1. Introduction

Hollow core photonic crystal fibers (HC-PCFs) are attracting a lot of attention, not least due to the prospect of achieving signal guidance with a propagation loss below the figure attainable in solid fibers. Rayleigh scattering and multi-phonon processes in solid fibers impose a loss floor of around 0.15 dB/km, but most of the light power guided in HC-PCFs resides in air so that bulk glass loss is strongly suppressed. The loss limitations in HC-PCFs are instead set by hole interface roughness scattering [1]. This roughness is primarily due to thermally excited surface capillary waves which become frozen-in when the glass solidifies.

Since the level of hole interface roughness appears to be an intrinsic property of the fiber drawing process and not amenable to substantial reduction, lowering the loss in HC-PCF guidance becomes an exercise is decreasing the field strength of the signal carrying mode at the hole interfaces. One approach to achieving this is to increase the size of the air core. The core size is, however, limited by macro- or micro-bend loss constraints imposed by the presence of unwanted guided modes within the band gap. As the core size is increased these modes become more plentiful and the detuning of the mode used for carrying the signal from the nearest of them decreases. This restricts the core diameter to being less than about 15 times the operational wavelength if robust low loss quasi-single mode guidance is to be maintained under practical usage conditions (which entails, for example, tolerance of a bend with a radius of order 10 cm).

For a given core size, there remains scope to alter the modal properties by careful design of the fiber geometry. Of particular importance is the glass arrangement that surrounds the air core and separates the core and cladding regions. It has been recognised for some time that antiresonant features around a core can lead to improved confinement. Anti-Resonant Reflective Optical Waveguides (ARROWs) of various geometries have been explored, beginning with work on planar waveguide systems [2]. Waveguides comprising arrangements of antiresonant cylinders with higher refractive index than the host material can show low leakage even if the number of cylinders is relatively small [3–8]. One or more antiresonant concentric rings which encircle a guiding core have also been shown to give rise to low-leakage guidance [9–12]. In contrast to these systems, confinement within a HC-PCF is in any case provided by the band gap cladding assuming it is of sufficient extent. For these waveguides, the field exclusion properties associated with antiresonance in features which surround the core can instead be used to decrease the field strength at the interfaces and hence the interface roughness loss. The exclusion effect also acts to reduce the fraction of light power that resides in the glass so that the fiber shows reduced nonlinearity [13–15]. This is important in the delivery of high intensity short pulse radiation, where nonlinear interactions are generally undesirable.

Silica HC-PCFs are generally fabricated using a stack and draw technique. The stack comprises a set of capillary tubes of equal outer diameter which is arranged in a triangular lattice arrangement around a larger central capillary tube, as shown schematically in Fig. 1(a). The simplest means of achieving antiresonance in the core surround of the fabricated fiber is to modify the thickness of the central capillary tube in the stack, as shown in Fig. 1(b). After the draw, the core surround of the fiber maintains an approximately even thickness t around its perimeter. The thickness of the central tube is chosen to result in a value of t which gives antiresonance within the air-guiding band gap of the cladding.

 

Fig. 1. Stacks for the fabrication of HC-PCF. The two cases (a) and (b) differ in the thickness of the central inserted core tube. In case (a), the central core tube is chosen so that the core surround of the fabricated fiber is similar to the strut width in the cladding. In case (b), the core tube is thicker so as to produce a core surround of antiresonant thickness.

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A good understanding of the improvements (in interface field strength and power-in-glass fraction) can be gained by first considering the core surround on its own, i.e. without the band gap cladding attached. This is described in section 2 for an annular core surround which is amenable to a simple analysis. A fiber design which incorporates an antiresonant core surround is then compared with one with a thinner, more conventional core surround in section 3, confirming that the beneficial properties persist even when a cladding is attached. Finally, conclusions are drawn in section 4.

2. The annular core surround geometry

The extent to which an annular tube of dielectric material (with air both inside its core and surrounding it) can confine light to its core is greatly affected by its thickness t. If the core diameter is several times the wavelength, the low order core-guided modes such as HE₁₁ propagate with a longitudinal wavevector component β along the tube axis which is very close to the free space wavenumber k=2π/λ. As a result, destructive interference takes place when the thickness t attains a value close to [11,12]

where n _gl is the index of the glass, λ is the wavelength and j is a positive integer or zero. This is simply the condition for a partial wave which emanates from one boundary of the layer, propagates to the other boundary where it is reflected, and then propagates back to the first boundary where it is again reflected, to collect a phase which is an odd multiple of π.

The electromagnetic properties of the annular geometry are readily calculated using a simple transfer matrix analysis [9]. Figure 2(a) shows the leakage loss of the HE₁₁ leaky mode of a silica tube as a function of t at a wavelength λ=1.55 μm, for two values of the core radius R. The core size R=6 μm is typical of the core in a HC-PCF operating at 1.55 μm in which the omission of seven capillary tubes in the stack defines the air core, whereas R=10 μm is typical of a core formed from 19 omitted tubes. In Fig. 2(a), the loss has been expressed both in terms of the imaginary part of the mode effective index n, and in dB/mm. It is apparent that the confinement is optimised for both core radii at a thickness near the value 0.37 μm estimated for antiresonance using Eq. (1) at j=0 (the index of silica takes the value 1.444 at a wavelength λ of 1.55 μm). The confinement associated with the larger core size is superior to that of the smaller one, but this comes at the expense of a larger number of higher order (leaky) modes associated with the core.

 

Fig. 2. (a) The confinement loss of a silica annulus as a function of its thickness t, for core radii R=6 μm and R=10 μm. The wavelength is 1.55 μm. (b) The spatial distribution of the H-field intensity at the antiresonant thickness t=0.37 μm (marked on Fig. 2(a) by the downwards arrow) for the annulus of inner radius R=10 μm. An intensity minimum occurs at the inner interface at this thickness.

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At antiresonance, the field intensity shows a minimum at the inner interface of the annulus, and the destructive interference within the silica annulus prevents the field from increasing by much outside of this interface so that radiative loss is minimized. The effect can be seen in the radial dependence of the field intensity associated with the HE₁₁ leaky mode, which is shown in Fig. 2(b) for an antiresonant annulus with core radius R=10 μm at 1.55 μm wavelength.

The scattering loss due to roughness at glass/air interfaces will be decreased as the field intensity at the interfaces is reduced. A relative measure of the scattering strength is provided by the factor F defined by

where E and H are the electric and magnetic fields propagating down the waveguide. In the numerator, the E-field is evaluated just inside each hole interface, i.e. within the air. Since the annulus supports leaky modes, the normalizing denominator term in fact involves a deformation of the integration over the transverse cross section into the complex plane [16]. For small scale Rayleigh scattering, the factor F can be shown to encapsulate the mode shape dependence of interface scattering loss. The scattering loss due to the scale-free roughness associated with frozen in capillary waves is not determined directly in terms of F, but the factor still provides a useful qualitative relative measure of the scattering loss that can be expected [1]. Figure 3(a) shows the dependence of F on the thickness t of the annular tube for the two core sizes. The antiresonance is signaled by a minimum in F, with the minimum value varying with the core radius R according to R ^-3.

The field exclusion associated with the anti-resonance also leads to a reduction in the fraction η of the modal power which propagates in the glass. For leaky waves, η will be defined by

Although for leaky modes strict association of η with a power fraction is not possible [16], for low attenuation η provides an accurate estimate of the power fraction which exists within the annulus if a perfect metal filled the region r>R_b, where r is the radial coordinate measured from the center of the annulus, and R_b satisfies ${\displaystyle {\int }_{R+t<r<R_b}}$ dS(E∧H)∙ẑ/${\displaystyle {\int }_{r<R_b}}$ dS(E∧H)∙ẑ≪1 and R_b ≫ R + t.

Figure 3(b) shows the variation of η with thickness t for the two core sizes at λ=1.55 μm. Over the plotted range of t, the normalized power attains a minimum value near the antiresonant thickness despite the reduction in the amount of glass associated with lower values of t.

 

Fig. 3. (a) The normalized interface field intensity F, defined by Eq. (2), for a silica annulus plotted as a function of its thickness t. The traces are for inner radii R=6 μm and R=10 μm. The wavelength is 1.55 μm. (b) The light-in-glass power fraction η, defined by Eq. (3), as a function of t for the same inner radius values. Both F and η show minima close to the antiresonance thickness.

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The core surrounds of fabricated HC-PCFs are deformed slightly from being perfectly circular in shape. However, as long as the thickness of the core surround is approximately constant around its perimeter, the condition given in Eq. (1) remains a good estimate for achieving antiresonance and optimal field exclusion. Calculations using a boundary element code [17], on a dodecagonal geometry corresponding to the core surrounds possessed by the fibers in Fig. 4 (i.e. the fibers stripped of their claddings), show that the properties of the HE₁₁-like leaky mode are little perturbed from those of a circular annulus with the same core area.

3. Comparison of fibers with and without antiresonant core surrounds

To test whether the favorable properties of the antiresonant core geometry persist when a cladding is attached, the mode properties of two fiber designs were compared. The first design, which is shown in Fig. 4(a), incorporates a core with a thickness t chosen to be 0.094Λ, where Λ is the cladding pitch. This thickness corresponds to the value estimated using Eq. (1) for the first (j=0) antiresonance at the normalised wavenumber kΛ=15, which lies within the band gap of the cladding. The second fiber, shown in Fig. 4(b), is identical except that its core surround thickness is 0.031Λ, which equals the thickness of the “struts” in the cladding region. The geometries of the two fibers represent an idealization of the geometries of fabricated fibers which result when 19 capillary tubes are omitted to form the basis of the core.

 

Fig. 4. Two designs of HC-PCF whose optical properties are to be compared. In case (a), the core surround has a thickness t=0.094Λ, where Λ is the cladding pitch, which gives antiresonance within the cladding band gap. In case (b), t=0.031Λ, which equals the strut-thickness within the cladding. Both the fibers (a) and (b) have identical cladding structures.

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The shape of the holes in the claddings of the model fibers in Fig. 4 was chosen to closely match the shape of the holes in fabricated fibers [18]. A unit cell for the cladding, which has an air filling fraction of approximately 92%, is shown in Fig. 5.

The mode computations were performed using a plane-wave scheme based on a variational formulation of Maxwell’s equations [19]. Each fiber structure was discretized on a 512×512 spatial grid in the calculations, based on the supercells in Fig. 4. Figure 6 shows the normalized interface field intensity F for HE₁₁-like modes, plotted as a function of normalized wavenumber for the two fiber designs. The value of F for the fiber with a thin core surround, shown by the dashed line, decreases rapidly as the band gap is entered and attains a minimum value of 0.089Λ^-1 at a normalised wavenumber kΛ=16.0. The peak occurring around kΛ=16.75 is due to the presence of an anti-crossing event occurring between the HE₁₁-like mode and a mode associated with the core surround [20–22]. A second anti-crossing occurs at kΛ=18.1, which is a frequency very close to the upper band gap edge. The continuous curve in Fig. 6(a) shows F for the fiber with a thicker core surround. Although four large peaks, each of which is associated with a mode anti-crossing event, are observed within the band gap region, the minimum value of F attained is 0.030Λ^-1 (occurring at kΛ=16.9). The thicker core surround has therefore succeeded in reducing the normalised interface field intensity by a factor of 3, suggesting that the loss of the fiber will be reduced by a similar amount if the losses are dominated by surface scattering. The reduction has come at the expense of a reduced operational bandwidth. For the idealised geometry of Fig. 4(b), the region of low F between mode anti-crossing events extends between about kΛ=16.5 and kΛ=17.5. For a fiber operating around 1550 nm, this corresponds to a bandwidth of around 90 nm.

 

Fig 5. The geometry of a cladding unit cell. The claddings of the fibers in Fig. 4 correspond to L ₁/L ₂=0.6 and w=0.031Λ, with Λ the lattice pitch.

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Figure 6(b) shows the light-in-glass power fraction η for the two fibers. Despite the substantial increase in the amount of glass associated with the thicker core surround, the minimum value attained for this fiber is 0.0012, which is 1.5 times lower than the minimum for the fiber with a thin core surround. This further confirms that the thicker core surround gives rise to antiresonant field exclusion. At such a low light-in-glass power fraction, the nonlinear phase shift is dominated by the field component in the air [13–15], despite the intrinsic nonlinearity of glass being nearly 3 orders of magnitude larger than air at atmospheric pressure.

 

Fig. 6. (a) The normalised interface field intensity F for HE₁₁-like modes, plotted as a function of normalised wavenumber kΛ for the two fibers shown in Fig. 4. (b) The light-in-glass power fraction, η, as a function of normalized wavenumber for the HE₁₁-like modes of these fibers.

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It is clear from Fig. 6(a) and Fig. 6(b) that both F and η show the same trend, rising sharply as a mode anti-crossing is approached. F is plotted against η in Fig.7 for the thick core fiber, each dot representing data at a different frequency within the band gap. It is seen that there exists an approximately linear relationship between the quantities. This indicates that discriminating the loss contribution due to interface roughness from the loss due to distributed inhomogeneities within the thin silica regions of the fiber needs to be handled carefully. Measurement of the scattered loss as a function of angle, as well as comparing the loss of fibers with different operating wavebands has resolved the issue, with the loss confirmed as being dominated by the interface roughness contribution [1].

Figure 8 shows the HE₁₁-like mode field intensity distribution for the fiber with the thick core surround at the wavenumber kΛ=16.9 corresponding to the position of the minimum in F. Near nulls of the field occur close to the inner face of the core surround, which is a manifestation of the destructive interference associated with antiresonance. The formation of the nulls is compromised at positions where the cladding attaches to the core surround, thus changing the local thickness of the glass. Throughout the fiber, the field intensity in glass is nonetheless maintained at least 18 dB lower than the peak intensity occurring at the centre of the core.

 

Fig. 7. The normalised interface field intensity F plotted against the light-in-glass power fraction, η for the fiber with an antiresonant core surround. Each dot corresponds to a different wavenumber within the band gap.

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Fig. 8. The mode field intensity distribution (log scale, 60 dB range shown), for the fiber shown in Fig 4(a), at the wavenumber kΛ=16.9 where the normalised interface field intensity F is minimized. Near nulls appear over much of the inner interface of the core surround.

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A route to further reducing the normalised interface intensity F entails decreasing the size of the nodes where the cladding connects onto the core surround ring. These features naturally diminish as the air filling fraction in the cladding is reduced. When these nodes are small, the beneficial field exclusion properties (associated with the antiresonant core surround) remain substantially intact and the role of the cladding is simply to complete the job of field confinement. Unfortunately, the thick antiresonant core possesses a myriad of whispering gallery-like modes, some of which interact with the HE₁₁-like mode (due to the deformation of the ring from circular and to residual interactions with the cladding) to give rise to anti-crossings within the band gap. The thickened core surround geometry is very different from any of the features within the band gap cladding and so acts as a sizeable defect. Hence the incursion of a number of defect modes into the band gap is to be expected. An increase in the number of core surround modes with core surround thickness has been reported in [23]. In that paper, it was suggested that the loss should be related to the number of these unwanted modes and their proximity to the HE₁₁-like modes. Since the inter-mode coupling decreases with decreasing interface field strength, the field exclusion effect of the antiresonance overrides the increase in the number and proximity of the unwanted guided modes and results in a lowering of the loss. A fuller understanding of the loss necessitates a proper treatment of the roughness spectrum which causes the coupling: this will be deferred to a later paper.

4. Conclusions

Numerical modeling results have been presented which confirm that antiresonance in the glass arrangement which surrounds the air core of a hollow core photonic crystal fiber (HC-PCF) acts to substantially reduce the field strength at the glass/air interfaces. Since roughness at the interfaces is the dominant loss mechanism in high quality HC-PCFs, this indicates that the loss of such fibers will be commensurately reduced. A comparison was made of the interface field intensity for two fibers which differ only in that one has an antiresonant core surround and the other has a more conventional core surround with a thickness equal to that of the glass struts in the cladding. This showed a predicted loss improvement of around a factor of 3 for the fiber which incorporates the antiresonant core surround.

A qualitative understanding of the role of the core surround geometry in a HC-PCF can be gained by first considering it in the absence of a band gap cladding. Simple modeling techniques can then be utilised to optimise the geometry. A core surround which approximates an annulus with an antiresonant thickness has been identified as an appropriate geometry which is easy to fabricate using the stack-and-draw technique. Such fibers show an increased number of mode anti-crossing events, but in the absence of deformations the operational bandwidth can still be around 90 nm when centred at a wavelength of 1550 nm. The measured loss of fabricated fibers with approximately antiresonant core surrounds has been reported to be as low as 1.2 dB/km [1], but with a bandwidth of just 25 nm [24]. Mode calculation based on the geometry of fabricated fibers (inferred from scanning electron micrograph images) suggest that the further lowering of the bandwidth is due mainly to the deformation of the holes in the vicinity of the core of the fabricated fiber.