### Abstract

Self-similarity is a ubiquitous concept in the physical sciences used to explain a wide range of spatial- or temporalstructures observed in a broad range of applications and natural phenomena. Indeed, they have been predicted or observed in the context of Raman scattering, spatial soliton fractals, propagation in the normal dispersion regime with strong nonlinearity, optical amplifiers, and mode-locked lasers. These self-similar structures are typically long-time transients formed by the interplay, often nonlinear, of the underlying dominant physical effects in the system. A theoretical model shows that in the context of the universal Ginzburg-Landau equation with rapidly-varying, mean-zero dispersion, stable and attracting self-similar pulses are formed with parabolic profiles: the zero-dispersion similariton. The zero-dispersion similariton is the final solution state of the system, not a long-time, intermediate asymptotic behavior. An averaging analysis shows the self-similarity to be governed by a nonlinear diffusion equation with a rapidly-varying, mean-zero diffusion coefficient. Indeed, the leadingorder behavior is shown to be governed by the porous media (nonlinear diffusion) equation whose solution is the well-known Barenblatt similarity solution which has a parabolic, self-similar profile. The alternating sign of the diffusion coefficient, which is driven by the dispersion fluctuations, is critical to supporting the zero-dispersion similariton which is, to leading-order, of the Barenblatt form. This is the first analytic model proposing a mechanism for generating physically realizable temporal parabolic pulses in the Ginzburg-Landau model. Although the results are of restricted analytic validity, the findings are suggestive of the underlying physical mechanism responsible for parabolic (self-similar) pulse formation in lightwave transmission and observed in mode-locked laser cavities.

Original language | English |
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Title of host publication | Proceedings of SPIE - The International Society for Optical Engineering |

Subtitle of host publication | Photonics North 2009 |

Volume | 7386 |

DOIs | |

Publication status | Published - 20 Nov 2009 |

Event | Photonics North 2009 - Laval, QC, United Kingdom Duration: 24 May 2009 → 27 May 2009 |

### Conference

Conference | Photonics North 2009 |
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Country | United Kingdom |

City | Laval, QC |

Period | 24/05/09 → 27/05/09 |

### Fingerprint

### Bibliographical note

Copyright 2009 SPIE. One print or electronic copy may be made for personal use only. Systematic reproduction, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited.### Keywords

- Dispersion-management
- Mode-locked lasers
- Self-similarity
- Similaritons

### Cite this

*Proceedings of SPIE - The International Society for Optical Engineering: Photonics North 2009*(Vol. 7386). [73860M] https://doi.org/10.1117/12.837774

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*Proceedings of SPIE - The International Society for Optical Engineering: Photonics North 2009.*vol. 7386, 73860M, Photonics North 2009, Laval, QC, United Kingdom, 24/05/09. https://doi.org/10.1117/12.837774

**Analytic theory of self-similar mode-locking with rapidly varying, mean-zero dispersion.** / Bale, Brandon G.; Kutz, J. Nathan.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Analytic theory of self-similar mode-locking with rapidly varying, mean-zero dispersion

AU - Bale, Brandon G.

AU - Kutz, J. Nathan

N1 - Copyright 2009 SPIE. One print or electronic copy may be made for personal use only. Systematic reproduction, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited.

PY - 2009/11/20

Y1 - 2009/11/20

N2 - Self-similarity is a ubiquitous concept in the physical sciences used to explain a wide range of spatial- or temporalstructures observed in a broad range of applications and natural phenomena. Indeed, they have been predicted or observed in the context of Raman scattering, spatial soliton fractals, propagation in the normal dispersion regime with strong nonlinearity, optical amplifiers, and mode-locked lasers. These self-similar structures are typically long-time transients formed by the interplay, often nonlinear, of the underlying dominant physical effects in the system. A theoretical model shows that in the context of the universal Ginzburg-Landau equation with rapidly-varying, mean-zero dispersion, stable and attracting self-similar pulses are formed with parabolic profiles: the zero-dispersion similariton. The zero-dispersion similariton is the final solution state of the system, not a long-time, intermediate asymptotic behavior. An averaging analysis shows the self-similarity to be governed by a nonlinear diffusion equation with a rapidly-varying, mean-zero diffusion coefficient. Indeed, the leadingorder behavior is shown to be governed by the porous media (nonlinear diffusion) equation whose solution is the well-known Barenblatt similarity solution which has a parabolic, self-similar profile. The alternating sign of the diffusion coefficient, which is driven by the dispersion fluctuations, is critical to supporting the zero-dispersion similariton which is, to leading-order, of the Barenblatt form. This is the first analytic model proposing a mechanism for generating physically realizable temporal parabolic pulses in the Ginzburg-Landau model. Although the results are of restricted analytic validity, the findings are suggestive of the underlying physical mechanism responsible for parabolic (self-similar) pulse formation in lightwave transmission and observed in mode-locked laser cavities.

AB - Self-similarity is a ubiquitous concept in the physical sciences used to explain a wide range of spatial- or temporalstructures observed in a broad range of applications and natural phenomena. Indeed, they have been predicted or observed in the context of Raman scattering, spatial soliton fractals, propagation in the normal dispersion regime with strong nonlinearity, optical amplifiers, and mode-locked lasers. These self-similar structures are typically long-time transients formed by the interplay, often nonlinear, of the underlying dominant physical effects in the system. A theoretical model shows that in the context of the universal Ginzburg-Landau equation with rapidly-varying, mean-zero dispersion, stable and attracting self-similar pulses are formed with parabolic profiles: the zero-dispersion similariton. The zero-dispersion similariton is the final solution state of the system, not a long-time, intermediate asymptotic behavior. An averaging analysis shows the self-similarity to be governed by a nonlinear diffusion equation with a rapidly-varying, mean-zero diffusion coefficient. Indeed, the leadingorder behavior is shown to be governed by the porous media (nonlinear diffusion) equation whose solution is the well-known Barenblatt similarity solution which has a parabolic, self-similar profile. The alternating sign of the diffusion coefficient, which is driven by the dispersion fluctuations, is critical to supporting the zero-dispersion similariton which is, to leading-order, of the Barenblatt form. This is the first analytic model proposing a mechanism for generating physically realizable temporal parabolic pulses in the Ginzburg-Landau model. Although the results are of restricted analytic validity, the findings are suggestive of the underlying physical mechanism responsible for parabolic (self-similar) pulse formation in lightwave transmission and observed in mode-locked laser cavities.

KW - Dispersion-management

KW - Mode-locked lasers

KW - Self-similarity

KW - Similaritons

UR - http://www.scopus.com/inward/record.url?scp=70449589692&partnerID=8YFLogxK

U2 - 10.1117/12.837774

DO - 10.1117/12.837774

M3 - Conference contribution

AN - SCOPUS:70449589692

SN - 9780819476685

VL - 7386

BT - Proceedings of SPIE - The International Society for Optical Engineering

ER -