A Large Mode Area Parabolic-Proﬁle Core Fiber with Modiﬁed Segmented in Cladding

: In this paper, a novel ﬁber with a parabolic-proﬁle core and eight segmented trenches in cladding is designed. The designed ﬁber consists of the segmented trench of low refractive index in cladding and parabolic-proﬁle of high refractive index in the core. The proposed ﬁber has good bending resistance. The eight segmented trenches in the cladding can decrease the refractive index of cladding to increase the difference index between the core and cladding to limit fundamental mode (FM) loss. The proposed ﬁber can offer an effective single mode (SM) operation with a large mode area (LMA) of 952 µ m 2 at the small bending radius (R = 10 cm). In addition, the ﬁber is also suitable under an 18 W/m heat load. The proposed ﬁber achieves good bending performance, which can be suitable for the compact high-power lasers.


Introduction
Compared with conventional gas and solid-state lasers, high-power fiber lasers have a series of advantages, such as lightweight, high beam quality, corrosion resistance, and low threshold [1][2][3][4]. In recent years, high-power fiber lasers have received much attention from scholars, and their power levels have exceeded the 10-kW class [5,6]. However, nonlinear effects [7,8] and optical damage become more severe with a further increase in power. An effective method commonly used to solve these problems is to use large mode field area fibers [9].
Large mode field area fibers will be bent and sometimes curled in practice due to space constraints. So, large mode field area fibers should be allowed to operate at a moderate bending radius. Moreover, large mode field area fibers must work in a single mode operation to ensure beam propagation quality in high-power lasers. However, the challenge is increasing the mode field area of the fiber and decreasing the loss of FM to maintain single-mode operation [10,11].
In recent years, people have proposed several fiber structures to achieve the requirement of single-mode operation in a large mode field area with good bending robustness, e.g., photonic crystal fibers [12][13][14][15][16][17][18], low numerical aperture (NA) step-index core fibers [19][20][21][22], Bragg fibers [23,24], multi-trench fibers [25][26][27][28][29], and leakage channel fiber [30][31][32][33] etc. However, these fibers have some problems. When photonic crystal fibers are spliced, the air holes in the fibers collapse and the stacking technology makes manufacturing more difficult. Low-NA step-index fibers achieve single-mode operation by reducing the cutoff wavelength. Although increasing the core diameter helps to reduce the NA, it is complicated to produce fibers with a NA of less than 0.06 during practical preparation [34]. Bragg fiber has shown good performance in enlarging the mode field area. However, Figure 1 shows the transverse cross-section of the proposed fiber. The grey region represents the silica material. The refractive index is n 1 . The blue region represents segmented trenches. The refractive index is n 2 and angular width is θ 2 . The angle width between two segmented trenches is θ 1 . The yellow region represents core in fiber and the refractive index is n 3 . The dn is differences in refractive index between refractive index of silica material and low refractive index. The dn1 is differences in refractive index between highest refractive index in the core and refractive index of silica material. The core radius is a, and the cladding radius is 72.5 µm. c is the thickness of optical fiber and perfect matching layer (PML), as shown in Figure 1a. ϕ is the bending orientation. The duty cycle of the segmentation is γ = θ 2 /(θ 1 + θ 2 ). Figure 1b,c show the refractive index along X and X directions, respectively. The incident wavelength is 1.064 µm. The bending radius R is defaulted to 10 cm. Bending orientation θ is fixed as 0 • . We use COMSOL Multiphysics commercial software based on the finite element method to perform numerical simulations. Adding a circular PML outside the cladding for calculating the bending loss, the loss of each mode in the fiber is constant when the thickness of the PML is greater than 5 μm. In this paper, the thickness of the PML is set to 10 μm. The Mesh generation of the proposed fiber as shown in Figure 2. The average unit mass is 0.88, and there are 41,425 degrees of freedom . We use COMSOL Multiphysics commercial software based on the finite element method to perform numerical simulations. Adding a circular PML outside the cladding for calculating the bending loss, the loss of each mode in the fiber is constant when the thickness of the PML is greater than 5 µm. In this paper, the thickness of the PML is set to 10 µm. The Mesh generation of the proposed fiber as shown in Figure 2. The average unit mass is 0.88, and there are 41,425 degrees of freedom. We use COMSOL Multiphysics commercial software based on the finite element method to perform numerical simulations. Adding a circular PML outside the cladding for calculating the bending loss, the loss of each mode in the fiber is constant when the thickness of the PML is greater than 5 µm. In this paper, the thickness of the PML is set to 10 µm. The Mesh generation of the proposed fiber as shown in Figure 2. The average unit mass is 0.88, and there are 41,425 degrees of freedom .

Theory and Structure
To calculate the bending loss of the proposed fiber, the refractive index of bending fiber can be expressed equivalently to the straight fiber structure. The refractive index equivalence equation is as follows [35]: where neq is equivalent refractive index of bending fiber. n(r) is the refractive index of the straight fiber. r is the coordinate axes of fiber. ρ is the elastic optical coefficient, θ is the bending azimuthal angle, and R is the bend radius. In this article, the value of ρ is set as 1.25 according to the literature [36]. The bending loss can be calculated using the following equation [35]: The proposed fiber has the RI in the core distribution of: To calculate the bending loss of the proposed fiber, the refractive index of bending fiber can be expressed equivalently to the straight fiber structure. The refractive index equivalence equation is as follows [35]: where n eq is equivalent refractive index of bending fiber. n(r) is the refractive index of the straight fiber. r is the coordinate axes of fiber. ρ is the elastic optical coefficient, θ is the bending azimuthal angle, and R is the bend radius. In this article, the value of ρ is set as 1.25 according to the literature [36]. The bending loss can be calculated using the following equation [35]: where λ is the incident wavelength, λ = 1064 nm in this paper. The effective mode area (A eff ) reflects the magnitude of the power density inside the fiber. Improving the A eff can reduce the nonlinear effects. A eff is calculated as [37,38]: where E(x,y) represents the fiber transverse electric field component. Because of the quantum defect, the rare earth doped fiber will generate heat load during use. The heat load will influence the equivalent refractive index of the proposed fiber. The relationship between heat load and refractive index can be calculated as [39]: where c is the radius of the fiber coating, β is the thermal optical coefficient, while k si and k ac are the thermal conductivity for the silica and coating material. h 1 is the convective coefficient between the coat and air. Parameters of value are shown by q is the heat load density, which can be expressed as [39]: where Q(z) is the heat load.

Discussion of Proposed Fiber
First, in order to demonstrate the superiority of the PPC fiber to resist bending, the traditional step-index fiber is analyzed to compare it with the proposed fiber. Figure 3 shows the electric field distribution of the proposed fiber and step-index fiber with the structure parameters a = 30 µm, b = 62.5 µm, dn = dn1 = 0.0006, γ = 0.65, λ = 1.064 µm, and R = 10 cm. Figure 3a,b plots the electric mode field of the LP 01 and LP 11v with proposed fiber. Figure 3c,d shows the electric field of the LP 01 and LP 11v with the step-index fiber. Figure 3a plots that the electric mode field of the parabolic core fiber, the mode of LP 01 is a circle. However, the LP 01 mode field of the step-index fiber is clearly compressed near the edge of the core, as shown in Figure 3b. The electric field lines of LP 11v mode have been radiated into the cladding, resulting in a large loss. The LP 01 and LP 11v loss of the parabolic core layer is 0.022 dB/m and 1.963 dB/m, respectively. The effective touch-field area is 751.8481 µm 2 . Figure 3d shows that the electric field lines of LP 11v mode were restricted to the core layer. The bending loss of LP 11v is small. The LP 01 and LP 11v loss of step-core fiber are 0.022 dB/m and 0.021 dB/m, respectively. The effective field area is 788.89 µm 2 . The proposed fiber can realize the single mode operation compared with step-index core fiber at the same bending radius.
To verify the bending performance of the PPC fiber and step-index fiber, the normalized electric field distribution has been analyzed, as shown in Figure 4. The normalized electric field of PPC fiber (red dot) is closer to the center compared with the step-index fiber (blue dot) at bending radius R = 10 cm. Therefore, the PPC fiber has better bending performance than step-index fiber. To verify the bending performance of the PPC fiber and step-index fiber, the normalized electric field distribution has been analyzed, as shown in Figure 4. The normalized electric field of PPC fiber (red dot) is closer to the center compared with the step-index fiber (blue dot) at bending radius R = 10 cm. Therefore, the PPC fiber has better bending performance than step-index fiber.

Effects of Core Radius
Firstly, the influence of core radius on fiber bending performance is analyzed. Figure  5 shows the bending loss, loss ratio, and effective mode area as a function of a. The parameters are: λ = 1064 nm, b = 62.5 µm, dn = dn1 = 0.0006, R = 10 cm, γ = 0.65, θ = 0°. Figure  5a shows that the bending loss of LP01 decreases at first and then increases with core radius increasing. The LP01 loss decreases from 0.0491 dB/m to 0.0136 dB/m with core radius ranging from 15 to 24 µm. With the core radius increasing from 24 to 35, the bending loss of LP01 increases from 0.0136 dB/m to 0.052 dB/m. When the core radius is small, the core  To verify the bending performance of the PPC fiber and step-index fiber, the normalized electric field distribution has been analyzed, as shown in Figure 4. The normalized electric field of PPC fiber (red dot) is closer to the center compared with the step-index fiber (blue dot) at bending radius R = 10 cm. Therefore, the PPC fiber has better bending performance than step-index fiber.

Effects of Core Radius
Firstly, the influence of core radius on fiber bending performance is analyzed. Figure  5 shows the bending loss, loss ratio, and effective mode area as a function of a. The parameters are: λ = 1064 nm, b = 62.5 µm, dn = dn1 = 0.0006, R = 10 cm, γ = 0.65, θ = 0°. Figure  5a shows that the bending loss of LP01 decreases at first and then increases with core radius increasing. The LP01 loss decreases from 0.0491 dB/m to 0.0136 dB/m with core radius ranging from 15 to 24 µm. With the core radius increasing from 24 to 35, the bending loss of LP01 increases from 0.0136 dB/m to 0.052 dB/m. When the core radius is small, the core

Effects of Core Radius
Firstly, the influence of core radius on fiber bending performance is analyzed. Figure 5 shows the bending loss, loss ratio, and effective mode area as a function of a. The parameters are: λ = 1064 nm, b = 62.5 µm, dn = dn1 = 0.0006, R = 10 cm, γ = 0.65, θ = 0 • . Figure 5a shows that the bending loss of LP 01 decreases at first and then increases with core radius increasing. The LP 01 loss decreases from 0.0491 dB/m to 0.0136 dB/m with core radius ranging from 15 to 24 µm. With the core radius increasing from 24 to 35, the bending loss of LP 01 increases from 0.0136 dB/m to 0.052 dB/m. When the core radius is small, the core has weak ability to limit light, which results in a high bending loss. When the core radius increases, the light capacity rises, and the light bending loss decreases. As the core radius continues to increase, it becomes increasingly sensitive to the bending radius. Therefore, the bending loss will increase. Photonics 2022, 9, x FOR PEER REVIEW 6 of 12 has weak ability to limit light, which results in a high bending loss. When the core radius increases, the light capacity rises, and the light bending loss decreases. As the core radius continues to increase, it becomes increasingly sensitive to the bending radius. Therefore, the bending loss will increase.
(a) (b) Figure 5. (a) The bending loss of LP01and LP11 in the core (b) loss ratio and effective mode area as the function of a. Figure 5b shows the loss ratio between the lowest high order modes (HOMs) and the highest FM and the effective mode area. The mode area of LP01 keeps increasing with core radius increasing. When core radius is 15 µm, the effective mode field area is 370.385 µm 2 . When a increases to 35 µm, the effective mode field area can reach 877.56 µm 2 . The overall trend of the loss ratio is down. With core radius ranging from 16 µm to 35 µm, the loss ratio reduces from 866 to 41.

Effects of Number of Segments
The number of segments with low-index cladding will affect the refractive index difference between the core and the cladding layer. The gap between the segments will also affect light energy leakage. Therefore, it is of great significance to analyze the influence of the number of segments in the cladding on the bending performance. The parameters of the fiber are a = 30 µm, λ = 1064 nm, b = 62.5 µm, dn = dn1 = 0.0006, R = 10 cm, γ = 0.65, θ = 45°.The bending loss of LP01 and LP11V fibers and the loss ratio influenced by different numbers of segments are shown in Figure 6a. With the number of segments increasing, the bending loss of LP01 will decrease. When the number is 0, 2, 4, 6, 8, and 10, the bending loss of the LP01 are 11.4687 dB/m, 0.3196 dB/m, 1.5156 dB/m, 0.1306 dB/m, 0.022177 dB/m, and 0.011 dB/m, respectively. The bending loss of LP11v is 120 dB/m, 58.52 dB/m, 0.092 dB/m, 9.4405 dB/m, 1.9636 dB/m, and 0.314 dB/m respectively. When N is 8, the loss ratio will be close to 90, the highest in all points. Figure 6b plots effective mode area with the number of segments in the cladding. The overall trend of effective mode area will decrease with N increasing. When N is 0 and 10, the effective mode areas are 804.35 µm 2 and 751.27 µm 2 , respectively. However, the effective mode area will be without big changes, except N = 0.  Figure 5b shows the loss ratio between the lowest high order modes (HOMs) and the highest FM and the effective mode area. The mode area of LP 01 keeps increasing with core radius increasing. When core radius is 15 µm, the effective mode field area is 370.385 µm 2 . When a increases to 35 µm, the effective mode field area can reach 877.56 µm 2 . The overall trend of the loss ratio is down. With core radius ranging from 16 µm to 35 µm, the loss ratio reduces from 866 to 41.

Effects of Number of Segments
The number of segments with low-index cladding will affect the refractive index difference between the core and the cladding layer. The gap between the segments will also affect light energy leakage. Therefore, it is of great significance to analyze the influence of the number of segments in the cladding on the bending performance. The parameters of the fiber are a = 30 µm, λ = 1064 nm, b = 62.5 µm, dn = dn1 = 0.0006, R = 10 cm, γ = 0.65, θ = 45 • .The bending loss of LP 01 and LP 11V fibers and the loss ratio influenced by different numbers of segments are shown in Figure 6a. With the number of segments increasing, the bending loss of LP 01 will decrease. When the number is 0, 2, 4, 6, 8, and 10, the bending loss of the LP 01 are 11.4687 dB/m, 0.3196 dB/m, 1.5156 dB/m, 0.1306 dB/m, 0.022177 dB/m, and 0.011 dB/m, respectively. The bending loss of LP 11v is 120 dB/m, 58.52 dB/m, 0.092 dB/m, 9.4405 dB/m, 1.9636 dB/m, and 0.314 dB/m respectively. When N is 8, the loss ratio will be close to 90, the highest in all points. Figure 6b plots effective mode area with the number of segments in the cladding. The overall trend of effective mode area will decrease with N increasing. When N is 0 and 10, the effective mode areas are 804.35 µm 2 and 751.27 µm 2 , respectively. However, the effective mode area will be without big changes, except N = 0. Figure 6c plots electric mode lines with the number of segments in the cladding. With N increasing, the refractive index of cladding will decrease, leading to the increase of the effective refractive index difference between core and cladding. The enhancement of the optical beam ability of the fiber core results in the reduction of the loss of LP 01 and LP 11V and the effective mode area. We suggest a PPC-SCF that consists of 8 periods of segmentation with a = 30 µm, b = 62.5 µm, dn = dn1 = 0.0006, and γ = 0.65, λ = 1.064 µm at R = 10 cm.

Effects of Bending Angle
The influence of bending angle on the bending loss, the effective mode area, and the electric mode lines of the proposed fiber are displayed in Figure 7. Due to the asymmetric structure of the fiber, the bending performance of the proposed fiber in different orientations from 0 • to 45 • needs to be analyzed. The parameters are: λ = 1064 nm, a = 30 µm, b = 62.5 µm, dn = dn1 = 0.0006, R = 10 cm, γ = 0.65, and R = 10 cm. From Figure 7a, although the bending loss of fiber LP 01 mode (FM) firstly increases and then decreases, it remains lower than 0.1 dB/m with the bending angle varying from 0 • to 45 • . The lowest bending loss LP 11v first decreases and then increases. When the bending angle is 23 • , the bending loss of LP 11v is lower than 1dB/m, which cannot be effective for single-mode operation. In a bending period, the bending angle from 13 • to 32 • cannot be transferred into a single mode. The bending angle range of the fiber has been greatly improved compared with N = 4 and 6. As Figure 7b shows, the effective mode area will remain unchanged with the increase of bending angle. The effective mode area is 751.85 µm 2 .
N increasing, the refractive index of cladding will decrease, leading to the increase of the effective refractive index difference between core and cladding. The enhancement of the optical beam ability of the fiber core results in the reduction of the loss of LP01 and LP11V and the effective mode area. We suggest a PPC-SCF that consists of 8 periods of segmentation with a = 30 µm, b = 62.5 µm, dn = dn1 = 0.0006, and γ = 0.65, λ = 1.064 µm at R = 10 cm.

Effects of Bending Angle
The influence of bending angle on the bending loss, the effective mode area, and the electric mode lines of the proposed fiber are displayed in Figure 7. Due to the asymmetric structure of the fiber, the bending performance of the proposed fiber in different orientations from 0° to 45° needs to be analyzed. The parameters are: λ = 1064 nm, a = 30 µm, b = 62.5 µm, dn = dn1 = 0.0006, R = 10 cm, γ = 0.65, and R = 10 cm. From Figure 7a, although the bending loss of fiber LP01 mode (FM) firstly increases and then decreases, it remains lower than 0.1 dB/m with the bending angle varying from 0° to 45°. The lowest bending loss LP11v first decreases and then increases. When the bending angle is 23°, the bending loss of LP11v is lower than 1dB/m, which cannot be effective for single-mode operation. In a bending period, the bending angle from 13° to 32° cannot be transferred into a single mode. The bending angle range of the fiber has been greatly improved compared with N = 4 and 6. As Figure 7b shows, the effective mode area will remain unchanged with the increase of bending angle. The effective mode area is 751.85 µm 2 .  Figure 8 shows the electric field lines with different bending angles. Combing Figures  7a and 8, we can see that when the bending angle increases to 22.5°, the LP11v mode is restricted to the fiber core. Compared with other bending angles, the electric field line radiated into the cladding is significantly reduced, resulting in a loss of less than 1 dB/m.

Effects of Duty Cycle
The influence of the duty cycle is shown in Figure 9. The parameters are: λ = 1064 nm, a = 30 µm, b = 62.5 µm, dn = dn1 = 0.0006, R = 10 cm, and θ = 0°. Figure 9a plots the effects of γ on the bending loss of the structure. The bending loss of LP01, LP11v, and LP11h

Effects of Duty Cycle
The influence of the duty cycle is shown in Figure 9. The parameters are: λ = 1064 nm, a = 30 µm, b = 62.5 µm, dn = dn1 = 0.0006, R = 10 cm, and θ = 0 • . Figure 9a plots the effects of γ on the bending loss of the structure. The bending loss of LP 01 , LP 11v , and LP 11h decreases sharply with the value of γ increasing. Because of the increasing value of γ, the index contrast between the core and cladding enhances. The proposed fiber can filter the higher order modes to realize single mode operation within 0. 5 < γ < 0.7.

Effects of Duty Cycle
The influence of the duty cycle is shown in Figure 9. The parameters are: λ = 1064 nm, a = 30 µm, b = 62.5 µm, dn = dn1 = 0.0006, R = 10 cm, and θ = 0°. Figure 9a plots the effects of γ on the bending loss of the structure. The bending loss of LP01, LP11v, and LP11h decreases sharply with the value of γ increasing. Because of the increasing value of γ, the index contrast between the core and cladding enhances. The proposed fiber can filter the higher order modes to realize single mode operation within 0. 5 < γ < 0.7.   Figure 9c shows the electric field lines with the duty cycle γ. Combing Figure 9a-c, it can be seen that with the increase of γ, the electric field line is gradually limited to the core, reducing the loss of each order mode and effective mode area.
In order to further increase the effective mode area, the core radius of the proposed is scaled to 38 µm, ensuring the cladding does not change. The parameters are: λ = 1064nm, a = 38µm, b = 62.5µm, dn = dn1 = 0.0006, R = 10 cm, θ = 0°. The bending loss of LP01, LP11v, and LP11h is 0.095 dB/m, 2.66 dB/m, and 6.98 dB/m, respectively. The proposed fiber still can realize the single mode operation. Moreover, the effective mode area can reach up to 952 µm 2 at a 10 cm bending radius.   Figure 9c shows the electric field lines with the duty cycle γ. Combing Figure 9a-c, it can be seen that with the increase of γ, the electric field line is gradually limited to the core, reducing the loss of each order mode and effective mode area.
In order to further increase the effective mode area, the core radius of the proposed is scaled to 38 µm, ensuring the cladding does not change. The parameters are: λ = 1064 nm, a = 38 µm, b = 62.5 µm, dn = dn1 = 0.0006, R = 10 cm, θ = 0 • . The bending loss of LP 01 , LP 11v , and LP 11h is 0.095 dB/m, 2.66 dB/m, and 6.98 dB/m, respectively. The proposed fiber still can realize the single mode operation. Moreover, the effective mode area can reach up to 952 µm 2 at a 10 cm bending radius. Figure 10 plots the bending loss and effective mode area of LP 01 , LP 11v , and LP 11h as a function of heat load. As can be seen from Figure 10, the bending loss of LP 01 , LP 11v , and LP 11h will decrease with heat load rising. The bending loss of LP 01 and LP 11v reduces from 0.095 dB/m to 0.031 dB/m, 2.66 dB/m to 0.98 dB/m with the heat load ranging from 0 W/m to 18 W/m, respectively. Therefore, the proposed fiber cannot realize single mode operation with heat load = 18 W/m. The effective mode area also diminishes with the heat load increasing. The effective mode field area decreased from 952.24 µm 2 to 929.69 µm 2 . Figure 9b plots effective mode area and loss ratio as a function of γ. The effective mode area decreases as the value of γ increases. With γ ranging from 0.2 to 0.75, the effec tive mode area decreases from 775.21 µm 2 to 748.58 µm 2 . Figure 9c shows the electric field lines with the duty cycle γ. Combing Figure 9a-c, it can be seen that with the increase o γ, the electric field line is gradually limited to the core, reducing the loss of each orde mode and effective mode area.
In order to further increase the effective mode area, the core radius of the proposed is scaled to 38 µm, ensuring the cladding does not change. The parameters are: λ = 1064nm, a = 38µm, b = 62.5µm, dn = dn1 = 0.0006, R = 10 cm, θ = 0°. The bending loss o LP01, LP11v, and LP11h is 0.095 dB/m, 2.66 dB/m, and 6.98 dB/m, respectively. The proposed fiber still can realize the single mode operation. Moreover, the effective mode area can reach up to 952 µm 2 at a 10 cm bending radius. Figure 10 plots the bending loss and effective mode area of LP01, LP11v, and LP11h as a function of heat load. As can be seen from Figure 10, the bending loss of LP01, LP11v, and LP11h will decrease with heat load rising. The bending loss of LP01 and LP11v reduces from 0.095 dB/m to 0.031 dB/m, 2.66 dB/m to 0.98 dB/m with the heat load ranging from 0 W/m to 18 W/m, respectively. Therefore, the proposed fiber cannot realize single mode opera tion with heat load = 18 W/m. The effective mode area also diminishes with the heat load increasing. The effective mode field area decreased from 952.24 µm 2 to 929.69 µm 2 .

Conclusions
In summary, this paper proposes a novel fiber design. The proposed fiber is analyzed by the FEM. The significant advantages of the proposed fiber over step-index fiber are demonstrated. For the proposed design, the mode area can scale to 952 µm 2 with effective SM operation at a 10-cm bending radius. The proposed fiber achieves certain advantages of practical fabrication, which can be suitable for compact high-power lasers.