A. Komarov 1,2, F. Amrani 2, A. Dmitriev 3, K. Komarov 1, D. Meshcheriakov 1,3, F. Sanchez 2 1 Institute of Automation and Electrometry, Russian Academy of Sciences, Acad. Koptyug Pr. 1, Novosibirsk, Russia 2 Laboratoire de Photonique d’Angers, Université d’Angers, 2 Bd Lavoisier, Angers, France 3 Novosibirsk State Technical University, K. Marx Pr. 20, Novosibirsk, Russia Mechanism of dispersive-wave soliton interaction in fiber lasers

22 Outlines Various lasing models Dispersive-wave soliton wings due to lumped saturable absorber Bound steady-states of a two-soliton molecule Various intersoliton bonds with 0 and π - phase differences High-stable noise-proof bound soliton sequences Coding of an information by such sequences

3 (1) G is saturated gain (2)(2) K.P. Komarov, Optics and Spectroskopy, v. 60, 231, 1986 E, ,  are dimensionless field amplitude, coordinate, and time, respectively; D r, D i are gain-loss and group velocity dispersions; a is pump power, b is gain saturation parameter, I = |E| 2 ; σ 0 is linear losses, p is nonlinearity of losses, q is Kerr nonlinearity α is frequency chirp, β is inverse pulse duration, K is correction of wave vector The simplest model for laser passive mode-locking Steady-state pulse for the Eq. (1) is The evolution equation for field in a laser

4 Fig. 1. In the simplest model under any initial conditions, for ξ, θ in area 1 the PML (single stationary pulse) is realized. In area 2 the cw-operation (filling of total laser resonator by radiation) is established. No other laser regimes are realized. ξ = q/p θ = D i /D r Established operation depending on nonlinear-dispersion parameters ξ, θ Passive mode-locking and cw operation Komarov A.K., Komarov K.P. Opt.Com., v. 183, № 1–4, 265, 2000 q is Kerr nonlinearity, p is nonlinearity of losses; D i is group velocity dispersion, D r is gain-loss dispersion 1 2

5 Fig. 2. (a) Transformation bell-shaped spectra into rectangle ones by changing of frequency chirp: (1) α = 0, (2) α = 1, (3) α = 3, (4) α = 5. (b) Arrows point the directions of maximal increasing of frequency chirp. Dependence of soliton spectrum I ν on frequency chirp α(ξ,θ) Soliton spectra “Gain-guided solitons” with rectangle spectrum : L.M. Zhao et al. Opt.Lett., v.32, 1581, 2007 β is inverse pulse duration, ν is detuning from the centre carrying frequency θ < 0 normal, θ > 0 anomalous dispersion; ξ > 0 focusing, ξ < 0 defocusing nonlinearity Exact analytical expression for soliton spectrum I v K.P. Komarov, Optics and Spectroskopy, v. 60, 231, 1986 (a)(a)(b)(b)

6 Multiple pulse operation, multistability, hysteresis Fig. 3. (a) Transient evolution of multiple pulse operation. (b) Multistability and hysteresis dependence of number of pulses N on pump power a. (c) Soliton amplification δΛ(I 0k ). Komarov A.K., Komarov K.P. Phys. Rev. E, v.62, № 6, R7607, 2000 Saturating nonlinearity of losses (a)(a) (b)(b) (c)(c)

77 Passive mode-locked fiber laser Fig. 4. Schematic representation of the studied passive mode-locked fiber laser.

88 The evolution equations for field in a laser (1) G is saturated gain (2)(2) A. Komarov, F. Armani, A. Dmitriev, K. Komarov, D. Meshcheriakov, F. Sanchez, Phys. Rev. A, v. 85, , 2012 K.P. Komarov, Optics and Spectroskopy, v. 60, 231, 1986 E, ,  are dimensionless field amplitude, coordinate, and time, respectively; D r, D i are gain-loss and group velocity dispersions; a is pump power, b is gain saturation parameter, I = |E| 2 ; q is Kerr nonlinearity; σ 0 is linear losses, σ nl is total unsaturated nonlinear losses, p is its saturation parameter; (1- η) is fraction of distributed nonlinear losses, η is fraction of lumped nonlinear losses. Distributed part Lumped part

99 Fig. 5. Zoom of soliton wing intensity I(τ) with varying lumped fraction of the saturable absorber η: (1) η = 1, (2) η = 0.75, (3) η = 0.50, (4) η = 0.25, and (5) η = 0. In the upper inset the soliton is shown entirely. a = 0.55, D r = 0.02, D i = 0.1 (anomalous dispersion), q = 1.5 (focusing nonlinearity), p = 1, σ 0 = 0.01, and σ nl = 0.8 (total unsaturated nonlinear losses). Dispersive-wave soliton wings Temporal distributions of pulse intensity I

10 Fig. 6. Spectrum of single soliton I ν with varying value of the lumped fraction of the saturable absorber η. Sidebandes of the soliton spectrum Spectral distributions of pulse intensity I ν

11 Fig. 7. The periodic change in a soliton pedestal during one pass (δζ = 1) through the laser resonator with the lumped nonlinear losses, η = 1. Dynamics of formation of dispersive-waves soliton wings Temporal distributions of pulse intensity I

12 Fig. 8. Soliton I(τ) (red color) and its phase evolution φ(τ) (blue color). a = 0.5, D r = 0.01, D i = 0.1, q = 1.5, p = 1, σ 0 = 0.01, and σ nl = 0.8. The additional lumped linear losses σ l0 = 0.1 in Fig (b). Powerful pedestal of ultrashort pulse Temporal distribution of pulse intensity I and its phase change φ (a) (b)

13 Fig. 9. Two bound steady-state solitons with different separations (red color) and their spectra (blue color). Alternation of the intersoliton phase differences: δφ = π and 0. a = 0.55, D r = 0.02, D i = 0.1, q = 1.5, p = 1, σ 0 = 0.01, and σ nl = 0.8. Bound steady-states of a two soliton molecule Lumped nonlinear losses η = 1

14 Fig. 10. Two sets of bound steady-state solitons with different separations. The green circle corresponds to the first soliton, the red squares and blue circles relate to the second soliton in the pair with the intersoliton phase change δφ = π and 0, respectively. (a) The rigorous parity alternation of 0 and π states. a = 0.55, D r = 0.02, D i = 0.1, q = 1.5, p = 1, σ 0 = 0.01, and σ nl = 0.8. (b) Breakdown of parity alternation (δτ ≈ 1-2) and occurrence of bands (δτ > 4). a = 0.5, D r = 0.01, the other parameters are the same as in Fig 7(a). Different sets of bound steady-states of a two soliton molecule (a) (b) Lumped nonlinear losses η = 1

15 Fig. 11. (a) Binding energy δJ n for first six steady-states (b) with varying values of the lumped fraction of the saturable absorber η. Where J n is the energy of two solitons in n-th bound steady-state, J ∞ is the energy of two far separated noninteracting solitons, and J p is the energy of one soliton. Without lumped losses, neither strong interaction between solitons nor set of bound states become possible. Binding energy Two soliton molecule (a) (b)

16 Fig. 12. (a) Binding energies δJ n of two soliton molecule in steady- states. J n is the energy of two solitons in bound steady states, J ∞ is the energy of two far separated noninteracting solitons, and J p is the energy of one soliton. Lumped fraction of nonlinear losses η = 1. (b) Temporal distributions of intensity in ground state I 0, first and second excited states I 1, I 2. (a)(b) Quantization of binding energy Two soliton molecule with δφ = 0, π

17 Passive mode-locked fiber laser Fig. 13. Schematic representation of fiber ring laser passively mode locked through nonlinear polarization rotation technique.

18 The evolution equations for field in a laser E, ,  are dimensionless field amplitude, coordinate, and time, respectively; D r, D i are the gain-loss and the group velocity dispersions; a is the pump power, b is the saturation parameter, q is Kerr nonlinearity. η is the transmission coefficient of the polarizer, I = |E| 2 ;  i are orientation angles of phase plates,  = 2  2 -  1 -  3, p = sin(2  3 )/3 (5) (6) Komarov A., Leblond H., Sanchez F. Phys. Rev. A, 71, pp , 2005 K.P. Komarov, Optics and Spectroskopy, v. 60, 231, 1986 Komarov A., Leblond H., Sanchez F. Phys. Rev. E, 72, pp (R), 2005 G is saturated gain Distributed part Lumped part

19 Fig. 14. (a) CW operation. Threshold self-start of PML. Multiple pulse operation, multistability and multihysteresis. (b) Transient evolution of multiple pulse operation. (c) Soliton amplification δΛ(I 0k ). Lasing regimes of passive mode-locked fiber lasers Haboucha A., Komarov A., Leblond H., Salhi M., Sanchez F. Jour. Optoelect. Adv. Mat., v. 10, 164, 2008 (a)(a)(b)(b) (c)(c)

20 Information sequence of bound solitons Multiple soliton molecules with δφ = 0, π Komarov A., Komarov K., Sanchez F. Phys. Rev. A, v.79, , 2009 Fig. 15. Stable sequence of bound solitons with ground (0) and first excited (1) states, in wich the number 2708 is coded in binary system The conversion of binary system into decimal one is 2708 = 1∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙2 0.

Information sequence of bound solitons – VI International Conference “Solitons, Collapses and Turbulence”

22 By numerical simulation and analytical treatment we have found: Powerful dispersive-wave soliton wings due to lumped saturable absorber Bound steady-states of a two soliton molecule with high binding energy Various intersoliton bonds with 0 and π - phase differences High-stable noise-proof bound soliton sequences Possibility of coding of an information by such sequences CONCLUSIONS