# Non-linear photonic crystals

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Non-linear photonic crystalsResumed by: D. Simeonov

PO-014 Photonic crystals

DefinitionNonlinear photonic crystals (NPC) are periodic structures whose optical response depends on the intensity of the optical field that propagates into the crystal. At low light densities:At high light densities:

Types of non-linear response in PCWith periodic modulation of the non-linear material propertiesNon linear response due to optical Kerr effect Modulated c(2) for quasi-phase matching (QPM)Applications: harmonic generation, wave mixing, optical parametric amplifiers etcWithout periodic modulation of the non-linear material properties

c(2) modulated NPCSecond harmonic generation (SHG) and phase matchingQuasi phase matching (QPM)Phenomenological approachAnalytical approachFabrication techniquesSome devices and applications2D QPM-NPCNatural QPM-NPC

SHGWhere 2deff = c(2) Non-linear polarization:Second harmonic polarization:Second harmonic polarization (vectorial representation):

SHGCoherence length:Dk=0SHG gained over the traveled distance (l):

QPM for SHGProposed by N. Bloembergen in 1962

QPM for SHGMaximal efficiency for 50/50 duty cycle and: The effective efficiency is reduced by factor of p/2

QPM for SHGWhereSecond harmonic of the electric field:c(2) susceptibility in Fourier representation:

QPM for SHGQPM when Dk=0After integration:The lattice reciprocal vectors can help for momentum conservation

QPM generalizedFor any frequency conversion process in media with periodic c(2) it can be generalized:Energy conservation law:Momentum conservation law:Such formalism can be derived for both 1D, 2D or 3D QPM-NPC crystals

Theory details

Some benefits of QPM

Methods and materialsPeriodic E field (via segmented electrode) + field-induced c(2)Frozen-in' field-induced c(2), in optical fibersPeriodic destruction/reduction of nonlinearity via ion-implantation through a maskOvergrowth on a template having periodic modulation of substrate orientation c(2) -c(2): semiconductor materials: GaAs, GaNPeriodic modulation of pump intensity (corrugated capillary waveguide for High Harmonic Generation)Periodic-poling of ferroelectrics, switching c(2) -c(2): LiBaNO3, etcMany more

Fabrication of PPLN~30 mmEasy to fabricateThe change could be either temporary or permanentReferences:

Fabrication of PPLN100 mmSEM top view of PPLN grating

PPLN tuning

Some results PPLN

Some results PPLNReview for different techniques:

Some results PPLN

Some results PPLN

Some results PPLN

Some results PPLN

Some results PPLN

Fabrication of GaAs QPM NPCWhy GaAs?Large nonlinearity, d14~ 100pm /VExtensive transparency, 0.9 m -17 mMature technology1st proposition stacking thin plates (wafers):A. Szilagyi, A. Hordvik, and H. Schlossberg, A quasi-phase matching technique for efficient optical mixing and frequency doubling, J. Appl. Phys., vol. 47, pp. 2025-2032, (1976) (2-5 plates, m = 3).2nd proposition growth inversion:Ex: O. Levi et al Optics Lett. 27, 2091, (2002)

Fabrication of GaAs QPM NPC

Some results on GaAs QPM NPC

GaN QPM NPCVery large transparency windowLow efficiency

2D QPM NPCInteresting for :Compensation of very large phase mismatchesSimultaneous phase matching of several parametric processesVery broad band OPOPioneering papers:TheoryExperiment

2D QPM NPCConstant linear dielectric constantPeriodically modulated c(2) constantWhere r is an in-plane vector

2D QPM NPCParametric process (SHG) in 2D:The periodically modulated c(2) constant can be represented as a Fourier series: Where G are the available vectors from the reciprocal lattice (RL), and kG is its corresponding Fourier coefficient

2D QPM NPCPhase matching condition (momentum conservation law):While deff ~ kGReciprocal lattice (RL) representation

2D QPM NPCNonlinear Ewald constructionIn the RL space:Draw 2.kw in the right direction finishing at an origin;Draw a circle with center Ce.s.;Where the circle passes trough an origin successful phase matching is possible.GmnIn 2D basis:Gmn = m Gx + n GyCan be generalized for of plane incident light.

Observation of SHG in 2D QPM NPCHexagonally Poled Lithium Niobate: A Two-Dimensional Nonlinear Photonic Crystalk2w - 2kw - Gmn = 0

Natural 2D QPM NPCExistence of natural structures 2D QPM NPCSr0.61Ba0.39Nb2O6 (SBN) At a Currie temperature the SBN crystal exhibit a phase transition to form random size (given distribution) of needle like domains with opposite sign c(2)Such crystals are natural 2D QPM NPC and for:Where p(L) is the probability of existence of domain size L=G/p

SHG in natural 2D QPM NPC

SHG in natural 2D QPM NPCInteresting but complicated analytically:Out of plane incident lightCentral symmetry due to the random size distribution:The G (kG) vector magnitudes are given by the domain size distributionAll possible G vectors exist in all directions perpendicular to the domains

Conical SHG

c(3) NPCDefinitionAnalytical considerationsPhotonic crystals with Kerr type defectsKerr effect super-prismKerr type PC - optical response

Non-linear modes, spatial optical solitonsAnalytical description

c(2) NPC conclusionUsed for assure the momentum conservation law for various non-linear parametric processes

Experimental techniques demonstrated it utility

Widely used and commercially available

A Fourier representation of c(2) gives both the available vectors in the reciprocal space and the efficiency coeficients

c(3) NPCDynamical switching of the optical response based on AC Kerr effect:Periodic modulation of the linear part of the refractive index as standard PCThe optical response is based on that of a linear PCTypes:Insertion of defects exhibiting Kerr type non-linearityThe material exhibits high Kerr non-linearityStudied phenomena:Switching of the properties of photonic crystal using high intensity control beamMode self generated changes of the optical properties: soliton wavesHigh order harmonic generation

Some literaturePhotonic Crystals with Kerr nonlinear effects: Existence of stable nonlinear localized modes in 2D & 3D PC S.John et al., PRL, 71 1168 (1993) Controlling transmission in 1D PC M.Scalora et al., PRL, 73 1368 (1994), P.Tran , Opt. Lett, 21 1138 (1996)Nonlinear guiding modes in 2D PC A.R. McGurn, Phys. Lett. A, 251 322 (1999) Tunable microcavity for fast switching P.R. Villeneuve, Opt. Lett., 21 2017 (1996)

Analytical considerationsKerr non-linearity is small:One of the materials is considered non-linear:Kerr non-linearity can be considered in perturbation theory

Diversity of Kerr type defectsA Symmetric optical filterB Asymmetric optical filterC Optical bendD Channel drop filterE Waveguide branchIn absence of high power excitation standard defect responseIn presence of high power excitation switched defect response due to changed refractive index

Some literatureS. F. Mingaleev and Yu.S.KivsharEffective equations for photonic-crystal waveguides and circuitsOpt. Lett. 27, 231 (2002)M Soljacic, C Luo, S Fan, and J. D. JoannopoulosNonlinear photonic crystal microdevices for optical integrationOpt. Lett. 28, 637 (2003)M Soljacic, M Ibanescu, S G Johnson, Y Fink, and J. D. JoannopoulosOptimal bistable switching in nonlinear photonic crystalsPhys. Rev. E 66, 055601R (2002)Theoretical proposals and descriptions:Experimental observations:Somebody should do them

Linear Drop-off filter2 waveguides2 high Q factor microcavitiesHigh index rodsFiling factor - 0.2In Out symmetric transmission given by:No power dependence

Bistable Drop-off filter1-4 Transmission for high intensity signal4-3 Transmission for the reflected weak signalRods from Non-linear Kerr materialFor carrier frequency:Expected bistability of the carrier transmission due to resonance shift

Bistable Drop-off filterNon-linear transmission:Where P0 is a characteristic power of the process

Feasibility of Bistable Drop-off filterDesign parameters:n2 = 1.5x10-17 m2/W (for GaAs n2 = 3x10-16 m2/W)Q = 4000 (compatible with 10 Gbit/s)l0 = 1.55 mmRequired conditions:P0 = 15 mWWorking power 25 mW

Kerr effect super-prismOptically tunable superprism effect in nonlinear photonic crystals, N. - C. Panoiu, M. Bahl, and R. M. Osgood, Jr., Opt. Lett. 28, 2503 (2003). GaAs-based PC slab:Kerr coefficient n2 = 3x10-16 m2/W. r/a 0.33Dependence of the diffraction angle on the signal powerControllable diffraction angle via pump pulse

Kerr type PC - optical responseCalculated band structure of 1D GaAs air PC (air gap DBR)Solid curves without switch beam

Dashed curves with intense switch beam

Kerr type PC - optical response

Solitons in NPCTemporal solitons:Kerr type PC (PC waveguide)Negative dispersion modeSpatial solitons:Can exist in almost any Kerr type PCCan design PC for their interactionCan use them for loss-less bends

Analytical descriptionSolution of the corresponding non-linear Schrdinger equation:Description in coupled-mode theory

Some literature

Some more literature

ConclusionNPC structures offer VERY wide range of possibilities:Harmonic generationsAll optically tunable PC optical responseSolitons and localized states Very nice theoretical approaches

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