# Design and Analysis of a Large Mode Field Area and Low Bending Loss Multi-Cladding Fiber with Comb-Index Core and Gradient-Refractive Index Ring

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}, and the bending loss is reduced to 8.452 × 10

^{−4}dB/m. Additionally, when the bending radius is smaller than 30 cm, there are two variations with low BL and leakage; one is a bending radius of 17 cm to 21 cm, and the other is from 24 cm to 28 cm (except for 27 cm). When the bending radius is between 17 cm and 38 cm, the highest bending loss is 1.131 × 10

^{−1}dB/m and the lowest mode field area is 1925 μm

^{2}. It has a very important application prospect in the field of high-power fiber lasers and telecom applications.

## 1. Introduction

^{2}and a BL of 8.452 × 10

^{−4}dB/m at a bending radius of 20 cm. The proposed fiber shows excellent performance and is expected to be used in a high fiber laser, fiber for the home, and so on. Additionally, by consulting relevant references on fiber manufacturing, we believe that the proposed fiber can be manufactured with existing technology [1,30].

## 2. Materials and Methods

#### 2.1. Structure

_{0}, n

_{1}, n

_{2}, n

_{r}, n

_{6}, n

_{3}, n

_{5}, and n

_{4}, respectively. CIF contributes to achieving a large MFA, so the fiber core is designed as a three-layer comb-index core; the width of the outermost two layers layer is r

_{d}and the radius of the core is r

_{1}. The width of the other layers is t, d, t

_{1}, t

_{1}, d

_{1}, t

_{1}, d

_{1}, and the radius of the fiber is r

_{2}.

_{0}, and the RIP can be expressed as follows:

_{2}is the lowest refractive index of the GRIR, r is the radius of the position, ∆ is the relative refractive index difference, and α is the refractive index distribution constant, set as 2 in this paper.

#### 2.2. Analysis Methods

_{eff}) can be expressed as follows [35]:

_{bend}is the bending radius, and n* is the refractive index in the bending state.

## 3. Numerical Simulations

_{0}= 1.44, n

_{1}= 1.4398, n

_{2}= 1.4395, n

_{3}= 1.4385, n

_{4}= 1.4386, n

_{5}= 1.4396, n

_{6}= 1.4397, r

_{1}= 38 μm, r

_{2}= 210 μm, r

_{bend}= 20 cm (r

_{bend}is the bending radius), r

_{d}= 5 μm, t

_{1}= 17 μm, d

_{1}= 18 μm, t = 16 μm, and d = 14 μm, respectively. Unless otherwise stated, the MFA and BL mentioned in the paper are all of the FMs, and the parameters are unchanged. The mode field distribution of the FM in straight and bending states is shown in Figure 2a,b, respectively. Additionally, we define a ratio of BL to MFA to find the best performance.

#### 3.1. Numerical Simulations of t_{1} and d_{1}

_{1}and d

_{1}; the range of t

_{1}was 17~18 μm and that of d

_{1}was 16.5~18 μm. The change in t

_{1}and d

_{1}has an excessive effect on the BL, which can lead to intense and irregular fluctuations in the BL, so the small variation range was selected to make the BL remain steady. Figure 3 shows the influence of the cladding’s size on the BL and MFA when d

_{1}is 16.5 μm, 17 μm, 17.5 μm, and 18 μm, respectively.

_{1}and d

_{1}at first, then becomes flat. Within the variation range of t

_{1}and d

_{1}, most values of BL are close to 1.000 × 10

^{−3}dB/m, and the lowest of them is 1.160 × 10

^{−3}dB/m when d

_{1}is 18 μm and t

_{1}is 17.7 μm. The BL nearly remains unchanged when t

_{1}changes from 17.5 μm to 18 μm, so the proposed fiber can obtain a steady performance of the BL. In Figure 3b, it can be considered that MFA has no relationship with d

_{1}and goes down about 100 μm

^{2}with t

_{1}. During the increase in t

_{1}from 17 μm to 18 μm, the MFA decreases from 2164 μm

^{2}to 2056 μm

^{2}. With the change in t

_{1}, the MFA is still greater than 2000 μm

^{2}and can reach the maximum of 2235 μm

^{2}when d

_{1}is 16.5 μm and t

_{1}is 17 μm. In conclusion, the BL and MFA can remain steady when t

_{1}changes from 17 μm to 18 μm and d

_{1}changes from 16.5 μm to 18 μm.

#### 3.2. Numerical Simulations of t and d

^{−4}dB/m when d is 14.5 μm and t is 17 μm, and the highest values of BL are less than 1.000 × 10

^{−2}dB/m. The BL can be reduced by one order of magnitude with the increase in d from 13 μm to 14.5 μm. Additionally, the change in t and d has a significant effect on the MFA from Figure 4b. The MFA decreases more slowly with t and d, and the largest MFA achieves 3232 μm

^{2}when d is 13 μm and t is 15 μm; the variety range of MFA is about 1200 μm

^{2}. Additionally, the ratio of BL to MFA is the lowest when d is 14.5 µm and t is 16.5 µm, and the BL is 8.452 × 10

^{−4}dB/m and the MFA is 2010 µm

^{2}. As a conclusion, the GRIR can affect the MFA and BL; both the MFA and BL are inversely proportional to t and d.

#### 3.3. Numerical Simulations of r_{1} and r_{d}

_{1}is 36 μm, 37 μm, 38 μm, and 39 μm, respectively. The simulations were performed by changing the values of r

_{1}and r

_{d}, and the range of r

_{d}was 3.5~5.5 μm.

_{d}; the lowest BL is 8.250 × 10

^{−4}dB/m when r

_{d}is 5.5 μm and r

_{1}is 37 μm. Only when r

_{d}is 3.5 μm and r

_{1}is 36 μm can the BL be larger than 1.000 × 10

^{−2}dB/m. As for r

_{1}, it can be regarded as having no significant effect on the BL. From Figure 5b, the MFA decreases and then becomes steady with r

_{d}and r

_{1}. The largest MFA is 2606 μm

^{2}when r

_{d}is 3.5 μm and r

_{1}is 39 μm, and the MFA is still larger than 2000 μm

^{2}. It can be concluded that with the increase in r

_{d}, the BL can be reduced by one order of magnitude, and the variation range of the MFA is close to 600 μm

^{2}.

## 4. Comparison and Analysis

#### 4.1. Importance of the CIC

_{1}from 34 μm to 39 μm.

^{−3}dB/m. Compared with the fiber with the CIC, the BL of the SIC structure rises slightly, in general. Additionally, with the increase in r

_{1}, the MFA increases slightly and it changes from 2170 μm

^{2}to 2311 μm

^{2}in Figure 7. When r

_{1}is 39 μm, the MFA of the fiber with the CIC can achieve 2606 μm

^{2}. Compared with this, the largest MFA of the fiber with an SIC decreases by 295 μm

^{2}. In conclusion, the SIC is beneficial to improve the performance of fiber by reducing the BL and increasing MFA to some extent, and the CIC can allow the smaller bending radius to transmit the FM.

#### 4.2. Importance of Multi-Cladding

_{1}is 17~18 μm and that of d

_{1}is 16.5~18 μm in the simulation. Figure 9 illustrates the BL and MFA of the three-cladding fiber when d

_{1}is 16.5 μm, 17 μm, 17.5 μm, and 18 μm, respectively.

_{1}and d

_{1}, and about half of the BL values are greater than 1 × 10

^{−1}dB/m, so the BL has been raised by two orders of magnitude compared with that in Figure 3a. From Figure 9b, it can be learned that the MFA remains unchanged with d

_{1}, and it is also not effectively affected with t

_{1}. Compared to the result in Figure 3b, the MFA in Figure 9b can be considered the same as it. When the number of claddings decreases, the BL will be increased. The reason is that the introduction of multiple refractive index claddings can improve the refractive index difference between the core and cladding, which contributes to the decrease in BL. Due to the long distance between the multi-cladding and the fiber core, the change in multi-cladding has almost no effect on the MFA. As a result, the multi-cladding can reduce the BL effectively, but has no effect on enlarging the MFA.

#### 4.3. Importance of GRIR

^{−3}dB/m and less than 1 × 10

^{−2}dB/m, so the result is close to that in Figure 4a. In Figure 11b, the MFA increases more and more quickly, because a larger and larger area of the FM leaks to the cladding instead of transmitting within the core with t. For example, under the condition that d is 14 μm, the mode field distribution of the FM is shown in Figure 12 when t is 16 μm and 17 μm. The phenomenon is not expected in the transmission of fiber. Additionally, most values of MFA in Figure 11b are close to 2000 μm

^{2}, except when d is 14 μm, and the largest MFA is 2891 μm

^{2}when d is 14 μm and t is 17 μm. Compared with this, the MFA in Figure 4b can achieve 3232 μm

^{2}at most, and the fiber can constrain FM transmission within the core well. As a conclusion, the GRIR has a significant effect on the MFA and supports the transmission of the FM better.

## 5. Bending Performance Research

#### 5.1. Bending Radius

_{bend}is greater than 30 cm and the fiber cannot transmit the FM when r

_{bend}is less than 17 cm, the change in the BL and MF with r

_{bend}from 17 cm to 38 cm is studied in Figure 13. Due to the complexity of the proposed fiber, the modes are prone to change suddenly when the bending radius is small, which leads to the BL and MFA fluctuating irregularly. In Figure 13, it can be learned that the BL fluctuates with r

_{bend}from 17 cm to 27 cm, and then still decreases when r

_{bend}is greater than 27 cm; the lowest BL is 1.231 × 10

^{−8}dB/m when r

_{bend}is 38 cm. Most values of BL are less than 1 × 10

^{−2}dB/m, and there are only four values of it greater than 1 × 10

^{−2}dB/m. The BL has approximately three peaks at an r

_{bend}of 18 cm, 21 cm, and 27 cm; the highest BL is 1.131 × 10

^{−1}dB/m when r

_{bend}is 27 cm. When the BL is less than 1 × 10

^{−1}dB/m, it is considered accepted. Therefore, when r

_{bend}is less than 30 cm, it can be considered that only when r

_{bend}is 27 cm can the BL be unacceptable.

_{bend}of 23 cm and 29 cm, the MFA achieves 4274 μm

^{2}and 5007 μm

^{2}, respectively, and the mode field distribution of them is shown in Figure 14a,b. Additionally, the lowest MFA is 1925 μm

^{2}when r

_{bend}is 18 cm. When r

_{bend}is greater than 30 cm, the performance of the proposed fiber goes steady, and the MFA finally remains about 2220 μm

^{2}because the FM does not leak into the cladding but transmits within the core in general, which is shown in Figure 14c. When the MFA does not exceed 1000 μm

^{2}of the steady state, the leakage is considered accepted. From Figure 13, when r

_{bend}is less than 30 cm, there are two ranges of MFA that are less than 3220 μm

^{2}; one is 17 cm to 21 cm, and the other is 24 cm to 28 cm.

_{bend}is smaller than 30 cm, but it can remain steady when the radius is greater than or equal to 30 cm. When r

_{bend}is less than 30 cm, there are two variations with low BL and leakage; one is an r

_{bend}of 17 cm to 21 cm, and the other is 24 cm to 28 cm (except for 27 cm).

#### 5.2. Wavelength

^{−3}dB/m, which is less than 1.000 × 10

^{−1}dB/m. Thus, the change in wavelength has little effect on the BL of the proposed fiber. Additionally, only when wl0 is 1500 nm can the MFA be less than 2000 μm

^{2}. Therefore, the analysis shows that the proposed fiber can maintain excellent performances of the BL and MFA when wl0 changes from 1500 nm to 1600 nm.

## 6. Conclusions

^{2}and the BL is always less than 1.000 × 10

^{−2}dB/m. The performance is best when d is 14.5 µm and t is 16.5 µm, corresponding to a BL of 8.452 × 10

^{−4}dB/m and an MFA of 2010 µm

^{2}.

_{bend}is smaller than 30 cm, there are two variations with low BL and leakage; one is an r

_{bend}of 17 cm to 21 cm, and the other is 24 cm to 28 cm (except for 27 cm). Additionally, by analyzing the dependence on the wavelength of the incident light, it indicates that the fiber can maintain excellent performance when the wavelength changes by 100 nm. As a result, the proposed fiber can achieve a low BL and large MFA. The proposed fiber is significant for fiber in the home, high-power lasers, and it also can be used in optical communications.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**The influence on BL and MFA with the change in t

_{1}and d

_{1}. (

**a**) BL of FM. (

**b**) MFA of FM.

**Figure 5.**The influence on BL and MFA with the change in r

_{d}and r

_{1}. (

**a**) BL of FM; (

**b**) MFA of FM.

**Figure 9.**The influence on BL and MFA of the three-cladding fiber with the change in t

_{1}and d

_{1}. (

**a**) BL of FM; (

**b**) MFA of FM.

**Figure 11.**The influence on BL and MFA of the step-refractive index ring fiber with the change in t and d. (

**a**) BL of FM; (

**b**) MFA of FM.

**Figure 14.**Mode field distribution of FM at (

**a**) r

_{bend}= 23 cm; (

**b**) r

_{bend}= 29 cm; (

**c**) r

_{bend}= 31 cm.

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## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; Lian, Y.
Design and Analysis of a Large Mode Field Area and Low Bending Loss Multi-Cladding Fiber with Comb-Index Core and Gradient-Refractive Index Ring. *Sensors* **2023**, *23*, 5085.
https://doi.org/10.3390/s23115085

**AMA Style**

Zhang Y, Lian Y.
Design and Analysis of a Large Mode Field Area and Low Bending Loss Multi-Cladding Fiber with Comb-Index Core and Gradient-Refractive Index Ring. *Sensors*. 2023; 23(11):5085.
https://doi.org/10.3390/s23115085

**Chicago/Turabian Style**

Zhang, Yining, and Yudong Lian.
2023. "Design and Analysis of a Large Mode Field Area and Low Bending Loss Multi-Cladding Fiber with Comb-Index Core and Gradient-Refractive Index Ring" *Sensors* 23, no. 11: 5085.
https://doi.org/10.3390/s23115085