Bend-resistant leaky multi-trench fiber with large mode area and single-mode operation

A novel structure of modified multi-trench fiber (MTF) with characteristics of bend-resistance and large mode-area is proposed. In this structure, each low refractive-index trench of traditional MTF is broken by two gaps up and down. Numerical investigations show that the mode field area of 840 μm2 can be achieved with effective single-mode (SM) operation when the bending radius is 15 cm. Moreover, the high order mode (HOM) suppression of the proposed design is better than that of standard MTF. The SM operation property can be enhanced with the decreases of bending radius. The proposed design shows great potential in high power fiber lasers with compact structure.


Introduction
Over the last decades, high power fiber lasers have developed rapidly due to their beam quality, heat dissipation, brightness, operating costs and efficiency [1][2][3]. However, with the further increase of output power, the nonlinear effect of fiber becomes the most important challenge. To eliminate the challenges induced by high power output, large mode area (LMA) fibers have become the preferred choice.
Rod MTFs can achieve large mode area and excellent high-order modes (HOMs) suppression capability [19]. However, when MTF is bent, the mode region must be less than 800 μm 2 in order to maintain the HOMs suppression capability. The mode area is about 410 μm 2 with a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 30 μm core diameter when bending radius is 20 cm. Sun et al. broke gaps on two outer trenches to improve the bending performance [21,22].
In this paper, we demonstrate that all trenches broken MTF can improve the SM operation outstandingly. The gap width can be adjusted to control the leakage losses of the fiber. The loss ratio between lowest-HOM and fundamental mode (FM) is more than 300 with the mode area of 840 μm 2 under bending radius of 15 cm. The propagation characteristics with different fiber parameters are also discussed in detail.

Optical fiber structure and theoretical model
The proposed modified MTF structure is shown in Fig 1. The leaky-MTF can be fabricated by carving grooves in MTF and inserting rods into these grooves [21,22]. The gray region represents the low refractive-index (RI) of n 2 = 1.444 at the wavelength of 1064 nm. The yellow region represents the high RI (n 1 ). The notations are also shown in Fig 1, where a stands for core radius. t 1 , t 2 and t 3 are the thickness of low RI trenches, respectively. d 1 and d 2 are the thickness of high RI rings, respectively. Δn = n 1 -n 2 is the RI difference between the core and the low RI trenches. t g1 , t g2 and t g3 are the gap width, respectively. F is the bending angle between the actual bending orientation and the reference bending orientation (AA'). It should be noted that, when we mention t, it represents all the low RI trenches (t 1 , t 2 and t 3 ). For example, when t = 3 μm, it represents t 1 = t 2 = t 3 = 3 μm. For t ranges from 3-9 μm, it denotes t 1 , t 2 and t 3 range from 3-9 μm simultaneously. Similarily, d represents d 1 and d 2 ; t gap represents t g1 , t g2 and t g3 .
The finite element method (FEM) is used in complex fiber structure analysis due to its high calculation precision. It is the most commonly used method in microstructure optical fiber simulation. The numerical simulations are calculated by using COMSOL Multiphysics software based on FEM, together with anisotropic perfectly matched layers (PMLs). For the proposed theoretical analysis of leaky-MTF, a 20-μm-thick circular PML is set outside the fiber cladding. Bending has an effect on the RI distribution of silica optical fiber. The bent fiber can be equivalent to a straight fiber through a proper mathematical transformation. After being modified with additional stress perturbations, the bent fiber RI distribution n 0 (x,y) can be expressed as [23,24]: where n(x,y) is the initial RI distribution of straight fiber, R is bending radius, F is the bending orientation angle (as shown in Fig 1) and ρ (here fixed to 1.25) is correction coefficient taking account of the stress factor. Bending loss and mode area A eff can be calculated by the following equations [25,26]: Imðn eff Þ ð2Þ Where n eff is the effective RI of modes, E is the electric field inside the fiber and λ is the operation wavelength, which is set as 1064 nm in this paper. Defaultly, the reference bending direction is AA' (F = 0˚), if not specially mention. In practical applications, FM loss less than 0.1 dB/m and HOMs loss more than 1 dB/m is considered as the basic condition of effective SM operation [27]. The loss ratio (LR) means ratio between lowest-HOM and FM in fiber, which is defined as Where Loss (lowest-HOM) is the loss of lowest-HOM and Loss (FM) is the loss of fundamental mode. In this paper, Loss(LP 01 ) refers to the loss of LP 01 mode, Loss(LP 11v ) refers to the loss of LP 11v mode, Loss(LP 11h ) refers to the loss of LP 11h mode.

Method verification
In order to confirm the accuracy of the simulation method and contrast the difference with our design, we simulate the standard MTF firstly. The standard MTF with structural parameters a = 15 μm, t = 2 μm, d = 8 μm and Δn = 0.005 as that in [18] is simulated at wavelength of 1064 nm. Fig 2(A) shows the numerically simulated losses of the FM and HOMs of standard MTF as a function of bending radius. Fig 2(B) shows the simulated A eff of the FM. The inset pictures show the simulated normalized electric field of LP 01 , LP 11v, LP 11h and LP 11like mode at a bending radius of 17 cm, 29 cm, 21 cm and 11 cm, respectively. When bending radius ranges from 10 cm to 14 cm, the lowest-HOM is LP 11like mode. When bending radius ranges from 14 cm to 30 cm, the lowest-HOM is LP 11 mode. The A eff remains larger than 400 μm 2 when bending radius ranges from 10-30 cm. The computed data are similar to the results given in Ref. 18. Therefore, the accuracy of the simulation method in this paper is reliable.
It can be seen from Fig 2(A) that, LP 11v and LP 11h mode are separated because of the birefringence caused by bending. LP 11h mode suffers more leakage loss than LP 11v mode. We propose leaky-MTF by breaking two gaps up and down on each low RI trench, as shown in Fig 1. The effects of gaps are shown in the following chapters. The A eff is shown in Fig 3(B). The losses of LP 01 and LP 11v mode increase when t gap enlarges. The loss of LP 11h mode remains stable when t gap changes from 0-6 μm but increases when t gap is larger than 6 μm. To illustrate this case, the contour line graphs of the mode field distribution of three modes with t gap = 0 μm, 4 μm, 6 μm, 7 μm and 8 μm are shown in Fig 4. LP 11h mode remains stable when t gap changes from 0-6 μm because the mode distribution is far from gap. At t gap = 4 μm, the mode leakage of LP 11v mode is equal to LP 11h mode (Loss(LP 11v ) = Loss(LP 11h )). It is the first peak value for lowest-HOMs. When t gap is larger than 6 μm, the mode distribution is close to gap and the leakage of LP 11h mode increase. However, At t gap = 7 μm, Loss(LP 11h ) has the peak value because the leakage and resonance reach the largest. It is the second peak value for lowest-HOMs. Corresponding with lowest-HOMs, it has optimum values at t gap = 4 μm and 7 μm.

Effects of gaps
For standard MTF (t gap = 0), the loss of LP 01 , LP 11v and LP 11h mode is 0.004, 0.56 and 3.8 dB/m, respectively. For t gap = 4 μm, the loss of LP 01 , LP 11v and LP 11h mode is 0.01, 3.6 and 3.9     dB/m, respectively. The LR increases by 150% from 139 to 350. For t gap = 7 μm, the loss of LP 01 , LP 11v and LP 11h mode is 0.15, 92 and 57 dB/m, respectively. The LR increases by 170% from 139 to 377. The introduction of gaps has an excellent improvement for SM operation. As shown in Fig 3, to achieve a high differential loss factor, a proper gap width is necessary. As the t gap increases, the A eff increases from 390 μm 2 to 460 μm 2 .
In order to further enlarge the effective mode area, we choose the parameters of fiber as a = 25 μm, t = 6 μm, d = 10 μm, t gap = 18 μm, Δn = 0.007 and R = 15 cm. The effects of various parameters of the structure are studied and summarized in Figs 5-17. Fig 5(A) shows the effect of the gap width on the loss of LP 01 , LP 11v and LP 11h modes and LR. Fig 5(B) shows the effective mode are of FM. The variation trend of loss curves and LR curve are similar to Fig 3. The loss of LP 01 and LP 11v mode increase with the increase of gaps. The loss of LP 11h mode keeps stable at first, and then increases. It can be seen from Fig 5(A) that, the LR has two peak values at t gap % 4 μm and t gap % 18 μm. For standard MTF (t gap = 0), the loss of LP 01 , LP 11v and LP 11h mode are 1.5×10 −11 , 1.2×10 −10 and 5×10 −9 dB/m, respectively. The LR is 8.4 and the mode area It is clear that, gap can enlarge the core modes' leakage losses thus it allows short fiber length to trip off HOMs. Meanwhile, LR can be tuned by adjusting to the gap width. By considering the largest loss of LP 11 mode and LR, we choose t gap = 18 μm. Fig 6 shows the variation of bending losses of the first three modes of fiber (LP 01 , LP 11v and LP 11h ) and A eff of FM on core radius (a) of the proposed structure. When core radius increases, the losses of LP 01 , LP 11v and LP 11h mode decrease, while A eff increases. By considering the trade-off between bending loss and mode area, we choose a = 25 μm to achieve both LMA and  Fig 7 shows the effect of low RI trench thickness (t) on SM operation and A eff of FM of the structure. It can be observed that the losses of LP 11v and LP 11h mode decrease slowly when t increases and the LP 01 mode decreases sharply. The A eff of FM also decreases when t increases. When t is in the range from 3.5 to 9 μm, the highest bending loss of LP 01 mode is lower than 0.1 dB/m and the lowest bending loss for LP 11 mode is higher than 4 dB/m, which is considered viable for SM operation. Meanwhile, the A eff is larger than 850 μm 2 . mode and A eff of FM have a slight increase when d increases. When d is in the range from 7 to 13 μm, the highest loss of LP 01 mode is lower than 0.02 dB/m and the lowest loss of LP 11 mode is large than 13 dB/m. Meanwhile, the A eff ranges from 866 to 972 μm 2 .

Effects of RI difference
The effects of the RI difference (Δn) are shown in Fig 9. The structural parameters are a = 25 μm, t gap = 18 μm, t = 6 μm, d = 10 μm and R = 15 cm. Fig 9 illustrates leakage losses of LP 01 , LP 11v and LP 11h mode and A eff of FM. From Fig 9, it can be seen that Δn has a slight effect on LP 01 , LP 11v and LP 11h mode and a serious effect on A eff of FM. When Δn is in the range from 0.2 to 0.7, the highest loss of LP 01 mode is lower than 0.06 dB/m and the lowest bending loss for LP 11 mode is higher than 17 dB/m. It is considered that the bending loss conforms effective SM operation. The A eff decreases from 925 to 869 μm 2 .

Effects of wavelength
The effect of the wavelength (λ) have been investigated and presented in Fig 10. The A eff of FM and bending losses of LP 01 and LP 11v modes increase with the increases of λ. When λ is in the Large mode area fiber with single-mode operation range from 1 to 1.7 μm, the highest loss of LP 01 mode is lower than 0.2 dB/m and the lowest bending loss of LP 11 mode is higher than 12 dB/m. The A eff of FM increases from 822 to 1157 μm 2 . The fiber can achieve effective SM operation and LMA in a wide transmission bandwidth from 1 μm to 1.7 μm.

Effects of bending
The effects of bending are shown in Figs 11-13. Fig 11 illustrates the leakage losses of LP 01 , LP 11v and LP 11h and mode area of FM at varying bending radius. The structural parameters are a = 25 μm, t = 6 μm, d = 10 μm, t gap = 18 μm, Δn = 0.007 and F = 0˚. When the bending radius ranges from 10 cm to 80 cm, the loss of LP 01 is less than 0.05 dB/m and the loss of LP 11 is larger than 5 dB/m. The A eff ranges from 700 to 1130 μm 2 . Fig 12 shows the contour line graphs of the mode field distributions of LP 01 , LP 11v and LP 11h modes of the leaky-MTF with R = 80 cm, 15 cm and 5 cm. Loss (LP 11h ) increases when R rangs from 80 cm to 5 cm. However, when the bending radius decreases, the losses of LP 01 and LP 11v mode decrease. When R = 80 cm, the LP 01 mode leaks from the gaps. When R = 5 cm, the mode distribution concentrates to the right side and the leakage loss reduced. So Loss (LP 01 ) decreases with reducing bending radius. It is the same for Loss (LP 11v ). When R = 80 cm, LP 11v mode suffers large loss because the mode leaks from gaps. LP 11v mode's loss leaks a lot to the fiber cladding. When R decreases, at R = 15 cm, the leaked energy decreases. When R = 5 cm, the mode moves to the right and less energy leaks to the cladding.
Since the gap breaks the circular symmetry of MTF, the discussion of bending direction is necessary. Fig 13 illustrates leakage losses of LP 01 LP 11v and LP 11h modes and mode area of FM at varying bending orientation. When the bending orientation increases from 0 to 45˚, Loss (LP 01 ) and Loss (LP 11v ) increases. However, Loss (LP 11h ) decreases and reaches a minimum value at F = 35˚. The inset picture shows the field distribution of LP 11h mode at F = 35˚. The mode area of FM has an obvious increase when bending orientation ranges from 0 to 30˚. Moreover, the effect of F on the performance is analyzed further, because F is a critical parameter when the fiber is bended.

Joint effects of gap and bending orientation
The joint effects of bending orientation and gap width are shown in Fig 14. The loss of FM, lowest-HOM, LR and mode area are plotted in Fig 14A-14D, respectively. Fig 14(A) shows that, the loss of FM increases with the increase of bending orientation and gap width. Fig 14  (B) shows that, the lowest loss of HOMs increases with the increases of gap width. However, the lowest loss of HOMs increases when F ranges from 0 to 10˚. Then it decreases when F ranges from 10˚to 30˚. It is because that, Loss(LP 11v ) is smaller than Loss(LP 11h ) with small bending orientation, while larger than Loss(LP 11h ) with large bending orientation. The lowest loss of HOMs increases and then decreases with bending orientation enlarges. It is obvious from Fig 13(C) that, the LR is large when the bending orientation is small. The LR increases with the increases of gap width. However, it has a maximum value with a proper gap width (at t gap = 18 μm). The characteristic is corresponding with the conclusion obtained from Fig 4. It can be seen that, when F is less than 10˚and t gap ranges from 15 to 20 μm, the LR is larger than 100, which indicates that it conforms SM operation conditions. The A eff increases when t gap and F increase. It is because the leakage of FM is easier with greater bending and larger gap.

Effects of one parameter changes
The discussion above of parameter (t) is assumed the thickness of low RI trenches change synchronously. So does the thickness of high RI rings (d) and the gap width (t gap ). Next, we discuss Large mode area fiber with single-mode operation the cases when there is only one parameter changing. Assume the fiber parameters are: a = 25 μm, t = 6 μm, d = 10 μm, t gap = 18 μm, Δn = 0.007 and R = 15 cm. Fig 15A-15C show the losses of LP 01 , LP 11v and LP 11h modes and the mode of FM with different low RI trench thickness t 1 , t 2 and t 3 , respectively. It can be seen that, t 1 , t 2 and t 3 has a small influence on the losses of LP 01 , LP 11v and LP 11h modes. t 1 has a big influence on A eff of FM. However, t 2 and t 3 has a small influence on A eff . The effect of one low RI trench is weak.   Fig 17A-17C show the losses of LP 01 , LP 11v and LP 11h modes and the mode are of FM with different gap width t g1 , t g2 and t g3 , respectively. t g1 , t g2 and t g3 have small influence on the losses of LP 01 , LP 11v and LP 11h modes and A eff of FM. t g1 has a big influence on A eff of FM. However, t g2 and t g3 have small influence on A eff . The effect of one gap width is weak.
Numerical analysis results demonstrate that when there is only one parameter changing, the fiber confirms to single mode operation.

Conclusion
A novel design of multi-trench fiber (MTF) with gaps is proposed and investigated in this paper. This fiber shows more excellent single-mode (SM) operation than standard MTF. The fiber can achieve mode area of 840 μm 2 with high loss ratio (>300) under a tight bending radius of 15 cm. It has a special character that leakage loss decreases with the decreases of bending radius. The fiber can achieve better SM operation with smaller bending radius. The fiber performance is resistant to one parameter variation that, it allows small errors during Large mode area fiber with single-mode operation Large mode area fiber with single-mode operation practical fabrication. This design shows the potential of mode field scaling and makes a contribution to compact high power fiber lasers.
Our present work is based on theoretical analysis. We are currently fabricating this fiber and the actual performance will be tested in the near future.