Numerical and theoretical study of photonic crystal fibers

In this work, we study a novel type of optical waveguide, whose properties derive from a periodic arrangement of fibers (not necessarily circular), and from a central structural defect along which the light is guided. We first look for propagating modes in photonic crystal fibers of high indexcore region which can be single mode at any wavelength [1-4]. Unlike the first type of photonic crystal fibers, whose properties derive from a high effective index, there exists some fundamentally different type of novel optical waveguides which consist in localizing the guided modes in air regions [4-5]. These propagating modes are localized in a low-indexstructural defect thanks to a photonic bandgap guidance for the resonant frequencies (coming from the photonic crystal cladding). We achieve numerical computations with the help of a new finite element formulation for spectral problems arising in the determination of propagating modes in dielectric waveguides and particularly in optical fibers [7]. The originality of the paper lies in the fact that we take into account both the boundness of the crystal (no Bloch wave expansion or periodicity boundary conditions) and the unboundness of the problem (no artificial boundary conditions at finite distance). We are thus led to an unbounded operator (open guide operator) and we must pay a special attention to its theoretical study before its numerical treatment. For this, we choose the magnetic field as the variable. It involves both a transverse field in the section of the guide and a longitudinal field along its axis. The section of the guide is meshed with triangles and Whitney finite elements are used, i.e., edge elements for the transverse field and node elements for the longitudinal field. To deal with the open problem, a judicious choice of coordinate transformation allows the finite element modeling of the infinite exterior domain. It is to be noticed that the discretization of the open guide operator leads to a generalized eigenvalue problem, solved thanks to the Lanczos algorithm. The code is validated by a numerical study of the classical cylindrical fiber for which the eigenmodes are known in closed form. We then apply the code to Low IndexPhotonic Crystal Fibers (LPCF) and to High IndexPhotonic Crystal Fibers (HPCF).