In conclusion, a considerable number of numerical methods are currently available for the modelling of MOFs. At the beginning of this project, a comparison between the methods previously illustrated was conducted, with the aim of determining the most efficient and versatile numerical method to be used, in prevalence, during the course of my PhD for tackling the simulation of different MOFs. Accuracy of the method was, of course, one of the first prerequisites, while its efficiency was also rather important, in view of its application in inverse simulations (see Chapter 4). In addition, the follow- ing requirements were also identified as highly desirable for a versatile, multi-purpose method:
• Full vector formulation, in order to allow the study of fibres with arbitrary air- filling fractions and refractive index contrast;
• Direct inclusion of the wavelength dependence of the refractive index for more accurate dispersion calculations;
• Capacity to calculate confinement losses, which can have a profound relevance in the design of MOFs (see Section 1.2.5);
• Capacity to simulate fibres with arbitrary cross section, in order to model the properties of fabricated fibres from their SEM images and support or complement measurement results;
• Symmetry exploitation, in order to take advantage of any possible structural sym- metry of the fibre to reduce the computational demand.
The main properties of the previously described methods are reported for comparison in Table 2.1:
As can be appreciated, all the methods analysed present a full vector formulation. The material dispersion cannot be included in the widely employed PWE implementation of Johnson et al. [129], although a modification of the method itself can be formulated to this aim [130, 131]. All other methods presents either an eigenvalue formulation, in which β is the eigenvalue and ε(ω) can be provided as an input, or a homogeneous matrix equation where, again, the solution – from which β can be derived – is found as a function ofε(ω).
PWE OFM ABC-FDM BPM MM SMT FEM Fully vectorial √ √ √ √ √ √ √ Material disp. ×(√) √ √ √ √ √ √ CL calculation × × √ ∼√ √ √ √ Arbitrary cross- section √ √ ×a √ ×b √ √ Symmetry
exploitation × partial partial × ×
√ √
Ref. [129]([131]) [134] [136] [146] [9] [143] [144] Limited to: aannular inclusions, bcircular inclusions.
Table 2.1: Comparison between the principal numerical methods for the study of
MOFs.
As was already mentioned, the CL can be evaluated by most methods except for the PWE and the OFM; the BPM can be rather inefficient for low-CL fibres.
All methods are generally suitable for the simulation of an arbitrary cross-section, with the exception of the ABC-FDM and the MM. These are limited, in their original formu- lation, to structures with annular or circular inclusions, respectively, even though the authors indicate that both techniques could be extended to include arbitrary inclusions. This can be done by combining the general features of the methods with numerical in- tegrations of the overlap integrals for the ABC-FDM or with a more general code (e.g. FEM) solving for the scattering matrix of a single inclusion, in the case of the MM. Finally, the FEM and the SMT are the only methods allowing the full exploitation of potential structural symmetries to reduce the physical computational domain. PEW, BPM and MM always require the full transverse structure to be simulated. A partial advantage can be gained from symmetries for the other two methods: in the OFM only even terms in the field expansion can be retained for symmetric structures. In the ABC- FDM the presence of radial symmetries are likely to allow for a reduction in the number of terms in the azimuthal Fourier decomposition.
In order to assess the accuracy of a particular method in solving for the fundamental or higher-order modes of MOFs, many comparisons among the various methods have been presented (see for example [136, 143]). A particularly interesting and rigourous comparison between four of the previous methods was recently reported by researchers of a consortium of Universities working within the framework of the European Science Foundation COST P11 [147]. Full vector PWE, OFM, FEM and SMT were applied to the study of the same fabricated birefringent HF from an SEM image of its cross- section. A convergence test on the accuracy of the effective index of the LPx
01 mode,
PWE and OFM were only stable up to the fourth/fifth digit. The residual differences in the value of neff between all these methods were lower than 4×10−4. Moreover,
since these discrepancies have a systematic character, they are further suppressed in the calculation of birefringence. A variation of between 0.1 and 1% was finally observed between the simulated values of GVD at 1550 nm. From the values reported, FEM and SMT, beside converging to a better precision, also seems generally quicker, even though a fair comparison of the computational time required was not entirely possible due to the use of machines with different CPUs.
From all of these considerations, the FEM and the SMT emerge as the most versatile and accurate methods for studying a vast range of MOFs. The SMT however was only proposed very recently, and it was not available at the beginning of this project, when the FEM was reputed to be the only method possessing all the desired requirements. For this reason the FEM was chosen as the preferred method to be developed and employed throughout the project.
Almost all the simulations reported in this thesis have been conducted with the FEM, for which we employed a commercial package, COMSOL MULTIPHYSICSTM1. The next section will present an overview of the method, the details of its implementation for the study of out-of-plane electromagnetic propagation and the specific adaptations that were required in order to study MOFs.