A. Komarov 1,2, F. Amrani 2, A. Dmitriev 3, K. Komarov 1, D. Meshcheriakov 1,3, F. Sanchez 2 1...
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Transcript of A. Komarov 1,2, F. Amrani 2, A. Dmitriev 3, K. Komarov 1, D. Meshcheriakov 1,3, F. Sanchez 2 1...
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