Foundations of Photonic Crystal Fibres -Second Edition, June 2012-:
website of the book

Introduction

The enhanced 2nd edition of our book Foundations of Photonic Crystal Fibres published by ICP is now available. It contains two new chapters including one dedicated to fabrication issues, and several others have been enlarged especially the one dealing with microstructured optical fibres properties. It contains 552 pages and several color figures. It is an hardcover book.

ISBN : 978-1-84816-728-5
ISBN 10 : 1-84816-728-8

Purposes of these web pages

This web site is intended to complete our book Foundations of Photonics Crystals Fibres 2nd Edition, it will contains errata, corrections, and also references which are missing in the present version of our book. Please do not hesitate to send an email if you have spotted an error or a typo in the book, it will be added to the corresponding list in this web site.

Web pages and emails of the authors

Gilles Renversez (Aix-Marseille Université & Institut Fresnel), email
Boris Kuhlmey (CUDOS & & University of Sydney, Australia), email
Sébastien Guenneau (Aix-Marseille Université & Institut Fresnel), email
André Nicolet (Aix-Marseille Université & Institut Fresnel)
Frédéric Zolla (Aix-Marseille Université & Institut Fresnel)
Didier Felbacq (Université de Montpellier II & Laboratoire Charles Coulomb)
Alexander Argyros (CUDOS & University of Sydney, Australia)
Sergio Leon-Saval (CUDOS & University of Sydney, Australia)

Our laboratories

Acknowledgements

We would like to thank our four distinguished colleagues who have kindly agreed to preface our book. In order of appearance:

Ross McPhedran, fellow of Australian Academy of Science, for giving us the honour of prefacing the first version of this book and also for all the friendly and inspiring conversations on the microstructured fibres and on wide-ranging subjects even including Shakespeare. This book has been made possible due to the Marseille-Sydney collaboration.

Roy J. Taylor, Head of Femtosecond Optics group at Imperial College London who accepted to preface the first version of Foundations of Photonic Crystal ``off the cuff''. Interestingly, his name made a disappearance act in the first printed version, which may be attributed to his proximity with Sir John Pendry's office...

The two versions of this book benefited from stimulating discussions with members of the Royal College of Science. Richard Craster, Head of Applied Mathematics and Mathematical Physics, was one of the first to make an acute observation regarding the strong interplay between the emerging topic of photonic crystal fibres and the development of modern mathematical tools for numerical and theoretical models of structured materials such as transformational optics, anomalous dispersion, leaky modes, ...

Finally, the Fourth Man is Fetah Benabid who is reknowned for his works on hollow core PCFs, and who now is lucky enough to share his time between Bath in England and Limoges in France. He has kindly agreed to write a good word on our prose.

We would also like to thank the host of people all of whom have made valuable contributions to different aspects of the book, some of which are mentioned below :

Our colleagues and friends from the ARC Centre for Ultrahigh Bandwidth Devices for Optical Systems (CUDOS) at the University of Sydney and University of Technology, Sydney, who developed the Multipole Method for MOFs with us. We are particularly indebted to Tom White who was audacious enough to dive into the -in these early days still quite hazardous- task of the very first simulations of MOFs using the Multipole Method; to the wizard of matrices (and more) Lindsay Botten; and to C. Martijn de Sterke - de facto and much appreciated co-supervisor of one of the authors. In this context we would also like to acknowledge the travel support received from the French and Australian governments and from the French embassy in Canberra and the University of Sydney under the PICS, IREX and cotutelle schemes, without which our very fruitful collaboration with our Australian colleagues would have remained wishful thinking.

Daniel Maystre, who initially put the founding writers of the book together and whose insight allowed the project to reach new depths.

Part of the book was developed whilst one author held a lectureship position in the group of Sasha Movchan in the Department of Mathematical Sciences at Liverpool University.

John Pottage who demonstrated that it is possible to read the first version of the book within a week and Tania Puvirajesinghe who took a bit longer for the second edition...

Niels Asger Mortensen for interesting discussions on modal cutoffs.

C. Geuzaine for his invaluable help as an infallible GetDP guru and, with him, all the present and former colleagues of the Department of Electrical Engineering of the University of Liège led by W. Legros: J.-F. Remacle, F. Henrotte, P. Dular, D. Colignon, H. Hédia, F. Delinc, A. Genon, B. Meys, R. Sabariego, Y. Gyselinck, J.-P. Adriaens, M. Um, P. Scarpa, the late J.-Y. Hody, T. Ledinh, J. Mauhin, V. Beauvois, L. Brokamp, N. Bamps, M. Paganini, W. Legros, and many others...

A. Bossavit, E. Tonti, and P. Kotiuga for fascinating conversations on numerical electromagnetism and particularly on the applications of differential geometry and algebraic topology.

A. Ferrando for helpul comments and discussions on nonlinear optics and especially on solitons.

We are indebted to several reseachers based in French entities in which microstructured optical fibres are drawn. On one side Géraud Bouwmans of the Phlam/IRCICA CNRS laboratory of the University of Lille I who provides us several figures of the silica microstructured optical fibres he has fabricated during the last decade. On the other Laurent Brilland of PERFOS (Technology platform dedicated to the research and development of specialty optical fibres) in Lannion and Johann Trolès of the Équipe Verres et Céramiques/Sciences Chimiques de Rennes CNRS laboratory of the University of Rennes I for giving us several figures and data from their unique chalcogenide microstructured optical fibres. Géraud Bouwmans and Laurent Brilland also made several useful comments and relevant remarks on a preliminary version of the second edition of this book.

And finally to Frédéric Désévédavy and Frédéric Smektala who now are in the Laboratoire Interdiciplinaire Carnot de Bourgogne (CNRS) of the Université de Bourgogne.

Incomplete list of typos

page 19, Fig 1.4, the symbol " $\mu$ " has disappeared in front of "m". One should read on the top of the figures "108 $\mu$ m" and "26 $\mu$ m".

Table of contents of the book

1 Introduction
    1.1 Conventional Optical Fibres
        1.1.1 Guidance mechanism
        1.1.2 Fibre modes
        1.1.3 Main properties
    1.2 Photonic Crystals
        1.2.1 One Dimension: Bragg Mirrors
        1.2.2 Photonic Crystals in Two and Three Dimensions
        1.2.3 Guiding Light in a fibre with Photonic Crystals
    1.3 High-index Core Photonic Crystal Fibres
        1.3.1 A bit of history
        1.3.2 Guidance mechanism
        1.3.3 Number of modes
        1.3.4 Endlessly Single Mode Fibres
        1.3.5 Dispersion
        1.3.6 Non-linearity
        1.3.7 Birefringence
        1.3.8 High Bandwidth Multimode fibres
        1.3.9 High NA fibres
    1.4 Low-index cores
        1.4.1 Bragg fibres
        1.4.2 Two dimensional bandgap guiding fibres
        1.4.3 Non-bandgap guiding fibres: inhibited coupling
        1.4.4 Application of bandgap and hollow-core fibres
        1.4.5 Applications (A hole is a hole until you fill it.)
    1.5 A Few Words on Leaky Modes
        1.5.1 Confinement Losses
        1.5.2 Modes of a Leaky Structure
        1.5.3 Heuristic Approach to Physical Properties of Leaky Modes
        1.5.4 Mathematical Considerations
        1.5.5 Spectral Considerations
2 PCF Fabrication and Post-processing
    2.1 Introduction
    2.2 Optical Fibre Materials
    2.3 Fabrication of PCFs
        2.3.1 Stacking
        2.3.2 Drilling
        2.3.3 Extrusion and Casting
            2.3.3.1 Extrusion
            2.3.3.2 Casting
        2.3.4 PCF drawing
    2.4 Post-processing of PCF
        2.4.1 Tapering PCFs
        2.4.2 Differential hole inflation in PCFs
    2.5 Splicing of PCFs
3 Electromagnetism - Prerequisites
    3.1 Maxwell's Equations
        3.1.1 Maxwell's equations in vacuo
        3.1.2 Maxwell equations in idealized matter
            3.1.2.1 Mesoscopic homogenization
            3.1.2.2 Dispersion relations - Kramers-Kronig relations
    3.2 The Monodimensional Case : (Propagation Modes and Dispersion Curves)
        3.2.1 A first approach
            3.2.1.1 A special feature of the 1D-case: the decoupling of modes
            3.2.1.2 Physics and functional spaces
            3.2.1.3 Spectral presentation
            3.2.1.4 Orthogonality of modes
        3.2.2 Localisation of constants of propagation
            3.2.3 How can one get practically the dispersion curves and the modes?
            3.2.3.1 Zerotic approach
            3.2.3.1.1 Modes in a simple slab
                3.2.3.1.2 Modes in a binary multilayered structure
                3.2.3.1.3 Some numerical results
        3.2.4 Spectral approach
                3.2.4.1 Strengths and weaknesses
                3.2.4.2 The variational formulation (weak formulation)
                3.2.4.3 An example of a Hilbert-basis in $L^2(\mathbb{R})$ : the Hermite polynomials
    3.3 The Monodimensional Case : (Leaky Modes and Dispersion Curves)
        3.3.1 Poles's hunting : the tetrachotomy method
                3.3.1.1 Setup of the problem
                3.3.1.2 How to compute efficiently the integrals ?
    3.4 A first foray into the realm of the finite element method
    3.5 Leaky modes of Pérot-Fabry structures
    3.6 The Two-Dimensional Vectorial Case (general case)
        3.6.1 $\rm\mathbf{curl}_\beta$ operator
        3.6.2 Three different kinds of modes : basic definitions
        3.6.3 Some useful relations between transverse and axial components
        3.6.4 Equations of propagation involving only the axial components
        3.6.5 What are the special features of isotropic microstructured fibres?
    3.7 The Two-Dimensional Scalar Case (weak guidance)
    3.8 Spectral Analysis for guided modes
        3.8.1 Preliminary remarks
        3.8.2 A brief vocabulary
        3.8.3 Posing of the problem
        3.8.4 Continuous formulation
    3.9 Non finiteness of energy of leaky modes
    3.10 Bloch Wave Theory
        3.10.1 The crystalline structure
        3.10.2 Waves in a homogeneous space
        3.10.3 Bloch modes of a photonic crystal
        3.10.4 Computation of the band structure
        3.10.5 A simple 1D illustrative example: the Kronig-Penney model
4 Finite Element Method
    4.1 Finite Elements: Basic Principles
        4.1.1 A one-dimensional naive introduction
        4.1.2 Multi-dimensional scalar elliptic problems
                4.1.2.1 Weak formulation of problems involving a Laplacian
                4.1.2.2 Generalisations
                4.1.2.3 The finite element method
        4.1.3 Mixed formulations
        4.1.4 Vector problems
        4.1.5 Eigenvalue problems
    4.2 The Geometric Structure of Electromagnetism and Its Discrete Analog
        4.2.1 Topology
        4.2.2 Physical quantities
        4.2.3 Topological operators
        4.2.4 Metric
        4.2.5 Differential complexes: from de Rham to Whitney
    4.3 Some Practical Questions
        4.3.1 Building the matrices (discrete Hodge operator and material properties)
        4.3.2 Reference element
        4.3.3 Change of coordinates
        4.3.4 Nédélec edge elements vs. Whitney 1-forms
        4.3.5 Infinite domains and leaky modes
                4.3.5.1 Transformation method for infinite domains
                4.3.5.2 Perfectly Matched Layer (PML)
    4.4 Propagation Modes Problems in Dielectric Waveguides
        4.4.1 Weak and discrete electric field formulation
        4.4.2 Numerical comparisons
        4.4.3 Variants
                4.4.3.1 Looking for $\beta$ with k₀ given
                4.4.3.2 Discrete magnetic field formulation
                4.4.3.3 Eliminating one component with the divergence
                4.4.3.4 E_z, H_z formulation
    4.5 Periodic Waveguides
        4.5.1 Bloch modes
        4.5.2 The Bloch conditions
        4.5.3 A numerical example
        4.5.4 Direct determination of the periodic part
    4.6 Perfectly Matched Layers (PMLs) and the computation of leaky modes
        4.6.1 Finite element method and PMLs
                4.6.1.1 How to choose the complex stretch coefficient ?
        4.6.2 Numerical results
                4.6.2.1 Comparison with the Multipole Method
                4.6.2.2 Leaky modes for gradient index MOF
                4.6.2.3 Leaky modes for elliptical hole MOF
    4.7 Conclusion
5 The Multipole Method
    5.1 Introduction
    5.2 The multipole formulation
        5.2.1 The geometry of the modelled microstructured optical fiber
        5.2.2 The choice of the propagating electromagnetic fields
        5.2.3 A simplified approach of the Multipole Method
                5.2.3.1 Fourier-Bessel Series
                5.2.3.2 Physical Interpretation of Fourier-Bessel Series (no inclusion)
                5.2.3.3 Change of Basis
                5.2.3.4 Fourier-Bessel Series and one Inclusion: Scattering Operator
                5.2.3.5 Fourier-Bessel Series and Two Inclusions: The Multipole Method
        5.2.4 Rigorous Formulation of the Field Identities
        5.2.5 Boundary Conditions and Field Coupling
        5.2.6 Derivation of the Rayleigh Identity
    5.3 Symmetry properties of MOF
        5.3.1 Symmetry properties of modes
    5.4 Implementation
        5.4.1 Finding modes
        5.4.2 Dispersion characteristics
        5.4.3 Using the symmetries within the Multipole Method
        5.4.4 Another way to obtain $\Im m(\beta)$
        5.4.5 Software and Computational Demands
    5.5 Validation of the Multipole Method
        5.5.1 Convergence and self-consistency
        5.5.2 Comparison with other methods
    5.6 First numerical examples
        5.6.1 A detailed C_6v example: the six hole MOF
        5.6.2 A C_2v example: a birefringent MOF
        5.6.3 A C_4v example: a square MOF
    5.7 Conclusion
6 Rayleigh Method
    6.1 Genesis of Baron Strutt's Algorithm
    6.2 Common Features Shared by Multipole and Rayleigh Methods
    6.3 Specificity of Lord Rayleigh's Algorithm
    6.4 Green's Function Associated with a Periodic Lattice
    6.5 Some Absolutely Convergent Lattice Sums
    6.6 The Rayleigh Identities
    6.7 The Rayleigh System
    6.8 Normalisation of the Rayleigh System
    6.9 Convergence of the Multipole Method
    6.10 Limit cases: Asymptotics for high-contrast, $\beta \; \Lambda \ll 1$ and $r_c \ll \Lambda$
        6.10.1 Effective boundary conditions for total internal reflection or high-contrast (arrow) fibres
        6.10.2 Estimate of the cutoff curve
        6.10.3 Dipole approximation and effective parameters (long wav
    6.11 Higher-order Approximations, Photonic Band Gaps for Out-of-plane Propagation
    6.12 Conclusion and Perspectives
7 À la Cauchy Path to Pole Finding
    7.1 A Simple Extension: Poles of Matrices
        7.1.1 Degenerate eigenvalues
        7.1.2 Multiple poles inside the loop
        7.1.3 Miracles sometimes happen
    7.2 Cauchy integrals for operators
    7.3 Numerical Applications
    7.4 Conclusion
8 Main properties of microstructured optical fibres
    8.1 Types of microstructured optical fibres or types of modes?
    8.2 Main linear properties of modes in solid core microstructured optical fibres with low index inclusion
        8.2.1 Solid core microstructured fibre with low index inclusions and band diagram point of view
        8.2.2 Basic properties of the losses
        8.2.3 Single-modedness of solid core C_6v MOF
                8.2.3.1 A cutoff for the second mode
                8.2.3.2 A phase diagram for the second mode
                8.2.3.3 Towards a generalized phase diagram for the second mode
        8.2.4 Modal transition without cutoff of the fundamental mode
                8.2.4.1 Existence of a new kind of transition
                8.2.4.2 A phase diagram for the fundamental mode
                8.2.4.3 Simple physical models below and above the transition region
        8.2.5 Chromatic dispersion
                8.2.5.1 Material and waveguide chromatic dispersion
                8.2.5.2 The influence of the number of rings N_r on chromatic dispersion
                8.2.5.3 The influence of the value of the refractive index on chromatic dispersion
                8.2.5.4 A more accurate MOF design procedure
    8.3 Two examples of hollow core MOFs with air-guided modes
        8.3.1 An core MOF made of silica and the band diagram point of view
                8.3.1.1 The photonic crystal cladding
                8.3.1.2 The finite structure
        8.3.2 An optimized hollow core MOF made of high index glass for the far infrared
                8.3.2.1 Getting the bandgap at the sought wavelength
                8.3.2.2 Finite structures: influence of the core diameter
                8.3.2.3 An optimized structure
    8.4 One detailed example of ARROW MOF to understand their properties
        8.4.1 Guiding in ARROW microstructured optical fibres and its interpretations
        8.4.2 The ARROW model and its application to MOFs
        8.4.3 ARROW MOFs and band diagrams
        8.4.4 ARROW MOFs and avoided crossings
                8.4.4.1 Some general properties of avoided crossings in ARROW MOFs
    8.5 Conclusion
9 Twisted Fibres
    9.1 Introduction
    9.2 Helicoidal coordinates
        9.2.1 Twisted PML
    9.3 Finite Element Modelling of Twisted Waveguides
    9.4 Quadratic eigenvalue problem
    9.5 Numerical Example
    9.6 Conclusion
10 Conclusion
Appendix A     From Change of Coordinates in Maxwell's Equations to Transformation Optics
        A.1 Change of Coordinates in Maxwell's Equations
        A.2 The Geometric Transformation - Equivalent Material Properties Principle
    A.3 Useful Jacobian matrices
    A.4 Transformation Optics
Appendix B     A Formal Framework for Mixed FEMs
Appendix C     Some details of the Multipole Method derivation
    C.1 Derivation of the Wijngaard identity
    C.2 Change of basis
C.2.1 Cylinder to cylinder conversion
C.2.2 Jacket to cylinder conversion
C.2.3 Cylinder to jacket conversion
    C.3 Boundary conditions: reflection matrices
Appendix D     Integration by Parts
Appendix E     Six hole plain core MOF example: Supercell's point of view
Appendix F     A Pot-Pourri of Mathematics