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Abstract
We numerically investigate the role of cladding geometries in two widely used anti-resonant hollow-core fiber designs with negative curvatures, the tubular negative-curvature fiber and ice-cream-cone negative-curvature fiber. The confinement loss governed by the inhibited coupling between the modes in the core and cladding is thoroughly examined systematically against the core-cladding curvature for both types. We show that, in addition to the mode-index mismatch, the mode-field overlap also plays a key role in determining the loss. Simultaneously, we find the ice-cream-cone negative-curvature fiber can exhibit better loss performance than the tubular design within a specific range of the curvature. This enhancement is achieved without sacrificing the transmission bandwidth and is relatively robust against the fabrication error.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Anti-resonant hollow-core fibers are attracting growing interest from the researchers in recent years. One of their main advantages is the guiding of the light in the hollow region with a very small overlap with the glass [1–3]. They not only provide broad transmission windows, but also permits fiber-based light guidance in spectral regions where conventional optical fibers do not work due to the intrinsic material properties [4,5]. They are highly promising for applications in high-power optical beam delivery, especially for beams in the mid-infrared and ultraviolet [6–10]. Moreover, when the hollow channels are filled with gas, these fibers present an ideal medium for investigating ultrafast nonlinear optical phenomena [11–13]. Their unique dispersion profile and nonlinearity which can be easily controlled by changing the gas pressure offer interesting opportunities for gas-based pulse compression [14,15], nonlinear frequency conversion [16] and supercontinuum generation [17,18].
The light guidance in anti-resonant hollow-core fibers is governed by the inhibited coupling theory [19,20]. Past studies have shown that index mismatch between the continuum of modes in the cladding and those in the core prevents the light leakage in the core [3,19,21,22]. The mismatch can be engineered through clever designing of the cladding structure [7,23]. Notably, the presence of negative curvature at the core-cladding interface has been shown to substantially enhance the light confinement, which led to the development of negative-curvature fibers with the transmission loss that can almost compete with the low-loss telecommunication fibers [19,24–26].
At present, there are two main base-designs of the negative-curvature fibers actively being investigated by various groups around the world due to their excellent transmission loss properties and relatively straightforward fabrication procedures. The first is the tubular negative-curvature fiber (TB-NCF), which is characterized by having thin circular cladding tubes that are not in contact with each other, surrounding the central hollow core [4,23]. This is the most promising base-design currently, and the record-low transmission loss so far has been achieved in a variant of TB-NCF that has an additional circular tube nested inside each cladding tube [24,27]. The other prominent design often being studied consists of cladding elements shaped in ice-cream-cones that are in contact with each other, and hence can be called the ice-cream-cone negative-curvature fiber (IC-NCF) [7].
Since the first introduction of the negative curvature in anti-resonant hollow-core fibers [23,28], many studies have been conducted to understand the role of the core-cladding shape on the light guiding properties. Reference [29] discusses the influence of the curvature on the confinement loss in a kagomé-lattice fiber that consists of negative curvatures surrounding the core. The study reports on a significant drop in the loss when the arc curvature is increased, which is attributed to the reduced mode overlap between the optical modes in the core and cladding [26,29]. In Ref. [30], numerical study on IC-NCF for a varying core-cladding curvature has been carried out, analyzing the confinement loss through the index mismatch between the core and cladding modes. Approximately four orders of magnitude reduction in the loss has been demonstrated when the cladding curvature is changed from flat to the highest that is attainable in IC-NCF [30]. Another notable work in this respect is Ref. [31] which presents a numerical investigation on TB-NCF with reference to an analytic expression for the loss based on the anti-resonant model. It presents a quantitative study on the variation of the confinement loss for varying cladding tube diameter [31].
In this work, we present a systematic numerical analysis on the role of cladding geometries in the confinement loss of both TB-NCF and IC-NCF. The loss determined by the inhibited coupling between the modes in the core and cladding is thoroughly examined using two quantities, namely, the mode index mismatch and mode-field overlap. We show that the latter, which has not been studied in detail until now, has an important role in dictating the strength of the interactions. It can describe some of the confinement loss features in anti-resonant hollow-core fibers that cannot be interpreted solely by the mode index mismatch. Moreover, we find that, within a specific range of the core-cladding curvature, IC-NCF can exhibit the loss that is approximately an order of magnitude lower than the lowest loss achievable in TB-NCF for the same core size. The enhancement is brought in without sacrificing its transmission bandwidth and is relatively robust against the fabrication errors.
2. Geometrical formulation and modeling
We formulate the geometries of the two fibers, TB-NCF and IC-NCF, as shown in Fig. 1. Namely, D is the diameter of the fiber core, which is defined as the diameter of the largest circle that fits in the central hollow region without overlapping with the cladding elements. In the case of TB-NCF, d is the outer diameter of the cladding tube that includes the tube thickness on each side, where the curvature at the core-cladding interface is then expressed using $d/D$. Here, the range of $d/D$ is dictated by the number of cladding elements, N. It is 0 < $d/D$ < ${({d/D} )_{\textrm{max}}}$, where ${({d/D} )_{\textrm{max}}}$ is
Fig. 1. Idealized cross sections of the two hollow-core fiber designs. The black and white shaded areas represent silica glass and hollow regions, respectively. (a) Tubular negative-curvature fiber (TB-NCF) with $N$ = 6. The ratio between the cladding element outer diameter d and the core diameter D determines the core-cladding curvature in TB-NCF. (b) Ice-cream-cone negative-curvature fiber (IC-NCF) with $N$ = 6. Here, the outer diameter $d^{\prime}$ of the virtual circular cladding element shown in dotted line defines the curvature $d^{\prime}/D$. In the fabrication, the cladding elements in IC-NCF are shaped by applying differential pressures on circular cane tubes that are in contact with each other. Therefore, the perimeter of each cladding element is given by $\pi D{({d/D} )_{\textrm{max}}}$.
We use a finite-element method for calculating the optical properties of the fibers. We note that most of the hollow-core fibers fabricated so far kept D around 25${\lambda _0}$–35${\lambda _0}$, where ${\lambda _0}$ is the wavelength that exhibits the lowest loss [1,5,7,33,34]. Therefore, we set $D$ = 30${\lambda _0}$ in the analyses to follow. Furthermore, we use the cladding-wall thickness $t = \; {\lambda _0}/\left( {3\sqrt {{n^2} - 1} } \right)$, where $n$ = 1.4497 is the refractive index of the waveguide material, and is fixed throughout this work. This choice ensures that the modes are calculated in the fundamental transmission band and far from the resonances [35].
3. Effect of the core-cladding curvature in TB-NCFs
The idealized cross-sectional geometries of TB-NCF at three different $d/D$ for $N$ = 6 are presented in Fig. 2(a) to illustrate how the curvature affects the overall shape of the fiber. Figure 2(b) presents the confinement loss of the fundamental core mode in TB-NCF as a function of the curvature in the range 0.3 $\le d/D < $ 1 for N from 5 to 9. For each N, the plot ends when ${({d/D} )_{\textrm{max}}}$ is reached except for $N$ = 5 where the upper limit is outside of the range studied. There is a minimum confinement loss for a given N at ${({d/D} )_{\textrm{opt}}}$, marked with vertical-dashed lines in Fig. 2(b), which decreases with increasing N. Note that ${({d/D} )_{\textrm{opt}}}$ for $N$ = 8 is ∼0.600 and that for $N$ = 7 is ∼0.606. All TB-NCFs feature a decreasing loss as $d/D$ is increased until the minimum loss is reached at ${({d/D} )_{\textrm{opt}}}$, it then rises with an increasing $d/D$. For N > 8, the minimum achievable loss is limited by the small ${({d/D} )_{\textrm{max}}}$. In general, a larger N leads to a lower confinement loss for a given $d/D$. It is worth noting, from the fabrication point of view, that TB-NCFs exhibit relatively low loss over a broad range of $d/D$. For example, the loss is below 20 dB·km−1 for 0.48 < $d/D$ < 0.75 when $N$ = 7, and 0.58 < $d/D$ < 0.76 when $N$ = 6. This allows for a large degree of freedom in the TB-NCF design, as well as a large fabrication tolerance. TB-NCF exhibits the lowest loss of 13.4 dB·km−1 when $N$ = 8 and $d/D$ = 0.6, given $D$ = 30${\lambda _0}$.
Fig. 2. (a) Idealized cross-sectional geometries of TB-NCF with $N$ = 6 at three different core-cladding curvature. (b) Confinement loss of the fundamental core mode in TB-NCF, as a function of $d/D$ for N from 5 to 9. For each N, the plot ends when ${({d/D} )_{\textrm{max}}}$ is reached except for $N$ = 5 where the upper limit is outside of the range shown. The minimum loss is achieved at ${({d/D} )_{\textrm{opt}}}$, each marked with a vertical-dashed line in (b) for given N. Note that ${({d/D} )_{\textrm{opt}}}$ for $N$ = 8 is ∼0.600 and that for $N$ = 7 is ∼0.606.
We investigate how the confinement loss in TB-NCF as observed in Fig. 2(b) is determined using the inhibited coupling guidance theory [19,20]. We look at three modes that mainly govern the confinement loss of the fundamental mode in TB-NCF to apply the inhibited coupling guidance theory. These are the HE11 mode in the core region (Mode A), the mode in the hollow regions in the cladding tubes (Mode B) and the mode that forms in the hollow region between the cladding elements (Mode C). The intensity profiles of these modes are shown in Fig. 3(a). Modes that interact dominantly with Mode A may appear in the glass strut in the cladding elements causing the structural resonance [3,20]. However, as we shall see, this is negligible with our choice of the core-cladding wall thickness.
Fig. 3. (a) Intensity profiles of the fundamental core mode (Mode A) and two cladding modes (Modes B and C) that interact strongly with the core mode in TB-NCF with $N$ = 6. (b) Effective indices of Modes A (blue lines), B (red lines) and C (yellow lines) for a varying $d/D$ when $N$ = 6 (solid lines), 7 (dashed lines) and 8 (dotted lines, only for Modes A and B). The index of Mode C for $N$ = 8 is omitted due to requiring computational time that is beyond reasonable. Note that the effective indices of Modes A and B do not change much when N is varied, and hence the lines appear to overlap with each other. (c) Confinement loss as a function of $d/D$ for $N$ = 6, 7 and 8.
We analyze two key quantities that governs the strength of the interactions between different modes. First is the index difference between the interacting modes which has been considered in past studies [3,21]. The other quantity is the field overlap between the interacting modes [19,29,36] which is analyzed here in detail for the first time.
Figure 3(b) plots the effective indices of Modes A and B as a function of $d/D$ for $N$ = 6, 7 and 8. The index of Mode B continues to increase as $d/D$ is increased while the index of Mode A remains mostly the same, and hence the difference between the two narrows. On the contrary, the opposite tendency is observed for the index difference between Modes A and C. It increases with increasing $d/D$ as shown in Fig. 3(b). Knowing that a larger index mismatch generally leads to a smaller loss, we can conclude that the interaction between Modes A and C is the dominant contributor to the confinement loss when $d/D$ is small, while it is the interaction between Modes A and B that governs the loss when $d/D$ is large. Furthermore, we observe from Fig. 3(b) that the index difference between Modes A and C at a given $d/D$ increases when N is increased, while that between Modes A and B remains almost the same. This indicates that the improved light confinement for large N at a given $d/D$ originates from the reduced interaction between Modes A and C, as placing more cladding tubes closes the gap between the tubes.
Apart from the index mismatch, another crucial parameter that dictates the degree of the interaction between two modes in the fiber is their spatial overlap. The intensity profiles of the two sets of the strongly interacting modes, i.e. Modes A and B and Modes A and C, are presented in Fig. 4(a) for $N$ = 6. They can be quantified by calculating the mode field overlap integrals, $\; \eta $, which is given by [37]:
Fig. 4. (a) Intensity profiles of the overlapping modes in TB-NCF with $N$ = 6: (top) Modes A and B; and (bottom) Modes A and C. (b) Mode field overlap integral, $\eta $, between Modes A and B (blue lines), and Modes A and C (red lines) as a function of $d/D$ for $N$ = 6 (solid lines) and 7 (dash lines) in TB-NCF. The vertical dashed lines indicate the curvatures that give the lowest losses, i.e. ${({d/D} )_{\textrm{opt}}}$, for $N$ = 6 and 7.
We also examine the influence of dielectric cladding modes on the fundamental core mode in TB-NCF using the index mismatch and mode field overlap. Figure 5(a) shows the effective indices of the dielectric cladding modes in the vicinity of low-loss $d/D$ for TB-NCF with $N$ = 6. The index of each dielectric cladding mode encountered is marked with ‘×’. We observe a continuum of dielectric cladding modes, although not many modes have their indices crossing the index of Mode A. Also, most of the light in these dielectric cladding modes are concentrated at the back of each tube with very little light close to the core. The profiles of two selected dielectric cladding modes (Modes D1 and D2) are presented in Fig. 5(b). The consequence of this is their much smaller overlap integrals with Mode A than those from the airy cladding modes as shown in Fig. 5(c). From this, it is evident that the dielectric cladding modes do not interact strongly with the fundamental core mode in TB-NCF in the transmission band and play only a minor role in shaping its confinement loss. In stark contrast, in the resonant band, we can clearly see in Fig. 5(d) that multitude of dielectric cladding modes have highly confined light close to the core leading to relatively large overlap integral. Similar observations on the dielectric cladding modes in TB-NCF in the resonant band have been reported in Ref. [22].
Fig. 5. (a) Effective indices of dielectric cladding modes, shown together with those of Modes A, B and C in the vicinity of low-loss $d/D$ for TB-NCF with $N$ = 6. The index of each dielectric cladding mode encountered is marked with ‘×’. (b) Mode profiles of two selected dielectric cladding modes (Modes D1 and D2) at $d/D$ = 0.65. (c) Mode overlap integrals, $\eta $, between Modes A and D1, and Modes A and D2. Those for Modes A and B, and Modes A and C are shown together for comparison. (d) Mode profiles, effective indices and overlap integral with Mode A of two example dielectric cladding modes in the first resonant band of IC-NCF, i.e. when $t = {\lambda _0}/\left( {2\sqrt {{n^2} - 1} } \right)$. The white arrows in (b) and (d) are pointing to the core center.
4. Effect of the core-cladding curvature in IC-NCFs
We now look at the transmission properties of the fundamental core mode in IC-NCF as a function of the curvature. The idealized cross-sectional geometries of IC-NCF at three different $d^{\prime}/D$ for $N$ = 7 are presented in Fig. 6(a) to illustrate how the curvature affects the overall shape of the fiber. Figure 6(b) shows the confinement loss for $N$ = 6, 7 and 8. The losses in TB-NCF, i.e. $d/D < {({d/D} )_{\textrm{max}}}$, with the same number of cladding elements are plotted in dashed lines together in Fig. 6(b) for comparison. The lowest loss reachable with IC-NCFs is around 0.37 dB·km−1, which is obtained when $N$ = 7 and $d^{\prime}/D$ = 0.793. This confinement loss is lower, by almost two orders of magnitude, than what is achievable with TB-NCF. However, the loss oscillates rapidly for varying core-cladding curvature which is in stark contrast to the smooth variation seen with TB-NCF. Despite the oscillation, the general tendency of rising loss for increasing $d^{\prime}/D$ is clear, with the presence of a relatively large window of $d^{\prime}/D$ near ${({d/D} )_{\textrm{max}}}$ that exhibits lower loss on average than the minimum loss achievable in TB-NCF. Another notable difference between TB-NCF and IC-NCF is that for IC-NCF, larger N generally leads to higher confinement loss.
Fig. 6. (a) Idealized cross-sectional geometries of $\; $IC-NCF with $N = 7$ at three different core-cladding curvatures. (b) Confinement loss of the fundamental core mode in IC-NCF as a function of $d^{\prime}/D$ for $N$ = 6, 7 and 8. For each N, the plot for the IC-NCF begin at ${({d/D} )_{\textrm{max}}}$, indicated by the vertical-dotted line. The plots to the left of ${({d/D} )_{\textrm{max}}}$ shown in dashed lines are for TB-NCF with the same N included for comparison. The horizontal-dotted line in (b) marks 13.4 dB·km−1, which is the lowest loss achievable in TB-NCF for $D$ = 30${\lambda _0}$. The curvature marked with a ‘×’ for $N$ = 7 is used later in Fig. 11.
Let us apply the same approach used for TB-NCF in Section 3 to explain the confinement loss profiles observed for IC-NCF in Fig. 6(b). This time we have three strongly interacting airy modes and continuum of dielectric cladding modes that determine the loss. They are presented in Fig. 7(a). Namely, Mode A′ is the HE11 mode in the core region, Mode B′ is the main mode that appears inside the ice-cream-cone-shaped cladding elements. Mode C′ is those that form in between the cladding elements. Mode D′ is an example dielectric mode that arises in the thin glass strut in the cladding elements. These dielectric modes arise due to the fiber having the contact points between the cladding elements, and they plays a significant role in IC-NCF [3].
Fig. 7. (a) Intensity profiles of the fundamental core mode (Mode A′), two cladding modes (Modes B′ and C′) and dielectric mode in the glass strut (Mode D′) in IC-NCF with $N$ = 7. (b) Effective indices of Modes A′ (blue lines) and B′ (red lines) and C′ (yellow lines) for a varying $d^{\prime}/D$ when $N = 7$. (c) Confinement loss as a function of $d^{\prime}/D$ for IC-NCF with $N$ = 7. (d) Effective indices of Modes A′, B′, C′ and D′ (purple line) at $d^{\prime}/D$ around one of the loss peaks as shown in (e).
Figure 7(b) plots the effective indices of the modes in the hollow regions, i.e. Modes A′, B′ and C′ in IC-NCF for $N$ = 7 when $d^{\prime}/D$ is in the low-loss region of 0.766 < $d^{\prime}/D$ < 0.78. Here, the index differences between the relevant modes in the hollow regions of IC-NCF (Modes A′, B′ and C′) are generally greater than those in TB-NCF (Modes A, B and C) at its lowest loss. This is the main reason why IC-NCF can exhibit lower confinement loss than TB-NCF within this range of $d^{\prime}/D$. One of the reasons for the general increase of the confinement loss for an increasing $d^{\prime}/D$ seen in Fig. 6(b) the increasing index of Mode C′ and therefore decreasing index mismatch between Modes A′ and C′. We can see that at a larger $d^{\prime}/D$, the index difference is now much smaller and hence the loss is higher as shown in Fig. 7(b). On the other hand, the mismatch between Modes A′ and B′ remains almost the same for all $d^{\prime}/D$. Shown in Fig. 7(d) is the effective indices of Modes A′ and D′ as a function of $d^{\prime}/D$. It demonstrates how two indices cross resulting in large loss peaks at these points [3,30]. Many loss peaks form due to such interactions between Mode A′ and multitude of dielectric modes. This leads to the strong loss oscillations observed in Fig. 6(b).
Again, the size of the interactions between the modes in IC-NCF is not determined solely by the index mismatch, but also by their spatial overlaps. In fact, the mode overlap plays a dominant role in shaping the overall loss of the fundamental core mode in IC-NCF. The intensity profiles of the two sets of the strongly interacting modes, i.e. Modes A′ and B′ and Modes A′ and C′, are illustrated for $N$ = 6 in Fig. 8(a). The mode field overlap integral between the two sets are show in Fig. 8(b) as a function of $d^{\prime}/D$ for $N$ = 6, 7 and 8. Here, the overlap integrals are calculated for the same ranges of $d^{\prime}/D$ as presented in Figs. 7(b) and 7(c). We observe that both the overlap integrals, particularly those between Modes A′ and B′, increases with $d^{\prime}/D$ for all N. This gives rise to the general increase in the confinement loss seen in Fig. 6(b). We note that the overlap between Modes A′ and B′ is much greater than that between Modes A′ and C′, which indicates that Mode B′ is the primary cause of the light leakage in IC-NCF. Moreover, Fig. 8(b) clearly shows that the mode field overlap increases as more cladding elements are added in IC-NCF. This is the reason why in Fig. 6(b) IC-NCF with larger N exhibits generally higher loss. The two curvature ranges plotted in Fig. 8(b), (i).e. 0.79 < $d^{\prime}/D$ < 0.80 and 1.025 < $d^{\prime}/D$ < 1.045, represent the regions of low loss in IC-NCF for $N$ = 7 and 6, respectively. Comparing the values between Modes A′ and B′ in these two ranges, we can see that the spatial overlap in the first range for $N$ = 7 is generally weaker than that in the second range for $N$ = 6. This is what makes it possible to achieve lower confinement loss with $N$ = 7 than $N$ = 6 in IC-NCF.
Fig. 8. (a) Intensity profiles of overlapping modes in $N$ = 6 IC-NCF: (top) Modes A′ and B′; and (bottom) Modes A′ and C′. (b) Mode field overlap integral, $\eta $, between Modes A′ and B′ (blue lines), and Modes A′ and C′ (red lines) as a function of $d^{\prime}/D$ for $N$ = 6 (solid lines), 7 (dashed lines) and 8 (dotted lines). The first range of $d^{\prime}/D$ represents the low-loss region for $N$ = 7, whereas the second range is that for $N$ = 6.
In Fig. 6(b), we observe a prominent loss peak at $d^{\prime}/D$ ≈ 1.2 for $N$ = 6 which does not seem to follow the general loss tendency. We examine the cause of this loss peak in Fig. 9. Namely, Modes A′ and C′ illustrated in Fig. 9(a) interact strongly when $d^{\prime}/D$ ≈ 1.2, because as shown in Fig. 9(b) the effective indices of the two modes intersect at this curvature. This results in the index matching between the two modes, leading to a sharp rise in the loss seen in Fig. 9(c). In fact, beyond this point, we find a clear evidence of the important role that the mode field overlap is playing in determining the confinement loss in the inhibited coupling guidance theory. We notice from Fig. 9(b) the index mismatch between A′ and C′ widens again from $d^{\prime}/D$ ≈ 1.2 onwards. However, Fig. 6(b) shows that the confinement loss in general is increasing again for $d^{\prime}/D$ > 1.4, despite the growing index mismatch. This rise is attributed to the increasing spatial overlap between Modes A′ and B′. Most of the features in the confinement loss exhibited in Fig. 6(b) is determined by the combined effect of the spatial overlap between Modes A′ and B′, and the index mismatch between Mode A′ and dielectric cladding modes exemplified as Mode D′. The former gives rise to the general increase in the loss with increasing $d^{\prime}/D$, while the latter induces the oscillations.
Fig. 9. (a) Intensity profiles of the fundamental core mode (Mode A′) and mode between the cladding elements (Mode C′) in IC-NCF with $N$ = 6. (b) Effective indices of Modes A′ (blue line) and C′ (yellow line) for a varying $d^{\prime}/D$. (c) Confinement loss as a function of $d^{\prime}/D$ for IC-NCF with $N$ = 6.
We further investigate the role of the dielectric modes in IC-NCF in Fig. 10. Namely, Fig. 10(a) plots the effective indices of the dielectric cladding modes in the vicinity of low-loss $d^{\prime}/D$ for IC-NCF with $N$ = 7. The index of each dielectric cladding mode encountered is marked with ‘×’. Similar to Fig. 5, we observe a continuum of dielectric cladding modes, but with a lot more of them crossing the index of Mode A′. The profiles of two selected dielectric cladding modes (Modes D′1 and D′2) presented in Fig. 10(b) indicate that the light is rather uniformly distributed in the dielectric strut, and hence their overlap integrals with the fundamental core modes are comparable to those from the airy cladding modes as shown in Fig. 10(c). As such, the dielectric modes in IC-NCF are responsible for the loss oscillations seen in Fig. 6(b). This account is also described in Figs. 7(d) and 7(e) using the index mismatch. In the resonant band, the dielectric cladding modes have much larger overlaps with the fundamental core mode as illustrated in Fig. 10(d) and become the dominant cause of the loss.
Fig. 10. (a) Effective indices of dielectric cladding modes, shown together with those of Modes A′, B′ and C′ in the vicinity of low-loss $d^{\prime}/D$ for IC-NCF with $N = 7$. The index of each dielectric cladding mode encountered is marked with ‘×’. (b) Mode profiles of two selected dielectric cladding modes (Modes D′1 and D′2) at $d^{\prime}/D$ = 1.056. (c) Mode overlap integrals, $\eta $, between Modes A′ and D′1, Modes A′ and D′2, Modes A′ and D′3, and Modes A′ and D′4. Modes D′3 and D′4 are two other chosen dielectric cladding modes that interact considerably with Mode A′ in 0.79 < $d^{\prime}/D$ < 0.8 range. The overlap integrals between Modes A′ and B′, and Modes A′ and C′ are shown together for comparison. (d) Mode profiles, effective indices and overlap integrals with Mode A′ of two example dielectric cladding modes in the first resonant band of IC-NCF, i.e. when $t = {\lambda _0}/\left( {2\sqrt {{n^2} - 1} } \right)$. The white arrows in (b) and (d) are pointing to the core center.
5. Transmission windows of TB-NCF and IC-NCF
One of the main advantages of the antiresonant-guiding hollow-core fibers is their broad transmission bandwidth. This, combined with the small light-glass overlap, offers an excellent opportunity for guiding light in extreme wavelengths where traditional fibers do not work [8,10,38]. We find out whether the exceptional light confinement property of the low-loss IC-NCF is maintained throughout its transmission windows across a wide spectrum. Let us consider IC-NCF having $N$ = 7 and $d^{\prime}/D$ = 0.78 which exhibits the loss of 3.4 dB·km−1 as marked with a ‘×’ in Fig. 6(b). Here, we deliberately avoid choosing the curvature that gives us the lowest loss in this highly oscillating region, as it is practically not realistic to realize the exact intended curvature that exhibits the lowest loss with a narrow margin of fabrication tolerance. Figure 11, presented as a function of the normalized wavelength, compares the loss spectrum over two transmission windows between IC-NCF and TB-NCF having $N$ = 7 and $d/D$ = 0.61 which is one that gives the lowest loss in the TB-NCF with $N$ = 7 geometry. In this calculation, the index of the glass is fixed at n ≈ 1.4497 and the material absorption is neglected.
Fig. 11. Confinement loss of the fundamental core mode TB-NCF and IC-NCF, both with $N$ = 7 over two transmission windows. For the TB-NCF, $d/D$ = 0.61 which gives the lowest loss for $N$ = 7 is used, and for the IC-NCF, $d^{\prime}/D$ = 0.78, which is within the low loss region as show in Fig. 6(b), is used.
We observe that, despite the presence of rapid oscillations, the confinement loss in IC-NCF is lower than that in TB-NCF across its entire two transmission bands. Moreover, the bandwidth of the transmission windows is preserved.
6. Fabrication tolerance in IC-NCF
The fabrication of the IC-NCF in its ideal form having a single contact point between the two adjacent cladding elements is not practical. As illustrated in Fig. 12(a), we will have edges forming at the contact points between the elements, with l defines as the edge of the length. We study the effect of these unintended edges on the confinement loss in IC-NCF. Figure 12(b) shows the confinement loss in IC-NCF with $N$ = 6 as a function of $d^{\prime}/D$, for three different l, i.e. the ideal single point contact ($l$ = 0), $l$ = 1.5$t$ and $l$ = 4.0$t$. In all cases, IC-NCF exhibits increasing loss with strong oscillations as $d^{\prime}/D$ is increased. We also examine the loss as a function of l for $d^{\prime}/D$ = 1, 1.02, 1.27 and 1.32 in Fig. 12(c). Our study shows that the losses display only some minor oscillations when l is changed. We observed similar patterns in the case of $N$ = 7 and 8 with no apparent degradation of the loss performance with increasing l. Our study suggests that the confinement loss of IC-NCF may not be affected much by the unintended formation of the edges illustrated in Fig. 12(a). Nevertheless, a full analysis on the impact of the fabrication error in IC-NCF calls for further investigations.
Fig. 12. (a) Illustration of the edge formation at the contact points between the adjacent cladding elements in IC-NCF. (b) Confinement loss of the fundamental core mode in IC-NCF with $N$ = 6 as a function of $d^{\prime}/D$, accounting for three different edge lengths. (c) Confinement loss of fundamental core mode in the same fiber as a function of l for four different $d^{\prime}/D$ values.
7. Conclusion
We present our numerical study on the effect of the core-cladding curvature in TB-NCF and IC-NCF on their confinement loss. The losses for the fundamental core mode in TB-NCF and IC-NCF exhibit different tendency along the changing curvature. These are governed by the combined effect of the mode index mismatching and mode field overlap integral. We show that within a certain range of $d^{\prime}/D$, IC-NCF can have confinement loss that is about an order of magnitude lower than the lowest loss that can be achieved in TB-NCF, while preserving the broad transmissions windows. It turns out that having seven cladding elements results in the best loss performance for both TB-NCF and IC-NCF. Finally, the study on the effect of the fabrication error in IC-NCF indicates that the formation of the edges at the contact points between the neighboring cladding elements may not have significant impact on the loss.
Funding
Ministry of Education - Singapore (2018-T1-001-048).
Acknowledgements
The authors acknowledge helpful discussions with Muhammad Rosdi Abu Hassan and Elizabeth Lee.
Disclosures
The authors declare no conflicts of interest.
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