presentation in the frame of photonic crystals course by r. houdre photonic crystal fibers georgios...
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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!