LOS NUEVOS MEDIOS DE COMUNICACIÓN LA APROPIACIÓN POR LASCOMUNIDADES Y LOS CIUDADANOS

Background

Predicting the bending losses of holey fibres is a challenging problem. Most of the meth- ods developed for conventional fibres (as summarised in Section 2.2) assume a circularly symmetric index profile and cannot be applied to holey fibres without first replacing the complex refractive index profile with that of an equivalent step-index (ESI) fibre. This approach has been used in a few studies on the modal properties of holey fibres with mixed results. In Refs. [4, 16], the authors demonstrate that the short wavelength bend loss edge in a holey fibre can be qualitatively described using an effective index approach together with the conventional formula for predicting pure bend loss from Ref. [128]. In another study, an ESI-based calculation of pure bend loss is shown to yield good agreement with experimental data for a fibre with Λ = 7.8 µm and d/Λ = 0.30, but not for a fibre with Λ = 10µm andd/Λ = 0.55 [32, 131]. This poor agreement is due in part to the difficulties in choosing certain ESI parameters, such as core radius, that are required to assign an

appropriate ESI profile [87].

In recent months, a different method based on a step-index analogy was proposed [132]. In this approach, an approximate formula for pure bend loss, developed for step-index single-mode fibres, is expressed solely in terms of AFM_eff, Ro, nFM and nFSM, where nFM is the effective index of the fundamental core mode and nFSM is the effective index of the fundamental cladding mode [129]. In this way, the conventional loss formula can be evaluated for a holey fibre without having to define an ESI profile. However, the modal properties AFM_eff, nFMand nFSMof the holey fibre still need to be evaluated. Various methods for calculating these properties in a holey fibre are discussed in Section 1.4. In general, these methods involve numerical solutions to the wave equation and are all fairly computationally intensive. The key to the simplicity of the method proposed in Ref. [132] is that the authors use approximate formulas to define AFM_eff and√nFM2−nFSM2in terms ofλ, Λ andd/Λ only. The approximate definitions are based on functions fitted to data generated for infinite, perfectly periodic triangular lattice structures using a plane-wave approach. Where nFM appears on its own in the approximate loss formula, the assumption nFM= nglass is made, resulting in a bend loss dependent only on nglass, λ, Λ and d/Λ. The results presented within Ref. [132] show good agreement with experimental data for the fibres considered. However, the assumption of n_FM = n_glass is only valid for large values of Λ/λ and d/Λ. These approximations lead to increasing inaccuracy as d/Λ decreases in the large-mode- area regime, as demonstrated in Section A.2, and also means that this method cannot be used to predict any properties of the long wavelength bend loss edge, which occur in holey fibres for Λ/λ¿1. In addition, this simple method is restricted solely to holey fibres with a perfect triangular arrangement of air holes in which the core is formed by the omission of a single hole due to the way in which the parameters AFM_eff and √n_FM2_−n

FSM2 are defined. This is a significant disadvantage since holey fibre geometry can vary significantly from this basic design, as illustrated in Fig. 1.6 in Section 1.2.4, including different air/glass geometries in addition to hybrid and solid microstructured fibre types. Furthermore, this method ignores the symmetry differences between holey and step-index fibres: holey fibres typically possess a 6-fold symmetric cladding geometry, but can be more complex, such as the case of a triangular core formed by three adjacent rods in the preform [34]. This may have an important influence on the bending losses of these novel fibres and needs to be considered. Also, since this technique is based on a formula derived for single-mode step- index fibres it cannot be used to evaluate the bending losses associated with higher-order

modes. However, this method has the virtues of being both quick and simple to evaluate and is accurate enough to gauge the practicalities of large-mode-area fibre design for simple triangular lattice geometries. Note that the model presented in Ref. [132] is discussed in more detail in Section A.2.

Bend loss model developed here

As mentioned in the introduction to this chapter, at the start of this project, there were no theoretical techniques that could be used to accurately model even the most basic aspects of bend loss in a holey fibre. Simple models based on the step-index analogy were in existence and could be used to show that the bending losses increase towards both long and short wavelengths in a holey fibre [4, 16], but were incapable of accurately assessing the magnitude of these losses. (Note that in recent months, one model has emerged that can be used to assess the magnitude of loss in large-mode-area holey fibres with large air holes, but this method has several limitations, as discussed above [132]). Since the ability to accurately predict bend loss in a holey fibre is essential to the aims of this thesis, the first step was to develop such a model. Furthermore, due to the fact that models based on approximating the holey fibre index profile by an equivalent step-index (ESI) fibre had so far proved unreliable, it was decided that the model developed here should avoid all such ESI approximations, and instead use the full geometrically complex refractive index profile of a holey fibre in all calculations. This approach would remove any inaccuracy resulting from a circularly symmetric representation of the transverse index profile, enabling the real mode shape to be considered and allowing the angular orientation of the bend to be considered in any calculation. The necessity for this is supported by experimental observations of bend loss, in which a dependence on the angular orientation of the fibre has been observed (see Section 4.5).

In addition, in the few studies of bend loss in holey fibres reported at the time of this study [4, 16, 32], the contribution from transition loss is ignored, and calculations based solely on the pure bend loss component are used to evaluate macro-bending losses. This may seem perfectly reasonable, since in studies of macro-bending losses in conventional fibres, pure bend loss is generally assumed to be dominant and transition loss is only considered to be an important contribution for very short lengths of curved fibre [105, 133]. Indeed, the effect of mode distortion is often ignored completely in the macro-bend regime [126, 128]. However, the relative contributions of these two components may well be different in a

Firstly, I would like to thank my supervisors; Tanya Monro and David Richardson; Tanya, thank you for your guidance and support, and for your contagious enthusiasm; and Dave, thank you for your advice, support and perspective throughout my PhD. It has been a pleasure to work with both of you.

I would also like to express my appreciation to Kentaro Furusawa, Marco Petrovich and John Hayes, who fabricated the beautiful holey fibers reported in this research. I would like to thank Vittoria Finazzi for her contributions to the theoretical work presented here and for the times spent puzzling over various numerical problems with me. I am also grateful to Matteo Fuocci, who carried out some of the experimental work presented here. My thanks go also to Elanor Tarbox for proof reading my thesis and to Arthur Longhurst, Dave Oliver and David Monro for their computer support and unfailing ability to make my computer well again. I would also like to thank the following people from the ORC for their time and help: Simon Butler, Chris Nash, Heather spencer, Christine stoner, and Eve Smith.

My thanks also go to my friends and fellow PhD students: Ami, for being a wonderful source of perspective, support, friendship and yummy Swedish recipes; to Anoma, for all those long lunches and pep talks and also to Ian and Malcolm; you all listened to me complain, you celebrated my small victories and you made me laugh when I needed to. Thank you. I would also like thank to Vir for her support, to Carla for the much appreciated Maltesers when writing up, to Jim for his assurances that the write-up will be over soon (it is!), to Mell for her understanding and willingness to listen to me rant, and to Heather and Marianne for the wonderful breaks in snowy Switzerland and sunny Dorset.

I am also deeply grateful to my parents and to my sister Ali, for their love and support. You are my best friends and I simply can’t thank you enough for everything you have done for both myself and Stu during our PhDs. I would also like to express my thanks to Anne and Bill for your support over these (many) years of student-dom.

And finally, my thanks go to Stu, for far more reasons than I can write here. Thank you. Now we are Drs together!

holey fibre. Indeed, mode distortion may play a more important part in the bend loss of a holey fibre due to the fact that there is no well defined boundary between core and cladding regions. Consequently, the core mode is free to distort out in the gaps between air holes, and this may result in a higher level of mode distortion than in conventional fibres. As a result, the model of bend loss developed here considers the contribution from both transition and pure bend loss components in a holey fibre. The techniques developed to predict these components of loss within a holey fibres are described in Chapter 3, within Sections 3.1, 3.4 and 3.5.

Numerical and experimental

methods

2.4

Overview

As mentioned in Section 1.5, the aims of the study presented here are threefold: (1) to develop methods of accurately predicting bend loss that can be applied to both holey and conventional fibres, (2) to use these techniques to explore the potential offered by holey fibres in the large-mode-area, single-mode regime, and (3) to place their performance in context against conventional step-index fibres. In order to fulfil these aims, it is also essential to be able to accurately predict the effective mode area and modedness of both fibre types. Furthermore, reliable methods of characterising the bend loss, the effective mode area and the bending losses of holey and conventional fibres are also essential to enable comparative studies and also to validate the theoretical techniques developed here.

The chapters in this part describe these experimental and numerical techniques. In Chapter 3 the numerical techniques used here to calculate the effective mode area, the bending losses and to assess the modedness of both holey and conventional fibres are de- scribed. Chapter 4 details the procedures that have been developed here to measure these modal properties in large-mode-area fibres. The numerical techniques presented here are also validated via comparison with the experimental results presented in Chapter 4. Note that the experimental and numerical techniques described in the following have been pur- posefully designed to be suitable for both holey and conventional fibres, to enable direct comparisons to be drawn between the two fibre types.

Modelling fibres

3.1

Introduction

As discussed in Section 2.4, in order to explore the potential offered by large-mode-area single-mode holey fibres, and to place their performance in context against conventional step-index fibres, methods of predicting the effective area, the modedness and the bending losses of both holey and conventional fibre types are required. The numerical techniques used here to model these three key properties are described in the following.

As discussed in Section 2.3.2, it was decided that the method of modelling bend loss developed here should ideally avoid any ESI approximation, and should instead use the full geometrically complex refractive index profile of a holey fibre in all calculations. In addition, since little was known about the bending losses of holey fibres at the time of this research, it was also decided that the model of bend loss developed here should consider the contribution from both transition and pure bend loss components in a holey fibre. In order to develop methods of predicting transition loss and pure bend loss in a holey fibre, we look back to the methods that have been developed for conventional waveguides and extract elements of these approaches that do not make any assumptions regarding the nature of the modal fields. If we then assume that the modal fields and propagation constants of the straight and bent holey fibres can be evaluated (as is discussed below), the components of bend loss can be evaluated in the following manner: the transition loss can be approximated as a splice loss between two different fibres, after the methods presented in Refs [100, 101, 102], and the pure bend loss can be evaluated by calculating the fraction of the modal field that cannot travel fast enough around the bend to maintain phase with the rest of the mode, after the method presented in Ref. [113]. The methods

from Refs [100, 101, 102, 113] are discussed in Section 2.2, and their adaptation to holey fibres is presented in detail in Sections 3.4 and 3.5.

The biggest challenge with the above approaches to modelling bend loss is therefore the calculation of the modal fields and propagation constants of the straight and bent holey fibre. A selection of techniques capable of modelling the modal fields of straight holey fibres and their associated propagation constants was presented in Section 1.4. However, the method chosen here must also be capable of modelling the modal properties of the bent fibre. In their current form, only methods based on a BPM approach would be capable of this, as all other approaches start from a two-dimensional representation of the transverse refractive index profile, which is assumed to be invariant in the direction of propagation. As mentioned in Section 1.4, BPM approaches are extremely computationally intensive. However, in conventional fibres and waveguides, a conformal transformation is often used to alter the refractive index profile of the straight guide to create an index profile that is invariant in the direction of propagation and that accurately mimics the modal properties of the bent waveguide [114]. This process is described in more detail in Sections 2.2 and 3.3, but essentially involves superimposing a gradient onto the refractive index profile of the straight waveguide. In studies on conventional waveguides, the transformed structure can then be used to evaluate the modal properties of the bent waveguide using a variety of techniques. However, holey fibres possess a complex transverse structure, and the addi- tional gradient present in a refractive index profile transformed in this way precludes the use of many models. For example, the multipole method is only capable of considering circular regions of uniform index embedded in a uniform background material. Techniques that are capable of modelling such a complex structure include those based on beam prop- agation, finite-element and expansion techniques, all of which are outlined in Section 1.4. However, bend loss is a sensitive function of the refractive index profile, which in a holey fibre contains many wavelength scale features. As such, it is necessary to use a high level of resolution in the numerical grid onto which the index profile is defined. Consequently, techniques based on beam propagation, finite-element and plane-wave methods can become prohibitively computationally intensive. Fortunately, the orthogonal function method de- scribed in Section 1.4.3 takes advantage of the fact that the core mode(s) of a fibre are localised to improve the efficiency of the calculations [86, 94, 95]. This technique is similar to the plane-wave approach, in which the modal fields and the refractive index profile are decomposed into plane waves. However, by using localised functions in the decomposition

of the modal fields, this method requires far fewer terms to form an accurate description and as a result can be computationally efficient. Since the modes of the bent fibre are localised for all bend radii of practical interest, this method is chosen here to model the modal fields and propagation constants of both straight and bent holey fibres, as discussed in Sections 3.2 and 3.3.

Of course, the bending losses are not the only modal properties that require evaluation in this study. In order to place any meaningful interpretation on the bending losses, knowledge of the mode area and the modedness of the straight fibre is also required. The mode area can be quite simply extracted from the modal field of the straight fibre via numerical integration, as explained in Section 3.2. However, evaluating the modedness of a holey fibre can be more complicated. Although the orthogonal function method calculates a spectrum of modes in any given fibre, it is necessary to determine the effective index of the holey fibre cladding (nFSM) in order to evaluate which of these modes are actually guided in the core. Since the orthogonal function technique uses localised functions, it is not capable of evaluating the effective index of the (extended) fundamental cladding mode, and another technique must be employed. Here, a commercial plane-wave method is chosen, which can be both accurate and efficient since only a single-unit cell of the cladding microstructure needs to be considered in this case (as explained in Sections 1.4 and 3.5.1) [134]. However, this combination of orthogonal function and plane-wave techniques is not sufficiently accurate for fibres close to cut-off. In this situation it becomes necessary to consider a single method to evaluate the effective indices of both core and cladding modes and the Multipole approach is used instead. Note that all calculations using this technique presented within this thesis are performed by Vittoria Finazzi at the ORC [54], and that the issue of determining modedness is discussed further in Section 3.6. The multipole approach has the added advantage of being able to calculate the confinement losses of all modes present in a holey fibre, something which the orthogonal function technique cannot perform, due to the periodic boundary conditions used. The confinement losses of higher- order modes can be high enough to render a fibre effectively single-mode, which is considered in Section 5.13.

In summary, three different methods of modelling the modal properties of holey fibres are