Non-linear photonic crystals Resumed by: D. Simeonov PO-014 Photonic crystals

Definition Nonlinear 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 PC With periodic modulation of the non-linear material properties Modulated c(2) for quasi-phase matching (QPM) Applications: harmonic generation, wave mixing, optical parametric amplifiers etc Without periodic modulation of the non-linear material properties Non linear response due to optical Kerr effect

c(2) modulated NPC Second harmonic generation (SHG) and phase matching Quasi phase matching (QPM) Phenomenological approach Analytical approach Fabrication techniques Some devices and applications 2D QPM-NPC Natural QPM-NPC

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

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

QPM for SHG Proposed by N. Bloembergen in 1962

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

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

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

QPM generalized For 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 materials Periodic E field (via segmented electrode) + field-induced c(2) ‘Frozen-in' field-induced c(2), in optical fibers Periodic destruction/reduction of nonlinearity via ion-implantation through a mask Overgrowth on a template having periodic modulation of substrate orientation c(2) →-c(2): semiconductor materials: GaAs, GaN Periodic modulation of pump intensity (corrugated capillary waveguide for High Harmonic Generation) Periodic-poling of ferroelectrics, switching c(2) →-c(2): LiBaNO3, etc… Many more…

Fabrication of PPLN ~30 mm References: Easy to fabricate The change could be either temporary or permanent

Fabrication of PPLN SEM top view of PPLN grating 100 mm

PPLN tuning

Some results PPLN

Some results PPLN Review for different techniques:

Some results PPLN

Some results PPLN

Some results PPLN

Some results PPLN

Some results PPLN

Fabrication of GaAs QPM NPC Why GaAs? ●Large nonlinearity, d14~ 100pm /V ●Extensive transparency, 0.9 μm -17 μm ●Mature technology 1st 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 NPC Very large transparency window Low efficiency

2D QPM NPC Interesting for : Compensation of very large phase mismatches Simultaneous phase matching of several parametric processes Very broad band OPO Pioneering papers: Theory Experiment

2D QPM NPC Constant linear dielectric constant Periodically modulated c(2) constant Where r is an in-plane vector

2D QPM NPC ~ Parametric 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 NPC Reciprocal lattice (RL) representation Phase matching condition (momentum conservation law): While deff ~ kG

2D QPM NPC Nonlinear Ewald construction In 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. Gmn In 2D basis: Gmn = m Gx + n Gy Can be generalized for of plane incident light.

Observation of SHG in 2D QPM NPC Hexagonally Poled Lithium Niobate: A Two-Dimensional Nonlinear Photonic Crystal k2w - 2kw - Gmn = 0

Natural 2D QPM NPC Existence of natural structures 2D QPM NPC 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) Sr0.61Ba0.39Nb2O6 (SBN) 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 NPC Interesting but complicated analytically: Out of plane incident light Central symmetry due to the random size distribution: The G (kG) vector magnitudes are given by the domain size distribution All possible G vectors exist in all directions perpendicular to the domains

Conical SHG

c(3) NPC Definition Analytical considerations Photonic crystals with Kerr type defects Kerr effect super-prism Kerr type PC - optical response Non-linear modes, spatial optical solitons Analytical description

c(2) NPC conclusion Used 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) NPC Periodic modulation of the linear part of the refractive index as standard PC The optical response is based on that of a linear PC Dynamical switching of the optical response based on AC Kerr effect: Types: Insertion of defects exhibiting Kerr type non-linearity The material exhibits high Kerr non-linearity Studied phenomena: Switching of the properties of photonic crystal using high intensity control beam Mode self generated changes of the optical properties: soliton waves High order harmonic generation

Some literature Photonic 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 considerations One of the materials is considered non-linear: Kerr non-linearity is small: Kerr non-linearity can be considered in perturbation theory

Diversity of Kerr type defects A – Symmetric optical filter B – Asymmetric optical filter C – Optical bend D – Channel drop filter E – Waveguide branch In absence of high power excitation – standard defect response In presence of high power excitation – switched defect response due to changed refractive index

Some literature Theoretical proposals and descriptions: S. F. Mingaleev and Yu.S.Kivshar Effective equations for photonic-crystal waveguides and circuits Opt. Lett. 27, 231 (2002) M Soljacic, M Ibanescu, S G Johnson, Y Fink, and J. D. Joannopoulos Optimal bistable switching in nonlinear photonic crystals Phys. Rev. E 66, 055601R (2002) M Soljacic, C Luo, S Fan, and J. D. Joannopoulos Nonlinear photonic crystal microdevices for optical integration Opt. Lett. 28, 637 (2003) Experimental observations: Somebody should do them …

Linear Drop-off filter 2 waveguides 2 high Q factor microcavities High index rods Filing factor - 0.2 In – Out symmetric transmission given by: No power dependence

Bistable Drop-off filter Rods from Non-linear Kerr material For carrier frequency: Expected bistability of the carrier transmission due to « resonance shift » 1-4 Transmission for high intensity signal 4-3 Transmission for the reflected weak signal

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

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

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

Kerr type PC - optical response Calculated 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 NPC Temporal solitons: Kerr type PC (PC waveguide) Negative dispersion mode Spatial solitons: Can exist in almost any Kerr type PC Can design PC for their interaction Can use them for loss-less bends

Analytical description Description in coupled-mode theory Solution of the corresponding non-linear Schrödinger equation:

Some literature

Some more literature

Conclusion NPC structures offer VERY wide range of possibilities: Harmonic generations All optically tunable PC optical response Solitons and localized states Very nice theoretical approaches

Thank you for Your patience

Introduction to solitons In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. There are two main kinds of solitons: Spatial solitons: the nonlinear effect can balance the diffraction. The electromagnetic field can change the refractive index of the medium while propagating, thus creating a structure similar to a graded-index fiber. If the field is also a propagating mode of the guide it has created, then it will remain confined and it will propagate without changing its shape Temporal solitons: if the electromagnetic field is already spatially confined, it is possible to send pulses that will not change their shape because the nonlinear effects will balance the dispersion. Those solitons were discovered first and they are often simply referred as "solitons" in optics.

Anomalous (negative) dispersion Temporal solitons Anomalous (negative) dispersion + Kerr effect = Temporal soliton Can propagate without changing form Does not change during collision Can interact with other solitons