Spontaneous Patterns in Nonlinear Optics Netwon Institute, Cambridge August 2005 William J Firth Department of Physics, University of Strathclyde, Glasgow, Scotland Acknowledgements Thorsten Ackemann, Gonzague Agez + many colleagues/collaborators Funding FP6 FunFACS Leverhulme Trust 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 2 nd Harmonic Generation nonlinearity very fast, but very weak Type II phase matching for SHG in KTP fundamental only input input beam ellipticity of 11:1 input peak intensity of 57 GW/cm 2 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 NLSE Envelope 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/l D, 1/ .

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 Crystal Lille 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 instability phase modulation infinitesimal fluctuation macroscopic modulation: a pattern  n = n (|E|) positive feedback ? fluctuation of refractive index homo- geneous phase and amplitude d modulation diffraction 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 ) Institut für Angewandte Physik Westfälische Wilhelms-Universität Münster T. Ackemann, M. Pesch, F. Huneus, J. Schurek, E. Schöbel, W. Lange Department of Physics University of Strathclyde Glasgow, Scotland, UK

Feedback mirror patterns in Na vapour medium: sodium vapor in nitrogen buffer gas pumping: in vicinity of D 1 -line nonlinearity: optical pumping between Zeeman sub-levels

Theoretical model m j =1/2m j =-1/2 2 P 1/2 2 S 1/2 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 pumpingsaturation precession thermal diffusion periodic patterns quasiperiodic boundarysolitonsspirals

Length scales 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 System Schäpers et al PRL (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 Solitons Ultanir et al, PRL (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

Nonlinearity sometimes N(E), but more usually through optical excitation of a medium within the cavity Optical Cavity Basics Reflected beam Incident beam Back mirror 6  m 5 cm Substrate Diffraction described by transverse Laplacian External field drives cavity close to resonance (  = 

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 Using exact numerical techniques, we have traced existence and stability of stripes as a function of wavevector and driving. Unstable to hexagons Unstable to hexagons. Zig-zag unstable Zig-zag unstable. Eckhaus unstable Eckhaus 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 (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 FunFACS 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 Patterns Coullet et al (PRL 84, ) 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.

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 Driving CS are usually composite light/excitation structures CS are usually composite light/excitation structures can create/destroy CS through the excitation component can create/destroy CS through the excitation component why not DRIVE CS through the excitation? why not DRIVE CS through the excitation? such a drive incoherent such a drive incoherent e.g. current drive - Cavity Soliton Laser e.g. current drive - Cavity Soliton Laser basis of new FunFACS EU project basis of new FunFACS EU project In Nice VCSEL experiment (left), CS were created and destroyed with a coherent address pulse, resp. in and out of phase with E in. In other systems switching (both on and off) has been achieved with incoherent 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) Reflected 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 k q k  k eff 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 scales experiment 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 waves Coordinate 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°CT= 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 K K K Scars of the wave functions of a quantum billard From Y.F. Chen et al, PRE 68, (20039: Huang et al. PRL 89, (2002); Chen PRE 66, (2002)

G. Robb (Strathclyde) & co-workers

CONCLUSIONS Some useful references and material from this talk on 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