dissipative solitons: the structural chaos and the...

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


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


Self-phase modulation

pulse profile frequency

deviation

faster

components

slower

components contraction

+

Classical (“Schrödinger”) soliton is a result of the phase balance


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


Action of linear and

nonlinear gain

VS.

linear and nonlinear

loss

H.A.Haus et al., J. Opt. Soc. Am. B 8, 2068 (1991)


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


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


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 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


𝛿, femtoseconds3


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.


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


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


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


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.


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,

[email protected])

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


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