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Abstract
We propose an antiresonant hollow-core fiber design that exhibits ultralow loss and exceptional single modedness at 1.55 µm. In this design, the confinement loss of less than 10−6 dB m−1 can be obtained with excellent bending performance even at a tight bending radius of 3 cm. At the same time, a record-high higher-order mode extinction ratio of 8 × 105 can be achieved in the geometry by inducing strong coupling between the higher-order core modes and cladding hole modes. These guiding properties make it an excellent candidate for applications in hollow-core fiber-enabled low-latency telecommunication systems.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Microstructured hollow-core fibers have received a great deal of attention in recent years [1]. Among them, antiresonant hollow-core fibers have made tremendous progress in the past decade [2–4]. With the right choice of cladding structure, antiresonant hollow-core fiber can tightly confine light in its central hollow region [2–4]. This gives the hollow-core fiber several advantages over conventional glass-guiding fibers, such as low dispersion, low nonlinearity, and broadband transmission of light [5–12]. These, combined with the fiber’s high power-handling capability, makes it suitable for a wide range of applications, including high-power beam delivery [13–15]. Moreover, recent studies have shown that further lowering of the loss, to below that in telecommunication fibers, can be achieved by carefully tailoring the cladding geometry [16,17]. This presents the possibility of using the hollow-core fiber as a physical channel in low-latency fiber-optic communication systems [16,17].
The first antiresonant hollow-core fiber appeared in the form of kagomé-lattice fiber, which had relatively high transmission loss of several dB m−1 [18,19]. In 2011, Wang et al. discovered that by introducing negative curvature in the dielectric strand surrounding the core can substantially reduce the loss [20]. Pryamikov et al. came up with a tubular-type hollow-core fiber, where only a single layer of thin-wall tubular cladding elements surrounds the hollow core for low-loss guidance [21]. It offers a simple geometry for easy fabrication while preserving the negative curvature. Here, the presence of dielectric nodes in the cladding, i.e., glass web connecting points, was one of the main contributors to the loss [22]. Hence an improved version of the tubular-type fiber, nodeless antiresonant hollow-core fiber, which has the cladding tubes that are separated from each other to eliminate the nodes, later appeared [23].
The loss can be further lowered by introducing additional thin-wall elements inside the cladding tubes, which reduces the coupling between the hollow modes in the core and cladding tubes via increasing the mode index mismatch. This led to the emergence of nested antiresonant nodeless fibers (NANFs) [17,24]. In particular, a NANF with five cladding elements demonstrated extremely low transmission loss of only 0.22 dB km−1 in the telecommunication band [25]. It was found that further nesting the tube with more thin-wall elements can cumulate the light confinement effect [17,24]. Based on this idea, a double-nested antiresonant nodeless fiber (DNANF) was fabricated, exhibiting remarkably low transmission loss of 0.174 dB km−1 [16]. Notably, the higher-order core mode, LP11, is well suppressed in the DNANF with the loss ratio between LP11 and LP01 larger than 1000, suggesting the possibility of achieving excellent single-modedness [16].
Antiresonant hollow-core fibers can support several higher-order core modes together with the fundamental core mode [5–7]. It is possible to suppress the higher-order modes by deliberately inducing their coupling to cladding hole modes [14,16,26]. In the tubular hollow-core fiber, this can be obtained by optimizing the ratio between the core size and cladding tube size [14,16]. Introducing the nested elements can also enhance the single modedness [27–30]. Especially, the NANF with five cladding elements featured excellent higher-order core mode suppression with the measured extinction ratio of 1.27 × 104 between LP11 and LP01 [29]. Likewise, it was found that the separation between the inner nested and outer nested elements can be adjusted to improve the single-mode guidance in DNANF [16]. In DNANF, the size, number, and position of the inner-nested tubes play a vital role in tuning its guiding properties [16,17]. Altering these changes effective indices of the cladding hole modes and their interactions with the core modes [27–30]. Hence for low-loss single-mode guidance, we need to work out the best cladding arrangement that can induce strong coupling between higher-order core modes and cladding hole modes, while at the same time allow low confinement loss in the fundamental core mode.
In this work, we propose a novel double-nested antiresonant hollow-core fiber design with dual inner-nested tubes, and present a comprehensive numerical study on general guiding properties of multi-nested antiresonant hollow-core fiber designs. The new geometry displays excellent guiding properties with ultralow loss, good single modedness, and outstanding bending performance. We numerically study the effect of varying the structural parameters, particularly the size of nested elements, on the guiding properties. We observe an order of magnitude reduction in the confinement loss compared with DNANF [16]. We also present how varying the structural parameters changes the cladding hole modes and their coupling to the core modes for both straight and bent fibers. The results show that the cladding hole modes are localized mainly in two hollow regions of the cladding, and their coupling to the core modes depends strongly on the size of the cladding hollow regions. In particular, a very strong coupling between LP11 core mode and cladding hole modes can be achieved, leading to a record-high higher-order mode extinction ratio of up to 8 × 105. Finally, fabrication tolerance of the proposed design is discussed.
2. Fiber geometry and numerical method
An idealized base design of the proposed fiber is presented in Fig. 1(a). Its cladding tubes have double nested elements with the inner one consisting of dual tubes. As such, we call this a dual-tube double nested antiresonant nodeless hollow-core fiber (DT-DNANF). The fiber geometry is characterized by the core diameter $D$, cladding element dielectric thickness t, the number of outermost cladding elements N. ${d_1}$, ${d_2}$, and ${d_3}$ are defined as the exterior diameters of the outermost-cladding tube, outer-nested tube, and inner-nested tubes, respectively. Hence, the nested element here includes both the outer and inner nested tubes. As we shall see, modes in the cladding holes are mainly located in two hollow regions, i.e., Regions A and B in Figs. 1(b) and 1(c), respectively. Namely, Region A is the hollow region between the outermost-cladding tube and nested element, whereas Region B is the hollow region inside the nested element between the inner-nested tubes.
Fig. 1. (a) Idealized cross-section of DT-DNANF. N is the number of outermost-cladding tubes; t is the wall thickness of the cladding tubes; D is the core diameter; ${d_1}$, ${d_2}$, and ${d_3}$ are the exterior diameters of the outermost-cladding tubes, outer-nested tubes, and inner-nested tubes, respectively. A penetration depth of $t/2$ is assumed at all silica nodes as shown in the inset. Schematic illustrations of (b) Region A and (c) Region B, which are the two major hollow regions in the cladding elements.
For all calculations presented in this work, we set $D$ = 30 µm and $t$ = 0.4 µm, which, at wavelength $\lambda $ of 1.55 µm, gives relatively low confinement loss, good bending performance, and single-modedness. Moreover, we use $N$ = 4, unless mentioned otherwise. This choice will be discussed in Section 3.1. It then leaves us with only three free parameters for tuning, namely ${d_1}$, ${d_2}$, and ${d_3}$. In our calculations, we normalize these three parameters to the diameters of their respective outer elements, i.e., ${d_1}/D$, ${d_2}/{d_1}$, and ${d_3}/{d_2}$. By doing so, the range of possible values of the three free parameters become ${d_1}/D$ < 2.4, ${d_2}/{d_1}$ < 1, and ${d_3}/{d_2}$ < 0.5. The penetration depth of $t/2$ is assumed for all connecting silica nodes as illustrated in the inset in Fig. 1(a).
3. Results
3.1 Confinement loss and modal property
As it is difficult to carry out full parameter sweeps with all three geometrical parameters as free variables, let us first determine ${d_1}$ that gives the lowest confinement loss (CL) for a fixed value of ${d_3}$. For this, we tie the value of ${d_3}$ to ${d_2}$ through the relation ${d_3} = ({d_2} - t/2)/2$. This ensures least amount of light leakage through the gap between the two inner nested cladding elements in Region B. It becomes apparent in the next section that the light leakage through this gap is one of the main sources of confinement loss. Figure 2(a) shows the confinement loss for varying ${d_1}/D$ and ${d_2}/{d_1}$ when ${d_3} = ({d_2} - t/2)/2$. The calculations are performed for ${d_1}/D$ ranging from 1.34 to 2.4 and for ${d_2}/{d_1}$ ranging from 0.291 to 0.97.
Fig. 2. (a) False colormap of the confinement loss in the fundamental core mode in DT-DNANF for varying ${d_1}/D$ and ${d_2}/{d_1}$ when ${d_3} = ({d_2} - t/2)/2$ and $\lambda $ = 1.55 µm. The insets show intensity profiles of the fundamental core modes at ⑤ and ⑥. (b–e) 3-dB of the $z$-component of Poynting vector in the fundamental core mode in DT-DNANF at ①, ②, ③, and ④, respectively.
We observe a broad range of ${d_1}/D$ and ${d_2}/{d_1}$ with ultralow confinement loss of less than 10−5 dB m−1. This corresponds to the area enclosed in white-dashed line in Fig. 2(a). The minimum confinement loss of 4.9 × 10−7 dB m−1 is observed at the point marked with ①, which is located at ${d_1}/D$ = 1.8 and ${d_2}/{d_1}$ = 0.86. The loss here is sufficiently low even for applications in long-distance fiber optic communication [17,28].
The ${d_1}/D$ ratio is an important parameter for realizing low loss guidance. When ${d_1}/D$ is too small, significant light leakage occurs through the gaps between adjacent cladding elements as evident in Figs. 2(b) and 2(c), which show the 3-dB contours of the $z$-component of Poynting vector in the fundamental core mode when ${d_1}/D$ = 1.8 and 1.4, respectively. The high-density contour lines in the central hollow core in Fig. 2(b) illustrates strong light confinement in DT-DNANF. On the other hand, the outermost contour line in Fig. 2(c) protrudes into the hollow areas between the outer-cladding tubes close to the jacket, indicating severity of light leakage. When ${d_1}/D$ is too large, however, the dielectric walls in adjacent tubes get too close to each other. This causes the light field to couple to silica layer of outermost-cladding tubes as shown in Fig. 2(d).
We also note that the confinement loss depends strongly on ${d_2}/{d_1}$. The loss generally becomes high when ${d_2}/{d_1}$ < 0.6. This is because the nested elements get situated far from the fundamental core mode when ${d_2}/{d_1}$ < 0.6, weakening its effect on the light confinement. The comparison of the 3-dB contour lines for ${d_2}/{d_1}$ = 0.86 and 0.6 are shown in Figs. 2(b) and 2(e), respectively. They indicate that the field penetrates through the outer-cladding tubes and leaks out more easily when ${d_2}/{d_1}$ is smaller due to the weaker confinement from the nested elements.
In Fig. 2(a), we notice several high loss bands around the points labeled ⑤ and ⑥. These loss bands appear due to coupling between the fundamental core mode and the hollow modes that are in Region A. This is depicted in the insets in Fig. 2(a), which presents the intensity profiles of the fundamental core mode at ⑤ and ⑥. We can observe the confining of the field in Region A. They can be either LP01-like cladding hole mode (⑤) or a higher-order cladding hole mode (⑥).
Since the confinement loss is low in the vicinity of ${d_1}/D$ = 1.8, we set ${d_1}/D$ = 1.8 in our subsequent analyses.
Figure 3(a) is a false colormap of the confinement loss in fundamental core mode of DT-DNANF when ${d_2}/{d_1}$ and ${d_3}/{d_2}$ are varied while ${d_1}/D$ = 1.8. The loss is below 10−5 dB m−1 in the area enclosed in the white-dashed line, indicating there is a broad region for ultralow-loss guidance in DT-DNANF.
Fig. 3. False colormaps of (a) confinement loss of the fundamental core (LP01) mode, (b) confinement loss of LP11 core mode, (c) confinement loss of higher-order core modes with the lowest loss, and (d) HOMER in DT-DNANF for varying ${d_3}/{d_2}$ and ${d_2}/{d_1}$ when ${d_1}/D$ = 1.8 and $\lambda $ = 1.55 µm. The two top insets in (a) are the 3-dB intensity contour plots at ① and ②, while the bottom inset is the intensity profile for the fundamental core mode at ③. The four insets in (b) are the intensity profiles of LP11 core mode at ④, ⑤, ⑥, and ⑦. The two insets in (c) are two higher-order core modes, namely, LP02 and LP21, at ⑧ and ⑨. The two insets in (d) are the intensity profiles of LP01 and LP11 core modes at ⑩. The color scale used for the normalized intensity profiles are the same as that presented in Fig. 2.
The size of inner-nested tube has a strong influence on the loss. Small inner-nested tubes leave large gap between the two inner-nested tubes leading to substantial light leakage [28]. This is evident in the insets in Fig. 3(a) that present the 3-dB intensity contour plots for two locations with different values of ${d_3}/{d_2}$ at ① and ② in Fig. 3(a). The contour lines protrude to the hollow region behind the inner nested tubes when ${d_3}/{d_2}$ is too small.
The high loss band around ③ in the mid-right section in Fig. 3(a) is caused by coupling between the fundamental core mode and hollow mode in Region B. Shown in the bottom inset in Fig. 3(a) is the intensity profile of the core mode at ③. There is confinement of the field in Region B, indicating simultaneous presence of the cladding hole mode and fundamental core mode.
We also study the properties of higher-order core modes in DT-DNANF. LP11 mode is generally—but not always—the higher-order core mode with the lowest loss. Figure 3(b) is a false colormap of the confinement loss in LP11 mode in DT-DNANF over the same ranges of ${d_2}/{d_1}$ and ${d_3}/{d_2}$. In the low loss area in Fig. 3(b), LP11 mode interacts only weakly with all cladding hole modes. On the other hand, the large high-loss region with the loss exceeding 0.02 dB m−1 bounded in black dashed-line, LP11 mode couples strongly with cladding hole modes. As the size of nested elements vary, the cladding hole mode that couples to the LP11 core mode changes. Specifically, when ${d_2}/{d_1}$ < 0.74 in the high-loss region passing through the points ④ and ⑤, the size of Region A is comparable to the core size, such that LP11 mode couples easily to the LP01-like cladding hole mode that is in Region A. The insets show the intensity profiles of LP11 core modes at ④ and ⑤. Such coupling disappears when ${d_2}/{d_1}$ > 0.74. This is because the effective index difference between LP11 and the cladding hole mode in Region A becomes larger as the hollow space is reduced. However, upon further increase in ${d_2}/{d_1}$, a narrow high-loss band appears around ⑥ in Fig. 3(b) for specific values of ${d_3}/{d_2}$. This is a result of interaction between LP11 core mode and LP01-like cladding hole mode that is in Region B. The intensity profile of LP11 core mode in this high-loss band at ⑥ is presented in the inset in Fig. 3(b). The field is strongly confined in Region B, indicating the coexistence of the LP01-like cladding hole mode and LP11 core mode.
In the high-loss area around ⑦ at the right bottom in Fig. 3(b), LP11 mode couples to a higher-order cladding hole mode in Region B. The intensity profile of LP11 core mode at ⑦ is shown in the inset in Fig. 3(b). It illustrates the co-presence of the higher-order cladding hollow mode in Region B and LP11 core mode.
We reiterate that LP11 core mode is not always the higher-order core mode with the lowest confinement loss. Figure 3(c) shows the false colormap of the loss in the regions where other higher-order core modes, namely LP21 and LP02 modes, exhibit lower loss than LP11 core mode. In these regions, LP11 core mode has high loss due to its strong coupling to cladding hole modes as discussed above. The intensity profiles of the two higher-order modes at ⑧ and ⑨ in Fig. 3(c) are shown in the insets. Namely, when ${d_2}/{d_1}$ = 0.765 and ${d_3}/{d_2}$ = 0.379, LP21 mode is the higher-order core mode with the lowest loss, and when ${d_2}/{d_1}$ = 0.81 and ${d_3}/{d_2}$ = 0.405, LP02 is the one with the lowest loss.
Single mode guidance in hollow-core fibers can be effectively realized by engineering a large confinement loss difference between the fundamental core mode and other higher-order modes. To this end, higher-order mode extinction ratio (HOMER) is a measure that can quantify the single-modedness of hollow-core fibers. HOMER is defined as the loss ratio between the higher-order mode with lowest loss and fundamental core mode. Here, HOMER > 103 can be considered a fiber with excellent single-mode guidance [16]. Figure 3(d) shows the false colormap of HOMER for varying ${d_2}/{d_1}$ and ${d_3}/{d_2}$ in DT-DNANF. There is a wide area with HOMER > 103, inside the black-dashed line in the parameter space. Its location overlaps with the high loss band of LP11 core mode shown in Fig. 3(b). DT-DNANF has the highest HOMER value of 8 × 105 at ${d_2}/{d_1}$ = 0.765 and ${d_3}/{d_2}$ = 0.37 (⑩), which is great for single mode guidance even over a short fiber length. The confinement loss also remains relatively low at 5.31 × 10−6 dB m−1 at this point, making it a sweet spot for both ultra-low loss and excellent single mode guidance. The intensity profiles of the LP01 and LP11 core mode at ⑩ are shown in the inset. Note that there is a large region of low HOMER of less than 1 bounded in the black solid line. This is due to the high loss in the fundamental core mode as studied in Fig. 3(a).
Figure 4(a) shows the effective indices of the fundamental core mode (LP01), higher-order core modes (LP11, LP21, LP31, LP02), and cladding hole modes (CM1, CM2, CM3) when ${d_3}/{d_2}$ is tuned in DT-DNANF. Note that ${d_1}/D$ and ${d_2}/{d_1}$ are set to be 1.8 and 0.765, respectively. The intensity profiles of the modes involved are shown next to the respective lines in Fig. 4(a). The effective indices of all the fundamental and higher-order core modes do not change much when ${d_3}/{d_2}$ is varied. This is because the core modes are influenced mostly by the core size, which are not affected by ${d_3}/{d_2}$. On the other hand, the cladding hole modes are significantly altered. There are the index matchings between the cladding hole modes and higher-order core modes (LP11, LP02, and LP21) at several points along ${d_3}/{d_2}$. For instance, the effective index of cladding hole mode in Region B matches that of LP11 core mode when ${d_3}/{d_2}$ = 0.379. The index matching leads to strong coupling between the modes resulting in high leakage loss. This is shown in Fig. 4(b) that plots the confinement loss of the fundamental core and higher-order core modes for the same range of ${d_3}/{d_2}$ in DT-DNANF. There are several apparent high loss peaks for the fundamental and higher-order core modes that are associated with their index matching with the cladding hole modes. For instance, a high loss peak exists for LP11 core mode at ${d_3}/{d_2}$ = 0.379. Likewise, there are corresponding index matchings for all high loss peaks for LP02 and LP21 core modes as evident in Figs. 4(a) and 4(b).
Fig. 4. (a) Effective indices of the fundamental core mode (LP01), high-order core modes (LP11, LP21, LP31, LP02), and cladding hole modes (CM1, CM2, CM3) as a function of ${d_3}/{d_2}$ in the DT-DNANF when ${d_1}/D$ and ${d_2}/{d_1}$ are 1.8 and 0.77, and $\lambda $ = 1.55 µm, respectively. The intensity profiles of the cladding hole modes involved are presented next to the respective lines. (b) Confinement loss of the fundamental and higher-order core modes as a function of ${d_3}/{d_2}$ in DT-DNANF with the same geometry as in (a).
In previous studies with single-nested tubes [4], fibers with five antiresonant reflection elements ($N$ = 5) exhibited the optimum performance. Based on this idea, the DNANF demonstration presented in [16] also used $N$ = 5. However, our finding here shows that this is not the case for antiresonant hollow-core fibers with multi-nested cladding elements. The effective indices and confinement losses of LP01 and LP11 core modes versus ${d_3}/{d_2}$ in $N$ = 4 DNANF when ${d_1}/D$ = 1.8 and ${d_2}/{d_1}$ = 0.77 are plotted in Figs. 5(a) and 5(b), respectively. The high loss peaks of the LP01 and LP11 core modes and corresponding index matching between these core modes and CM4, CM5, and CM6 cladding hole modes are evident. The intensity profiles of the respective cladding hole modes involved in the couplings are shown in Fig. 5(a). The index matching between the LP11 higher-order core mode and cladding hole modes occur at ${d_3}/{d_2}$ ≈ 0.28, 0.62, and 0.85. They enable the higher-order core mode to couple to the cladding hole modes CM4, CM5, and CM6, each resulting in a high loss peak. The loss in the LP11 mode is the highest when ${d_3}/{d_2}$ = 0.62, which originates from its coupling to CM5 mode. The mode coupling to Region A (the hollow region between the outer-most and outer-nested tubes) does not occur in DNANF due Region A’s small size. This is a common feature to both $N$ = 4 DNANF and $N$ = 4 DT-DNANF. Here, we emphasize that a DNANF with $N$ = 4 exhibits about an order of magnitude lower confinement loss than that with $N$ = 5 [16]. Furthermore, the confinement loss in the most prominent LP11 higher-order mode is higher by a factor of two in the four-element design, representing a better single-mode guidance when $N$ = 4.
Fig. 5. (a) Effective indices of the fundamental core mode (LP01), higher-order core mode (LP11), and cladding hole modes (CM4, CM5, and CM6) as a function of ${d_3}/{d_2}$ in $N$ = 4 DNANF when $\lambda $ = 1.55 µm, ${d_1}/D$ = 1.8, and ${d_2}/{d_1}$ = 0.77. The intensity profiles of the cladding hole modes involved are presented next to the respective lines. (b) Confinement loss versus ${d_3}/{d_2}$ for LP01 and LP11 core modes in DNANF and DT-DNANF when $N$ = 4. The two cladding modes, i.e., CM5 in the DNANF and CM1 in the DT-DNANF, are also included in the plot to understand the origin of the higher LP11 confinement loss observed in DT-DNANF. The inset shows the cross-sectional geometry of the $N$ = 4 DNANF used in the calculations.
Figure 5(b) also includes the confinement losses of the fundamental (LP01) and higher-order (LP11) core modes of DT-DNANF to demonstrate its superior single mode performance against the DNANF. Here we note, overall, much higher loss in the LP11 mode in DT-DNANF than in DNANF, indicating more efficient higher-order mode suppression. This is because CM1 in DT-DNANF sits near the jacket tube, making it highly leaky and leading to quick dissipation of the energy when LP11 mode couples to CM1. On the contrary, CM5 in DNANF is tightly confined inside the cladding elements via multiple reflecting layers, increasing the likelihood of light in CM5 coupling back to LP11 core mode.
We compare the performance of DT-DNANF with other high-performance antiresonant hollow core fiber with nested elements. The confinement loss of the fundamental core mode and HOMER in nested antiresonant nodeless fiber (NANF, [30]), double nested antiresonant nodeless fiber (DNANF, [16]), and DT-DNANF are plotted in Figs. 6(a) and 6(b) for the spectral range 1–2.5 µm. The structural parameters of the NANF are ${d_1}/D$ = 1.053 and ${d_2}/D$ = 0.547; $N$ = 5 DNANF are ${d_1}/D$ = 1.047, ${d_2}/D$ = 0.803, and ${d_3}/D$ = 0.22; $N$ = 5 DT-DNANF are ${d_1}/D$ = 1.24, ${d_2}/{d_1}$ = 1.082, and ${d_3}/{d_2}$ = 0.345; and $N$ = 4 DT-DNANF are ${d_1}/D$ = 1.8, ${d_2}/{d_1}$ = 0.765, and ${d_3}/{d_2}$ = 0.37. For all fibers, $D$ = 30 µm and $t$ = 0.4 µm. The parameters of $N$ = 5 NANF and DNANF are chosen from their numerically optimized designs in Refs. [30] and [16], respectively. Figure 6(a) shows that $N$ = 4 DT-DNANF exhibits superior confinement loss performance over the other types of nested geometries.
Fig. 6. (a) Confinement loss of LP01 core mode and (b) HOMER in NANF, $N$ = 5 DNANF, $N$ = 4 DT-DNANF, and $N$ = 5 DT-DNANF as a function of wavelength for the spectral range 1–2.5 µm. The geometrical parameters of these fibers are chosen from their numerically optimized designs. The insets show the cross-sectional geometries of each. The structural parameters used in the calculations are: ${d_1}/D$ = 1.053 and ${d_2}/D$ = 0.547 for NANF; ${d_1}/D$ = 1.047, ${d_2}/D$ = 0.803, and ${d_3}/D$ = 0.22 for $N$ = 5 DNANF; and ${d_1}/D$ = 1.8, ${d_2}/{d_1}$ = 0.765, and ${d_3}/{d_2}$ = 0.37 for $N$ = 4 DT-DNANF and ${d_1}/D$ = 1.24, ${d_2}/{d_1}$ = 1.082, and ${d_3}/{d_2}$ = 0.345 for $N$ = 5 DT-DNANF.
As we are entering the ultralow confinement loss regime with the multi-nested geometries, surface scattering loss (SSL) may become the dominant transmission loss mechanism in the fiber. The surface roughness arises when the surface capillary wave solidifies at the interface of the silica layer during the thermal drawing process. The loss due to surface scattering is calculated using [17]:
At 1.55 µm wavelength, the surface scattering loss in the DT-DNANF and DNANF are around 1 × 10−4 dB m−1, which is comparable to that of NANF. It suggests that in the telecommunication spectral range (1.5–1.7 µm), surface scattering is the most prominent factor contributing to the total transmission loss in these fibers. However, the same cannot be said at longer wavelengths. In particular, the confinement loss in DNANF increases much more rapidly than in DT-DNANF as the wavelength increases, and eventually exceeds the surface scattering loss at 1.7 µm to become the dominant loss factor. On the other hand, DT-DNANF maintains excellent confinement loss until 2.3 µm is reached, meaning that DT-DNANF can be designed to exhibit outstanding loss performance over a broad spectral range of up to and beyond 2 µm.
We should also mention that $\eta $ is associated with surface quality of the silica interface, and we highlight some promising recent advances in the fiber fabrication front for suppressing the surface roughness. Notably, [31] successfully reduced the surface roughness value by introducing shear flow in their fabrication of hollow-core fibers. Based on their method, $\eta $ is reduced by a factor of 7.27. We believe this new technique, combined with the low confinement loss offered in the four-element DT-DNANF design, can potentially bring meaningful reduction in the overall transmission loss in antiresonant hollow-core fibers.
We observe oscillations in the confinement loss spectrum of DT-DNANF in Fig. 6(a). This appears due to the presence of dielectric nodes between adjacent layers, where the fundamental core mode couples to dielectric cladding modes [22]. Despite the oscillations, the confinement loss in the telecommunication spectral range in DT-DNANF is still well below 1 × 10−5 dB m−1.
Here, we reaffirm that $N$ = 4 is the optimum choice for the number of cladding elements in multi-nested antiresonant hollow-core fibers. Shown in Fig. 6 is the confinement loss and higher-order mode extinction ratio comparisons between the DT-DNANFs with $N$ = 4 and $N$ = 5. We can clearly notice significantly better guiding properties when $N$ = 4. All in all, we emphasize that $N$ = 4 DT-DNANF has the highest HOMER among the multi-nested antiresonant hollow-core fibers in Fig. 6(b). It is around two orders of magnitude higher than the other fibers. This makes it an excellent candidate for low-loss, single-mode-guiding fiber that transmits over a broad spectral range covering near- and mid-infrared spectral ranges.
3.2 Bending loss
Sensitivity to bending is another important characteristic in optical fibers. To quantify the loss induced by bending, we apply a conformal mapping technique—a widely used technique to calculate the bending loss of antiresonant hollow-core fibers [32,33]. Here, the refractive index of the fiber under mechanical bending $\boldsymbol{n^{\prime}}$ along the direction of bending axis is given by:
where x is the distance from the fiber center in the direction of bending, R is the bending radius, and n is the refractive index in the absence of mechanical stress. The bending-induced loss is then calculated by subtracting the confinement loss of the straight fiber from that of the bent fiber [34,35].We first investigate the effect of altering ${d_1}$. Figure 7(a) is a false colormap of the bending loss for varying ${d_1}/D$ and ${d_2}/{d_1}$ at R of 3 cm. Here, we first set ${d_3}$ to be $({d_2} - t/2)/2$. The bending-induced loss is high when ${d_2}/{d_1}$ < 0.75, indicating that strong coupling between fundamental core mode and cladding hole modes occurs here at $R$ = 3 cm. The intensity profiles of the fundamental core mode at ① and ② shown in Fig. 7(a) insets suggest that $R$ = 3 cm bending induces strongly coupling between the fundamental core mode and cladding hole modes in Region A. When ${d_2}/{d_1}$ is further increased, the bending-induced loss becomes comparably low. In the region where ${d_2}/{d_1}$ > 0.75, several bands with high bending-induced loss appear around ③, ④, and ⑤. These are caused by the bending-induced resonant coupling between the fundamental core mode and a variety of cladding hole modes. The intensity profiles are presented in the insets in Fig. 7(a) for ③, ④, and ⑤ to illustrate such couplings. From these, we can observe $R$ = 3 cm bending causes the fundamental core mode to couple to the cladding hole modes inside the inner nested tubes (③, ⑤) or Region B (④).
Fig. 7. False colormaps of the bending-induced loss in the fundamental core mode in DT-DNANF at $R$ = 3 cm for (a) varying ${d_1}/D$ and ${d_2}/{d_1}$ when ${d_3} = ({d_2} - t/2)/2$, and (b) varying ${d_3}/{d_2}$ and ${d_2}/{d_1}$ when ${d_1}/D$ = 1.8 and $\lambda $ = 1.55 µm. The inset show the intensity profiles at the corresponding points.
No bending-induced couplings are seen when 1.6 < ${d_1}/D$ < 1.9. This means there is a broad range of ${d_1}/D$ for low bending-induced loss. DT-DNANF has the lowest $R$ = 3 cm bending-induced loss of around 1.4 × 10−4 dB m−1 when ${d_1}/D$ = 1.8 and ${d_2}/{d_1}$ = 0.946. Thus, we continue to set ${d_1}/D$ = 1.8 for our subsequent analysis. We recall that this value is also the optimal choice for the lowest confinement loss.
We further investigate the effect of altering the nested elements on the bending-induced loss. Figure 7(b) is a false colormap of the bending-induced loss for varying ${d_2}/{d_1}$ and ${d_3}/{d_2}$ in DT-DNANF when $R$ = 3 cm. There are several high bending-induced loss bands around ⑥, ⑦, ⑧, and ⑨. When ${d_2}/{d_1}$ = 0.7, the high loss band around ⑥ forms. The intensity profile at ⑥ is shown in the inset in Fig. 7(b). It is evident that this band is caused by the bending-induced resonant coupling to a cladding hole mode in Region A. The $R$ = 3 cm bending of DT-DNANF causes the index matching between the fundamental core mode and cladding hole mode in Region A, leading to high bending-induced loss.
As ${d_2}/{d_1}$ becomes larger, the size of Region A becomes smaller. This makes $R$ = 3 cm bending-induced index difference between cladding hole mode in Region A and fundamental core mode larger. On the other hand, as ${d_2}/{d_1}$ further increases, the size of the outer nested tube increases. Correspondingly, the size of Region B increases. Thus, $R$ = 3 cm bending of DT-DNANF easily satisfies the index matching between various cladding hole modes in the nested element and fundamental core mode, resulting in the appearance of the three high loss bands around ⑦, ⑧, and ⑨ when ${d_2}/{d_1}$ is increased. The intensity profiles at ⑦, ⑧, and ⑨ are shown in the insets in Fig. 7(b). The bending causes the fundamental core mode to couple to the cladding hole modes located in the outer cladding tubes along the bending direction. The cladding hole modes at play are either in Region B (⑦, ⑧) or inside the inner nested tubes (⑨).
High bending-induced loss bands do not show up when ${d_3}/{d_2}$ > 0.45 and ${d_2}/{d_1}$ > 0.7. This indicates enlarging the size of nested elements, particularly the inner nested tubes, substantially reduces the bending-induced couplings. As ${d_3}/{d_2}$ and ${d_2}/{d_1}$ increase, Regions A and B become smaller. This creates a larger size difference and hence the index mismatch between the core and Regions A and B. It ensures that bending-induced index matching hardly occurs when ${d_3}/{d_2}$ > 0.45 and ${d_2}/{d_1}$ > 0.7 when DT-DNANF is bent at $R$ = 3 cm.
The lowest bending-induced loss of around 1.42 × 10−4 dB m−1 is realized at $R$ = 3 cm when ${d_2}/{d_1}$ = 0.815 and ${d_3}/{d_2}$ = 0.49. This is the optimum structural parameter for achieving the low bending-induced loss with $R$ = 3 cm. The above analysis shows DT-DNANF has excellent bending-insensitivity even under tight bending condition. As R is reduced, the effective index of the hollow regions in nested elements will be modified more severely [34,36,37]. Thus, the coupling between the core and cladding hole modes will occur more easily even when Regions A and B become smaller. As such, Regions A and B must become smaller to prevent such couplings. When R is increased, the trend will be the opposite.
Figure 8 shows the bending-induced loss as a function of R for DT-DNANF with ${d_1}/D$ = 1.8, ${d_2}/{d_1}$ = 0.815, and ${d_3}/{d_2}$ = 0.49. There is a loss peak at $R$ = 4 cm. The lowest loss is below 1 × 10−3 dB m−1 when bending radius R is greater than 2 cm. DT-DNANF has low bending-induced loss of around 10−2 dB m−1 even at an extreme bending condition with bending radius of only 1 cm. The intensity profile of fundamental core mode at $R$ = 1 cm is shown in the inset, exhibiting an excellent mode profile.
Fig. 8. Bending-induced loss of the fundamental core mode as a function of R in DT-DNANF with ${d_1}/D$ = 1.8, ${d_2}/{d_1}$ = 0.8148, ${d_3}/{d_2}$ = 0.49 when $\lambda $ = 1.55 µm. The inset is the intensity profile of the fundamental core mode at $R$ = 1 cm.
3.3 Fabrication tolerance
In practice, the rotation of inner nested cladding tubes around the center of the outer nested tubes is one of the main fabrication errors [38]. We study the effect of such fabrication error on the guiding performance. As illustrated in Fig. 9(a), $\theta $ is the rotational angle between the inner nested tubes and outer nested tube. We assume $\theta $ < 0 when the inner nested element is rotated towards the core, and $\theta $ > 0 for rotation in the opposite direction. In our numerical study, other parameters are set to be ${d_1}/D$ = 1.8, ${d_2}/{d_1}$ = 0.765, and ${d_3}/{d_2}$ = 0.37. The calculation ends when the two inner nested tubes are in contact with each other. We should mention that in practice, $\theta $ is usually within ± 5 deg [16]. When $\theta $ < 0, i.e., the inner nested tubes are closer to the core, the confinement loss of the LP01 does not change much. This indicates that the proximity of the inner cladding tubes to the core do not affect the light confinement. When $\theta $ > 0, the loss increases significantly with increasing $\theta $. For the single-modedness, DT-DNANF retains high HOMER >1000 over a large range of $\theta $.
Fig. 9. (a) Schematic illustration of the definition of angle $\mathrm{\theta }$. (b) Confinement loss of the fundamental core mode and higher-order (LP11) mode as a function of $\mathrm{\theta }$ for DT-DNANF with ${d_1}/D$ = 1.8, ${d_2}/{d_1}$ = 0.765, and ${d_3}/{d_2}$ = 0.37 when $\lambda $ = 1.55 µm. HOMER is plotted together for reference.
Regarding the DT-DNANF’s fabricability, we believe that the stack-and-draw method is a promising technique for realizing DT-DNANF. References [39,40] report successful fabrication of four-element simple tubular and nested-tubular antiresonant hollow-core fibers. These demonstrations suggest potential pathways to fabricate the geometry proposed in this work. In addition, these articles show that the four-fold symmetry does not affect the light coupling in and out of the fibers. The laser beam with a four-fold symmetry mode profile was successfully delivered in these works, and no coupling issues were reported.
4. Conclusions
We proposed a new antiresonant hollow-core fiber design, i.e., DT-DNANF, which is based on a multi-nested cladding geometry. We carried out comprehensive numerical analyses to identify common mode couplings that occur in DT-DNANF, studied their dependence on the structural parameters, and determined the optimal geometry. The confinement loss can be as small as 4 × 10−7 dB m−1, while HOMER of more than 105 can be achieved, outperforming the other low-loss antiresonant hollow-core fibers that rely on the nested cladding geometry. Moreover, DT-DNANF can exhibit excellent bending performance even at tight bending conditions.
Funding
Ministry of Education - Singapore (RG135/20).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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