Foundations of Photonic Crystal Fibres

The following sections are included:

• Foreword by Professor Ross C. McPhedran

• Foreword by Professor Roy J. Taylor

• Contents

• Preface

The following sections are included:

• Conventional Optical Fibres
  • Guidance mechanism
  • Fibre modes
  • Main properties
    • Number of modes
    • Losses
    • Dispersion
    • Non-linearity

• Photonic Crystal
  • One dimension: Bragg mirrors
  • Photonic crystals in two and three dimensions

• Guiding Light in a Fibre with Photonic Crystals
  • Bragg fibres
  • Photonic crystal fibres and hollow core microstructured optical fibres
    • Hollow core MOFs

• Solid Core MOFs
  • Guidance mechanism
  • Main properties and applications

• Leaky Modes
  • Confinement losses
  • Modes of a leaky structure
  • Heuristic approach to physical properties of leaky modes
  • Mathematical considerations
  • Spectral considerations

The following sections are included:

• Maxwell Equations
  • Maxwell equations in vacuo
  • Maxwell equations in idealized matter
    • Mesoscopic homogenization
    • Dispersion relations – Kramers-Kronig relations

• The Mono dimensional Case (Modes, Dispersion Curves)
  • A first approach
    • A special feature of the 1D-case: the decoupling of modes
    • Physics and functional spaces
    • Spectral presentation
    • Orthogonality of modes
  • Localisation of constants of propagation
  • How can one practically get the dispersion curves and the modes?
    • Zerotic approach
  • Spectral approach
    • Strengths and weaknesses
    • The variational formulation (weak formulation)
    • An example of a Hilbert-basis in L²(ℝ): the Hermite polynomials

• The Two-Dimensional Vectorial Case (general case)
  • Some useful relations between transverse and axial components
  • Equations of propagation involving only the axial components
  • What are the special features of isotropic microstructured fibers?

• The Two-Dimensional Scalar Case (weak guidance)

• Spectral Analysis
  • Preliminary remarks
  • A brief vocabulary
  • Posing of the problem
  • Continuous formulation
  • Discrete finite element formulation

• Bloch Wave Theory
  • The crystalline structure
  • Waves in a homogeneous space
  • Bloch modes of a photonic crystal
  • Computation of the band structure
  • A simple 1D illustrative example: the Kronig-Penney model

The following sections are included:

• Finite Elements: Basic Principles
  • A one-dimensional naive introduction
  • Multi-dimensional scalar elliptic problems
    • Weak formulation of problems involving a Laplacian
    • Generalizations
    • The finite element method
  • Mixed formulations
  • Vector problems
  • Eigenvalue problems

• The Geometric Structure of Electromagnetism and Its Discrete Analog
  • Topology
  • Physical quantities
  • Topological operators
  • Metric
  • Differential complexes: from de Rham to Whitney

• Some Practical Questions
  • Building the matrices (discrete Hodge operator and material properties)
  • Reference element
  • Change of coordinates
  • Nédélec edge elements vs. Whitney 1-forms
  • Infinite domains
    • Transformation method for infinite domains
    • Perfectly Matched Layer (PML)

• Propagation Modes Problems in Dielectric Waveguides
  • Weak and discrete electric field formulation
  • Numerical comparisons
  • Variants
    • Looking for β with k₀ given
    • Discrete magnetic field formulation
    • Eliminating one component with the divergence
    • E_z, H_z formulation

• Periodic Waveguides
  • Block modes
  • The Bloch conditions
  • A numerical example
  • Direct determination of the periodic part

• Twisted Fibres

• Conclusion

The following sections are included:

• Introduction

• The Multipole Formulation
  • The geometry of the modelled microstructured optical fiber
  • The choice of the propagating electromagnetic fields
  • A simplified approach of the Multipole Method
    • Fourier-Bessel series
    • Physical interpretation of Fourier-Bessel series (no inclusion)
    • Change of basis
    • Fourier-Bessel series and one inclusion: scattering operator
    • Fourier-Bessel series and two inclusions: the Multipole Method
  • Rigourous formulation of the field identities
  • Boundary conditions and field coupling
  • Derivation of the Rayleigh Identity

• Symmetry Properties of MOF
  • Symmetry properties of modes

• Implementation
  • Finding modes
  • Dispersion characteristics
  • Using the symmetries within the Multipole Method
    • Another way to obtain ℑm(β)
  • Software and computational demands

• Validation of the Multipole Method
  • Convergence and self-consistency
  • Comparison with other methods

• First Numerical Examples
  • A detailed C_6v example: the six hole MOF
  • A C_2v example: a birefringent MOF
  • A C_4v example: a square MOF

• Six Hole Plain Core MOF Example: Supercell Point of View

• Conclusion

The following sections are included:

• Genesis of Baron Strutt's Algorithm

• Common Features Shared by Multipole and Rayleigh Methods

• Specificity of Lord Rayleigh's Algorithm

• Green's Function Associated with a Periodic Lattice

• Some Absolutely Convergent Lattice Sums

• The Rayleigh Identities

• The Rayleigh System

• Normalization of the Rayleigh System

• Convergence of the Multipole Method

• Higher-order Approximations, Photonic Band Gapsfor Out-of-plane Propagation

• Conclusion and Perspectives

The following sections are included:

• A Simple Extension: Poles of Matrices
  • Degenerate eigenvalues
  • Multiple poles inside the loop
  • Miracles sometimes happen

• Cauchy integrals for operators

• Numerical Applications

• Conclusion

The following sections are included:

• Basic Properties of the Losses

• Single-Modedness of Solid Core C_6v MOF
  • A cutoff for the second mode
  • A phase diagram for the second mode

• Modal Cutoff of the Fundamental Mode
  • Existence of a new kind of cutoff
  • A phase diagram for the fundamental mode
  • Simple physical models below and above the transition region

• Chromatic Dispersion
  • Material and waveguide chromatic dispersion
  • The influence of the number of rings Nr on chromatic dispersion
  • A more accurate MOF design procedure

• A Hollow Core MOF with an Air-Guided Mode
  • The photonic crystal cladding
  • The finite structure

• Conclusion

We believe that the terminology “new optical waveguides” is fully justified to qualify microstructured optical fibres (MOFs) even though leaky modes were already encountered in conventional W-fibres. MOF unique properties such as single-mode guidance (now clearly defined) and photonic band gap propagation (revisited for finite structures) make them much more than a simple generalization of conventional optical fibres…

The following sections are included:

• Appendix A: A Formal Framework for Mixed Finite Element Methods

• Appendix B: Some Details of the Multipole Method Derivation
  • Derivation of the Wijngaard Identity
  • Change of Basis
    • Cylinder to cylinder conversion
    • Jacket to cylinder conversion
    • Cylinder to jacket conversion
  • Boundary Conditions: Reflection Matrices

• Appendix C: A Pot-Pourri of Mathematics

• Bibliography

• Index