## Abstract

In this paper, the material loss of anti-resonant hollow-core fiber (AR-HCF) and its properties are studied. We revisit the formula of power attenuation coefficient for the index-guiding optical fiber described by Snyder and Love in the 1980s and derive the modal overlap factor that governs the material loss of hollow-core fibers (HCF). The modal overlap factor formula predicts the material loss of AR-HCF, which agrees with numerical simulations by the finite element method. The optimization of silica-based AR-HCF design for the lowest loss at 4 µm wavelength is numerically discussed where the silica absorption reaches over 800 dB/m. Our work would provide practical guidance to develop low-loss AR-HCF at highly absorptive wavelengths, e.g. in the vacuum UV and mid/far-infrared spectral regions.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

By guiding light in the vacuum/air, hollow-core fiber (HCF) makes it possible to minimize the influence of host material on guiding properties and significantly bring down the fiber dispersion, optical nonlinearity and absorption induced loss. A low-loss HCF can be an ideal medium for high-power/ultrashort laser pulse delivery [1,2] as well as a highly compact and efficient gas cell for gas-light interaction [3]. Unfortunately, the failure of total internal reflection naturally leads to a leaky tendency of light escaping from the hollow core. Consequently, loss reduction has been a priority in the development of HCF in which photonic bandgap hollow-core fiber (PBG-HCF) and anti-resonant hollow-core fiber (AR-HCF) were proposed and demonstrated great successes.

In 2011, AR-HCF emerged which is featured by low-loss and broadband transmission properties [4]. Different from PBG-HCF, typically AR-HCF has a simpler sing-ring cladding design and a larger core-diameter-to-wavelength ratio [5,6]. By making use of the thin wall of capillaries as a Fabry-Perot resonator around the core, AR-HCF has achieved a loss record of 0.65 dB/km [7] better than that achieved in PBG-HCF [8] and shows a more promising potential [9] to eventually surpass the Rayleigh scattering limit of traditional silica optical fiber in future [10].

The loss of HCF is mainly comprised of surface scattering loss, leakage loss and material loss. Surface scattering loss is induced by the surface roughness of core wall formed in the thermodynamic process during fiber fabrication and it determines the loss limit of HCF [8,9]. Leakage loss is originated from the leakage nature of HCF and dominates in the spectral region where the fiber host material is transparent. Many efforts have been devoted in the understanding of leakage loss of AR-HCFs. In 1993, Archambault *et al.*, presented expressions for modal losses in the simplest AR-HCF with concentric multilayered structure [11]. In 2017, Bird further analytically modeled the AR-HCF and derived the leakage loss formula for fundamental and higher-order modes [12]. Besides, Zeisberger and Schmidt gave an analytical description for the leakage loss of thin capillary AR-HCF in 2017 [13]. Based on extensive numerical studies, Vincetti presented leakage loss estimation with high accuracy for nodeless AR-HCF which has achieved many successes in laser delivery and gas nonlinear optics study [14,15]. Recently, Bird and colleagues quantitatively analyzed the leakage loss of nodeless AR-HCFs and its correlation with concentric-ring model, and revealed the unique function of the negative-curvature core shape in the reduction of leakage loss in AR-HCF [16].

Material loss of AR-HCF becomes a major contribution to the overall fiber loss when the host material becomes opaque and highly absorptive. In AR-HCF, the host material absorbs/interacts with the light field via the limited spatial overlap of mode distribution with the cladding. As a result, for a large core HCF such as AR-HCF, such modal overlap is trivial so that the material loss is often negligible for a moderate material absorption. When the material absorption rises up to hundreds of dB/m and above, the material loss of AR-HCF starts to compete with and dominate over the leakage loss eventually [17–21]. By using silica glass, AR-HCFs have been demonstrated tens of dB/km losses in the mid-IR [17,21–23] where the material absorption reaches up to nearly a thousand dB/m. Loss records of 18 dB/km at 3.16 µm and 40 dB/km at 4 µm have been recently demonstrated in silica-based AR-HCF [21] where the silica absorption are 59 dB/m and 863 dB/m respectively. It is noted that such loss performance of AR-HCF is even better than the state-of-the-art fluoride glass solid fibers at same wavelengths [24], where the fluoride glass is almost transparent.

In spite of a quick development of AR-HCF at highly absorptive wavelengths e.g. in the mid-IR and UV [25], there have been few works focusing on the mechanism of material loss of AR-HCF [17,19,21]. In this paper, we explore the fundamentals of material loss of AR-HCF and its properties. We revisit Snyder and Love’s work on the general formula of power attenuation coefficient in an index-guiding optical fiber [26] and derive the formula of material loss of AR-HCF. The explicit form of modal overlap factor *η* in AR-HCF is obtained which governs the contribution of material absorption to the fiber loss. The prediction of formulas agrees well with numerical simulations by finite-element method for different AR-HCF design. Next, we numerically examine all the roles of fiber design elements of AR-HCF played in the material loss and discuss the optimized designs of AR-HCFs for minimum material loss. Our conclusions can provide practical guidelines for low-loss AR-HCF design at highly absorptive wavelengths, e.g. in the vacuum UV and mid/far-infrared spectral regions.

## 2. Material loss and modal overlap factor in AR-HCF

Figure 1(a) presents a model of six-capillary nodeless AR-HCF. Figure 1(b) presents the simulated total loss for different material absorption conditions. The core of AR-HCF is defined as the circle inscribed with cladding capillaries. The outer diameter *d* and wall thickness *t* of cladding capillaries remain constant while the core diameter *D* and the gap *g* varies.

In Fig. 1(b), the total loss of AR-HCF is calculated for different material absorptions by finite element method using COMSOL software. When the material absorption equals zero, the total loss is namely the leakage loss and the difference of total loss and leakage loss defined as the material loss in Eq. (1). Note that enlarging *D* also leads to the increase of *g* due to the mutual effect of geometric parameters in AR-HCFs. Too large gap is detrimental to the confinement of central mode field and that’s why the confinement loss rises instead when *D* increases from 120 µm to 150 µm. Nevertheless, as the material absorption rises, the total loss is found to increase linearly for different core diameters. The slopes as constants for different AR-HCFs are calculated as 5.395×10^{−5}, 2.294×10^{−5} and 1.187×10^{−5} when *D* is 90 µm, 120 µm and 150 µm, respectively. The slopes in Fig. 1(b) are found dependent on the structure of AR-HCF only which governs the contribution from material absorption to the fiber loss. Here we define the slope as the modal overlap *η* [6,21] that decides the material loss of AR-HCF,

*α*is the material loss induced by material absorption and

_{m}*α*the material absorption,

_{abs}*α*the leakage loss due to the leakage nature of hollow-core fibers,

_{L}*α*the total fiber loss namely the sum of

_{total}*α*and

_{m}*α*, and all corresponding units are dB/m. Here surface scattering loss is ignored due to the low field strengths at the glass-air interfaces in AR-HCFs [9].

_{L}To deduce a semi-analytical expression of Eq. (1), we revisit the general formula of power attenuation coefficient proposed by Snyder and Love for an arbitrary solid-core optical waveguide of longitudinal uniformity in [26]. We derivate the material loss of AR-HCF,

*ɛ*and

_{0}*µ*denote free-space dielectric constant and free-space permeability, respectively,

_{0}*k*is the free-space wavevector,

*n*and

^{r}*n*are real and imaginary parts of refractive index of fiber material respectively, Re denotes real part, $\overrightarrow e $ and $\overrightarrow h $ are the electric field and magnetic field distribution of mode, $\widehat z$ is the unit vector parallel to the waveguide axis, Ω

^{i}_{1}represents the whole cross-section of fiber including the hollow and solid material region and Ω

_{2}the region of absorbing material.

By substituting Eq. (2) into Eq. (1), the modal overlap factor *η* is presented as

*η*is calculated as 5.407×10

^{−5}, 2.296×10

^{−5}and 1.191×10

^{−5}, for three core diameters respectively, which agrees well with the results calculated by Eq. (1).

*η*was firstly proposed in [21], however in the first place it was approximated by using the ratio of power in the host material, which are 4.137×10

^{−5}, 1.765×10

^{−5}and 9.153×10

^{−6}for our AR-HCFs.

According to Eq. (3), the modal overlap is decided by the distribution of electric field strength resident in the host material only and a quick drop of electric field strength in the cladding is the key to reach a lower material loss in principle. In Fig. 2, we compare the electric field distributions of fundamental mode in the tube waveguide and two AR-HCFs at the same wavelength. The mode in the tube waveguide spreads out outside the core and slowly declines which results in a high modal overlap. In both AR-HCFs, the field strength quickly ‘cuts off’ at the core boundary and in the cladding the modal field has a much steeper declining rate than in the tube waveguide, which we contribute to the multiple-beam interference thanks to the anti-resonance design. By comparing Fig. 2(b) with Fig. 2(c), it clearly shows that a larger core is favorable to achieve a lower material loss due to a faster drop of electric field strength in the absorbing host material.

The detailed derivation of Eq. (2) is found in the Appendix. We start from the basic assumption proposed by Snyder and Love on page 234 in [26] and present a new derivation process. Equation (2) is identical as the conclusion reached by Snyder and Love on page 233, but an approximated form of our new formula of Eq. (13) in the Appendix. The typo and flaw of derivation in [26] are discussed in the Appendix. Note that Eq. (2) is used to predict the material loss in our paper.

## 3. Optimization of AR-HCF design for low material loss

In this section, we focus on the nodeless type of AR-HCFs which have been extensively studied and achieved success in many applications. We numerically investigate functions of all fiber design elements played in the material loss. Optical constants of F300 silica glass as host material are used in the simulation [27]. By numerically evaluating the loss performance at 4 µm (silica absorption of 863 dB/m), we discuss and summarize optimized designs of AR-HCF for the lowest material loss. All the simulations are carried out by using COMSOL.

#### 3.1 Effect of core diameter on material loss

The advantages of using a larger core to reduce the material loss are shown in Figs. 1 and 2. However, the expansion of core diameter in the practical fabrication is limited. The core-diameter-to-wavelength ratio in AR-HCFs has been reported in the range from 20 to 50 generally [5]. It is noted that a larger core also gives rise to a higher bending induced leakage and material losses of AR-HCF [6,28]. For a fixed (*d*-2*t*)/*D*, the critical bend radius where bend loss becomes significant was demonstrated to scale with *D*^{3} [29].

#### 3.2 Effect of anti-resonance on material loss

Transverse phase is used as a measure to describe the anti-resonance condition of cladding layers in AR-HCF. Following Bird’s model [12], the transverse phase lags of air and glass layer regions can be approximated by

*d*. Besides, ${w^{(g)}}$ is the width of glass layers approximated to

*t*. ${n_{glass}}$ is the refractive index of glass at the wavelength $\lambda $. ${\mathop{\rm Re}\nolimits} ({n_{eff}})$ is the real part of complex effective index and the classical formula derived by Marcatili and Schmeltzer is used for approximation namely ${\mathop{\rm Re}\nolimits} ({n_{eff}}) \approx \sqrt {1 - {{({u_0}\lambda /\pi D)}^2}}$, where ${u_0}$ is the zero root of first kind of Bessel function.

According to Eqs. (4) and (5), the phase lag in the air region is mainly decided by the ratio *d*/*D* while the phase lag in the glass is closely related to the core wall thickness *t*. Therefore, to investigate the effect of cladding resonance on the material loss of AR-HCFs, the structural parameters, *d*/*D* and *t*, are swept for different AR-HCF designs. Meanwhile, we fix the core diameter *D* at 120 µm and the wavelength *λ* at 4 µm, corresponding to a core-diameter-to-wavelength ratio of 30. In the COMSOL model, the refractive index of air is assumed to be 1 while *n _{glass}* 1.389 following the Sellmeier formula [30]. The gap between the cladding capillaries

*g*= (

*D*+

*d*)sin(π/

*N*)-

*d*, where

*N*is the number of cladding capillaries.

With optimized mesh size and perfectly matched layer settings, the leakage loss of AR-HCF is numerically simulated [9]. In Fig. 3(a) series, the leakage loss *α _{L}* as function of

*d*/

*D*and core wall thickness

*t*for

*N*= 6, 7 and 8 shows strongly dependence on transverse phases of air and silica layers. α

*changes by 2-3 orders of magnitude by tuning*

_{L}*d*and by 1-2 orders of magnitude by tuning

*t*. Minimum leakages losses in three AR-HCFs are listed in Table 1. According to Eq. (4) and Eq. (5), the anti-resonance condition corresponds to

*d*/

*D*and

*t*as 0.33 and 1.04 µm. In Fig. 3(a) series, all minimum leakage loss appears when

*d*/

*D*equals 0.6/0.65 rather than 0.33, which we attribute to the difference between the Bird’s model and nodeless AR-HCF.

The Fig. 3(b) series shows the effect of cladding resonance on material loss. In the Fig. 3(b) series, material loss α* _{m}* for all

*N*are calculated by the modal overlap

*η*which is defined in Eq. (3) and can be extracted by using pre- or post processing method in COMSOL. Here α

*is found less sensitive to the phase lags in the air cladding regions than the leakage loss. For all AR-HCFs, the lowest material loss all appears near the anti-resonant core wall thickness, which agrees with [21]. Too thick or too thin thickness will lead to the change of phase lags in the glass core wall, failing to minimize the electromagnetic field distributed there and increasing the material loss.*

_{m}The Fig. 3(c) series are total loss *α _{total}* as a sum of leakage and material losses. Near resonance, the extremely high leakage loss dominates the overall loss. When off resonance, the material loss

*α*determines the floor level of overall loss at the highly absorption wavelength.

_{m}Table 1 summarizes the lowest *α _{L}*,

*α*and

_{m}*α*in the whole simulated parameter space for

_{total}*N*equals 6, 7 and 8. 6-capillary AR-HCF design is preferred for the lowest overall total loss performance where the optimized ratio of

*d*/

*D*also approaches the single mode condition [31].

Next, we extend our analysis of Fig. 3(b-1) by further increasing *t* so that the simulated wavelength can enter the second-order transmission band. As shown in Fig. 4, material loss *α _{m}* in the high order band follows almost the same dependence on resonance condition as in the fundamental band. Moreover, because of a thicker core wall, the material loss clear rises, as similarly found in [21].

#### 3.3 Effect of nested resonance design on material loss

Adding extra reflective interfaces in the cladding has been demonstrated effective in the enhancement of multiple beam interference and significantly reduces the leakage loss in AR-HCF [7,32]. Numerical simulations suggest proper nested design of AR-HCF could reduce the leakage loss to less than 0.1 dB/km, totally surpassing the Rayleigh scattering limit of solid-core fibers [9,20,33]. Recently a loss of 0.65 dB/km was measured in a nested type of silica-based AR-HCF at the wavelength around 1550 nm [7].

Figure 5 compares the overall loss performance of silica-based AR-HCFs with and without nested designs at wavelengths from 2.6 to 5 µm. All AR-HCFs have six-fold symmetry with the same core diameter *D* and core wall thickness *t* as 120 µm and 1.25 µm, respectively. The diameters of capillaries in the cladding are *d* = 100 µm, *d _{1}* = 50 µm,

*d*= 25 µm.

_{2}When the wavelength is swept from 2.6 to 4 µm, the silica material absorption rises from 3.64 dB/m and approaches to 863 dB/m. As Fig. 5 shows, the overall loss almost follows the same trend as the silica absorption and increases rapidly due to the contribution from the material absorption. At short wavelengths, the material loss starts to compete with the leakage loss at 3.55 µm and 2.65 µm for simple and nested AR-HCFs. At 4.3 µm and beyond, the material losses far exceed the leakage losses and dominate regardless of fiber design. At these long wavelengths, nested AR-HCF designs exhibit no advantage in terms of low material loss and overall loss.

## 4. Conclusion

In this paper, we derive the formulas of material loss and modal overlap factor of AR-HCF and demonstrate that the contribution of material absorption to the fiber loss in AR-HCF is ultimately determined by the amount of electric field strength distribution in the cladding only. We numerically explore the dependence of material loss on AR-HCF design elements, including core diameter *D*, core wall thickness *t*, external diameter of cladding capillary *d*, the gap between capillaries *g*, capillary number *N* and nested elements. The optimized designs of AR-HCFs for minimum loss at highly absorptive wavelengths are summarized. The complex nested design of AR-HCF although effective to reduce the leakage loss, is demonstrated no benefit to reduce the material loss. Our work suggests a simple six-capillary AR-HCF design is preferred to achieve the lowest material loss and best overall loss performance at the mid/far-infrared and other highly absorptive wavelengths.

## Appendix

In the appendix, we extend Snyder and Love’s work for an arbitrary solid-core waveguide in [26] to the field of hollow-core fiber. Due to the typo and flaw of derivation in [26], we present a different derivation process for material loss and modal overlap in hollow-core fibers.

According to Poynting’s vector theorem, the time-averaged power produced by a current density $\overrightarrow J $ within volume *V* is

By introducing an imaginary part to the refractive index in an absorption-free waveguide model, the material absorption gives rise to an equivalent induced current expressed as,

*ɛ*and

_{0}*µ*denote free-space dielectric constant and free-space permeability, respectively,

_{0}*k*is the free-space wavevector, $n^{\prime}$ is the refractive index of the absorbing medium and

*n*the refractive index of the corresponding non-absorbing medium. It is noted that $\overrightarrow E$ is the electric field strength of the non-absorptive waveguide. Here we follow the assumption in Snyder and Love’s work in [26] that the imaginary part of refractive index is too small to alter the electric field distribution. In the case of AR-HCF, we assume such small perturbation would not make a change to the leakage loss property, too.

We set $n = {n^r}$ and $n^{\prime} = {n^r} + i{n^i}$. Then the absorbed power by material can be obtained by combining Eq. (6) with Eq. (7)

On page 597 in [26], the time-averaged Poynting vector form of Eq. (6) was not used, which made it impossible to split the real part of ${n^{\prime2}}$ to reach Eq. (8).

Because of material absorption, an imaginary part $\beta _m^i$ is introduced to the propagation constant so that the electric field is attenuated by $\exp ( - \beta _m^iz)$, and the power attenuation coefficient $\gamma = 2\beta _m^i$. Take $\overrightarrow e $ as the electric field distribution of a mode of waveguide, then electric field along the waveguide $\overrightarrow E = \overrightarrow e \cdot \exp ( - \gamma z/2)$. Then power absorption by material for z distance along the waveguide follows,

The power flow ${P_z}^\prime $ along the absorptive waveguide is defined as,

According to the small perturbation assumption above, we let $\overrightarrow e = {\overrightarrow e ^\prime }$ and the same leakage loss are shared. Marcatili and Schmeltzer demonstrated that in the hollow-core waveguide, the electric and magnetic fields satisfy [28],

where ${Z_0}$ is the impedance of free space. In this case, the perturbation of refractive index must change the magnetic field accordingly giving rise to ${\overrightarrow h ^\prime } = (n^{\prime}/n)\overrightarrow h$. So we obtain,If we let $\Delta P = \Delta {P_m}$, then the formula of material loss in the hollow-core fiber is derived as,

In [26], ${\overrightarrow h ^\prime } = \overrightarrow h$ was simply assumed which must lead to the unchanged Poynting flow before and after the perturbation of refractive index and contradict with the introduction of material absorption.

At highly absorptive wavelength, most materials have ${n^i}$ less than 1 by several orders of magnitude. At 4 µm wavelength, 863 dB/m material absorption corresponds to ${n^i} =$ 6.325×10^{−5}. In the case of ${n^i} \ll {n^r}$, Eq. (13) can be simplified as,

_{1}is the whole cross-section of fiber and Ω

_{2}the region of host material.

We must point out that as *n ^{i}* increases approaching to

*n*, the small perturbation assumption becomes no longer valid while the modal electric field would be redistributed by introducing an extremely large

^{r}*n*. Therefore, the predicted modal overlap factor by Eq. (14) would eventually deviate from the numerical simulation as shown in Fig. 6. However, it is noted that such error is still found less than 10% even when

^{i}*n*is close to 0.055 (the corresponding material absorption is about 10

^{i}^{6}dB/m), implying a good validity of Eq. (14) in a broad range of material absorption.

To further explore the accuracy of Eq. (14), we simulate AR-HCF model in Fig. 1(a) with different fiber parameters and wavelengths and summarize the calculated modal overlap factors by using three different methods, as shown in Table 2. We demonstrate that the accuracy of prediction by Eq. (3) is maintained for different *N*, *D*, *d*, *t*, *n* and *λ*, whereas the ratio of power in the host material deviates from the FEM simulated modal overlap factor by over 20%.

## Funding

Key Technologies Research and Development Program (2018YFB0504500); International Science and Technology Cooperation Programme (2018YFE0115600); National Natural Science Foundation of China (51972317, 61307056, 61705244, 61875052); International Cooperation and Exchange Programme (61961136003); Natural Science Foundation of Shanghai (17ZR1433900, 17ZR1434200, 18ZR1444400); Chinese Academy of Sciences (Pioneer Hundred Talents Program); Chinese Academy of Sciences (ZDBS-LY- JSC020).

## Disclosures

The authors declare no conflicts of interest.

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