Geometry of Chalcogenide Negative Curvature Fibers for CO₂ Laser Transmission

Abstract

We study the impact of geometry on leakage loss in negative curvature fibers made with As${}_2$Se${}_3$ chalcogenide and As${}_2$S${}_3$ chalcogenide glasses for carbon dioxide (CO${}_2$) laser transmission. The minimum leakage loss decreases when the core diameter increases both for fibers with six and for fibers with eight cladding tubes. The optimum gap corresponding to the minimum loss increases when the core diameter increases for negative curvature fibers with six cladding tubes. For negative curvature fibers with eight cladding tubes, the optimum gap is always less than 20 $\mathsf{\mu }$m when the core diameter ranges from 300 $\mathsf{\mu }$m to 500 $\mathsf{\mu }$m. The influence of material loss on fiber loss is also studied. When material loss exceeds 10${}^2$ dB/m, it dominates the fiber leakage loss for negative curvature fiber at a wavelength of 10.6 $\mathsf{\mu }$m.

1. Introduction

Carbon dioxide (CO${}_2$) lasers have been widely used in surgery, medicine, and material processing [1,2,3]. Step index fibers are commonly used to transmit CO${}_2$ laser light. The material loss of silica glass in the mid-infrared limits the transmission of mid-infrared light using silica step-index fibers. However, it is possible in principle to obtain a lower loss in hollow-core fiber than in step-index fiber because air does not contribute to material loss [4,5]. In addition, the nonlinearity in the glass sets a limit to the transmitted power. Hollow-core fibers have low nonlinearity, because the light is mostly transmitted in air, which does not contribute to the nonlinearity. Recently, hollow-core negative curvature fibers have drawn a large amount of interest due to their attractive properties including low loss, broad bandwidth, and a high damage threshold [6,7,8,9,10,11,12]. The delivery of mid-infrared radiation has also been demonstrated using chalcogenide negative curvature fibers for a CO${}_2$ laser at a wavelength of 10.6 $\mathsf{\mu }$m [13,14,15]. Previous study shows that chalcogenide glass should be used for wavelength larger than 4.5 $\mathsf{\mu }$m [16]. The relative simplicity of the negative curvature structure could enable the fabrication of fiber devices for mid-IR applications using non-silica glasses, such as chalcogenide [13,14,15].

The guiding mechanism in negative curvature fibers is inhibited coupling [10,17,18]. A large amount of research [10,19] has been carried out to determine the impact of fiber parameters on leakage loss [20] in negative curvature fibers and then optimize these parameters to minimize the loss. These parameters include the curvature of the core boundary, the thickness of the tubes, the number of cladding tubes, and the nested cladding tubes [17,18,21,22,23,24]. By introducing a gap between cladding tubes, the loss can be decreased in negative curvature fibers [24,25]. When the tubes touch, modes exist in the localized node area. A gap between the cladding tubes removes the additional resonances due to the localized node. Fibers with a gap between tubes are also expected to be easier to fabricate, since surface tension would assist to maintain the circular shape of the tubes [22]. On the other hand, when the gap is too big, the core mode can leak through the gaps, which increases the loss in negative curvature fibers [26]. Therefore, an optimum gap exists. The optimal gap corresponding to the minimum loss in a fiber with six cladding tubes is three times as large as the optimal gap in fibers with eight or ten cladding tubes [26]. In a fiber with six cladding tubes, a larger gap is needed to remove the weak coupling between the core mode and tube modes [26].

In previous studies, the optimum gap was found in negative curvature fibers with a fixed core diameter [26]. Chalcogenide negative curvature fibers with different core diameters of 170 $\mathsf{\mu }$m to 380 $\mathsf{\mu }$m have been fabricated [13,14,15]. In this paper, we find optimal structures of chalcogenide negative curvature fibers for CO${}_2$ laser transmission, in which we minimize the loss in the two-dimensional parameter space that consists of the core diameter and the gap size. In previous studies, the optimum gap was found in negative curvature fibers with a fixed core diameter [26]. We find that the minimum leakage loss decreases when the core diameter increases both for fibers with six and for fibers with eight cladding tubes. The optimum gap increases when the core diameter increases for negative curvature fibers with six cladding tubes. The optimum gap is always less than 20 $\mathsf{\mu }$m when the core diameter increases for negative curvature fibers with eight cladding tubes when the core diameter ranges from 300 to 500 $\mathsf{\mu }$m. We find optimal structures of chalcogenide negative curvature fibers for ${\mathrm{CO}}_2$ laser transmission, in which we minimize the loss in the two-dimensional parameter space that consists of the core diameter and the gap size.

2. Geometry

Negative curvature fibers with six and eight cladding tubes have been fabricated by several research groups [17,25,27,28]. Figure 1 shows schematic illustrations of negative curvature fibers with six and eight cladding tubes. The white regions represent air, and the gray regions represent glass. The inner tube diameter, $d_{\mathrm{tube}}$, the core diameter, $D_{\mathrm{core}}$, the tube wall thickness, t, the minimum gap between the cladding tubes, g, and the number of tubes, p, are related by the expression: $D_{\mathrm{core}}=(d_{\mathrm{tube}}+2t+g)/\sin (\pi /p)-(d_{\mathrm{tube}}+2t)$ [29]. We calculate the leakage loss for negative curvature fibers using Comsol Multiphysics, a commercial full-vector mode solver based on the finite-element method. Perfectly matched layers are added outside the cladding region in order to reduce the size of the simulation window [30]. The wavelength of 10.6 $\mathsf{\mu }$m for a CO${}_2$ laser is used in our simulation.

3. As${}_2$Se${}_3$ Chalcogenide Glass

In this section, we study the loss in negative curvature fibers made with As${}_2$Se${}_3$ chalcogenide glass. We use a refractive index of 2.8 and a material loss of 10.6 dB/m for As${}_2$Se${}_3$ chalcogenide glass in our simulations [31]. The tube thickness, t, is fixed at 5.2 $\mathsf{\mu }$m corresponding to the third antiresonance. A glass thickness corresponding to the third antiresonance has been drawn in the past [15]. A thicker tube wall with a higher-order antiresonance makes fabrication easier. Geometries that use tube thicknesses corresponding to the first, second, or third antiresonance have similar minimum losses in the transmission band [16,26]. We first study negative curvature fibers with six cladding tubes. We define $d_{6\max }$ as the maximum possible tube diameter for the fiber with 6 cladding tubes, which equals $D_{\mathrm{core}}-2t$. Figure 2a shows the contour plot of loss as a function of core diameter, $D_{\mathrm{core}}$, and normalized tube diameter, $d_{\mathrm{tube}}/d_{6\max }$. For a fixed $D_{\mathrm{core}}$, the loss decreases and then increases when $d_{\mathrm{tube}}/d_{6\max }$ increases from 0.2 to 1.0. The minimum loss occurs when $d_{\mathrm{tube}}$/$d_{6\max }$ = 0.62, and it does not change when $D_{\mathrm{core}}$ increases from 300 to 500 $\mathsf{\mu }$m. The loss decreases when $D_{\mathrm{core}}$ increases. In addition, we show the loss as a function of the core diameter, $D_{\mathrm{core}}$, and the gap, g, in Figure 2b. The loss first decreases and then increases as the gap, g, increases. When there is no gap, a mode exists in the node that is created by the two touching tubes [25]. When the gap is too large, core mode leaks through the gap [17,26]. Previous study shows that the electric field intensity in the middle of the gap between cladding tubes can increase by a factor of 15 when the gap increases from 5 to 10 $\mathsf{\mu }$m in a silica negative curvature fiber with a glass index of 1.45 and a core diameter of 30 $\mathsf{\mu }$m at a wavelength of 1 $\mathsf{\mu }$m [10]. Here, we study chalcogenide negative curvature fibers with a glass index of 2.8 at a wavelength of 10.6 $\mathsf{\mu }$m. The electric field intensity in the middle of the gap between cladding tubes increases by a factor of 15 when the gap increases from 50 to 100 $\mathsf{\mu }$m in a negative curvature fiber with a core diameter of 300 $\mathsf{\mu }$m. We also plot the loss as a function of gap, g, for different core diameters in Figure 3a. In order to quantify the minimum loss and the corresponding optimum gap for different core diameters, we also plot the minimum loss and the corresponding optimum gap, g, using blue solid curve and red dashed curves, respectively, in Figure 3b. When the core diameter increases from 300 to 500 $\mathsf{\mu }$m, the minimum loss decreases by more than one order of magnitude and the corresponding optimum gap, g, increases from 60 to 90 $\mathsf{\mu }$m. Hence, a larger gap is needed for a fiber with a larger core diameter to decrease the loss in negative curvature fibers with six cladding tubes.

We next carry out the same loss analysis on negative curvature fibers with eight cladding tubes. Figure 4a shows the contour plot of loss as a function of core diameter, $D_{\mathrm{core}}$, and normalized tube diameter, $d_{\mathrm{tube}}$/$d_{8\max }$, where $d_{8\max }$ is defined as the maximum possible tube diameter for the fiber with 8 cladding tubes, which is $D_{\mathrm{core}}\sin (\pi /8)/[1-\sin (\pi /8)]-2t$ [32]. Figure 4b shows the contour plot of loss as a function of core diameter, $D_{\mathrm{core}}$, and gap, g. The minimum loss occurs at a larger value of $d_{\mathrm{tube}}$/$d_{8\max }$, or a smaller value of g, than is the case for negative curvature fibers with six cladding tubes. In Figure 5a, we show the loss as a function of the gap, g, for different core diameters. The optimum gap corresponding to the minimum loss is less than 20 $\mathsf{\mu }$m for fibers with different core diameters and the loss increases slowly when gap further increases. The minimum loss and the corresponding gap, g, are plotted using blue solid curve and red dashed curves, respectively, in Figure 5b. The minimum loss decreases by around one order of magnitude when the core diameter increases from 300 to 500 $\mathsf{\mu }$m. Different from fibers with six cladding tubes, the corresponding optimum gap, g, is much smaller and is always less than 20 $\mathsf{\mu }$m when the core diameter increases from 300 to 500 $\mathsf{\mu }$m in fibers with eight cladding tubes. There is a wide range of gaps that realize low loss in the fibers with eight cladding tubes, as shown in Figure 5a. The loss is less sensitive to the gap in the region between 10 and 50 $\mathsf{\mu }$m. Since the tube diameter is much smaller than the diameter of core, the coupling between the core mode and tube modes is weak. It has been shown that the power ratio in the air region of cladding tubes is always less than $0.1\%$ in the negative curvature fiber with eight cladding tubes, while the power ratio in tube air could be more than $0.8\%$ for fibers with six cladding tubes [26]. In negative curvature fibers with six cladding tubes, a larger gap is needed to remove the weak coupling between the core and cladding tube modes.

4. As${}_2$S${}_3$ Chalcogenide Glass

In this section, we carried out the same loss analysis in negative curvature fibers made with As${}_2$S${}_3$ chalcogenide glass. We use a refractive index of 2.4 and a material loss of 500 dB/m for As${}_2$S${}_3$ chalcogenide glass in our simulations [15,16]. The tube thickness, t, is fixed at 6.1 $\mathsf{\mu }$m corresponding to the third antiresonance. Figure 6a shows the loss as a function of gap, g, when the core diameter increases from 300 to 500 $\mathsf{\mu }$m in As${}_2$S${}_3$ chalcogenide fiber with six cladding tubes. Compared with the loss in Figure 3a, the losses in the fiber using As${}_2$S${}_3$ chalcogenide glass, shown in Figure 6a, are higher and have a flatter minimum. In Figure 6b, we show the minimum loss and the corresponding gap, g, as blue solid curve and red dashed curve, respectively. We also study the fiber leakage loss with and without material loss in an As${}_2$S${}_3$ chalcogenide fiber with six cladding tubes. In Figure 7a, we show the results in order to explain the broad, low-loss region in Figure 6a. The core diameter is fixed at 300 $\mathsf{\mu }$m. The solid curve shows the fiber loss with material loss of 500 dB/m for As${}_2$S${}_3$ chalcogenide glass, which is the same as the blue solid curve in Figure 6a. The dashed curve shows the fiber loss without material loss, which is similar to the curve in Figure 3a. The high material loss of As${}_2$S${}_3$ chalcogenide glass dominates and leads to a flat minimum in the fiber loss curve, as shown by the blue solid curve in Figure 7a.

In order to better illustrate the influence of the material loss on the total fiber loss, we study the fiber loss as a function of material loss both for As${}_2$S${}_3$ chalcogenide glass and As${}_2$Se${}_3$ chalcogenide glass, shown in Figure 7b as the red dashed and blue solid curves, respectively. The core diameter is 300 $\mathsf{\mu }$m and the gap is 60 $\mathsf{\mu }$m. The fiber loss changes little when the material loss increases from 0.1 to 10 dB/m, and the fiber loss is dominated by the confinement loss in the blue region for both curves. The loss of fiber that is made with As${}_2$Se${}_3$ chalcogenide glass is located in the blue region, which is marked with the blue circle on the blue solid curve. The fiber loss begins to increase when the material loss increases from 10 to 10${}^2$ dB/m, and the influence of the material loss becomes visible. When the material loss further increases, the fiber loss increases sharply, and the fiber loss is dominated by the material loss in the red region for both curves, when the material loss is higher than 10${}^2$ dB/m. The loss of fiber made with As${}_2$S${}_3$ chalcogenide glass is located in the red region, which is marked with the red triangle on the red dashed curve. Due to the inhibited coupling between the core mode and glass modes, the power ratios in the glass of negative curvature fibers for the two points marked by circle and triangle in Figure 7b are 0.0016% and 0.002%, respectively. With this low power ratio in glass [33], the fiber leakage loss in negative curvature fibers is more than three orders of magnitude lower than the material loss of glass, as shown in Figure 7b.

Figure 8a shows the loss as a function of gap, g, in As${}_2$S${}_3$ chalcogenide fiber with eight cladding tubes. In Figure 8b, we show the minimum loss and the corresponding gap, g, using a blue solid curve and a red dashed curve, respectively. The minimum loss decreases by less than one order of magnitude and the corresponding optimum gap, g, is always less than 20 $\mathsf{\mu }$m, which agrees with the results in the As${}_2$Se${}_3$ chalcogenide fiber with 8 cladding tubes. Small loss variation near zero gap occurs due to the glass modes existed near the node area between two tubes in Figure 8a.

Chalcogenide negative curvature fibers with eight cladding tubes have been successfully fabricated. The fiber loss was measured to be 2.1 dB/m at 10 $\mathsf{\mu }$m for a fiber with a core diameter of 172 $\mathsf{\mu }$m and a gap of 9 $\mathsf{\mu }$m. Due to the structure distortion during fabrication, the losses of fabricated fibers are two orders of magnitude higher than the losses in simulation, indicating there are room to improve the fabrication [34]. The distortion of the negative curvature fiber structure has an evident impact on the transmission window and the leakage loss [34]. We also observed higher-order modes in the negative curvature fibers [35].

5. Conclusions

In this paper, we optimize the structure of negative curvature fibers for CO${}_2$ laser transmission. We investigate the impact of the size of the gap between cladding tubes on the loss of negative curvature fibers made with As${}_2$Se${}_3$ and As${}_2$S${}_3$ chalcogenide glasses. For As${}_2$Se${}_3$ chalcogenide fibers with six cladding tubes, the minimum loss decreases by an order of magnitude and the corresponding optimum gap, g, increases from 60 to 90 $\mathsf{\mu }$m when the core diameter increases from 300 to 500 $\mathsf{\mu }$m. A greater gap is needed for a fiber with greater core diameter to reduce the coupling between the core mode and tube mode. For a fiber with eight cladding tubes, the optimum gap, g, that corresponds to the minimum loss is always less than 20 $\mathsf{\mu }$m when the core diameter ranges from 300 to 500 $\mathsf{\mu }$m. We also study As${}_2$S${}_3$ chalcogenide fibers, which has a higher material loss at a wavelength of 10.6 $\mathsf{\mu }$m. It is found that material loss dominates the fiber leakage loss. The fiber loss is dominated by the material loss, when the material absorption loss is higher than 10${}^2$ dB/m.