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Presentation in the frame of Photonic Crystals course by R. Houdre Photonic Crystal Fibers Georgios Violakis EPFL, Lausanne June 2009 Outline Introduction to Photonic Crystal Fibers Fiber types / classification Common Fabrication Techniques Properties of Microstructured Optical Fibers Properties of Photonic Bandgap Fibers Modeling of Photonic Crystal Fibers Applications of PCFs Optical Fibers An optical fiber is a glass structure specially designed in order to efficiently guide light along its length (long distances) Step Index Optical Fibers Light guidance by means of total internal reflection. Widely utilized in telecommunications Polymer jacket Fiber cladding c arcsin(n2 / n1 ) Fiber core Photonic Crystal Fibers 2 main classes of PCFs High Index core Fibers High N.A. Large Mode Area Highly non linear Photonic Bandgap Fibers Low Index Core Hollow Core Bragg Fiber High Index Core Fibers High Index Fiber core Light guidance by means of modified total internal reflection. Introduction of low index (e.g. air) capillaries in the “cladding” area, effectively reduces the refractive index of the core surrounding area, allowing TIR Low index capillaries, e.g. air channels High index material, e.g. silica glass Photonic Bandgap Fibers In most cases the core has a lower refractive index than the cladding area “True” photonic crystal fibers. Light guidance by means of light trapping in the core, due to the photonic bandgap zones of the “cladding” Cladding structure must be able to exhibit at least one photonic bandgap at the frequency of interest Fabrication Techniques I a) Preparation of each capillary b) Assembly of capillaries to the desired structure c) Preparation of the preform d) Fiber drawing Fabrication Techniques II Variations of technique depending on the preform material Chalcogenide fibers Polymer fibers Compound glass fibers Variations of technique depending on the fiber layout Honeycomb structure Hollow core fibers etc… Properties of Microstructured Optical Fibers I Optical properties affected by: d a) Geometry of the fiber b) Core/cladding/defect materials d/Λ typically varies between a few % - 90% Λ Λ typically varies between 1 – 20 μm Core size usually between 5 – 20 μm Core usually made of the same material as the cladding (high quality fused silica), but in some cases it can contain dopants By adjusting the geometrical features of the fibers one can adjust the light propagation properties from highly linear performance to highly non-linear propagation Properties of Microstructured Optical Fibers II In standard optical fibers the number of modes supported is calculated by: 2 Veff k a nco ncl2 2 2 a nco ncl2 “Effective” refractive index of cladding is wavelength dependant 2 2 Veff k 2 nco fsm As the frequency is increased, the effective index of the cladding ncl is approaching nco and equation Veff can reach a stationary value, determined by the d/Λ ratio Endlessly Single Mode Fibers Possibility to design a fiber with d/Λ below a certain value, ensuring that the Veff value does not exceed the second order mode cutoff value over the desired wavelength range (dashed line) Properties of Microstructured Optical Fibers III Dispersion properties Dispersion is calculated using the full vectorial plane wave approximation Possible to have broadband near zero dispersion flattened behavior Cladding morphology has a great effect on dispersion properties Triangular hole structure Λ = 2.3μm, various d Larger pitch results in reduced dispersion for fixed λ and d Properties of Photonic Bandgap Fibers I Optical properties affected by: a) Geometry of the fiber b) Core/cladding/defect materials Numerical methods applied to achieve bandgap diagram 1st forbidden frequency domain: ω/c = kz/neq where neq: equivalent index of silica + holes and it is λ dependant Grey area corresponds to the classical guiding in fibers by TIR for which as long as k z/neq ≥ ω/c (=kfree space) the wave propagating in the core is confined there (no refraction) 2nd forbidden frequency domain: The four narrow bands caused by the photonic crystal structure and are associated with Bragg reflections Properties of Photonic Bandgap Fibers II Core – cladding design The cladding must exhibit photonic bandgaps that cross the air line (requirement for hollow core fibers) Number of modes in the core region: N PBG 2 2 2 (k 2 neff )( D / 2 ) ,co L eff 4 NPBG is the number of PBG-guided modes, Deff: effective core diameter for PC, βL is the lower propagation constant of a given PBG Core determination by the above equation for desired number of modes Properties of Photonic Bandgap Fibers III Losses Losses decrease exponentially with the number of air hole rings in the cladding For hollow core fibers it is also crucial the shape of the core Dispersion And area of mode Higher leakage for first two core geometries d/Λ ration also important as well as the air-silica filling ration Theoretically predicted attenuation: 0.13dB/km at 1.9μm Experimentally measured attenuation : 1.2dB/km at 1.62μm Properties of Photonic Bandgap Fibers IV Dispersion Λ = 1.0μm, dcl = dco = 0.40Λ Λ = 2.3μm, dcl = 0.60Λ Anomalous dispersion can be used for dispersion management (dispersion compensation in optical transmission links) By adjusting core size and cladding properties it is possible to achieve broadband, near zero dispersion flattened behavior Properties of Photonic Bandgap Fibers V Special properties By inducing “defects” in the cladding area (for example a change of size of two of the holes in the first ring outside the core area) it is possible to induce birefringence in the fiber (two polirazationstates experience different β/k values) Possibility to design fibers with the second order mode confined and the fundamental leaky (mode propagation manipulation – sensing) Simulations reveal the presence of ring shaped resonant modes between the core-cladding interface (issue of ongoing research) Modeling of PCFs I The effective index approach I Simple numerical tool Evaluates the periodically repeated cladding structure an replaces it with an neff. Core refractive index usually same as matrix material (e.g. fused silica) Analogy to step index fibers and use of calculation tools readily available Determination of neff Determination of cladding mode field, Ψ, by solving the scalar wave equation within a simple cell centered on one of the holes Approximation by a circle to facilitate calculations Application of boundary conditions (dΨ/ds)=0 Propagation constant of resutling fundamental mode, βfsm used in: neff fsm k Modeling of PCFs I The effective index approach II neff fsm k nco = nsilica ncl = neff rcore = 0.5*Λ or 0.62*Λ Full analogy to a step index fiber realized Use of tools for step index fibers Refractive index in matrix material can be also described as being wavelength dependent using the Sellmeier formula Simple Minimum computational requirements 2 A n2 1 2 i i 1 Bi Qualitative method Cannot compute photonic bandgaps 3 Modeling of PCFs II Plane-wave expansion method I First theoretical method to accurately analyze photonic crystals Takes advantage of the cladding periodicity: E (r , t ) E (r )e jt Bloch’s E ( r ) V k ( r ) e j k r H (r , t ) H (r )e jt theorem H ( r ) U k ( r ) e j k r V and U in reciprocal space Fourier expansion in terms of the reciprocal lattice vectors G E (r ) E k (G )e j ( k G )r Fourier transformation H (r ) H k (G )e j ( k G )r Maxwell’s equations G G Wave equation in the reciprocal space Can be re-written in matrix form and solved using standard numerical routines as eigenvalue problems Once the wave equation has been solved for one of the fields (e.g. H) E (r ) 1 H (r ) j r (r ) 0 Modeling of PCFs II Plane-wave expansion method II 2 dimensional photonic crystals with hexagonal symmetry (x y 3) 2 R2 (x y 3) 2 R1 2 (x y 2 G2 (x y G1 R1, R2: real space primitive lattice vectors G1, G2: reciprocal lattice vectors Ri G j 2i , j 3 ) 3 3 ) 3 Solutions for k vectors restricted in the 1st Brillouin zone Calculation of the εr-1(G) which is required to set up the matrix equation Solution of E and H Calculates PBGs Good agreement with experiments Widely used Unsuitable for large structures Unsuitable for full PCF analysis Modeling of PCFs III Multipole method I Method used to calculate confinement losses in PCFs Similar to other expansion methods, but: uses many expansions, one for each of the fiber holes in the fiber cladding Does not require periodicity Calculation of complex propagation constant (confinement losses) Around a cylinder l the longitudinal E-field component Ez is: Ez (l ) e (l ) (l ) e [ a J ( k r ) b H ( k m m l m m rl )]exp( jml ) exp( jz) m with ke k02 ne2 2 being the transverse wave number in silica Inside the cylinder where ni=1, Ez is: Ez (l ) i [ c J ( k m m rl )]exp( jml ) exp( jz) m i 2 2 2 where k k0 ni Application of Boundary conditions: am(l ) bm(l ) cm(l ) Modeling of PCFs III Multipole method II In order to describe leaky modes, cladding is surrounding by jacketing material with nj = ne-jδ, δ<<1 Without jacket, expansions lead to fields that diverge far away from the core, because the modes are not completely bound Confinement loss determined by the multipole method. Λ = 2.3μm, λ=1.55μm Modeling of PCFs III Multipole method IIΙ Calculates confinement loss Does not require symmetrical boundary conditions Does not make the assumption that the cladding area is infinite Computational intensive Cannot analyze arbitrary cladding configurations (applies only for circular holes) Modeling of PCFs IV Fourier decomposition method Calculates confinement losses in PCFs that do not have circular holes Computational domain D with radius of R is used to encapsulate the centre of the waveguide Mode field inside D is expanded in basis functions Polar-coordinate harmonic Fourier decomposition of the basis functions Initial guess of neff Leakage loss prediction Iterations Improved estimate of neff Requires adjustable boundary condition Modeling of PCFs V Finite Difference method I Finite Difference Time Domain method Maxwell’s equations can be discretized in space and time (Yee-cell technique) Field components of the mesh could be the discrete form of x-component of Maxwell’s first curl equation: H x 1 E E y z t y z H 1 n 2 x i, j H 1 n 2 x i, j t E z i , j n Ez i , j 1 y n i, j n j E y i, j n: discrete time step i,j: discretized mesh point Δt: time increment Δx, Δy: intervals between 2 neighboring grid points Modeling of PCFs V Finite Difference method II Finite Difference Time Domain method Boundary conditions using in most cases the Perfeclty Matched Layer (PML) technique Fields in time domain Artificial initial field distribution -> non physical components disappear in the time evolution and physical components (guided modes) remain Fourier transformation General approach Fields in frequency domain Requires detailed treatment of boundaries Describes variety of structures Computationally intensive Modeling of PCFs VI Finite Element method I The most generally used method for various physical problems Method has been used for the analysis of standard step index fibers and it was later (2000) applied for photonic crystal fibers Maxwell’s differential equations are solved for a set of elementary subspaces Subspaces are considered homogenous (mesh of triangles or quadrilaterals) Maxwell’s equations applied for each element Boundary conditions (continuity of the field) neff, E- and H- field can be numerically calculated Modeling of PCFs VI Finite Element method II Propagation mode results indicate that modes exhibit at least two symmetries Introduction of Electric and Magnetic Short Circuit. Study of ¼ of the fiber area – decrease in computational time Reliable (well-tested) method Accurate modal description Complex definition of calculation mesh Can become computationally intensive Modeling of PCFs VII Other methods Finite Difference Frequency Domain General approach, well tested, analyses any structure Computationally very intensive, detailed boundary conditions Beam propagation method Reliable method, can use complex propagation constant Also computationally intensive Equivalent Averaged Index method Simple and efficient (fast method) Qualitative results Modeling examples of two PCFs ESM-12-01 Blaze photonics (Crystal Fibre A/S) Core diameter: 12μm Holey region diameter: 60μm Cladding diameter: 125μm LMA-10 Crystal Fibre A/S Mode field calculations using the multipole method Calculation of the fundamental mode using the freely available CUDOSMOF tools which are based on the multipole method White holes represent air holes and blue background the silica matrix http://www.physics.usyd.edu.au/cudos/mofsoftware/ Mode field calculations using the FDTD method Calculation of the fundamental mode using commercially available FDTD software. (OptiFDTD) Higher order modes, though calculated, are leaky and are not supported by the fiber which is endlessly single mode http://www.optiwave.com/ Mode field calculations using the FEM method Calculation of the fundamental mode using commercially available FEM software. (COMSOL multiphysics) Higher order modes were not found to be supported for this kind of optical fiber http://www.comsol.com/ Photonic Crystal Fiber Applications Light guidance for λ that silica strongly absorbs (IR range) High power delivery Gas-filling the core (sensing, non-linear processes) Gas-lasers (hollow core) / Fiber lasers (doped core) In-fibre tweezers (nanoparticle transportation in the hollow core) Tunable sensors (liquid crystals in PCFs) Thank you!