A. Komarov1,2, F. Amrani2, A. Dmitriev3, K. Komarov1, D. Meshcheriakov1,3, F. Sanchez2

1 Institute of Automation and Electrometry, Russian Academy of Sciences,Acad. Koptyug Pr. 1, 630090 Novosibirsk, Russia

2 Laboratoire de Photonique d’Angers, Université d’Angers,2 Bd Lavoisier, 49000 Angers, France

3 Novosibirsk State Technical University, K. Marx Pr. 20, 630092 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)

i

iK

ch

eEE

10

K.P. Komarov, Optics and Spectroskopy, v. 60, 231, 1986

E, , are dimensionless field amplitude, coordinate, and time, respectively; Dr, Di are gain-loss and group velocity dispersions;a is pump power, b is gain saturation parameter, I = |E|2;σ0 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

EEiqpIdb

aEiDD

Eir

2

02

2

1

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 = Di/Dr

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

-4 -2 0 2 4

-15

-10

-5

0

5

10

15

2

2

1

256

512

160320

480640

800

0

0,2

0,4

0

0,2

0,4

I(z,t)

256

512

200

6000

7000

0

0,1

0,2

0

0,1

0,2

I(z,t)

q is Kerr nonlinearity, p is nonlinearity of losses;Di is group velocity dispersion, Dr 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 Iν on frequency chirp α(ξ,θ)

gr

chch

shII

20

2

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

0 256 512

0

1,5

3

I

1

2

3

4

θ < 0 normal,θ > 0 anomalous dispersion;ξ > 0 focusing,ξ < 0 defocusing nonlinearity

021

32

Exact analytical expression for soliton spectrum Iv

K.P. Komarov, Optics and Spectroskopy, v. 60, 231, 1986

-5 -2,5 0 2,5 5

-5

-2,5

0

2,5

5

= 0

2

2

(a) (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(I0k).

256

512 z50

150250t

0

0,75

1,5

0

0,75

1,5

I(z,t)

0 1 2 3 4 5 6 7 8

a

1

2

3

4

5

6

7

8

9

10

N

I

pp

10

Komarov A.K., Komarov K.P. Phys. Rev. E, v.62, № 6, R7607, 2000

Saturating nonlinearity of losses

(a) (b)

(c)

I0k

0

I2crI1cr 1

77

Passive mode-locked fiber laser

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

Lumped saturable absorber

Pump power

88

The evolution equations for field in a laser

(1)

G is saturated gain

(2)

A. Komarov, F. Armani, A. Dmitriev, K. Komarov, D. Meshcheriakov, F. Sanchez, Phys. Rev. A, v. 85, 013802, 2012K.P. Komarov, Optics and Spectroskopy, v. 60, 231, 1986

E, , are dimensionless field amplitude, coordinate, and time, respectively; Dr , Di are gain-loss and group velocity dispersions;a is pump power, b is gain saturation parameter, I = |E|2; q is Kerr nonlinearity; σ0 is linear losses, σnl 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

EpI

E nl

1

Lumped part

,

1

1

10

2

2

2

EpI

EiqIdb

aEiDD

E nlir

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, Dr = 0.02, Di = 0.1 (anomalous dispersion), q = 1.5 (focusing nonlinearity), p = 1, σ0 = 0.01, and σnl = 0.8 (total unsaturated nonlinear losses).

Dispersive-wave soliton wings

Temporal distributions of pulse intensity I

0 1 2 3 4 5

(a.u.)

0

I (a.u.) 0

1

2

3

45

0.18

0.09

1010

Fig. 6. Spectrum of single soliton Iν with varying value of the lumped fraction of the saturable absorber η.

Sidebandes of the soliton spectrum

Spectral distributions of pulse intensity Iν

I (a.u.)

-10

0

10 (a.u.)

0

1

0

400

800

0

400

800

0.25

0.750.5

1111

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

I (a.u.)

-5

0

5 (a.u.)

300

301

0

0

300.2300.4

300.6300.8

0.2

0.1

Temporal distributions of pulse intensity I

1212

Fig. 8. Soliton I(τ) (red color) and its phase evolution φ(τ) (blue color). a = 0.5, Dr = 0.01, Di = 0.1, q = 1.5, p = 1, σ0 = 0.01, and σnl = 0.8.The additional lumped linear losses σl0 = 0.1 in Fig (b).

Powerful pedestal of ultrashort pulse

Temporal distribution of pulse intensity I and its phase change φ

(a)

(b)

-15 -10 -5 0 5 10 15

(a.u.)

-4

0

4

8

12

16

20

I (a.u.) (rad)

-15 -10 -5 0 5 10 15

(a.u.)

-4

0

4

8

12

16

20

I (a.u.) (rad)

0

0

0

0

1313

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, Dr = 0.02, Di = 0.1, q = 1.5, p = 1, σ0 = 0.01, and σnl = 0.8.

Bound steady-states of a two soliton molecule

Lumped nonlinear losses η = 1

-3 -1,5 0 1,5 3

0

6

12

I

-3 -1,5 0 1,5 3

0

6

12

I

-3 -1,5 0 1,5 3

0

4000

8000

I

-3 -1,5 0 1,5 3

0

4000

8000

I

0

-3 -1,5 0 1,5 3

0

6

12

I

-3 -1,5 0 1,5 3

0

4000

8000

I

1414

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, Dr = 0.02, Di = 0.1, q = 1.5, p = 1, σ0 = 0.01, and σnl = 0.8.(b) Breakdown of parity alternation (δτ ≈ 1-2) and occurrence of bands (δτ > 4). a = 0.5, Dr = 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

1515

Fig. 11. (a) Binding energy δJn for first six steady-states (b) with varying values of the lumped fraction of the saturable absorber η. Where Jn is the energy of two solitons in n-th bound steady-state, J∞ is the energy of two far separated noninteracting solitons, and Jp 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)

p

nn J

JJJ

0 1

-24

-12

0

Jn (%)

1

345

2

0

0.25 0.750.5

16

Fig. 12. (a) Binding energies δJn of two soliton molecule in steady-states.Jn is the energy of two solitons in bound steady states, J∞ is the energy of two far separated noninteracting solitons, and

Jp is the energy of one soliton. Lumped fraction of nonlinear losses η = 1.(b) Temporal distributions of intensity in ground state I0, first and

second excited states I1, I2.

(a) (b)

Quantization of binding energy

Two soliton molecule with δφ = 0, π

p

nn J

JJJ

-25

-20

-15

-10

-5

0 5

0

43

2

1

Jn ,% -3 -1,5 0 1,5 3

0

6

12

I0

-3 -1,5 0 1,5 3

0

6

12

I1

0

-3 -1,5 0 1,5 3

0

6

12

I2

17

Passive mode-locked fiber laser

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

Polarizing Isolator

1 23

Pump power

4

4

2

18

The evolution equations for field in a laser

E, , are dimensionless field amplitude, coordinate, and time, respectively; Dr, Di 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, = 22-1-3, p = sin(23)/3

EiqIIdb

aEiDD

Eir

12

2

nnnn EpIipIE 31311 sinsincoscos

(5)

(6)

Komarov A., Leblond H., Sanchez F. Phys. Rev. A, 71, pp. 053809, 2005K.P. Komarov, Optics and Spectroskopy, v. 60, 231, 1986Komarov A., Leblond H., Sanchez F. Phys. Rev. E, 72, pp. 025604(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(I0k).

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

0 1 2 3 4

a

0

1

2

3

4

5

6

N

ath aML

PML

CW operation

(a) (b)

I0k

0

Icr1 Icr2 Icr3

100

300

500

2040

60

0

5

0

5

I

2.5

2.5

(c)

2020

Information sequence of bound solitons

Multiple soliton molecules with δφ = 0, π

0 256 512

0

3

6

I

Komarov A., Komarov K., Sanchez F. Phys. Rev. A, v.79, 033807, 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 101010010100. The conversion of binary system into decimal one is 2708 = 1∙211 + 0∙210 + 1∙29 + 0∙28 + 1∙27 + 0∙26 + 0∙25 + 1∙24 + 0∙23 + 1∙22 + 0∙21 + 0∙20.

Information sequence of bound solitons

4.06.2012 – 1111011111101100111100VI International Conference “Solitons, Collapses and Turbulence”

2222

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