Presentation in the frame ofPresentation in the frame ofPhotonic Crystals coursePhotonic Crystals course

by R. Houdreby R. Houdre

Photonic Crystal FibersPhotonic Crystal Fibers

Georgios Violakis

EPFL, Lausanne June 2009

Outline

Introduction to Photonic Crystal Fibers

Properties of Microstructured Optical Fibers

Applications of PCFs

Fiber types / classification

Properties of Photonic Bandgap Fibers

Common Fabrication Techniques

Modeling of Photonic Crystal Fibers

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

Fiber core)/arcsin( 12 nnc

Photonic Crystal Fibers

2 main classes of PCFs

High Index core Fibers Photonic Bandgap Fibers

High N.A. Highly non linear

Large Mode Area

Low Index Core

Bragg FiberHollow

Core

High Index Core Fibers

Light guidance by means of modified total internal reflection.

High Index Fiber core

Low index capillaries, e.g. air channels

High index material, e.g. silica glass

Introduction of low index (e.g. air) capillaries in the “cladding” area, effectively reduces the refractive index of the core surrounding area, allowing TIR

Photonic Bandgap Fibers

“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

In most cases the core has a lower refractive index than the cladding area

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:

By adjusting the geometrical features of the fibers one can adjust the light propagation properties from highly linear performance to highly non-linear propagation

a) Geometry of the fiber

b) Core/cladding/defect materials

Λ

d

d/Λ typically varies between a few % - 90%

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

Properties of Microstructured Optical Fibers II

In standard optical fibers the number of modes supported is calculated by:

“Effective” refractive index of cladding is wavelength dependant

2222 2clcoclcoeff nnannakV

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

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)

Endlessly Single Mode Fibers

222fsmcoeff nkV

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

Triangular hole structureΛ = 2.3μm, various d

Larger pitch results in reduced dispersion for fixed λ and d

Cladding morphology has a great effect on dispersion properties

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

2nd forbidden frequency domain: The four narrow bands

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 kz/neq ≥ ω/c (=kfree space) the wave propagating in the core is confined there (no refraction)

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:

Core determination by the above equation for desired number of modes

NPBG is the number of PBG-guided modes, Deff: effective core diameter for PC, βL is the lower propagation constant of a given PBG

4

)2/)(( 222,

2effLcoeff

PBG

DnkN

Properties of Photonic Bandgap Fibers III

Losses decrease exponentially with the number of air hole rings in the cladding

Losses

Higher leakage for first two core geometries

For hollow core fibers it is also crucial the shape of the core

DispersionAnd area of mode

Theoretically predicted attenuation: 0.13dB/km at 1.9μm

Experimentally measured attenuation : 1.2dB/km at 1.62μm

d/Λ ration also important as well as the air-silica filling ration

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

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)

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 inducebirefringence in the fiber (two polirazationstates experience different β/k values)

Modeling of PCFs IThe 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)

Determination of neff

Analogy to step index fibers and use of calculation tools readily available

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:

fsmeffn k

Modeling of PCFs IThe effective index approach II

fsmeffn 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

232

21

1 i

i i

An

B

Simple

Minimum computational requirements

Qualitative method

Cannot compute photonic bandgaps

Modeling of PCFs IIPlane-wave expansion method I

( , ) ( )

( , ) ( )

j t

j t

E r t E r e

H r t H r e

( ) ( )

( ) ( )

j k rk

j k rk

E r V r e

H r U r e

First theoretical method to accurately analyze photonic crystals

Takes advantage of the cladding periodicity:

Bloch’s

theorem

V and U in reciprocal space Fourier expansion in terms of the reciprocal lattice vectors G

Fourier transformation

Maxwell’s equations

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)

0

1( ) ( )

( )r

E r H rj r

( )

( )

( ) ( )

( ) ( )

j k G rk

G

j k G rk

G

E r E G e

H r H G e

Modeling of PCFs IIPlane-wave expansion method II

)3

3(

2

)3

3(

2

2

1

yxG

yxG

)3(2

)3(2

2

1

yxR

yxR

2 dimensional photonic crystals with hexagonal symmetry

R1, R2: real space primitive lattice vectorsG1, G2: reciprocal lattice vectors

Solutions for k vectors restricted in the 1st Brillouin zone

jiji GR ,2

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 IIIMultipole method I

mll

elm

lml

em

lmz zjjmrkHbrkJaE )exp()exp()]()([ )()()(

mll

im

lmz zjjmrkJcE )exp()exp()]([ )(

2220 e

e nkk

Method used to calculate confinement losses in PCFs

220

2i

i nkk )(lma

)(lmb

)(lmc

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:

with being the transverse wave number in silica

Inside the cylinder where ni=1, Ez is:

where

Application of Boundary conditions:

Modeling of PCFs IIIMultipole 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 IIIMultipole method IIΙ

Calculates confinement loss Computational intensive

Does not require symmetrical boundary conditions

Does not make the assumption that the cladding area is infinite

Cannot analyze arbitrary cladding configurations (applies only for circular

holes)

Modeling of PCFs IVFourier 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 Improved estimate of neff Iterations

Leakage loss prediction Requires adjustable boundary condition

Modeling of PCFs VFinite Difference method I

z

E

y

E

t

H yzx

1

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:

n

jiy

n

jiz

n

jiz

ji

n

jix

n

jixEj

y

EEtHH

,

,1,

,

2

1

,2

1

,

n: discrete time stepi,j: discretized mesh point

Δt: time incrementΔx, Δy: intervals between 2

neighboring grid points

Modeling of PCFs VFinite Difference method II

Finite Difference Time Domain method

Boundary conditions using in most cases the Perfeclty Matched Layer

(PML) technique

Artificial initial field distribution -> non physical components disappear in the

time evolution and physical components (guided modes) remain

Fields in time domain Fields in frequency domainFourier transformation

General approach Requires detailed treatment of boundaries

Describes variety of structuresComputationally intensive

Modeling of PCFs VIFinite 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 VIFinite 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 Complex definition of calculation mesh

Accurate modal description Can become computationally

intensive

Modeling of PCFs VIIOther methods

General approach, well tested, analyses any structure

Computationally very intensive, detailed boundary conditions

Finite Difference Frequency Domain

Beam propagation method

Equivalent Averaged Index method

Reliable method, can use complex propagation constant

Also computationally intensive

Simple and efficient (fast method) Qualitative results

Core diameter: 12μm

Holey region diameter: 60μm

Cladding diameter: 125μm

ESM-12-01 Blaze photonics (Crystal Fibre A/S)

Modeling examples of two PCFs

LMA-10 Crystal Fibre A/S

Mode field calculations using the multipole method

Calculation of the fundamental mode using the freely available CUDOS-MOF tools which are based on the multipole method

http://www.physics.usyd.edu.au/cudos/mofsoftware/

White holes represent air holes and blue background the silica matrix

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!