1. Introduction

The phase requirement for Fresnel waveguides leads to constructive generation of the mode profile in the propagation direction. The mode profile is therefore defined by the chosen zone cross-section (ZC), which is equivalent to the zone plates of Fresnel lenses [1]. Consequently, all waveguides are described on the basis of coherent scattering leading to such interference, which is a generic concept that can be shaped by adjusting the spatial geometry of the zones over the fibre cross-section. Since this property is associated generally with the phase of the interfering light, where in practice matter is used to affect this phase within the waveguide, there is a fundamental correlation with free space beams [1, 2]. As result of this description waveguides are defined by their particular zone cross-section (ZC). For example, a Bragg fibre is the binary solution of the linear Fresnel relationship where each periodic region represents the zones of the fibre. This particular Fresnel fibre is thus described as having a Bragg zone cross-section (BZC). Chirping the Bragg period extends this to the simplest recognised Fresnel fibre with a chirped BZC [1, 3]. The chirped solution for bound mode generation implicitly recognises the 2-D circular symmetry of the waveguide. Not all solutions need to be radially symmetric and appropriate optical localisation is possible with hyperbolic profiles, for example, defined by a hyperbolical zone cross-section (HZC), a direct analogy to hyperbolic zone plates (HZP) of Fresnel lenses [4]. In practice, the production of these fibres has recently been possible by the improved fabrication of structured optical fibres with capillaries running along the fibre defining the zone cross-sections [5]. This approach has led to the first experimental demonstration of Fresnel fibre with a cross section made up of holes placed along the zones of the chirped BZC Fresnel fibre [6]. Characteristic features such as multiple foci in the far field as well as peak intensity light inside a central air hole have been demonstrated experimentally [6] and supported numerically [7], proving the generic concept of the Fresnel waveguide. The zones can therefore literally be shaped to make any arbitrary mode profile.

In this paper we examine through numerical simulation the possibility of generating Fresnel fibre propagation based on taking the simplest zone cross-section – the BZC – and dividing every alternate zone into increasingly number of finer zones. The purpose is to incorporate the well known angular bandgap increases found in similar omnidirectional filters [8], including recently demonstrated omnidirectional fibres [9], and to create what we define as an omnidirectional zone cross section – OZC (or in the case of a lens an omnidirectional zone plate). We expect this approach to generate very wide bandgap waveguides. To implement a practical approach to this problem a fundamental link between omnidirectional filter design and fractals is proposed: the self-imaging of fractals leads directly to omnidirectional filter-like properties and we believe this explains why fractals have diffractive properties that satisfy the basic Fresnel requirement for coherent scattering. To verify this concept, as well as demonstrate the potency of a new generation of Fresnel fibres with these omnidirectional-like properties, we numerically simulate a Fresnel fibre with a zone cross-section described by a basic triadic Cantor fractal [10]. These fractal structures have the property of self-imaging such that an overall Bragg period is maintained whilst alternate zones are divided, exactly one type of structure falling into the more generic category of OZC. If the interpretation is valid then the most obvious feature that will be observed is an increasing photonic bandgap with increasing level of self-imaging.

To our knowledge, the study of diffraction in fractal zone structures was first proposed by Berry [11] and has attracted a great deal of interest in the ensuing years, particularly within antenna engineering [12]. Fractal zone plates, for example, have been theoretically proposed and demonstrated [13, 14]. In spite of the apparent separation in many papers devoted to fractals, these solutions are a subset of the underlying coherent scattering phenomena first described as Fresnel lenses. Consequently, it is not surprising similar properties can be found, including multiple foci [15]. The reason why they diffract has not been clearly elaborated in part because of the often very complex nature of the fractal description compared to that of the more traditional Fresnel diffraction. However, from our arguments above the very property of fractals, that of self-imaging, leads to structures which for all intensive purposes resemble omnidirectional filters and hence the same mechanism is invoked, despite their complexity. This explains why they diffract so efficiently. Thus there is a logical extension of the work on Fresnel fibres to examine those with more complex zone cross-sections such as a fractal zone cross-section (FZC). Solid core fractal fibres for assisting tapering have been recently proposed and demonstrated by scaling down a series of air hole rings towards the central core of the structured fibre [16]. Work which has explored Fractal zones for optical localisation include the reflection and transmission selectivity modelled and experimentally observed in fractal arranged multilayer arrays [17], and the localisation of electromagnetic fields demonstrated in 3-D fractal zone structures [18]. The work here extends Fresnel fibre technology and focuses on generating optical localisation in 2-D and waveguide propagation using a simple triadic Cantor FZC in place of the commonly used chirped BZC. An outcome of this work is to realise wide bandgap, large mode area hollow core fibres for sensing and other applications that can in principle be fabricated much more easily than conventional fibre bandgap technologies.

2. Cantor based FZC

The fractal zone cross-section (FZC) structure employed is defined by the self-similar triadic Cantor fractal [10] illustrated in Fig. 1. Self-similar fractals are made of n scaled-copies of themselves. The scale factor (1/r) is related to the number of copies, n, through the fractal dimension, d, such that [10]:

The Cantor bar fractal shown in Fig. 1 corresponds to the refractive index profile of the studied fibres at different exponent orders of growth (S) and it is equivalent to the number of zones within the cross section. At this stage we note that as for the Bragg condition often used to define Bragg fibres, or linear periodic Fresnel fibres, this is a solution that is optimized for 1-D and not necessarily for 2-D. Therefore, similar to the BZC, which is surpassed in performance by the chirped BZC, further improvements in performance are anticipated when these structures are fully optimized for the 2-D cross-section. In the specific case examined here n = 2 and 1/r = 3, consequently d = 0.631. As a result of the fractal growth there will be 2^S zones of thickness σ= (1/3)^S, which essentially indicates that the Cantor fractal has increasingly smaller constituents, but the overall Bragg structure is maintained. This resembles simple omnidirectional filters [19].

If the contrast between zones is assumed to be that of silica and air, then in practice there is a limit to the number of zones one can employ before they are quickly sub-wavelength potentially raising other complex interactions. From a theoretical perspective, this is not such a problem since the vector wave equation solution for a diffractive waveguide depends on the complete phase shift of the confined electric field at each point of reflection at the various zone interfaces. For the purposes of this work we restrict the number of zones employed to low level order FZC of not more than S = 4, and do not consider other interactions, including optical tunneling and plasmon excitation, that complicate the analysis when the zone boundaries become significantly less than the wavelength of propagating light. It is worth noting that at S→∞ the waveguide solution approaches free-space diffraction-free beam generated by self interference [1, 2, 20].

 

Fig. 1. Generation of Cantor bar with five stages of growth (S).

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The resulting FZC fibres are presented in Fig. 2. The fibre core diameters are chosen to be 22 μm, similar to other large air core diffractive fibres already demonstrated [9, 21]. A traditional periodic Bragg zone cross-section (BZC) is obtained for S = 2 (Fig. 2). The second and third fibres correspond to S = 3 and S = 4. As S increases the thickness of the self imaged silica layers becomes thinner, ranging from 3.7 μm (S = 2) to 0.41 μm (S = 4). What is interesting to observe is that the basic overall Bragg structure is maintained whilst each zone is increasingly divided into self replicated images. This is equivalent to the addition of finer layers within alternate zones so that an omnidirectional filter is created. It should therefore give rise to a much wider angular and spectral bandgap as a result of the increased phase conditions imposed by the finer structure [19].

 

Fig. 2 The cross-section of a Fresnel fibre with triadic Cantor fractal zones at different exponent orders representing increasing stages in the evolution of the fractal.

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3. Numerical simulation and results

The guidance properties of the fundamental-like mode are evaluated using a full vectorial algorithm for 2-D structures which has been successfully used to design diffractive fibres [22]. The algorithm solves Maxwell’s equations based on the adjustable boundary condition -Fourier decomposition method (ABC-FDM) [22]. It calculates the effective index and the confinement loss of modes of fibres with arbitrary structures from the vector wave equation. Finite differences are used in the radial direction while the Fourier decomposition method is used in the angular direction. Since all designs studied in this paper are circularly symmetric, the computational demand is low and the simulations are run on a desktop computer.

Generally, finite cladding diffractive fibres have no bound modes (only leaky modes)-the definition of guidance is therefore open to variation depending on the acceptable limits of a particular application; e.g. fibre sensors do not require ultra low loss over a km. Fig. 3 shows the confinement loss of the fundamental-like mode for the Fresnel fibres with Cantor fractal cross section. The fibre corresponding to S = 2 (Bragg fibre) has four transmission bandgaps over 850-1600nm, with bandwidths varying between 75 and 115 nm. As S (from S = 2 to S = 4) goes up, the bandwidth increases to over 690 nm and the confinement losses decrease substantially (0.1dB/km), values useful for long haul telecommunications applications ignoring other factors that add to loss. This low loss is due in part to the increased number of reflecting interfaces of the omnidirectional like structure of the waveguide cladding. The specific increase in bandwidth supports this description of the triadic Cantor FZC.

 

Fig. 3. Confinement loss of the fundamental-like mode for Fresnel fibres with Cantor fractal zones at different orders of evolution stages.

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Fig. 4. Modal field profile of the lowest order air guided mode for the Fresnel fibres with fractal zones cross sections.

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The corresponding calculated modal field profiles were calculated at 1.4 μm and are shown in Fig. 4. For all fibre designs the light is confined in the hollow core due to the coherent scattering at the omnidirectional–like cladding structure, or equivalently, the Fresnel fibres with fractal zone cross-sections (FZC).

4. Discussions

Numerical simulation has supported the arguments proposed in the introduction. Further, the results obtained using a very simple triadic Cantor FZC indicates that these structures offer a real and important alternative to current fibre bandgap technology. Many other fractal designs exist. The ability to widen the bandgap within simple circularly symmetric structures is extremely attractive for a number of sensing, optical transport and other applications. Further, we have shown that very basic expressions defining the practical parameters required to make these fibres are readily employed so for simple structures numerical computation from a design perspective can be avoided. This is supported by the availability of clearly defined omnidirectional filters that may be adopted as an alternative to fractal structures [19].

The actual fabrication of these fibres, in practice, can be done using existing structured fibre technology based on holes. For example, the zone cross-sections can be built up using a series of hollow fibres or capillary and rods which make up each zone [23], or alternatively drilled [5]. In the case of soft glasses and polymers, other approaches such as extrusion and casting can also be used. These fibres could also be manufactured using solid materials with different refractive indices [9]. From a simulation perspective, a more complex holey version of the Fresnel fibre demands significant more computational time because the required mesh is substantially larger than that required for the circularly symmetric ring structures described in this work. This increased computational time is made worse if any asymmetry in the structure exists.

Ignoring technical fabrication limitations, the ultimate loss reduction obtained will depend in part on an optimised omnidirectional structure as well as the feasible number of layers able to be incorporated. We note that the improved bandgap performance with increasingly thin bridges is also consistent with previous reports on low loss photonic bandgap fibre with ultra thin walls [23, 24] suggesting a strong physical link for diffraction in the bandgap fibres that is related to the proposed omnidirectional-like interpretation of fractal waveguide propagation.

Conclusions

A physical explanation of fractal diffraction based on the generation of essentially omnidirectional filters through self-imaging has been proposed. By way of example numerical simulation of a Fresnel fibre with a triadic Cantor fractal zone cross section (FZC) has been demonstrated. The results confirm the prediction that very large bandgap Bragg-like fibres can be designed using simple algorithms of common structures offering a new and alterative approach to making wide bandgap fibres comparable with the transmission bands of some existing Fresnel fibres. Confinement losses are also reduced below that of Bragg fibres demonstrating that fractal zones can be used to generate strong waveguide propagation. They can also be used to enhance or custom design fibres for specialty applications, such as nonlinearity, where materials are incorporated.

Future work will focus on the fabrication of these waveguides based on structured fibre technologies where holes are used to define each of the zones of the Fresnel fibres.