Abstract

A simple analytical model of the mechanism responsible for the formation of bound states of pulses in the stretched-pulse fiber laser is given. The proposed model is based on a noncoherent interaction occurring between the pulses near their position of maximum stretch within the dispersion-managed cavity, where the pulses possess a large linear chirp. This nonlinear interaction is due to the combined effects of the cross-phase modulation and the cross-amplitude modulation caused by the nonlinear gain associated with the mode-locking mechanism used in the laser. This model predicts the existence of a single bound state with a separation of the order of the pulsewidth at maximum stretch, a result consistent with simulations and experiments.

© 2009 Optical Society of America

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  1. N. Akhmediev, J. M. Soto-Crespo, M. Grapinet, and Ph. Grelu, "Dissipative soliton interactions inside a fiber laser cavity," Opt. Fiber Technol. 11, 209-228 (2005).
    [CrossRef]
  2. D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, "Soliton interaction in a fiber ring laser," Phys. Rev. E 72, 1-10 (2005).
    [CrossRef]
  3. M. Stratmann, T. Pagel, and F. Mitschke, "Experimental Observation of Temporal Soliton Molecules," Phys. Rev. Lett.,  95, 143902 (2005).
    [CrossRef] [PubMed]
  4. Ph. Grelu, J. Béal, and J. M. Soto-Crespo, "Soliton pairs in a fiber laser: from anomalous to normal average dispersion regime," Opt. Express 11,2238-2243 (2003).
    [CrossRef] [PubMed]
  5. Ph. Grelu, and N. Akhmediev, "Group interactions of dissipative solitons in a laser cavity: the case 2+1," Opt. Express 12, 3184-3189 (2004).
    [CrossRef] [PubMed]
  6. G. Martel, C. Chédot, V. Réglier, A. Hideur, B. Ortaç, and Ph. Grelu, "On the possibility of observing bound soliton pairs in a wave-breaking-free mode-locked fiber laser," Opt. Lett. 32, 343-345 (2007).
    [CrossRef] [PubMed]
  7. B. Ortaç, A. Hideur, M. Brunel, C. Chédot, J. Limpert, A. Tünnermann, and F. Ilday, "Generation of parabolic bound pulses from a Yb-fiber laser," Opt. Express 14, 6075-6083 (2006).
    [CrossRef] [PubMed]
  8. M. Olivier, V. Roy, M. Piché, and F. Babin, "Pulse collisions in the stretched-pulse fiber laser," Opt. Lett. 29, 1461-1463 (2004).
    [CrossRef] [PubMed]
  9. M. Olivier, V. Roy, and M. Piché, "Influence of the Raman effect on bound states of dissipative solitons," Opt. Express 14, 9728-9742 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  11. M. Hofer, M. E. Fermann, F. Haberl, and M. H. Ober, "Characterization of ultrashort pulse formation in passively mode-locked fiber lasers," IEEE J. Quantum Electron. 28, 720-728 (1992).
    [CrossRef]
  12. A. E. Siegman, Lasers (University Science Books, 1986), Chap. 27.
  13. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995), Chap. 7.
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    [CrossRef]
  15. M. Olivier, V. Roy, and M. Piché, "Third-order dispersion and bound states of pulses in a fiber laser," Opt. Lett. 31, 580-582 (2006).
    [CrossRef] [PubMed]
  16. H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, "Stretched-pulse additive pulse mode-locking in fiber lasers: theory and experiment," IEEE J. Quantum. Electron. 31, 591-598 (1995).
    [CrossRef]
  17. A. Komarov, H. Leblond, and F. Sanchez, "Quintic complex Ginzburg-Landau model for ring fiber lasers," Phys. Rev. E 72, 1-4 (2005).
    [CrossRef]
  18. F. M. Mitschke, and L. F. Mollenauer, "Experimental observation of interaction forces between solitons in optical fibers," Opt. Lett. 12, 355-357 (1987).
    [CrossRef] [PubMed]
  19. J. N. Kutz, B. C. Collings, and W. H. Knox, "Stabilized pulse spacing in soliton lasers due to gain depletion and recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
    [CrossRef]

2007

2006

2005

N. Akhmediev, J. M. Soto-Crespo, M. Grapinet, and Ph. Grelu, "Dissipative soliton interactions inside a fiber laser cavity," Opt. Fiber Technol. 11, 209-228 (2005).
[CrossRef]

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, "Soliton interaction in a fiber ring laser," Phys. Rev. E 72, 1-10 (2005).
[CrossRef]

M. Stratmann, T. Pagel, and F. Mitschke, "Experimental Observation of Temporal Soliton Molecules," Phys. Rev. Lett.,  95, 143902 (2005).
[CrossRef] [PubMed]

A. Komarov, H. Leblond, and F. Sanchez, "Quintic complex Ginzburg-Landau model for ring fiber lasers," Phys. Rev. E 72, 1-4 (2005).
[CrossRef]

2004

2003

1998

J. N. Kutz, B. C. Collings, and W. H. Knox, "Stabilized pulse spacing in soliton lasers due to gain depletion and recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

1995

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

1993

1992

M. Hofer, M. E. Fermann, F. Haberl, and M. H. Ober, "Characterization of ultrashort pulse formation in passively mode-locked fiber lasers," IEEE J. Quantum Electron. 28, 720-728 (1992).
[CrossRef]

1987

Akhmediev, N.

N. Akhmediev, J. M. Soto-Crespo, M. Grapinet, and Ph. Grelu, "Dissipative soliton interactions inside a fiber laser cavity," Opt. Fiber Technol. 11, 209-228 (2005).
[CrossRef]

Ph. Grelu, and N. Akhmediev, "Group interactions of dissipative solitons in a laser cavity: the case 2+1," Opt. Express 12, 3184-3189 (2004).
[CrossRef] [PubMed]

Babin, F.

Béal, J.

Brunel, M.

Chédot, C.

Collings, B. C.

J. N. Kutz, B. C. Collings, and W. H. Knox, "Stabilized pulse spacing in soliton lasers due to gain depletion and recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

Fermann, M. E.

M. Hofer, M. E. Fermann, F. Haberl, and M. H. Ober, "Characterization of ultrashort pulse formation in passively mode-locked fiber lasers," IEEE J. Quantum Electron. 28, 720-728 (1992).
[CrossRef]

Grapinet, M.

N. Akhmediev, J. M. Soto-Crespo, M. Grapinet, and Ph. Grelu, "Dissipative soliton interactions inside a fiber laser cavity," Opt. Fiber Technol. 11, 209-228 (2005).
[CrossRef]

Grelu, Ph.

Haberl, F.

M. Hofer, M. E. Fermann, F. Haberl, and M. H. Ober, "Characterization of ultrashort pulse formation in passively mode-locked fiber lasers," IEEE J. Quantum Electron. 28, 720-728 (1992).
[CrossRef]

Haus, H. A.

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

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

Hideur, A.

Hofer, M.

M. Hofer, M. E. Fermann, F. Haberl, and M. H. Ober, "Characterization of ultrashort pulse formation in passively mode-locked fiber lasers," IEEE J. Quantum Electron. 28, 720-728 (1992).
[CrossRef]

Ilday, F.

Ippen, E. P.

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

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

Knox, W. H.

J. N. Kutz, B. C. Collings, and W. H. Knox, "Stabilized pulse spacing in soliton lasers due to gain depletion and recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

Komarov, A.

A. Komarov, H. Leblond, and F. Sanchez, "Quintic complex Ginzburg-Landau model for ring fiber lasers," Phys. Rev. E 72, 1-4 (2005).
[CrossRef]

Kutz, J. N.

J. N. Kutz, B. C. Collings, and W. H. Knox, "Stabilized pulse spacing in soliton lasers due to gain depletion and recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

Leblond, H.

A. Komarov, H. Leblond, and F. Sanchez, "Quintic complex Ginzburg-Landau model for ring fiber lasers," Phys. Rev. E 72, 1-4 (2005).
[CrossRef]

Limpert, J.

Lushnikov, P. M.

Martel, G.

Mitschke, F.

M. Stratmann, T. Pagel, and F. Mitschke, "Experimental Observation of Temporal Soliton Molecules," Phys. Rev. Lett.,  95, 143902 (2005).
[CrossRef] [PubMed]

Mitschke, F. M.

Mollenauer, L. F.

Nelson, L. E.

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

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

Ober, M. H.

M. Hofer, M. E. Fermann, F. Haberl, and M. H. Ober, "Characterization of ultrashort pulse formation in passively mode-locked fiber lasers," IEEE J. Quantum Electron. 28, 720-728 (1992).
[CrossRef]

Olivier, M.

Ortaç, B.

Pagel, T.

M. Stratmann, T. Pagel, and F. Mitschke, "Experimental Observation of Temporal Soliton Molecules," Phys. Rev. Lett.,  95, 143902 (2005).
[CrossRef] [PubMed]

Piché, M.

Réglier, V.

Roy, V.

Sanchez, F.

A. Komarov, H. Leblond, and F. Sanchez, "Quintic complex Ginzburg-Landau model for ring fiber lasers," Phys. Rev. E 72, 1-4 (2005).
[CrossRef]

Soto-Crespo, J. M.

N. Akhmediev, J. M. Soto-Crespo, M. Grapinet, and Ph. Grelu, "Dissipative soliton interactions inside a fiber laser cavity," Opt. Fiber Technol. 11, 209-228 (2005).
[CrossRef]

Ph. Grelu, J. Béal, and J. M. Soto-Crespo, "Soliton pairs in a fiber laser: from anomalous to normal average dispersion regime," Opt. Express 11,2238-2243 (2003).
[CrossRef] [PubMed]

Stratmann, M.

M. Stratmann, T. Pagel, and F. Mitschke, "Experimental Observation of Temporal Soliton Molecules," Phys. Rev. Lett.,  95, 143902 (2005).
[CrossRef] [PubMed]

Tam, H. Y.

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, "Soliton interaction in a fiber ring laser," Phys. Rev. E 72, 1-10 (2005).
[CrossRef]

Tamura, K.

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

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

Tang, D. Y.

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, "Soliton interaction in a fiber ring laser," Phys. Rev. E 72, 1-10 (2005).
[CrossRef]

Tünnermann, A.

Zhao, B.

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, "Soliton interaction in a fiber ring laser," Phys. Rev. E 72, 1-10 (2005).
[CrossRef]

Zhao, L. M.

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, "Soliton interaction in a fiber ring laser," Phys. Rev. E 72, 1-10 (2005).
[CrossRef]

IEEE J. Quantum Electron.

M. Hofer, M. E. Fermann, F. Haberl, and M. H. Ober, "Characterization of ultrashort pulse formation in passively mode-locked fiber lasers," IEEE J. Quantum Electron. 28, 720-728 (1992).
[CrossRef]

J. N. Kutz, B. C. Collings, and W. H. Knox, "Stabilized pulse spacing in soliton lasers due to gain depletion and recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

IEEE J. Quantum. Electron.

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

J. Opt. Soc. Am. B

Opt. Express

Opt. Fiber Technol.

N. Akhmediev, J. M. Soto-Crespo, M. Grapinet, and Ph. Grelu, "Dissipative soliton interactions inside a fiber laser cavity," Opt. Fiber Technol. 11, 209-228 (2005).
[CrossRef]

Opt. Lett.

Phys. Rev. E

A. Komarov, H. Leblond, and F. Sanchez, "Quintic complex Ginzburg-Landau model for ring fiber lasers," Phys. Rev. E 72, 1-4 (2005).
[CrossRef]

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, "Soliton interaction in a fiber ring laser," Phys. Rev. E 72, 1-10 (2005).
[CrossRef]

Phys. Rev. Lett.

M. Stratmann, T. Pagel, and F. Mitschke, "Experimental Observation of Temporal Soliton Molecules," Phys. Rev. Lett.,  95, 143902 (2005).
[CrossRef] [PubMed]

Other

A. E. Siegman, Lasers (University Science Books, 1986), Chap. 27.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995), Chap. 7.

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Figures (8)

Fig. 1.
Fig. 1.

The laser cavity considered in our model. The PAPM system is made up of two waveplates and a polarizer.

Fig. 2.
Fig. 2.

Pulse power profile and pulse spectrum in the single-pulse regime.

Fig. 3.
Fig. 3.

On the left, the pulse profile at the locations where its duration is maximum and minimum within the laser cavity. On the right, the pulse profile and instantaneous frequency at its point of maximum duration, i.e. at the output of the erbium-doped fiber.

Fig. 4.
Fig. 4.

Evolution of the separation and phase difference between two pulses for different initial conditions. The evolution from given initial conditions (separation and phase difference) is identified by the same color on top and bottom.

Fig. 5.
Fig. 5.

Intracavity dynamics of the bound state and pulse profile at the positions of maximum compression and maximum stretch within the laser cavity.

Fig. 6.
Fig. 6.

Mechanism of the non-coherent XPM interaction.

Fig. 7.
Fig. 7.

Mechanism of the interaction due to the nonlinear gain. The asymmetric nonlinear gain resulting from the presence of a second pulse combined with the linear chirp of the pulse during the interaction leads to a modification of its average frequency and thus its speed.

Fig. 8.
Fig. 8.

Rate of change of the frequency difference between the two pulses as a function of their separation according to the analytical model. Due to the normal average GVD of the laser cavity, it leads to a repulsion for small separation and an attraction for large separation with a stable equilibrium point near 1.9 ps which corresponds to a single bound state.

Tables (1)

Tables Icon

Table 1. Parameters of the fibers used in the laser cavity considered above.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

Ert=12 [x̂FxyAxzteiβ0ziω0t+ŷFxyAyzteiβ0ziω0t] ,
A±z=gsat2A±i2β22A±t2+iγ3(A±2+2A±2)A±.
Az=iγA2A+δA2AvA4A ,
δAjz=iγAk2AjXPMinteraction+δAk2Ajv(Ak4+2Aj2Ak2)AjNonlineargaininterction .
ω̄j=12Ej (Aj*AjtAjAj*t) d t ,
d(Δω̄)dz=i2Ej dt { 4 i γ A12 t (A22)
+ (2δA224vA12A22)(A1*A1tA1A1*t)
(2δA124vA12A22) (A2*A2tA2A2*t) } .
A1,2(t)=P0 exp [(1+iC)2(t±T/2)2T02] ,
d(Δω̄)dz=P0TT02 exp [T22T02] {2(δCγ)893vCP0exp[T26T02]} .
Tboundstate=6ln[9283(δCγ)vCP0] T0 .

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