Download Kinetic equations

Wave Turbulence in nonintegrable and integrable
optical (fiber) systems
Pierre Suret et Stéphane Randoux
Phlam, Université de Lille 1
Antonio Picozzi
laboratoire Carnot, Université de Bourgogne, Dijon
Séminaire CEMPI, Décembre 2012
Wave Turbulence in optical fibers
Motivations
Physics of multimode continuous wave (Raman fiber) lasers
Nonlinear Dynamics / statistical Physics
Nonlinear propagation of incoherent waves
Experiments
Theory
Numerical simulations
Wave Turbulence in optical fibers
Introduction
Lasers 1960s
Nonlinear optics
Optical fibers 1970s
2nd harmonic generation
Telecommunications (linear operation)
Nonlinear fiber optics
intense cw/pulsed coherent light waves
Photonic Crystal fibers 1995
supercontinuum
Incoherent nonlinear (fiber) optics
Wave Turbulence in optical fibers
Outlines
1
Wave turbulence in optical fibers : some fundamentals
2
Anomalous thermalization (non integrable system)
Non trivial degeneracy of resonance conditions
3
Irreversible evolution in Nonlinear Schrödinger equation
(NLS 1D)
No resonance conditions
4
Open questions
Wave Turbulence in optical fibers
1) Principles
WaveTurbulence
Hydrodynamics, Mechanics, Plasmas physics, BEC, Optics
Wave Turbulence Theory :
incoherent wave
weak nonlinearity (HNL << HL) (~ linear dispersion curve !)
A(t, k) A* (t, k ') = n(t, k) d (k - k ')
closure of moments
kinetics equations ( Boltzmann equation)
Dissipatives systems
Kolmogorov-Zakharov cascade
2 ö
æ
ln ç A(k) ÷
è
ø
forcing
k -a
damping
ln(k)
Hamiltonian systems
Wave Thermalization
Entropy
Microcanonics
Energy equipartition
Condensation of waves
Wave Turbulence in optical fibers
1) Principles
Example in mechanics
vibrating plate
Miquel B., N. Mordant
Phys. Rev. Lett. 107(3), 034501 (2011)
Cobelli P. et al.
Phys. Rev. Lett. 103 204301 (2009)
WaveTurbulence
Wave Turbulence in optical fibers
1) Principles
Vibrating plate
WaveTurbulence
Wave Turbulence in optical fibers
1) Principles
Experiments with optical fibers
Nonlinear fiber optics : how does it look like in real life?
Wave Turbulence in optical fibers
1) Principles
What do we observe in optics ?
narrow optical spectrum
E(r, t) = A(r,t) ei(w0t-k0z)
E(w )
l
e(r, t) = Re(E(r, t))
2
w
w0
T
l
0
w
Detector / intensity
= low pass filter Dt ³10-10 s
l » 1. /1.6mm
t
Iinst (t) µ [ e(t) ]
A(w )
2
Dw << w0
slow varying Amplitude
carrier wave
2
I 0 (r, t) = E(r, t) = A(r, t)
2
2
T=
l
» 10-15 s
c
Wave Turbulence in optical fibers
1) Principles
Optical (power) spectrum
Optical spectrum : « measurement » of fast dynamics
Optical power
A(t)
2
~1 ps
Slow detector
t
Dispersive setup
Optical spectrum = power spectral density
= number of particles (kinetic equation)
˜A(z,w ) A˜ * (z,w ') = n(z,w ) d(w - w')
A(w )
2
w
Wave Turbulence in optical fibers
1) Principles
Interaction light / matter
What is non linear ?
Dielectric Polarization
P = e0 éë c (1) E + c (2) E 2 + c (3) E 3 ùû
Electromagnetic waves
Maxwell equations (E & B)
1 ¶2 E(r, t)
1 ¶2 P(r, t)
DE(r, t) - 2
= 2
2
c
¶t
c e0 ¶t 2
Generalized NLS
k
æ ¶2 ¶ 2 ö
¶A
k+1 b k ¶ A
= -A + i ç 2 + 2 ÷ A + åi
+ NL
k
¶z t-z/vg
k! ¶t
è ¶x ¶y ø
k³1
losses
diffraction
dispersion
Kerr
Raman
Gain
…
Wave Turbulence in optical fibers
1) Principles
Single mode optical fiber
l0 » 1. /1.6mm
Optical cladding
core
LOW losses
(0.1-1% /100m)
n0
L » 10cm < - >100km
n(r)
single-mode fibers
Fiber core diameter 6-9µm
L
E(x, y, z, t) = Y(x, y) A(z, t)ei(w0t-k0z)
1 or several Waves
Wave Turbulence in optical fibers
1) Principles
Propagation in single mode fiber (1D)
Spatiotemporal system (scalar 1D Nonlinear Schrödinger equations)
1
2
i ¶z A(z, t) = - b2 ¶t2 A(z, t) + c (3) A(z, t) A(z, t)
2
group velocity dispersion
Kerr effect
z
Distance z
t
|A|2
t
Time t
« time »
«space»
Wave Turbulence in Optical systems
Optical spectrum of cw lasers
1) Principles
Strongly multimode Continuous Wave laser = incoherent waves
101-106 modes !
complex dynamics
A(t) = å A˜ (w k ) e iw k t e ij k
A(t)
2
k
j k j k' = 2p d(k - k')
t
random phases
˜A(w,z = 0) 2
2
˜
A(w,z = L)
w
L
?
w
Wave Turbulence in optical fibers
1) Principles
Wave Turbulence / kinetic Theory
Weak nonlinear interaction among spectral components
i ¶z A1 (z, t) = - b1 ¶t2 A1 (z,t) + ( A1 + k A2 ) A1 (z,t)
2
1
2
i ¶z A(z,t) = - b 2 ¶2t A(z,t) + c (3) A(z,t) A(z,t)
2
2
i ¶z A2 (z,t) = - b2 ¶2t A2 (z,t) + ( A2 + k A1 ) A2 (z,t)
2
2
(3) / Four Waves Mixing
˜A(w ) 2
Initial condition
101-108 modes !
incoherent waves /
random phases
w3
w1 w 2
Phase matching conditions (
w4
w1
w3
w2
w4
w1 + w 2 = w 3 + w4
w
k(w1) + k(w2 ) = k(w3 ) + k(w4 )
Wave Turbulence in Optical systems
Motivations, context
Thermodynamics
ex gas : collisions
1
2
3
4
E1 + E 2 = E 3 + E 4
p1 + p2 = p3 + p4
Kinetic Theory of gases
Wave Turbulence Theory
Nonlinear Optics
Four Waves Mixing
w1 + w2 = w3 + w4
k1 + k2 = k3 + k4
Wave Kinetic theory
Wave Turbulence
Propagation of incoherent waves in optical fiber
1
Wave turbulence in optical fibers : some fundamentals
2
Anomalous thermalization
3
Irreversible evolution in Non Linear Schrödinger equation
(NLS 1D)
Wave Turbulence in optics
Anomalous Thermalization
A simple example of wave thermalization
i ¶z A1(z,t) = - b1 ¶ A (z,t) + ( A1 + k A2 ) A1(z,t)
2
2
i ¶z A2 (z,t) = - b2 ¶ A (z,t) + ( A2 + k A1 ) A2 (z,t)
2
2
t
1
2
t
2
2
b1 ¹ b 2
Cross 4 Waves mixing ()
w4
w1
w1 + w2 = w3 + w4
Phase-Matching
b1 w + b2 w 2 = b 2 w 3 + b1w 4
2
2
k =2
A˜1(w )
Energy
2
1
XPM
A˜ 2 (w )
2
w3
w2
Wave Turbulence in optics
A simple example of waves thermalization
Anomalous Thermalization
Kinetic equations
H-Theorem
*
optical spectrum : A˜ j (z,w) A˜ j (z,w ') = n j (z,w) d(w - w ')
¶z n j (w,z) = Coll [ n j ,ni ]
entropy
¶ zS ³ 0
S(z) = å ò log[ ni (w )] dw
i
Thermodynamical equilibrium
Number of particules
global
invariants
Momentum
Kinetic Energy
Nj =
ò n (w) dw
j
é2
ù
P = ò êåw n j (w )ú dw
êë j=1
úû
é2
ù
E = ò êå b jw 2 n j (w )ú dw
êë j=1
úû
Wave Turbulence in optics
Anomalous Thermalization
b1 ¹ b 2
k j (w ) = b jw 2
A simple
example of
thermalization
Distribution
dewave
Rayleigh-Jeans
Energy equipartition
Rayleigh-Jeans
n
RJ
j
»w
-2
é2
ù
E = ò êå b jw 2 n j (w )ú dw
êë j=1
úû
T
n (w )=
b jw 2 + lw - m j
eq
j
Numerical simulations
1 l -m j
, ,
T T T
Lagrange parameters / E, P, Nj
Wave Turbulence in optics
Anomalous Thermalization
Numerical simulations
b1 ¹ b 2
k j (w ) = b jw 2
Rayleigh-Jeans
n
RJ
j
»w
-2
b1 = b 2
k1 (w ) = k2 (w) = bw 2
NO energy equipartition
Wave Turbulence in optics
Phase-matching conditions
Anomalous Thermalization
A˜1(w )
4 waves mixing ()
w1 + w2 = w3 + w4
w1
b1 w + b2 w 2 = b 2 w 3 + b1w 4
2
1
2
2
2
A˜ 2 (w )
w3
b1 = b 2
degeneracy
w4
w2
w2 = w4
A˜1(w )
w1 = w 3
A˜ 2 (w )
Wave Turbulence in optics
Anomalous Thermalization
Anomalous
thermodynamical
equilibrium
Distribution
de Rayleigh-Jeans
¶z n j (w,z) = Coll [ n j ,ni ]
é 1
1
1
1 ù
¶z n1(w,z) = òòò dw1-3 W n1 (w )n 2 (w1 )n 2 (w 2 )n1 (w 3 )ê
+
ú
ë n1(w ) n 2 (w1 ) n2 (w 2 ) n1 (w 3 )û
W=
1
p
k 2 d(w + w1 - w 2 - w 3 ) d(k1(w ) + k2 (w1 ) - k2 (w 2 ) - k1(w 3 ))
ò dw
w3 = w + w1 - w2
k1 (w ) = w 2
k2 (w ) = w 2
3
¶z n1(w,z) =
1
2
òò dw
1-2
k1(w) + k2 (w1) - k2 (w2 ) - k1(w3 ) = 2(w2 - w1)(w - w2 )
n1 (w )n 2 (w1 )n 2 (w 2 )n1 (w + w1 - w 2 )
é 1
ù éd (w 2 - w1 ) d (w - w 2 ) ù
1
1
1
+
+
ú
ê
úê
n
(
w
)
n
(
w
)
n
(
w
)
n
(
w
+
w
w
)
w
w
w 2 - w1 û
ë 1
2
1
2
2
1
1
2 ûë
2
ò dw
2
¶z n1(w,z) =
1
2
ò dw1
G[J,n1 ]
w - w1
Wave Turbulence in optics
Anomalous Thermalization
New invariant
Jw LOCAL invariant
(for each w)
H-theorem
Anomalous
thermodynamical
equilibrium
Distribution
de Rayleigh-Jeans
¶Jw
=0
¶z
Jw = n1(w,z) + n2 (w,z)
¶Sloc (z)
>0
¶z
Sloc (z) =
local equilibrium spectrum n1loc (w ) = Jw /2 -
[
ò Log( n (w)[Jw - n (w)]) dw
1
1
]
1+ ( lJw /2) 2 -1 / l
 : Lagrange parameter associated to N1 =
ò n (w)dw
1
NO energy equipartition
local equilibrium state preserves a memory of the initial condition
Wave Turbulence in optics
Anomalous Thermalization
Anomalous
thermodynamical
equilibrium
Distribution
de Rayleigh-Jeans
N1 =
particular case
ò n (w)dw = N = ò n (w)dw
1
2
n1(w )
n1(w )
n1loc (w ) = Jw /2 -
2
[ 1+ (lJ
w
]
/2) 2 -1 / l
l=0
n2 (w )
w
w
n1loc (w ) = n2loc (w ) = Jw /2
=
n1(w,z = 0) + n 2 (w,z = 0)
2
Wave Turbulence in optics
Anomalous Thermalization
Experiments
Distribution
de Rayleigh-Jeans
Polarization maintaining
fiber (PMF)
WDM
Raman
QWP
Q-Switch Nd/YAG
Laser =1064 nm
QWP
trigger electronics
Isotropic (spun) fiber
1.6
meters
OSA
AOM
10ns
t
HWP
Wave Turbulence in optics
Anomalous Thermalization
Experiments
Distribution
de Rayleigh-Jeans
Wave Turbulence in optics
Anomalous Thermalization
z=0
n1(w )
Experiments
Distribution
de Rayleigh-Jeans
z=L
n1(w )
z=0
z=L
n2 (w )
n1(w )
n1(w )
Wave Turbulence in optics
Anomalous Thermalization
a general phenomenon
another example :
Conclusion
Distribution
de Rayleigh-Jeans
Non-trivial degenerate resonances
Local invariant
¶Jw
=0
Jw = n1(w,z) + n2 (w,z)
¶z
Breakdown of Energy equipartition
Memory of the initial condition
i ¶z A(z,t) = -ia ¶3t A(z, t) - ¶t2 A(z, t) + A(z, t) A(z,t)
2
Suret et al., PRL 104, 054101 (2010)
C. Michel et al., Opt. Lett. 35, 2367-2369 (2010)
C. Michel et al., Lett. In Math. Phys., 96, p 415 (2011)
Wave Turbulence in optical fibers
1
Wave turbulence in optical fibers : some fundamentals
2
Anomalous thermalization (non integrable system)
3
Irreversible evolution in Nonlinear Schrödinger equation
(NLS 1D)
Wave Turbulence in optics
1D NLS
Experiments / numerical simulations
Irreversible evolution toward a steady state
defocusing case : no BF
B. Barviau, S. Randoux, and P. Suret , Optics Letters, 31, pp. 1696-1698 (2006)
Wave Turbulence in optics
1D NLS
1D nonlinear Schrödinger equation
(3) / 4 waves interaction
1
2
i ¶z A(z,t) = - b 2 ¶2t A(z,t) + c (3) A(z,t) ) A(z,t)
2
w3
w1 w 2
w4
2>0 normal dispersion (<1300nm) : no modulationnal instability
w1 + w2 = w3 + w4
k(w1) + k(w2 ) = k(w3 ) + k(w4 )
1
k( w ) = b 2w 2
2
NLS1D : integrable equation
usual Wave Turbulence theory
w1 = w 3
Trivial interaction !
infinity of motion constants
quasi-periodic behavior
¶z n(w, z) = 0
w
Wave Turbulence in optics
1D NLS
1D NLS
What does the kinetic theory say ?
Wave Turbulence in optics
1D NLS
What does the kinetic theory say ?
Oscillatory terms neglected in the usual treatment
1DNLS : NO Phase matched interactions
Oscillatory terms ?
Transient regime ?
¶z nw (z) = 0
Wave Turbulence in optics
1D NLS
HNL/HL=0.05
Numerical simulations
HNL/HL=0.5
Wave Turbulence in optics
1D NLS
HNL/HL=0.05
Numerical simulations
HNL/HL=0.5
Wave Turbulence in optics
1D NLS
HNL/HL=0.05
Numerical simulations
HNL/HL=0.5
Good approximation N(z) = N(z=0) !!
Wave Turbulence in optics
1D NLS
Dominant
contributions
A last approximation…
Wave Turbulence in optics
Dominant contribution
1D NLS
-
w1
3
+
éæ w ö2
ù éæ w ö2 æ w ö2 ù
2
1
Dk = êç ÷ + w1 ú - êç 1 ÷ + ç 1 ÷ ú
êëè 3 ø
úû êëè 3 ø è 3 ø úû
w1
3
w1
8 2
Dk = w1
9
HNL/HL=0.5
Wave Turbulence in optics
1D NLS
Damped oscillations toward steady state
comparison with numerical integration 1D NLS

The period of oscillations
is given by the dominant contribution
Wave Turbulence in optics
Experiments
1D NLS
P »12mW
=1064nm
Experiments / numerical simulations (NLS)
Simulations (NLS / Kinetic equations)
0.15nm
nw = e
0
-w 4
Propagation of incoherent waves in optical fiber
1DNLS (single pass) Open Questions
• Complete characterization
of the equilibrium state
(exponential tails)
• General wave turbulence theory
HNL << HL
• Which theory at high optical power
HNL ~ HL and HNL >> HL
Propagation of incoherent waves in optical fiber
1DNLS (single pass) Open Questions
- Gas of solitons ?
- Numerical Inverse Scattering Transform (incoherent initial conditions)
i yt = y xx + 2 y 2 y
Periodic defocusing 1D NLS equation
y R (x, t) = Re(y ) = 0.30
1
x j + åh j (x, t)
å
2 j
j
0.20
Hyperelliptic functions
Eigenvalues of the IST problem
0.10
0.3
0.2
-0.00
0.1
-0.10
0
−0.1
-0.20
−0.2
−0.3
-40.00
-30.00
-20.00
-10.00
0.00
10.00
20.00
30.00
40.00
−40
−30
−20
−10
0
10
20
30
40
Propagation of incoherent waves in optical fiber
Open Questions
Non-local / non-instantaneous (experiments / NLS ?)
i yt = y xx + y (x) ò R(x') y 2 (x - x') dx'
Single pass : toward thermodynamical equilibrium ?
Initial conditions
?
2D (multimode fibers)
Dissipative systems
Optical cavity (forcing / losses)
Gain
Propagation of incoherent waves in optical fiber
Open Questions
T
T= slow time
« time »
t
|A|2
Mean-field model
t
t =fast time (round trip)
«space»
Ginzburg-Landau eq.
Braggs : -ln(R1(w)R2(w))=0+2w2
Dispersion / effet Kerr : 2 / 
Raman Gain : g
losses : S
Statistical properties of Raman
fiber lasers
Dissipative systems : open Questions
Recent experimental results about extreme statistics (Raman fiber lasers)
S. Randoux and P. Suret, Opt. Lett. 37, 500-502 (2012)
Off-centered
filter
Model /Numerical integration : ok
Which theory (mechanisms, PDF) ?
Statistical properties of Raman
fiber lasers
Open Questions
Optical power spectra transmitted by
the narrow-bandwidth (5GHz-2pm) optical filter
Statistical properties of Raman
fiber lasers
Dynamics at the output of the narrow-bandwidth optical filter
Optical filter at central Stokes wavelength Off-centered optical filter (detuned from 1.5 nm
from the central Stokes wavelength)
Total (not filtered) Stokes power
Statistical properties of Raman
fiber lasers
Statistics at the output of the narrow-bandwidth optical filter
Centered
filter
Off-centered
filter