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The aims of this thesis, which are discussed more fully in Section 1.5, are essentially 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. In order to do this, methods of predicting the effective area, the modedness and the bending losses of both holey and conventional fibre types are required. At the time of this study no suitable techniques existed to model the bending losses of holey fibres, and so the first step was to develop such a method. In addition, since little was known about the bending losses of holey fibres in general, it was also decided that any model of bend loss should include contributions from both transition and pure bend loss components. The methods that have successfully been developed to model both of these components of bend loss in holey fibres are outlined in the above sections of this chapter. Note that these techniques are also suitable for conventional fibre types.

The models of transition loss and pure bend loss presented in this chapter have been constructed by combining elements from methods developed for conventional waveguides with established techniques that have been developed for modelling the modal properties of straight holey fibres. Transition loss is modelled by approximating the mode conversion at points of changing curvature as a splice loss between the straight and bent fibre [100, 101, 102], as described in Section 3.4. The modal fields required in this calculation are evaluated using the orthogonal function technique developed by Tanya Monro [13, 86, 94, 95], with the addition of a conformal transformation in the case of the bent fibre (see Sections 2.2 and 3.3). A description of this technique is presented in Section 3.2, and the adaption to bent holey fibres is discussed in Section 3.3. This approach to modelling the modal field of the bent fibre is also employed in the model of pure bend loss developed here. This method is described in Section 3.5 and is based on the method presented in Ref. [113], in which the bend loss is 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. The effective cladding index of the straight fibre is required in this calculation and this is evaluated using a commercially available plane-wave technique [134].

The methods of modelling bend loss developed here involve several assumptions. The assumptions inherent to the conformal transformation routinely used in conventional fibres restricts modal calculations to bend radii in the macro-bend regime. The macro-bend regime is defined for bend radii that are much larger than the core radius of the fibre. However, via inspection of calculations, we find that this model breaks down within the macro-bend regime. Fortunately, this break-down occurs at bend radii that are far smaller than any required to evaluate the critical bend radius for all of the fibres considered here (see Section 3.3.4). Furthermore, in the method developed for modelling transition loss, the loss due to mode conversion is assumed to take place instantaneously, when in reality, the mode changes shape over a finite length scale. In addition, mode coupling is ignored in both the model of transition and pure bend loss. These assumptions mean that both components of bend loss will be overestimated. For the case of transition loss, the model presented here serves as an estimate of the maximum possible loss that can be attributed to this component of loss. Indeed, as it turns out, this contribution is small and can be neglected, as demonstrated experimentally in Section 4.5. In addition, we find that the overestimation in the method of pure bend loss can be compensated for by introducing a scaling factor. This is discussed in more detail in Sections 3.5 and 4.5.

For reasons of simplicity and computational efficiency it is preferable to use a scalar version of the orthogonal function method to evaluate the modal fields and propagation constants of holey fibres. However, the refractive index profile in a holey fibre contains features with a large refractive index contrast and the conformal transformation, used to model a bent structure, imposes an asymmetry on this profile. However, for large- mode-area holey fibres with reasonably small holes, the effective cladding index creates a low NA fibre and, in the macro-bend regime, the asymmetric distortion imposed by the conformal transformation represents only a slight perturbation. As a result, we find that the scalar version of the orthogonal function technique can indeed be used to model the modal properties of both straight and bent holey fibres with large-mode-areas. For example, for a fibre with Λ = 15.0 µm and d/Λ = 0.63, which represents the most highly multi- mode structure considered within this thesis, the propagation constant for thex-polarised mode differs only in the 10th _{significant figure between scalar and vector calculations at}

1064 nm. In addition, there is less than 5% difference between the critical bend radius of the fundamental mode for this fibre calculated using the scalar and vector version of the model presented above at 1064 nm (see also Sections 4.5.3 and 4.5.5). However, the fully vectorial version of the orthogonal function technique outlined in Section 3.2 is used to study the effect of polarisation in the bent fibre in Section 4.5.5.

As mentioned above, 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 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. In Section 3.6, the three methods used here to evaluate the modedness of holey fibres are discussed: (1) The orthogonal function technique can be used in conjunction with the calculation of the effective cladding index nFSM using a plane-wave technique, (2) The V-parameter of an equivalent step-index (ESI) fibre can be determined, and (3) the multipole method can be used. This last approach is by far the most accurate and is used when considering fibres that are close to cut-off. Note that all multipole calculations included in this thesis are performed by Vittoria Finazzi [54]. In conclusion, the methods listed in the above sections of this chapter can be used to model mode area, pure bend loss, transition loss, and modedness, in both holey and conventional fibres. In the next chapter, the experimental techniques developed to measure these properties are presented, together with a comparative study between experimental and numerically derived values that are used to validate these numerical models.

Experimental methods

4.1

Introduction

As discussed in Section 2.4, in order to explore the potential offered by large-mode area single-mode holey fibres, methods of characterising the effective area, the modedness and the bending losses of both holey and conventional fibre types are required. Experiments designed to measure these three key properties are described in the following.

As discussed briefly in Section 1.4.2, the effective mode area of holey fibres can be rea- sonably well approximated by an equivalent step-index fibre at a given wavelength due to the fact that the modal field can be approximated by a Gaussian function [17]. Indeed, all the fibres considered in this thesis, which have 7µm<_∼Λ<_∼20µm and 0.2<_∼d/Λ<_∼0.5, have an overlap with a Gaussian function of optimum width that is greater than 95% in the wavelength range of 300 to 1600 nm (this overlap is defined in Eq. 3.11, in which E_b

is replaced by a Gaussian function). As a result, we find that conventional methods for measuring effective mode area can be adapted for the large-mode-area holey fibres consid- ered in this study, as described in Section 4.3. However, we have found that conventional techniques for measuring bending losses are not transferable to holey fibres. In part, this is due to the fact that most reports of bend loss are vague in their description, but also results from the fact that these measurements are designed for conventional fibres with mode areas less than approximately 100µm2 _{at telecommunications wavelengths. For such fibres, the} critical bend radii are typically less than 1 cm and the majority of techniques induce tension on the fibre to ensure that the bent fibre describes a perfect circle [127, 136, 137, 122]. We have found that using tension on large-mode-area fibres is not only unnecessary, since the bend radii of interest are typically large, but that it can result in non-repeatable results

for holey fibres. We suspect that this non-repeatability for the case of holey fibres results from the fact that they are more sensitive to strain than their conventional counterparts. Indeed, we have observed that the strain induced by the small magnets traditionally used to hold the fibre in a V-groove can severely distort the modal field. However, it should be noted that the holey fibres studied here have been shown to possess comparable levels of mechanical strength to conventional solid fibres [138]. In addition, since step-index fibres are typically circularly symmetric, conventional bend loss measurements do not consider the effect of the angular orientation of the fibre. In contrast, holey fibres typically possess a six fold symmetry, and this may influence the bend loss. In response to these requirements, we have developed several techniques to measure the bending losses of large-mode-area ho- ley fibres as a function of bend radius and the angular orientation of the fibre that ensure minimal strain and tension on the fibre. In the following sections detailed descriptions are presented for the experiments designed to study the following (1) the bend loss as a function of bend radius, (2) the bend loss as a function of wavelength, (3) the relative impact of transition loss and pure bend loss and (4) the effect of the fibre geometry on the bend loss. Note that these experiments are also suitable for conventional fibres, enabling comparative measurements.

Whilst the general procedures for each experiment are identical for both fibre types, some basic experimental techniques such as fibre end preparation and measurements of the refractive index profile must be approached differently for holey fibres. The details of these basic techniques are briefly discussed in Section 4.2. A brief outline of the methods used to launch light into the fibres studied here is also included in this section.