1 Introduction

Recently, the development of high power fibre lasers has been so rapid that the maximum output power of a single mode continuous wave laser has reached 10 kW with diffraction-limited beam quality (Nilsson 2011). It have numerous advantages compare with solid state lasers and semiconductor lasers such as thermal management, easy manufacturing, high reliability, robust, scalable output power, and high beam quality. However, a number of issues, such as mode instability, substantial thermal damage and nonlinear effects, have an impact on the beam quality and output power for increasing power level. The fibre nonlinear effect turns out to be the most significant difficulty as laser output power increases (Agrawal 2000). Large mode area (LMA) fibres have drawn a lot of interest because of its potential to address high power’s problems which reduce the nonlinear effects. Further, to maximize the effective mode-area for lasing application, single-mode condition must be used to obtain effective beam quality. The potential applications of LMA fiber lasers are optical communication, industrial, medical and spectroscopy (Jeong et al. 2004; Richardson 2010; Limpert et al. 2002). Numerous LMA fibre lasers have recently been produced and put through numerical analysis. It can be obtained by enlarging the core, but several modes propagate in the fibre and also reduce the numerical aperture (NA) which very difficult to maintain SM operation. LMA fibre bends in order to get compact fibre in practical situation, which reduces mode area and beam quality. In 2015, LMA fibre with minimum NA was proposed by Jain et al. and manufactured by the modified chemical vapour deposition method (MCVD) with core diameter 35 μm. The bending characteristics were examined using a 16 cm bending radius to obtain a modal area of 700 µm2 (Jain et al. 2015). Further, photonic crystal fibre also (PCF) is supported to maximum LMA with SM operation for alternative to conventional step index fibres. For instance, negligible bending loss has been found for PCF with huge core diameters of 100 μm (Brooks and Di Teodoro 2006). But at practical bending radius of 5–10 cm, mode-area declines, which is the constraint for LMA PCF designs (Wang et al. 2013). A substitute for PCF that prevents mode-area shrinking is leakage channel fibre (LCF). LCFs are made of one or a few low-index rods in cladding. By optimizing the pitch and diameter of low index rods in LCF, which provide flexibility in managing the confinement loss for fundamental mode and higher order modes for attaining endlessly single mode operation. Initially, LCF developed with 100 μm core diameter by Linag dong and obtained mode area of > 1000 µm2at 6.7 cm bending radius (Dong et al. 2007). However, the big air holes cause a variety of practical issues such as air holes collapsed and heat dissipation. To overcome above mentioned issues, LCFs were changed to have an all-glass structure with fluorine-doped rods in place of the air holes. Nevertheless, it is challenging to bend the all-glass with fluorine-doped rods LCF into a tight bending radius and also weak confinement capability than air holes. In 2014, LCF with microstructure cladding air holes designed for obtaining fundamental mode mode area of 1400 µm2and bend loss of < 3 dB/turn at the bending radius of 20 cm (Saitoh et al. 2011). However, due to a rather high device size, the intended bend radius of 20 cm is not optimal for compact laser systems.

Additionally, single and multi-trench fibres (MTF) are becoming more popular for obtaining SM condition and LMA at the 15–20 cm bending radii which can be fabricated by MCVD technique (Jain et al. 2014). Trench-assisted designs have the issue of satisfying endlessly single mode operation by effectively suppressing higher order modes. The main strategy to solve this problem is resonant coupling. It means higher order modes linked into resonate ring or cladding mode at particular thickness of resonant ring. So, higher order modes loss is high for tight bending condition (Jain et al. 2014; Jain et al. 2013). The bending orientation influences the coupling of higher order modes into the cladding ring to some extent. In 2016, multi trench (MT) in core LMA fiber designed and bending performances analyzed at 15 cm bending radius (Wang et al. 2015). At a relaxed bending condition of 20 cm, the mode-area of the gap-MTF is limited to 920 µm2, and the reported differential bending loss ratio is 100 (Sun et al. 2015). These designs are not appropriate for the 7.5 cm practical packaging radius. For such fibre designs, the effective mode-area drastically decreases and difficult to maintain SM condition at the compact bending condition. In 2019, asymmetric clad MTF with high index arc proposed and attained mode area of 1300 µm2at the compact bending radii of 7.5 cm (Kurade et al. 2019). But, it is very difficult to fabricate high index arc in the trench region. In addition, chirally coupled core (CCC) fibres (Towe 2017), gain-guided and index anti-guided (GG + IAG) optical fibres (Hageman 2010), segmented cladding fibres (SCF) (Millo et al. 2007), and multilayer cladding fibres (Kumar and Rastogi 2007)are used to accomplish LMA with single-mode operation. SCFs stand out among these well-designed fibres due to their SM operation over a wide range of wavelengths and their special skills of LMA and higher order modes suppression at the compact bending radius. SCF is made up of an extensive high index core and alternative arrangement of low and high index in cladding. It is supported to attenuate higher order modes with large mode field region at short range of length. Further, it is overcome the problems of conventional LMA fiber with better beam quality (Rastogi and Chiang 2004). In 2020, Guanli Wang et al., proposed pixelated trench assistant SCF for obtaining mode area of 914 µm2 at relaxed bending radius of 20 cm (Wang et al. 2020). Further, resonant ring assisted LMA SCF designed with high index ring in the core region and obtained mode area of 835 µm2 at lasing wavelength of 1.06 μm (Guo et al. 2021).

In this paper, we have proposed numerical analysis of SCF with four layer of high index circular multi trench in the core by finite element method. According to duty cycle, bending orientation angle, high index circular multi-trenches, low index trenches, the suggested fiber’s bending loss and mode area performance are examined. At compact bending radius of 7.5 cm, minimum bending loss of 0.051 dB/m and mode area of 1006 µm2 for mode is attained with lasing wavelength of 1.06 μm. The paper is organized as follows; Sects. 2, 3 and 4 described proposed model design and characteristics, properties of proposed LMA fiber and analysis of numerical bending characteristics by varying structural parameters (bending loss, mode area and loss ratio).

2 Design and modal characteristics

The cross sectional view of LMA fiber with high index circular multi trench segmented cladding fiber (MTSCF) and refractive index profile is shown in Fig. 1a and b, respectively. The proposed fiber is contained three regions. The dark brown regions indicates the pure silica layer whose refractive index (nL) is n = 1.45. The outer region radius is Rd = 200 μm, which includes 45 μm of segmented cladding region. Two low index trenches (nL) are created around the circular core region, namely M1 and M3, which made by pure silica material for reducing fabrication difficulty. The light brown region is represented as M2 and M4 region with high refractive index region (nc) whose RI = 1.451. It is worked as a resonant ring for leaking higher order modes in the segmented cladding region with help of resonant coupling. The four rings of circular MT are introduced with high index of 1.452 (nH) in the core area. The diameter of the circular MT’s is r, Rc means radius of the core region, and r1, r2, r3 and r4 are the gap from the center of the high index circular MT to the inner most of the core region (see expanded Fig. 1). The circular MT of high index region is used to maintain maximum fundamental mode area (LP01) in the core region at the highly bend condition of 7.5 cm. In cladding region, low index (nL) and high index medium (nc) are angularly and regularly segmented for obtaining large mode area with SM operation. 2θ1 and 2θ2 is denoted by angular width of the low index (nL) and high index (nc) medium, respectively.

Fig. 1
figure 1

a cross section and b refractive index profile for the high index circular MTSCF

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Generally, bending loss is controlled greater than 1dB/m for higher order modes and less than 0.1 dB/m for fundamental mode for attaining SM operation (Li et al. 2009). The distortion of the refractive index profile (\({n}_{\text{eq}}\)) caused by bend can be expressed by following formula (Jain et al. 2013),

$$n_{{{\text{eq}}}} \left( {r,\phi } \right) = n_{0} \left( {r,\phi } \right)*\sqrt {1 + \frac{{2r}}{{\rho R_{b} }}\cos \phi }$$
(1)

Where, \({R}_{b}\) is the bending radius, Φ is the bend orientation angle, \({n}_{0}\left(r,\phi \right)\) is RI in straight case, and ρ is represented by stress factor whose value is 1.25. The proposed high index circular MTSCF can be fabricated by stack and draw method.

3 Properties of the MTSCF

In this study, we use the fully vectorial finite element method to analyze the bending characteristics of the suggested high index circular MTSCF model with suitable perfectly matched layer (PML). The essential characteristics for LMA fibers such as effective mode area (EMA) and confinement loss (CL) is calculated by (Wang et al. 2013),

$${\text{EMA}} = \frac{{\left( {\iint {\left| E \right|^{2} }{\text{d}}x{\text{d}}y} \right)^{2} }}{{\iint {\left| E \right|^{4} }dx{\text{d}}y}}$$
(2)
$${\text{CL}} = \frac{{20}}{{{\text{ln}}10}}k_{0} {\text{Im}}\left( {n_{{{\text{eff}}}} } \right)$$
(3)

where \({k}_{0}=\frac{2\pi }{\lambda }\) and \(\text{I}\text{m}\left({n}_{eff}\right)\)is the imaginary part of effective refractive index of the mode. The optimized parameters of the proposed fiber is r1 = 3 μm, r2 = 6 μm, r3 = 10 μm, RC= 36 μm, r = 1 μm, M1= 9 μm, M2 = 10 μm, M3 = 8 μm, M4 = 8 μm, Rd = 109 μm. Another one important parameter for SM operation is loss ratio. It is stated as the ratio between lowest loss of the higher order mode and loss of the fundamental mode. Typically loss ratio (LR) is greater than 100 sufficient for SM operation. Meanwhile, shorter fibre lengths are sufficient to strip off higher order modes because the lowest loss of higher order mode is > 1 dB/m.

$${\text{Loss}}\;{\text{ratio}}\left( {{\text{LR}}} \right){\text{ = }}\frac{{{\text{Lowest}}\;{\text{loss}}\;{\text{of}}\;{\text{higher}}\;{\text{order}}\;{\text{modes}}}}{{{\text{Loss}}\;{\text{off}}\;{\text{undamental}}\;{\text{mode}}}}$$
(4)

4 Results and analysis

4.1 Effect of multi trench

In the proposed fiber, we have achieved bending loss constraints through low index trenches (M1 and M3). Figure 2 shows the variation of bending loss and effective mode area with respect to M1. The black, green and the red curves are represented the bending loss for the LP01, LP11-even and LP11-odd mode, respectively. The effective mode area for LP01 mode is represented by dotted blue line. When the M1 thickness increases from 7 to 11 μm, the LP01 mode decreases upto 9 μm and then increases. Similarly, LP11-even and LP11-odd mode increases upto 9 μm and then decreases. The effective single mode operation obtained at M1 thickness of 9 μm, such as loss of LP01 is 0.015 dB/m, LP11-even is 2.435 dB/m and LP11-odd is 16.45 dB/m. And also the effective mode area of ~ 1000 µm2 is attained over the range of 8 to 10 μm.

Fig. 2
figure 2

Bending loss of LP01, LP11-even and LP11-odd and mode area of LP01 by varying M1

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Similarly, bending characteristics are analyzed with respect to M3 as shown in Fig. 3. The M3 also is supported to effective light fundamental mode confinement in the core region with BL constrains. With help of M3 low index trench, we can easily achieved suppression of higher order modes. In Fig. 3, SM operation obtained at M3 of 8 μm with maintained LMA of ~ 1000 µm2. Further, the proposed fiber obtained LR of 162 for effective single mode propagation.

Fig. 3
figure 3

Bending loss of LP01, LP11-even and LP11-odd and mode area of LP01 by varying M3

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4.2 Effect of high index circular MT

We have considered four layer of high index circular MT in the core area for effective mode field distribution in the core area at the bending condition of 7.5 cm. We have indicated in the Sect. 2, r1, r2, r3 and r4 represent distance from high index circular ring to the center of the fiber and also its initial values.

Fig. 4
figure 4

Intensity distribution of LP01 mode under various layers with different bending conditions. a r3 and r4 b r2, r3, r4 and c r1, r2, r3, r4 at 15 cm bending radius. c r3 and r4 d r2, r3, r4 and e r1, r2, r3, r4 at 7.5 cm bending radius

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Figure 4 represents the intensity distribution of LP01 mode under various layers with different bending conditions. At 15 cm and 7.5 cm bending radius, maximum amount of light is distributed in the four layer of high index circular MT than two and three layer of circular MT (See Fig. 4c and f). The effective mode area for a two, three, and four-layer of high index circular ring, respectively, is 1045 µm2, 1008 µm2, and 915 µm2 (see Fig. 4a, b, and c). Similarly, the mode area for tight bending radius of 7.5 cm is 911 µm2, 950 µm2 and 1006 µm2 for two, three and four layer of high index circular MT (see Fig. 4d, e, and f). Since, we have obtained maximum effective area of 1000 µm2 at compact bending radius of 7.5 cm with help of high index circular MT. Further, we examined the effects of the placement of the circular high index MT. The innermost layer of the high index circular core trench is crucial for mode field scaling in terms of mode field area because it can direct the mode field to the left, increasing mode field area as r1 decreases. The mode area is 1038 µm2 at a bend radius of 7.5 cm when r1 = 2 μm, and it decreases to 998 µm2 when r1 = 5 μm, as can be seen in Fig. 5. But, difficult to leak out the Lp11-odd mode from the core region at r1 = 2 μm. Finally, we have obtained SM condition at r1 = 3 μm with optimum effective area of 1006 µm2.

Fig. 5
figure 5

Bending loss of LP01, LP11-even and LP11-odd and effective mode area of LP01 by varying r1

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Then, we have analyzed bending characteristics for fundamental mode and higher order modes with respect to r2 and r3 as shown in Fig. 6. When r2 and r3 is varied from 6 to 7.5 μm and 8 to 12 μm, respectively, the bend loss is < 0.1 dB/m for LP01 and > 1 dB/m for LP11-odd, indicating that the bend loss complies with the requirements of single mode operation. Then, LP11-even mode is 2.45 dB/m for r2 = 6 μm and r3 = 10 μm. Further, mode area of fundamental mode decreases with increases of r2 and r3. So, we have considered optimum effective area of 1006 µm2 with SM condition at r2 = 6 μm and r3 = 10 μm.

Fig. 6
figure 6

Bending loss of LP01, LP11-even and LP11-odd and effective mode area of LP01 by varying a r2 and b r3

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Fig. 7
figure 7

Bending loss of LP01, LP11-even and LP11-odd and effective mode area of LP01 by varying Rc

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Additionally, Fig. 7 illustrates the effect of core radius (Rc) on bending loss and mode area at the 7.5 cm bending radius. When Rc range between 34.5 and 36 μm, the fundamental mode and higher order modes are controlled for obtaining maximum mode area with SM operation. The direction of bending causes the mode area to increase as Rc rises from the lowest to the maximum. When proposed fibre is bent along the + x axis, its refractive index typically increases along the + x axis and decreases along the −x axis.

4.3 Effect of bending orientation angle

In general, more complex fibre coiling derives from the greater freedom over bend orientation angle. It is crucial to keep in mind that the proposed fibre is suggested for the constrained practical bending radius of 7.5 cm. In order to achieve LMA with bending loss constraints, the suggested fibre is worked up to a 10° bending orientation angle. Within 10o, bending loss of LP01 and LP11-odd is 0.061 dB/m and 1.321 dB/m, respectively as shown in Fig. 8a. Further, optimum mode area and power fraction in core for LP01 is 1006 µm2 and 97.48% with bending loss constrains as shown in Fig. 8b.

Fig. 8
figure 8

a bending loss for fundamental mode and higher order modes and b power in core and mode area for fundamental mode by varying bending orientation angle

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4.4 Effect of duty cycle

By adjusting the duty cycle in SCF, the leakage losses of the core modes are changed. The investigation of how the duty cycle affects the characteristics of the high index circular MTSCF is crucial for design improvement. Duty cycle can be calculated by γ = 1-ϴ/(360/N), where N is represented number of segments. In the proposed fiber, we have considered N = 8. Figure 9 represents the bending loss and mode area with respect to duty cycle. When γ range between 0.4 and 0.8, the LP01 loss is < 0.1 dB/m and LP11-odd is > 1dB/m and mode area is ~ 1000 µm2. Hence, the proposed fiber provides adequate segment fabrication tolerance. Further, loss ratio of > 40 with SM condition can be achieved when \( 0.4 \le \gamma \le 0.8 \).

Fig. 9
figure 9

Variation of bending loss and effective mode area by varying duty cycle

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Finally, we have obtained large mode area of 1006 µm2, Loss ratio of 162 and SM operation at tight bending radii of 7.5 cm and optimized parameters are r1 = 3 μm, r2 = 6 μm, r3 = 10 μm, RC=36 μm, r = 1 μm, M1 = 9 μm, M2 = 10 μm, M3 = 8 μm, M4 = 8 μm, Rd=109 μm. Further, regarding bending radius, loss of fundamental and lowest higher order mode, loss ratio, SM operation and mode-area, Table 1 shows the comparison of various existing LMA fibers with proposed MTSCF fiber in detail.

Table 1 Comparison of various existing LMA fibers with proposed MTSCF fiber
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5 Conclusion

We have proposed bend resistant segmented cladding fiber with four layer of high index circular multi trench in the core region to obtain SM operation with maximum effective mode area at practical bending condition. The performance of the proposed fiber is investigated in relation to high index circular multi trenches, low index trenches, duty cycle and bending orientation angle. The obtained mode area is 1006 µm2 with SM condition at tight bending radius of 7.5 cm. Moreover, loss ratio between fundamental mode (LP01 = 0.015 dB/m) and lowest higher order mode (LP11-odd = 2.435 dB/m) is 162, which is confirmed effective single mode operation. It has been shown to have excellent higher order mode and bending-immune suppression performance. To simplify fabrication, pure silica-based low refractive index trenches are added around the core region. The suggested high index circular MTSCF designs are an appropriate option for high-power compact fibre laser systems.