1. Introduction

Due to their unique light guidance, hollow-core anti-resonant fibers (HC-ARFs) have emerged as a key enabler for ultra-low loss transmission across the telecommunications bands [1–3], with transmission losses approaching or even surpassing those of standard single-mode fibers (SSMFs) [4,5]. Unlike conventional optical fibers, HC-ARFs offer unprecedented optical properties such as low latency [1], low non-linearity [2], negligible power overlap with silica glass [3,6], exceptional polarization purity [7], and ultra-low backscattering [8]. These fibers find numerous applications in the broad spectrum, including high-power delivery [9,10], gas-based non-linear optics [11,12], extreme UV light generation [13], and short-reach data transmission [2], to mention a few. In the past few years, numerous HC-ARF architectures were studied [14–24] that exploit the guiding principle of inhibited coupling to achieve low loss. Among the various designs, the tubular nested HC-ARF architecture is probably the most promising since adding nested tubes significantly enhances inhibited coupling between the core and cladding modes and hence reduces the propagation loss and bend loss [16]. Recently, tremendous progress was reported in the development of nested HC-ARFs with the lowest reported loss to date of 0.174 dB/km in the C-band [25]. The outstanding optical properties of nested HC-ARFs open up opportunities for short-reach optical communication and also possibly for long-haul communication if the loss is further reduced. The key to this advancement is a geometry that is tuned down to the smallest detail, which inherently leaves the optical properties highly susceptible to structural perturbations introduced in the fabrication process. The effects of fiber parameters such as gap separation and normalized tube ratio on the propagation loss are investigated in [20]; however, they are considered as a single, process-invariant variable. In reality, structural perturbations in fibers occur with random variations, and these effects on the propagation loss have not been investigated before to the best of our knowledge.

Imperfections can be observed per lot, meaning that separately drawn fibers exhibit different imperfection realizations. However, the geometry can also vary along the same fiber as a function of its length, with changes occurring at approximately 100’s of meters. This work examines case 1 and therefore assumes a perturbed but static fiber geometry along its length.

The goal of this work is to investigate the impact of several random geometrical imperfections in both the outer and nested cladding elements on the overall loss and guiding performance of nested HC-ARF structures. This study aims to assist fiber fabrication manufacturers in identifying the geometric imperfections with the highest loss contribution for realistic fibers and improve specific structural properties that promise the highest increase in transmission quality. Based on practical observations, four different fiber imperfections, which are clearly observable in the microscopic image, namely random angle offsets of the outer and nested cladding elements as well as random anisotropic shape deformations in radial and axial directions of the nested tubes, are selected and analysed. Both effects are studied in an isolated and joint fashion in a Monte-Carlo manner. The article is organized as follows: Section 2. recalls the principle nested HC-ARF structure and introduces the mentioned imperfections. Moreover, realistic statistical parameters based on practical observations are given. Section 3. studies the impact of those effects on the transmission loss for FM and HOM propagation particularly at $\lambda = {1.55}\;\mathrm{\mu}\textrm{m}$ and provides an overview in the broad-band domain. The article is concluded in Section 4.

2. Fiber geometry and structural perturbations

The nested HC-ARF geometry as well as the structural perturbations considered in this article are displayed in Fig. 1. The five-tube structure is chosen following a preceding 2D parameter optimization considering a constant tube thickness and tube gap separation. In comparison to similar six-, seven-, and eight-tube nested HC-ARF designs, the five-tube structure exhibits the widest low-loss transmission window as a function of both parameters. For this reason, its expected susceptibility to random parameter variations in the cladding elements is low, promising good realistic performance. The ideal geometry, characterized by its symmetry and constant geometric parameters, is displayed in Fig. 1(a). For this study the core diameter $D_\text {c}= {35}\;\mathrm{\mu}\textrm{m}$, gap separation $g= {5.25}\;\mathrm{\mu}\textrm{m}$, and outer and nested tube wall thicknesses $t_\text {o}= {393}\;\textrm{nm}$ and $t_\text {n}= {432}\;\textrm{nm}$, respectively, are chosen. These parameters are the result of a numerical transmission loss optimization for a nested tube diameter ratio $d/D = 0.5$, providing a low loss of ${0.17}\;\textrm{dB/km}$ while ensuring effectively single-mode operation.

 

Fig. 1. 5-tube nested HC-ARF geometries with random perturbations: (a) ideal geometry with circular cladding elements, (b) misalignment of outer tubes, (c) misalisgnment of nested tubes, (d) combination of (b,c), (e) ideal geometry with anisotropic nested cladding elements, defined by different radial and axial nested diameters $d_\text {r}$ and $d_\text {a}$, (f) elliptically deformed nested cladding elements. Illustrations are to scale with design parameters, except outer / nested tube wall thicknesses for enhanced visibility.

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However, imperfections to this structure are frequently observed for fabricated fibers. Particularly the cladding elements, which are made out of sub-micrometer geometry features, show pronounced deviations to the ideal geometry, easily identifiable with the human eye under the microscope. The two visually most pronounced perturbations are misalignments of the outer and the nested tubes. Both effects are illustrated in Fig. 1(b) and 1(c). Although relatively minor in practice, positional inaccuracy of the outer tubes leads to different gap separations, which are well-known to impact the transmission loss and the ’single-modeness’ of HC-ARFs significantly. Much more pronounced, however, are the angular misalignments of the nested tubes. This creates an effect of a rolling nested tube inside the outer capillary shell. Figure 1(d) shows both effects applied concurrently.

Furthermore, another imperfection that can be widely observed for realistic fibers is a deformation of the circular shape of the nested tubes to an elliptical shape. Interestingly, radially elongated nested tubes can lead to lower losses compared to perfectly circular shapes, as shown in [26]. Incorporating those degrees of freedom into the geometry leads to a modified lowest-loss geometry exhibiting 0.08 dB/km with an outer tube by radial nested tube diameter ratio $d_\text {r}/D=0.65$ and a so-called ellipticity $l=d_\text {p}/d_\text {r}=0.7$. Note that it is preferred to use the ellipticity parameter over the eccentricity because of its simple relationship between both radii. Nevertheless, they are, of course, interconnected with the eccentricity $e=\sqrt {1-l^2}$. The ideal geometry, considering anisotropic nested tubes, is illustrated in Fig. 1(e). A sample geometry with each radii $d_\text {a,i}$ and $d_\text {r,i}$ interpreted as an independent random variable is shown in Fig. 1(f). Both effects, namely random outer/nested tube angle offsets and random anisotropically shaped nested tubes, are investigated separately in the course of this article.

Other imperfections, particularly related to the silica wall thicknesses of the nested and outer cladding elements, are very common and difficult to control in the fiber fabrication process. However, preliminary analysis has shown a very low susceptibility of the hollow-core fiber guidance performance to individual tube thickness variations. Therefore, these effects are not examined in this article but are expected to be evaluated in further research.

2.1 Experimental fabrication characteristics

To study the impact of the previously introduced imperfections with practically feasible intensities, multiple cross-sections of fabricated nested HC-ARFs are analyzed via scanning electron microscopy (SEM) with the aim of deriving key geometrical characteristics and their corresponding statistical properties. The fibers were fabricated in CREOL at the University of Central Florida. The SEM fiber analyses are restricted to six-tube nested HC-ARFs since at the time of this writing only those were available. However, the characteristics of the imperfections are not expected to vary significantly for five-tube fibers since the fabrication process is the same. The collected data is summarized in Table 1, and the corresponding SEM images are shown in Fig. 2. The measurements show that the average standard deviation of the nested angle offsets with 13.4° is approximately one magnitude greater than for the outer tubes with 1.08°. Therefore, ranges of 0° to 10° and 0° to 2° for nested and outer tube angle offset standard deviations, respectively, are chosen for the numerical simulations.

 

Fig. 2. Scanning electron microscrope (SEM) photographs of fabricated 6-tube nested HC-ARFs. Measured averaged fiber parameters: Fiber diameter $D= {79.1}\;\mathrm{\mu}\textrm{m}$, core diameter $D_\text {c}= {30.3}\;\mathrm{\mu}\textrm{m}$, gap separation $\overline {g}= {3.4}\;\mathrm{\mu}\textrm{m}$, outer [nested] tube diameter $\overline {d_\text {o}}= {24.1}\;\mathrm{\mu}\textrm{m}$ [$\overline {d_\text {n}}= {10.4}\;\mathrm{\mu}\textrm{m}$], outer [nested] tube thickness $\overline {t_\text {o}}= {714}\;\textrm{nm}$ [$\overline {t_\text {n}}= {664}\;\textrm{nm}$]. The fiber exhibits a minimum transmission loss of $ {36}{dB/km}$ at $\lambda = {576}\;\textrm{nm}$.

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Table 1. Structural imperfections of the fabricated imperfect 6-tube HC-ARFs. Data corresponds to fiber crosssections displayed in Fig. 2.

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On the other side, the standard deviations of the ellipticity for both outer and nested tubes, which are given in $\%$ relative to the respective mean diameters, only differ by approximately a factor of two, with the nested tubes being more affected. Furthermore, radial and axial directions seem to be equally perturbed by about 1.8 % and 3.5 % for outer and nested tubes, respectively. However, those values are observed for fibers with a circular nested target shape. Anisotropic nested tubes, which are necessary to achieve the lowest losses, have not yet been implemented because of their great difficulty to fabricate. Because of this, the analyzed range of the ellipticity standard deviations is extended by approximately a factor of five, resulting in 0 % to 20 %.

3. Numerical simulations and discussion

All simulations are performed using finite-element-based COMSOL software in combination with MATLAB-Livelink. The simulation environment (8-layer-deep perfectly matched layer, mesh-element sizes of $\lambda /6$ and $\lambda /4$ in the silica and air regions, respectively, etc.) is configured similarly to [16,20,27]. To reduce the mode-searching time, the solver is supplied with a refractive mode index approximation using an analytical capillary model [28]: $n_\text {guess}=\sqrt {1-\left (\frac {U_{mn}\lambda }{2\pi R_c}\right )^2}$, where, $R_c$ is the core radius, $\lambda$ is the wavelength, and $U_{mn}$ is the nth zero of the mth-order Bessel function of the first kind. However, due to individual random structural perturbations introduced to each cladding element, symmetry can no longer be exploited. Therefore the full fiber structure has to be solved, which leads to a greatly increased memory consumption and simulation time. Since propagation loss is the main fiber parameter analyzed in this article, confinement/leakage loss (CL) and surface scattering loss (SSL) are included in the calculations. SSL calculation is based on the $\lambda ^{-1.24}$ dependency [29] by multiplying the $F$-factor [30], which represents the normalized electric field intensity at the silica-air boundaries [31] with a calibration factor $\eta$. Wavelength-dependent estimations are implemented by [16]: $\alpha _\text {SSL} [\textrm{dB/km}]=\eta {}F\left (\frac {\lambda }{\lambda _0}\right )^{-3}$, with calibration factor $\eta =300$ at $\lambda _0= {1.55}\;\mathrm{\mu}\textrm{m}$. Effective material loss (EML) is neglected because of the insignificant power overlap with silica glass $<10^{-4}$. The operating wavelength is chosen to 1.55 µm unless otherwise noted.

Both fundamental mode and higher-order modes are considered, whereas for higher-order modes the mode with minimum loss (limited to $\text {LP}_{11}$ and $\text {LP}_{21}$) is used for data analysis.

The structural imperfections studied in this article occur at a random nature. Observations show that small perturbations occur more frequently than strong perturbations. This behavior is approximated by modeling and sampling the random variables from Gaussian distributions. Their standard deviations represent the intensities of the effects, with higher values corresponding to a worse fabrication quality.

Available analytic fiber models are not suitable for an imperfection analysis, since they on the one hand lack a detailed hollow-core geometry description with parameters used in this work, i.e., anisotropic nested tube radii ($d_\text {a}$, $d_\text {r}$). On the other hand, individual tube-dependent parameters are not supported, e.g., individual tube angle offsets $\alpha _1 \cdots \alpha _N$. For those reasons, inference on the realistic fiber characteristic is gained by generating and evaluating multiple perturbed HC-ARF geometries using a Monte-Carlo technique by sampling from the infinite continuous sample space. Decoupling the statistical analysis from the fiber model allows for the highest individualization in the fiber’s geometry while enabling to imprint arbitrary statistical profiles on the fiber. Sufficient sample sizes are chosen based on result convergence to practically negligible values. The utilized sample sizes are described in each subsection separately.

In the following simulations, median loss increase values are calculated for each simulated standard deviation (or combination of two) based on 20 samples. The edge values, which are emphasized by box plots, represent the impact on the transmission loss of each imperfection applied isolated to the fiber geometry. Those are calculated using 100 samples. The median value is chosen over the mean value because of its resilience to one-sided outliers, which results in a skewed mean value. This effect depends on the imperfection and can be clearly observed in the edge-located box plots. Moreover, the median denotes, per definition, the center of symmetric confidence intervals, marking the expected "average" performance highly relevant for fiber manufacturers.

However, it can be noticed that for the described sample sizes, the medians can still vary in an erratic manner, under- and overshooting their neighbored values even several quantization units apart. Further increasing the sample size limits this effect only partially at the expense of total computation time. To combat this problem, the data is post-processed using a custom dual-axis regression-based smoothing algorithm. In short words, this custom algorithm interprets the data as a spatially ordered 2D matrix with respect to both axes and then carries out a row- and column-wise polynomial regression of second degree. The predicted values of the dual-axis regression each produce another matrix with an identical structure to the original, which are then averaged, producing a single intermediate matrix with smoothed up data. This process can be applied repetitively until a certain degree of smoothing is reached. The resulting data converges with a rising iteration count. Data interpolation is implemented by evaluating the last regression pair on arbitrary points and averaging them. The graphs shown in subsections 3.1 and 3.2 are post-processed using three iterations and interpolated by a factor of 32.

3.1 Impact of random misalignments of circular nested and outer cladding elements

In this subsection the susceptibility of the transmission loss to random angular misalignments of the outer and nested circular cladding elements for FM and HOM propagation is analyzed. Both effects are applied to the five-tube nested HC-ARF geometry shown in section 2. as a function of their respective standard deviations. A 2D sweep for standard deviations in the ranges of 0° to 10° and 0° to 2° is performed for the nested and outer tubes, respectively. This results in a sample size of $20\times {}10\times {}10=2000$ (${\#}_\text {samples}\times {\#}_\text {x}\times {\#}_\text {y}$). The calculated median increase in propagation loss is given in $\%$ with respect to the ideal geometry loss of 0.17 dB/km. The result is displayed in Fig. 3. For fundamental mode propagation (Fig. 3(a)) the loss increases with either standard deviation rising by a maximum of $5\%$ at the end of both ranges. However, random outer tube angle offsets show a more pronounced impact by a factor of $\approx 10$ than random nested tube angle offsets. This difference can also be observed when applying both effects individually to the structure, shown in both box plots, located on the lefthand-side and below the graph. The white and green markers hereby indicate the median and mean value of each sample group, respectively. Not only do the scales of the box plots vary greatly, but especially the possible worst-case loss values caused by random outer tube angle offsets, marked by the 75$\%$-quartile, extend much further away from the group median, especially for high standard deviations. In other words, significant perturbations in the outer tube angles can lead to massively spread loss increase values of a multiple of the median.

 

Fig. 3. Calculated median (a) FM and (b) HOM propagation loss increase in $\%$ and (c) resulting HOMER for perturbed 5-tube nested HC-ARF geometries as a function of imperfection standard deviations. Imperfections: Random angular misalignment of outer and nested cladding elements. Sample size: $20\times {}10\times {}10=2000$ (${\#}_\text {samples}\times {\#}_\text {x}\times {\#}_\text {y}$). Box plots detail the impact on the loss of isolatedly applied imperfections, located at the left and bottom edge. White and green markers denote the median and mean values, respectively. Additional design parameters: Core diameter $D_\text {c}=$35 µm; Outer [nested] tube thickness $t_o= {393}\;\textrm{nm}$ [$t_n= {432}\;\textrm{nm}$]; mean outer tube gap separation $\overline {g}= {5.25}\;\mathrm{\mu}\textrm{m}$; outer by nested tube diameter ratio $d/D=0.5$. All simulations are performed at wavelength $\lambda = {1.55}\;\mathrm{\mu}\textrm{m}$.

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Similar behavior can be observed for HOM propagation, shown in Fig. 3(b). Perturbations of the outer tube angles dominate the impact on the loss by approximately an order of magnitude compared to perturbed nested tube angles. In addition, the overall guiding performance decreases much more rapidly for HOM propagation, with a maximum joint loss increase of 82$\%$ at the end of the analyzed range at $\sigma _{\alpha,n}= {10}^{\circ}$ and $\sigma _{\alpha,o}= {2}^{\circ}$. This strong susceptibility of HOM propagation to random perturbations is, of course, reflected in the higher-order-mode extinction ratio (HOMER), which is defined as the ratio of minimum HOM propagation loss by FM propagation loss, displayed in Fig. 3(c). The HOMER increases from $\approx 1400$ for the ideal structure to $\approx 2300$ for strongly perturbed structures. In other words, the ’single-modeness’ of five-tube nested HC-ARFs increases for realistic fibers.

In summary, FM propagation for a five-tube nested HC-ARF shows a rather small susceptibility to imperfect angle offsets. In contrast, HOM propagation can be highly affected by random angle offsets in the outer tubes, resulting in nearly doubled propagation losses for realistic fiber geometries. A possible explanation for the high HOM susceptibility lies in the net phase matching condition to couple cladding modes with core modes, which is higher for HOMs compared to the FM.

3.2 Impact of random anisotropic deformations of nested cladding elements

This subsection studies the impact of randomly perturbed nested cladding element shapes. As described in subsection 2.1, the shapes of realistic fibers’ outer and inner tubes tend to get deformed to an ellipse in the fabrication process. The nested tubes are observed to be stronger affected than the outer tubes, which is why the following analysis specifically focuses on those. Since now the nested tube ellipticity $d_\text {a}/d_\text {r}$ is incorporated as an additional degree of freedom into the fiber model, the propagation characteristics are not optimized anymore. Therefore, optimization results are adapted from recent publications by Habib and Petry [20,26], leading to an optimized ellipticity of 0.7 and a modified nested tube by outer tube diameter ratio $d_\text {r}/D=0.65$, providing a low loss of 0.08 dB/km. The drawback of nearly halfing the transmission loss is a much-decreased HOMER from 1350 to only 15, greatly decreasing the ’single-modeness’ of the fiber.

In this study, the perturbations are implemented as individual offsets to the radial and axial nested tube diameters, $d_\text {r,1} {\cdots }d_\text {r,5}$ and $d_\text {a,1} {\cdots }d_\text {a,5}$, respectively. Those offsets are sampled from zero-mean Gaussian distributions, whose standard deviations are in the range of 0 % to 20 % relative to both respective ideal diameters. A 2D scan of both standard deviations is performed similarly as in subsection 3.1. The same sample size of $20\times {}10\times {}10=2000$ (${\#}_\text {samples}\times {\#}_\text {x}\times {\#}_\text {y}$) is utilized. The loss increase is given in percent with respect to the prior determined loss of the ideal anisotropic structure. The simulation result is displayed in Fig. 4. FM propagation loss (Fig. 4(a)) shows to be equally impacted by either standard deviation, reaching a maximum loss increase of $62\%$ for both radial and axial diameter standard deviations of $20\%$. The box plots, located at the bottom and left edges of the graphs, confirm similar statistical properties, i.e., the mean (denoted in green) exceeds the median (denoted in white) by multiple factors, which indicates the existence of high-loss outliers for both effects. HOM propagation loss (Fig. 4(b)) shows a similar equal susceptibility on both radial and axial nested diameter perturbations. However, the transmission loss increases with much higher rates compared to the previous perturbations, reaching extreme levels of $>800\%$ for joint maximum standard deviations. Even higher HOM losses are possible, as indicated by the box plots 75$\%$-quartiles.

 

Fig. 4. Calculated median (a) FM and (b) HOM propagation loss increase in $\%$ and (c) resulting HOMER for perturbed 5-tube nested HC-ARF geometries as a function of imperfection standard deviations. Imperfections: Random elliptical nested tube shape deformation in radial and axial direction. Sample size: $20\times {}10\times {}10=2000$ (${\#}_\text {samples}\times {\#}_\text {x}\times {\#}_\text {y}$). Box plots detail the impact on loss of isolatedly applied imperfections, located at the left and bottom edge. White and green markers denote the median and mean values, respectively. Design parameters: Mean outer by nested tube diameter ratio $\overline {d_\text {r}/D}=0.65$; mean axial by radial nested tube diameter ratio $\overline {d_\text {a}/d_\text {r}}=0.7$. Rest: Refer to Fig. 3.

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As a result of the massively increased HOM losses caused by nested tube shape deformations, HOMER reaches its six-fold value of $108$ for the highest perturbation intensities, compensating for the small value of 15 for the ideal structure caused by allowing anisotropic nested shapes. However, HOMER still ranges an order of magnitude smaller compared to the non-elliptic nested tube geometry.

In conclusion, the loss increase caused by elliptically perturbed nested shapes is about a factor of $10$ more pronounced for both FM and HOM propagation in the studied ranges compared to the impact on the transmission loss of random outer and nested tube angle offsets studied in subsection 3.1, making fabrication optimization very important for the considered geometry features.

3.3 Wavelength analysis

In this final subsection, the impact of both previously introduced random structural perturbations on the transmission loss is analyzed over a broader wavelength spectrum of 1 to 2 µm. Therefore, two groups of 20 randomly perturbed geometries are sampled. The first group exhibits random angular outer and nested tube misalignments with standard deviations $\sigma _{\alpha {}\text {,outer}}=$1° and $\sigma _{\alpha {}\text {,nested}}=$10°, respectively. The second group, whose ideal geometry features anisotropic nested tubes as described in subsection 3.1, exhibits elliptically deformed nested cladding elements with an equal radial and axial tube diameter standard deviation of 10 % with respect to their respective mean diameters. A sample size of $2\times {}41\times {}20=1640$ (${\#}_\text {plots}\times {\#}_\text {samples}\times {\#}_\text {x}$) is utilized. All geometries are evaluated for FM and HOM propagation loss over the specified wavelength spectrum, and the median transmission characteristic is displayed in Fig. 5. FM and HOM transmission loss are denoted by blue- and red-colored lines, respectively, whereas the ideal and perturbed structure loss is displayed by solid and solid-dotted lines, respectively. For enhanced visibility, the area between respective ideal and perturbed structure losses is shaded by a color gradient, with high loss increase regions denoted by red colors and low loss increase regions by blue colors. Moreover, the wavelength-dependent HOMER, related to the respective y-axes on the right-hand side, is denoted by a dashed black line.

 

Fig. 5. Calculated median FM (blue) and HOM (red) propagation loss for ideal (solid) and perturbed (solid+dotted) 5-tube nested HC-ARF geometries and HOMER (black dashed) for wavelengths from 1 to 2 µm: (a) Random angular misalignment of outer and nested tubes with standard deviations ${\sigma }_{\alpha {}\text {,outer}}= {1}^{\circ}$ and ${\sigma }_{\alpha {}\text {,nested}}= {10}^{\circ}$, (b) random elliptical deformations of nested tubes in axial and radial direction with standard deviations ${\sigma }_{\text {d,radial}}={\sigma }_{\text {d,axial}}= {10}\%$ of mean nested tube diameters. Sample size: $2\times {}41\times {}20=1640$ (${\#}_\text {plots}\times {\#}_\text {samples}\times {\#}_\text {x}$). Design parameters: Circular nested tubes with $d/D=0.5$ for (a), anisotropic nested tubes with $d_\text {a}/d_\text {r}=0.7$ and $d_\text {r}/D=0.65$ for optimized loss for (b). Rest: Refer to Fig. 3.

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The results in Fig. 5(a) show that the impact of misaligned outer and nested tubes on the FM propagation loss remains fairly low (between 0 % to 8.7 %) over the studied range and shows no significant fluctuations. In contrary to that, a much more pronounced susceptibility of the HOM propagation can be observed with loss increases up to 64 %. Moreover, the rainbow-colored area enclosed between the ideal and perturbed HOM fiber loss curves denotes a highly wavelength-dependent impact on the propagation loss. A transmission window with negligible impact can be observed at $\lambda = {1.2}\;\mathrm{\mu}\textrm{m}\pm {50}\;\textrm{nm}$, whereas the highest impact is found around $\lambda = {1.45}\;\mathrm{\mu}\textrm{m}\pm {100}\;\textrm{nm}$. Interestingly, the wavelength-dependent HOM propagation loss of perturbed fibers shows a ’flattened’ behavior which compensates for high-loss regions (small additional loss increase) and amplifies the loss increase for low-loss regions. The dynamic of FM and HOM loss is reflected in the HOMER (dashed black), which remains surprisingly constant in the range of 1452 and 2223 while exhibiting minor fluctuations.

On the other side, a whole different pattern can be observed for nested five-tube HC-ARF structures whose nested tubes are perturbed by anisotropic deformations, as shown in Fig. 5(b). FM propagation loss increase shows a highly asymmetric dependence on the operating wavelength with negligibly small values for $\lambda < {1.55}\;\mathrm{\mu}\textrm{m}$ and a rapidly rising loss increase up to 316 % on the other side of the spectrum. These values even surpass the HOM loss increase, which ranges fairly stable between 22 % to 191 % over the whole spectrum. This characteristic shows great dissimilarity to previous observations in subsection 3.2, where HOM loss increase is about one magnitude stronger affected. The rising FM loss is naturally reflected in the HOMER, which fluctuates between 11 and 44 with its maximum at $\lambda = {1.7}\;\mathrm{\mu}\textrm{m}$ and rapidly declines to its minimum in the upper end of the analyzed spectrum.

Finally, it is worth emphasizing the different statistical properties of both perturbed fiber geometries regarding their spread around the median loss. A small spread is hereby linked to a constant transmission loss over multiple fabricated fibers and is therefore desired. As indicated by the box plots in Fig. 5, which are located on top of each calculated median value for both perturbed HOM fiber loss graphs, the spread of the resulting sample groups around the median is very different. For the angular outer and nested tube misalignment effect, the average interquartile range (25%- and 75%-quartile) with respect to each median is very small with −0.8 % to 1.1 % for FM and −3.5 % to 4.9 % for HOM and can be considered acceptable for quality assessment. For the anisotropic deformation effect, however, the average interquartile range is much greater with −9 % to 14 % for FM and −26 % to 32 % as clearly seen by the larger box plots. This results in a highly fluctuating transmission performance of perturbed fabricated fibers.

In summary, both analyzed imperfections exhibit a highly wavelength-dependent impact on the propagation loss of the fiber. At specific wavelengths, the FM loss can be more affected than HOM loss. Therefore, it is highly advised to study the specific configuration combined with operating conditions in order to predict the expected transmission performance after fiber fabrication correctly.

4. Conclusion

The overall guiding performance of perturbed nested HC-ARF structures with fabrication-based imperfections in the cladding elements is investigated in this article. Based on physical observations, two effects, namely randomly misaligned outer and nested cladding tubes as well as anisotropic shape deformations of the nested tubes, are thoroughly analyzed. The susceptibility of the propagation loss follows a similar pattern for both effects, i.e., HOM loss is about one magnitude stronger affected than FM loss. However, the absolute impact in terms of relative loss increase to the ideal (unperturbed) fiber differs by over a factor of 10 in the analyzed range, with anisotropically deformed nested tubes showing a clear domination over misaligned tubes with increased median losses at $\lambda = {1.55}\;\mathrm{\mu}\textrm{m}$ by up to 62% and 850% for FM and HOM propagation, respectively. Furthermore, the HOMER increases monotonically with the intensity of either imperfection, denoting a rising single-modeness of a perturbed fiber. Nevertheless, the impact on the transmission performance is very wavelength-dependent and fluctuates between negligible small loss increases in the region of $\lambda = {1.2}\;\mathrm{\mu}\textrm{m}\pm {50}\;\textrm{nm}$ to a maximum value of 316%.

The most important point to reduce the imperfections in the drawn fiber is to precisely fabricate a perturbation-free fiber-preform, since the drawn fiber geometry reflects the preform’s geometry including imperfections exactly. Additionally, we anticipate that improved process control, monitoring, and automated process adjustments of the drawing process will reduce in-fiber variability, ultimately improving FM loss.

The data provided in this article paves the way for achieving a constant and improved transmission quality of fabricated hollow-core fibers.