1. Photonic Crystal Fibers
Presentation in the frame of
Photonic Crystals course
by R. Houdre
Photonic Crystal Fibers
Georgios Violakis
EPFL, Lausanne
June 2009

2. 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

3. 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

4. Photonic Crystal Fibers
2 main classes of PCFs
High Index core Fibers
Photonic Bandgap Fibers
High N.A.
Low Index Core
Bragg Fiber
Highly non linear
Large Mode Area
Hollow Core

5. 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

6. 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

7. Fabrication Techniques I
a) Preparation of each capillary
b) Assembly of capillaries to the desired structure
c) Preparation of the preform
d) Fiber drawing

8. 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…

9. 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 typically varies between 1 – 20 μm Λ
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

10. 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
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

11. 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
Larger pitch results in reduced dispersion for fixed λ and d
Triangular hole structure
Λ = 2.3μm, various d

12. 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 kz/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

13. 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:
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

14. 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

15. 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

16. 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)

17. 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:

18. Modeling of PCFs I The effective index approach II
nco = nsilica
ncl = neff
rcore = 0.5*Λ or *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
Qualitative method
Minimum computational requirements
Cannot compute photonic bandgaps

19. Modeling of PCFs II Plane-wave expansion method I
First theoretical method to accurately analyze photonic crystals
Bloch’s
theorem
Takes advantage of the cladding periodicity:
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)

20. Modeling of PCFs II Plane-wave expansion method II
2 dimensional photonic crystals with hexagonal symmetry
R1, R2: real space primitive lattice vectors
G1, G2: reciprocal lattice vectors
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
Calculates PBGs
Unsuitable for large structures
Good agreement with experiments
Unsuitable for full PCF analysis
Solution of E and H
Widely used

21. 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:
with being the transverse wave number in silica
Inside the cylinder where ni=1, Ez is:
Application of
Boundary conditions:
where

22. 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

23. Modeling of PCFs III Multipole method IIΙ
Calculates confinement loss
Computational intensive
Does not require symmetrical boundary conditions
Cannot analyze arbitrary cladding configurations (applies only for circular holes)
Does not make the assumption that the cladding area is infinite

24. 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
Iterations
Improved estimate of neff
Requires adjustable boundary condition
Leakage loss prediction

25. 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:
n: discrete time step i,j: discretized mesh point
Δt: time increment
Δx, Δy: intervals between 2 neighboring grid points

26. Modeling of PCFs V Finite Difference method II
Finite Difference Time Domain method
Artificial initial field distribution -> non physical components disappear in the time evolution and physical components (guided modes) remain
Boundary conditions using in most cases the Perfeclty Matched Layer (PML) technique
Fourier transformation
Fields in time domain
Fields in frequency domain
General approach
Requires detailed treatment of boundaries
Describes variety of structures
Computationally intensive

27. 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

28. 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
Complex definition of calculation mesh
Accurate modal description
Can become computationally intensive

29. 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
Qualitative results
Simple and efficient (fast method)

30. Modeling examples of two PCFs
ESM Blaze photonics (Crystal Fibre A/S)
Core diameter: 12μm
Holey region diameter: 60μm
Cladding diameter: 125μm
LMA-10 Crystal Fibre A/S

31. 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
White holes represent air holes and blue background the silica matrix

32. 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

33. 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

34. 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)

35. Thank you!