## Abstract

A major limitation to attaining low-loss single-mode guidance in hollow core photonic crystal fibre (PCF) is surface guided modes that are trapped in the core surround. This is particularly severe when high index (*n* > 2) glasses are used. By modelling a structure that has the characteristic features of a realistic fibre we show that, by tuning the thickness of the core wall, the influence of these ‘surface’ modes can be minimised. For a refractive index of 2.4 we predict power-in-air fractions of over 95% over a fractional bandwidth of ~ 5%, peaking at over 98%. The designs are appropriate for mid- to far-IR PCFs for which suitable glasses (e.g., tellurites and chalcogenides) have high refractive indices.

©2005 Optical Society of America

## 1. Introduction

A photonic crystal fibre (PCF) is a fine strand of glass with a periodic array of air channels along its length [1]. The microstructure of these air holes can give rise to a full two-dimensional photonic bandgap, and if this crosses the “air-line” (*β* = *k*
_{0}, where *β* is the component of the wavevector along the length of the fibre, and *k*
_{0} the vacuum wavevector) it is possible to guide light in a hollow core passing along the central axis of the fibre. As the fraction of power in air in these fibres can be very high, they provide excellent potential for applications such as power delivery [2] and nonlinear optics in gases [3,4].

Most modelling and experimental work on hollow-core PCF (HC-PCF) has so far focussed on silica fibres [2–6]. These fibres are made from glasses with *n* ~ 1.5 at visible wavelengths, and the bandgap utilised is that between the 4^{th} and 5^{th} bands (counting from the band with the largest *β*). This bandgap, which is appreciable at high air-filling fractions of ≳ 80% [2,7], is denoted a “type-I” bandgap following the notation of [7]. However, there is much interest in developing HC-PCF for use at mid- to far-infrared wavelengths (approximately 2-10 *μ*m), for applications such as power delivery, and fibre sensors and devices. Transmission of light in this wavelength region is not possible with silica HC-PCF: the desired wavelengths fall outside the window of transparency of silica, so the fraction of light in glass (although small) gives rise to large losses. Glasses which are transparent at these wavelengths have higher refractive indices but, at higher refractive index, the type-I bandgap closes up and shifts to the high-*β* side of the air-line.

It has been demonstrated, however, that a robust photonic bandgap exists in high-index glass (*n* ≳ 2.0) when the air-filling fraction is relatively low (~ 60%) [7]. This is a different bandgap from that usually exploited in silica, occurring instead between the 8^{th} and 9^{th} bands of the PCF cladding bandstructure, and is hereafter referred to as a “type-II” bandgap. Tellurite [8] and chalcogenide [9] glasses are ideal candidates for the manufacture of such structures, being transparent and having appropriate refractive indices over the required wavelength range.

Both experimental evidence [2,6,10] and theoretical studies [11,12] have shown that the anticrossings between surface modes (modes associated with the core surround) and core modes of the same symmetry play a major role in causing the transmission loss of hollow-core PCF. In order to reduce these losses, it is therefore necessary to consider core designs that suppress surface modes. This problem has been considered in silica glass for ‘undistorted’ cores, i.e., cores formed by superposing an air circle on an otherwise periodic cladding, either with or without a thin circular ring of glass forming a core wall. If there is no core wall, core radii exist such that no surface modes are supported [13, 14]. If a core wall does exist, then the core radius should be carefully selected and the wall thickness should be as thin as possible to suppress surface modes [15]. However, in fabricating real fibres, it is not possible to create an ‘undistorted’ core. Distortion inevitably occurs in practice and all real fibres have a core wall. There has been some study of distorted cores in silica, including suggestions for suppressing surface modes such as including ‘fingers’ of glass to reduce the distortion [11,14], using a thin core wall [12], or designing an ‘anti-resonant’ wall [16,17]. Our work in this paper concerns the design of realistic (i.e., distorted) cores in HC-PCF made from high-index glass which guides using the type-II bandgap.

In the same way that more surface modes are found in thicker core walls of silica fibres [15], it could reasonably be expected that the potential for surface modes to exist increases when using the greater filling fraction of glass and higher refractive index that is needed to utilise the type-II bandgap. In order to study this effect, we have carried out calculations using the fixed-frequency plane-wave method. Perhaps surprisingly, our calculations show a ‘clean’ fundamental mode (free of anticrossings) over a frequency range of ~ 5% of the central gap frequency, even for a realistic core design guided by what we believe can be fabricated.

In Sec. 2 we outline the computational method used to perform our calculations, then describe our core design and the principles on which it is based in Sec. 3. Results of our calculations are given in Sec. 4, and we conclude in Sec. 5.

## 2. Method

In this section, the fixed-frequency plane-wave method used to perform calculations is summarised briefly. We then describe a modification to the method that has enabled efficient study of the high-index glass PCF described in this work.

The propagation of light in PCF, assumed to be translationally invariant in the *z*-direction, is governed by [18]:

where *n*
^{2}(*x*,*y*) is the dielectric constant, **h**
_{t} = (*h _{x}*,

*h*) is the transverse magnetic field,

_{y}*β*is the axial component of the wavevector, and ∇

_{t}is the gradient operator in the transverse direction. All other components of the electric and magnetic fields can be obtained from

**h**

_{t}[18].

The use of a plane-wave expansion of the magnetic field has become common to solve Maxwell’s equations in photonics because of its flexibility and speed. Although the fixed-wavevector method is often used [19], we use instead the fixed-frequency method which is more convenient when studying PCF. This requires the use of an iterative solver to locate interior eigenvalues (i.e., those near to the air-line) of a non-Hermitian system. Our fixed-frequency plane-wave method is described in detail in Ref. [20], although in this work we do not use the extension to adaptive curvilinear coordinates.

When using the fixed-frequency method, the reciprocal-space representation of Eq. (1) is a non-Hermitian matrix eigenproblem of the form **Mv** = *β*
^{2}
**v**, where the vector **v** comprises reciprocal-space components of both *h _{x}* and

*h*. In practice this is solved by an iterative eigen-solver, which requires only the result of the matrix-vector multiplication (

_{y}**M**-

*σ*

**I**)

^{-1}

**u**for arbitrary vectors

**u**, where

*σ*is a constant chosen to shift the eigenvalues such that those of interest become extremal. Each matrix inversion required by the iterative eigensolver is equivalent to solving a set of linear equations (

**M**-

*σ*

**I**)

**w**=

**u**. In Ref. [20] we explain how this is done using the GMRES linear solver [21] and a simple preconditioner

**P**consisting of a combination of an exactly-inverted block of (

**M**-

*σ*

**I**) and the Jacobi preconditioner. This amounts to solving the system

Calculations involving only a PCF cladding, which do not need a supercell dielectric function, are sufficiently small for this preconditioning method to be effective. However, application of this method to high-index PCF structures when using supercells tends to be problematic, with the ineffectiveness of the simple ‘block/Jacobi’ preconditioner manifesting itself as slow or failed (‘stagnated’) convergence of GMRES. We believe this is a consequence of the greater index contrast and the increased amount of glass present compared to usual silica PCF structures.

It is important to note that the storage requirement of GMRES when solving a system with *N* plane-wave coefficients is *O*(*mN*) after *m* iterations, and the amount of work required for each iteration increases as *m*
^{2}. In practice this places an upper limit on the number of iterations. One solution is to restart GMRES after a fixed number of iterations, using as an initial guess for the next set of iterations the best estimate of **w** obtained so far, but discarding the Krylov subspace built up from previous applications of (**M**-*σ*
**I**). This loss of information dramatically reduces the rate of convergence immediately after restarting [22], and we have found that it is important to avoid the need to restart the algorithm wherever possible.

In the case of modelling high-index glass structures the slow convergence of GMRES brought about by the ineffective preconditioner creates a need for repeated restarts. Consequently, determining **w** is expensive. One proposed way to improve the preconditioner is to use “flexible” GMRES (FGMRES) [21], which is a modification to GMRES allowing the preconditioner to vary between iterations. It has been suggested [21] that a few iterations of GMRES could be used in the FGMRES algorithm as a preconditioner. However, we have not found any significant improvement using this method.

One requirement of preconditioned GMRES is the result **y** of a preconditioner acting on an arbitrary vector **x**, i.e., **y** = **Px**. The perfect preconditioner would be P = (M-sI)-1, but consider instead the approximation given by **P** = (**M̃** - *σ*
**I**)^{-1}, where **M̃** is the matrix corresponding to a system similar to that described by **M**, but somehow easier to invert. Determining **y** is then equivalent to solving a new linear system given by:

where **Q** can be the simple ‘block/Jacobi’ preconditioner, but created using the matrix **M̃** rather than **M**. Again, we use GMRES to solve Eq. (3); we denote this the ‘inner’ GMRES loop, and the iterative solution of Eq. (2) the ‘outer’ loop.

When using the plane-wave method to describe PCF structures, smoothing of sharp dielectric interfaces is needed to ensure convergence [7,20]. We use Gaussian smoothing for this purpose. In practice, we find that, when studying high-index glass structures, applying a greater amount of smoothing to a structure improves the rate of convergence of GMRES; the unfortunate side-effect is that it can perturb the eigenvalues *β*
^{2} from their true values. However, this observation shows that a structure with more smoothing provides an ideal candidate for the matrix **M̃** : it is an easily-controllable approximation to **M** (since **M̃** → **M** as the smoothing is reduced to that of **M**), and empirically we find that smoother structures converge more quickly. Convergence of the inner GMRES loop for this structure is relatively fast, and we find that the preconditioner obtained enables the outer loop to converge much more quickly than with the ‘block/Jacobi’ preconditioner alone. Neither the inner nor outer GMRES loop ever requires restarting. Typical numbers of iterations required are given in Sec. 4.

## 3. Fibre Design

The design of PCF involves a trade-off between developing structures that have desirable properties (such as wide bandgaps and ‘clean’ core modes without anticrossings with surface modes), and ensuring that these structures are physically realisable. In this section we describe the design of a realistic PCF made from high-index glass that guides light using the type-II bandgap.

For a glass of refractive index *n* = 2.4, the type-II bandgap at the air-line is at its greatest width in frequency when the radius of air holes is *r* ≈ 0.4Λ, where Λ is the pitch of the triangular lattice. (A map of the photonic density of states for this structure is given in Ref. [7].) A reasonable starting point for the design of a hollow-core fibre to use this bandgap is therefore a cladding of air holes with *r* = 0.4Λ.

A 19-cell PCF core was designed, for ease of modelling, using only geometrical shapes (rectangles and circles), as shown in Fig. 1. The details of the core design were chosen using currently existing silica fibres with 19-cell cores as a guide; the similarity between our geometrical core and an existing silica fibre is shown in Fig. 2. In particular, we have aimed to reproduce the relatively constant core wall thickness of the silica fibre and the fact that the cladding structure returns to being undistorted beyond one ring of holes around the core. Because a much lower air-filling fraction is needed for the type-II bandgap, we use circular holes rather than rounded hexagons in the cladding region.

In designing a PCF structure, it is desirable both to minimise leakage out of the core and to reduce the fraction of light present in the glass. One possible way to achieve this is to use the principle behind antiresonant reflective optical (ARROW) waveguides [16,17,23]: the core wall thickness is selected such that it forms an antiresonant Fabry-Perot ‘reflector’ for the radial component of the wavevector. In silica, this approach has been shown to be highly effective in designing low-loss HC-PCF [16, 17]. However, the core wall thickness required for antireso-nance in silica is significantly thicker than the strut thickness in the cladding. If we apply the same criterion to *n* = 2.4, we find antiresonance at the air-line for wall thicknesses *t* in the range 0.124Λ < *t* < 0.135Λ for values of *k*
_{0} within the type-II bandgap. This gives a thinner core wall than the strut thickness, which is of the order 0.2Λ. In this case glass in the distorted region becomes more important than in the core ring itself and our numerical results show that the antiresonant thickness is far from optimal (it supports many surface modes). Instead, we have found that thinner core walls are more suitable and have determined an optimal wall thickness by a trial-and-error process, the results of which are given in Sec. 4.

## 4. Results

We have carried out calculations of the photonic bandstructure of the PCF design shown in Fig. 2 for core wall thicknesses *t*/Λ = 0.03,0.05,0.07. In this section we describe how the calculations were carried out, then present and comment on the results obtained.

In order to employ the plane-wave method, we use a 9×9 supercell description of the PCF dielectric function and a fast Fourier transform (FFT) grid of size 512×512. The Gaussian smoothing (see Ref. [20]) of the dielectric function corresponds to a full-width half-maximum of 0.03Λ for the mode-solver, and 0.1Λ for the smoothed preconditioner as described in Sec. 2. Typically we find that 3-6 matrix-vector operations are required per eigenvalue, each requiring 20-50 outer GMRES operations. Each outer operation requires an application of the iterative preconditioner, which for the highly smoothed structure converges within 15-30 inner GM-RES operations. Note that typical silica PCF structures require 3-8 matrix-vector operations per eigenvalue, each requiring 20-40 outer GMRES operations, but the simple ‘block/Jacobi’ preconditioner is sufficient to give this rate of convergence without the need for an inner GM-RES loop.

To distinguish between modes, we determine modal symmetry by examination of the magnetic field in reciprocal space, where field components are labelled by reciprocal lattice vectors **G**. The 7 lowest-|**G**| components of the magnetic field (**G** = **0** and the first **G**-vector star) are sufficient to determine the symmetry. The fibre has *C*
_{6v} symmetry leading to 8 possible field symmetries (of which 4 are non-degenerate and 4 form two degenerate pairs); we label these *p* = 1…8 following the notation of [24].

Figure. 3(a) shows a selected region of the type-II bandgap for the structure with core wall thickness *t* = 0.03Λ. The horizontal axis shows (*β*-*k*
_{0})Λ, extending from the air-line (the right-hand edge of the plot) to *β*Λ = *k*
_{0}Λ - 0.4; the vertical axis shows frequency *k*
_{0}Λ. Air-guided modes can be seen in this figure as near-vertical lines, whereas surface modes are visible as lines with a shallower slope. The fundamental air-guided mode can be seen as the rightmost near-vertical line; it is followed as *β* decreases by a group of the three modes of higher order (a non-degenerate mode, a doubly-degenerate pair, and another non-degenerate mode).

In order to extend the range over which the fundamental mode is free from anticrossings, it is necessary to shift the surface modes visible in the lower-right corner of Fig. 3(a) to higher *β*.

This may be achieved by the addition of glass, i.e., by increasing the core wall thickness slightly. In Fig. 3(b) we show the same region of the bandstructure but for a wall thickness of *t* = 0.05Λ. At this thickness the fundamental mode is ‘clean’ over almost the entire range shown, and its anticrossings with surface modes are weaker. Further increasing the wall thickness, as shown in Fig. 3(c) for *t* = 0.07Λ, continues to sweep the surface modes to higher *β* away from the fundamental mode (they can still be seen in the lower-right corner of Fig. 3(c)), but also leads to the appearance of new surface modes introduced at the upper edge of the bandgap. These new modes reduce the range over which the fundamental mode is ‘clean’. We have considered a range of cases and find that thicker core walls significantly worsen the problem of surface modes. This is consistent with our experience with silica glass, and that of other authors [15]. We show an expanded bandstructure for our optimal thickness *t* = 0.05Λ in Fig. 4.

The nature of the surface modes supported by the core with wall thickness *t* = 0.05Λ is shown in Fig. 5. We note that the surface modes at the high-frequency edge of the ‘clean’ region tend to have intensity localised in the glass struts supporting the core wall and in the core wall itself, whereas those towards the low-frequency edge of the ‘clean’ region appear to be peaked further from the core wall in the glass surrounding the ring of distorted air holes. However, this is not a general feature of all of the surface modes and we have not been able to determine any specific physical reason for the existence of the large ‘clean’ region.

Figure 6 shows the six lowest-order air-guided modes of the PCF structure with wall thickness *t* = 0.05Λ at *k*
_{0}Λ = 5.5. We also give the fraction of power in air, which we define to be:

where **S** is the Poynting vector, **ẑ** is the axial unit vector, and d*A* is an area element of the unit cell. We observe that *P*
_{air} of the fundamental mode remains above 95% over a frequency range of approximately 5.4 ≤ *k*
_{0}Λ ≤ 5.7, comprising ≈ 5% of the central gap frequency at the airline. The maximum fractions of power in air of over 98% occur in the range 5.5 ≤ *k*
_{0}Λ ≤ 5.6. Higher-order modes also have a relatively high fraction of power in air, with the six modes shown in Fig. 6 all having *P*
_{air} > 95% at *k*
_{0}Λ = 5.5.

## 5. Conclusions

We have demonstrated computationally that hollow-core guidance in PCF made from high-index glass is possible with a fundamental air-guided mode having a fraction of power in air of up to 98%. This is a particularly high fraction given that the core diameter *D* is rather small relative to the wavelength of guided radiation (*D* ~ 4.5*λ*). A similar fraction of power in air can be obtained in 7-cell silica fibres, but in this case the core diameter is relatively larger (*D* ~ 7*λ*) [25].

Our HC-PCF design has the characteristic features of a realistic fibre, including distortion around the air core. We therefore believe that it should be possible to fabricate a structure similar to that in Fig. 2, and that similar designs would produce comparable results over a refractive index range of approximately 2.0–2.8.

## Acknowledgments

This work was funded by the UK Engineering and Physical Sciences Research Council; and Dr Jas S. Sanghera and Dr Ishwar D. Aggarwal of the US Naval Research Laboratory, through the Office of Naval Research International Field Office (ONRIFO).

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