Efficient modelling of holey fibers

Abstract

Recently great interest has been generated by the development of what have become known in the literature as (PCFs), i.e. single material optical fibers with a regular array of air holes along their entire length (see Fig. 1). Fig. 1. Holey fiber cross-section; air holes are arranged in a hexagonal lattice in the cladding region.
The large, highly controllable, periodic variations of refractive index provided by these fibers offers exciting new opportunities for the control and guidance of light, promising the development of fibers with unique transmission characteristics. For example, for a hexagonal arrangement of holes with a large air fraction, it should be possible to guide light using the formation of a full 2D photonic band gap. Guidance by this mechanism has yet to be demonstrated experimentally due to current difficulties in fabricating the required fiber.
Another means of guiding light in such structures has been demonstrated requiring significantly less air within the fiber (i.e. d/lambda is small), and which are more readily manufactured. By introducing a high index defect into the periodic structure (i.e. eliminating a hole), guidance is achieved due to an effective volume average index difference between the defect region and the periodic cladding region. The effective index difference between core and cladding is a strong function of wavelength, since at longer wavelengths the mode extends further into the holes thereby reducing the effective cladding index. This results in unique and potentially useful properties for such fibers including, amongst others, single-mode operation over broad wavelength ranges, large mode sizes and unusual dispersion characteristics, as demonstrated herein.
The basic operation of these fibers does not depend on having a periodic array. In principle, other arrangements of holes may serve a similar function. For this reason we refer to fibers operating in such a fashion as (HFs) to differentiate them from fibers operating due to photonic band gap effects and for which we retain the PCF label.
Whilst considerable experimental advances have been made in the fabrication and basic understanding of HFs, a fully satisfactory theoretical model has yet to be developed to enable reliable, accurate predictions for their propagation characteristics. Such tools are essential for the successful development of HF technology. Effective index models1, which largely ignore the complex refractive index profiles within HFs provide some insight into their operation but are unable to predict modal characteristics such as for example, dispersive properties3. In order to be able to derive such information a full numerical model is required.
The modal properties of HF can be modelled using an adaptation of the full-vector technique developed by Silvestre et al. In this technique, the modal fields are decomposed into plane waves, and the wave equation is then solved to find the propagation constants and modes. As this approach accounts for the complicated cladding structure, it can accurately model HF. However it is not efficient as it does not take advantage of the localisation of the guided modes, and so many terms are needed for an accurate description. This technique involves defining the refractive index profile over a restricted region and using periodic boundary conditions to extend the structure over all space. Hence an additional periodicity is imposed on the system (i.e. a periodic distribution of defects), which therefore restricts its applicability.
An alternate scalar approach was recently developed by Mogilevtsev et al., in which both the refractive index structure and the electric field are described by localised Hermite-Gaussian functions. Although this technique takes advantage of mode localisation, and so is more efficient than the plane-wave method, Hermite-Gaussians provide a poor description of the non-localised transverse refractive index profile in these fibers, once again severely limiting the applicability of the approach. In this work we describe the index defect and the air hole lattice independently. By decomposing the electric field and the index defect into Hermite-Gaussians, and the lattice of holes using cosines, each quantity can be represented efficiently and accurately. The technique is significantly more efficient than previous approaches since it takes advantage of the localisation of the guided modes, and is extendable to the full vector case.

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ORCID for D.J. Richardson: orcid.org/0000-0002-7751-1058

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