dissipative solitons: the structural chaos and the...
TRANSCRIPT
Dissipative Solitons:
The Structural Chaos And
The Chaos Of Destruction
V.L. Kalashnikov, E. Sorokin
Institut für Photonik, TU Wien, Gusshausstr.
27/387,
A-1040 Vienna, Austria
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
Outlook
Dissipative solitons
Concept of the chirped dissipative soliton
Generalized complex nonlinear Ginzburg-Landau equation and
variational approach to the soliton analysis
Resonant excitation of vacuum
Vacuum as a temporal continuum and a soliton-independent
sector
Vacuum as a resonant mode of soliton
Chaotization of dynamics due to interaction with vacuum
Soliton spleeting
Soliton pulsations
Soliton dying and revival
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
Nonlinear self compression
050
100150
200
Distance2
1
0
1
2
Time
00.050.1
0.150.2
050
100150
200
Distance
Dispersion spreading
020
4060
80100
Distance 5
0
5
Time
00.0250.05
0.075
020
4060
80100
Distance
Propagation in a dispersive nonlinear medium
Anomalous dispersion
pulse profile frequency
deviation
faster
components
slower
components
expansion
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
Self-phase modulation
pulse profile frequency
deviation
faster
components
slower
components contraction
+
Classical (“Schrödinger”) soliton is a result of the phase balance
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
First order soliton
050
100150
200
Distance 4
2
0
2
4
Time
00.0250.05
0.0750.1
050
100150
200
Distance -4
-2
0
2
4-5
-2.5
0
2.5
5
0
0.5
1
1.5
-4
-2
0
2
4t
w
Wigner function
-4
-2
0
2
4-5
-2.5
0
2.5
5
0
0.5
1
1.5
-4
-2
0
2
4
t
w
Chirped dissipative soliton (squeezed soliton)
Pulse is chirped (squeezed) due to
self-phase modulation
and normal dispersion
t -4 -2 0 2 4
0
0.5
1
1.5
2
2.5
Pulse shortening due to
spectral cutoff
-4
-2
0
2
4
-2
0
2
0
0.5
1
1.5
-4
-2
0
2
4
w
t cutoff
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
Action of linear and
nonlinear gain
VS.
linear and nonlinear
loss
H.A.Haus et al., J. Opt. Soc. Am. B 8, 2068 (1991)
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
Variational approach to the soliton theory I Schrödinger soliton
Lagrangian for the non-dissipative factors [ 𝛾 is the self-phase modulation coefficient , 𝛽 is the net-
group-delay dispersion coefficient; 𝐴(𝑧, 𝑡) is the slowly-varying field envelope (|𝐴|2 is the power), 𝑡 is the local time, 𝑧 is the cavity round-trip number (for a distributed model)]:
𝕷 =𝟏
𝟐𝒊 𝑨∗
𝝏𝑨
𝝏𝒕− 𝑨
𝝏𝑨∗
𝝏𝒕− 𝜷
𝝏𝑨
𝝏𝒕
𝝏𝑨∗
𝝏𝒕+ 𝜸 𝑨 𝟒 .
Equations of motion:
𝜹 𝕷𝒅𝒕′∞−∞
𝜹𝐟−𝒅
𝒅𝒛
𝜹 𝕷𝒅𝒕′∞−∞
𝜹𝐟= 𝟎.
Result is the nonlinear Schrödinger equations:
𝜸 𝑨 𝒛, 𝒕 𝟐𝑨∗ 𝒛, 𝒕 +𝜷
𝟐
𝝏𝟐
𝝏𝒕𝟐𝑨∗ 𝒛, 𝒕 + 𝒊
𝝏
𝝏𝒛𝑨∗ 𝒛, 𝒕 = 𝟎; 𝜸 𝑨 𝒛, 𝒕 𝟐𝑨 𝒛, 𝒕 +
𝜷
𝟐
𝝏𝟐
𝝏𝒕𝟐𝑨 𝒛, 𝒕 − 𝒊
𝝏
𝝏𝒛𝑨 𝒛, 𝒕 = 𝟎.
This trial function gives the exact soliton solution:
𝑨 𝒛, 𝒕 = 𝑨𝟎 𝒛 𝐞𝐱𝐩 𝒊𝝓 𝒛 𝐬𝐞𝐜𝐡𝒕
𝑻 𝒛, 𝐟 ≡ 𝑨𝟎, 𝑻, 𝝓 .
D. Anderson et al., Pramana J. Phys. 57, 917–936 (2001)
Variational approach to the soliton theory II
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
Dissipative soliton
Driving forces for the dissipative factors [ for a laser: 𝜎 is the net-loss, 𝜌 is the small-
signal gain, 𝜗 is the inverse gain saturation energy, 𝛼 is the squared inverse spectral filter
bandwidth, 𝜇 is the self-amplitude modulation strength, 𝜁 is the saturation power of a self-
amplitude modulation]
𝑸 = 𝒊 −𝝈𝑨 +𝒈𝟎
𝟏 + 𝝑 𝑨 𝟐𝒅𝒕′∞
−∞
𝑨 + 𝜶𝝏𝟐𝑨
𝝏𝒕𝟐+ 𝝁 𝑨 𝟐 𝟏 − 𝜻 𝑨 𝟐 𝑨 + 𝒆𝒕𝒄.
appear in the Euler-Lagrange equations:
𝜹 𝕷𝒅𝒕′∞
−∞
𝜹𝐟−𝒅
𝒅𝒛
𝜹 𝕷𝒅𝒕′∞
−∞
𝜹𝐟= 𝟐𝕽 𝑸
𝝏𝑨
𝝏𝐟
∞
−∞
.
The underlying equation is the famous complex nonlinear Ginzburg-Landau equation:
𝜸 𝑨 𝒛, 𝒕 𝟐𝑨 𝒛, 𝒕 +𝜷
𝟐
𝝏𝟐
𝝏𝒕𝟐𝑨 𝒛, 𝒕 − 𝒊
𝝏
𝝏𝒛𝑨 𝒛, 𝒕 = 𝑸
with sole known exact soliton-like solution:
𝑨 𝒛, 𝒕 =𝑨𝟎(𝒙)
𝜽 𝒙 + cosh(𝒕𝑻(𝒙)
)
𝒆𝒊 𝝓 𝒙 +𝝍(𝒙) ln(𝜽 𝒙 +cosh(
𝒕𝑻(𝒙)
))
B.G.Bale et al., JOSA B 25, 1763 (2008).
Solitonic metamorphosis vs. chaos
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
Extension of the Schrödinger solitonic sector into the dissipative solitonic one has a
simplest representation:
𝑨 𝒛, 𝒕 = 𝑨𝟎 𝒛 𝐞𝐱𝐩 𝒊𝝓 𝒛 𝐬𝐞𝐜𝐡𝒕
𝑻 𝒛
𝟏+𝒊𝝍(𝒛),
where 𝜓 is the squeezing parameter (or “chirp”). This case correspond to 𝜃 = 1 in the
solution of the Ginzburg-Landau equation.
Its stability against continuum excitation is defined by following diagram (𝐸 is the
soliton energy).
soliton is stable
soliton is unstable
log10 𝛼𝛾 𝛽𝜇
log10𝐸𝛾𝜇𝜁𝛼 /𝜇
𝛾
𝜇> 5
𝛾
𝜇=2
asymptotic
V.L.Kalashnikov, A.Apolonski, Optics Express 18, 25757 (2010); V.L.Kalashnikov et al. APB 83, 503 (2006).
But the chaos does not appear in the
vicinity of shown stability border
because new solitonic branch with
𝜃 ≠ 1 develops.
-0,10 -0,05 0,00 0,05 0,100
2
4
spec
tral
inte
nsi
ty, ar
b. u.
w, fs-1
Resonant excitation of the vacuum
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
The simplest modification of the Lagrangian due higher-order derivative term
(physically, higher-order dispersion):
𝕷 = 𝕷𝟎 +𝒊𝜹
𝟐
𝝏𝟐𝑨
𝝏𝒕𝟐𝝏𝑨∗
𝝏𝒕.
𝛛𝑨
𝝏𝒛= −𝝈𝑨 + 𝜶 − 𝒊𝜷
𝝏𝟐𝑨
𝛛𝒕𝟐+ 𝜹
𝝏𝟑𝑨
𝝏𝒕𝟑+ 𝝁 + 𝒊𝜸 𝑨 𝟐𝑨 − 𝝁𝜻 𝑨 𝟒𝑨.
That corresponds to the generalized complex nonlinear Ginzburg-Landau equation:
Now, the resonant interaction of the soliton with the vacuum is possible, i.e. dispersive
wave generation: the resonance condition for the dispersive wavenumber 𝑘(𝜔) and the
soliton wavenumber 𝑞 is 𝒌(𝝎) ≡ 𝜷𝝎𝟐 + 𝜹𝝎𝟑 = 𝒒.
If the corresponding resonant frequency 𝜔𝑟 shifts inside the soliton spectrum (i.e.
𝜔𝑟 ≤ Δ ≈ 𝜁𝐴02 𝛽 for the chirped dissipative soliton developing in the normal
dispersion regime, Δ is the soliton spectrum halfwidth). Because 𝑞 is small, the condition
becomes |𝜔𝑟| ≈ | 𝛽 𝛿 | ≤ Δ, i.e. that zero-dispersion wavelength reaches the spectrum edge
at Δ.
Dispersion wave excitation
V.L.Kalashnikov et al., Optics Express 16, 4206 (2008).
Resonance frequency and
strength of soliton-continuum binding
Resonance frequency 𝜔𝑟 (in femtoseconds-1)
shifts inside the soliton spectrum
Binding strength 𝛿𝜔𝑟3
enhances
𝛿, femtoseconds3
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
𝛿, femtoseconds3
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
Transformation of the soliton spectrum
4
spec
tral
pow
er
-40 -30 -20 -10 0 10 20
-1000
-500
0
500
1000
2
w, THz
1
2
13
4
w
(femto
secon
ds
2)
3
3900 4000 4100 4200 4300
KLM Cr:ZnSe
130-150 mW, 91 MHz
net-g
rou
p d
elay d
ispersio
n (fs
2)
Spec
tral
inte
nsi
ty
Frequency (cm-1
)
-1000
0
1000
2000
2550 2500 2450 2400 2350 2300 nm
Increasing third-order dispersion 𝛿 transforms
initially rectangular spectrum (1) to trapezoid (2)
and then triangular (3). Simultaneously, an
intensive dispersive component appears in the
region of anomalous dispersion. The main
spectrum acquires strong modulation (4).
Finally, the spectrum would become completely
fragmented.
-0.1 -0.05 0 0.050
2
4
6
8
x 10-7
w, fs-1
sp
ectr
al
po
wer,
arb
. u
n.
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
Chaos production With increasing 𝜹, the soliton develops strong perturbations. The resonance frequency shifts
towards the soliton spectrum. The same effect can be achieved by increasing the pulse power 𝐴02, thus
expanding the spectral width 2Δ. As a result, the chaos develops through the central frequency jitter.
6,0 6,1 6,2 6,3 6,4 6,5
SH
G i
nte
nsi
ty C
entr
al w
avel
ength
Time (ms)
0 2 4 6 8 10
Round-trips (x104)
6,20 6,21 6,22 6,23 6,24 6,25
SH
G i
nte
nsi
ty C
entr
al w
avel
ength
Time (ms)
0 2000 4000 6000 8000 10000
Round-trips
-4 -2 0 2 40
2
4
6
8
Chaotic
regime
SH
G i
nte
nsi
ty (
rel.
u.)
Time delay (ps)
Regular
chirped
regime
-0,03 -0,02 -0,01 0,00 0,01 0,020
2
4
averaged
step N+100
Spec
tral
po
wer
, ar
b.
un
.
w,fs-1
step N
Resonant interaction with the
dispersive wave perturbed
strongly the soliton spectrum and
causes its structurization and the
central frequency jitter. As a
result, the soliton behaves
chaotically although the energy
remains almost constant.
Nonresonant excitation of the vacuum
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
Mechanism of excitation is the growth of spectral dissipation for a soliton. As a result, its energy
𝐸 decreases that reduces the gain saturation so that the net-gain becomes positive:
𝒈𝟎
𝟏 + 𝝑 𝑨 𝟐𝒅𝒕′∞
−∞
− 𝝈 > 𝟎.
10000 7500 5000 2500 0 -2500
-0,010
-0,005
0,000
three solitons
two solitons
single
soliton
single soliton net
-gai
n o
ut
of
soli
ton
, femtoseconds2
Thus, the vacuum becomes exited. New solitons appear from such an excitation.
10000 5000 0100
150
200
250
, femtoseconds2
sol
iton
wid
th, f
emto
seco
nds
three
solitons
two solitons
single soliton
Soliton width contraction with 𝛽 → 0
increases the spectral loss
As a result, the positive net-gain excites the
vacuum and new solitons appear
V.L.Kalashnikov et al., IEEE J. Quantum Electron. 39, 323 (2003).
-2220 -2210 -2200 -2190 -21800
2
4
6
Pow
er, a
rb.
un
.
t, piscoseconds
Multi-soliton complexes
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
A sequence of such excitations
results in an appearance of the
multi-soliton complexes. These
complexes are stable if the
interaction between
neighboring solitons is week.
Otherwise, the energy exchange
between solitons begins that
destabilizes the complex.
𝑧
𝑡 𝑡
0 20000 400000
20
40
60
Pea
k p
ow
er, a
rb.
un
.
z
Structural chaos
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
𝑡 𝑡
𝑧
Strong interactions inside the complex leads to
the structural chaotization. The field remains
localized on a picosecond scale, but chaotically
structured on a femtosecond one.
Spontaneous creation of the stable soliton
complexes from a chaotic “soup” is possible, as
well.
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)
Macro-structural chaos
0 50 100 150 2000,00
0,01
0,02
0,03
0,04
0,05
-10 -5 0 5 100,0
0,5
1,0
1,5
2,0
po
wer
, ar
b.
un
.
time, ns
pea
k p
ow
er, ar
b.
un
.
time, s
There exist the long-range interactions in a dissipative system containing a resonant saturable
medium with the recovery time 𝑇𝑟𝑒𝑙 ≫ 𝑇. For instance, the gain (𝑔) evolution in a laser obeys
𝝏𝒈
𝝏𝒕= 𝑷 𝒈𝟎 − 𝒈 − 𝝑𝒈 𝑨 𝟐 −
𝒈
𝑻𝒓𝒆𝒍.
The gain dynamics can result in the vacuum excitation far from the soliton. Energy exchange
through the gain leads to macro-structural chaos.
by courtesy of O. Pronin (University of Munich,
Concept of a dissipative soliton is reviewed in brief on the basis of the
variational method.
Two main sources of the solitonic chaos production are considered: resonant
and nonresonant vacuum excitations.
Resonant excitation causes the soliton spectral jitter with the subsequent
chaotic dynamics and even the soliton destruction.
Nonresonant excitation of vacuum forms the multi-soliton complexes. Strong
interactions inside such complexes cause the structural chaos.
Long-range interactions in a system can be additional source of the
nonresonant vacuum excitation that leads to macro-structural solitonic chaos.
Acknowledgements
This work is supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, Project P20293
Conclusions
The 4nd Chaotic Modeling and Simulation Conference (Agios Nikolaos, Crete, Greece, May 31 - June 3, 2011)