## Abstract

A novel design, two-layer low-index trench fiber with parabolic-profile core, is proposed and investigated numerically in this paper. Based on scalar FD-BPM algorithm, the excellent performance over other types of structures and great potential in mode area enlargement are demonstrated. The effective mode area of our design (D = 100μm) is approximately 890 μm^{2}. Both the high order mode (HOM) suppression and bending resistance of our design are better than that of Multi-Trench Fiber (MTF). The mode loss ratio and effective mode area are independent on the bending radius. Due to the circular symmetry of our proposed configuration design, the bending property is not varied with the changing of bending directions.

© 2014 Optical Society of America

## 1. Introduction

For the past decades, high power fiber lasers and amplifiers have shown the potential to be as highly attractive coherent light sources due to their compact design, high-power capability, output stability, robustness, cost effectiveness, and power scalability [1, 2]. 10kW beam power in continuous wave regime has been reported in a diffraction limited beam in 2010 [3]. However, optical non-linear effects are some of the key limiting factors in power scaling [4]. An effective method for mitigating these non-linear effects is the effective mode-area scaling of the fundamental mode, while maintaining the single-mode output.

In recent years, several fiber structure have been proposed to achieve single-mode operation and large mode area (LMA), such as low-NA step-index fibers [5], chirally-coupled-core (CCC) fibers [6], tailored cladding fibers for HOMs suppression [7], Photonic Band-gap Fibers [4], Photonic Crystal Fibers (PCF) [8], parabolic-profile fiber [9], gain-guided and index anti-guided (GG + IAG) optical fibers [10], Leakage Channel Fibers (LCF) [11] and Multi-Trench Fiber(MTF) [12, 13]. However, the application limits of these fibers are the complex and expensive fabrication and detrimental bending effects [13].

In this paper, a novel all-solid structure, two-layer low-index trench fiber with parabolic-profile core, has been presented. The great potentials of our design in single-mode operation, mode area enlargement and resistance to bending are demonstrated. In the following, the simulation method, Multi-Trench Fiber (MTF), parabolic-profile fiber and our design are introduced orderly.

## 2. Scalar finite-difference beam propagation method (scalar FD-BPM)

A scalar finite-difference beam propagation method (FD-BPM) is used in our simulation, since our structures in this work are low-contrast and some studies have also demonstrated the efficient potential of scalar BPM for the modeling of such structures [14]. FD-BPM algorithm is a most widely used propagation technique for modeling integrated and fiber optic photonic devices. First, the main advantage of BPM algorithm is undoubtedly its ability to handle both the guided and the radiating parts of the field with the same simple formalism [15]. Besides, FD-BPM algorithm is more efficient and stable compared to FFT-BPM [16].

The boundary conditions used in this work is the simple implementation of the transparent boundary condition which effectively lets the radiation pass through the boundary and leave the computational domain. In [17], the simulated HOM suppression of a LCF with high index coating seems to provide a good agreement with the experimental HOM suppression of a double clad LCF with low index coating. In [18], a suitable boundary condition in low-index coated fibers leads to the measured mode losses being very close to the simulations based on infinite cladding. So the transparent boundary condition is accredited.

Bending is modeled here using the equivalent index model [19]. In order to save computation, the equivalent index profile is simplified,

where n_{eff}is the equivalent index profile of the bent fiber, n is the index profile of the unbent fiber, x is the transverse position, and R is the bending radius. This simplification is accurate because of x

_{max}<<R in all our simulation, where x

_{max}is the transverse domain boundary.

Actually, the practical limit of bending radius is about 0.25m. So here the simulated bending radius is fixed at 0.25m if not specially mentioned. And in all our simulations, cladding refractive index n_{c} is 1.444 and wavelength λ is 1.05μm. Given the simulation errors, the too-low mode loss (<5x10^{−5} dB/m) approximates at 5x10^{−5} dB/m.

## 3. Multi-trench fiber (MTF)

Figure 1(a) shows the cross section of the MTF structure and Fig. 1(b) shows the refractive index profiles of the unbent and bent MTF as well as the notations used in this paper, where D is the core diameter, t is the thickness of all the low-index rings (trenches), d is the thickness of the all high-index rings, n_{c} is the cladding index and Δn is the refractive index difference between the core and the trenches.

In the papers [12, 13], the width of the low-index trenches and the index difference between the core and trenches can be used to adjust the leakage loss of the fundamental mode (LP_{01}) and HOMs, but these are slightly effective on the loss ratio between low-loss HOM and LP_{01}. In this paper, the result of the HOM loss (dB/m) divided by the FM loss (dB/m) is defined as the ratio of mode losses, so the ratio unit is just 1. In this paper, the total loss of LP_{01} is tried to adjust below 1 dB/m but it is not fixed. The effects of the width of the low-index trenches and the index difference between the core and trenches are not the point researched here.

Figure 2 shows the performance of an unbent and bent MTF with structural parameters D = 50μm, t = 7μm, Δn = 0.0012 and bending radius R = 0.25m. Figure 2(a) plots the leakage loss of the LP_{01} and HOMs and loss ratio between the low-loss HOM and LP_{01} at a range of the thickness d (4-38μm) in this unbent MTF. Similarly, Fig. 2(b) plots the total loss of LP_{01} and HOMs, including the leakage and bending loss, and loss ratio between the low-loss HOM and LP_{01} in this bent MTF. Figures 2(c) and 2(d) plot the effective index of the corresponding core modes in this unbent and bent MTF, respectively. And the index of the first leaky cladding mode is also plotted in Figs. 2(c) and 2(d), since the effect of first leaky cladding mode is important and representative.

And our unbent MTF results are concordant with those reported in [12]. Compared with Fig. 2(a), the total loss curves in Fig. 2(b) like to be compressed in the range of d, 4-10μm and confused in the range of d, 10-38μm. And the vertical and horizontal polarizations of LP_{11} are separated, named LP_{11v} and LP_{11h} [as shown in Fig. 2(b)] in this paper, respectively. Just near d = 9μm and d = 20μm, the loss ratio between the low-loss HOM and LP_{01} in the bent MTF is more than 30.

The first leakage loss peak of LP_{01} is at d = 30μm, LP_{11} at d = 17μm and LP_{21} at d = 12μm [as shown in Fig. 2(a)], that are near the cross points of the index curves of LP_{01}, LP_{11}, LP_{21} and the first leaky cladding mode [as shown in Fig. 2(c)], respectively. The similar phenomenon is also revealed in Fig. 2(d). These are corresponding to the principle of mode suppression through index-matched mixing with leaky cladding modes reported in [20]. And this principle can help us to understand the MTF performance well.

The bend-induced increase of the index of leaky cladding modes causes the compressions (4μm<d<10μm) and confusions (10μm<d<38μm) of all cross points of the index curves of core modes and leaky cladding modes. So the poor HOM suppression of the bent MTF occurs accordingly. Besides, since the bend-induced birefringence of this MTF causes the separation of LP_{11v} and LP_{11h}, the HOM suppression of the bent MTF becomes worse. Therefore, this MTF is sensitive to bending.

## 4. Parabolic-profile fiber

The parabolic-profile fiber has the bending resistance [9] and slower degradation of the injected beam quality [21]. Figure 3(a) shows index profiles of unbent (blue) and bent (red) parabolic-profile fiber and parameters, where core diameter is D, and index difference of the top of the parabolic profile and cladding is Δn. Figures 3(b)–3(d) plot the total loss of LP_{01} and the low-loss HOM in the unbent and bent parabolic-profile fiber with core diameter D = 50μm, 75μm and 100μm, respectively. Given the simulation errors, the too-low mode loss (<5x10^{−5} dB/m) approximates at 5x10^{−5} dB/m. Obviously, bending the parabolic-profile fiber causes the increase of the total loss of LP_{01} and HOMs and the increase of the loss ratio between low-loss HOM and LP_{01}. These results indicate that the bent parabolic-profile fiber is excellent for HOM suppression. However, in the bent parabolic-profile fibers, even considering the points of the 10dB/m total loss of the low-loss HOM [as marked in yellow in Figs. 3(b)–3(d)], the index difference Δn is 0.0006 at core diameter D = 50μm, 0.00055 at D = 75μm and 0.0005 at D = 100μm, respectively. Since a fiber core NA lower than 0.05 (Δn = 0.0087 at cladding index n_{c} = 1.444) is generally difficult to achieve in conventional manufacturing processes [22], the parabolic-profile fibers with such low index differences are hardly fabricated.

## 5. Two-layer low-index trench fiber with parabolic-prolife core

Based on the aforementioned analyses, a novel design, two-layer low-index trench fiber with parabolic-profile core is presented in this paper. Figures 4(a) and 4(b) display its index profile and the key parameters, respectively. D stands for core diameter, Δn for index difference of trenches and cladding, d for the thickness of the high-index rings, t_{1} for the thickness of the inner low-index trench and t_{2} for the thickness of the outside low-index trench. The inner trench and parabolic core are conjoint, which design is easily fabricated.

In Fig. 5, we present the performance of unbent and bent two trench fiber with parabolic-profile core with structural parameters D = 50μm, t_{1} = 3μm, t_{2} = 6μm, Δn = 0.0012 and bending radius R = 0.25m.

Figure 5(a) plots the leakage loss of LP_{01} and HOMs and loss ratio between the low-loss HOM and LP_{01}, and Fig. 5(c) displays the effective index of core modes and the first leaky cladding mode in the unbent two trench fiber with parabolic-profile core. It is found in Fig. 5(a) that the loss ratio between the low-loss HOM and LP_{01} is more than 30 at 2μm<d<9μm in this unbent design. So the HOM suppression of this unbent design is excellent. Figure 5(b) plots the total loss of LP_{01} and HOMs and loss ratio between the low-loss HOM and LP_{01}, and Fig. 5(d) displays the effective index of core modes and first leaky cladding mode in this bent design. From Fig. 5(b), the loss ratio between the low-loss HOM and LP_{01} is more than 30 at 2μm<d<7μm. Clearly, in this bent design, the compression (d<9μm) and confusion (d>9μm) of total loss curves is weaker than that in the bent MTF above. The HOM suppression is better than that in the bent MTF above.

As shown in Figs. 5(c) and 5(d), the cross points of index curves of core modes and the first leaky cladding mode are also near the loss peaks of the corresponding core modes. Because of the parabolic index core, the effective-index difference of LP_{01} and LP_{11} is about twice that in the bent MTF above and two polarizations of LP_{11}, LP_{11v} and LP_{11h}, are not separated. The performance of our design is evidently improved. The effective mode area of LP_{01} is 446μm^{2} in our unbent design and 448μm^{2} in our bent design, respectively.

## 6. Mode area scaling

For scaling the effective mode area, the simulated core diameter of our design is scaled to 100μm. Figure 6(a) shows the leakage loss of LP_{01}, low-loss HOM and loss ratio between the low-loss HOM and LP_{01} in the unbent design with D = 100μm, t_{1} = 1μm, t_{2} = 3μm and Δn = 0.0012. And Fig. 6(b) shows the corresponding case of bent design with bending radius R = 0.25m. As shown in Fig. 6(a), the performance of our unbent design is very well. From Fig. 6(b), the loss ratio between the low-loss HOM and LP_{01} in this bent design is more than 30 in the range of d, 3-7μm, 16-20μm and 29-33μm, respectively. The range of d with the high loss ratio in our bent design (D = 100μm) is wider than that in the bent MTF (D = 50μm), obviously.

Figure 7 plots the influence of the bending radius and the thickness (d) of the high-index ring on the total loss of the low-loss HOM and LP_{01} in our bent design (D = 100μm, t_{1} = 1μm, t_{2} = 3μm and Δn = 0.0012). From Fig. 7, the HOM suppression and bending resistance at 4μm<d<6μm keep well when the HOM total loss is below 10dB/m (marked by red arrow dotted lines) in our bent design. In consideration of the 3.3dB/m average gain in the final-stage power fiber amplifier [23] and the average gain below 1dB/m in high power fiber lasers [24, 25], this HOM suppression in our design is sufficient.

As a comparison, Fig. 7 also plots the influence of the bending radius on the total loss of the low-loss HOM and LP_{01} in the parabolic-profile fiber (D = 100μm and Δn = 0.0012). As seen in Fig. 7, the simulated HOM-total-loss curves in our design have smaller slopes than the curve in this parabolic-profile fiber. So the bending sensitivity of our design is weaker than that of the parabolic-profile fiber with the same index difference.

Figure 8(a) plots the influence of bending radius on the effective mode area of LP_{01} in our design (D = 100μm) and MTF (D = 50μm). With the decrease of the bending radius, the effective mode area of LP_{01} in our bent design (D = 100μm) which is about 890μm^{2}, keeps constant, while the one in the bent MTF (D = 50μm) reduces. Figure 8(b) displays the LP_{01} surface profiles of our design (D = 100μm) and MTF (D = 50μm). Obviously, the bend-induced LP_{01} distortion of our design is lower than that of MTF. And the bend-induced distortion of fiber modes can severely impact the amplifier performance [26].

## 7. Discussions and conclusions

In this paper, a novel design, two-layer low-index trench fiber with the parabolic-profile core, is proposed and investigated numerically. Based on scalar FD-BPM algorithm, the excellent performance over other types of structures and great potential in mode area enlargement are demonstrated. The range of the thickness d with high loss ratio in our bent design (D = 100μm) is larger than the bent MTF (D = 50μm). The mode loss ratio and effective mode area of our design (d = 6μm, D = 100μm) are 890 μm^{2} and independent on the bending radius. Due to the circular symmetry of our proposed configuration design, the bending property is not varied with the changing of bending directions. Like the parabolic-profile fiber, our design also has the slower degradation of the injected beam quality [21] and little bend-induced distortion of fiber modes. Compared with the parabolic profile fiber, the performance of our design isn’t limited by the index difference of the parabolic profile. Given the fabrication, our design can be fabricated by conventional processes, such as MCVD with careful control, and the rare-earth ions can also be easily doped simultaneously, while other designs, like PCF and LCF, are difficultly fabricated, due to the requirement of stack and draw technique. In conclusion, the HOM suppression and bending resistance of our design are excellent.

## Acknowledgments

I thank Junhua Ji (at Nanyang Technological University), Seongwoo Yoo (at Nanyang Technological University) and Johan Nilsson (at University of Southampton) for all the useful discussions. This work was supported by the Major State Basic Research Development Program of China granted No. 2010CB328206.

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