Design and Dispersion Control of Microstructured Multicore Tellurite Glass Fibers with In-Phase and Out-of-Phase Supermodes

High nonlinearity and transparency in the 1–5 μm spectral range make tellurite glass fibers highly interesting for the development of nonlinear optical devices. For nonlinear optical fibers, group velocity dispersion that can be controlled by microstructuring is also of great importance. In this work, we present a comprehensive numerical analysis of dispersion and nonlinear properties of microstructured two-, four-, six-, and eight-core tellurite glass fibers for in-phase and out-of-phase supermodes and compare them with the results for one-core fibers in the near- and mid-infrared ranges. Out-of-phase supermodes in tellurite multicore fibers are studied for the first time, to the best of our knowledge. The dispersion curves for in-phase and out-of-phase supermodes are shifted from the dispersion curve for one-core fiber in opposite directions; the effect is stronger for large coupling between the fields in individual cores. The zero dispersion wavelengths of in-phase and out-of-phase supermodes shift to opposite sides with respect to the zero-dispersion wavelength of a one-core fiber. For out-of-phase supermodes, the dispersion can be anomalous even at 1.55 μm, corresponding to the operating wavelength of Er-doped fiber lasers.

When designing nonlinear optical fiber devices, significant attention is paid to the control of group velocity dispersion, which is often of great importance in the development of laser sources based on supercontinuum generation, Raman solitons, and long-wavelength dispersive waves [1,16]. Microstructuring is widely used for the development of fibers with required dispersion. Ordered structures with characteristic sizes of microns or submicrons are created using air holes or glass rods [15][16][17][18][19][20][21][22][23][24][25][26][27]. Waveguide modes in microstructured fibers are formed as a result of the interference of waves arising from reflection and refraction at the refractive-index microinhomogeneities [18]. In addition to the widespread photonic crystal fibers, in which several regular layers of air holes or glass rods are located around the core [16,26,28], there are suspended-core fibers, in which a thin core is surrounded by one row of air holes with thin walls between them [14,26,[29][30][31][32][33][34][35]. In addition, the manufacture of solid microstructured fibers from low-temperature chalcogenide and tellurite glasses simultaneously was reported [36].
Microstructuring can also be used to obtain multicore fibers, which is an emerging technology in fiber photonics. There exist structures formed by several regularly distributed cores with characteristic diameters of a few microns, in which the modes of individual cores physicochemical glass properties [57]. Modern technologies of deep dehydration of TeO 2 -WO 3 -La 2 O 3 glasses make it possible to obtain an extremely low content of hydroxyl groups (5.5 × 10 15 cm −3 , the corresponding volume absorption coefficient is~0.001 cm −1 [57]), which significantly reduces optical losses in the wavelength range of about 3 µm. In addition, TeO 2 -WO 3 -La 2 O 3 glasses have a large value of the nonlinear refractive index (n 2 = 6.9 × 10 −19 m 2 /W), more than 20 times higher than that for silica glass [56], which is important for the nonlinear optical conversion of radiation in the mid-IR range, including the generation of supercontinuum, Raman solitons, long-wavelength dispersive waves, etc. [1]. Moreover, TeO 2 -WO 3 -La 2 O 3 glasses are compatible with many other glasses with significantly different refractive indices (have similar thermal properties), which makes it possible to produce fibers with dispersion strongly different from material dispersion [56] due to a large waveguide contribution.
We studied two cases of core/cladding glasses: in the first case, the difference between the refractive indices of the cores and the cladding (dn) was small, equal to 0.3% of the refractive index of the core, and in the second case, the difference was very large, amounting to almost 20%. For a small difference in the refractive indices between the core and the cladding, the waveguide contribution leads to a shift of the zero-dispersion wavelength (ZDW) to a longer wavelength compared to the material ZDW, while for a large difference in the refractive indices, the waveguide contribution can lead to a ZDW shift to a shorter wavelength [56] and also to the formation of group velocity dispersion curves with two ZDWs [58]. The use of microstructuring, including the creation of multicore structures, gives additional opportunities for controlling the dispersion dependences. In both cases, we considered the core glass with the 69TeO 2 -23WO 3 -8La 2 O 3 (TWL) composition [56]. In the first case (dn = 0.3%), a glass with a composition close to TWL was taken as the cladding glass; we denoted it TWL2. In the second case for dn ≈ 20%, the 47.5GeO 2 −17.5TeO 2 −20ZnO−15Na 2 O (GTZN) glass [56] was used. The choice of this cladding glass was due to its physical compatibility with TWL in producing step-index fibers, which was experimentally demonstrated in [56]. TWL and GTZN glasses have glass transition temperatures T g of 425 and 401 • C, onset of crystallization temperatures T x of 660 and 565 • C, thermal expansion coefficients of 12.8 and 11.8 ppm/ • C, and softening points T s of 412 and 442 • C, respectively [56]. Refractive indices n for TWL and GTZN glasses depending on wavelength λ were determined by the Sellmeier formula [56]: with constants B, C 1 , C 2 , D 1 , and D 2 given in Table 1. The refractive index of TWL2 was set to be 0.997 of the TWL refractive index. The refractive indices of the considered glasses are shown in Figure 1a, and the calculated material dispersions (β 2 = ∂ 2 (n·ω/c)/∂ω 2 , where ω = 2πc/λ is the angular frequency, and c is the speed of light [1]) in Figure 1b. The material dispersion of the glasses is also shown in Figure 1c but at a magnified scale near ZDWs. Note that the ZDWs of TWL and GTZN are about 2.2 µm and 2.3 µm, respectively.
The refractive index of TWL2 was set to be 0.997 of the TWL refractive index. The refractive indices of the considered glasses are shown in Figure 1a, and the calculated material dispersions (β2 = ∂ 2 (n•ω/c)/∂ω 2 , where ω = 2πc/λ is the angular frequency, and c is the speed of light [1]) in Figure 1b. The material dispersion of the glasses is also shown in Figure 1c but at a magnified scale near ZDWs. Note that the ZDWs of TWL and GTZN are about 2.2 μm and 2.3 μm, respectively.  (1) using the Sellmeier constants from Table 1. Calculated dispersion of TWL and GTZN glasses in a wide wavelength range (b) and near ZDWs (c).

Fiber Geometry and Mode Structures
Let us now study step-index one-core fibers, two-core fibers, and multicore fibers with 4, 6, and 8 cores arranged in a ring. The diameter of each individual core is denoted by d and the distance between the cores by L (Figure 2). The diameter of the cladding was not very important, but it should be at least 120-150 μm not to affect the distribution of the mode fields in the cores and to ensure sufficient mechanical strength of the fiber. For step-index one-core fibers, we considered only the fundamental HE11 mode. For N-core fibers (N = 2,4,6,8), we studied supermodes with in-phase and out-of-phase field distributions. We did not analyze the possibility of exciting higher modes in each individual core (which is possible if the V-parameter is larger than 2.405 [1]). The problem of one-core fibers can be of independent interest since, for their certain parameters, the dispersion can be changed significantly in comparison with the material dispersion. In addition, by comparing the results obtained for multicore fibers with the ones obtained for one-core fibers, we could analyze the contribution of core coupling to dispersion characteristics and formulate fairly general conclusions.  Table 1. Calculated dispersion of TWL and GTZN glasses in a wide wavelength range (b) and near ZDWs (c).

Fiber Geometry and Mode Structures
Let us now study step-index one-core fibers, two-core fibers, and multicore fibers with 4, 6, and 8 cores arranged in a ring. The diameter of each individual core is denoted by d and the distance between the cores by L ( Figure 2). The diameter of the cladding was not very important, but it should be at least 120-150 µm not to affect the distribution of the mode fields in the cores and to ensure sufficient mechanical strength of the fiber. For stepindex one-core fibers, we considered only the fundamental HE 11 mode. For N-core fibers (N = 2, 4, 6, 8), we studied supermodes with in-phase and out-of-phase field distributions. We did not analyze the possibility of exciting higher modes in each individual core (which is possible if the V-parameter is larger than 2.405 [1]). The problem of one-core fibers can be of independent interest since, for their certain parameters, the dispersion can be changed significantly in comparison with the material dispersion. In addition, by comparing the results obtained for multicore fibers with the ones obtained for one-core fibers, we could analyze the contribution of core coupling to dispersion characteristics and formulate fairly general conclusions. Examples of the fundamental mode field for a one-core fiber and fields of supermodes for two-core, four-core, six-core, and eight-core fibers. All calculations were made for TWL/TWL2 core/cladding glasses, d = 10 μm, L = 15 μm at a wavelength of 2 μm. The color bar shown near the six-core fiber is the same for all fibers.
For an N-core fiber, there are N supermodes [38], the lowest (fundamental) of which with the maximum effective refractive index corresponds to the in-phase field distribution, and the highest one with the minimum effective refractive index corresponds to the Figure 2. Examples of the fundamental mode field for a one-core fiber and fields of supermodes for two-core, four-core, six-core, and eight-core fibers. All calculations were made for TWL/TWL2 core/cladding glasses, d = 10 µm, L = 15 µm at a wavelength of 2 µm. The color bar shown near the six-core fiber is the same for all fibers. For an N-core fiber, there are N supermodes [38], the lowest (fundamental) of which with the maximum effective refractive index corresponds to the in-phase field distribution, and the highest one with the minimum effective refractive index corresponds to the out-ofphase field distribution. Examples of field distributions in fibers with a different number of cores are demonstrated in Figure 2. Note that many supermodes, differing from the fundamental and the highest ones, were degenerate with respect to the effective refractive index (for example, II and III for a four-core fiber) [38], so their use was impractical. In addition, for supermodes with intermediate values of effective refractive indices, the field intensities in some cores can differ significantly from the maximum and even be equal to zero. For in-phase and out-of-phase supermodes, the intensities in each core were maximized, which is most practical for applications.

Numerical Methods
For one-core step-index fibers, we used the following simple approach to find effective refractive indices n eff and field distributions of the fundamental propagating modes. It is well-known that for an axially symmetric core, the characteristic equation obtained based on Maxwell's equations using the boundary conditions for the tangential field components for the HE 11 mode can be written in the form [59]: where J 1 is the Bessel function of the 1st kind of the 1st order, K 1 is the modified Bessel function of the 2nd kind (Macdonald function) of the 1st order, the prime denotes differentiation with respect to the argument, k 0 = ω/c, β = n eff ω/c, n Core and n Clad are refractive indexes of core and cladding glasses described in Section 2.1. The results obtained by this method for a set of fiber parameters were compared with the results obtained by a finite element simulation in the program COMSOL. We found a perfect agreement between them, which confirmed the correctness of our simulation based on homemade computer code.
To investigate the properties of microstructured multicore fibers described within the framework of the Maxwell system of equations [59], we performed numerical simulations in COMSOL. We found effective refractive indices n eff for all supermodes and their field distributions.
The dispersion β 2 for one-core and microstructured multicore fibers was calculated as where Re(n eff ) is the real part of the effective refractive index. We also calculated nonlinear fiber characteristics. We calculated the effective mode field areas as where the z-component of the Poynting vector is P z , and r is the coordinate in the plane normal to the z-direction [59]. The nonlinear Kerr coefficient is defined as here n 2 is the nonlinear refractive index; n 2 = 6.9 × 10 −19 m 2 /W for TWL and n 2 = 2.1 × 10 −19 m 2 /W for GTZN glasses [56].

One-Core Fibers
First of all, we analyzed one-core step-index fibers. In the case of TWL/TWL2 core/cladding glasses, we considered core diameters in the 7-12 µm range, reasonable at a small dn. Figure 3a shows the simulated dispersion. For this case, there was only one ZDW, slightly shifted from the material ZDW. The ZDW position for the considered fiber diameters is shown in Figure 3a by a solid black line. The thinner the core, the farther the ZDW was shifted to longer wavelengths due to the larger waveguide contribution. The thicker the core, the closer the fiber dispersion to the material one. For example, ZDW = 2.34, 2.25, 2.23 µm for d = 7, 10, 12 µm, respectively.

One-Core Fibers
First of all, we analyzed one-core step-index fibers. In the case of TWL/TWL2 core/cladding glasses, we considered core diameters in the 7-12 µ m range, reasonable at a small dn. Figure 3a shows the simulated dispersion. For this case, there was only one ZDW, slightly shifted from the material ZDW. The ZDW position for the considered fiber diameters is shown in Figure 3a by a solid black line. The thinner the core, the farther the ZDW was shifted to longer wavelengths due to the larger waveguide contribution. The thicker the core, the closer the fiber dispersion to the material one. For example, ZDW = 2.34, 2.25, 2.23 μm for d = 7, 10, 12 μm, respectively.  In the case of TWL/GTZN core/cladding glasses with a large dn value, we investigated thinner core diameters in the 1.5-6.5 µm range. The simulated dispersion is demonstrated in Figure 3b. For minimal diameters of about 1.5 µm, the dispersion was completely normal in the considered wavelength range of 1-5 µm. Starting from a diameter of 1.7 µm, two ZDWs appeared one of which was always shorter than the material ZDW. The second ZDW for small core diameters was shorter than the material ZDW, but with an increase in d, it shifted significantly to the long-wavelength range. So, for d = 3.1 µm, the second ZDW was 4.75 µm. For d ≥ 3.2 µm, only the first ZDW remained in the wavelength range of 1 to 5 µm. Note that at a wavelength of about 2 µm, where standard Tm:fiber lasers operate, the dispersion was anomalous for d ≥ 1.8 µm, which may be important for nonlinear optical pulse conversion and design of photonic devices.
Next, we calculated effective mode field areas and nonlinear Kerr coefficients. The wavelength dependences are plotted in Figure 4a,b for TWL/TWL2 core/cladding glasses and in Figure 4c,d for TWL/GTZN core/cladding glasses. In the near-IR range, the mode field for both TWL/TWL2 and TWL/GTZN core/cladding glass fibers was fairly well localized near the core. Therefore, the larger the core diameter, the larger the effective mode field area and the lower the nonlinear Kerr coefficient were. However, with an increase in the wavelength, differences in the qualitative behavior of these functions for two kinds of fibers were observed. For TWL/TWL2 core/cladding fibers, starting from λ~2.5 µm, a relatively thin core with a small dn could no longer confine the mode sufficiently well, so its field area started to increase significantly with an increase in λ. Whereas for larger values of d, the mode was still localized near the core. Therefore, at the wavelengths λ > 3 µm for TWL/TWL2 core/cladding fibers, the following behavior was observed: the smaller the core diameter, the larger the effective mode area, and the lower the nonlinear Kerr coefficient is (Figure 4a,b). For TWL/GTZN core/cladding fibers, the mode is localized quite well, even for d ≥ 3 µm, which can be seen in Figure 4c. But at smaller d at the wavelengths λ > 4 µm, the mode is poorly confined, and the effective mode area begins to increase quickly, crossing the curves for other d.
μm for TWL/TWL2 core/cladding fibers, the following behavior was observed: the smaller the core diameter, the larger the effective mode area, and the lower the nonlinear Kerr coefficient is (Figure 4a,b). For TWL/GTZN core/cladding fibers, the mode is localized quite well, even for d ≥ 3 μm, which can be seen in Figure 4c. But at smaller d at the wavelengths λ > 4 μm, the mode is poorly confined, and the effective mode area begins to increase quickly, crossing the curves for other d.

Mode Coupling in Two-Core Fibers
Now consider the mode coupling of optical signals propagating in two closely spaced cores. Hereinafter, we investigated TWL/TWL2 core/cladding fibers with d = 10 µm (for other diameters, the patterns are similar). For TWL/GTZN core/cladding fibers, we studied two options for the diameter d = 2 µm and d = 3 µm, since in some cases discussed below, there were qualitative differences between them. The schematic diagram is shown in Figure 5a. The fundamental modes propagating in cores with diameters d and distance L between the centers of the cores overlapped and interacted. The coupling coefficients were calculated as in [1] and plotted as functions of wavelength for different L/d ratios in Figure 5b for TWL/TWL2 core/cladding fibers and in Figure 5c,d for TWL/GTZN core/cladding fibers. The coupling coefficients were higher at longer wavelengths due to larger mode field areas (Figure 4a,c). As expected, the closer the cores, the higher the coupling coefficient.

Multicore Fibers with Different Numbers of Cores and Constant L/d Ratio
When considering multicore fibers, the mode field in each core is coupled with mode fields propagating in the remaining N-1 cores. For each core, the coupling is maximal with the two nearest neighbors (or with one in the case of two-core fibers) since the distance to them is minimal. The coupling coefficient, as shown in Section 3.2, was strongly dependent on a distance between the cores. For a four-core fiber, the distances from the center of the selected core to the others are L, L, and 2 1/2 ·L. For the six-core fibers, the distances are L, L, 3 1/2 ·L, 3 1/2 ·L, and 2 L. With an increase in the number of cores, coupling with the other neighbors becomes increasingly less significant. We investigated multicore fibers with N = 2, 3, 6, and 8. Six-core fibers are quite popular, and fibers with N ≥ 8 are rarely used. For N-core fibers with N = 10, 12, . . . cores arranged in a ring, it is expected that the dispersion properties will be practically the same as for eight-core fibers, since the ied two options for the diameter d = 2 μm and d = 3 μm, since in some cases discussed below, there were qualitative differences between them. The schematic diagram is shown in Figure 5a. The fundamental modes propagating in cores with diameters d and distance L between the centers of the cores overlapped and interacted. The coupling coefficients were calculated as in [1] and plotted as functions of wavelength for different L/d ratios in Figure 5b for TWL/TWL2 core/cladding fibers and in Figure 5c,d for TWL/GTZN core/cladding fibers. The coupling coefficients were higher at longer wavelengths due to larger mode field areas (Figure 4a,c). As expected, the closer the cores, the higher the coupling coefficient. Figure 5. Schematic diagram demonstrating mode field overlapping in a two-core fiber (a). Calculated coupling coefficients for two-core fibers with different L/d ratios for TWL/TWL2 core/cladding, d = 10 μm (b); TWL/GTZN core/cladding, d = 2 μm (c); TWL/GTZN core/cladding, d = 3 μm (d).

Multicore Fibers with Different Numbers of Cores and Constant L/d Ratio
When considering multicore fibers, the mode field in each core is coupled with mode fields propagating in the remaining N-1 cores. For each core, the coupling is maximal with the two nearest neighbors (or with one in the case of two-core fibers) since the distance to them is minimal. The coupling coefficient, as shown in Section 3.2, was strongly dependent on a distance between the cores. For a four-core fiber, the distances from the center of the selected core to the others are L, L, and 2 1/2 •L. For the six-core fibers, the distances are L, L, 3 1/2 •L, 3 1/2 •L, and 2 L. With an increase in the number of cores, coupling with the other neighbors becomes increasingly less significant. We investigated multicore fibers with N = 2,3,6, and 8. Six-core fibers are quite popular, and fibers with N ≥ 8 are rarely used. For N-core fibers with N = 10,12, ... cores arranged in a ring, it is expected that the dispersion properties will be practically the same as for eight-core fibers, since the distances to cores located across one are quite similar and coupling with farther cores can be neglected. We simulated the dispersion for the in-phase and out-of-phase supermodes of multicore fibers with L = 1.5d ( Figure 6) and compared with the simulated results for the one-core fiber plotted by bold red lines. The results for TWL/TWL2 core/cladding with d = 10 µm in a wide wavelength range and at a magnified scale near ZDWs are shown in Figure 6a,b, respectively. It was observed that the coupling between the modes of each individual core, in this case, made only a minor contribution. For in-phase supermodes, the dispersion curves were shifted slightly upwards, and the ZDW was slightly shifted towards longer wavelengths relative to the dispersion curve for the one-core fiber. For out-of-phase supermodes, the dispersion curves were shifted slightly downwards, and the ZDW was slightly shifted towards shorter wavelengths relative to the dispersion curve for the one-core fiber. The results obtained for six-and eight-core fibers practically coincided (the corresponding curves were indistinguishable in the graphs). For the in-phase supermode with N = 4, the result also coincided with the results for N = 6 and 8, but for the out-of-phase supermode with N = 4, there was a slight difference from the results obtained for out-of-phase supermodes with N = 6 and 8. The fact that the results for N = 2 differed from the others can be explained very simply by the presence of only one nearest neighboring core rather than two, in contrast to the other N.
one-core fiber. The results obtained for six-and eight-core fibers practically coincided (the corresponding curves were indistinguishable in the graphs). For the in-phase supermode with N = 4, the result also coincided with the results for N = 6 and 8, but for the out-ofphase supermode with N = 4, there was a slight difference from the results obtained for out-of-phase supermodes with N = 6 and 8. The fact that the results for N = 2 differed from the others can be explained very simply by the presence of only one nearest neighboring core rather than two, in contrast to the other N. Figure 6. The calculated dispersion in wide wavelength ranges (left column) and at magnified scales near ZDWs (right columns) for fibers with a different number of cores with L = 1.5d for TWL/TWL2 core/cladding with d = 10 μm (a,b); for TWL/GTZN core/cladding with d = 2 μm (c,d); and for TWL/GTZN core/cladding with d = 3 μm (e,f). In the right column, the dispersion curves for out-of-phase supermodes are located in the gray areas under the dispersion curves for one-core fibers.
Next, the dispersion curves for TWL/GTZN core/cladding fibers with d = 2 μm and N = 1,2,4,6, and 8 in a wide wavelength range and at a magnified scale near ZDWs are potted in Figure 6c,d, respectively. Here, the dispersions for multicore fibers were significantly different from the dispersion for one-core fibers since the coupling coefficients, as Figure 6. The calculated dispersion in wide wavelength ranges (left column) and at magnified scales near ZDWs (right columns) for fibers with a different number of cores with L = 1.5d for TWL/TWL2 core/cladding with d = 10 µm (a,b); for TWL/GTZN core/cladding with d = 2 µm (c,d); and for TWL/GTZN core/cladding with d = 3 µm (e,f). In the right column, the dispersion curves for out-of-phase supermodes are located in the gray areas under the dispersion curves for one-core fibers.
Next, the dispersion curves for TWL/GTZN core/cladding fibers with d = 2 µm and N = 1, 2, 4, 6, and 8 in a wide wavelength range and at a magnified scale near ZDWs are potted in Figure 6c,d, respectively. Here, the dispersions for multicore fibers were significantly different from the dispersion for one-core fibers since the coupling coefficients, as demonstrated in Section 3.2, were an order of magnitude higher than the values for the TWL/TWL2 core/cladding fiber. At wavelengths λ < 3 µm, the curves for six-and eight-core fibers practically coincided for both in-phase and out-of-phase supermodes, respectively. At λ > 3.5 µm, the mode fields were poorly localized near each core and strongly interacted not only with the nearest neighbors but also with more distant cores; therefore, the difference in the behavior of the curves was more pronounced. Atv λ < 3.5 µm, the dispersion curves for in-phase supermodes lay above, and for out-of-phase supermodes below the dispersion curve for the one-core fiber. In Figure 6d, the curves for out-of-phase supermodes are in the gray area. For in-phase supermodes, the distance between ZDWs decreased: The first ZDW was red-shifted, and the second ZDW was blueshifted. An interesting feature was that in the wavelength range of 1.65 µm < λ < 1.85 µm, the dispersion for in-phase supermodes in four-, six-, and eight-core fibers was flat and almost zero.
The dispersion curves for TWL/GTZN core/cladding fiber with d = 3 µm in a wide wavelength range and at a magnified scale near ZDWs are plotted in Figure 6e,f, respectively. The dispersion curves for six-and eight-core fibers were located very close to each other for the same supermodes over the entire wavelength range since the fields were confined strongly compared to the fibers with d = 2 µm. As before, the gray area in Figure 6f contains the dispersion curves for out-of-phase supermodes. The shortwave ZDWs, as seen in Figure 6f, was about 1.66 µm. For out-of-phase supermodes, there was only one ZDW in the considered range of 1.2 to 4.2 µm, while for in-phase supermodes, there were two ZDWs.

Six-Core Fibers
We examined the influence of the distance between the cores on the dispersion and nonlinear characteristics of six-core fibers. We took the values L/d = 1.2, 1.5, and 2. The results of dispersion calculations for TWL/TWL2 core/cladding fibers with d = 10 µm are presented in Figure 7a. It was observed that for all L/d ratios, the curves were located quite close to each other. The dispersion curves for TWL/GTZN core/cladding fibers with d = 3 µm are shown in Figure 7b. Here, the difference between the results for different L/d was quite pronounced. The lower the L/d ratio, the more the dispersion curves for the six-core fiber differ from the calculated results for the one-core fiber. The dispersions for TWL/GTZN core/cladding fibers with d = 2 µm over a wide wavelength range and at a magnified scale near the ZDWs are plotted in Figure 7c,d, respectively. Here, the difference in the behavior of the curves was even more pronounced in comparison with Figure 7b. For L/d ≤ 1.5 for an out-of-phase supermode, the dispersion was anomalous even around 1.55 µm, where standard erbium-doped fiber lasers can be used. At the same time, for an in-phase supermode, for small L/d, the dispersion was normal in the entire considered range. The effective mode field areas (left column) and nonlinear Kerr coefficients (right column) for TWL/TWL2 core/cladding fibers with d = 10 μm are demonstrated in Figure  8a,b), for TWL/GTZN core/cladding fibers with d = 2 μm in Figure 8c,d) and for TWL/GTZN core/cladding fibers with d = 3 μm in Figure 8e,f). The differences in the curve behavior were more noticeable for longer wavelengths. The effective mode field areas (left column) and nonlinear Kerr coefficients (right column) for TWL/TWL2 core/cladding fibers with d = 10 µm are demonstrated in Figure 8a,b, for TWL/GTZN core/cladding fibers with d = 2 µm in Figure 8c,d and for TWL/GTZN core/cladding fibers with d = 3 µm in Figure 8e,f. The differences in the curve behavior were more noticeable for longer wavelengths.
length range (c) and at a magnified scale near ZDWs (d).
The effective mode field areas (left column) and nonlinear Kerr coefficients (right column) for TWL/TWL2 core/cladding fibers with d = 10 μm are demonstrated in Figure  8a,b), for TWL/GTZN core/cladding fibers with d = 2 μm in Figure 8c,d) and for TWL/GTZN core/cladding fibers with d = 3 μm in Figure 8e,f). The differences in the curve behavior were more noticeable for longer wavelengths. To qualitatively explain the dependences observed in the left column in Figure 8, we considered the structure of the supermode fields at two different wavelengths (2 and 4 µm) for two different distances between the cores (L = 1.2d and L = 2d). We chose TWL/GTZN core/cladding fibers with d = 2 µm. In Figure 9, the top row corresponded to the field structures of the in-phase and out-of-phase supermodes simulated at a wavelength of 2 µm. Figure 9 also shows for comparison the structure of the field in a one-core fiber. It is observed that the fields were well localized near the core, and their structure was practically preserved even at L = 1.2d. Therefore, the mode fields calculated for six-core fibers with different L/d values almost coincided with the results of the calculations for six independent cores. For λ < 2 µm, the mode was localized even better, so the graphs of the functions A eff (λ) became indistinguishable. At longer wavelengths, the structure of the fields in each core was strongly deformed. The field structures of the in-phase and out-of-phase supermodes simulated at a wavelength of 4 µm are demonstrated in the bottom row in Figure 9. It is observed that the change in the out-of-phase supermode structure was more significant than for the in-phase one. The field was pushed towards the periphery of the structure. Therefore, the largest deviations from the effective field of six independent cores were observed for the out-of-phase supermode at the minimal L/d ratio.
of the functions Aeff(λ) became indistinguishable. At longer wavelengths, the structure of the fields in each core was strongly deformed. The field structures of the in-phase and outof-phase supermodes simulated at a wavelength of 4 μm are demonstrated in the bottom row in Figure 9. It is observed that the change in the out-of-phase supermode structure was more significant than for the in-phase one. The field was pushed towards the periphery of the structure. Therefore, the largest deviations from the effective field of six independent cores were observed for the out-of-phase supermode at the minimal L/d ratio. Figure 9. The fields of in-phase and out-of-phase supermodes for six-core fibers with different L/d ratios and the fundamental mode field of a one-core fiber calculated at a wavelength of 2 μm (upper row) and a wavelength of 4 μm (lower row). All results are obtained for TWL/GTZN core/cladding glasses, d = 2 μm.

Discussion and Conclusions
In this work, we calculated the dispersion and nonlinear characteristics of tellurite glass microstructured multicore fibers in the near-and mid-IR using the full-vector finiteelement method. The choice of glasses for cores and cladding was based on the compatibility of their physicochemical properties suitable for manufacturing fibers [56]. The numerical analysis presented in this work showed that the dispersion of multicore fibers could be effectively controlled by fitting the following parameters: core diameters, distance between centers, number of cores, as well as by choosing the operating supermode (in-phase or out-of-phase). Note that in the previous works devoted to multicore tellurite glass fibers, the out-of-phase supermode was not investigated, but numerical studies of dispersion of in-phase supermode have been reported [53,54].
The fields propagating in individual cores overlapped and interacted, thus making a contribution to the dispersion of supermodes. Coupling coefficients were larger at longer wavelengths due to larger mode field areas. The closer the cores, the larger the coupling coefficient. So, dispersion as a function of wavelength for microstructured multicore fibers Figure 9. The fields of in-phase and out-of-phase supermodes for six-core fibers with different L/d ratios and the fundamental mode field of a one-core fiber calculated at a wavelength of 2 µm (upper row) and a wavelength of 4 µm (lower row). All results are obtained for TWL/GTZN core/cladding glasses, d = 2 µm.

Discussion and Conclusions
In this work, we calculated the dispersion and nonlinear characteristics of tellurite glass microstructured multicore fibers in the near-and mid-IR using the full-vector finite-element method. The choice of glasses for cores and cladding was based on the compatibility of their physicochemical properties suitable for manufacturing fibers [56]. The numerical analysis presented in this work showed that the dispersion of multicore fibers could be effectively controlled by fitting the following parameters: core diameters, distance between centers, number of cores, as well as by choosing the operating supermode (in-phase or out-of-phase). Note that in the previous works devoted to multicore tellurite glass fibers, the out-of-phase supermode was not investigated, but numerical studies of dispersion of in-phase supermode have been reported [53,54].
The fields propagating in individual cores overlapped and interacted, thus making a contribution to the dispersion of supermodes. Coupling coefficients were larger at longer wavelengths due to larger mode field areas. The closer the cores, the larger the coupling coefficient. So, dispersion as a function of wavelength for microstructured multicore fibers can differ significantly compared to the one-core fibers. Effective mode field areas and nonlinear Kerr coefficients of N-core fibers did not dramatically differ from the same characteristics of N independent one-core fibers (the corresponding values differ by no more than 1.5 times). The differences in the curve behavior were more noticeable for longer wavelengths.
For fibers with similar refractive indices of cores and cladding glasses (in this work, dn was 0.3%), the dispersion of individual one-core fibers did not differ much from the material dispersion. The microstructured multicore geometry made the following contribution: a single ZDW shifted toward longer wavelengths for in-phase supermodes and toward shorter wavelengths for out-of-phase supermodes compared to one-core fibers. The value of this ZDW shift was several tens of nm (up to~100 nm).
With the use of fibers made of glasses with strongly different refractive indices of cores and cladding (in this work dn was almost 20%) with sufficiently thin cores of 2-3 µm, it was possible to achieve different qualitative and quantitative values of the group velocity dispersion in the near-and mid-IR ranges. For one-core fibers with such core properties, the dispersion already contained two ZDWs: The first ZDW was <2 µm and the second ZDW was >2.5 µm. For in-phase supermodes, the wavelength difference between the ZDWs decreased: The short-wavelength ZDW shifted to a longer wavelength, and the long-wavelength ZDW shifted to a shorter wavelength. For thin, closely spaced cores, there may arise a situation when ZDWs disappear, and the dispersion becomes all-normal. For out-of-phase supermodes, the wavelength difference between the ZDWs increased: the short-wavelength ZDW shifted to a shorter wavelength, and the long-wavelength ZDW shifted to a longer wavelength (or completely vanished). For out-of-phase supermodes, anomalous dispersion can be attained even at 1.55 µm, where standard Er-doped fiber lasers operate. This may be useful for the development of mid-IR sources utilizing a standard near-IR pump for supercontinuum generation, soliton self-frequency shift, etc.
The results of this work can be useful both for the design and development of dispersioncontrolled microstructured multicore tellurite fibers and for the development of fibers based on other glasses. The presented qualitative conclusions about the behavior of dispersion curves for in-phase and out-of-phase supermodes can be used as a basis for choosing parameters of other pairs of glasses to obtain the desired dispersion characteristics.

Conflicts of Interest:
The authors declare no conflict of interest.