1. Introduction

Hollow-core fibres that are characterised by a relatively large core and simple cladding design have been the focus of much recent attention, due to their broad transmission windows and low loss (see [1–3] and references therein). The simplest structure consists of a single ring of thin-walled, circular capillaries distributed uniformly around an outer jacket (as in Fig. 1(a)), although refinements of this basic design (for example, nested capillaries [4–6] or an additional glass membrane within the capillaries [7]) have attracted attention because of the potential for a further reduction of the loss. There is no bandgap in the cladding of these fibres and therefore core-guided modes are leaky and have an intrinsic confinement loss. A variety of names has been used to describe this new generation of fibres (inhibited-coupling, anti-resonant, negative-curvature, revolver, single-ring) which, for the first three, reflects an ongoing discussion about the nature and relative contribution of different aspects of the guidance mechanism.

 

Fig. 1. Cross sections of model hollow core fibres, where white and grey areas represent air and glass respectively. (a) A jacketed single-ring structure with $M=7$ and with the cladding capillaries just touching the outer glass jacket. (b) An unjacketed single-ring structure with $M=7$. (c) The reference concentric rings structure with $N=3$. Key structural parameters are indicated: core radius, $r_c$; inner capillary radius, $r_{in}$; gap between capillaries, $g$; glass width, $w^{(gl)}$; air width, $w^{(air)}$. Paths $A$ and $B$ are described in the text.

Download Full Size | PDF

The inhibited-coupling argument is discussed in detail in [3] and [8]. Because there is no bandgap, states localised in the cladding can exist with the same propagation constant as a core-guided mode. However, the nature of the states is very different. Away from the edges of the transmission bands, the cladding modes have a rapid spatial variation along the thin glass structures of the cladding capillaries, and therefore have a small overlap with the relatively slowly-varying core modes. An approximate model for the cladding states is presented in [9] that clearly demonstrates this effect. There must be avoided crossings between the core and cladding states, but the coupling is very weak. In [3] and [8] it is argued that anti-resonance cannot be the dominant guidance mechanism, because the loss of single-ring fibres is too small to be explained in this way. However, they consider only the anti-resonant effect of the glass that immediately surrounds the core. Other authors (for example, [10–15]) have shown that anti-resonant reflection from the air regions in the cladding can also contribute to the strong confinement of core modes. Another important feature of the guidance is the shape of the core wall, and a negative curvature (where the core surround is convex with respect to the fibre axis) has been shown experimentally and computationally to lead to a lower loss (for example, [14,16–21]). Other intrinsic and extrinsic loss mechanisms (surface scattering, material loss, bending loss) can also be relevant in practice, but we concentrate here on the confinement loss.

The premise of this paper is that inhibited coupling is a prerequisite for low loss in single-ring fibres, but that anti-resonance is the physical mechanism that determines the underlying magnitude of the confinement loss and its dependence on the wavelength and the structural parameters of the fibre. We aim to show this by building on the analysis presented by Bird [13]. In [13] a set of of highly simplified structures consisting of $N$ concentric glass and air regions was analysed, and an analytic expression derived for the confinement loss. The important point about these model structures is that, by symmetry, there is no coupling between core-guided modes and modes localised in the glass layers. Inhibited coupling is effectively perfect, but the confinement loss is not zero. It is shown in [13] that anti-resonance is the physical guidance mechanism in operation and that this provides a quantitative explanation of the confinement loss. An example $N=3$ structure with three complete anti-resonant layers (i.e. glass-air-glass) is shown in Fig. 1(c); the $N=2$ case is similar, but with the outer glass ring replaced with a glass jacket that extends to infinity.

The confinement loss incurred in propagation through a distance of one wavelength $\lambda _0$, for the fundamental HE$_{11}$ mode of the concentric layers model, is given by [13]

Equation (1) is not exact, but it is shown in [13] that it provides an excellent approximation of the true confinement loss of concentric layer structures, provided $r_c/\lambda _0$ is large and $\phi ^{(gl)}$ and $\phi ^{(air)}$ do not approach their resonant values at integer multiples of $\pi$. It provides an analytic expression for the dependence of the loss on key structural parameters including the ratio of the wavelength to the core radius, the width of the glass and air regions, and the dielectric constant. The $(\lambda _0/r_c)^{N+3}$ factor shows that anti-resonant reflection (defined by half integer multiples of $\pi$ of $\phi ^{(gl)}$ and $\phi ^{(air)}$) in successive glass and air layers can reduce the confinement loss to a very low value. We note that the parameter $U_{glass}$ used in [14] is equivalent to $\phi ^{(gl)}$, except that the factor $\sqrt {\epsilon -1}$ in Eq. (4a) is replaced with $\sqrt {\epsilon -\Re (n_{eff})^2}$, where $n_{eff}$ is the effective index of the fundamental mode. However, this difference is insignificant for large $r_c/\lambda _0$.

The question addressed in this paper is: to what extent does Eq. (1) provide a quantitative description of the confinement loss of single-ring fibres? Our aim is to understand which aspects of the loss can be captured by the simple anti-resonant model, and which cannot. Two fibre structures are considered: the archetypal single-ring design shown in Fig. 1(a) and the unjacketed single-ring model shown in Fig. 1(b) (a similar model is used in [14] and [22]). The reason for including the unphysical structure of Fig. 1(b) is that it provides a useful stepping stone between the more realistic structure of Fig. 1(a) and the analytically tractable concentric layers model. The unjacketed structure shares important features with Fig. 1(c), with the cladding in both cases having two anti-resonant glass layers and a single anti-resonant air region between the core and the outside. On the other hand, while Figs. 1(a) and 1(b) are clearly related, the relationship between the jacketed structure of Fig. 1(a) and the concentric layers model is less obvious. Parts of Fig. 1(a) (where the cladding capillaries touch the outer jacket, i.e. path $A$ in Fig. 1(a)) resemble the $N=2$ model of [13], while other parts (where there is an air gap between the capillaries and the jacket, i.e. path $B$ in Fig. 1(a)) could be considered to be closer to $N=4$, although in this case the width of the air regions is sub-optimal. Our hypothesis is that a comparison of the jacketed and unjacketed structures, and an analysis of both in terms of the scalings implicit in Eq. (1), will provide the basis for a better quantitative understanding of the confinement loss of real fibres.

In section 2 the relationship between the single-ring structures of Figs. 1(a) and 1(b) and the concentric layers model of Fig. 1(c) is analysed, and a scaled confinement loss is introduced that accounts for the variation with respect to the radius of the cladding capillaries. A systematic analysis of how the confinement loss of the fundamental mode depends on the number of capillaries and the size of the gap between them is presented in section 3. Previous authors have carried out a similar survey (for example, [21,23–25]), but not in the context of analysing a specific physical model. The dependence of the loss on structural parameters and the wavelength is analysed with reference to Eq. (1) in section 4. Section 5 is the conclusion and Section 6 is an appendix in which we present a simple tunnelling model that describes the loss due to leakage through the gaps between capillaries in single-ring structures. All the data shown in sections 3 and 4 is available at [26].

2. Theory and computation

The structure of the idealised single-ring fibres considered in this paper is characterised by the variables shown in Fig. 1: the number of capillaries, $M$; the core radius, $r_c$; the inner capillary radius, $r_{in}$; the glass width, $w^{(gl)}$; and the minimum gap between capillaries, $g$. From these, we define dimensionless geometrical parameters normalised to the the core radius: $\rho =r_{in}/r_c$; $\omega =w^{(gl)}/r_c$; $\gamma =g/r_c$. Together with $M$ and $r_c/\lambda _0$, these parameters fully define light propagation in the fibres, including the confinement loss. For a given $M$ and $\omega$, $\rho$ and $\gamma$ are not independent variables and we can define the structures using either of the sets $(M,\rho ,\omega )$ or $(M,\gamma ,\omega )$. These are related by

Table 1. Values of normalised inner capillary radius, $\rho$, for single-ring structures characterised by the number of capillaries, $M$, and normalised gap, $\gamma$, for $r_c/\lambda _0=15$, $\epsilon =2.25$, and scaled glass width $\Omega =1$.

View Table

It is also convenient to define the glass width $w^{(gl)}$ relative to its value at the first anti-resonance condition, that is, when $\phi ^{(gl)}=\pi /2$. We define

As stated in Section 1, our aim is to compare the confinement losses of jacketed and unjacketed single-ring structures and to analyse these in terms of the analytic prediction of Eq. (1). The quantities $r_c/\lambda _0$, $\epsilon$ and $\phi ^{(gl)}$ in Eq. (1) have an obvious equivalent in the single-ring models, and it is natural to replace $x_0$ by the actual value of the normalised transverse wavevector of the fundamental core-guided mode, which we write as $x_{FM}$. While the value of $x$ is very close to the Bessel function zero, $x_0$, in the concentric layers model of [13], this is not the case for single-ring fibres. The fundamental core-guided mode spreads out further in these structures than the inscribed circle that defines $r_c$, which means that $x_{FM}$ is different for different structures and is always less than $x_0$. The results of the next section show that $x_{FM}$ is typically 4 to 7% smaller than $x_0$.

In deriving an expression for $\phi ^{(air)}$ for single-ring structures, a quantity equivalent to $w^{(air)}$ is needed. Although this is expected to be proportional to the radius of the cladding capillaries, the precise relationship is less clear. We therefore consider the resonance condition between the fundamental modes of the core and the cladding capillaries, and define $\phi ^{(air)}$ relative to this. At resonance the propagation constants are equal, which implies from Eq. (2) that the $x/r$ values must also match [27]. It follows that

In order to compare different single-ring fibres, it is useful to define a scaled loss $L^{(sc)}_{\lambda _0}$ by

The confinement loss of the fundamental mode of jacketed and unjacketed single-ring fibres has been calculated using both the commercial COMSOL Multiphysics package and an in-house boundary element (BE) code. The BE code is similar in design and function to that described in [28] and, being constructed using outwardly propagating Green’s functions, is particularly well suited to the calculation of the confinement loss of leaky modes. A key advantage is that convergence of the solutions depends effectively on a single parameter; provided the numerical integrals are accurately calculated, the results of a BE simulation depend only on the density of points that describe the discretisation of the glass-air boundaries. The BE code therefore provides a robust test of the accuracy of the simulations; its own accuracy has been tested against the benchmark calculations discussed in [28]. Finite element calculations using COMSOL are generally more efficient than BE calculations, but more care needs to be taken to ensure accurate results. Extensive tests have been carried out of the mesh density and the parameters of the perfectly matched layers (PMLs) required for convergence for a variety of single-ring structures. By using large mesh densities (particularly in the glass regions, where a maximum grid spacing of $1/8$ of the transverse wavelength is required), a carefully tuned PML, and benchmarking results against BE simulations, we are confident that results for the confinement loss are accurate to better than 1%. All the calculated points shown in Figs. 2 to 5 are COMSOL results.

 

Fig. 2. Calculated properties of the fundamental mode of unjacketed ((a),(c),(e)) and jacketed ((b),(d),(f)) single-ring fibres as a function of the normalised gap $\gamma$, for $M=6$ to $M=10$. $M$ values are indicated by different colours, as shown in (a). (a) and (b) show the confinement loss $L_{\lambda _0}$ incurred in propagation through a distance of one wavelength. (c) and (d) show the normalised transverse wavevector $x_{FM}$ defined in Eq. (2). (e) and (f) show the scaled confinement loss $L^{(sc)}_{\lambda _0}$ defined in Eq. (10). In all cases $r_c/\lambda _0=15$, $\epsilon =2.25$ and $\Omega =1$. Bullets and triangles are calculated points for unjacketed and jacketed structures respectively, and full curves are cubic spline fits through these points. For $M=10$, calculations have not been performed for $\gamma \,> \,0.325$ because the loss is very high. The loss in dBm$^{-1}$ for a particular wavelength is obtained by dividing $L_{\lambda _0}$ by the wavelength (in metres).

Download Full Size | PDF

3. Variation of confinement loss with $M$ and $\gamma$

Figures 2(a) and 2(b) show the confinement loss incurred in propagation through one wavelength $\lambda _0$ for the fundamental mode of unjacketed and jacketed single-ring structures respectively. Results for the number of capillaries, $M$, from 6 to 10 are shown, as a function of the normalised gap $\gamma$. In all cases, $r_c/\lambda _0=15$, $\epsilon =2.25$ and the width of the glass capillaries is anti-resonant (i.e. $\Omega =1$). The figures show some clear similarities. In both cases $L_{\lambda _0}$ rises dramatically as $\gamma$ increases beyond a certain value, with the loss increasing more rapidly for larger values of $M$. For $M=6$ and $7$, the confinement loss for both unjacketed and jacketed structures also rises as $\gamma$ approaches zero, leading to an optimal value for the normalised gap. There are, however, also clear differences between the two panels. First, the scales differ by a factor of five which means that the jacket has the overall effect of increasing the confinement loss. For example, for $\lambda _0=1\ \mu$m, the minimum loss for the unjacketed case is 2.1 dBkm$^{-1}$, while for the jacketed structure it is 12.2 dBkm$^{-1}$. Second, the increase in loss with increasing gap size is clearly shifted to higher values of $\gamma$ for the jacketed fibres in Fig. 2(b). Finally, the variation with $\gamma$ is smooth for the unjacketed structures, while there is a larger variability in Fig. 2(b) as $\gamma$ approaches zero, particularly for higher values of $M$. The first two points are discussed below; we do not have a clear physical reason for the third point, but it is perhaps not surprising that the greater geometrical complexity of jacketed structures can result in a more rapid variation of $L_{\lambda _0}$.

The normalised transverse wavevector $x_{FM}$ defined in Eq. (2) is shown in Figs. 2(c) and 2(d), where the values can be compared to the baseline given by $x_0=2.4048$. It can be seen that unjacketed and jacketed cases are almost identical, with $x_{FM}$ decreasing as the normalised gap $\gamma$ increases and being smaller for lower values of $M$; these trends are consistent with the amount of space the fundamental mode has to spread into as $M$ and $\gamma$ change. The similarity between the panels is also to be expected, with the size of the mode being primarily determined by the boundary between the core and the cladding capillaries, which is the same in both cases. For sufficiently large values of $\gamma$ the jacketed and unjacketed structures must eventually differ, with $x_{FM}$ monotonically decreasing for the unjacketed case, while the limit of the jacketed case is the $N=0$ structure of [13], which has $x=x_0$. However, for the range of $\gamma$ that has been calculated, these differences are not apparent.

Figures 2(e) and 2(f) show the results of applying the scaling of Eq. (10) to the losses of Figs. 2(a) and 2(b). The general effect is to flatten out the variation of the confinement loss at smaller values of $\gamma$, and to bring the curves for different $M$ to a similar absolute level. Our interpretation of this is that the $1/\sin ^2\phi ^{(air)}$ term in Eq. (1) accurately describes the variation with capillary radius of the confinement loss of both the unjacketed and jacketed structures, provided the gap between the capillaries does not exceed a given value. This implies that anti-resonance of the air regions in the cladding is a fundamental part of the guidance mechanism of single-ring fibres and, for sufficiently small $\gamma$, provides a quantitative explanation of the variation of the confinement loss with the inner radius of the capillaries.

The value of $\gamma$ above which this ”universal” behaviour breaks down is different for different values of $M$ and, for a given $M$, is larger for jacketed than unjacketed structures. The most obvious explanation for the shape of the curves seen in Figs. 2(e) and 2(f) is that, as $\gamma$ increases, leakage of the light through the gaps between the cladding capillaries provides an additional loss channel that rapidly overwhelms anti-resonance. This loss mechanism has been noted in several papers, for example [3,8,21,22,24,25]. The difference between the unjacketed and jacketed structures can be explained, at least in part, by the higher overall loss in Figs. 2(b) and 2(f) compared to Figs. 2(a) and 2(e). In jacketed structures, the gap between the capillaries must be larger for leakage through the gaps to exceed the underlying, anti-resonant loss.

To provide a more quantitative picture, we note that light guided in the fundamental mode cannot escape freely through the gaps. The transverse wavelength in the core is larger than the gap between capillaries and, if it is assumed that the wave amplitude is small at the air-capillary boundary (as can be seen in [8]), the gap acts effectively as a barrier through which the fundamental mode must tunnel in order to escape. A toy model of this tunnelling process is presented in the Appendix. The results for the tunnelling probability (Fig. 6(b)) show a clear similarity to Figs. 2(e) and 2(f); in particular Fig. 6(b) replicates the pattern where the point at which leakage starts to dominant shifts to a larger $\gamma$ as $M$ decreases. The tunnelling model provides a simple explanation for this behaviour; for a given minimum gap (i.e. $\gamma$) the tunnel barrier is longer for a smaller value of $M$, which leads to a larger reduction in amplitude as the barrier is traversed.

Our overall conclusion from Fig. 2 is that there are two regimes of confinement loss in single-ring structures. For larger values of $\gamma$ leakage between the cladding capillaries starts to dominate. This regime is clearly not captured by the concentric layers model of Eq. (1). However, because real fibres are fabricated for low loss, this part of the confinement loss is relatively unimportant. For smaller values of $\gamma$ (which cover a significant range for smaller $M$ values) the scaled loss is approximately constant, which is consistent with the anti-resonant guiding mechanism. In the remainder of the paper we will focus on this regime, and analyse the extent to which the other terms of Eq. (1) apply to single-ring fibres.

4. Further tests of Eq. (1)

In this section five representative points are selected from the data shown in Fig. 2 and the variation of the scaled loss of the fundamental mode with respect to $r_c/\lambda _0$, $w^{(gl)}$ and $\epsilon$ is considered. The chosen points, denoted by their $(M;\gamma )$ values, are $(7;0.1)$, $(7;0.2)$, $(7;0.3)$, $(9;0.1)$ and $(9;0.2)$. For the unjacketed case, $(7;0.1)$, $(7;0.2)$ and $(9;0.1)$ represent structures that are in the flat part of Fig. 2(e); they would therefore be expected to have their confinement loss characterised by anti-resonance and to be described by Eq. (1). In contrast, $(7;0.3)$ and $(9;0.2)$ lie in the part of Fig. 2(e) where leakage through the gaps starts to dominate (more so for $(7;0.3)$) and Eq. (1) is not expected to hold. For the jacketed case, the $(7;0.3)$ and $(9;0.2)$ structures are also in the relatively flat part of the scaled loss in Fig. 2(f). The selected points therefore encompass a set of structures which allow for a comparison both between the jacketed and unjacketed cases, and with the anti-resonant, concentric layers model of Fig. 1(c).

In order to single out the dependence of the loss on $r_c/\lambda _0$, $w^{(gl)}$ and $\epsilon$ respectively, it is important to keep the other parameters in Eq. (1) constant in each case. Equation (6) shows that the anti-resonant glass thickness depends on $r_c/\lambda _0$ and $\epsilon$; for variation with respect to these parameters we therefore adjust the thickness of the glass capillaries to maintain anti-resonance, so $\Omega =1$ and $\phi ^{(gl)}=\pi /2$ in all cases. Because the actual glass width changes as $r_c/\lambda _0$ and $\epsilon$ are varied, for a given $\gamma$ the radius ratio $\rho$ also changes (as shown in Eq. (5)). For a fixed $\gamma$, $\rho$ also changes as $w^{(gl)}$ is varied, with $r_c/\lambda _0$ and $\epsilon$ held constant. To compensate for this $\rho$ variation, the scaled confinement loss $L^{(sc)}_{\lambda _0}$ is used for all the representative single-ring structures discussed in this section. However, the range of $\rho$ variation is small and the trends discussed below are not significantly different if the unscaled loss is used (both scaled and unscaled data are available at [26]).

The behaviour of the loss of the $N=3$ reference structure of Fig. 1(c) is also considered. In this case the width of the air layer is held at its anti-resonant value, so $\phi ^{(air)}=\pi /2$. For variation with respect to $r_c/\lambda _0$ and $\epsilon$ the width of the glass layers, as for the single-ring structures, is adjusted so that $\phi ^{(gl)}=\pi /2$. Results are shown both for the effectively exact, numerically calculated confinement loss, $L_{\lambda _0}$, and for the analytic expression of Eq. (1).

4.1 Variation of confinement loss with $r_c/\lambda _0$

Results for $r_c/\lambda _0$ varying from 10 to 20 are shown for the five selected single-ring structures in Fig. 3. For all the calculated points $\epsilon =2.25$ and $\Omega =1$. The full line in each panel is the exact loss of the $N=3$ concentric layers model and the dot-dashed line is the analytic expression of Eq. (1). As discussed in [13], Eq. (1) is not exact, but it provides an excellent approximation to the confinement loss of the model structure over the full range of $r_c/\lambda _0$ values.

 

Fig. 3. Variation of scaled confinement loss $L^{(sc)}_{\lambda _0}$ with $r_c/\lambda _0$ for five single-ring structures characterised by number of capillaries, $M$, and normalised gap, $\gamma$, as shown on each panel. Bullets and triangles are calculated points for unjacketed and jacketed structures respectively (with colours consistent with Fig. 2). In all cases $\epsilon =2.25$, and for every calculated point the glass width is anti-resonant, i.e. $\Omega =1$. Dotted and dashed lines are functions of the form $(r_c/\lambda _0)^{-5}$ and $(r_c/\lambda _0)^{-6}$ respectively, fitted to pass through calculated points at $r_c/\lambda _0=15$. Also shown are results for the reference $N=3$ concentric layers structure of Fig. 1(c).

Download Full Size | PDF

Figure 3 shows that the unjacketed $(7;0.1)$, $(7;0.2)$ and $(9;0.1)$ structures follow the $(r_c/\lambda _0)^{-6}$ dependence of the $N=3$ reference model very closely. As would be expected, this scaling begins to break down for the unjacketed $(7;0.3)$ and $(9;0.2)$ structures where leakage between the glass capillaries starts to dominate. For example, for $(7;0.3)$ the overall dependence is closer to $(r_c/\lambda _0)^{-5}$, but the slope is steeper for smaller values of $r_c/\lambda _0$ and less steep for larger values. The jacketed cases also have an approximately power law dependence, with an overall scaling close to $(r_c/\lambda _0)^{-5.5}$, although the variation is less smooth for $M=9$ as $r_c/\lambda _0$ becomes smaller. This dependence is similar to the $\lambda _0^{4.5}$ scaling found in [23] (where the loss in dB/m rather than in a distance of one wavelength was considered). In contrast to the unjacketed structures, it can be seen that the $r_c/\lambda _0$ dependence is similar in all the panels of Fig. 3; as discussed above, this is in accordance with the different positions of the selected structures within Figs. 2(e) and 2(f). A $-5.5$ power law, in terms of Eq. (1), indicates that the jacketed single-ring structures have a behaviour that is intermediate between the $N=2$ and $N=3$ concentric layers model. In Section 1. we argued that the anti-resonant pathways in Fig. 1(a) could be consistent with a value of $N$ between 2 and 4. Figure 3 clearly indicates a value towards the lower end of this range, which is also consistent with the greater overall loss for jacketed structures found in Fig. 2.

The final, and particularly notable, feature of Fig. 3 is the absolute value of the scaled loss of the single-ring structures, in relation to the $N=3$ reference model. At $r_c/\lambda _0=15$, for the unjacketed cases where leakage through the gaps between the capillaries is not significant, the scaled loss is a factor of between 16 and 20 times smaller than for the concentric layers model. This remarkable difference is, we assume, a manifestation of the negative curvature effect that has been widely reported in single-ring fibres. This is discussed in more detail in Section 5.

4.2 Variation of confinement loss with $w^{(gl)}$

Results for the glass width varying from 0.5 to 1.5 times its anti-resonant thickness are shown in Fig. 4, where $\epsilon =2.25$ and $r_c/\lambda _0=15$ for all calculations. In this case Eq. (1) predicts a variation of the form $1/\sin ^4(\pi \Omega /2)$ for the $N=3$ reference model; it can be seen that this provides a very good approximation to the exact results, but with the difference between exact and analytic confinement losses increasing towards the first resonance value (i.e. as $\Omega \rightarrow 2$).

 

Fig. 4. Variation of scaled confinement loss $L^{(sc)}_{\lambda _0}$ with scaled glass width $\Omega$ for five single-ring structures characterised by number of capillaries, $M$, and normalised gap, $\gamma$, as shown on each panel. Bullets and triangles are calculated points for unjacketed and jacketed structures respectively. In all cases $r_c/\lambda _0=15$ and $\epsilon =2.25$. Dotted and dashed lines are functions of the form $\sin ^{-2}(\pi \Omega /2)$ and $\sin ^{-4}(\pi \Omega /2)$ fitted to pass through calculated points at $\Omega =1$. Also shown are results for the reference $N=3$ concentric layers structure of Fig. 1(c).

Download Full Size | PDF

As in Fig. 3, the unjacketed $(7;0.1)$, $(7;0.2)$ and $(9;0.1)$ single-ring structures closely follow Eq. (1) with $N=3$. The $1/\sin ^4(\pi \Omega /2)$ variation also holds well for the $(9;0.2)$ structure but is seen to break down for $(7;0.3)$ for smaller values of $\Omega$.

The jacketed structures behave very differently. For values of $\Omega$ greater than 1 the different structures have a similar $\Omega$ dependence with a variation close to the $1/\sin ^2(\pi \Omega /2)$ scaling that would be expected for $N=2$ in Eq. (1). The confinement loss rises more steeply for smaller values of $\Omega$, and shows a rapid variation with $\Omega$, particularly in the finer scans used for the $M=9$ cases. This more erratic behaviour is characteristic of a stronger interaction between the fundamental core-guided mode and the cladding modes. An enhancement of the coupling is consistent with the model described in [9], where the width of the bands derived from the glassy modes of individual cladding capillaries is shown to increase towards the lower edge of the transmission band which, in our case, corresponds to a reduction of the glass thickness. The wider bands arise from the less rapid azimuthal variation of the capillary mode and an increased interaction between neighbouring capillaries which, in turn, can lead to a greater coupling between the core and cladding modes. What is less clear, however, is why this ”uninhibited coupling” is greater for the jacketed than the unjacketed case, or why the effect is more significant for the $M=9$ examples than for those with $M=7$. This remains an open question that we intend to return to in a future paper.

4.3 Variation of confinement loss with $\epsilon$

Results for the dielectric constant varying from 1.5 to 3.0 are shown in Fig. 5. In all cases, $r_c/\lambda _0=15$ and $\Omega =1$. The $\epsilon$ dependence predicted by Eq. (1) for the fundamental mode of the $N=3$ reference model is of the form $(\epsilon ^4+1)/2(\epsilon -1)^2$; similarly to Figs. 3 and 4, Eq. (1) provides a good approximation to the exact result for the $N=3$ concentric layers model, although the exact loss does not rise as rapidly as the analytic expression for larger values of $\epsilon$.

 

Fig. 5. Variation of scaled confinement loss $L^{(sc)}_{\lambda _0}$ with $\epsilon$ for five single-ring structures characterised by number of capillaries, $M$, and normalised gap, $\gamma$, as shown on each panel. Bullets and triangles are calculated points for unjacketed and jacketed structures respectively. In all cases $r_c/\lambda _0=15$ and $\Omega =1$. Dashed lines are functions of the form $(\epsilon ^4+1)/2(\epsilon -1)^2$ fitted to pass through calculated points at $\epsilon =2.25$. Also shown are results for the reference $N=3$ concentric layers structure of Fig. 1(c).

Download Full Size | PDF

In contrast to the behaviour seen in Figs. 3 and 4, the unjacketed structures in Fig. 5 do not closely follow the prediction of Eq. (1) for $N=3$. In all cases, the scaled confinement loss rises more rapidly than the analytic expression for smaller values of $\epsilon$ and less rapidly for larger values. At first sight, it is surprising that Eq. (1) could capture the dependencies with respect to $\rho$, $r_c/\lambda _0$ and $w^{(gl)}$ so convincingly, and yet fail to predict the $\epsilon$ dependence. If the anti-resonance mechanism underlying Eq. (1) provides an accurate description of the physics (as it appears to do from Figs. 2 to 4), why is Fig. 5 different? We suggest that the answer lies in the difference in the core-cladding boundary between the single-ring structure of Fig. 1(b) and the concentric layers model of Fig. 1(c). The cylindrical symmetry of Fig. 1(c) means that TE, TM and HE/EH mode labels have precise meanings in terms of which field components are non-zero and the azimuthal dependence of these components. As shown in [13], the different boundary conditions for modes of pure TE and TM symmetry lead to expressions for the confinement loss that differ only in their $\epsilon$ dependence; all other factors are identical for the two modes. For the $N=3$ case, the $\epsilon$ dependent terms are $1/(\epsilon -1)^2$ and $\epsilon ^4/(\epsilon -1)^2$ for TE and TM modes respectively (which, for $\epsilon =2.25$, differ by a factor of 25.6). As shown in [13] (and given in Eq. (1)), the loss of HE/EH modes is given by the average of the TE and TM expressions, which reflects an equal admixture of TE- and TM-like components in the doubly-degenerate modes. In single-ring structures, where cylindrical symmetry is broken, the modes are no longer purely of TE, TM and HE/EH character and the geometry of the core-cladding boundary is very different. Given this, it is reasonable to assume that there is a more complex admixture of TE- and TM-like boundary conditions in the HE-like fundamental mode, and that this could lead to a different $\epsilon$ dependence, particularly given the large difference between the magnitudes of the TE and TM expressions. However, despite this difference, the variation with respect to $\rho$, $r_c/\lambda _0$ and $w^{(gl)}$ would still be expected to follow Eq. (1), because both the TE and TM components have the same dependence on these parameters.

As in Figs. 3 and 4, the jacketed structures in Fig. 5 all behave in a similar way, and we observe that the $\epsilon$ dependence is rather closer to the $N=3$ concentric layers model than for the unjacketed cases.

5. Conclusions

The results shown in Figs. 2 to 4 demonstrate that, provided leakage through the gaps between the capillaries is suppressed, the confinement loss of unjacketed single-ring fibres has the same dependencies with respect to $r_c/\lambda _0$, $\phi ^{(air)}$ and $\phi ^{(gl)}$ as the $N=3$ concentric layers model of Fig. 1(c). This shows conclusively that anti-resonance in both glass and air regions is the underlying guidance mechanism for unjacketed structures. However, this is not the end of the story; we have also shown that the absolute value of the loss is a order of magnitude lower in unjacketed single-ring fibres than in the reference model of Fig. 1(c), and that the dependence of the loss on the dielectric constant does not follow the prediction of Eq. (1). We believe that both of these differences are linked to the geometry of the core-cladding interface and are a manifestation of a negative-curvature effect. Although there have been several computational demonstrations of this effect, there has been very little work to date on developing a clear physical understanding of why the shape of the core wall matters, and why the effect on the confinement loss is so dramatic. Reference [14] takes some useful steps in this direction, but there is considerable scope for further analytical work.

For jacketed structures, the relationship with the predictions of Eq. (1) is less certain, but there is still strong evidence that anti-resonance is key to understanding the guidance mechanism. Figure 2 shows that the variation of the confinement loss with the radius of the cladding capillaries has the $1/\sin ^2\phi ^{(air)}$ dependence that is characteristic of anti-resonance in the air regions. This dependence would be expected in both the $N=2$ and $N=3$ reference models. Figure 3 shows a dependence for jacketed structures that is intermediate between $N=2$ and $N=3$, while Fig. 4 (for larger values of the glass thickness) has a dependence characteristic of $N=2$. We do not at this stage have a clear understanding of why these differences arise, or of the behaviour seen in Fig. 4 for smaller values of $\Omega$. These points also provide scope for future studies.

Our conclusion that anti-resonance is the key guidance mechanism will not be a surprise to many working in the field, and the recent demonstrations of lower loss fibres [6,7] are founded on the inclusion of additional anti-resonant elements in the cladding structure. However, we believe that, by providing a quantitative framework for how anti-resonant effects in both glass and air regions influence the confinement loss, our work will be valuable in the analysis and design of novel fibres, for example, by using Eq. (1) to understand the dependence of the loss of different modes as a function of the fibre’s structural parameters or the wavelength.

Appendix: Tunnelling model

In this appendix, the leakage of the core-guided fundamental mode through the gaps between the cladding capillaries is considered in terms of a highly simplified tunnelling model. The geometry is defined in Fig. 6(a), and it is straightforward to show that the gap, $G$, between the capillaries can be written as

(11)$$G(x^\prime) = g + 2\left( r_{out} - \sqrt{ r_{out}^2 - {x^\prime}^2 } \right)$$

where $g$ is the minimum gap defined in Section 2. We define $\xi$ as a dimensionless length scaled to the outer radius of the cladding capillaries, $r_{out}$, so $x^\prime =\xi r_{out}$ and the gap extends from $\xi =-1$ to $\xi =1$. By using Eq. (5) the gap function becomes

(12)$$\frac{G(\xi)}{r_c}= 2a \left(1-b\sqrt{1-\xi^2}\right)$$

with

(13)$$a= \frac{(1-\gamma /2)\sin\frac{\pi}{M} }{1-\sin\frac{\pi}{M}} \qquad \textrm{and} \qquad b= \frac{\sin\frac{\pi}{M}-\gamma/2} {(1-\gamma/2)\sin\frac{\pi}{M}} .$$

It is assumed that leakage through the gaps can be considered as a two-dimensional problem with a cylindrical scalar wave with in-plane wavevector $k_c$ incident on the ring of cladding capillaries. $k_c$ is defined in Eq. (2) and we approximate $k_cr_c=x_0$. For simplicity, the cylindrical geometry is reduced to a planar geometry with only one gap, and it is assumed that a wave $\psi =\exp (ik_c x^\prime )$ is incident from the left in Fig. 6(a). The governing Helmholtz equation is

(14)$$\frac{\partial^2\psi}{\partial {x^\prime}^2} + \frac{\partial^2\psi}{\partial {y^\prime}^2} + k_c^2 \psi = 0$$

and we look for solutions of the form

(15)$$\psi(x^\prime,y^\prime)=F(x^\prime)\cos\left(\frac{\pi y^\prime}{G(x^\prime)}\right)$$

for which the wavefunction $\psi$ goes to zero at the glass-air boundary within the gap. This boundary condition is key to the model and is justified by the effect of anti-resonance acting to exclude light from the glass region. Equation (15) is substituted into Eq. (14) and all derivatives of $G$ are neglected. This assumption, that the gap function $G(x^\prime )$ is slowly varying with $x^\prime$, is clearly not justified at the edges of the gap, but it will be a better approximation towards the centre of the gap, where the gap is at its smallest and leakage is most constrained. The function $F(x^\prime )$ is determined by the Schrödinger-like equation

(16)$$-\frac{d^2F}{d{x^\prime}^2}+\frac{\pi^2}{G^2(x^\prime)}F = k_c^2 F$$

where it can be seen that if the ”potential” term $\pi ^2/G^2$ exceeds the incident ”energy” $k_c^2$ then the gap acts as a tunnelling barrier. We follow the standard WKB approach to quantum tunnelling and the barrier penetration factor, giving the ratio of the probability density at the two ends of the barrier, becomes [29]

(17)$$B=\exp({-}2I) \qquad \textrm{with} \qquad I=\int_{{-}x_1}^{x_1}dx^\prime \, \sqrt{\frac{\pi^2}{G^2(x^\prime)}-k_c^2} .$$

By using Eq. (12) and (13), this can be expressed as

(18)$$I = b\pi \int_0^{\xi_1} d\xi \, \left( \sqrt{ \frac{1}{\left( 1-b\sqrt{1-\xi^2}\right)^2} - c^2 } \right)$$

where $c=2ax_0/\pi$. The limit $\xi _1$ is either the value of $\xi$ at which the integrand of Eq. (18) becomes zero, or 1 if it does not. It can be shown that the condition for a turning point to exist is $c\ge 1$, which occurs only for $M=6$ or $7$. Numerical values of $B$ are shown in Fig. 6(b), as a function of the normalised gap, $\gamma$, and for $M=6$ to $10$.

 

Fig. 6. (a) Geometry of the tunnelling model, showing the origin of the $(x^\prime ,y^\prime )$ coordinate system at the mid-point of the gap between cladding capillaries. $r_{out}$ is the outer radius of the capillaries, $\theta =2\pi /M$, and the gap function $G(x^\prime )$ is described in the text. (b) Barrier penetration factor, $B$, as a function of the normalised gap, $\gamma$, for different numbers of cladding capillaries, $M$. Colours are the same as in Fig. 2.

Download Full Size | PDF

The range of $B$ values shown in Fig. 6(b) has been chosen to provide a qualitative match to the scaled confinement losses shown in Fig. 2(e). However, a simple model based on estimating the number of tunnelling “attempts” can be used to provide quantitative support to this range. We consider a ray model where light is incident on the core-cladding boundary at a grazing angle $\theta$ given by

(19)$$\theta \approx k_c/k_0 \approx \frac{x_0}{2\pi} \frac{\lambda_0}{r_c}.$$

The ray undergoes multiple reflections across the core, with the number of reflections in a distance $z$ given by $\theta z / 2r_c$. If the power decreases by a factor $(1-T)$ at each reflection, it is straightforward to show that the light intensity decays along the length of the fibre as $\exp (-T\theta z / 2r_c)$ (a similar bouncing ray model is used in [14]). The loss (in dB) incurred in propagation through one wavelength then becomes

(20)$$L_{\lambda_0} = 8.686 \, \frac{T}{4} \left( \frac{x_0}{2\pi} \right) \left( \frac{\lambda_0}{r_c} \right)^2.$$

If this model is applied to propagation in a thick-walled capillary (i.e. the case analysed by Marcatili and Schmeltzer [30], and the $N=0$ model of [13]) then $(1-T)$ is the intensity reflection coefficient for grazing angle reflection at the air-glass boundary. For small $\theta$, $T=4\theta \nu$, where $\nu =1/\sqrt {\epsilon -1}$ and $\nu =\epsilon /\sqrt {\epsilon -1}$ for TE and TM reflections respectively. Substitution of $T$ into Eq. (20) gives exactly the Marcatili & Schmeltzer result for the loss.

In the tunnelling model of confinement loss it is reasonable to associate $T$ with the tunnelling probability at each reflection, and so $B$ and $T$ would be expected to be of a similar order of magnitude. For $r_c/\lambda _0=15$ (i.e. the value used in Fig. 2), Eq. (20) becomes $L_{\lambda _0}=3.7\times 10^{-3}\, T$ dB. To match the magnitude of the confinement loss of Fig. 2(e) a value of $T$ between $10^{-6}$ and $10^{-5}$ is therefore required, which is consistent with the range of $B$ shown in Fig. 6(b).

Acknowledgements

We thank Tim Birks and William Wadsworth for helpful discussions and a critical reading of the manuscript, and Jonathan Knight for hosting PS’s visit to Bath. PS was supported by a scholarship awarded by the Shandong Provincial Education Department and KYP was supported by an Undergraduate Research Internship awarded by the University of Bath’s Institute for Mathematical Innovation.

References

1. F. Yu and J. C. Knight, “Negative curvature hollow-core optical fiber,” IEEE J. Sel. Top. Quantum Electron. 22(2), 146–155 (2016). [CrossRef]  

2. C. Wei, R. J. Weiblen, C. R. Menyuk, and J. Hu, “Negative curvature fibers,” Adv. Opt. Photonics 9(3), 504–561 (2017). [CrossRef]  

3. B. Debord, F. Amrani, L. Vincetti, F. Gérôme, and F. Benabid, “Hollow-core fiber technology: the rising of gas photonics,” Fibers 7(2), 16 (2019). [CrossRef]  

4. W. Belardi and J. C. Knight, “Hollow antiresonant fibers with reduced attenuation,” Opt. Lett. 39(7), 1853–1856 (2014). [CrossRef]  

5. F. Poletti, “Nested antiresonant nodeless hollow core fiber,” Opt. Express 22(20), 23807–23828 (2014). [CrossRef]  

6. T. D. Bradley, J. R. Hayes, Y. Chen, G. T. Jasion, S. R. Sandoghchi, R. Slavik, E. N. Fokoua, S. Bawn, H. Sakr, I. A. Davidson, A. Taranta, J. P. Thomas, M. N. Petrovich, D. J. Richardson, and F. Poletti, “Record low-loss 1.3dB/km data transmitting antiresonant hollow core fibre,” in European Conference on Optical Communication (2018).

7. S. F. Gao, Y. Y. Wang, W. Ding, D. L. Jiang, S. Gu, X. Zhang, and P. Wang, “Hollow-core conjoined-tube negative-curvature fibre with ultralow loss,” Nat. Commun. 9(1), 2828 (2018). [CrossRef]  

8. B. Debord, A. Amsanpally, M. Chafer, A. Baz, M. Maurel, J. M. Blondy, E. Hugonnot, F. Scol, L. Vincetti, F. Gérôme, and F. Benabid, “Ultralow transmission loss in inhibited-coupling guiding hollow fibers,” Optica 4(2), 209–217 (2017). [CrossRef]  

9. R. F. Ando, A. Hartung, B. Jang, and M. A. Schmidt, “Approximate model for analysing band structures of single-ring hollow-core anti-resonant fibers,” Opt. Express 27(7), 10009–10021 (2019). [CrossRef]  

10. J.-L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11(3), 416–423 (1993). [CrossRef]  

11. F. Poletti, J. R. Hayes, and D. J. Richardson, “Optimising the performances of hollow antiresonant fibres,” in European Conference on Optical Communication (2011).

12. J. R. Hayes, F. Poletti, M. S. Abokhamis, N. V. Wheeler, N. K. Baddela, and D. J. Richardson, “Anti-resonant hexagram hollow core fibers,” Opt. Express 23(2), 1289–1299 (2015). [CrossRef]  

13. D. Bird, “Attenuation of model hollow-core, anti-resonant fibres,” Opt. Express 25(19), 23215–23237 (2017). [CrossRef]  

14. Y. Y. Wang and W. Ding, “Confinement loss in hollow-core negative curvature fiber: a multi-layered model,” Opt. Express 25(26), 33122–33133 (2017). [CrossRef]  

15. S. F. Gao, Y. Y. Wang, W. Ding, and P. Wang, “Hollow-core negative-curvature fiber for UV guidance,” Opt. Lett. 43(6), 1347–1350 (2018). [CrossRef]  

16. Y. Y. Wang, N. V. Wheeler, F. Couny, P. J. Roberts, and F. Benabid, “Low loss broadband transmission in hypocycloid-core Kagome hollow-core photonic crystal fiber,” Opt. Lett. 36(5), 669–671 (2011). [CrossRef]  

17. W. Belardi and J. C. Knight, “Effect of core boundary curvature on the confinement losses of hollow antiresonant fibers,” Opt. Express 21(19), 21912–21917 (2013). [CrossRef]  

18. B. Debord, M. Alharbi, T. Bradley, C. Fourcade-Dutin, Y. Y. Wang, L. Vincetti, F. Gérôme, and F. Benabid, “Hypocycloid-shaped hollow-core photonic crystal fiber part I: arc curvature effect on confinement loss,” Opt. Express 21(23), 28597–28608 (2013). [CrossRef]  

19. W. Ding and Y. Y. Wang, “Analytic model for light guidance in single-wall hollow-core anti-resonant fibers,” Opt. Express 22(22), 27242–27256 (2014). [CrossRef]  

20. A. D. Pryamikov, G. K. Alagashev, A. F. Kosolapov, and A. S. Biriukov, “Impact of core-cladding boundary shape on the waveguide properties of hollow core microstructured fibers,” Laser Phys. 26(12), 125104 (2016). [CrossRef]  

21. L. D. van Putten, E. N. Fokoua, S. A. Mousavi, W. Belardi, S. Chaudhuri, J. V. Badding, and F. Poletti, “Exploring the effect of the core boundary curvature in hollow antiresonant fibers,” IEEE Photonics Technol. Lett. 29(2), 263–266 (2017). [CrossRef]  

22. A. D. Pryamikov and G. K. Alagashev, “Strong light localisation and a peculiar feature of light leakage in the negative curvature hollow core fibers,” Fibers 5(4), 43 (2017). [CrossRef]  

23. L. Vincetti, “Empirical formulas for calculating loss in hollow core tube lattice fibers,” Opt. Express 24(10), 10313–10325 (2016). [CrossRef]  

24. C. Wei, C. R. Menyuk, and J. Hu, “Impact of cladding tubes in chalcogenide negative curvature fibers,” IEEE Photonics J. 8(3), 1–9 (2016). [CrossRef]  

25. C. Wei, C. R. Menyuk, and J. Hu, “Geometry of chalcogenide negative curvature fibers for CO$_2$ laser transmission,” Fibers 6(4), 74 (2018). [CrossRef]  

26. University of Bath Research Data Archive https://doi.org/10.15125/BATH-00685.

27. P. Uebel, M. C. Günendi, M. H. Frosz, G. Ahmed, N. N. Edavalath, J.-M. Ménard, and P.St.J. Russell, “Broadband robustly single-mode hollow-core PCF by resonant filtering of higher-order modes,” Opt. Lett. 41(9), 1961–1964 (2016). [CrossRef]  

28. W. Lu and Y. Y. Lu, “Efficient boundary integral equation method for photonic crystal fibers,” J. Lightwave Technol. 30(11), 1610–1616 (2012). [CrossRef]  

29. L. I. Schiff, Quantum Mechanics, 3rd Edition (McGraw-Hill, 1968).

30. E. A. J. Marcatili and R. A. Schmeltzer, “Hollow metallic and dielectric waveguides for long distance optical transmission and lasers,” Bell Syst. Tech. J. 43(4), 1783–1809 (1964). [CrossRef]