Open Access
Add to BibSonomy
Abstract
A systematic analysis of the dependence of the confinement loss of an anti-resonant capillary on the curvature of the core surround is presented. The core boundary is described by circular arcs and the construction allows for a wide range of core shapes to be considered. It is found that both negative and positive curvatures substantially reduce the confinement loss relative to that of a circular anti-resonant capillary and that this effect is insensitive to the size of the core relative to the wavelength and to the properties of the glass capillary wall. In contrast, for a solid core surround there is a small increase in the confinement loss with curvature. Results of scalar and vector calculations are shown to be similar. A qualitative explanation of the results is proposed based on azimuthal confinement of the wave fields generated by the curved boundaries.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Hollow-core fibres that are characterised by a relatively large core and a simple cladding structure have been the focus of much recent attention, due to their broad transmission windows and low loss (see [1–4] and references therein). The shape of the core is a key feature of these fibres and an inwardly-curving core surround (i.e. negative curvature) is a ubiquitous element of fibre designs (as seen, for example, in fibres with a cladding consisting of a single ring of capillaries [5,6], nested-capillary fibres [7], conjoined-tube fibres [8] and low-loss Kagome fibres [9]). Numerous computational studies have been reported that demonstrate that a negatively curved core surround has a strong effect in reducing the confinement loss. For example, in [10] and [11] the curvature of the core surround was systematically varied in “ice cream cone” and Kagome fibres respectively and in both cases the confinement loss was shown to be substantially lowered as the negative curvature increases. In [12] the authors investigated model fibres with a cladding consisting of semi-elliptical elements and showed that an increased curvature of the core surround leads to a reduction of the confinement loss. Similarly, in [13] and [14] a range of differently shaped cores was analysed and a negatively curved core-cladding boundary was again shown to substantially reduce the confinement loss relative to a circular boundary.
In analysing the physical origin of this negative curvature effect, two distinct elements need to be considered. First, a characteristic of the new generation of hollow-core fibres is that there are modes, localised primarily in the glass regions of the cladding structure, that have the same effective index as the core-guided mode. This opens up the possibility of coupling between core-guided and cladding modes, which can provide a loss channel between the core and the outer jacket. To reduce the confinement loss is it therefore vital to suppress this coupling as far as possible. Glassy nodes that are formed where cladding elements touch are particularly susceptible to enhancing the interaction between core and cladding modes, and it follows that it is important to keep such nodes as far from the core boundary as possible. As discussed for example in [11], negative curvature of the core surround acts to “hide” the nodes, and this is likely to be a key element in the results for ice cream cone and Kagome fibres [10,11].
However, there is a second effect that is intrinsically related to curvature and for which there is currently no clear physical explanation. It is best illustrated by cases where there is a reference structure with cylindrical symmetry and where the underlying loss mechanism is well understood. In [15] we analysed a set of model hollow-core fibres, consisting of $N$ concentric layers of air and glass (see Fig. 1(a) for a schematic of the $N=0$ and $N=1$ structures). It was shown that anti-resonance in both glass and air regions is the physical guidance mechanism in operation, and an analytical expression was derived for the confinement loss. In this case there is no coupling between core-guided modes and cladding modes localised in the anti-resonant glass layers. This is because the relevant core-guided modes have a low-order azimuthal variation whereas glassy modes with a similar effective index to the core-guided modes must have a high azimuthal order, and in cylindrical symmetry coupling between modes of different azimuthal orders is forbidden. This work was used in [16] as a reference case to analyse the confinement loss of fibres whose cladding structures consist of a single ring of thin-walled capillaries. A concentric-layers model with $N=3$ anti-resonant regions (i.e. glass-air-glass) was compared with a model consisting of an unjacketed single ring of glass capillaries. A path from the core to the outside region in the single-ring structure again passes through three glass-air-glass regions, but the core surround in this case is negatively curved. We showed that the dependence of the confinement loss of the single-ring model with respect to the radius and glass thickness of the capillaries, and the ratio of the wavelength to the core radius, closely follows the prediction of the analytic model. However, the absolute value of the confinement loss for single-ring structures was shown to be an order of magnitude lower than that of the equivalent concentric-layers model. This cannot be due to suppression of coupling between core-guided and cladding modes, because symmetry forbids such coupling in the reference concentric-layers model whereas, in principle, there can be coupling in the lower loss structure with a negatively curved core surround.
Fig. 1. (a) Schematics of the circular reference structures used in the paper, where air regions are white and glass regions are shaded. $N=0$ corresponds to a hollow tube with a solid glass core surround that extends to infinity; $N=1$ represents an anti-resonant, thin-walled capillary with glass thickness $w_{\textrm {gl}}$. The reference structures have a core radius $r_{\textrm {c}}$. (b) Construction of the wavy boundary. $\Omega$ is determined by the symmetry order $M$ ($\Omega =2\pi /M$), $f$ is the fraction of the sector occupied by the major arc and $r_0$ is the radius of the circle that governs the overall size of the core (and is varied to keep the real part of the effective index constant). The major and minor arcs have local origins that are distances $r_{\textrm {maj}}$ and $r_{\textrm {min}}$ respectively from the global origin $0$, and they have radii $R_{\textrm {maj}}$ and $R_{\textrm {min}}$ respectively. $C$ is the curvature parameter that determines the radius of the major arc and $\Delta \theta$ is the opening angle of the major arc segment. The boundary shown by the thick line corresponds to $M=7$, $f=0.75$, $C=-2.0$.
A similar picture emerges from the work of other groups [13,14] where the confinement loss of a circular anti-resonant capillary (i.e. the $N=1$ case of Fig. 1(a)) is compared to that of capillaries with a variety of different shapes. Negative curvature of the core surround is again shown to reduce the loss relative to that of a circular capillary; interestingly, in [14] it is also shown that a capillary with a primarily positive curvature with respect to the circular case again shows a reduction in the confinement loss. The results of [13,14,16] clearly show that there is a curvature effect that is separate from considerations of the coupling between core-guided and cladding modes. This is reinforced by the results of [12] where the cladding structure is nodeless, and the analysis shows that coupling into glassy modes is a minor contribution to the overall leakage. It is also useful to note that [12] and [14] analysed structures with a solid, as well as an anti-resonant, core surround (i.e. corresponding to the $N=0$ model of Fig. 1(a)). In both papers it was shown that curvature has little effect on the confinement loss for a solid core surround, indicating that anti-resonance is an important component of the curvature effect.
A theoretical understanding of this effect remains elusive. In [17] a model was proposed to calculate the confinement loss of anti-resonant capillaries of different shapes, based on the formation of an equiphase surface at the outer boundary of the capillary wall and subsequent free-space propagation governed by the Helmholtz equation. This model was used in [14] to argue that radiation to the far field is suppressed by destructive interference arising from different parts of a negatively or positively curved boundary. Although this provides a useful picture of why a non-circular boundary can lead to a reduced confinement loss, it does not provide a fully quantitative explanation.
Our aim in this paper is to provide a comprehensive analysis of the dependence of the confinement loss on the curvature of the core wall in $N=0$ and $N=1$ structures. It builds on the work of [13] and [14] by considering a substantially wider range of core shapes and by investigating the dependence on the ratio of the wavelength to the core radius and the thickness and dielectric constant of the glass. The construction of the curved boundaries is discussed in section 2 and results for the variation of confinement loss with curvature are presented in section 3. The effects of interactions between the core-guided mode and modes localised in the anti-resonant glass surround can be seen in these results; this coupling is analysed in the Appendix. The results are discussed in section 4 and an interpretation of the effect of curvature is proposed that could form the basis of more complete theoretical explanation. Section 5 is the conclusion.
2. Structures and computation
The core boundaries used throughout the paper are constructed from arcs of circles, as shown in Fig. 1(b). This construction allows for a wide range of smooth boundaries to be investigated, and circular arcs provide a good approximation to the shape of the core boundary in realistic hollow-core fibres. Each boundary is characterised by a symmetry order $M$ that defines a repeating sector with an angle $\Omega =2\pi /M$ at the origin (i.e. the fibre axis). As shown in Fig. 1(b), the curved boundary consists of alternating major and minor arcs, with the major arc occupying a fraction $f$ of the fundamental sector. Without loss of generality it can be assumed that $f\ge 0.5$. An underlying circle of radius $r_0$ determines the overall size of the core, and the final parameter that defines the core shape is the curvature $C$, where $C=1$ corresponds to a circular core. A set of core shapes with $M=7$ and $f=0.75$ and with a range of $C$ values is shown in Fig. 2.
Fig. 2. Core shapes (thick blue lines) for $M=7$ and $f=0.75$ for different curvature parameters $C$. The black circle represents the reference core of radius $r_{\textrm {c}}$, corresponding to $C=1$, and is the same in all cases. The red circle has radius $r_0$, as used in Fig. 1. For $C<0.0$ the major arc of the boundary has negative curvature and the minor arc positive curvature. For $0.0<C<1.31$ both major and minor arcs have positive curvature. For $C>1.31$ the major and minor arcs have positive and negative curvature respectively.
The construction of the curved boundary starts with a major arc of radius $R_{\textrm {maj}}=r_0/\vert C \vert$ whose extent is fixed by the two points at which it touches the underlying circle (Fig. 1(b)). If $C>0$ the major arc is outwardly curving relative to the origin and if $C<0$ it is inwardly curving. $C=0$ represents the intermediate case where the major arc is a straight line. The position of the local origin of the major arc and the angle subtended by the arc at its local origin are fully determined by this construction. For example, in the negative curvature case shown in Fig. 1(b), it is straightforward to show that the local origin is (in polar coordinates) situated at $(r_{\textrm {maj}},\Omega /2)$ where
It is assumed that the major arc does not exceed a semi-circle, which places an upper limit $C_{\textrm {max}}$ on the curvature parameter (this restriction ensures that a radial line from the origin will never cross the core boundary twice). By putting $\Delta \theta =\pi$ in Eq. (2) we find
Equations (2) and (3) also hold for a major arc with positive curvature, while in Eq. (1) the plus sign before the square root is replaced with a minus.
The geometry of the minor arc segments is determined by the condition that the major and minor arcs join smoothly, as shown in Fig. 1(b). When the arcs touch at the underlying circle their tangents must be parallel, and this fully determines the properties of the minor arc. For the $C<0$ example of Fig. 1(b), the minor arc is always positively curved; it is centred at $(r_{\textrm {min}},\Omega )$ where
The $C>0$ case has a slightly more complicated geometry because the minor arc can have either a positive or a negative curvature. The transition between these regimes occurs when the minor arc becomes a straight line, i.e. when the tangent of the major arc as it touches the underlying circle at angular coordinate $\Omega /2 - f\Omega /2$ is perpendicular to the $x$-axis. This condition becomes $\Delta \theta =\Omega$ and so from Eq. (2) the critical curvature is
For $C_{\textrm {crit}}<C<C_{\textrm {max}}$ the minor arc has negative curvature and for $0<C<C_{\textrm {crit}}$ both major and minor arcs have positive curvature. In both cases, it is straightforward to derive expressions equivalent to Eqs. (4) and (5) for $r_{\textrm {min}}$ and $R_{\textrm {min}}$.
The analysis so far fully defines the shape of a core with an infinite glass surround (i.e. the $N=0$ case shown in Fig. 1(a)). For a thin-walled capillary ($N=1$) it defines the inner core boundary. The outer glass-air boundary is constructed from arcs with the same local origins as for the inner boundary, but with the radius of the major and minor arcs being adjusted by $\pm w_{\textrm {gl}}$ as appropriate, where $w_{\textrm {gl}}$ is the thickness of the glass. The anti-resonant thickness $w_{\textrm {AR}}$ is given by
where $\epsilon$ is the relative dielectric constant of the glass and $\lambda _0$ is the free space wavelength.It is well known that the loss of anti-resonant hollow-core fibres is strongly dependent on the ratio of the core radius to the wavelength. For example, in a circular thin-walled capillary of core radius $r_{\textrm {c}}$ the loss incurred in propagation through a distance of one wavelength is proportional to $(\lambda _0/r_{\textrm {c}})^4$ [15]. In analysing the effect of curvature in cores with a non-circular boundary it is therefore important to separate out the effects of the shape of the core and its overall size [14]. To do this, the real part of the effective index, $n_{\textrm {eff}}$, of the guided mode is used as a measure of the effective core size. For a given set of parameters ($\lambda _0$, $r_{\textrm {c}}$, $\epsilon$, $w_{\textrm {gl}}$, $M$, $f$), and for a particular mode, $\Re (n_{\textrm {eff}})$ is calculated for a circular waveguide, which corresponds to $C=1$. The curvature parameter $C$ is then varied, keeping other parameters fixed but allowing $r_0$ to vary. For each $C$ the value of $r_0$ is found that has the same $\Re (n_{\textrm {eff}})$ as for $C=1$. The variation of the imaginary part of $n_{\textrm {eff}}$ then reflects the influence of the shape of the core on the confinement loss, free from the obscuring effect of a change in the effective core size [14].
Figure 2 shows a set of core shapes for a thin-walled capillary with $M=7$, $f=0.75$, $r_{\textrm {c}}/\lambda _0=15$, $\epsilon =2.25$ and $w_{\textrm {gl}}=w_{\textrm {AR}}$. In this case $C_{\textrm {max}}=3.028$ (so the figure spans nearly the full range of valid $C$ values) and $C_{\textrm {crit}}=1.314$. Each of the set has the same value of $\Re (n_{\textrm {eff}})$ and hence the same effective core size. It can be seen that the value of $r_0$ required to meet this constraint varies considerably across the set, particularly for larger negative curvatures. It is also apparent that the construction produces a wide range of smooth core shapes which allows for a systematic analysis of the effect of both positive and negative curvature on the confinement loss.
As in [16], calculations have been performed both with the finite element based COMSOL Multiphysics package and with an in-house boundary element (BE) code. It is found that $\Im (n_{\textrm {eff}})$ differs by less than $0.2\%$ between the methods for the great majority of the calculated points. The calculations become more sensitive close to the mode crossings that are discussed in the Appendix, but even here the two methods agree to better than $2\%$. As discussed above, as the curvature parameter $C$ is varied, the radius $r_0$ is adjusted to keep $\Re (n_{\textrm {eff}})$ constant at its $C=1$ value. This is done by interpolation from a set of calculations with uniformly spaced $r_0$ values. With the BE method we are able to maintain $\Re (n_{\textrm {eff}})$ constant to better than $1$ part in $10^9$; with COMSOL this becomes $1$ part in $10^8$. One advantage of the BE method is that it is straightforward to change the code from a full, vector simulation to a scalar wave calculation [18], which allows us to investigate whether the effect of curvature of the core wall is specifically related to electromagnetic boundary conditions or if it is a more general wave phenomenon. The results shown in Figs. 3–5 in the following section are from BE calculations and those in Figs. 6 and 7 are from COMSOL. All the data presented is available at [19].
Fig. 3. Confinement loss for non-circular cores, scaled to the loss for $C=1$, as a function of the curvature parameter $C$ for $M=7$ and $f=0.75$ and with parameters $r_{\textrm {c}}/\lambda _0=15.0$, $\epsilon =2.25$. (a) and (b) show results for vector and scalar wave equations respectively, for $N=0$ structures with an infinite glass surround. (c) and (d) show results for vector and scalar wave equations respectively, for $N=1$ structures (i.e. a thin-walled capillary) with $w_{\textrm {gl}}=w_{\textrm {AR}}$. In (a) and (c) results are for the HE$_{11}$ mode. In (b) results are for the LP$_{01}$ mode and in (d) both LP$_{01}$ and the next lowest order mode (LP$_{11}$) are shown. Actual values of $n_{\textrm {eff}}$ in each case for $C=1$ are given in Table 1. Shaded regions in each plot show the range of curvatures for which both the major and minor arcs are positively curved. In (c) vertical lines give the positions of mode interactions predicted by Eq. (10), with $m$ values as shown.
3. Variation of confinement loss with curvature
Results for the confinement loss as a function of the curvature parameter $C$ are shown in Figs. 3, 5 and 6 for a range of different systems and structures. In each case the confinement loss is scaled relative to its value for a circular capillary (i.e. $C=1$); the actual values of the real and imaginary parts of the effective index for $C=1$ are given in Table 1. All of the data points shown are actual calculations; there is no interpolation or curve fitting. In all but one of the plots the results are for the fundamental core-guided mode (i.e. HE$_{11}$ for vector and LP$_{01}$ for scalar calculations respectively); results for the LP$_{11}$ mode are also shown in Fig. 3(d). Figures 3 and 5 refer to core shapes with $M=7$ and $f=0.75$, as demonstrated in Figs. 1 and 2, while Fig. 6 includes results for a wider range of $M$ and $f$ values.
Figures 4(a) and (b) show the value of $r_0$ required to keep $\Re (n_{\textrm {eff}})$ constant for an anti-resonant capillary with $M=7$ and $f=0.75$, together with the corresponding radius of curvature of the major arc: $r_0/\left | C \right |$. These values are all scaled relative to the underlying core radius, $r_{\textrm {c}}$, for $C=1$. Plots of $r_0$ against $C$ for the other cases shown in Figs. 3 and 5 are very similar to those in Fig. 4; these results are not shown but the data are available at [19]. It can be seen that as $C$ approaches its extremal value near $C=-3.0$ the radius of curvature of the major arc is approximately $0.5\times r_{\textrm {c}}$. This case therefore has a core shape that strongly resembles the single-ring fibres with the lowest confinement loss analysed in [16], which explains the choice of $M=7$, $f=0.75$ as a reference structure.
Fig. 4. Results for the $N=1$ vector case (corresponding to Fig. 3(c)) for $M=7$, $f=0.75$. (a) Radius $r_0$ as a function of curvature parameter $C$. (b) Radius of curvature of the major arc, i.e. $r_0/\vert C \vert$. In both (a) and (b) the radii are scaled relative to $r_{\textrm {c}}$, their value at $C=1$.
The confinement loss for thin-walled capillaries (i.e. the $N=1$ case in Fig. 1(a)) shows a number of anomalies, best seen in the high-resolution data of Fig. 3(c). These are the signature of an avoided crossing between the core-guided mode and a glassy mode localised in the anti-resonant core surround. As the curvature $C$ is varied, the length of the perimeter of the core changes and this allows for coupling to occur at particular values of $C$. A simplified model that can be used to identify where these interactions are expected to arise is presented in the Appendix. In this section, we ignore the mode-coupling features and focus on the overall shape of the confinement loss plots.
Figures 3(a) and 3(b) show that both positive and negative curvature of the core away from $C=1$ increases the confinement loss of a hollow core with an infinite glass surround (i.e. the $N=0$ case of Fig. 1(a)). As discussed in the introduction, the absence of a strong curvature effect in fibres with a solid core surround has been previously noted [12,14]. In contrast, Figs. 3(c), 3(d) and 5 show a very strong curvature dependence for the loss of a thin-walled capillary. While the increases for the $N=0$ case over the full range of curvatures is only about $20\%$, $N=1$ structures show a dramatic decrease in the loss for both negative and positive curvatures of the major arc. Towards the extreme of negative curvature, the confinement loss is over an order of magnitude smaller than for the circular $C=1$ case. The decrease on the positive curvature side is not as large, but the reduction in the loss is still about $80\%$.
As shown in Fig. 2, core shapes for larger values of $\left | C \right |$ always have negatively curved parts of the core wall immediately facing the core, coming from the major arc for negative $C$ and the minor arc for positive $C$. It might therefore be argued that the substantial reduction in confinement loss is essentially a negative curvature effect. However, the confinement loss calculations for $N=1$ structures clearly show that all curvature away from $C=1$ reduces the loss. For example, for $0<C<1.31$ the core wall has only positive curvature (see the shaded regions in Fig. 3) and the loss is clearly reducing in this region in Figs. 3(c) and 3(d). The plots also show a smooth variation as negative curvature sets in, indicating that the observed reduction is an effect of departure from circularity, rather than a specifically negative curvature effect.
Comparison of Figs. 3(a,c) with 3(b,d) shows that the curvature effect for vector and scalar calculations is almost identical. Similarly, a comparison of each of the plots in Fig. 5 with Fig. 3(c) shows that the effect of curvature is essentially independent of the ratio $r_{\textrm {c}}/\lambda _0$, the dielectric constant $\epsilon$ or the thickness of the glass $w_{\textrm {gl}}$; the overall shapes of the curves are almost indistinguishable in each case. The only curve that shows a significant difference for $N=1$ is that for the higher-order LP$_{11}$ mode in Fig. 3(d). The confinement loss still decreases strongly with curvature away from $C=1$, but the drop is not as large, particularly for negative values of $C$.
Fig. 5. Confinement loss, scaled to the loss for $C=1$, for $M=7$ and $f=0.75$, as a function of the curvature parameter $C$ for the $N=1$ vector case. In each panel the small black dots are the same results as shown in Fig. 3(c). (a) shows the variation with respect to $r_{\textrm {c}}/\lambda _0$, (b) the variation with respect to the dielectric constant of the glass, and (c) the variation with respect to the glass width. Actual values of $n_{\textrm {eff}}$ in each case for $C=1$ are given in Table 1.
Figure 6 shows results for a wider range of $M$ and $f$ values, for an anti-resonant, thin-walled capillary with $r_{\textrm {c}}/\lambda _0=15$. A greater variation between the different data sets can be observed, both for the confinement loss and the radius $r_0$. However, the plots are similar to each other in their overall form, and to the $M=7$, $f=0.75$ result in Fig. 3(c). The main difference between the confinement loss plots is that the range of curvatures is different in each case. The $C_{\textrm {max}}$ values for each $(M,f)$ pair are: $(6,0.5)=3.86$; $(6,0.6)=3.24$; $(6,0.7)=2.79$; $(6,0.8)=2.46$; $(8,0.5)=5.13$; $(8,0.6)=4.28$; $(8,0.7)=3.68$; $(8,0.8)=3.24$. It can be seen that the different plots in Fig. 6 are basically stretched and compressed relative to the $M=7$, $f=0.75$ result according to the respective values of $C_{\textrm {max}}$.
Fig. 6. Top: confinement loss, scaled to the loss for $C=1$, as a function of the curvature parameter $C$ for the $N=1$ vector case with parameters $r_{\textrm {c}}/\lambda _0=15.0$, $\epsilon =2.25$, $w_{\textrm {gl}}=w_{\textrm {AR}}$. Bottom: radius $r_0$, scaled relative to $r_{\textrm {c}}$, required to keep $\Re (n_{\textrm {eff}})$ constant. The left and right panels shows results for $M=6$ and $M=8$ respectively; the different data sets correspond to different values of $f$, as indicated in the upper right panel.
4. Discussion
Any theory of the effect of curvature of the core wall on the confinement loss must be able to explain the results presented in section 3, in particular: the difference between $N=0$ and $N=1$ structures; the large size of the effect for anti-resonant capillaries and the relative insensitivity to the dielectric constant or thickness of the core wall; the difference between the fundamental and higher order modes seen in Fig. 3(d). The fact that the results of vector and scalar calculations show a very similar dependence on the curvature is potentially useful, because a scalar wave theory is likely to be substantially easier to construct.
In this paper we do not attempt a full theoretical explanation of the results. Instead, we present a plausible argument based on the analysis presented in [16] of the loss caused by leakage between the cladding capillaries in an anti-resonant, single-ring fibre. There it was shown that the lateral confinement imposed on the wave field in the gaps between the capillaries leads to an exponential-like decay of the field; the gap acts as a tunnelling barrier and, provided it is sufficiently narrow, prevents leakage that would lead to a higher confinement loss. The idea behind the analysis in [16] is simple. Core-guided modes in a hollow-core fibre with core radius $r_{\textrm {c}}$ have a transverse component of their wavevector, $k_{\textrm {tr}}$, that is determined by the core radius and the mode index:
$x$ is a mode-dependent factor that for the fundamental core-guided mode is close to the first zero of the $J_0$ Bessel function; Eq. (8) also shows the relationship between $x$ and the effective index $n_{\textrm {eff}}$. Any confinement of the mode in the azimuthal coordinate will have an associated wavevector, $k_\phi$, that is determined by the length scale of the confinement; for example, if at a radius $R$ there is an azimuthal confinement of order $\Delta \phi$, then $k_\phi \approx \pi /R\Delta \phi$. If $k_\phi$ exceeds $k_{\textrm {tr}}$ then the radial dependence becomes evanescent, which leads to the tunnelling behaviour discussed in [16]. The significance of the transverse component of the wavevector is also emphasised in the model of confinement loss presented in [17].We speculate that a similar mechanism occurs within the cladding capillaries of a single-ring fibre and, for $N=1$ structures, within the negatively and positively curved regions of the core boundaries shown in Fig. 2. Figure 7 shows the radial dependence of the $z$ component of the time-averaged Poynting vector, $S_z$, for $C=-3.0$ (for azimuthal angles corresponding to the centres of the negatively-curved major and positively-curved minor arcs), and for $C=1.0$. The $C=1.0$ plot provides a baseline against which the $C=-3.0$ results can be compared; it shows that $S_z$ falls relatively slowly with radius for $r>r_{\textrm {c}}$. However, the behaviour for $C=-3.0$ is very different. For the major arc, $S_z$ decays considerably more rapidly after the anti-resonant, core-facing wall. The mode is confined in the azimuthal direction by the non-circular capillary such that $k_\phi$ exceeds $k_{\textrm {tr}}$, and this induces an evanescent decay until the radius at which $k_\phi \approx k_{\textrm {tr}}$. This effect is even more pronounced for the minor arc, where azimuthal confinement occurs before the capillary wall is encountered and where, as can be seen in Fig. 2 for $C=-3.0$, the capillary wall provides a tighter confinement than for the major arc. This produces a strong exponential decay (note that the vertical axis of Fig. 7 is on a logarithmic scale), leading to a very small value of $S_z$ close to the capillary wall. The overall effect is that the fields, and therefore $S_z$, decay much more rapidly for $C=-3.0$ than for a circular capillary. The same idea can be applied to the other core shapes in Fig. 2, and we note that, for example, a similar effect can be observed in Fig. 5 of [12].
Fig. 7. $z$ component of the time-averaged Poynting vector for the $N=1$ vector case with $M=7$, $f=0.75$ (i.e. corresponding to Fig. 3(c)) for $C=1.0$ and $C=-3.0$, as a function of the radial coordinate $r$. The Poynting vector is scaled to $1$ at $r=0$ and $r$ is scaled to $r_{\textrm {c}}$. The two curves for $C=-3.0$ correspond to radial directions passing through the mid-points of the major (maj) and minor (min) arcs, i.e. the $-x$ and $+x$ directions respectively in Fig. 2. The vertical bars indicate the position of the core wall: for $C=-3.0$ between $r/r_{\textrm {c}}=0.9370$ and $0.9519$ and between $1.6403$ and $1.6552$ for the major and minor arcs respectively; for $C=1.0$ between $1.0000$ and $1.0149$.
This model makes two major assumptions: (i) that the core wall does indeed produce a sufficiently strong azimuthal confinement to lead to evanescent decay, and (ii) that this decay feeds through into a reduction of the confinement loss. These assumptions clearly need to be properly tested, but the picture has a number of attractive features. First, the model is based on wave propagation in air regions of the structures; the details of the anti-resonant core wall are not so important, provided it produces sufficient confinement. This could explain the observation that the reduction of confinement loss with curvature is relatively insensitive to the dielectric constant and thickness of the glass. Second, an exponential-like decay of the wave fields could explain the very large curvature effect seen in section 3; it is hard to imagine another functional form that could produce an order of magnitude reduction in the confinement loss. Third, the magnitude of the transverse wavevector $k_{\textrm {tr}}$ is greater for higher order modes than for the fundamental mode. This would tend to reduce the decay rate of the radial evanescent decay and hence lead to higher-order modes having a less pronounced curvature effect, as observed in Fig. 3(d). Finally, as discussed in [14], for $N=0$ structures the value of $k_{\textrm {tr}}$ in the outer glass region is large and evanescent decay will not be induced by azimuthal confinement. This would explain the absence of a strong curvature effect in Figs. 3(a) and (b) but, at present, we do not have an explanation of the weak increase of confinement loss seen in these figures.
5. Conclusions
Our results show that there is a strong curvature effect for anti-resonant capillaries that is absent in structures with a solid core surround. This effect is an intrinsic property of the curvature of the core wall and is not related to the strength of the coupling between core-guided and cladding modes with the same effective index. It is clear in Figs. 3, 5 and 6 where any such coupling occurs; away from these localised regions the interaction is suppressed and yet the confinement loss remains strongly dependent on the curvature. For anti-resonant capillaries, any deviation away from a circular core boundary, and not only negative curvature, leads to a reduction of the confinement loss. The similarity between the results of vector and scalar calculations indicates that the curvature effect is a general wave phenomenon and is not specific to electromagnetic boundary conditions. We have proposed a model of evanescent decay caused by azimuthal confinement in air regions of the structures that provides a qualitative explanation of some key features of our results, including the insensitivity to the details of the anti-resonant core surround and the difference between the fundamental and higher order modes. We believe that these ideas have considerable potential for explaining the beneficial effect of a (negatively) curved core boundary on the confinement loss for anti-resonant capillaries and other hollow-core fibres. Work is in progress to develop the qualitative model into a fully formulated theory.
Appendix: coupling between core-guided and glassy modes
A thin, infinitely extended planar glass waveguide with dielectric constant $\epsilon$ and thickness $w_{\textrm {pl}}=t w_{\textrm {AR}}$ has guided modes with an effective index $n_{\textrm {pl}}$ given by
In the first expression $k_0$ is the free space wavevector and in the second expression quantities are scaled with respect to the reference core radius $r_{\textrm {c}}$ to make a connection with the results above. This equation for $n_{\textrm {gl}}$ is an approximation because in practice there is not a clear separation of the normal and tangential dependence of glassy modes in the wavy capillaries under investigation. However, Eq. (10) gives a good approximation for the effective indices of glassy modes with $n_{\textrm {gl}}\sim 1$ (both guided and leaky modes) of a circular, thin-walled capillary, and it should provide a useful model to interpret the observed anti-crossings. A similar approach has been used, for example, in [11] and [6]. Note that, in Eq. (10), the perimeter length $L$ varies with curvature $C$ both because of the intrinsic change of the core shape and because the length scale $r_0$ changes.
Given any structure and value of $C$, a set of $n_{\textrm {gl}}$ values can be obtained from Eqs. (9) and (10) for different values of $m$. When these match the real part of the effective indices of core-guided modes given in Table 1 there is a possibility of a mode interaction. However, it is not sufficient only to match the effective index; the symmetries of the core-guided and glassy modes must also be the same. As shown in [20], the HE$_{11}$ mode of fibres with a C$_{M\textrm {v}}$ symmetry has azimuthal components of the form $\exp [i(pM\pm 1)\phi ]$ where $p$ is an integer and $\phi$ is the azimuthal coordinate. A glassy mode will have the same symmetry provided the integer $m$ in Eq. (10) is also of the form $m=pM\pm 1$. For scalar wave equations the TE expression of Eq. (9) can be used for $n_{\textrm {pl}}$ and the symmetry condition for the fundamental LP$_{01}$ mode becomes $m=pM$.
For $\epsilon =2.25$ and $w_{\textrm {gl}}=w_{\textrm {AR}}$, Eq. (9) gives $n_{\textrm {pl}}^{(\textrm {TE})}=1.19901$ and $n_{\textrm {pl}}^{(\textrm {TM})}=1.08196$ for TE and TM solutions respectively. It follows from Eq. (10) that a lower value of $m$ is required for mode-matching for glassy modes with a TM-like transverse profile, compared with TE-based modes. A lower value of $m$ means a less rapid spatial variation around the perimeter of the capillary, leading to a stronger interaction between the core-guided and glassy modes [3,11]. It would therefore be expected that TM-based glassy modes dominate in producing the observed anti-crossings. The strongest features seen in Fig. 3(c) correspond to coupling with TM-like glassy modes with $m=41$ and $m=43$. It can be seen that the prediction of Eq. (10) for the position of the anti-crossing is not perfect, but it is quite good, given the sensitivity of the predicted $C$ value to small changes in the perimeter length. For example, if $L$ is calculated for the inner or outer boundary (rather than the mid-point) of the glass core surround, the variation in the predicted value of $C$ is of the order of $\pm 0.1$ to $\pm 0.2$. The difference between the actual and predicted anti-crossing points is of a similar magnitude. The lowest value of $m$ that would satisfy Eq. (10) for TE-like glassy modes in Fig. 3(c) is $m=64$. No anti-crossing points have been clearly identified with this or higher $m$ values in Fig. 3(c), indicating that the circumferential variation is sufficiently rapid in this case for mode interaction to be suppressed. The weaker features seen in Fig. 3(c) near $C=-1.75$ and $C=-2.00$ are associated with TM-like glassy modes with $m=48$ and $m=50$ respectively. Interactions with TE-like glassy modes are, however, faintly observable in Fig. 3(d). As shown in Table 1, the actual confinement loss of the scalar LP$_{01}$ mode is significantly smaller than for the HE$_{11}$ mode of Fig. 3(c), which allows for weaker coupling to be seen. The weak features near $C=-1.1$ and $C=2.6$ in Fig. 3(d) correspond to interactions with glassy modes with $m=70$.
A similar analysis can be applied to the results shown in Figs. 5 and 6. The strongest anomalies are observed for $w_{\textrm {gl}}=0.5 w_{\textrm {AR}}$ in Fig. 5(c). In this case $n_{\textrm {pl}}^{(\textrm {TM})}=1.01960$ and the interaction of the core-guided mode with TM-like glassy modes can occur for values of $m$ as low as $20$ and $22$. The enhanced mode coupling that this produces strongly affects the local details seen in Fig. 5(c), but it does not change the overall shape of the variation of confinement loss with curvature.
Acknowledgements
We thank Tim Birks and William Wadsworth for a critical reading of the manuscript. WW’s visit to Bath was supported by a scholarship awarded by the China Scholarship Council.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available at [19].
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. Ding, Y. Y. Wang, S. F. Gao, M. L. Wang, and P. Wang, “Recent progress in low-loss hollow-core anti-resonant fibers and their applications,” IEEE J. Sel. Top. Quantum Electron. 26(4), 1–12 (2020). [CrossRef]
5. A. D. Pryamikov, A. S. Biriukov, A. F. Kosolapov, V. G. Plotnichenko, S. L. Semjonov, and E. M. Dianov, “Demonstration of a waveguide regime for a silica hollow-core microstructured optical fiber with a negative curvature of the core boundary in the spectral regime >3.5µm,” Opt. Express 19(2), 1441–1448 (2011). [CrossRef]
6. 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]
7. H. Sakr, Y. Chen, G. T. Jasion, T. D. Bradley, J. R. Hayes, H. C. H. Mulvad, I. A. Davidson, E. N. Fokoua, and F. Poletti, “Hollow core optical fibres with comparable attenuation to silica fibres between 600 and 1100 nm,” Nat. Commun. 11(1), 6030 (2020). [CrossRef]
8. 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]
9. M. Maurel, M. Chafer, A. Amsanpally, M. Adnan, F. Amrani, B. Debord, L. Vincetti, F. Gérôme, and F. Benabid, “Optimised inhibited-coupling Kagome fibers at Yb-Nd:Yag (8.5 dB/km) and Ti:Sa (30 dB/km) ranges,” Opt. Lett. 43(7), 1598–1601 (2018). [CrossRef]
10. 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]
11. 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]
12. 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]
13. 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]
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. D. Bird, “Attenuation of model hollow-core, anti-resonant fibres,” Opt. Express 25(19), 23215–23237 (2017). [CrossRef]
16. P. Song, K. Y. Phoong, and D. Bird, “Quantitative analysis of anti-resonance in single-ring, hollow-core fibres,” Opt. Express 27(20), 27745–27760 (2019). [CrossRef]
17. 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]
18. L. Chen, “Modelling of photonic crystal fibres,” PhD thesis, University of Bath (2009)
19. W. Wang and D. Bird, “Dataset for “Confinement loss of anti-resonant capillaries with curved boundaries”,” University of Bath Research Data Archive, 2021, https://doi.org/10.15125/BATH-01014.
20. D. Bird, “Mode symmetry in microstructured fibres revisited,” Opt. Express 26(24), 31454–31463 (2018). [CrossRef]
