Spontaneous Patterns in Nonlinear OpticsNetwon Institute, Cambridge

August 2005William J Firth

Department of Physics,University of Strathclyde, Glasgow, Scotland

willie@phys.strath.ac.uk

Acknowledgements

Thorsten Ackemann, Gonzague Agez + many colleagues/collaboratorsFundingFP6 FunFACS 2005-08 Leverhulme Trust 2005-08

Scottish Universities Physics Alliance

Abstract

“Spontaneous patterns in optics usually involve diffraction, rather than diffusion, as the primary spatial coupling mechanism. The simplest and most successful system involves a nonlinear medium with a single feedback mirror. The basic theory and experimental status of that system will be reviewed, along with discussion of other systems such as semiconductor micro-resonators, and the closely related topic of dissipative solitons in such systems.”

Spontaneous Patterns in Nonlinear Optics

Modulational Instability in 2nd Harmonic Generation

nonlinearity very fast, but very weak

Type II phase matching for SHG in KTPfundamental only inputinput beam ellipticity of 11:1input peak intensity of 57 GW/cm2

Fuerst et. al., Phys. Rev. Lett, 78, 2760 (1997)

Input

Output

“Traditional” Nonlinear Optics

• need to accumulate or concentrate nonlinearity • e.g. use material excitation• then material determines bandwidth• and the light has to be essentially monochromatic• leading to envelope patterns (and solitons)

Inertial NLSEEnvelope E of a quasi-monochromatic optical field,

coupled to a material excitation N(r) evolves like

N is a refractive index perturbation. Suppose it diffuses and relaxes, and is driven by |E|2 (optical intensity):

• In steady state, NL Schrödinger type equation

• Strength of nonlinearity scaled by (good rule of thumb)

• Spatial and temporal bandwidth scaled by 1/lD, 1/.

iE

z

1

2(2E

x 2

2E

y 2) NE 0

lD22N N

t N E

2

Using noise speckle pattern for the measurements of director reorientational relaxation time and diffusion length of aligned liquid crystals,G. Agez, P. Glorieux, C. Szwaj, and E. Louvergneaux, Opt. Comm. 245, 243 (2005)

Scaling Confirmation – Nematic Liquid CrystalLille Group

Other materials and response times: photorefractive ms; Na vapour µs; semiconductor ns; glass fs.

Kerr-like Nonlinearity of Nematic Liquid Crystal Lille Group

Refractive index change of 1% (large!) at intensity levels eight orders of magnitude lower than in SHG modulational instability

Spontaneous Pattern Formation

Needs NONLINEARITY and SPATIAL COUPLING In NL Optics coupling usually diffractive.

NL and diffraction can be separate, in a feedback configuration ....

… or occur together in an optical cavity

Reflected beam

Incident beam

Back mirror

6 m

5 cm

Substrate

Mechanism of instabilityphase

modulation

infinitesimalfluctuation

macroscopic modulation:

a pattern

n = n (|E|)

positivefeedback

?

fluctuationof refractive

index

homo-geneous

phase

and amplitude

damplitude

modulation

diffractiondiffraction

length scale ~ ( d) 0.5

• Instability lobes at Talbot intervals

• Diffusion raises high-K threshold • Interleaved lobes for N>0 and N<0.

Patterns in Feedback Mirror System

Liquid Crystal Patterns – Lille

Quasi-pattern due to effect of higher lobe.

Liquid Crystal Patterns – Lille

Tilting mirror:Hexagons give way to drifting rolls, then to static rolls via squares, then “diamonds”.

Self-organization phenomena in nonlinear optical systems: High-order spatial solitons and dynamical phenomena

(ENOC, Aug 8-12 2005)

Institut für Angewandte PhysikWestfälische Wilhelms-Universität

Münster

Email: thorsten.ackemann@strath.ac.uk

T. Ackemann, M. Pesch, F. Huneus, J. Schurek, E. Schöbel, W.

Lange

Department of PhysicsUniversity of Strathclyde

Glasgow, Scotland, UK

Feedback mirror patterns in Na vapourplanem irror

polarizationcontro l

po larizationanalyzer

hom ogeneous hold ing beam

focused addressing

beam

B

• medium: sodium vapor in nitrogen buffer gas

• pumping: in vicinity of D1-line

• nonlinearity: optical pumping between Zeeman sub-levels

Theoretical modelmj=1/2mj=-1/2

2P1/2

2S1/2

P

• modeled as homogeneously broadened J=1/2 -> J=1/2 transition

• optical pumping by circularly polarized light

• optical properties (absorption coefficient and index of refraction) dependent on z-component of magnetization

collisions pumping saturation precessionthermal diffusion

m

periodic patterns quasiperiodic boundary solitons spirals

Length scales

d

• scaling of length like square-root of cell-to-mirror distance

expected for single-mirror scheme

• size of solitons related to pitch of hexagons

indicates relationship between solitons and modulational instability

Targets and spirals

Multistability

• switch-on experiments: power is switched from zero to a value beyond threshold and a snapshot is taken (200 cycles)

• dynamical targets and spirals with opposite chiralities and different numbers of arms are observed for one set of parameters

• most frequent number of arms is obtained from histogram

T. Ackemann et al, Münster

Na vapor feedback scheme: polarization-sensitive.

Soliton Clusters in Na Vapour Feedback Mirror SystemSchäpers et al PRL 85 748 (2000)

• Circular polarisation holding beam• Spontaneous over a small range• Clusters show preferred distances

Experimental confirmation that CS exist as stable/unstable pairs (LCLV feedback system)

Unstable branch identified with marginal switch-pulse

Propagating Dissipative SolitonsUltanir et al, PRL 90 253903-1 (2003)

Peak field of solitons versus gain in alternate gain/loss waveguide (inset). Current assumes 300 µm width contact patterns on a 1 cm long device. (a) Images from output facet when the measured

input is 160 mW and 16.5 µm FWHM.

(b) Numerical simulation of the output profile

Spontaneous Pattern Formation

Needs NONLINEARITY and SPATIAL COUPLING In NL Optics coupling usually diffractive.

NL and diffraction can be separate, in a feedback configuration ....

… or occur together in an optical cavity

Reflected beam

Incident beam

Back mirror

6 m

5 cm

Substrate

E

t (1 i)E Ein i

2 E N(E)

Nonlinearity sometimes N(E), but more usually through optical excitation of a medium within the cavity

Optical Cavity BasicsReflected beam

Incident beam

Back mirror

6 m

5 cm

Substrate

E

t (1 i)E Ein i

2 E

Diffraction described by transverse Laplacian

External field drives cavity close to resonance (=

E

t (1 i)E Ein

Experimental Cavity Patterns

VCSEA, external injection, two different wavelengths (Nice)

Incoherent light, photorefractive (Segev group, Israel). (a) linear (b) NL, no cavity (c) NL, cavity.

Reflected beam

Incident beam

Back mirror

6 m

5 cm

Substrate

Patterns in a Saturable Absorber Cavity

E

t E iE Ein i2E

2C

1 E 2E

Using exact numerical techniques, we have traced existence and stability of stripes as a function of wavevector and driving.

Unstable to hexagonsUnstable to hexagons.Zig-zag unstableZig-zag unstable.

Eckhaus unstableEckhaus unstable.White region: stable.

Fourier Control of Optical Patterns Natural patterns are imperfect May also have wrong symmetry Both problems fixed by Fourier feedback control

Martin et al PRL (1997), Harkness et al PRA 58 2577(1998)

Negative feedback of unwanted Fourier components (mask) Stabilizes existing but unstable states by "subtle persuasion"

Fourier Control of Optical Patterns

Numerics Experiment(LCLV)

"Optical turbulence" stabilized to any of three unstable steady patterns

Cavity Solitons

Seems possible to create and control regular optical patterns.

For image and informatic applications of patterns, it should be possible to selectively create or remove any single element of the pattern.

Requires that a single isolated spot be stable.

Such a structure now called a CAVITY SOLITON.

Practical Definition of a Cavity Soliton

A cavity soliton:

• is exponentially self-localized transverse to its propagation direction

• can be present or absent under the same conditions - sub-critical

• has freedom of movement in the localization dimension(s)

• IS BOUNDARY-LOCALIZED IN PROPAGATION DIRECTION

• has losses, needs driving, hence has fixed amplitude (is an attractor)Reflected beam

Incident beam

Back mirror

6 m

5 cm

Substrate

VCSR device for cavity solitons in semiconductors.PIANOS 1998-2002FunFACS 2005-08.

Experiment INLN (Nice)

Cavity Soliton Pixel Arrays

Stable square cluster of cavity solitons which remains stable with several solitons missing – pixel function.

John McSloy, private commun.

Cavity Solitons linked to PatternsCoullet et al (PRL 84, 3069 2000) argued that n-peak cavity solitons generically appear and disappear in sequence in the neighbourhood of the “locking range” within which a roll pattern and a homogeneous state can stably co-exist.

We have verified this in general terms (in both 1D and 2D), but find much more complexity than Coullet et al imply.

We have tested Coullet’s theory for the bifurcation structure

of Kerr cavity solitons.

• This theory seems to properly describe the bifurcation structure, but is incomplete:

• We find a much higher level of complexity than predicted

• Additional homoclinic and heteroclinic intersections between the manifolds of fixed points and periodic orbits should be considered

• As a consequence new types of localized states are found

• Existence of arbitrary “soliton-bit” sequences not proven.

D. Gomila, W.J. Firth, and A.J. Scroggie

Bifurcation Structure of Kerr Cavity Solitons

Applications of Cavity Solitons?

binary “soliton-1, no-soliton -0” logic but not viable vs Intel transverse mobility may be the key e.g. optical buffer memory for serial-parallel.

“normal” beam also moves, but diffracts away.

QuickTime™ and aAnimation decompressor

are needed to see this picture.

Pinning of Cavity Solitons

Experiment (left) and simulation (right) of solitons and patterns in a VCSEL amplifier agree provided there is a cavity thickness gradient and thickness fluctuations.Latter needed to stop the solitons drifting on gradient.

A cavity soliton is self-localized transverse to its propagation direction, but not self-located …

What determines its location?

• boundary/background effects – then at best a “dressed CS”

• control beam – informatics, tweezers

• other CS – interactions and dynamics

• local imperfections (as in experiments) – CS microscope?

CS may move (due to any of above)

• parameter gradients couple to, and excite, translational mode

• velocity proportional to gradient force

• no force, no motion – CS normally has no inertia

Coherent/Incoherent Switching and DrivingCoherent/Incoherent Switching and Driving

• CS are usually CS are usually compositecomposite light/excitation structures light/excitation structures• can create/destroy CS through the excitation componentcan create/destroy CS through the excitation component• why not why not DRIVEDRIVE CS through the excitation? CS through the excitation? • such a drive incoherentsuch a drive incoherent• e.g. current drive - e.g. current drive - Cavity Soliton LaserCavity Soliton Laser

• basis of new basis of new FunFACSFunFACS EU project 2005-08 EU project 2005-08

In Nice VCSEL experiment (In Nice VCSEL experiment (leftleft), ), CS were created and destroyed with CS were created and destroyed with a a coherentcoherent address pulse, resp. in address pulse, resp. in and out of phase with Eand out of phase with Einin..

In other systems switching (both on In other systems switching (both on and off) has been achieved with and off) has been achieved with incoherentincoherent pulses. pulses.

• main FunFACS aims relate to cavity solitons in semiconductor laser systems

• related to pattern formation in these systems

• links to other work by Thorsten Ackemann (while at Münster)

www.funfacs.orgReflected beam

Incident beam

Back mirror

6 m

5 cm

Substrate

Tilted Waves

gain

0,28 nm/K

0,07 nm/K

gain

tilt

Change in temperature shifts gain curve and resonance detuning

in VCSELs:temperature

controls detuning

m /2 eff

m /2L

optica laxis

k

qk

keff

if resonator is too long for emission in gain maximum, L > m /2 tilted wave favored, since projection of tilted waves fits into resonator, effective wavelength eff>

Emission wavelength lower than longitudinal resonance, “off-axis”

emission

Length scalesexperiment

scaling exponent: 0.49

theory

scaling exponent: 0.5

w/o dispersion

with dispersion

• confirmation of predicted scaling behavior• good qualitative agreement of length scales

(„cold“ cavity theory: propagation through spacer layer and Bragg reflectors)

Patterns and tilted wavesCoordinate space (near-field)

Fourier space (far-field)

• Infinite laser: traveling wave, homogeneous emission

• Laser with boundaries: reflection creates standing wave, line pattern

• Four wave vectors: stripe-like, wavy pattern

• ... and more complex cases possible!

Spatial structures

T= -10.3°C T= 18.3°C

I = 12 mA

I = 18 mA

I = 15 mA

I = 23 mA

I = 20 mA

I = 17 mA

Other aspects: Quantum billiard• For low temperatures: patterns with a very high

wave number, well defined wave vectors• Pattern bears resemblance to trajectory of a

quantum particle in a 2d potential well

270-280K 260-270K 240-260K

Scars of the wave functions of a quantum billard

From Y.F. Chen et al, PRE 68, 026210 (20039:

Huang et al. PRL 89, 224102 (2002); Chen PRE 66, 066210 (2002)

G. Robb (Strathclyde) & co-workers

CONCLUSIONS

Some useful references and material from this talk on www.funfacs.org

Spatial patterns and cavity solitons can be found in many nonlinear optical media

Potential CS applications as pixel arrays, but more likely using their transverse mobility

Micro/nano structured media, and time domain, are interesting future directions

Spontaneous Patterns in Nonlinear Optics