1. Introduction

Since the low-loss property of Kagome hollow-core fiber(HCF) was discovered, exploring its ultimate loss with simplified geometries has emerged as an important research field. Antiresonant hollow-core fiber(ARF), which is characterized by its significant and succinct negative curvature boundary, has become a research focus.Attenuation in ARF is often attributed to surface scattering loss(SSL), confinement loss (CL) and material absorption [1–3]. When the thin membranes adhere to the antiresonance condition, the electromagnetic field inside and on the surface of the membranes is minimized [3–6]. Therefore, SSL and material absorption hardly dominate loss in the C-band. Instead, CL, which is structurally dependent, is often the main limitation.

Some structural design principles of ARF, such as tubes arrangement [7], avoidance of Fano resonances [8,9], and predicting effective index curves [7,10], have roots in the design of PBGF. Analytical theories and adjustability studies have also been developed for single-wall ARF to achieve lower CL and improve design efficiency [11–14]. However, as the low-loss records were broken by the conjoined-tube fiber (CTF) and the Nested Antiresonant Nodeless Fiber (NANF) successively, the boundary conditions of emerging ARF structures have become challenging to analyze [15–17]. The finite element method(FEM) is capable of decomposing complex boundary conditions for model solving, but the enormous computational demand often necessitates empirical and parameter optimization approaches in research based on this method [18–21].

The development and application of artificial intelligence and neural networks provide a new way for the design of ARF. Studies in recent years have reported the use of neural networks, k-nearest neighbor (k-NN) and decision tree algorithms to predict CL in HCF [18,22,23]. Additionally, swarm intelligence algorithms and reinforcement learning have been proposed as efficient methods to optimize the structure of ARFs [19,24]. However, these approaches still exhibit limitations when dealing with newly emerging structures.

One limitation is that inverse design of ARFs using reinforcement learning faces challenges in constraining local curvature, making the resulting structure difficult to fabricate. Another limitation is that current swarm intelligence-based ARF design research is often based on a certain kind of structure, and lacks universality. Besides, simplifying the structure into parameters often makes it challenging to adjust local features, leading to a lack of specificity for additional objectives. As a result, significant trade-offs in multi-objective optimization problems may occur.

We propose that the fundamental issue lies in the underutilization of the similar topological structure of a large number of well-designed classic structures, specifically the double-ring topology. Stacking geometric components does not provide sufficient design freedom to express different basic structures, and some common features of classic structures have not been fully expressed. Consequently, it is a challenge to determine whether the topological potential of current sleeve components has been fully exploited. In other words, it remains unclear whether adding a topological loop to reduce losses is necessary even if it presents manufacturing difficulties.

In this work, we propose an adaptive particle swarm optimization (PSO) algorithm to design the structure of ARF by decomposing various classical double-ring topological structure parts into scatter points for optimization and reconstruction into new structures. This flexible adjustment not only achieves lower CL but also allows for targeted adjustments of additional optimization objectives, such as higher order mode extinction ratio(HOMER) at a lower cost for multi-objective optimization. Due to the substantial calculation power demands of the numerical computation-driven algorithms, it is impractical to execute these iterations on a single PC. Moreover, deploying the algorithms on commercial or public interest high-performance computation platforms also faces challenges due to the various software interaction process. Therefore, our algorithm is designed not limited to local computation but can adaptively allocate multiple computers through a broadcasting channel. Simultaneously, considering the random availability of these devices, dynamic allocation of computational tasks becomes necessary. Any computer or server with a configured computing environment can join the cluster via LAN, the internet, or wired connections to obtain computational tasks . This method provides a low-cost and reliable way to conduct large-scale numerical computations . As a result, we have obtained new structures with very low CL and achieved ultra-high HOMER at a lower cost of degradation when adding HOMER as an additional optimization objective. A comparison of the electric field distributions suggests that adjusting the HOMER can be achieved by adjusting the effective index of boundary mode, which represents a novel approach to addressing this issue.

2. Decomposition and reconstruction of ARF structures

Spacing between the sleeves in ARFs is typically designed to be close but not in contact. This is done to avoid extra losses due to Fano resonances generated by sharp edges at contacts, as well as to reduce light leakage through gaps. Besides, to minimize light leakage at the azimuthal position of the sleeves, their shapes need to be designed to reduce the coupling between internal air modes and core modes, especially the fundamental mode. There are currently many cleverly designed structures that can reduce such leakage, and many designs with ultra-low CL can be traced back to basic dual-ring structures. Examples of such basic fibers include NANFs, CTFs, and U-shaped fibers. Their sleeve structures share similar topologies and constraints, which lead us to believe that they can be decomposed in a similar manner and extract common features.

Figure 1 shows four kinds of classic structures, which are (a) CTF [15], (b) NANF [16], (d) Anti-resonant layer assisted ARF [20] and (e) U-shaped nested ARF [21]. Besides, the new structure obtained by our algorithm is also presented as (c) to illustrate the segment correspondence, and is called as the bulb-shaped ARF. To obtain some effective structures as the initial population, as is shown in (a) and (d), we limit the overall size of the fiber. The radius of the air core and the radius of the cladding boundary are $10\mu m\le r\le 20\mu m$ and $45\mu m\le R\le 55\mu m$ respectively. Benefiting from the rotational symmetry of ARF, it is only necessary to describe half of a sleeve, i.e. the brown sector in (a), to convey the entire structure.

 

Fig. 1. Classification of edges and points in different structures. (a) CTF. (b) NANF. (c) Bulb-shaped ARF. (d) Anti-resonant layer assisted ARF. (e) U-shaped nested ARF and (f) partial enlarged drawing containing free points.

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The premise of the universality is to use a universal method to describe the different infrastructures mentioned above. Therefore, we segment the glass wall according to the relative position and constraint conditions. These glass segments are respectively expressed by a series of scattered points, and will be reconstructed into ARF sections when numerical solutions are required. Since these scattered points are derived from the constrained segments, the constraints are inherited to these points. Based on these constraints, we classify the points into four categories. The first type is the fixed point, represented by the purple points in Fig. 1. These points serve as the connection between the sleeve and outer clad, and determine the position of the sleeve. The second type is the semi-fixed point, represented by the red points. Located on the sleeve’s axis of symmetry excluding the fixed points, these points can only move radially. The third type is the free point, represented by the orange dots in (f). These dots can be flexibly adjusted and each connects with only two adjacent points. The fourth type is the semi-free point, represented by the green points. These points are special free points since they connect to three adjacent points during structure reconstruction, but have no explicit constraints on their movement direction. Using the semi-free point as the endpoint, the segments are divided into two types: the first type is the fixed edge with the fixed point as the midpoint, i.e. the yellow edges; the other is the free edges with the semi-fixed point as the midpoint. To avoid structural confusion and redundant calculations, it is crucial to differentiate each free edges. We suggest classifying the free edges into paraxial and off-axis free edges based on their distance from the fiber center, as shown in Fig. 1 with cyan and green colors respectively. This auxiliary methods remains applicable even as the tubes have more topological loops along the axial direction.

Due to the fact that the computational overhead is associated with solving the model rather than updating parameters, we suggest using more free points to represent glass fragments to reduce interpolation errors caused by reconstruction. In this work, we used 61 points to represent half of a tube. These points are reconstructed into curves by quadratic Lagrange interpolation polynomial before numerical solution, and then the curves are converted into ARF skeleton by mirroring and rotation. We set the tube thickness to $0.372\mu m$, which is obtained by $t=(m-0.5) \lambda /\left [2\left (n_{1}^{2}-n_{0}^{2}\right )^{1 / 2}\right ]$ [2], where $t$ is the tube thickness corresponding to the antiresonant condition when the wavelength is $\lambda$ and the refractive index of the silica glass and air are $n_1$ and $n_0$ respectively. The refractive index of silica glass at different wavelengths is calculated by Sellmeier equation. We use the commercial software COMSOL to calculate the CL, and the loss is used as the fitness function for the PSO algorithm to optimize the locations of the points.

This method can also be applied to the design of the refractive index along the propagation direction, such as the index period of a grating. By discretizing different sinusoidal segments into a series of points and utilizing this method, the index distribution can theoretically converge to any desired shape of a periodic function through a finite term Fourier series. This enables various task requirements to be solved, including stress-strain sensitivity [25], output wavelength characteristics, and energy efficiency ratio of the inscribed grating [26]. Additionally, interpolation methods are employed to ensure accurate approximation and prevent artifacts like the Gibbs phenomenon. Ultimately, this approach aims to improve the targeted performance of grating sensors and enhance the optimization efficiency of corresponding intelligent algorithms.

3. Scalable and adaptive PSO for distributed ARF design

FEM is a feasible method to solve the complex boundary conditions of ARF by decomposing them into simpler fragments [18–21]. To ensure the effectiveness of algorithm iteration, it is necessary to guarantee computation accuracy. Therefore, the maximum element sizes for air and glass regions are set at $\lambda /4$ and $\lambda /6$ respectively [3]. As these simulations can be computationally intensive, we proposed a scalable and adaptable algorithm using a distributed architecture across multiple devices. The advantage of this approach is that the hardware performance of a single device is not a requirement to reduce computation time, as any device can be added or removed from the cluster and the workload can be distributed dynamically.

In the computational system shown in Fig. 2, we have designed a signal containing workload and population information and broadcast it through an encrypted channel. Each device can access the channel to join the computing cluster, download workloads, release signals and update the local population.

 

Fig. 2. Efficient distributed computing of PSO-based ARF Design across multi-devices.

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In Fig. 2, $i$ is the individual index, $N$ is the population size, $iter$ is the current iteration number, $ger$ is the maximum iteration number, $x$ is a set of scatter points coordinates for individuals, $xm$ is a set of coordinates representing each individual’s historical best values, and the values are represented by $fxm$, $ym$ is the set of coordinates representing the population’s best value, and the value is $fym$, and $S(i)$ is the signal, specifically referring to the local signal when in the flowchart, which can either be downloaded from the channel or an updated signal that has not been uploaded. Besides, to minimize communication data, we devised an encoding function $h_1$ and an array reshaping function $h_2$. As $h_1$ has a one-to-one mapping, any device added during the process can be informed of the current $iter$ and its assigned workload ID by computing the inverse of $h_1$. Using the inverse of $h_2$, a points set for the certain workload is constructed and then reconstructed into the profile of an ARF using the method mentioned in the previous section.

The left side of Fig. 2 illustrates how our algorithm uses $S(i)$ to perform multi-devices collaborative iteration. The algorithm initializes the parameters first and then confirms there is no broadcast signal in the channel before initializing the signal. If an identifier is found and not in the first frame of the signal, the algorithm extracts it to obtain workload information and sends the remaining frames as a new signal into the channel. After reconstructing the ARF profile, FEM is used to solve steady-state equations and obtain CL or other objectives as fitness values to update $fx_{local}$, which is the local fitness set. The algorithm repeatedly monitors the channel, downloads $S(i)$, and updates $fx_{local}$ until the signal no longer contains an identifier or the identifier is in the first frame. When the identifier is in the first frame, the algorithm uploads $fx_{local}$ as a new signal without the identifier and resumes monitoring the channel. If no identifier is found in the signal, each device repeats the process of downloading $S$, merging $S$ and $fx_{local}$, and uploading $fx_{local}$ until either a device’s collection time exceeds the deadline or all $fx$ are collected, in which case a new signal containing an identifier is broadcasted. In the former case, missing workloads with an identifier will be broadcasted, assuming that some devices may go offline during the current iteration; while in the latter case, the parameters will be updated and the new population is broadcasted as a signal. The new population is obtained by Eq. 1 and Eq. 2.

The aforementioned steps are repeated until the specified number of cycles reaches a certain number.

4. Results and disscusion

Figure 3 illustrates the Pareto sets obtained from optimization with equal and unequal weights assigned to HOMER and CL objectives. The former aims to explore the limits of each objective, and the latter seeks the lower limit of CL premising a large HOMER. We suggest setting $c_2>c_1$ to increase the diversity of Pareto front when solving the former problem, and setting $c_2<c_1$ while multiplying a fitness-related adaptive attraction coefficient, such as Sigmoid, when solving the latter. Although the solution set for the latter problem typically converges to a subset of the solution set for the former problem, this approach significantly accelerates convergence and reduces superfluous computations. We observe certain hotspots on the Pareto front, indicated by circles in (a), which implies the emergence of subpopulations. The structures within each subpopulation are similar, and we present their typical structures in (b)-(d).

 

Fig. 3. Iterative Pareto set and structural samples. (a) CL-HOMOR correspondence plot. (b) Ho-ARF. (c) Compromise structure. (d) Co-ARF.

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The three structures depicted in Fig. 3 differ from classic structures. In the three new structures, the topological ring connected with the outer cladding is small and not fully enclosed by another topological ring, while the far-axis free edge exhibits negative curvature characteristics. We postulate that the first two features primarily arise from CTF, whereas the latter feature mainly stems from NANF. Figure 4 illustrates the CL-wavelength characteristics for five samples generated during the iterative process. The five ARFs included in the figure are NANF and CTF from the initial population, CL-oriented ARF (Co-ARF) and HOMER-oriented ARF (Ho-ARF) located at the Pareto front, and the compromise structure of CL and HOMER (CH-ARF).

 

Fig. 4. Variation of CL with wavelength for five different ARF structures.

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The new structures exhibit significant improvements in reducing CL compared to the initial samples. Co-ARF effectively reduces CL to $2.21\times 10^{-5}dB/m$ at 1300nm, and this achievement enables CL values below $0.001dB/m$ within the wavelength range of 1 to 1.9 micrometers. However, the loss spectra of Co-ARF and CH-ARF exhibit significant loss peaks beyond the resonance wavelength. This phenomenon is because of the abrupt variation in the glass boundary, leading to a geometric overlap of a portion of the membrane. As a result, a curved-edge trapezoidal junction is formed, causing continuous thickness variation within this small section of the glass wall. The electric field within the junction extends to air modes beyond the resonance wavelength, resulting in lowered effective indices for a series of membrane modes, even approaching that of the fundamental mode at some wavelengths. This can lead to substantial leakage and contribute to the observed loss peaks.

When it comes to fabrication, the sharp turning points of Co-ARF are likely not the desired features and may experience expansion due to the surface tension of softened glass, resulting in broadening of loss peaks. Therefore, in practice, we highly recommend adopting Ho-ARF, which exhibits a structurally fuller geometry with minimal loss disadvantages. This not only implies reduced loss fluctuations but also indicates manufacturing feasibility comparable to well-established structures like CTF in the field of optics. However, the algorithm-generated fiber still exhibits lower CL compared to the original individuals To investigate the origin of the reduction in CL, we analyze the normalized electric field distributions of the fundamental modes in the five different optical fibers and plot them in Fig. 5. Subfigure (a) illustrates the field distributions along the sleeve’s symmetry axis (y-axis) and the gap’s symmetry axis (x-axis). Subfigure (b) demonstrates the distributions at the connection between the sleeve and the clad. Subfigure (c) displays contour plots of the field distributions. By studying these distributions, we aim to gain insights into the mechanisms underlying the observed CL reduction.

 

Fig. 5. Electric field distributions of the fundamental modes in five distinct ARFs.

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Subfigure (a) reveals an interesting observation for NANF and CTF: the electric field at the cladding boundary is lower along the x-axis than along the y-axis. This implies that in these two initial fibers, light predominantly escapes from the core through the sleeves rather than directly leak through the gaps. However, in the case of three new fibers, the opposite behaviors are observed.

In subfigure (b), the boundary electric field of NANF exhibits a distinct shape compared to other fibers. A notable characteristic is the splitting of the electric field inside the sleeves into three peaks. This phenomenon arises from the positioning of the connection at the cladding boundary, which fails to effectively prevent the leakage from the corners. To address this issue, inspiration can be drawn from the structural features of the CTF by moving the connection point of the two topological rings away from the cladding and eliminating the containment relationship between them.

However, the direct connection of the rings in CTF may not an ideal feature as it may lead to a high electric intensity inside the glass membranes. Besides, an oversize cladding connection may result in wide-ranging electric field leakage. The algorithmic design results demonstrate the potential for trade-offs and optimization of these structural features.

By canceling the containment relationship between the topological rings and moving the nodes beyond the cladding boundary, the algorithm-generated bulb-shaped sleeves ensure that the electric field inside the topological ring formed by the two free edges needs to pass through at least one membrane to leak into the cladding. As a result, leakage of the core modes is effectively suppressed, as indicated by the smaller central peak in subfigure (b). Moreover, compared to those in CTF, the electric field variations on the surface of the membranes are smooth, indicating a successful optimization of the shape and therefore reduced the leakage along it. As a result, the bulb-shaped sleeves effectively mitigate the leakage through the sleeves, while maintaining electric field intensity in the sleeve gap comparable to that of NANF. We believe that the slender and succinct sleeve-cladding junctions found in bulb-shaped sleeves are instrumental in minimizing leakage. Interestingly, this feature does not emerge in the initial population, leading us to speculate that it originated from CTF and underwent refinements inspired by NANF and U-shaped nested ARF.

Ho-ARF is distinguished by its exceptional HOMER over 14,000 at 1550nm. However, the HOMER of the Co-ARF, which has a similar structure and approximate fundamental mode shape to the Ho-ARF, falls short of 100. This implies a unique regulation mechanism for higher-order modes in Ho-ARF although its CL performance may not be optimal. Given that the electromagnetic field mainly leaks through the gaps, substantial opportunities likely exist for continued CL reduction. In Fig. 6, the comparative analysis of mode loss origin differences includes the electric field distributions of the fundamental mode and the lowest loss higher order mode (LP11), their power leakage density at the outer geometric boundary, and the streamlines of the LP11 mode’s Poynting vector as a function of angular position.

 

Fig. 6. Loss visualization of the fundamental mode and LP11 mode in Ho-ARF through the density of power leakage at the outer boundary of the geometry and streamlines following the transverse Poynting vector.

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By comparing the fundamental mode and LP11 mode, it is evident that power leakage through the gap cannot be neglected as a significant source of loss in Ho-ARF, especially for the LP11 mode, which exhibits significantly high leakage compared to the leakage at the contact area. This results in substantial loss of higher-order modes and an increase in HOMER.

In other low-HOMER ARFs, we have not observed such pronounced leakage of LP11 mode, nor such pronounced intensity at the boundary region. This suggests the coexistence of the LP11 mode and another air mode that is concentrated near the boundary in Ho-ARF. We calculated the normalized effective index of the fundamental mode, LP11 mode, and modes with effective indices similar to LP11 in the range of $0.5\mu m$ to $2\mu m$. The three modes with the closest effective indices to LP11 are shown in Fig. 7, where $f=2 t \sqrt {n^{2}-1} / \lambda$.

 

Fig. 7. (a) Normalized effective index curves of (b) Fundamental mode; (c) LP11 mode; (d) Boundary mode; (e) Airy mode; and partial membrane modes at (f) $0.7\mu m$ and (g) $0.78\mu m$.

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The normalized effective index curves in subfigure(a) exhibit a typical anticrossing around $0.8\mu m$ corresponding to the loss peak in Fig. 4. In subfigures (f) and (g), we illustrate the membrane modes corresponding to wavelengths of $0.7\mu m$ and $0.78\mu m$ respectively. In comparison to the vicinity of $0.78\mu m$, the membrane modes exhibit a slow change in phase, indicating an increased overlap integral value with the core modes. Consequently, the core modes readily resonate with the membrane modes, which is manifested by the significant deformation of the normalized effective index curves at the resonance wavelength. Subfigures (g) visually presents this phenomenon through the significant expansion of the membrane modes. Due to the concentrated occurrence of resonance different core modes and these membrane modes, a peak in loss is observed near this wavelength. Additionally, the presence of airy mode concentrated within the sleeves is observed, which is also a common feature among most ARFs.

However, the distinctive feature of Ho-ARF lies in the LP11 mode, which is a higher order mode with the lowest-loss and shares a close effective refractive index with the boundary modes that readily couple to the outer cladding. This results in significant coupling of the LP11 mode with the high-loss mode. While high-loss boundary modes can be readily found in some ARFs, we believe that it is challenging to adjust the effective refractive index of these boundary modes to be close to that of the LP11 mode through conventional design approaches, as it requires a trade-off with the CL. By employing a scatter-based reconstruction approach, we shaped the boundary modes, thereby adjusting their effective refractive index while balancing the considerations of the fundamental mode’s CL. We believe that such a novel method of adjusting the HOMER in ARFs provides a solution to the challenge of promising a significant loss of higher-order modes while achieving a larger mode field diameter of the fundamental mode. It offers promising prospects for the development of ARFs with enhanced HOMER, representing a new way to regulate the HOMER.

In addition to HOMER and CL, the bending loss(BL) is also an important feature for ARF. The index profile under the influence of bending is modified by Eq. 3 [27].

This is a well-known technique called “conformal mapping”, and widely used to calculate the modes for the modified index profile of the fiber under bending conditions. The parameter $\chi$ compensates for the elastic coefficient of the material, with a value of -0.22 for silica and zero for air. The bending radius and the index distribution after bending are denoted by $R_{b}$ and $n_{b}$, respectively. We depict the fiber bending in Fig. 8. Considering that the ARF is not a uniform optical waveguide, it is essential to examine the loss resulting from its bending along various directions. We designate the bending direction as the x-axis and define the acute angle formed by the midline of the gap and the x-axis as $\theta$.

 

Fig. 8. Schematic illustration of ARF fiber bending.

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Fig. 9 demonstrates the bending loss variation of the fundamental mode in Ho-ARF with respect to $R_{b}$ and $\theta$. A extra loss of 3 dB occurs at approximately $R_{b} = 9cm$, indicating the onset of significant bending-induced losses. As the bending radius decreases, BL increases sharply, accompanied by the coupling of the fundamental mode to the boundary mode. It is noteworthy that although the bending threshold is similar in different bending directions, the specific bending direction also influences the loss. The electric field distribution around the boundary of the fundamental mode in Ho-ARF, as depicted in Fig. 4 and Fig. 6, reveals that bending along the midline of the gaps has a more pronounced impact on leakage. This phenomenon is supported by the contour lines marked for three bending radii near the bending threshold. When bending occurs along the midline of gaps, the coupling from the fundamental mode to the boundary mode is more significant than that along the midline of the sleeve, resulting in a faster increase in loss. However, the situation becomes difficult to analyze when $R_{b}$ is further reduced and leads to significant coupling of the fundamental mode to the membrane modes.

 

Fig. 9. Variation of bending loss for the fundamental mode in Ho-ARF with respect to bending radius and bending direction.

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The preservation of the exceptional HOMER performance in Ho-ARF under bending conditions is also pivotal to its practical viability in optical applications. In Fig. 9, we illustrate the variation of bending loss for the mode with the second lowest loss with respect to the bending radius and bending direction.

In contrast to the behavior of the fundamental mode, the loss in Fig. 10 exhibits a distinct step-like pattern. The first step signifies the coupling of the LP11 mode to the boundary modes. Because of their similar effective indices, when the bending leads to a partially spatial overlap between them, the coupling occurs rapidly, resulting in the cutoff of the LP11 mode. Consequently, the boundary mode becomes a new second lowest loss mode.

 

Fig. 10. Variation of bending loss for the mode with the second lowest loss in Ho-ARF with respect to bending radius and bending direction.

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The leakage path of the LP11 mode, as revealed by the Poynting vector in Fig. 6, differs from that of the fundamental mode. For the LP11 mode, power initially leaks into sleeves and subsequently escapes from the side of sleeves towards the cladding boundary. The leakage direction of the LP11 mode differs from that of the fundamental mode, resulting in a completely opposite response to the bending direction. The LP11 mode exhibits sensitivity to bending along the midline of a sleeve, with a bending threshold of approximately $81 cm$, while it is insensitive to bending along the midline of the gap, with a bending threshold of about $50 cm$. As $R_{b}$ decreases further, the LP11 mode is cutoff, and the mode with the second lowest loss becomes the boundary mode, resulting in a loss exceeding $1000 dB/m$. This implies that even under the most unfavorable condition of bending along the midline of the gap, all core modes other than the fundamental mode can be completely suppressed within the bending radius range of $10 cm$ to $50 cm$. Consequently, high-quality and low-loss single-mode transmission can be achieved. The second bending threshold occurs at $R_{b}=5cm$, where the boundary mode is coupled with the membrane mode, which is similar to that of the fundamental mode.

5. Conclusions

In this work, we employed a decomposition, optimization, and reconstruction approach to flexibly adjust the shape of the ARFs for specific optimization objectives. To address the computational demands arising from multi-objective optimization and the fusion of various classic structure features, we have developed a scalable, adaptive, and cross-device multi-objective PSO framework. As a result, we achieved a Co-ARF with a CL of $2.21\times 10^{-5}dB/m$ at 1300nm and a Ho-ARF with a CL of $6.83\times 10^{-5}dB/m$ at 1350nm, while maintaining a HOMER exceeding 14,000. Both sleeve structures are based on a simple dual-ring topology, which makes them easily manufacturable and of practical significance.

Furthermore, through the analysis of other modes in the Ho-ARF, we have discovered the principle behind its exceptionally high HOMER, which involves introducing high-loss boundary modes with an effective index close to that of the LP11 mode to leak the LP11 mode. This high degree of freedom in design, enabled by the proposed approach, allows for the realization of such a HOMER regulation mechanism. According to the calculations of the bending loss, Ho-ARF demonstrates the ability to effectively suppress core modes except the fundamental mode, ensuring high-quality and low-loss transmission transmission when the bending radius exceeds $9cm$. This method resolves the challenge of simultaneously achieving large mode field diameters and high HOMER, holding significant value in applications such as high-power laser transmission, spatial light coupling, and gas lasers utilizing hollow-core fibers.