Geometry of chalcogenide negative curvature fibers for CO 2 laser transmission

We study impact of geometry on leakage loss in negative curvature fibers made with As2Se3 chalcogenide and As2S3 chalcogenide glasses for carbon dioxide (CO2) 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 μm when the core diameter ranges from 300 μm to 500 μm. The influence of material loss on fiber loss is also studied. When material loss exceeds 102 dB/m, it dominates the fiber leakage loss for negative curvature fiber at a wavelength of 10.6 μm.

Step index fibers are commonly used to transmit CO 2 laser light.The nonlinearity in the silica glass limits the transmitted power.Hollow-core fibers have low nonlinearity, because the light is mostly transmitted in air, which does not contribute to the nonlinearity.In addition, 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].Recently, hollow-core negative curvature fibers have drawn a large amount of attention 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 successfully demonstrated using chalcogenide negative curvature fibers for a CO 2 laser at a wavelength of 10.6 µm [13][14][15].Previous study shows that chalcogenide glass should be used for wavelength larger than 4.5 µm [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 the fiber parameters on the 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 number of cladding tubes, the thickness of the 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 separation 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 naturally 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].A larger gap is required to remove the weak coupling between the core mode and tube modes in a fiber with six cladding tubes [26].
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 µm when the core diameter increases for negative curvature fibers with eight cladding tubes when the core diameter ranges from 300 µm to 500 µm.

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 gray regions represent glass, and the white regions represent air.
The inner tube diameter, d tube , the core diameter, D 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: [29].The wavelength of 10.6 µm for a CO 2 laser is used in our simulation.

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.The material loss of 10.6 dB/m for As 2 Se 3 chalcogenide glass is included in the simulation [30].
The tube thickness, t, is fixed at 5.2 µm corresponding to the third antiresonance.A glass thickness corresponding to the third antiresonance has been drawn in the past [15].We first study negative curvature fibers with six cladding tubes.We define d 6max as the maximum possible tube diameter for the fiber with 6 cladding tubes, which equals D core − 2t. Figure 2(a) shows the contour plot of loss as a function of core diameter, D core , and normalized tube diameter, d tube /d 6max .For a fixed D core , the loss decreases and then increases when d tube /d 6max increases from 0.2 to 1.0.The minimum loss occurs when d tube /d 6max = 0.62, and it does not change when D core increases from 300 µm to 500 µm.The loss decreases when D core increases.In addition, we show the loss as a function of the core diameter, D core , and the gap, g, in Fig. 2(b).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].We also plot the loss as a function of gap, g, for  different core diameters in Fig. 3(a).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 Fig. 3(b).When the core diameter increases from 300 µm to 500 µm, the minimum loss decreases by more than one order of magnitude and the corresponding optimum gap, g, increases from 60 µm to 90 µm.Hence, a larger gap is needed for a fiber with a larger core diameter to lower 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.core diameters.The optimum gap corresponding to the minimum loss is less than 20 µm 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 Fig. 5(b).The minimum loss decreases by around one order of magnitude when the core diameter increases from 300 µm to 500 µm.Different from fibers with six cladding tubes, the corresponding optimum gap, g, is much smaller and is always less than 20 µm when the core diameter increases from 300 µm to 500 µm 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 Fig. 5(a).The loss is less sensitive to the gap in the region between 10 µm and 50 µm.Since the tube diameter is much smaller than the diameter of core, the coupling between the core mode and tube modes is weak.A larger gap is needed for fibers with six cladding tubes to remove the weak coupling between the core mode and cladding tube modes in negative curvature fibers with six cladding tubes.

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.The material loss of 500 dB/m for As 2 S 3 chalcogenide glass is included in the simulation [15,16].The tube thickness, t, is fixed at 6.1 µm corresponding to the third antiresonance.
Figure 6(a) shows the loss as a function of gap, g, when the core diameter increases from 300 µm to 500 µm in As 2 S 3 chalcogenide fiber with six cladding tubes.Compared with the loss in Fig. 3(a), the losses in the fiber using As 2 S 3 chalcogenide glass, shown in Fig. 6(a), are higher and have a flatter minimum.In Fig. 6(b), 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 Fig. 7(a), we show the results in order to explain the broad, low-loss region in Fig. 6(a).The core diameter is fixed at 300 µm.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 Fig. 6(a).The dashed curve shows the fiber loss without material loss, which is similar to the curve in Fig. 3(a).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 Fig. 7(a).
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 Fig. 7(b) as the red dashed and blue solid curves, respectively.The core diameter is 300 µm and the gap is 60 µm.The fiber loss changes little when the material loss increases from 0.1 dB/m 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 dB/m 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.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 Fig. 8(a).

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 µm to 90 µm when the core diameter increases from 300 µm to 500 µm.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 µm when the core diameter ranges from 300 µm to 500 µm.We also study As 2 S 3 chalcogenide fibers, which has a higher material loss at a wavelength of 10.

Figure 1 .
Figure 1.Schematic illustration of negative curvature fibers with (a) six and (b) eight cladding tubes.

Figure 2 .
Figure 2. (a) Contour plot of loss as a function of core diameter and normalized tube diameter.(b) Contour plot of loss as a function of core diameter and gap.The number of cladding tube is six.

Figure 3 .
Figure 3. (a) Loss as a function of gap in fibers with different core diameters.(b) Minimum loss and the corresponding optimum gap in fibers with different core diameters.The number of cladding tube is six.

Figure 4 (Figure 4 .
Figure 4(a) shows the contour plot of loss as a function of core diameter, D core , and normalized tube diameter, d tube /d 8max , where d 8max is defined as the maximum possible tube diameter for the fiber with 8 cladding tubes, which is {D core sin(π/8) − 2t[1 − sin(π/8)]}/[1 − sin(π/8)] [31].Figure 4(b)shows the contour plot of loss as a function of core diameter, D core , and gap, g.The minimum loss occurs at a larger value of d tube /d 8max , or a smaller value of g, than is the case for negative curvature fibers with six cladding tubes.In Fig.5(a), we show the loss as a function of the gap, g, for different

Figure 5 .
Figure 5. (a) Loss as a function of the gap in fibers with different core diameters.(b) Minimum loss and the corresponding gap in fibers with different core diameters.The number of cladding tubes is eight.

Figure 8 (Figure 6 .
Figure 8(a) shows the loss as a function of gap, g, in As 2 S 3 chalcogenide fiber with eight cladding tubes.In Fig. 8(b), 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 µm, which agrees with the results in the

Figure 7 .
Figure 7. (a) Loss as a function of gap in fibers with and without material loss.(b) Fiber loss as a function of material loss in As 2 Se 3 chalcogenide glass fiber and As 2 S 3 chalcogenide glass fiber with six cladding tubes, a core diameter of 300 µm, and a gap of 60 µm.

Figure 8 .
Figure 8.(a) Loss as a function of gap in fibers with different core diameters.(b) Minimum loss and corresponding gap in fibers with different core diameters.The number of cladding tube is eight.