1. Introduction

Hollow-core photonic crystal fibre (HC-PCF) guides light by means of a full two-dimensional photonic band gap that crosses the “air-line” β=k ₀ (the boundary between propagation and evanescence in vacuum) into the region β<k ₀, where β is the wavevector component along the fibre axis and k ₀ the vacuum wavevector [1]. A considerable amount of work has already been done on both the modelling and fabrication of HC-PCF, but, largely for historical reasons, the vast majority of this has been concerned with fibres made from silica glass (refractive index ~1.46). There is considerable interest in developing HC-PCF for low-loss transmission in the mid-IR (λ>2 µm), outside the transparency window of silica glass. Suitable glasses include tellurites [2] and chalcogenides [2], which are transparent in the mid/far-IR. These glasses have relatively high refractive indices (nominally 1.8–2.3 and 2.4–2.7 respectively), and for realistic air-filling fractions this causes the bandgap so far used in silica-based HC-PCFs to close up and shift to the high-index side of the air-line. This rules out the possibility of mid/far-IR HC-PCFs based on these band gaps, which we will designate ‘type-I’.

In this paper, we report an unexpected new type of bandgap that exists at β≤k ₀, and which becomes substantial in a triangular lattice of holes at refractive indices of ≳2.0 and air-filling fractions of ~60%. This air-filling fraction is much lower than that required (around 80%) to open a substantial type-I gap in a silica HC-PCF. This new band gap, which we will call ‘type-II’, makes low loss mid/far-IR guidance in HC-PCF a practical possibility.

2. Method

The photonic density of states (DOS) was calculated for uniform triangular array of cylindrical holes in a glass matrix. Calculations were performed for a wide range of refractive indices, from n=1.5 to n=3.6, and for a wide range of hole radii, from 30%Λ to 49%Λ, where Λ is the lattice pitch (i.e. the air-filling fraction varies from 33% to 87%). The DOS is obtained directly from the calculated photonic bandstructure of the cladding. In this section, we outline our method for calculating this bandstructure and the associated density of states. The fundamental equation that governs electromagnetic waves in a PCF (which is assumed to be uniform along its length in the z direction) is [3]:

where h(x,y) is the transverse component of the magnetic field, n²(x,y) represents the dielectric function of the cladding structure, ck ₀ is the angular frequency of the radiation, and we call β the ‘propagation constant’. We use a plane-wave representation of h(x,y), which is a natural basis for the calculation of the bandstructure of the periodic cladding, and is often used for defective structures (e.g. hollow-core PCF) in conjunction with a supercell [4] description of the dielectric function.

The eigenvectors of the matrix eigensystem corresponding to Eq. (1), in the plane-wave basis, describe modes which travel with different propagation constants β along the fibre, at fixed frequency. However, it is more common to use an eigen-solver which fixes β and calculates the values of k ₀ [5]. The reason for this is that the resulting equation is Hermitian, so plane-wave-based iterative eigen-solvers, developed for large-scale electronic structure calculations, can simply be adapted to the photonic case [6]. However, Eq. (1) is non-Hermitian, so fixed-frequency plane-wave codes have been forced to rely on exact matrix diagonalisation [4], which is impractical for large supercells because the scaling with matrix size is so poor (of order N ³, where N is the number of plane waves).

We have developed an iterative eigen-solver that solves Eq. (1) for a plane-wave basis. Here we give a brief description of the method; full details will be provided elsewhere [7]. The eigensystem that results from Eq. (1) can be expressed in the form Mx=ξ x, where x is the vector of the Fourier amplitudes of h, M is a large, non-Hermitian matrix, and the eigenvalue ξ=β ². For hollow-core guidance we are interested in states near the air-line, where β≈k ₀. It is therefore the interior eigenvalues, close to ξ=$k_0^2$, that are required. Efficient iterative eigensolvers exist for extremal eigenvalues [8], so we transform the eigensystem to (M-σI)^-1 x=µ x. The eigenvectors x are identical to those of the original system, and the eigenvalues µ are given by µ=1/(ξ-σ). If we choose σ=$k_0^2$, then the required interior ξ eigenvalues are transformed to extremal µ eigenvalues. To solve the transformed system we use the ARPACK package [9], which has efficient routines for both non-Hermitian and Hermitian matrices. ARPACK requires only the results of (M-σI)^-1 v operations, where v is an arbitrary vector. Each matrix inversion is equivalent to the solution of the set of linear equations (M-σI)u=v. To solve these equations we use the standard iterative solver GMRES [10], which again works for non-Hermitian as well as Hermitian systems. GMRES requires only matrix-vector multiplications of the form (M-σI)w and, if fast Fourier transforms (FFTs) are used for these, then the computation of selected eigenvalues near the air-line is fast and economical [7]. The scaling of computation time with respect to the number of plane waves is reduced to order NlogN.

To calculate the density of states of the PCF cladding at a fixed frequency, values of β ² are computed on a uniform grid in transverse wavevector space which covers the one-twelfth-area irreducible wedge of the first Brillouin zone. The DOS, ρ(k ₀Λ,βΛ), is defined to be the number of states between βΛ and βΛ+δ(β Λ):

where k represents the transverse wavevectors and i labels the eigenvalues for each k. The weights, w_k , are normalised to Σ_k w_k =1. In practice the delta-function in Eq. (2) is replaced with a triangular function of unit area, where the width of the function must be small enough to avoid excessive blurring of the DOS, but large enough to produce a smooth DOS. We have found that using 64 k-points in the irreducible wedge, together with a triangular function of FWHM ΔβΛ=0.2, gives a well-converged DOS. In the continuum limit of Eq. (2), the vacuum density of states under our definition becomes ρ_vac =2πβΛ/(3^1/2). Throughout this paper the calculated DOS is normalised to this vacuum value so that we can immediately see whether the DOS at a particular combination of k ₀Λ and βΛ is enhanced or suppressed relative to the vacuum. To produce DOS plots that are continuous along the frequency axis we use linear interpolation, based on the β ² eigenspectra calculated for every point on a fixed sampling grid of k ₀Λ values, for each transverse wavevector in turn.

Because of the high refractive index contrast in some of our structures, we have made a careful check of the convergence of the plane-wave calculations. A Gaussian smoothing of the Fourier components of the refractive index is applied, such that the Gaussian function falls to e^-4 at the edge of the chosen reciprocal space FFT grid. Fourier components of the h-field are included within a circular cut-off that extends to the edge of the same FFT grid. Convergence of the calculated β ² values was then analysed as a function of the FFT grid size (note that as the grid size is increased, the extent of smoothing in real space is reduced). For comparison, values of β ² which are known to be exact to within a certain number of decimal places were obtained for several cases that span the range of refractive indices, air-hole radii and frequencies for which the DOS has been calculated. These ‘exact’ values were calculated using a KKR-based method [11] with sufficient basis states to obtain convergence to at least three decimal places in β Λ. Excellent agreement between the plane-wave and KKR results is obtained for an FFT grid of size 64×64 across the whole range of structures, and this grid size has been used for the results in the next section.

 

Fig. 1. Normalised DOS maps for different hole radii, for three different glass indices: (a) 1.5, (b) 2.4, and (c) 3.3. The hole radii are (from left to right): 48%Λ, 44%Λ, 40%Λ, 36%Λ, and 32%Λ. The Roman numerals indicate the types (I, II, or III) of the most significant gaps, under our classification scheme.

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3. Results and interpretation

3.1. Evolution of the DOS with respect to glass-index and air-filling-fraction

Many DOS maps were calculated, for various glass indices and hole radii (expressed as a percentage of the lattice-pitch, Λ), for a cladding structure consisting of an infinite triangular lattice of cylindrical holes. In Fig. 1 we present only a selection of these graphs, with the intention of illustrating the main features that are present in the DOS, in particular the various types of band gaps that arise. A more complete picture of the evolution of these band gaps with glass index and hole radius is given in Fig. 2.

 

Fig. 2. (a) Schematic diagram of the quantities used in the definitions of the normalised gap width and the normalised gap depth. (b) Normalised percentage depths and (c) widths of the lowest-frequency air-line-bandgaps for a range of glass indices and hole radii.

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Each plot in Fig. 1 is a colour-map of the computed normalised density of states (DOS) as a function of normalised frequency k ₀Λ and excess propagation constant (β-k ₀)Λ. White indicates zero DOS, and grey represents a high DOS. Regions of ((β-k ₀)Λ, k ₀Λ) space in which the DOS is very low but nevertheless nonzero are coloured red. The red vertical line running through the centre of each DOS map is the ‘air-line’, (β-k ₀)Λ=0.

Before discussing these plots, we state our definitions of the gap-width and gap-depth. A schematic diagram of the quantities used in these definitions is to be found in Fig. 2(a). We define the ‘width’ of an air-line-bandgap to be its extent along the frequency axis, and the term ‘depth’ will be reserved for describing its extension along the (β-k ₀)Λ axis. Both the gap-width and the gap-depth are normalised by dividing by the central frequency k ₀Λ, and the normalised gap measurements thus obtained are expressed as percentages.

Fig. 1(a) refers to a glass index of 1.5, which approximately corresponds to that of silica. In this case, the conventional ‘type-I’ air-line gaps predominate, and these only begin to appear for hole radii in excess of approximately 44%Λ. Type-I band gaps are found between the 4^th and 5^th bands of the cladding bandstructure (numbering from the band with the highest β ²). Note that although the type-I air-line-gap is quite wide (about 10%) at a hole-radius of 48%Λ, the gap-depth is small, and to make it visible we have stretched the horizontal axis so that it extends from -1 to 1. The type-I gaps for an index of 1.5 are therefore wide but shallow and are sensitively-dependent on the hole-radius. It is this gap that practical, low-loss hollow core fibres have been based on [12]. In addition to the type-I gaps, another higher order gap is present at this index for hole radii of 40%Λ and 44%Λ, which we call ‘type-II’. For an index of 1.5 these gaps are extremely narrow.

The situation is very different at a glass index of 2.4, as can be seen from Fig. 1(b). Here, the type-I gap has moved to the high β side of the air-line and the higher order type-II gaps are now pre-eminent. These type-II gaps have maximum widths at a hole-radius of about 40%Λ, and they close up slowly with increasing extent of departure from this optimal value. The impressive robustness of a type-II gap is illustrated by the fact that it still exists even if the hole-radius is 4%Λ above or below 40%Λ (see Fig. 1(b)). The optimal hole-radius of 40%Λ corresponds to an air-filling fraction of approximately 60%. This fraction is surprisingly low, and it is also fortunate because various practical difficulties in HC-PCF fabrication become more manageable at lower air-filling-fractions. The lowest-frequency type-II gap is found between the 8^th and 9^th bands of the cladding bandstructure and this gap remains the widest and deepest type-II gap over the full range of glass indices and hole radii, as will be shown in Fig. 2.

For very high glass indices (between 3.0 and 4.0) the type-II gaps become more numerous, and their proliferation is accompanied by the appearance of yet another type of air-line-gap: ‘type-III’ (see Fig. 1(c)). This type-III gap is not new; it is merely a continuation of the familiar ‘in-plane’ (i.e. β=0) gap [5,13]. The type-III air-line gap probably cannot be used for HC-PCF guidance, because commercial optical-fibre glasses do not have refractive indices higher than about 2.7.

3.2. Demonstration of the robustness of the new type-II air-line gap

From the extensive set of DOS maps referred to above, we have extracted the widths, depths, and central frequencies of all of the air-line-gaps, and we have used this data to generate Figs. 2(b) and 2(c), which show the normalised depths and widths (respectively) of each type of gap having the lowest central frequency, for many combinations of index and radius. Type-II gaps occupy the majority of parameter-space in Figs. 2(b) and 2(c). They are therefore very resilient to perturbations in both index and hole-radius. Type-I gaps exist only in the lower right-hand corners (i.e. at the extremes of large hole-radius and low glass-index), whilst type-III gaps exist in the upper right-hand corner (i.e. at large hole-radius and very high glass-index).

 

Fig. 3. Axial Poynting-vector profiles of selected core-guided and cladding-guided modes calculated using an 8Λ×8Λ supercell in conjunction with an FFT grid size of 512×512. The Poynting vector magnitude is normalized to unity over the supercell. The photonic crystal cladding has holes of radii 40%Λ in a glass matrix of index 2.4, the frequency k ₀Λ is set to 5.4, and the hollow core has a radius of 1.0Λ.

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Additional confirmation of the tenaciousness of the type-II gap was obtained by perturbing the shape of the holes. If the cylindrical holes are replaced by holes having hexagonal cross-sections, thus forming a honeycomb lattice, then strong type-II gaps still exist at an air-filling fraction of approximately 60% for refractive indices n≳2.0.

3.3. Practical fibre parameters for type-II hollow-core guidance

As a concrete example, we will now give the cladding parameters for an experimentally-viable HC-PCF based on a type-II gap. The most substantial gap for an index of 2.4, which is attainable with a chalcogenide glass of appropriate composition, appears at a normalised frequency (k _oΛ) of approximately 5.4 (see the corresponding DOS plot for 40%Λ hole-radius in Fig. 2(b)), which implies that for guiding mid-IR light at a wavelength of 3.5µm (i.e. in the middle of the mid-IR spectroscopy band 2-5µm), the lattice pitch of the cladding must be ~3.01µm, and the cladding hole-radius must be ~1.20µm. The eigenmodes of an HC-PCF having this cladding structure were computed using the method outlined in Sec. 2, assuming that the hollow core has a radius of 1.0Λ (i.e. ~3.01µm in this case). Fig. 3 shows the modes closest to the air-line. The computation parameters are given in the caption, and each mode profile is labeled with its corresponding axial propagation constant, βΛ. The modes at βΛ=5.735 and βΛ=4.104 represent the edges of the bandgap; all other modes are gap-states that are localized in or near the central defect. They can be characterised either as ‘surface states’ (for example the modes at βΛ=5.550 or βΛ=4.882) or as ‘air-guided states’ (for example the pair of degenerate fundamental modes at βΛ=5.041). The apparent breaking of symmetry of the degenerate modes is due to the arbitrary choice of superposition of degenerate eigenvectors.

4. Conclusion

By numerical simulations, we have discovered a new type of air-line bandgap, of remarkable robustness, that is of considerable importance in the design of practical HC-PCF for guidance in the mid/far-IR. The normalised gap-depth and gap-width of this type-II band gap are both maximised at an air-filling fraction of ~60%, for any given glass index beyond ~2.0. Our results show that it is possible to obtain satisfactory guidance in hollow-core PCF made from high-index glass. There are various practical issues to be dealt with regard to the fabrication of soft-glass HC-PCF, but recent work [14,15] suggests that these can be overcome. The existence of the wide, deep, and robust type-II air-line gap implies that, for example, mid-IR spectroscopy, LIDAR, and CO₂ laser-beam guidance can now be added to the ever-growing list of potential HC-PCF applications.