## Foundations of Photonic Crystal FibresThis book aims to provide expert guidance to researchers experienced in classical technology, as well as to those new to the field. A variety of perspectives on Photonic Crystal Fibres (PCFs) is presented together with a thorough treatment of the theoretical, physical and mathematical foundations of the optics of PCFs. The range of expertise of the authors is reflected in the depth of coverage, which will benefit those approaching the subject for a variety of reasons and from diverse backgrounds. The study of PCFs enables us to understand how best to optimize their applications in communication or sensing, as devices confining light via new mechanisms (such as photonic bandgap effects). It also assists us in understanding them as physically important structures which require a sophisticated mathematical analysis when considering questions related to the definition of effective refractive index, and the link between large finite systems and infinite periodic systems. This book offers access to essential information on foundation concepts of a dynamic and evolving subject. It is ideal for those who wish to explore further an emerging and important branch of optics and photonics. |

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### Contents

Introduction | 1 |

Electromagnetism Prerequisites | 27 |

components | 57 |

components | 60 |

model | 81 |

Finite Element Method | 89 |

The Multipole Method | 157 |

Rayleigh Method | 205 |

A la Cauchy Path to Pole Finding | 225 |

Basic Properties of Microstructured Optical Fibres | 247 |

Conclusion | 281 |

Appendix B Some Details of the Multipole Method Derivation | 287 |

A PotPourri of Mathematics | 295 |

323 | |

337 | |

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### Common terms and phrases

1-forms algorithm annulus associated band gap Bloch waves boundary conditions chapter chromatic dispersion cladding coefficients complex components computed consider convergence coordinates corresponding curl curlt cylinder defined denoted diagram differential dimensional discrete dispersion curves domain dxdy edge effective index eigenvalues electric field equation exterior derivative finite element method formulation Fourier frequency function fundamental mode geometrical given grad gradt guided modes Hilbert space hollow core MOF inclusions infinite integral lattice leaky modes linear losses magnetic field material mathematical matrix mesh mode classes modulus Multipole Method neff nodes notation obtained OOOOOOOOO operator optical fibres parameter permittivity photonic band gap photonic crystal photonic crystal fibres poles problem propagation constant properties Rayleigh refractive index region scalar product silica single-mode solid core MOF solution structure symmetry tangential theorem transform transverse vector field vector space waveguide wavelength Whitney zero