping, and so the stable field will be some kind of combination of a gaussian and a sech.

*arg*(

*α*) = -

*π*/2. A similar result was seen in an analysis of soliton dynamics in the presence of phase modulators

^{16}, which suggests that soliton shaping has become the dominant effect in our system. Indeed, when we examine the stable solutions we find that the addition of the Kerr lens has produced a compression in both space and time. The temporal profiles are shown in figures 3 (amplitude modulation) and 4 (phase modulation). Clearly these are now sech-like, becoming more so as the gain is increased. A similar result is also found for the spatial profiles.

## 8. Conclusion

^{33. D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 694 (1970). [CrossRef] }. Stable pulses are produced for

*α*= ±

*i*|

*α*| and

*α*= -|

*α*|.

*α*= -

*i*|

*α*| and

*α*= -|

*α*|.

*π̂*

_{t}.

## Acknowledgements

## References

1. | T. Deutsch, Appl. Phys. Lett. |

2. | D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. |

3. | D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. |

4. | H. A. Haus, IEEE J. Quantum Electron. |

5. | T. Brabec, C. Spielmann, P. F. Curley, and F. Krausz, Opt. Lett. |

6. | V. Magni, G. Cerullo, and S. DeSilvestri, Opt. Commun. |

7. | A. Agnesi, IEEE J. Quantum Electron. |

8. | G. Cerullo, S. DeSilvestri, V. Magni, and L. Pallaro, Opt. Lett. |

9. | H. A. Haus and Y. Silberberg, IEEE J. Quantum Electron. |

10. | H. A. Haus, J. G. Fujimoto, and E. P. Ippen, J. Opt. Soc. Am. B |

11. | O. E. Martinez, R. L. Fork, and J. P. Gordon, J. Opt. Soc. Am. B |

12. | A. M. Dunlop, W. J. Firth, E. M. Wright, and D. R. Heatley, Opt. Lett. |

13. | P. Baues, Opto-Electron. |

14. | A. M. Dunlop, W. J. Firth, and E. M. Wright, Opt. Commun. |

15. | J. V. Roey, J. van der Donk, and P. Lagasse, J. Opt. Soc. Am. |

16. | N. J. Smith, W. J. Firth, K. J. Blow, and K. Smith, Opt. Commun. |

**OCIS Codes**

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

(190.3270) Nonlinear optics : Kerr effect

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 16, 1998

Revised Manuscript: November 3, 1997

Published: March 2, 1998

**Citation**

Alison Dunlop, William Firth, and Ewan Wright, "Pulse shapes and stability in Kerr and Active Mode-Locking (KAML)," Opt. Express **2**, 204-211 (1998)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-2-5-204

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### References

- T. Deutsch, Appl. Phys. Lett. 7, 80 (1965).<br> [CrossRef]
- D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 709 (1970).<br> [CrossRef]
- D. J. Kuizenga and A. E. Siegman, IEEE J. Quantum Electron. QE-6, 694 (1970).<br> [CrossRef]
- H. A. Haus, IEEE J. Quantum Electron. QE-11, 323 (1975).<br> [CrossRef]
- T. Brabec, C. Spielmann, P. F. Curley, and F. Krausz, Opt. Lett. 17, 1292 (1992).<br> [CrossRef] [PubMed]
- V. Magni, G. Cerullo, and S. DeSilvestri, Opt. Commun. 96, 348 (1993).<br> [CrossRef]
- A. Agnesi, IEEE J. Quantum Electron. 30, 1115 (1994).<br> [CrossRef]
- G. Cerullo, S. DeSilvestri, V. Magni, and L. Pallaro, Opt. Lett. 19, 807 (1994).<br> [CrossRef] [PubMed]
- H. A. Haus and Y. Silberberg, IEEE J. Quantum Electron. QE-22, 325 (1986).<br> [CrossRef]
- H. A. Haus, J. G. Fujimoto, and E. P. Ippen, J. Opt. Soc. Am. B 8, 2068 (1991).<br> [CrossRef]
- O. E. Martinez, R. L. Fork, and J. P. Gordon, J. Opt. Soc. Am. B 2, 753 (1985).<br> [CrossRef]
- A. M. Dunlop, W. J. Firth, E. M. Wright, and D. R. Heatley, Opt. Lett. 21, 770 (1996).<br> [CrossRef] [PubMed]
- P. Baues, Opto-Electron. 1, 37 (1969).<br> [CrossRef]
- A. M. Dunlop, W. J. Firth, and E. M. Wright, Opt. Commun. 138, 211 (1997).<br> [CrossRef]
- J. V. Roey, J. van der Donk, and P. Lagasse, J. Opt. Soc. Am. 71, 803 (1981).<br> [CrossRef]
- N. J. Smith, W. J. Firth, K. J. Blow, and K. Smith, Opt. Commun. 102, 324 (1993).<br> [CrossRef]

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