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Abstract
The development of wideband guided hollow-core anti-resonant fiber (HC-ARF) that covers the sensitive range of the human eye's visible spectrum is progressing rapidly. However, achieving low-loss wideband transmission with a small bending radius remains a challenging issue to be addressed. In light of this, we propose a novel, to our knowledge, HC-ARF with a nested double-semi-elliptical cladding structure in the visible spectral region. By employing finite element method simulations, we investigate the confinement loss, bending loss, and single-mode performance of this fiber design. The result shows that the confinement loss of this new fiber exhibits below 10−5 dB·m-1 across almost the entire visible band range, with a minimum loss of 1.55 × 10−7 dB·m-1 achieved for λ = 650 nm. Furthermore, this fiber demonstrates excellent resistance to bending and can maintain an ultra-low bending loss as low as 3 × 10−7 dB·m-1 even under extreme bending conditions with a radius of only 3 cm. Notably, its 3-dB bending radius reaches just 3.5 cm for λ = 532 nm. Additionally, it exhibits outstanding single-mode conductivity under various bending scenarios and achieves a high extinction ratio of up to 104 for higher-order modes after parameter optimization for specific wavelengths.
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1. Introduction
Over the past decade, microstructured hollow-core fibers (HCFs) have experienced rapid development [1]. Compared to traditional silica fibers, HCFs offer numerous advantages, including low backscattering, low nonlinearity, low dispersion, and wide spectral transmission capabilities [2–4]. Among these, the hollow-core anti-resonant fiber (HC-ARF) has particularly attracted widespread attention from researchers [5–7]. The material selection for HC-ARF significantly influences its optical performance. In the near-infrared and visible light bands, silica glass is commonly used. However, the material absorption coefficient of silica glass is extremely high in the mid-infrared band, limiting its application for single-mode transmission. To address this, researchers have conducted studies on HC-ARF based on soft glass materials such as sulfide glass and fluoride glass, achieving significant advancements [8]. In the terahertz range, due to significant transmission losses caused by silica glass, HC-ARF typically utilizes polymer materials such as PMMA, Topas, and Zeonex, which offer low cost and effective material loss [9]. In terms of the optical bands guided by HC-ARF, research efforts have mainly focused on the near-infrared and mid-infrared (NIR and MIR) regions [10–13]. The design and manufacturing processes for HC-ARF in these two bands have become increasingly refined. However, the broad-spectrum guided HC-ARF covering the visible light spectrum has gradually attracted researchers’ interest, emerging as a new research hotspot [14–17].
Within the visible spectrum range sensitive to the human eye, HC-ARF have found widespread application. Surface plasmon resonance (SPR) refractive index sensors made using visible light-guided HC-ARF can be applied in medical diagnostics, bioimaging, and bioanalysis [18,19]. In applications such as biological fluorescence spectroscopy [20,21], antibody detection [22,23], secretory gland cell detection [24], and endoscopy [25,26], HC-ARF in the visible light spectrum play a major experimental role. Benefiting from the natural advantages of their structure, HC-ARF can also serve as high-quality gas chamber structures in gas detection systems based on stimulated Raman scattering and photoacoustic spectroscopy [27–32], acting as effective media for enhancing measurement data. In the field of laser processing, visible light-guided HC-ARF, with their good dispersion resistance, enable the transmission of ultrafast laser pulses and high-power pulses [33,34]. Additionally, HC-ARF have been used in experiments measuring nanoscale magnetic fields [35] and in fiber optic lighting systems [36]. These applications place high demands on the bending resistance of HC-ARF to achieve low-loss transmission under small bending radius. Currently, due to limitations in the selection of cladding structures, most circular-cladding anti-resonant hollow-core fibers still need improvement in terms of bending performance.
The correct selection of cladding structure is crucial for enhancing the bending resistance of HC-ARF. The light-guiding principle of HC-ARF can be explained by suppressing mode coupling [37]. Through careful design of the cladding structure, a refractive index mismatch between the cladding and the core effectively confines the modal energy within the core region. HC-ARF typically employ circular cladding and its nested structures, which have garnered widespread attention [5,13–15]. Among these reports, in 2022, Gregory T. Jasion and others successfully fabricated a double-nested anti-resonant hollow-core fiber (DNANF), which achieved a record-low loss of 0.174 dB/km in the C-band [13]. However, recent research shows that a circular cladding structure is not essential for enhancing the fiber's optical performance [12,38,39]. Instead, the light-guiding properties of ARFs are mainly influenced by the negative curvature at the fiber core boundary. Therefore, non-circular cladding elements can also achieve similar functionality. The semi-elliptical cladding element is one of the structures proposed in the study of the impact of the curvature at the fiber core boundary [40]. The report analyzed different degrees of ellipticity and the number of cladding elements, finding that the semi-elliptical structure of ARF exhibits good optical performance in light guidance and low loss. Compared to circular elements, semi-elliptical elements could provide additional structural stability and more precise positioning of the cladding elements, because each semi-elliptical element has two anchor points connected to the outer sheath. Furthermore, the adjustable position of the anchor points allows for a larger negative curvature at the fiber core, reducing the overlap area between the core mode and the glass mode, suppressing effective refractive index matching, reducing the coupling between the core and glass modes, optimizing the fundamental mode confinement ability, and enhancing the fiber's resistance to bending.
In this work, we present a novel nested double-semi-elliptical anti-resonant hollow-core fiber that covers almost the entire visible light spectrum. We conduct a comprehensive numerical study of this new hollow-core fiber, investigating the impact of changes in the structural parameters of the cladding elements on the fiber's optical characteristics and comparing it with typical double-nested nodeless anti-resonant hollow-core fiber. The introduction of a large negative curvature double semi-elliptical shape and nested smaller circle suppresses core mode leakage into the cladding, resulting in ultra-low confinement loss, excellent bending performance, and single-mode performance under bending conditions across almost the entire visible spectrum.
2. Fiber geometry and numerical method
The design of the proposed fiber is shown in Fig. 1(a). Its cladding consists of a double semi-elliptical structure with an innermost nested circle. Therefore, it is referred to as a nested double-semi-elliptical anti-resonant nodeless hollow-core fiber (NDSE-ANF). The two semi-elliptical tubes of the NDSE-ANF have different curvature degrees, and the innermost nested circle has an adjustable radius. It retains the negative curvature at the three air-glass boundaries typical of conventional DNANF, as shown in Fig. 1(b), (c), and (d). When light attempts to leak from the core, it encounters six different dielectric layers (including three glass dielectric layers of thickness t and three air layers in region A, B, and C), which effectively confining light within the core region. Compared to DNANF, NDSE-ANF offers a more flexible and adjustable negative curvature range. This suggests that NDSE-ANF is likely to maintain its special properties of ultra-low optical loss, wide optical bandwidth, and ultra-low overlap between core and glass modes. Therefore, we believe that introducing this nested double-semi-elliptical cladding element in fiber design is worthwhile, as it provides a high degree of design flexibility while introducing a large negative curvature. Furthermore, in the context of the known ARF manufacturing technique, which is based on stacking cylindrical tubes, fusing the tubes together at contact points, and drawing the resulting prefabricated component into a fiber, researchers have also explored drawing techniques for non-circular cladding elements and have successfully manufactured negative curvature HCFs with conjoined tubes or quadruple rotational symmetry in cladding thicknesses [7,12]. This indicates that, based on the cylindrical tube stacking technique and the ideal curvatures of nested semi-ellipses and the innermost small circle obtained from simulation, multiple polymer tubes with varying curvatures can be selectively provided. These tubes are inserted into each other to form a nested group or array, with the tube having the highest curvature positioned at the innermost location. By moving the nested group outward, the curvature is gradually reduced. The nested group is inserted into the cylindrical outer sheath, and the nested polymer tubes are fixed inside the outer sheath to form the required prefabricated component. Finally, the prefabricated component is drawn into a fiber, forming the finished NDSE-ANF structure.
Fig. 1. (a) Idealized cross-section of NDSE-ANF. t represents the thickness of the cladding element, $D_{c}$ is the core diameter, $r_{a1}$ is the major semi-axes of the outer semi-ellipse, $r_{b1}$ is the minor axes of the outer semi-ellipse, $r_{a2}$ is the major semi-axis of the inner semi-ellipse, $r_{b2}$ is the minor axes of the inner semi-ellipse and r is the radius of the nested innermost circle. (b), (c), and (d) correspond respectively to the three hollow regions within the cladding elements.
The geometry of the NDSE-ANF is characterized by the core diameter $D_{c}$, the dielectric layer thickness of the cladding elements t, and the number of cladding elements N. Parameters $r_{a1}$ and $r_{a2}$ define the major semi-axes of the outer and inner semi-ellipse, Parameters $r_{b1}$ and $r_{b2}$ define the minor axes of the outer and inner semi-ellipse. Parameter r represents the radius of the nested innermost circle. The core diameter $D_{c}$ refers to the maximum diameter of the inscribed circle in the fiber core, as shown by the dashed circle in Fig. 1(a). We note that most hollow-core fibers manufactured to date maintain the core diameter $D_{c}$ at around $25\lambda_{0} - 35\lambda_{0}$, where $\lambda_{0}$ is the optimized target wavelength [38]. Here, we choose $\lambda_{0} = 532\,nm$. Therefore, we set $D_c \approx 30\lambda_{0} \approx 16\,\mu m$ in the subsequent analysis. The glass refractive index ${n_{gl}}$ can be calculated using Eq. (1) [41].
A tube thickness $t = 210\,nm$ that generates the first resonance peak at approximately $\lambda = 400\,nm$ is selected for analysis in the following section. This configuration exhibits relatively low confinement loss for the visible band range. To achieve wideband transmission and excellent bending performance, we chose $N = 8$, which will be explained in detail in subsequent sections. To maximize the adjustability of the innermost circle, we set $r_{b1} = r_{b2} - 2t$, ensuring that the anchor points of the inner semi-ellipse are adjacent to those of the outer semi-ellipse.
After the above definitions, there are four parameters left for adjustment, namely $r_{a1}$, $r_{b1}$, $r_{a2}$,and r. In subsequent calculations, we normalized the changes in $r_{a1}$ and $r_{b1}$ as $\alpha_{1} = 2r_{b1}/D_{c}$ and $\alpha_{2} = 2r_{a1}/D_{c}$, and the changes in $r_{a2}$ and r as $\beta_{1} = r/2r_{a1}$ and $\beta_{2} = r_{a2}/r_{a1}$. This normalization allows us to explore the parameter space without generating nodes, resulting in the following ranges for the four parameters: $0.35 \le \alpha_{1} \le 0.45$, $1.0 \le \alpha_{2} \le 2.0$, $0.08 \le \beta_{1} \le 0.17$, and $0.65 \le \beta_{2} \le 0.95$.
All simulations in this study were performed using the finite-element-method solver (COMSOL) to calculate the losses and mode indices for all structural parameters. To obtain convergent simulation results, the maximum grid size for the quartz glass regions, which contribute to the anti-resonant reflection, should not exceed $\lambda /5.8$, and for the air regions, it should not exceed $\lambda /4$ [5]. Additionally, perfect matching layers (PML) boundaries were used for the outermost layer of the geometric shape.
3. Simulation and results
3.1 Confinement loss
Confinement loss is the key character of the proposed NDSE-ANF. It refers to the power loss caused by the leakage of the optical field within the core of the fiber, which is essentially determined by the inherent structure of the fiber. The confinement loss can be calculated using Eq. (3) [43].
Here, c represents the speed of light in free space, f represents the operating frequency, and $Im({{n_{eff}}} )$ represents the imaginary part of the effective refractive index.
Due to the complexity of simultaneously varying all four geometric parameters as independent variables, we simplified the process by initially setting $\beta_{1} = 0.12$ and $\beta_{2} = 0.75$, which means that $r_{a2}$, r, and the major semi-axis of the outer ellipse, $r_{a1}$, vary in proportion. This ensures that the relative distances between adjacent cladding elements remain constant. Then, we parameterized the scan for $\alpha_{1}$ and $\alpha_{2}$, which represent the major and minor axes of the outer semi-ellipse, to find the optimal structural parameters. As shown in Fig. 2(a), under the condition of $\lambda = 532\textrm{ }nm$, we observed smooth variations in the confinement loss within the range of $0.35 \le \alpha_{1} \le 0.45$ and $1.0 \le \alpha_{2} \le 2.0$.
Fig. 2. (a) False colormap of the LP01 mode's confinement loss for NDSE-ANF as $\alpha_{1}$ and $\alpha_{1}$ vary, with $\lambda = 532\,nm$. The insets illustrate the intensity distribution of the fundamental core mode at positions ① and ②. (b), (c), and (d) respectively represent the z-components of the Poynting vector of the LP01 mode in NDSE-ANF at positions ①, ③, and ④.
From an overall perspective, keeping $\alpha_{1}$ constant while decreasing $\alpha_{2}$ leads to a significant increase in the LP01 mode loss in the fiber core. This is due to the reduction in the major semi-axis of the outer semi-ellipse, which decreases the cladding area and increases the ratio of core area to cladding area, resulting in significant gaps between the cladding tubes, leading to leakage of the additional light field. Figures 2(b) and 2(c) show 3-dB contour plots of the z-component of Poynting vector for the LP01 mode of the core when $\alpha_{2} = 1.3$ and $\alpha_{2} = 1.9$, respectively. The contour lines around the core in Fig. 2(b) indicate strong confinement of the LP01 mode by the cladding elements, while in Fig. 2(c), due to the decreased major semi-axis of the outer semi-ellipse, some light leaks into the gaps between the cladding elements.
In the range of $1.0 \le \alpha_{2} \le \textrm{1}\textrm{.6}$, keeping $\alpha_{2}$ constant while decreasing $\alpha_{1}$ also leads to a significant increase in the LP01 mode loss. This is because when the minor axis of the outer semi-ellipse is small, significant light leakage occurs between adjacent cladding elements, as shown in Fig. 2(c) and (d). In Fig. 2(d), compared to Fig. 2(c), the outermost contour lines extend closer to the outermost jacket, indicating a higher degree of light leakage.
In the wider range of $1.6 \le \alpha_{2} \le 2.0$, as indicated by the white dashed line encompassing the area in Fig. 2(a), the fiber exhibits an ultra-low confinement loss of less than 10−6 dB·m-1. In this range, keeping $\alpha_{2}$ constant while increasing $\alpha_{1}$ results in a small initial decrease followed by increase in the confinement loss. The area marked with a purple circle ② in Fig. 2(a) shows the lowest confinement loss of 3.46 × 10−7 dB·m-1 at $\alpha_{1} = 0.36$, $\alpha_{2} = 1.9$. Another point marked with a purple circle ① shows a slightly higher but still low confinement loss of 7.23 × 10−7 dB·m-1. The slight increase in the confinement loss at point ① can be attributed to the additional resonance caused by the cladding tubes approaching each other just before contact.
While point ② represents the lowest confinement loss, it's worth noting that previous studies have reported a narrowing of the transmission band with increasing curvature of circular and elliptical cladding elements [40]. This phenomenon is attributed to changes in the cutoff frequency of the guided modes within the cladding elements [43]. Additionally, due to the small minor axis of the outer semi-ellipse at point ②, which limits the freedom to adjust the nested innermost circle. Based on the above analysis, in subsequent studies, we chose to sacrifice a minimal increase in confinement loss at point ① ($\alpha_{1} = 0.45,\textrm{ }\alpha_{2} = 1.9$) to gain more freedom to adjust the nested innermost circle, and widen the transmission band.
Figure 3(a) shows the false colormap of the LP01 mode confinement loss in NDSE-ANF when $0.08 \le \beta_{1} \le 0.17$ and $0.65 \le \beta_{2} \le 0.95$. The region enclosed by the white dashed lines shows areas where the loss is below 10−6 dB·m-1, indicating the presence of a broad region for ultra-low loss guidance in NDSE-ANF. The influence of the inner semi-ellipse and the nested innermost circle on the confinement loss fluctuates by approximately two orders of magnitude. The lowest confinement loss occurs at position ① ($\beta_{1} = 0.11,\textrm{ }\beta_{2} = \textrm{0}\textrm{.7}$), where the loss is 5.90 × 10−7 dB·m-1. In comparison, at point ③, the increase in the inner semi-ellipse major semi-axis leads to an oversized Region B, causing significant leakage and a reduction in confinement loss, as shown in the insets (b) and (d) in Fig. 3. Similarly, at point ②, the increase in the radius of the nested innermost circle leads to an oversized Region C, as depicted in the insets (b) and (c) in Fig. 3, where the contour lines extend into the hollow area behind the nested innermost circle.
Fig. 3. (a) False colormap of the LP01 mode's confinement loss for NDSE-ANF as $\beta_{1}$ and $\beta_{2}$ vary, with $\lambda = 532\,nm$. Insets (b), (c), and (d) depict the z-components of the Poynting vector for the LP01 mode at positions ①, ②, and ③, respectively.
To intuitively compare the individual effects of the components in the cladding of NDSE-ANF on confinement loss, we conducted a simulation scan of NDSE-ANF and its substructures in the wavelength range of 450–1000 nm, corresponding to the optimal structural parameters $\alpha_{1} = 0.45,\textrm{ }\alpha_{2} = \textrm{1}\textrm{.9 },\beta_{1} = 0.11,\textrm{ }\beta_{2} = \textrm{0}\textrm{.7}$.As shown in Fig. 4, the confinement loss of different structural components of the fiber is distinguished by different curve colors. The semi-elliptical anti-resonant nodeless hollow-core fiber (SE-ANF) exhibits the flattest loss curve but also the highest loss among all structures. For the NDSE-ANF, we show two curves representing further optimized structures corresponding to points ① and ② in Fig. 2(a). NDSE-ANF-1 corresponds to point ① and provides a wider low-loss transmission band. NDSE-ANF-2 corresponds to point ②, representing the structure with the lowest loss at $\lambda = 532\,nm$. This also demonstrates the influence of the parameter $r_{b1}$ on the transmission band, which is consistent with the previous discussion.
Fig. 4. Confinement loss of the LP01 fundamental core mode for NDSE-ANF and its substructures. Insets depict the z-components of the Poynting vector for the LP01 mode in NDSE-ANF and its substructures.
In the wide visible light range of 450–800 nm, the confinement loss of NDSE-ANF-1 is lower than that of double-semi-elliptical anti-resonant nodeless hollow-core fiber (DSE-ANF) and nested semi-elliptical anti-resonant nodeless hollow-core fiber (NSE-ANF), reaching a minimum of 1.55 × 10−7 dB·m-1 at $\lambda = 650\,nm$. The Insets of Fig. 4 depict the z-components of the Poynting vector for the LP01 mode in NDSE-ANF and its substructures when $\lambda = 650\,nm$, the high-density contour lines in the central hollow core illustrates strong light confinement in NDSE-ANF-1, while other substructures exhibit varying degrees of light leakage. This indicates the inner semi-ellipse and the nested innermost circle reflect the penetrating field back to the core through Fresnel reflection, confining more field inside the core. In the range of 800–1000 nm, the confinement loss curve of NDSE-ANF-1 overlaps with that of DSE-ANF and NSE-ANF, as the wavelength guided in the fiber is closer to the resonance window, the nested structure reduces the constraint ability of the core mode, leading to a significant increase in confinement loss. Due to coupling effects, the loss curve of NDSE-ANF-1 also exhibits some oscillations, which seem to be inherent to the nested structure. Due to its excellent limitation performance, the NDSE-ANF-1 displayed here shows a leakage loss within the range of 10−5 dB·m-1, or even lower, across almost the entire visible light spectrum.
To assess the ultra-low confinement loss and wideband transmission characteristics of the NDSE-ANF structure in the visible spectrum with $N = 8$ cladding elements, we have plotted the confinement loss curves of the LP01 mode against wavelength for N values of 6, 8, and 10, and compared them with a typical nodeless DNANF structure, as shown in Fig. 5. All the anti-resonant hollow-core fibers have the same core diameter ($D_{c} = 16\mu m$) and dielectric layer thickness ($t = 210\,nm$). The parameters for the double-nested anti-resonant nodeless fiber (DNANF) are $r_{1}/D_{c} = 0.5235$, $r_{2}/D_{c} = 0.4015$, and $r_{3}/D_{c} = 0.11$, which were optimized based on reference literature [13]. The $N = 6$ and $N = 10$ NDSE-ANF structures have also undergone parameter optimization.
Fig. 5. Confinement loss of the LP01 fundamental core mode for the DNANF and NDSE-ANF structures with 6, 8, and 10 cladding elements.
We observed that the $N = 8$ NDSE-ANF, compared to the DNANF, exhibits a ∼13 dB decrease in confinement loss at $\lambda = 650\,nm$, attributed to the introduction of the double-elliptical structures, which increase the negative curvature at the fiber core boundary, causing effective refractive index mismatch between the core mode and the cladding modes. Additionally, the $N = 6$ NDSE-ANF has a relatively flat transmission band, which becomes narrower with an increase in cladding elements N. Due to its lowest LP01 fundamental core mode's confinement loss between 520 nm and 650 nm among the three structures, the $N = 8$ NDSE-ANF is selected for further analysis.
3.2 Bending loss
Bending loss is another important characteristic of anti-resonant hollow-core fibers. Here, we studied the bending loss characteristics of the NDSE-ANF. The bending of the fiber can be simulated using a straight fiber with a modified refractive index distribution. We used the conformal transformation method to calculate the bending loss [44,45], which can be obtained by substituting ${n^\prime }(x,y)$ into Eq. (3) as follows:
where $n^{\prime}({x,\textrm{ }y} )$ is the effective refractive index of the bent fiber, $n({x,\textrm{ }y} )$ is the refractive index distribution of the straight fiber, $R_{b}$ is the bending radius, and x is the lateral distance from the center of the fiber.We first study the influence of the major and minor axes of the outer semi-ellipse on the bending loss. As shown in the coordinate axes of the insets in Fig. 6(a) and Fig. 6(b), the bending direction of all fibers in this study is chosen along the x-axis. Figure 6(a) shows the false colormap of the bending loss as a function of $\alpha_{1}$ and $\alpha_{2}$ for a bending radius $R_{b} = 3\,cm$. Here, we maintain $\beta_{1} = 0.12$ and $\beta_{2} = 0.75$. Under the condition of $\lambda = 532\textrm{ }nm$, we can still observe the smooth variation of the bending loss within the range of $0.35 \le \alpha_{1} \le 0.45$, $1.0 \le \alpha_{2} \le 2.0$. This is due to the small overlap between the double-semi-ellipse and the fiber core mode, making it difficult for the fiber core mode to effectively match the modes in the cladding. Even within a wide range of $\alpha_{1}$ and $\alpha_{2}$ variations, the bending loss can still be maintained below 10−5 dB·m-1, as indicated by the area enclosed by the white 10−5 contour line in Fig. 6(a), demonstrating the strong bending resistance of this structure. We obtained the lowest bending loss at point ②, which is 1.86 × 10−6 dB·m-1. We recall that this value corresponds to the point of lowest confinement loss. However, as mentioned in Section 3.1, to obtain a wider transmission band, we still choose the structure parameters corresponding to point ①, i.e., $\alpha_{1} = 0.45$ and $\alpha_{2} = 1.9$. The increase in bending loss at points ③ and ④ is consistent with the discussion in Section 3.1, and will not be reiterated here.
Fig. 6. (a) False colormap of the LP01 mode's bending loss for NDSE-ANF as $\alpha_{1}$ and $\alpha _{2}$ vary, with $R_{b} = 3\,cm$. (b) False colormap of the LP01 mode's bending loss for NDSE-ANF as $\beta_{1}$ and $\beta_{2}$ vary, with $R_{b} = 3\,cm$. Insets show the intensity distribution at the corresponding points.
We further studied the effect of changing the structural parameters of the nested elements on the bending loss. From Fig. 6(b), it can be seen that the change in $\beta_{1}$ and $\beta_{2}$ has a significant impact on the bending loss. Several high-loss bands are observed around points ⑥, ⑦, and ⑧. When $\beta_{1} = 0.085$ and $\beta_{2} = 0.90$, a high-loss band is formed around point ⑥, as shown in the intensity distribution at point ⑥ in the inset of Fig. 6(b). Due to the large size of Region B (the area between the inner semi-ellipse and the nested innermost circle), under the bending condition of $R_{b} = 3\,cm$, an effective refractive index matching occurs between the fiber core mode and the cladding hole mode, resulting in high bending loss. As $\beta_{1}$ increases, the size of Region C (the area of the nested innermost circle) gradually increases, and the size of Region B decreases. This reduces the coupling efficiency between the cladding hole mode and the fiber core mode induced by the bending at $R_{b} = 3\,cm$, resulting in a low-loss band around $\beta_{1} = 0.11$. The bending loss decreases to a minimum of 2.75 × 10−6 dB·m-1 at point ⑤ ($\beta_{1} = 0.11,\textrm{ }\beta_{2} = 0.7$). When $\beta_{1}$ increases to 0.16, the size of Region C approaches the size of the fiber core, making it easy to match the refractive indices between the cladding hole mode in the nested innermost circle and the fiber core mode. This leads to high-loss regions near points ⑦ and ⑧.
To characterize the excellent bending insensitivity of the proposed NDSE-ANF, we plotted the relationship between bending loss and bending radius for DNANF, NDSE-ANF, and its substructures, as shown in Fig. 7. SE-ANF is not considered here due to its significantly higher loss compared to other structures. The structural parameters used in this section's calculations are the same as those shown in Fig. 5. For DSE-ANF, there are two loss peaks at small bending radius of $R_{b} = 4\,cm$ and $R_{b} = 6\,cm$. This phenomenon has also been observed in similar single elliptical core structures [40]. A possible explanation is that at specific bending radius, the field distribution undergoes displacement due to bending, leading to phase matching between the core mode and the cladding mode. In contrast to DSE-ANF, the introduction of the nested innermost circles in the NDSE-ANF and NSE-ANF significantly reduces bending loss and eliminates the peak effect. The fiber's resistance to bending can be characterized by its 3-dB bending radius, defined as the bending radius at which the bending loss, in units of dB·m-1, doubles from its minimum value. Calculations show that the 3-dB bending radius for DNANF at $\lambda = 532\,nm$ is approximately 9 cm, for NSE-ANF it is approximately 8 cm, and for NDSE-ANF it is further reduced to approximately 3.5 cm, as indicated by the black triangles in Fig. 7. This indicates that NDSE-ANF has excellent resistance to bending.
To characterize the light-guiding ability of NDSE-ANF under extreme bending conditions across the visible spectral range, we performed a parametric scan of bending loss at wavelengths ranging from 450 nm to 1000 nm for bending radius of 1 cm, 2 cm, and 3 cm, as shown in Fig. 8. As expected, for $450\,nm < \lambda < 750\,nm$, the bending loss increases as the bending radius decreases. At a bending radius of 3 cm, indicated by the gray area in the figure, the bandwidth with bending loss less than 10−5 dB·m-1 is approximately 250 nm, covering the wavelength range from 510 nm to 760 nm, demonstrating excellent bend resistance. For $750\,nm < \lambda < 1000\,nm$, the bending loss remain constant with the variation of bending radius, even under extreme bending with a radius of 1 cm, it maintains low loss comparable to that of a straight fiber. The insets in Fig. 8 illustrates that under tightly bent conditions, the mode field of the NDSE-ANF fiber shifts in the bending direction, but remains fully confined within the core, indicating the excellent bend insensitivity of the NDSE-ANF structure. To explain the phenomenon that bending loss is almost unaffected by bending radius in the range of 750–1000 nm, we plotted the z-components of the Poynting vector for the LP01 mode when $\lambda = 650\,nm$ and $\lambda = 800\,nm$ for $R_{b} = 1\,cm$ and $R_{b} = \infty $, as shown in the insets of Fig. 8. From the insets, it can be seen that when $\lambda = 650\,nm$, the curved fiber's outer contour line protrudes to the innermost small circle near the protective sheath compared to the straight fiber, indicating significant leakage caused by bending. When $\lambda = 800\,nm$, both the core mode of the curved fiber and the straight fiber couple laterally through the capillary wall into the capillary, leading to leakage losses. This phenomenon arises from the fact that when $\lambda > 750\,nm$, well beyond the central region of anti-resonance, the light field within the fiber core leaks into the capillary wall due to coupling effects, irrespective of the fiber's bending status. As a result, the bending loss curves almost overlap at different bending radius.
Fig. 7. The relationship between bending loss and bending radius for DNANF, NDSE-ANF, DSE-ANF, and NSE-ANF at $\lambda = 532\,nm$. The black triangles represent the 3-dB bending radius. The inset shows the intensity distribution of the cores of different structures at $R_{b} = 3\,cm$.
Fig. 8. Bending loss for the LP01 fundamental core mode of NDSE-ANF, with bending radius of 1 cm, 2 cm, and 3 cm. The insets show the z-components of the Poynting vector for the LP01 mode at $\lambda = 650\,nm$, $R_\textrm{{b}} = \infty $, $\lambda = 650\,nm$, $R_\textrm{{b}} = 1\,\textrm{cm}$, $\lambda = 800\,nm$, $R_\textrm{{b}} = \infty $, and $\lambda = 800\,nm$, $R_\textrm{{b}} = 1\,cm$.
3.3 Single-mode performance
Benefiting from the exceptional bending resistance of NDSE-ANF, this fiber can increase the coupling efficiency of cladding modes with higher-order core modes through bending and twisting, effectively suppressing higher-order modes. By designing a significant difference in confinement losses between the LP01 core mode and other higher-order modes, effective single-mode guidance can be achieved in NDSE-ANF. Therefore, we also investigated the characteristics of the LP11 mode in NDSE-ANF under a bending radius of $R_{b} = \,cm$.
Figure 9(a) shows a false colormap of the confinement loss of the LP11 mode in NDSE-ANF as $\beta_{1}$ and $\beta_{2}$ change. In the low-loss region of Fig. 9(a), the interaction between the LP11 mode and all cladding modes is weak. Conversely, the high-loss region, defined by the black dashed line where the loss exceeds 0.01 dB·m-1, indicates strong coupling between the LP11 mode and cladding modes. As the dimensions of the nested elements change, the cladding modes coupled to the LP11 core mode also change.
Fig. 9. (a) False colormap shows the variation of confinement loss of the LP11 mode in NDSE-ANF with changes in $\beta_{1}$ and $\beta_{2}$ at $\lambda = 532\,nm$ for $R_{b} = 3\,cm$. (b) False colormap shows the variation of the HOEMR in NDSE-ANF with changes in $\beta_{1}$ and $\beta_{2}$ at $\lambda = 532\,nm$ for $R_{b} = 3\,cm$.
Specifically, near point ① ($\beta_1 = 0.115,\textrm{ }\beta_2 = 0.9$) in Fig. 9(a), a high-loss band appears. At this point, the Region B (the area enclosed by the inner semi-ellipse and the nested innermost circle) has a large area, leading to interaction between the LP11 core mode and the cladding modes resembling LP01 in this area, resulting in high loss for the core LP11 mode. This coupling can be easily observed in the inset of Fig. 10. When $0.135 < \beta_1 < 0.145$, a high-loss band for the LP11 mode emerges. In this range, the area of the nested circle is comparable to that of the core, causing coupling between the cladding modes resembling LP01 in the nested circle and the LP11 mode, resulting in increased loss for the LP11 mode.
Fig. 10. Confinement loss and HOMER of the LP01 and LP11 modes in NDSE-ANF at wavelengths ranging from 450 nm to 800 nm when $\beta_1 = 0.115$ and $\beta_2 = 0.9$. The insets show the intensity distribution of the LP01 and LP11 mode at $\lambda = 532\,nm$ for $R_b = 3\,cm$.
The extinction ratio of higher-order modes (HOMER) can be used to evaluate the single-mode performance of NDSE-ANF. HOMER is defined as the ratio between the loss of HOM (higher-order-mode), which have the lowest loss, and the loss of fundamental core mode. Research indicates that, $HOMER > {10^3}$ is considered indicative of excellent single-mode guidance in fibers [13]. Figure 9(b) displays a false colormap of HOMER in NDSE-ANF corresponding to different $\beta_{1}$ and $\beta_{2}$ values. There is a wide region within the black dashed lines in the parameter space where $HOMER > {10^3}$. Its position overlaps with the high-loss band of the LP11 core mode shown in Fig. 9(a). NDSE-ANF exhibits the highest HOMER value at $\beta_{1} = 0.115$ and $\beta_{2} = 0.9$, corresponding to point ① in Fig. 9(a).
Additionally, we performed a parametric scan of wavelengths for the NDSE-ANF structure corresponding to this point ($\beta_{1} = 0.115$, $\beta_{2} = 0.9$). We obtained the confinement loss and HOMER plots for the LP01 and LP11 modes, as shown in Fig. 10. NDSE-ANF maintains a HOMER greater than 102 over a bandwidth of approximately 225 nm, and the parameter optimization for $\lambda = 532\,nm$ specifically causes strong coupling between the LP11 mode and the cladding modes, significantly increasing the loss of the LP11 mode. This results in the highest HOMER value of 3.19 × 104 at $\lambda = 532\,nm$. The extremely high extinction ratio of higher-order modes allows NDSE-ANF to maintain single-mode guidance even over short fiber lengths, with the loss remaining at a relatively low level of 1.60 × 10−5 dB·m-1. It should be noted that we have only shown the optimal parameter optimization for $\lambda = 532\,nm$. Similar methods can be applied to other wavelengths to achieve ultra-high HOMER at specific wavelengths.
4. Conclusions
In this paper, a nodeless anti-resonant hollow-core fiber with a nested double-semi-elliptical structure as the anti-resonant unit is proposed. Comprehensive simulations using finite element modeling were performed for the new fiber's confinement loss, bending loss, and single-mode performance. The results indicate that employing the semi-elliptical dielectric tube as the outer cladding layer introduces a flexible and adjustable negative curvature. By using inner semi-ellipse and nested innermost circle, the addition of anti-resonance layers is achieved while effectively avoiding the generation of cladding nodes. The proposed NDSE-ANF achieves ultra-low confinement loss below 10−5 dB·m-1 across almost the entire visible light spectrum, with a minimum confinement loss of 1.55 × 10−7 dB·m-1. Moreover, NDSE-ANF can maintain ultra-low bending loss on the order of 10−7 dB·m-1 even under extreme bending conditions with a radius of curvature $R_{b} = 3\,cm$, and its 3-dB bending loss radius can be as low as 3.5 cm. Furthermore, NDSE-ANF demonstrates excellent single-mode guidance under bending conditions. After parameter optimization for specific wavelengths, a very strong coupling between the LP11 core mode and cladding mode can be achieved, resulting in a high extinction ratio of up to 104 for higher-order modes. These characteristics endow the fiber with significant potential for miniaturized applications in visible spectral range.
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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