Foundations of Photonic Crystal Fibres

The following sections are included:

• Forewords to the First Edition

• Forewords to the Second Edition

• Preface

• Acknowledgements

• Contents

This chapter introduces the field of optical fibres, photonic crystals and microstructured optical fibres (MOFs). These last are also named photonic crystal fibres (PCFs).¹ A number of concepts are introduced in a somewhat heuristic manner, with the aim of assembling all of the notions required for the different forms of microstructured optical fibres and their guidance mechanisms, for comparing them with standard optical fibres and for introducing groundbreaking applications of these structured fibres that should serve as a motivation for the rest of the book. Subsequent chapters will introduce much more rigorously many aspects only presented heuristically here. While no particular knowledge of photonic crystal fibres is assumed, some background knowledge of waveguide physics will be useful for a full understanding of this chapter (and indeed the remainder of the book) …

The following sections are included:

• Introduction

• Optical Fibre Materials

• Fabrication of PCFs

• Post-processing of PCF

• Splicing of PCFs

The following sections are included:

• Maxwell's Equations

• The Monodimensional Case: Propagation Modes and Dispersion Curves

• The Monodimensional Case: Leaky Modes and Dispersion Curves

• A First Foray into the Realm of the Finite Element Method

• Leaky Modes of Péerot–Fabry Structures

• The Two-dimensional Vectorial Case (General Case)

• The Two-dimensional Scalar Case (Weak Guidance)

• Spectral Analysis for Guided Modes

• Non-finiteness of Energy of Leaky Modes

• Bloch Wave Theory

The following sections are included:

• Finite Elements: Basic Principles

• The Geometric Structure of Electromagnetism and its Discrete Analogue

• Some Practical Questions

• Propagation Mode Problems in Dielectric Waveguides

• Periodic Waveguides

• Perfectly Matched Layers (PMLs) and the Computation of Leaky Modes

• Conclusion

The following sections are included:

• Introduction

• The Multipole Formulation

• Symmetry Properties of MOF

• Implementation

• Validation of the Multipole Method

• First Numerical Examples

• Conclusion

The following sections are included:

• Genesis of Baron Strutt's Algorithm

• Common Features of the 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

• Normalisation of the Rayleigh System

• Convergence of the Multipole Method

• Limit Cases: Asymptotics for High-contrast, βΛ ≪ 1 and r_c ≪ Λ (Long-wave Limit and Dilute Composite)

• Higher-order Approximations, Photonic Bandgaps for Out-of-plane Propagation

• Conclusion and Perspectives

In the preceding chapters, it has been shown that the modes guided inside a photonic crystal fibre were associated with poles of the scattering matrix. The determination of the dispersion curves of these modes amounts to finding the poles of the scattering matrix. The numerical computation of poles of functions of a complex variable is by no means a simple problem. It cannot be considered as a simple extension of the same problem for functions of one real variable. Indeed, the description of a function of one real variable amounts, roughly speaking, to going along the real axis and looking at the values of the function: it is just a one-dimensional walk, along which the accidents of the function (its poles, asymptotes or zeros) are easily found. For complex variables, the walk should be two dimensional: we can of course define a grid that serves as a discrete version of the function, the poles of which are then sought over this grid. Such an operation is extremely time consuming. This is why one would like to have procedures to reduce as much as possible the number of numerical evaluations of the function at issue. There are two principal ways of finding the zeros or poles of a function of a complex variable. The first is iterative: it has its roots in Newton methods for real-variable functions. Given an a priori estimate of the value of the zero, a sequence of successive approximations is generated that hopefully converges towards the correct value (such as, for instance, the Müller algorithm, see Press et al. (1986)). The main drawback of this method lies in its instability, and its incapacity to give the a priori estimate. The second way, which we shall take, is the use of Cauchy integrals for holomorphic functions. The starting point is precisely the Cauchy theorem, which we give with no attempt at mathematical rigour (for the theory of functions of one complex variable, we refer the reader to Rudin (1986)) …

The following sections are included:

• Types of Microstructured Optical Fibres or Types of Modes?

• Main Linear Properties of Modes in High-index Microstructured Optical Fibres with Low-index Inclusions

• Two Examples of Hollow-core MOFs with Air-guided Modes

• A Detailed Example of an ARROW MOF

• Conclusion

The following sections are included:

• Introduction

• Helicoidal Coordinates

• Finite Element Modelling of Twisted Waveguides

• Quadratic Eigenvalue Problem

• Numerical Example

• Conclusion

We believe it is fully justified that MOFs are described as 'novel optical waveguides' even though they were fabricated as early as 1973 and leaky modes were encountered in conventional W-fibres. The unique properties of MOFs, such as single-mode guidance (now clearly established) and photonic bandgap propagation (revisited for finite structures) make them much more than a simple generalisation of conventional optical fibres. Their properties have led to several applications including supercontinuum generation, high-power transmission and fibre lasers…

The following sections are included:

• Appendix A From Change of Coordinates in Maxwell's Equations to Transformation Optics

• Appendix B A Formal Framework for Mixed FEMs

• Appendix C Some Details of the Multipole Method Derivation

• Appendix D Integration by Parts

• Appendix E Six-hole Plain-core MOF Example: Supercells

• Appendix F A Potpourri of Mathematics

• Bibliography

• Index