Conceptos Básicos para la Gestión de Riesgos

Introduction

Over the years, numerous different approaches for calculating the modal fields and the losses associated with bent waveguides and fibres have emerged. In the following, a brief summary of some of these techniques is presented, which also serves to illustrate some of the bend loss characteristics of conventional fibres. Note that single-mode fibres are the focus of this study and macro-bend loss is considered to represent the major limiting factor for practical mode areas in this regime (see Section 2.2.1). Consequently, all the approaches described below deal only with macro-bend losses in single-mode waveguides and, unless otherwise indicated, rely on analytical solutions to the wave equation with a weak guidance

approximation. Within these approaches, transition loss is most often ignored due to the fact that pure bend loss is found to dominate in the macro-bend regime. Furthermore, the effect of field deformation is often ignored completely in the calculation of pure bend loss in conventional fibres and waveguides in order to simplify the descriptions of the modal fields. This approach is approximately valid for gentle bends, in which the field of the bent waveguide is not greatly different from that of the straight waveguide, but does, obviously, lead to some degree of inaccuracy in the predictions of bend loss.

Summary of theoretical approaches

The separation of macro-bend loss in an optical waveguide into the two components de- scribed above was first proposed in Ref. [113], where these components are described as: (1) A dissipative loss along the curved waveguide section (referred to in later literature as pure bend loss), which results from the inability of the tails of the modal field to negotiate the bend, and (2) A mode conversion loss (also referred to as transition loss), incurred by the modal field on entering and leaving the bend, which is related to the fact that the modal field distorts radially outwards in a bent waveguide. Approximate formulas to predict the contribution from both of these bend loss components are calculated using analytical so- lutions to Maxwell’s equations for slab waveguides. The pure bend loss is calculated by evaluating the fraction of the guided mode that cannot travel fast enough around the bend to keep phase with the rest of the mode and the transition loss is calculated via a gener- alised coupled wave approach. However, in this approach, the field deformation induced by the curved waveguide is ignored in the model of pure bend loss and the modal fields are approximated by those of the straight waveguide.

One popular method of defining the modal fields of a bent waveguide is to transform the refractive index of the straight waveguide such that it mimics the modal properties of the bent waveguide. This approach is called conformal transformation and was first proposed as a solution to this problem in 1975 [100]. In this conformal transformation approach, a curved waveguide is approximately represented by a straight waveguide with an effective refractive index distribution by expressing the wave equation in terms of a local coordinate system that follows the curvature of the waveguide. Once defined, the transformed index profile can then be used to evaluate the modal fields and the attenuation of the bent structure using a variety of techniques. This technique was first applied to slab waveguides, where the modal fields and complex propagation constants are found by breaking the transformed

index profile into a series of constant index steps and applying a Wentzel-Kramer-Brillouin (WKB) approach [100]. The pure bend loss is then obtained from the imaginary part of the propagation constant and the transition loss via an overlap integral used to calculate the mode mismatch between straight and bent waveguide sections. Although no comparison was made to experimental results in this first study, results from another report in which the same technique was used showed excellent agreement with experimental results [101], demonstrating the validity of the conformal transformation.

This conformal transformation technique can also be applied to optical fibres for which the weak guidance approximation is valid [114], and was first implemented to calculate the modal fields of bent step-index fibres in 1976 [115]. The effective refractive index distribution for a bent optical fibre is given by Eq. 3.9, and essentially acts to superimpose a gradient on the refractive index profile of the fibre, rising in the direction of the bend. The full derivation for the conformal transformation for an optical fibre is given in Ref. [114], and is discussed in detail in Section 3.3.1. A conformal transformation is a popular first step to modelling the modal properties of bent step-index fibres and some of the various approaches that can evaluate both pure bend and transition losses in this way are explored in the following [111, 106, 108, 109, 116, 117, 114, 102].

For example, a simple approach to model the transition losses of step-index fibre is proposed in Ref. [102]. In this technique, a first order perturbation method is used to find approximate analytical expressions for the modal fields of the transformed refractive index profile. The transition loss is then calculated as a splice loss using an overlap integral that calculates the modal overlap between the mode of the straight and bent fibre. In another method, coupled mode theory is used to evaluate the losses incurred due to mode coupling within the transition region of a step-index fibre via the energy in, and interaction between, the fundamental core mode and the lowest-order radiation mode as a function of distance along the bend [118]. Using this model, the authors show that the power lost in the transition region of a bend in an optical fibre oscillates as a function of distance and that radiation is emitted in discrete beams in this region. This behaviour had previously been observed experimentally for bent fibres immersed in index-matching liquid [119]. In a following paper [120], oscillations in bend loss as a function of bend radius are studied with the same technique, showing reasonable agreement with experimental results in terms of peak positions. Note that various other approaches have also been used to successfully predict the oscillations in bend loss that occur as a function of wavelength, which have

been shown to arise from the wavelength dependent coupling between core and cladding modes [121, 122].

A couple-mode technique together with a conformal transformation is also employed in Ref. [111], to demonstrate how the rate of change of curvature in the transition regions can influence the bend loss. In this technique, the power in the radiation field is calculated from integrating the Poynting vector over the appropriate solid angle for a step-index fibre with an infinite cladding. The authors demonstrate that the contribution from transition losses to the overall bend loss decreases as the change in curvature becomes less abrupt.

Other techniques apply beam propagation methods (BPM) to model the bending losses in both step-index and parabolic-index fibres [108, 106, 123]. Using BPM techniques, the pure bend loss is evaluated via the imaginary part of the propagation constant of the fundamental mode and the transition losses can be calculated by considering coupling to the radiation modes of the bent fibre and the associated attenuation coefficients. In Refs [108] offsetting the core and/or introducing a dip in the refractive index profile along the outside of the bend are shown to considerably improve the losses. A similar approach to improving bending losses in step-index fibres and waveguides is also proposed in Ref. [123], where BPM techniques are used to show that introducing both depressed and increased index regions on the outside of the core can reduce bend losses beyond that achievable with the presence of a depressed region alone. Note that in Ref. [106], the authors show that an equivalent step-index (ESI) approach is not an accurate method of assessing bending losses in a graded-index fibres due to the arbitrary nature in which certain parameters, such as the core radius of the ESI parameter, must be defined.

Other approaches towards modelling pure bending losses involve Fourier decompositions of the modal field. In ref. [124] the fields near and far from the core are assumed to be unperturbed, and the modal field between these two regions are expanded into a Fourier series containing Airy functions. The coefficients of the backward propagating field are determined by matching boundary conditions and the attenuation coefficient for pure bend loss is then derived from the amplitude of the backward propagating field. In Ref. [125] a model including the finite coating and cladding of a step-index fibre is developed, which is essentially an extension of [124] and this approach shows reasonable comparison with experimental results. Ref. [116] extends the theory in [125] so that the expansion includes the whole of the cladding. In Ref. [117] a more generalised theory based on the same approach is developed, which shows excellent agreement with experimental results.

Approaches that do not use a conformal transformation typically ignore the modal distortion in the bent fibre altogether. As mentioned before, this approach is approximately valid for gentle bends, in which the field of the bent waveguide is not greatly different from that of the straight waveguide, but does, obviously, lead to some degree of inaccuracy in the predictions of bend loss. However, this approximation enables the problem of calculating the bend loss to be greatly simplified and permits simple analytical models of bend loss to be derived. For example, in Ref. [126], a loss formula for the fundamental mode of infinitely clad step-index fibres is derived for weakly-guiding fibres. In this derivation, the loss coefficient is determined by calculating the power outflow from the field in the cladding, which is expressed in terms of a superposition of cylindrical outgoing waves. This technique was later adapted to include the field deformation via conformal mapping [115]. However, a comparison of results from these two techniques show that for single-mode fibres, the effect of the field-deformation has a minimal effect on the predicted loss. Comparison with experimental results shows that both techniques overestimate the bend loss in optical fibre, although they do predict the correct parametric dependencies [127]. The approach of Ref. [126] is also used in Ref. [128] in which an analytical expression for bend loss in optical fibres with axially symmetric refractive index profiles is evaluated by approximating the index profile by a staircase function and expressing the fields in a matrix representation. A simplified version of this approach is also presented in Ref. [129].

One novel method uses a surface current analogy to determine pure bend loss in a step- index fibre by replacing the effect of the core with fictitious currents on its surface [130]. The formula for the attenuation coefficient derived using this technique is remarkably similar to those derived using more conventional techniques [126], although the predicted losses are significantly overestimated.

2.3

Bending losses in holey fibres

2.3.1 Differences between bend loss in holey and conventional fibres

The bending losses of holey fibres differ qualitatively from those of conventional step-index fibres. Like conventional fibres, holey fibres exhibit a bend loss edge at long wavelengths due to the fact that the mode extends further into the cladding, resulting in a more weakly guided mode that will suffer a greater perturbation in response to bending. Holey fibres also possess an additional bend loss edge at short wavelengths as a direct consequence of

their novel cladding structure [4]. In a holey fibre the effective cladding index is strongly wavelength dependent and increases towards short wavelengths (see Section 1.2.2). This acts to decrease the NA towards short wavelengths, resulting in a more weakly guided mode that becomes more susceptible to bend induced distortion and loss. This explanation is sufficient to explain this phenomenon, however, since bending losses in holey fibres represent the main focus of the work presented here, these losses are explained in a little more detail in the following.

In a bent fibre, power is lost as a result of bend induced coupling from the core mode(s) to any or all of leaky higher-order, cladding, radiation and backward propagating modes. The amount of power lost in a bent fibre is dependent on the severity of the bend, the fibre structure, and the wavelength of light, each of which influence the strength of coupling between the modes present. To first approximation, the strength of coupling between two modes can be gauged from the effective index difference of those modes: the closer they are in effective index the more power is coupled from one to the other. Here this approximation is used to illustrate the spectral dependence of bend loss in holey and conventional fibres. To keep matters simple, we consider the coupling between the fundamental core mode and the lowest-order cladding mode only. Note that in a step-index fibre the effective index of the lowest-order cladding mode is taken to be the refractive index of the solid cladding, nclad, and in a holey fibre is defined as the effective cladding index of the microstructured region, n_FSM. The relative change in bend loss with respect to wavelength can thus be evaluated by considering the relative spacing of the cladding index (nclad or nFSM) and the effective index of the fundamental core mode, n_FM. This is illustrated in Figs 2.1 (a) and (b), in which sketches of nFMand the cladding index are shown as a function of wavelength for a holey and conventional fibre respectively. A sketch of the bend loss is shown as a function of wavelength for these two fibre types in Figs 2.1 (c) and (d) respectively. These graphs are explained in the following.

As can be seen in Figs 2.1 (a) and (b), in both fibre types, the effective index of the core mode decreases towards long wavelengths as the mode extends further into the lower index cladding, and tends asymptotically towards the refractive index of the core towards short wavelengths. The differences in bend loss between the two fibre types arise from the fact that the cladding indices in these two fibre types have different functionalities. In a conventional fibre, the cladding index is weakly dependent on wavelength and can be considered to be constant. The difference between the effective index of the core mode

n

n

n

n

Bend loss

Bend loss

n

n

Holey fibre

Step−index fibre

n

n

Holey fibre

Step−index fibre

n

(c)

(d)

(b)

(a)

λ

λ

λ

λ

∼Λ/2

n

Figure 2.1: Schematic of the effective index for the fundamental core mode and the lowest-order cladding mode in (a) a holey fibre and (b) a conventional step-index fibre. Here, nsilicais the refractive index of silica

glass, nav represents the volume average index of a holey fibre cladding and ncoreand ncladare the core and

cladding indices of a step-index fibre respectively. nFMis the effective index of the fundamental core mode

and nFSMis the effective refractive index of a holey fibre cladding, which is also referred to as the effective

index of the fundamental space filling mode (FSM) of the cladding. A sketch of the bend loss is shown in (c) and (d) for holey and conventional fibres respectively.

and the cladding index thus only ever decreases towards long wavelengths, and thus the bending losses only ever increase with wavelength, as shown in Fig. 2.1 (d). In a holey fibre, the fundamental core mode and the fundamental cladding mode share the same parametric dependency on wavelength. For both of these modes, the effective index increases towards short wavelengths as the mode becomes more confined to the silica regions, and decreases towards long wavelengths as the light samples more of the air holes. The bending losses therefore increase towards both long and short wavelengths, as shown in Fig. 2.1 (c).

Consequently, although a holey fibre can be single-mode at all wavelengths, the two bend loss edges limit the bandwidth of useful operation and thus define the maximum practical mode size for each wavelength in that range. It has been shown empirically that the mid-point in wavelength between the long and short bend loss edges in holey fibres with a

triangular arrangement of air holes is approximately given by Λ/2 [32]. For large-mode-area holey fibres, in which 5µm< Λ <25µm, all wavelengths of light that are transparent in silica lie on the short-wavelength side of this mid-point. As such, the bending losses of large- mode-area silica holey fibres are expected to decrease with wavelength, for all wavelengths of interest. This is in direct contrast to conventional fibres, in which bending losses increase towards long wavelengths only. As such, it is not obvious to see how these two fibre types will compare in the large-mode-area single-mode regime. Consequently, the ability to accurately predict these losses is essential in order to assess the potential offered by holey fibres, relative to conventional fibres. In addition, another important difference between conventional and holey fibres is the angular symmetry of the index profile. Conventional fibres are typically circularly symmetric, whereas holey fibres are not. Holey fibres usually possess a six fold symmetry, and this may be reflected in the bending losses. As a result, it is important that any model of bending losses in holey fibres can incorporate the effect of the complex refractive index profile.

In the following section, some of the models that have been developed to evaluate bending losses in holey fibres are discussed. These models are all based on analogy with conventional single-mode fibres and the limitations associated with this are explored. The requirements for an accurate model of bend loss in a holey fibre is also outlined.