# Geometrical Scaling of Antiresonant Hollow-Core Fibers for Mid-Infrared Beam Delivery

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}at 3.1 μm and 40 dB∙km

^{−1}at 4 μm. Ref. [11] demonstrates low-loss guidance over a broad mid-IR spectral band spanning 2.5–7.9 μm.

## 2. Design and Its Geometrical Formalism

^{−1}[10,14]. An idealized cross-sectional structure of a six-element THCF is presented in Figure 1a, where gray shades correspond to the regions of dielectric medium, and white regions are hollow. D is the core diameter which is defined as the diameter of the largest circle that fits in the middle hollow region; d is the exterior diameter of the tubular cladding elements; and t is their wall thickness. The size of the tubular cladding elements in THCF is often characterized in terms of the ratio d/D. The range of possible d/D is then set by the number of cladding elements, N, i.e., $0<d/D<\mathrm{sin}\left(\pi /N\right)/\left(1-\mathrm{sin}\left(\pi /N\right)\right)$ [15].

_{1}, and that of the inner tubes, d

_{2}. The core size, D, and dielectric wall thickness, t, are defined in the same way as in THCF.

_{1}= 2d

_{2}= 39 μm in both fibers. Their loss spectra consist of low and high loss regions, where in the low-loss regions, NANF outperforms THCF by approximately two orders of magnitude. The locations of the high-loss bands in AR-HCFs are dictated by the dielectric wall thickness of cladding elements. This gives rise to presence of the peak seen at around 1.25 μm in both geometries. At this wavelength, dielectric wall essentially acts as a Fabry–Perot resonator, and light can easily escape from the core resulting in leakage loss. The resonant wavelengths are given by [20,21]:

^{−1}and $\Im \left({n}_{\mathrm{eff}}\right)$ are related through [22]:

## 3. Results and Discussions

^{−1}and wavelength, we plot, in Figure 3b, the corresponding confinement loss in the case of two different t values, 0.6 and 0.8 μm, with the equivalent wavelengths for the two cases shown in the upper horizontal axis. For long wavelength operation, light guidance in the first transmission band, i.e., F < 1, is of high relevance. In particular, the confinement loss at small F values—in the gray-shaded regions in Figure 3a,b—reveals how THCF is expected to operate in mid-IR. For instance, F < 0.4 corresponds to the wavelength longer than 3.15 μm when t = 0.6 μm. One striking observation which we discuss for the first time is how the resonance-like loss starts to show up below F ≈ 0.4, in a much similar fashion to how it happens around the resonances at F = 1, 2, and so on. In other words, after the minimum confinement loss is reached at F = 0.5, the loss increases again as if there is another t-induced resonance located at F = 0. Even the small peaks at the red-edges of the transmission bands, marked with vertical-dotted lines at F = 2.26 and 1.23, are replicated in the first transmission band at F = 0.21. Considering that F corresponds to the number of half wavelengths that fits in the thickness of the dielectric cladding tube wall, it is interesting to note that a Fabry–Perot-like resonance still exists when the thickness of the dielectric wall approaches zero. This suggests that the dielectric wall thickness in THCF must be increased accordingly to ensure F > 0.2 and achieve low-loss guidance in the long wavelength limit.

_{1}/D = 0.65. The ratio between the exterior diameters of the outer and inner cladding elements, d

_{1}/d

_{2}= 2 is used in all calculations. In Figure 4b, we set t = 0.6 μm to evaluate the corresponding confinement loss and wavelength. The same quantities for THCF, identical to those presented in Figure 3, are plotted together in Figure 4 for comparison. The reduction of over two orders of magnitude is evident in the transmission bands, clearly demonstrating its superior performance. Towards both edges of each transmission band in NANF, the loss rises rapidly and reaches the same level as in THCF beyond the vertical-dotted lines. This happens when F < m − 0.75 in the long- and F > m in the short-wavelength edges. On the other hand, NANF does not further reduce the light-glass overlap as shown in Figure 4c. Therefore, NANF does not offer additional benefit when it comes to mitigating the high material absorption in mid-IR. Furthermore, NANF also exhibits a resonance-like loss in the long-wavelength side of its fundamental transmission band, i.e., as F approaches 0, just like that seen in THCF in Figure 3. This appears to be a feature that is universal to all AR-HCFs.

_{0}. The loss per unit length in a dielectric capillary can then be obtained through Equation (4). It indicates that the confinement loss in a dielectric capillary scales with the inverse cube of D. For a thin wall dielectric capillary, Zeisberger and Schmidt analytically showed that the loss becomes proportional to D

^{−4}[25]. This scaling changes to D

^{−4.5}in THCF when the tubular cladding elements are in contact with each other, i.e., $d/D=\mathrm{sin}\left(\pi /N\right)/\left(1-\mathrm{sin}\left(\pi /N\right)\right)$, as demonstrated numerically in Reference [26]. In this regard, we highlight that the two of the cases discussed above—the capillary with infinitely thick dielectric medium and THCF with touching cladding elements—represent two extreme values of d/D in THCF. On one end, we have the dielectric capillary when d/D = 0 where the loss scales with D to the power −3, while on the other end, we arrive at THCF with touching cladding elements when d/D is at its maximum where the loss scales with D to the power −4.5. An interesting question is then how this scaling changes between the two limiting cases. Knowing that the confinement loss of a six-element THCF remains relatively flat and low when 0.5 < d/D < 0.8 [27,28], understanding how the loss scales with the core size in this range is of particular interest.

^{−5.4}. This translates to the confinement loss being proportional to ${\lambda}^{4.4}/{D}^{5.4}$ via Equation (4). Similarly, the changes in $\Im \left({n}_{\mathrm{eff}}\right)$ as a function of G for THCF with different number of cladding elements, while fixing F = 0.5, i.e., at the center of the first transmission band, are presented in Figure 5b. For each $N$, the cladding tube size that exhibits low loss, as determined in Reference [15], is used in the calculations. These are d/D = 0.79, 0.65, 0.6, 0.5, 0.45 for N = 5, 6, 7, 8, 9, respectively. We find that the rate of reduction against $G$ does not vary much with change in N when d/D that gives low loss is chosen in each N. In fact, we obtain the same loss decay rate of 5.4 from Figure 5b. This demonstrates that the dielectric wall thickness and number of cladding elements do not have significant impact on the rate of reduction in $\Im \left({n}_{\mathrm{eff}}\right)$ with increase in the core size.

^{−3}for the former and G

^{−4.5}for the latter.

^{−x}, which implies through Equation (4) that ${\alpha}_{\mathrm{dB}}$ is proportional to ${\lambda}^{x-1}/{D}^{x}$. At each data point in Figure 6, x is numerically calculated by fitting the $\Im \left({n}_{\mathrm{eff}}\right)$ versus G plot using the least squares method in the range 20 < G < 40. Moreover, since the possible range of d/D in THCF depends on N, i.e., from 0 to $\left(\mathrm{sin}\left(\pi /N\right)\right)/\left(1-\mathrm{sin}\left(\pi /N\right)\right)$, we introduce a new parameter (d/D)′ to standardize the cladding size for all N for easy comparison. This is defined as:

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Idealized cross-sections of (

**a**) a six-element tubular hollow-core fiber (THCF) and (

**b**) a six-element nested antiresonant nodeless fiber (NANF). Gray shades correspond to the regions of dielectric medium, and white regions are hollow. D is the core diameter which is defined as the diameter of the largest circle that fits in the central hollow region. t is the dielectric wall thickness of the cladding elements. d is the exterior diameter of the cladding elements in THCF, and d

_{1}and d

_{2}are those of the outer and inner cladding tubes, respectively, in NANF. (

**c**) Confinement loss of silica-based six-element THCF and NANF calculated using finite-element method. The structural parameters are D = 60 μm, t = 0.6 μm, and d = d

_{1}= 2d

_{2}= 39 μm.

**Figure 2.**(

**a**) Imaginary part of effective index of the fundamental core mode, $\Im \left({n}_{\mathrm{eff}}\right)$, in a six-tube THCF versus f when the fiber structure depicted in (

**b**) and wavelength are both scaled by f. The parameters used in the calculation are λ = 1.06 μm, D = 26.5 μm, d = 17.2 μm, and t = 0.26 μm.

**Figure 3.**(

**a**) Imaginary part of the fundamental core mode index, $\Im \left({n}_{\mathrm{eff}}\right)$, as a function of F in a six-tube THCF when G = 25 and d/D = 0.65. The calculation is carried out for four different t values, 0.4, 0.6, 0.8, and 1 μm, to illustrate the scalability of the results. (

**b**) Corresponding confinement loss versus wavelengths when t = 0.6 and 0.8 μm. Gray-shaded regions in (

**a**,

**b**) mark the long wavelength region, e.g., wavelength longer than 3.15 μm when t = 0.6 μm. (

**c**) Fractional optical power in the dielectric medium in the same THCF.

**Figure 4.**(

**a**) Imaginary part of the fundamental core mode index, $\Im \left({n}_{\mathrm{eff}}\right)$, as a function of F in a six-tube NANF when G = 25 and d

_{1}/D = 0.65, and d

_{1}= 2d

_{2}. (

**b**) Corresponding confinement loss versus wavelength when t = 0.6 μm. (

**c**) Fractional optical power in the dielectric medium in the same NANF. The same quantities for THCF with the identical geometrical parameters are plotted in gray-dashed lines for comparison.

**Figure 5.**$\Im \left({n}_{\mathrm{eff}}\right)$ versus G in THCF for different structural parameters. (

**a**) For F = 0.4, 0.5, 0.6, 0.8, 1.5, while N = 6 and d/D = 0.65. (

**b**) For N = 5, 6, 7, 8, 9, while F = 0.5 and d/D is set at a value that gives low loss in each N as identified in Reference [15]. (

**c**) For d/D = 0.4, 0.5, 0.6, 0.65, 0.7, 0.8, 0.9, 0.99, while N = 6 and F = 0.5.

**Figure 6.**Confinement loss decay rate, x, versus the standardized cladding size, (d/D)′, for THCF when N = 6, 7, 8, and F = 0.5. The decay rate, x, means ${\alpha}_{\mathrm{dB}}$ is proportional to ${\lambda}^{x-1}/{D}^{x}$. At each data point, x is calculated by fitting the $\Im \left({n}_{\mathrm{eff}}\right)$ versus G plot using the least squares method in the range 20 < G < 40. The standardized cladding size, (d/D)′, is obtain by normalizing d/D to its respective maximum, $\mathrm{sin}\left(\pi /N\right)/\left(1-\mathrm{sin}\left(\pi /N\right)\right)$. Example geometries at three different (d/D)′ values when N = 7 are illustrated in the insets for reference. The colored crosses mark (d/D)′ values that give the lowest loss in each N which coincides with the largest decay rate.

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**MDPI and ACS Style**

Deng, A.; Chang, W.
Geometrical Scaling of Antiresonant Hollow-Core Fibers for Mid-Infrared Beam Delivery. *Crystals* **2021**, *11*, 420.
https://doi.org/10.3390/cryst11040420

**AMA Style**

Deng A, Chang W.
Geometrical Scaling of Antiresonant Hollow-Core Fibers for Mid-Infrared Beam Delivery. *Crystals*. 2021; 11(4):420.
https://doi.org/10.3390/cryst11040420

**Chicago/Turabian Style**

Deng, Ang, and Wonkeun Chang.
2021. "Geometrical Scaling of Antiresonant Hollow-Core Fibers for Mid-Infrared Beam Delivery" *Crystals* 11, no. 4: 420.
https://doi.org/10.3390/cryst11040420