# Optimization and Dispersion Tailoring of Chalcogenide M-Type Fibers Using a Modified Genetic Algorithm

^{1}

^{2}

^{*}

## Abstract

**:**

_{02}mode possessing unusual dispersion properties, namely, a zero-dispersion wavelength (ZDW) shifted to the short wavelength region relative to the material ZDW. The LP

_{02}mode can be selectively excited since it is predominantly localized near the core, while the fundamental LP

_{01}and other higher modes are localized near the ring (for proper fiber parameters). In this paper, we present a comprehensive theoretical analysis of effective dispersion tailoring for the HE

_{12}mode of highly nonlinear chalcogenide glass fibers (for which the LP mode approximation fails due to large refractive index contrasts). We demonstrate fiber designs for which ZDWs can be shifted to the spectral region < 2 μm, which is of great interest for the development of mid-IR supercontinuum sources and frequency-tunable pulse sources with standard near-IR pumping. We obtained the characteristic equation and solved it numerically to find mode fields and dispersion characteristics. We show the possibility of achieving dispersion characteristics of the HE

_{12}mode with one, two, three, and four ZDWs in the wavelength range of 1.5–5.5 μm. We used a modified genetic algorithm (MGA) to design fibers with desired dispersion parameters. In particular, by applying an MGA, we optimized four fiber parameters and constructed a fiber for which HE

_{12}mode dispersion is anomalous in the 1.735–5.155 μm range.

## 1. Introduction

_{2}) 2–3 orders of magnitude higher than n

_{2}of silica glass [4,5,6,7]. Chalcogenide glasses are composed of chalcogens (S, Se, and Te) and other chemical elements such as As, Ga, Sb, Ge, and/or others [5]. Many chalcogenide glassy systems are characterized by high chemical stability, resistance to atmospheric moisture, relatively low optical losses in the near and mid-IR ranges, and a transparency band up to ~10 μm (and even significantly longer for some compositions) [4,5]. One of the features of chalcogenide glasses is that their zero dispersion wavelength (ZDW) is located in the mid-IR range. For example, the ZDW is ~4.9 μm for As

_{2}S

_{3}glass and ~7.2 μm for As

_{2}Se

_{3}glass [4]. At wavelengths shorter than the ZDW, the glass dispersion is normal, while at longer wavelengths, it is anomalous. To implement many nonlinear optical transformations (generation of frequency-tunable Raman solitons, supercontinuum generation in certain regimes, higher-order soliton compression, etc. [1]), anomalous fiber dispersion at a pump wavelength is required, which is a challenge when using a standard near-IR pump.

_{1}is surrounded by a thin ring layer having an outer radius b with a refractive index n

_{2}> n

_{1}, and then the second cladding follows with a refractive index n

_{3}< n

_{1,2}(Figure 1). Thus, unlike step-index fibers, M-type fiber has an additional ring layer around the core with thickness d = (b − a), whose refractive index is greater than the refractive index of the core.

_{2}− n

_{1}) can be used to selectively excite higher-order modes, such as LP

_{02}, localized near the core [15]. In this case, the fundamental LP

_{01}mode is localized near the ring and has a small overlap integral with the core. The LP

_{02}mode can have the following specific properties: (1) anomalous dispersion in the range of shorter wavelengths than for the fundamental LP

_{01}mode of step-index and M-type fibers made of the same glasses and than for the glasses themselves; (2) the possibility of matching with the LP

_{01}mode of standard fibers [16,17].

_{1m}modes (since the LP approximation is violated at the considered contrasts of refractive indices >10%). This approach is faster than finite element modeling, especially when it is necessary to perform optimization on a large number of parameters (up to four in our case). A comprehensive theoretical analysis of the dispersion properties of the HE

_{12}mode was carried out, and a fiber profile was designed using a specially implemented modified genetic algorithm (MGA). In particular, anomalous dispersion in the 1.735–5.155 μm range was numerically demonstrated.

## 2. Materials and Methods

#### 2.1. Fiber Model

_{40}Se

_{60−x1}Te

_{x1}/As

_{40}Se

_{60−x2}Te

_{x2}glasses and cladding made of As

_{40}S

_{60}glass (Figure 2a).

_{40}Se

_{60−x1}Te

_{x1}glasses can be varied in a wide range by varying tellurium content x

_{1}(examples are shown in Figure 2b) [18]:

_{40}Se

_{60}and As

_{40}S

_{60}glasses on wavelength are taken from [19] (“AMTIR-2” and “AMTIR-6”, respectively). Note that the technologies for the synthesis of As

_{40}Se

_{60−x}Te

_{x}and As

_{40}S

_{60}glasses are mature [20]. These glasses have suitable physicochemical properties and are compatible with the manufacture of optical fibers, which was demonstrated experimentally [20], but their ZDWs are located in the mid-IR (Figure 2c). However, thanks to the strong waveguide contribution, the first ZDW of the fibers made of these glasses can be shifted to significantly shorter near-IR wavelengths, which is a subject of our study. We investigate numerical dispersion tailoring by varying four fiber parameters: a, b, x

_{1}, and x

_{2}.

#### 2.2. Characteristic Equation

_{1n}mode from the Maxwell equations with allowance for the boundary conditions using the well-known approach [1,21]. For a cylindrical symmetry of an M-type fiber, it is convenient to write the wave equation for the electric and magnetic fields in the cylindrical coordinates (r, φ, z) and, after that, apply the well-described method of separation of variables [1,21]. The electric and magnetic z-field components are as follows [1,21]:

_{0}= ω/c, c is the speed of light in vacuum, and J

_{l}and N

_{l}are the Bessel and the Neumann functions, respectively. A(ω), B(ω), C(ω), and D(ω) are determined from the boundary conditions. It is also necessary to take into account the natural conditions of the field finiteness at r = 0 and the field decrease at infinity. The expressions for E

_{z}and H

_{z}with allowance for the natural conditions at r = 0 and r = ∞ (after renaming A, B, C, and D to X

_{j}) become

_{eff}= β/k

_{0}is the effective refractive index. The radial and φ-components of the fields are found from the Maxwell equations [21]:

_{z}, H

_{z}, E

_{φ}, and H

_{φ}for r = a and r = b) and rewriting the equations in matrix form, we obtain

_{eff}depending on the parameters of the fiber and the wavelength.

_{eff}into the coefficients of the system (14), we find the relation X

_{j}, j = 1,…,8. Thus, we determine the structure of the field of the selected mode by substituting the found constants into (4)–(13).

_{eff}:

_{z}distributions for the HE

_{11}and HE

_{12}modes are shown in Figure 3. So, Figure 3 demonstrates almost ideal coincidence of the results of calculations obtained employing two different approaches. It should be emphasized that Figure 3 also shows that the fundamental mode HE

_{11}is localized near the ring, while the HE

_{12}mode is localized near the core.

#### 2.3. Modified Genetic Algorithm

_{2}(λ) for any set of fiber parameters (a, d, x

_{1}, and x

_{2}). This opens up an opportunity for dispersion optimization by selecting parameters that allow obtaining the required dependence for a particular problem. Let us introduce the error function F(β

_{2}(a, d, x

_{1}, x

_{2}), β

_{2}

^{target}), which quantitatively characterizes the difference between the calculated function β

_{2}and the target function β

_{2}

^{target}. The smaller the value of the error function, the closer the found solution β

_{2}is to the target β

_{2}

^{target}. In general, the error function can be written as follows:

_{λ}is the number of wavelength points (β

_{2}

^{(i)}= β

_{2}(λ

_{i})), μ

_{i}is the weight coefficients, and G(β

_{2}, β

_{2}

^{target}) is a special component of the error function (the specific form of G(β

_{2}, β

_{2}

^{target}) is selected, taking into account the problem to be solved).

- The formation of an initial population of N individuals, where each individual is a set of randomly generated parameters in the search range: I
_{j}= (a_{j}, d_{j}, x_{1j}, x_{2j}) and j = 1…N; - The calculation of the error function F
_{j}= F(I_{j}) for each individual and sorting the population in ascending order of F_{j}values (from the best solutions to the worst ones); - Algorithm iteration:
- (1)
- Random division of the best 2k individuals into k pairs and crossing in each pair with the formation of 2k new individuals that are added to the population;
- (2)
- Random selection of s individuals and applying the mutation operator to them, changing their parameters randomly. This is performed to provide the widest coverage of the parameter space and to prevent premature stagnation of the algorithm in the local optimum, bypassing the global one;
- (3)
- Sorting the updated population of N + 2k individuals and removing the 2k worst ones (with the largest value of the error function F).

- Repeating steps (1)–(3) until the maximum number of iterations n
_{max}is reached or the convergence criterion is met.

_{1}and I

_{2}with the formation of new individuals I

_{1′}and I

_{2′}is performed using the operator $\widehat{X}$:

_{j}

^{max}− I

_{j}or σ = I

_{j}− I

_{j}

^{min}with equal probability (I

_{j}

^{max}and I

_{j}

^{min}are the largest and smallest value of the variable parameter in the current population, respectively). This type of mutation operator allows us to make the process uneven: at the initial steps of the algorithm, the degree of mutation is the highest, which is important for the most complete search in the parameter space and the reduction of the probability of premature convergence; at the final stage, the mutations are minimal, which makes it possible to reduce the scatter of individuals upon convergence to the final solution. The parameter h allows us to set the degree of this unevenness of the mutation process; the larger the h, the more the nature of the process changes during the operation of the algorithm.

## 3. Results

#### 3.1. Dependences of Dispersion on Parameters

_{12}mode in detail. Before optimizing the dispersion curves using MGA with respect to four parameters (a, d, x

_{1}, and x

_{2}) for achieving the desired dispersion profile, we performed a series of numerical simulations in which some parameters were fixed. This was required for a better understanding of the tendencies and for finding patterns in the behavior of dispersion curves when changing each parameter. This also allowed us to determine the range of parameters for executing MGA.

#### 3.1.1. Fixed d, x_{1}, and x_{2}; Varied a

_{1}= 10, d = 0.3, 0.6 μm, and x

_{2}= 30, 50. Thus, there were four possible combinations. We calculated β

_{2}for each combination (d = 0.3 μm and x

_{2}= 30; d = 0.3 μm and x

_{2}= 50; d = 0.6 μm and x

_{2}= 30; and d = 0.6 μm and x

_{2}= 50) by varying a in the range from 2 to 3 μm. The results are presented in Figure 4. Hereinafter, the black dotted lines correspond to ZDWs. It is seen that for the chosen parameters, anomalous dispersion can start from 2 to 2.7 μm. This result was obtained thanks to the significant waveguide contribution. The smaller a is, the shorter the first ZDW. Therefore, to achieve the minimum ZDW, it is necessary to reduce the size of the core. However, this narrows the range of anomalous dispersion (the second ZDW appears for small a). It is also seen that the thicker the ring layer, the longer the first ZDW.

#### 3.1.2. Fixed a, x_{1}, and x_{2}; Varied d

_{1}= 10, a = 2, 2.5 μm, and x

_{2}= 30, 50. Figure 5 demonstrates β

_{2}as a function of wavelength for four combinations of parameters. We found an interesting feature, namely, that the dispersion dependences can be more complex than in Figure 4. For the considered parameters, they can have from one to four ZDWs in the 1.5–5.5 μm range (see Figure 5, the lowest subplot). However, the dispersive curves with four ZDWs exist in a narrow range of d. Such exotic dispersion was known for fundamental modes of fibers and waveguides of special design [24,25], but it was not reported for M-type fibers. For some problems of nonlinear optical conversion, β

_{2}with four ZDWs may be required. So, the HE

_{12}mode of M-type fibers can also be optimized for this purpose.

#### 3.1.3. Fixed a, d, and x_{1}; Varied x_{2}

_{1}and varied the refractive index difference between the core and the ring layer. The dispersion β

_{2}calculated as a function of λ is presented in Figure 6. The dispersion characteristics have one or two ZDWs in the considered spectral range. We note an interesting feature for this series of numerical simulations: the dependence of the ZDW on (x

_{2}− x

_{1}) is nonmonotonic; there is the value of (x

_{2}− x

_{1}) corresponding to the minimum of the first ZDW. It is seen in Figure 6 that the first ZDW can be shifted to wavelengths shorter than 2 μm. The second ZDW shifts to longer wavelengths with increasing (x

_{2}− x

_{1}).

#### 3.1.4. Fixed a, d, and (x_{2} − x_{1}); Varied x_{1}

_{2}− x

_{1}) assuming varied x

_{1}. The corresponding dispersion is shown in Figure 7. We can conclude from this figure that the first ZDW is almost independent of x

_{1}, but the smaller x

_{1}is, the shorter the second ZDW.

#### 3.2. Dispersion Profiles Tailored with Modified Genetic Algorithm

#### 3.2.1. Minimizing the First ZDW with MGA

_{12}mode. As can be seen in Figure 5 (right column), there are fiber parameters providing a ZDW~1.6 μm. However, for some tasks, it might be useful to have a fiber with anomalous dispersion at wavelengths near 1.55 μm corresponding to standard erbium-doped fiber lasers. Therefore, further optimization is required. As shown in Section 3.1, the short-wavelength ZDW shift can be achieved by decreasing the core radius a. However, the core size can only be reduced to a certain value limited by a cutoff wavelength λ

_{cutoff}for HE

_{12}mode. As a result, it is necessary to set a condition on λ

_{cutoff}and search for the minimum of the ZDW, taking this condition into account. Further optimization was carried out for λ

_{cutoff}> 3 μm.

_{cutoff}belongs to the interval from 3 to 4 μm depending on other parameters. We made optimization for three parameters, as follows: d $\in $ (0.1 μm, 0.4 μm), x

_{1}, x

_{2}$\in $ {[0, 50]|x

_{2}> x

_{1}}. To minimize the computational complexity of the MGA algorithm, the dispersion was calculated at four points: λ = 1.3 μm, 1.4 μm, 1.5 μm, and 1.6 μm. A large absolute value of the anomalous dispersion was set as the objective function. The MGA parameters providing optimal balance between the speed of finding a solution and minimizing the tendency to premature convergence are as follows:

- N = 400 is the number of individuals in the population;
- k = 40 is the number of pairs of individuals that give offspring at each iteration (the same number of individuals was eliminated at each iteration);
- n = 5 is the parameter of the crossing operator;
- T = 100 and h = 5 are mutation operator parameters.

_{1}, and x

_{2}); the color of the point shows the value of the error function for this individual (Figure 8). The algorithm converged fairly quickly. A little more than 10 iterations were sufficient for this problem.

_{1}= 11.48; and x

_{2}= 47.53. The dispersion dependence on wavelength for these M-type fiber parameters is plotted in Figure 9a. The second ZDW is 2.317 μm. Thus, the width of the anomalous dispersion region is 0.92 μm. The dispersion is almost flat (near −450 ps

^{2}/km) in the 1.55–2.1 μm range. Since the effective mode area is also very important for the nonlinear optical pulse conversion, we plotted A

_{eff}in Figure 9b. The Poynting vector, depending on radial coordinate and wavelength, is also demonstrated in Figure 9c.

#### 3.2.2. Obtaining Anomalous Dispersion in the 2–5 μm Range with MGA

_{12}mode in the entire spectral range of 2–5 μm. The main difference in comparison with the previous case is that here, all four parameters were optimized (the core radius a was not fixed). The ranges of parameter variation were selected based on the results of the simulation presented in Section 3.1. We also set the requirement λ

_{cutoff}> 5 μm. We assumed a ∈ (1.5 μm, 2.5 μm); d ∈ (0.1 μm, 0.6 μm); and x

_{1}, x

_{2}∈ {[0, 50]|x

_{2}> x

_{1}}.

_{2}< 0 over the entire interval λ ∈ (2 μm, 5 μm), priority should be given to the set of parameters providing the smaller ZDW. Thus, the values of β

_{2}near the ZDW should be taken into account with the highest weight. This is achieved by introducing weight coefficients μ

_{i}in the expression (19). They were calculated as follows:

_{2}(λ), and the optimal value of the parameter q is determined empirically based on the results given by the MGA for various q. The second important condition is the need to cut off sets of parameters for which the dispersion is anomalous only in part of the 2–5 μm interval. However, it is not advisable to discard them during the MGA operation since such individuals that are not suitable as a solution to the problem can generate suitable ones due to crossing and mutations. Therefore, for each value β

_{2}> 0 in the wavelength range of 2–5 μm, a special large addition G(β

_{2}, β

_{2}

^{target}) in the expression (19) was taken into account.

_{1}= 29.46, and x

_{2}= 49.46. The corresponding dispersion dependence on wavelength is demonstrated in Figure 11a. For the found solution, the first ZDW is 1.735 μm, and the second ZDW is 5.155 μm. The effective mode area and Poynting vector for these fiber parameters are also plotted in Figure 11b,c, respectively. Note that A

_{eff}changes very slowly at the wavelength of 2.5–5 μm, which is an advantage for nonlinear pulse conversion. Note that found x

_{2}= 49.46 is close to the upper limit of the considered range. The glass with a similar tellurium concentration was demonstrated in [18]. However, if, for fiber manufacturing, specific technological conditions do not allow the creation of glass with the desired physical and chemical properties at such a concentration, the optimization problem should be solved anew, taking into account technical capabilities.

## 4. Discussion and Conclusions

_{40}Se

_{60−x1}Te

_{x1}/As

_{40}Se

_{60−x2}Te

_{x2}/As

_{40}S

_{60}glass core/ring layer/cladding. These glasses have suitable physicochemical properties and are compatible with the manufacture of optical fibers [20]. By varying the tellurium content in the glass composition, it is possible to strongly affect the refractive index difference between the core of radius a and the ring layer of thickness d [18]. We considered the HE

_{12}mode localized near the core rather than the fundamental mode HE

_{11}localized near the ring layer. We demonstrated that, by controlling four fiber parameters (a, d, x

_{1}, and x

_{2}), the ZDW of HE

_{12}mode can be effectively shifted to the near-IR (while for As-Se-Te glasses, material ZDW > 7 μm). Note that it was previously demonstrated through numerical simulation by the finite-element method that for M-type Ge-As-Se/As-Se chalcogenide fibers, the ZDW can be only slightly shorter than 3 μm [17]. Here, we proposed smart designs and optimization of M-type As-Se-Te/As-S chalcogenide fibers, providing the first ZDW shorter than 2 μm and even shorter than 1.55 μm. This will make it possible to develop broadband light converters in the anomalous dispersion regimes using standard fiber laser pump sources at 2 μm and even at 1.55 μm.

_{1n}modes of M-type fiber from the Maxwell equations. We developed a numerical code for its solution and for the calculation of the group velocity dispersion, the Poynting vector, and the effective mode area. The developed model was verified by comparing the results with the data obtained by finite element modeling. We studied in detail the properties of the HE

_{12}mode. We performed a series of numerical simulations by varying some parameters to understand the tendencies and to find patterns in the dispersion behavior. We showed that thanks to the strong waveguide contribution, it is possible to obtain from one to four ZDWs in the 1.5–5.5 μm range.

_{1}, and x

_{2}) and shifted the first ZDW of the HE

_{12}mode to a wavelength shorter than 1.55 μm. We found the parameters providing a ZDW = 1.395 μm and the 0.92 μm width of the anomalous dispersion range. In the second problem solved with MGA, we optimized four fiber parameters (a, d, x

_{1}, and x

_{2}) and tried to design M-type fiber with anomalous dispersion in the spectral range wider than 2–5 μm. We found parameters providing the first ZDW of 1.735 μm and the second ZDW of 5.155 μm (3.4 μm width of the anomalous dispersion range).

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**(

**a**) Cross section scheme of considered M-type fibers. Refractive indices (

**b**) and dispersions (

**c**) of chalcogenide glasses.

**Figure 3.**Examples of calculated Poynting vector P

_{z}for HE

_{11}(

**top**) and HE

_{12}(

**bottom**) modes for M-type fiber parameters a = 2 μm, b = 2.4 μm, x

_{1}= 10, and x

_{2}= 30.

**Figure 5.**Dispersion as a function of wavelength λ and ring thickness d (

**top**and

**middle**). Examples of dispersion with one, two, and four ZDWs (

**bottom**) corresponding to horizontal lines in the right panel of the middle row.

**Figure 8.**Visualization of MGA execution for minimizing the first ZDW. For red dots the error is maximum, for blue dots the error is minimum.

**Figure 9.**Dispersion with the first ZDW at 1.395 μm (

**a**) and effective mode area (

**b**) of HE

_{12}mode calculated for a = 1 μm; b = 1.201 μm; x

_{1}= 11.48; and x

_{2}= 47.53. (

**c**) Poynting vector of HE

_{12}mode depending on radial coordinate and wavelength.

**Figure 11.**Dispersion with the first ZDW at 1.735 μm and the second ZDW at 5.155 μm (

**a**) and effective mode area (

**b**) of HE

_{12}mode calculated for a = 1.8484 μm, b = 2.204 μm, x

_{1}= 29.46, and x

_{2}= 49.46. (

**c**) Poynting vector of HE

_{12}mode depending on radial coordinate and wavelength.

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**MDPI and ACS Style**

Salnikov, N.I.; Andrianov, A.V.; Anashkina, E.A.
Optimization and Dispersion Tailoring of Chalcogenide M-Type Fibers Using a Modified Genetic Algorithm. *Fibers* **2023**, *11*, 89.
https://doi.org/10.3390/fib11110089

**AMA Style**

Salnikov NI, Andrianov AV, Anashkina EA.
Optimization and Dispersion Tailoring of Chalcogenide M-Type Fibers Using a Modified Genetic Algorithm. *Fibers*. 2023; 11(11):89.
https://doi.org/10.3390/fib11110089

**Chicago/Turabian Style**

Salnikov, Nikolay I., Alexey V. Andrianov, and Elena A. Anashkina.
2023. "Optimization and Dispersion Tailoring of Chalcogenide M-Type Fibers Using a Modified Genetic Algorithm" *Fibers* 11, no. 11: 89.
https://doi.org/10.3390/fib11110089