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