Disorder, Gain & NonlinearityDisorder, Gain & NonlinearityFrom mirrorless lasers to speckle instabilitiesFrom mirrorless lasers to speckle instabilities
Patrick SebbahPatrick SebbahLaboratoire de Physique de la Matière CondenséeLaboratoire de Physique de la Matière Condensée
CNRS, Université de Nice, FranceCNRS, Université de Nice, France
OutlineOutline
• Light scattering in random media :An overview
• Multiple scattering in the presence of gain :Nature of the lasing mode in a random laser
• Nonlinear scattering in a Kerr disordered medium :Experimental evidence of speckle instabilities
IntroductionIntroduction
Ballistic Regime
Free propagation
Single scattering approximation
Weak Scattering
Diffusion approximation
Strong Scattering
Strong Localization of Light ?
2D Model 2D Model
Scatterers
• n1 = from 3 to 1.05• Ф = 40%• Ø = 120 nm
Matrix
• n0 = 1
L = 5.5 µm
Open boundary conditions (PML)
Maxwell Equations
µ0∂Hx/∂t = - ∂Ez/∂y
µ0∂Hy/∂t = ∂Ez/∂x
ε0 εi ∂Ez/∂t = ∂Hy/∂x - ∂Hx/∂y
Diffusive Regime: L>Diffusive Regime: L>ℓℓ>>λλ
• n0 = 1• n1 = 1.5
Localized Regime : Localized Regime : ℓℓ ~ ~ λλ
• n0 = 1• n1 = 3.0
Spectral signature in transmissionSpectral signature in transmission
Degree of Spectral Overlap : Degree of Spectral Overlap : δδ = < = <δνδν>/<>/<∆ν∆ν>>
0.000
0.001
0.002
10.20 10.25 10.30 10.35
δν
∆ν
Localized RegimeLocalized Regime
δδ <1<1
GHz17.2
17.0 17.1Diffusive RegimeDiffusive Regime
Δν
δν
δδ >1>1
GHz
Localized StatesLocalized States
a b
c d
e f
Wave Propagation in Random MediaWave Propagation in Random Media
• Some Fundamental questions:– Anderson Localization “Absence of diffusion in certain random lattices”,
P.W. Anderson, Phys. Rev 109(1958).
Topolancik & al., PRL 2007• Some applied problems
– Imaging in turbid media (seismology, medical imaging)– Focusing through random media– Quantum communication– Photonic crystals– …
–Transport in mesoscopic systems (weak localization, long range correlation, …)–Random lasing–Nonlinear scattering (NL optics in weakly disordered media / multiple scattering with small NL)–…
Roati & al. Nature 2008 Schwartz & al. Nature 2007 Laurent & al., PRL 2007Hu & al., Nature P 2008 Störzer & al., PRL 2006
OutlineOutline
• Light scattering in random media :An overview
• Multiple scattering in the presence of gain :Nature of the lasing mode in a random laser
• Nonlinear scattering in a Kerr disordered medium :Experimental evidence of speckle instabilities
Random LasingRandom Lasing
Wiersma et al., Nature 2000
From Noginov, Nature (2008)
H. Cao, PRL82 (1999)
ZnO Powder
Fundamental questionsFundamental questions
• How scattering provides with the necessary feedback to observe sharp peaks in the emission spectrum ?
• How lasing modes are selected in random systems ?
• How is it possible to predict which mode will lase first ?
→ Simple answer in the regime of strong localizationVanneste et al. PRL01 , Sebbah et al. PRB02
Laser EquationsLaser Equations
yEtH zx ∂∂−=∂∂0µ
xEtH zy ∂∂=∂∂0µ
yHxHtPtE xyzi ∂∂−∂∂=∂∂+∂∂0εε
dN1 dt = N2 τ21 − WpN1
dN2 dt = N3 τ32 − N2 τ21 − Ez hωl( )dP dt
( ) dtdPENNdtdN lz ωττ �+−= 3234343
dN4 dt = − N4 τ43 + Wp N1
zll ENPdtdPdtPd ..²² 2 ∆=+∆+ κωω
Maxwell Equations Population Equations
Polarization Equation
Wp
Pump Laser Transition ωl
4
3
2
1
Vanneste et al. PRL01 , Sebbah et al. PRB02
Laser Field Amplitude
Min
Max
Localized (∆n =1)
Threshold Lasing Mode: localized regimeThreshold Lasing Mode: localized regime
Vanneste et al., PRL, 183903 (2001)
The lasing mode is a mode of the passive system
Fundamental questionsFundamental questions
• How scattering provides with the necessary feedback to observe sharp peaks in the emission spectrum ?
• How lasing modes are selected in random systems ?
• How is it possible to predict which mode will lase first ?
→ Simple answer in the regime of strong localization
→ Not so simple in the diffusive regime
430 440 450 460
1010
1015
1020
Spec
tral
Inte
nsity
(a.u
.)
Wavelength (nm)
Passive SpectrumPassive Spectrum
430 440 450 460
1010
1015
1020
Spec
tral
Inte
nsity
(a.u
.)
Wavelength (nm)
Localisé (∆n =1) Diffusif (∆n =0.25)
Origin of the lasing modes Origin of the lasing modes in the diffusive regime?in the diffusive regime?
• Donut Modes (Shapiro, PRL02)
• Long-lived modes (Chabanov, PRL03)
• Lucky photons (Wiersma, PRL04)
We showed that there is no need to search for exotic mechanisms
Lasing with resonant feedback is observed not only in strongly scattering RL, where localization provides with necessary feedback, but also in « badly » scattering systems.
Vanneste et al., PRL98, 143902 (2007)
Threshold Lasing Mode: diffusive regimeThreshold Lasing Mode: diffusive regime
Laser Field AmplitudeMin
Max
Diffusive (∆n =0.25)
Vanneste et al., PRL98, 143902 (2007)
The lasing mode is « similar » to a quasimode/resonance of the passive system
Threshold Lasing ModeThreshold Lasing Mode
Localized (∆n =1) Diffusive (∆n =0.25)
+ + + +++ + +
ReIm
+
++
+
++
++
+ +
++
+
ImRe
P
P
Quasimodes vs. Constant Flux ModesQuasimodes vs. Constant Flux Modes
Quasimodes CF Modes
Complex Real
H. Tureci et al., PRA 06
Lasing Mode vs Passive QuasimodeLasing Mode vs Passive Quasimode
n=1.75
passiveactive
A. Asatrian et al, « The nature of lasing modes in Random lasers: a Review », in progress
Lasing Mode vs Passive QuasimodeLasing Mode vs Passive Quasimode
n=1.5
passiveactive
A. Asatrian et al, « The nature of lasing modes in Random lasers: a Review », in progress
Lasing Mode vs Passive QuasimodeLasing Mode vs Passive Quasimode
n=1.25passiveactive
A. Asatrian et al, « The nature of lasing modes in Random lasers: a Review », in progress
In SummaryIn Summary• Lasing with coherent feedback is possible, ,not only in the localized regime,
but even in open weakly-scattering media (no need of exotic mechanism)
• In contrast to conventional cavity laser, the threshold lasing mode of an open random active medium is not a quasimode of the passive system.
• Quasimodes, which diverge outside the system, must be replaced by Constant Flux Modes.
H.Cao et al, Phys.RevA 2006
H. Tureci et al., Science 320 (2008)
OutlineOutline
• Light scattering in random media :An overview
• Multiple scattering in the presence of gain :Nature of the lasing mode in a random laser
• Nonlinear scattering in a Kerr disordered medium :Experimental evidence of speckle instabilities
Speckle Pattern SensitivitySpeckle Pattern Sensitivity
Re{E}
Im{E}
βφβ
ieA+
β
γ
γφγ
ieA+=E
c
ss νπλπφ αα
α22 ==
α
αφα
ieA )exp( ϕiA=+...
Nonlinear Scattering : PrincipleNonlinear Scattering : Principle
Fluctuation of |ψ(r,t)|2 local change of refractive index, ∆n = n2 |ψ(r,t)|2
Alter Phases & interferences
Kerreffect
λπφ α
αs2=
Nonlinear Scattering & Optical LimitationNonlinear Scattering & Optical Limitation
Intensity I(r,t) Index n(r,t)
LASER
Sebbah et al., Nonlinear Optics, Vol. 27, p. 377 (2001)
NLNL
NLtNLL
ntrnnnn
ττ2),(
1 with =
+∂+=
Scatterers with nL=n0 & a non instantaneous Kerr NL : n2 , τNL
Speckle InstabilitySpeckle Instability
Positive feedback
Fluctuation of |ψ(r,t)|2 local change of refractive index, ∆n = n2 |ψ(r,t)|2
Alter Phases & interferences
Kerreffect
λπφ α
αs2=
Earlier Theoretical PredictionsEarlier Theoretical Predictions
Instability threshold p=<n2I>2 (L/ℓ)3 > 1 (ℓ = the mean free path)
No experimental observation reported to date
B. Spivak and A. Zyuzin, Phys. Rev. Lett. 84, 1970 (2000).S. E. Skipetrov and R. Maynard, Phys. Rev. Lett. 85, 736 (2000)S. E. Skipetrov Phys. Rev. E (2001); Optics Lett. 28, 646 (2003); Phys. Rev. E67, 016601 (2003); J. Opt. Soc. Am. B21, 168 (2004).
The fundamental timescale of the speckle pattern dynamics is set by the larger of τNL and TD=L2/D
Slow NL :τNL > TD Fast NL : τNL < TD
Photorefractive Liquid-Crystal Light-ValvePhotorefractive Liquid-Crystal Light-Valve
Reorientational Kerr effectRelaxation time:
τNL=γ/K d² ≈ 550 ms > TD (~ 1µs)
BSO crystalPhotoconductive
Liquid crystalE48
d=50 µm planar anchoring
∆n = 0.2306 L
≈ 20 mm
Disorder ProjectionDisorder Projection
Diameter 17 µm
Spatial light modulator
Spatial resolution down to 1 µm
Experimental SetupExperimental Setup
Experimental Observation Experimental Observation of Speckle Instabilityof Speckle Instability
Spectral AnalysisSpectral Analysis
1/τNL
Spectral AnalysisSpectral Analysis
Instability thresholdInstability threshold
Disorder dependenceDisorder dependence
0 50 100 1500
0.5
1
1.5
Averaged Intensity
Spec
tral P
eak
Am
plitu
de
s12
2.22Hz∅50 µm
ℓ~100µm
0 50 100 1500
0.5
1
1.5
Averaged Intensity
Spec
tral P
eak
Am
plitu
de
s22
2.33Hz∅20 µm
ℓ~40µm
Threshold depends on disorder mfp, not frequency
Kerr effect as a Gain MechanismKerr effect as a Gain Mechanism
Parametric process
Energy transfers from ωp to ωm when ωp≠ωm
Silberberg and Bar-Joseph, JOSA B (1984)
ωm
ωp
NonlinearKerr Medium
( ) 21
2
NL
NL
τδωτδω⋅+⋅
Gain ∝ with δω=ωp-ωm
Frequency SelectionFrequency Selection
τNL < TD
Linear spectrum
τNL > TD
0 0.005 0.01 0.015 0.020
0.2
0.4
0.6
0.8
1
Frequency ω
Gai
n cu
rve
ωp -τ
NL-1
ωp
( ) 21
2
NL
NL
τδωτδω⋅+⋅
Frequency oscillation = δω= 1/τNL
δω depends only on τNL
Frequency oscillation = δω= ωP - ωm
δω depends on τNL & disorder
In SummaryIn Summary
• Speckle instability demonstrated in 2D random scattering medium in a transverse geometry for a slow Kerr effect τNL>τD
• Similar to self phase modulation of the speckle pattern, However, the nonlinear phase φNL is not deterministic (kn2I0L) but randomly fluctuating
• Threshold occurs when fluctuations of φNL ~1– Depends on disorder (?)– Does not depend on the sign of the NL (??)
• Oscillation Frequency : beating between the pump ωP and – ωm = 1/τNL if τNL>τD
– ωm=eigenmode of the passive system if τNL<τD
AcknowledgementsAcknowledgements
Nice LPMC : C. Vanneste, L. Labonté
Nice INLN : S. Résidori, U. Bortolozzo, F. Haudin
Yale University : H. Cao, D. Stone, L. Ge, J. Andreasen
ETH Zurich : H. Tureci
UT Sydney : A. Asatryan
