mathematical ideas in nonlinear optics: guided waves in inhomogenous nonlinear media july 19-23,...

July 19-23, 2004, EdinburghJuly 19-23, 2004, Edinburgh

Mathematical Ideas in Nonlinear Optics: Guided

Waves in Inhomogenous Nonlinear Media

Nonlocal SolitonsNonlocal Solitons

Ole Bang, Nikola I. Nikolov, and Peter L. ChristiansenOle Bang, Nikola I. Nikolov, and Peter L. ChristiansenCOM Centre, Informatics and Mathematical Modelling, and Physics DepartmentCOM Centre, Informatics and Mathematical Modelling, and Physics Department

Technical University of Denmark, Lyngby, DenmarkTechnical University of Denmark, Lyngby, Denmark

Wieslaw Krolikowski, Darran Edmundson, and Dragomir NeshevWieslaw Krolikowski, Darran Edmundson, and Dragomir NeshevLaser Physics Centre, ANU Supercomputer Facility, and Nonlinear Physics Group Laser Physics Centre, ANU Supercomputer Facility, and Nonlinear Physics Group

Australian National University, Canberra, AustraliaAustralian National University, Canberra, Australia

John WyllerJohn Wyller Department of Mathematical Sciences, Agricultural University of NorwayDepartment of Mathematical Sciences, Agricultural University of Norway

Jens Juul RasmussenJens Juul Rasmussen

Risoe National Laboratory, Roskilde, DenmarkRisoe National Laboratory, Roskilde, Denmark




Physical systems exhibiting Physical systems exhibiting

nonlocal nonlinear responsenonlocal nonlinear response Systems involving transport effectsSystems involving transport effects– heat conduction in materials with thermal nonlinearityheat conduction in materials with thermal nonlinearity- light-induced diffusion of molecules or atoms in atomic light-induced diffusion of molecules or atoms in atomic

vapoursvapours- Drift/diffusion of photoexcited charges in photorefractivesDrift/diffusion of photoexcited charges in photorefractives

Propagation of electromagnetic waves in Propagation of electromagnetic waves in plasma plasma Many body interaction with finite scattering Many body interaction with finite scattering parameter in Bose-Einstein condensatesparameter in Bose-Einstein condensates

Molecular re-orientation in liquid crystalsMolecular re-orientation in liquid crystals

Parametric wave-mixingParametric wave-mixing




Impact of nonlocality on beam Impact of nonlocality on beam propagationpropagation

Theoretical predictions: Theoretical predictions: – nonlocality arrests collapse (catastrophic self-focusing) of optical beams in self-nonlocality arrests collapse (catastrophic self-focusing) of optical beams in self-

focusing media and enables formation of stable 2D solitons [Turitsyn, 1985]focusing media and enables formation of stable 2D solitons [Turitsyn, 1985]– Nonlocality-induced long-range interaction enables attraction of out-of-phase Nonlocality-induced long-range interaction enables attraction of out-of-phase

bright solitons [Kolchugina, Mironov, Sergeev, 1980]bright solitons [Kolchugina, Mironov, Sergeev, 1980]– Nonlocality can suppress MI in focusing media [Litvak, Mironov, Fraiman, Nonlocality can suppress MI in focusing media [Litvak, Mironov, Fraiman,

Yunakovskii, 1975]Yunakovskii, 1975]

Experimental observations:Experimental observations: – Stabilization of 2D beams in atomic vapors due to atomic diffusion Stabilization of 2D beams in atomic vapors due to atomic diffusion

removing excitation from the interaction region [Suter, Blasberg,1993]removing excitation from the interaction region [Suter, Blasberg,1993]– Assanto and his group observed this attraction in nematic liquid crystals Assanto and his group observed this attraction in nematic liquid crystals

[Peccianti, Brzdkiewicz, Assanto, 2002][Peccianti, Brzdkiewicz, Assanto, 2002]– Assanto and his group observed suppression of MI in nematic liquid Assanto and his group observed suppression of MI in nematic liquid

crystals [Peccianti, Conti, Assanto, 2003]crystals [Peccianti, Conti, Assanto, 2003]




Nonlocal modelNonlocal model

u(x,y,z) – slowly varying field amplitudeu(x,y,z) – slowly varying field amplitude

0)()('|)'(||)'(|)()()( 22 rurVrdrurrRrsuruz

rui

diffraction nonlocal nonlinearity ext. confinement

|)(|)(

)/||exp(2

1)(

)/exp(1

)( 22

xrR

xrR

xrR

1D1D

V(r) - confining potential (waveguide)V(r) - confining potential (waveguide)

s =±1 determines the type of nonlinearitys =±1 determines the type of nonlinearity

R(r) - nonlocal response functionR(r) - nonlocal response function Determined by the physical processDetermined by the physical process

1)( rdrR




Nonlocal modelNonlocal modelThe relative width of the response function and the The relative width of the response function and the intensity profile determines 4 regimesintensity profile determines 4 regimes

0)(|)'(|)()( 22 rurusruz

rui

Local NLS

rdruP

rursPRruz

rui

2

2

|)(|

0)(|)(|)()(

Strongly nonlocal

rdrurD

rururusruz

rui

22

222

|)(|2

1

0)()|)'(||)'((|)()(

Weakly nonlocal




Modulational instability in nonlocal Modulational instability in nonlocal mediamedia

Modulational instability (MI) signifies the exponential growth of a Modulational instability (MI) signifies the exponential growth of a weak perturbation of the amplitude of a plane wave as it weak perturbation of the amplitude of a plane wave as it propagatespropagates

The gain leads to amplification of sidebands, which breaks up the The gain leads to amplification of sidebands, which breaks up the otherwise uniform wave front - filamentation otherwise uniform wave front - filamentation

MI may act as a precursor for the formation of bright solitonsMI may act as a precursor for the formation of bright solitons

Stable dark solitons requires absence of MI of the constant Stable dark solitons requires absence of MI of the constant intensity backgroundintensity background

MI has been identified in fluids, plasma, nonlinear optics, discrete MI has been identified in fluids, plasma, nonlinear optics, discrete nonlinear systems, such as molecular chains and waveguide nonlinear systems, such as molecular chains and waveguide arrays arrays

Gaussian responseGaussian response

Self-focusing (s=+1)Self-focusing (s=+1)σσ=0.1=0.1

σσ=1.0=1.0




ionnormalizat todue 10

~spectrumFourier

~MI andgrowth lexponentia 02

R

kR


Linearization of propagation equation around plane wave solution Linearization of propagation equation around plane wave solution

zirkizru

00 exp),(

Plane wave solution and dispersion relation Plane wave solution and dispersion relation

)exp()],([),( 010 zirkizrazru

Equation for the perturbation growthEquation for the perturbation growth

0202

1 sk

0

2

022

4

~

k

kRsk

Sign indefinite spectrumSign indefinite spectrum– S = +1 => MIS = +1 => MI– S = -1 => S = -1 => Possibility for MIPossibility for MI

Positive definite spectrumPositive definite spectrum– S = +1 => MIS = +1 => MI– S = -1 => StabilityS = -1 => Stability





Self-focusingSelf-focusingGaussian Gaussian

responseresponse

Self-defocusing Self-defocusing rectangular responserectangular response




Beam collapse in nonlocal Beam collapse in nonlocal mediummedium

Conserved Power P and Hamiltonian H Conserved Power P and Hamiltonian H

Fourier approachFourier approach

rNIdrNIdH

2

1

2

1 2

2

2

2

PrdrIrdrkirIkIkdkIkRrNIdD

)()exp()()(

~ ,)(

~)(

~

)2(

1 2

')'()'(N ,2

1 ,)(

2

2rdrIrrRrNIduHrdrIP

kdkRRRPkdkIkRrNIdDD

)(

~

)2(

1 ,)(

~)(

~

)2(

100

22

20

2

2 2

1PRH

Gradient norm bounded from aboveGradient norm bounded from above

=> No collapse=> No collapse




Beam stabilization via nonlocalityBeam stabilization via nonlocalityStabilizing role of the nonlocality can be illustrated by considering properties of stationary solutions (2D). Stabilizing role of the nonlocality can be illustrated by considering properties of stationary solutions (2D).

Using Gaussian response function and gaussian soliton ansatz: Using Gaussian response function and gaussian soliton ansatz:

the variational approach gives the relation between soliton power and propagation constant the variational approach gives the relation between soliton power and propagation constant

u(r )Aexp( r 2 / 2 )exp( iz)

P4 (1 2 )




Interaction of dark nonlocal solitonsInteraction of dark nonlocal solitons

2.0 1.0, 0.1, :ynonlocalit of Degree

:2.5by separated 0.95 of jumps phase opposite Two

2.5 and 4.0, 5.5, :separationInput

:2for of jumps phase opposite Two

Dark solitons in local nonlinear media Dark solitons in local nonlinear media always repelalways repel

Nonlocality induces long-range attraction of dark solitons Nonlocality induces long-range attraction of dark solitons and leads to the formation of their bound statesand leads to the formation of their bound states

CW beam passing through phase mask with 2 opposite CW beam passing through phase mask with 2 opposite phase jumps (a-b) and through a thin wire (c)phase jumps (a-b) and through a thin wire (c)

6.0 3.0, 0.1, :ynonlocalit of Degree

:7.5 width of Wire




Nonlocality-assisted stabilization of Nonlocality-assisted stabilization of vorticesvortices

Vortex beams in local nonlinear media Vortex beams in local nonlinear media always disintegratealways disintegrate

Nonlocality induces long-range attraction keeps the vortex togetherNonlocality induces long-range attraction keeps the vortex together

Gaussian response and a charge 1 vortex:Gaussian response and a charge 1 vortex:

σσ=0=0

σσ=1=1

σσ=10=10

)exp(/exp)( 20

2 irrrru




Nonlocality-assisted vortex Nonlocality-assisted vortex stabilization - IIstabilization - II

More accurate variational solution:More accurate variational solution:– Stable over exceptionally long distances.Stable over exceptionally long distances.

Nonlocality =1 Vortex width 0 =10




Solitons in weakly nonlocal mediaSolitons in weakly nonlocal media

Both types of solitons are stableBoth types of solitons are stable

The weakly nonlocal The weakly nonlocal 1D model can be 1D model can be solved analytically solved analytically

iuz

2u

x2 u | u |2 2 | u |2

x2

0

weak nonlocality




ConclusionsConclusionsNonlocality in nonlinear media Nonlocality in nonlinear media

arrests collapse of multidimensional beams and stabilizes solitonsarrests collapse of multidimensional beams and stabilizes solitons stabilizes propagation of vortex beams in focusing mediastabilizes propagation of vortex beams in focusing media suppresses MI of plane waves in focusing media and may induce MI suppresses MI of plane waves in focusing media and may induce MI

in defocusing mediain defocusing media induces long-range attraction of dark and out-of-phase bright solitons induces long-range attraction of dark and out-of-phase bright solitons

and enables formation of bound statesand enables formation of bound states

Nonlocal nonlinear media support formation of many novel stable Nonlocal nonlinear media support formation of many novel stable bright and dark solitons and their bound statesbright and dark solitons and their bound states

Nonlocality provides nice physical picture of parametric wave Nonlocality provides nice physical picture of parametric wave interaction, for example predicting novel quadratic soliton solutionsinteraction, for example predicting novel quadratic soliton solutions

mathematical ideas in nonlinear optics: guided waves in inhomogenous nonlinear media july 19-23,...

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