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Abstract
The objective of the study is to optimize the optical fiber structure for mode-division multiplexing systems using nanostructurization. The nanostructuring technique allows to fabricate fibers with arbitrarily designed (free-form) refractive index distribution based on two glasses. Three optimization schemes have been proposed. The nanostructuring method allows for designing fibers with optical properties similar and even better parameters impossible to produce by other methods. In this proposal, we examined four linearly polarized (LP) few-mode fibers. We report a high effective refractive index difference between modes while maintaining other important parameters for the weakly coupled approach.
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1. Introduction
A constant increase in bite rate in single-mode network systems has led researchers to develop space-division multiplexing (SDM) [1,2]. This concept of encoding and decoding signals using spatially different channels is currently the latest technology for increasing data capacity. In the multi-mode or multi-core operation mode, optical fibers can act without any limitations when multiplexing techniques (e.g., wavelength, polarization, time) are adopted, allowing for bandwidth multiplication within several supported spatial modes [3]. Both types of optical fibers, i.e., multi-mode and multi-core, are of great interest to researchers. Still, a few-mode fiber (FMF) is considered an elementary concept, while the multi-core fiber is the result of solution parallelization. Therefore, the full potential of SDM generally lies in single-core FMFs, which later can be applied to multiple structures.
For single-core fibers, mode-division multiplexing (MDM) [4] is commonly used to reinforce the importance of guided spatial modes representing independent channels. Each channel could be adapted to transmit data. However, under the long-distance transmission, an output signal in the form of modes is time-delayed or generally degraded by inter-modal crosstalk. To reduce these undesirable phenomena, it is essential to properly design and fabricate the FMF for long-distance transmission. Nowadays, two main approaches to designing FMFs for MDM systems are used [5,6].
The first approach is called strongly-coupled and is focused on reducing the differential mode delay (DMD) by appropriately modifying the refractive index distribution in the gradient-index FMFs [7]. While the FMF is adequately designed, the coding concept is based on the fact that all transmitted modes can be simultaneously detected on a complex 2N×2N (2 polarizations × N spatial modes) MIMO technique [8], regardless of the mode coupling. The second preference is the weakly coupled approach, which reduces the mode coupling by maximizing the separation of adjacent LP modes’ propagation constants. This makes it possible to simplify the receiving circuits, reduce the computational requirements of the MIMO technique, or even allow full resignation in the MIMO technique (MIMO-free) [9]. Which ultimately results in lower complexity and power consumption in prospective data centers.
The most critical parameters for the weakly coupled approach have been estimated based on numerous scientific works [5]. The first one, the effective refractive index difference (Δneff) between adjacent LP modes, should reach values higher than 0.8 × 10−3 (preferably 1 × 10−3) [5]. This property is inversely proportional to the mode-coupling coefficient, and when the minimum Δneff for all propagating modes exceeds 1.7 × 10−3, the coefficient becomes constant (7.4 × 10−5 1/km) [10]. The latest experimental record achieved in the weakly coupled 6-LP FMF is 1.49 × 10−3 of Δneff [11]. Another important parameter is the effective mode area (Aeff), which should have a value above 80 µm2 (preferably >100 µm2) [5] for all guided LP modes to limit both intra- and inter-mode nonlinearities. Otherwise, a small Aeff can lead to high energy density, with nonlinear effects introducing mode instabilities. The last parameter strictly refers to the practical application of optical fibers, quantifying their resistance to unavoidable bending. Thus, to reduce unwanted attenuation, an upper limit commonly assumed in numerical models is 1 dB/turn (in any guided LP mode) for a 10 mm bending radius [5]. All requirements mentioned above have to be met while designing weakly coupled FMFs for MDM systems to ensure long-range transmission for all supported LP modes.
In recent years, the most efficient solutions recognized for MDM are few-mode optical fibers based on step-index fiber [12] or composed of a step-index core divided into rings with different refractive indices [13–15,11]. Examples of such fibers and their basic parameters are presented in Table 1. The limitations of circular symmetry are mainly related to the fabrication method, which is the widely known method of modified chemical vapor deposition (MCVD) [16]. However, a novel methodology with a high potential to enhance FMF performance in SDM applications may be the nanostructuring method [17]. This fiber design and development technique allows for the tailoring of optical properties by free-form arrangement (not necessarily in circular symmetry) of internal nanostructures made of at least two glasses differing in the refractive index. An all-solid and arbitrary refractive index distribution is obtained by the well-developed stack-and-draw technique, mainly used to fabricate photonic optical fibers. The refractive index pattern in the core and/or cladding prepared in such a way is averaged by incident light according to Maxwell-Garnett mixing theory [18] and can be used to effectively modulate previously specified optical properties. The broad range of flexibility not only allows for optimizing the FMF core to meet the requirements of the weakly coupled approach but also gets to the physical limits of this approach. Since 2015, our group has several times experimentally proved the feasibility of nanostructured optical fibers and their high application potential [19,20]. We have also shown that nanostructured fibers can be spliced into standard fibers and the joint losses are similar to those for non-nanostructured fibers [19].
Table 1. Selected theoretical and experimental weakly coupled FMFs optimized at 1550 nm
In this article, we study the implementation of the nanostructurization method to improve the performance of weakly coupled 4-LP FMF. We present an optimization method to freely design FMF with selected geometrical parameters (forced symmetry) and selected optical properties (forced equal distance between the effective refractive indices of the guided modes). Although the presented analysis is for a structure supporting only six spatially distinguishable modes (4-LP), its extension to FMFs with more modes is relatively simple [21] due to its high design and optimization freedom. To our knowledge, the current study is the first implementation ever of the nanostructurization method in weakly coupled FMFs for MDM systems.
2. Concept of weakly coupled FMFs
The great majority of weakly coupled FMFs proposed are step-index structures with additional core modifications. The modification involves adding different structural elements in the core, such as low-index nanopores or high- or low-index rings, to achieve a higher minimum Δneff between adjacent LP modes. Many optical fibers are additionally trenched around the core to ensure lower bending losses of guided modes, below 1 dB/turn at a bending radius of 10 mm. A complete chronological overview of theoretical and experimental weakly coupled FMFs is presented in Table 1. The table includes the FMF type with reference and year, the kind of publication result, the number of guided modes in the optimized fiber, the minimum Δneff,min, and Aeff,min, as well as the maximum bending loss for a 10 mm radius (L10mm) and the maximum differential group delay relative to the LP01 mode (DGDLP01). The extracted parameters indicate the particular LP modes that address these values in brackets. All structures shown in Table 1 have circular symmetry. Symmetry is a consequence of the fact that this type of refraction index distribution can be fabricated with the commonly used MCVD technology. The advantage of the fabricated radial core symmetry is that it maintains low transmission losses and compatibility with other fibers used.
The record FMF in terms of inter-mode separation (Δneff) and guiding 4-LP modes is the design by Jiang et al. [14], which achieved a minimum Δneff value between adjacent LP modes of 1.8 × 10−3 by adding a higher refractive index ring in the core. In the rest of this paper, we will explore the possibility of improving the FMF properties for 4-LP modes using the nanostructuring technique.
3. Concept of the nanostructurization
In a nutshell, the process of fiber fabrication consists of two main steps. In the first one, a preform is prepared, and in the second, the preform is thinned on an optical tower. The preform is a cylindrical element in which the transverse distribution of the refraction index is an enlarged version of the distribution expected to obtain in the target fiber. Such a preform is produced, for example, by using the standard MCVD technique [16]. Drawn at high temperature, the preform reduces its diameter, forming the target fiber, while the transverse refractive index distribution is maintained (Fig. 1(a)). However, due to the use of the MCVD, fibers with circular refractive index symmetry are the easiest to produce.
As opposed to the MCVD, the nanostructurization technique offers a chance to divert from the circular symmetry and fabricate fibers with a free-form (arbitrary) refractive index distribution [30,31]. Compared to the standard method described above, the preform is not fabricated directly but is arranged from thousands of smaller elements - rods, made of at least two thermally matched glasses of different refractive indexes. After drawing on the optical tower, the individual rods merge into the final fiber structure with the proper parameters of the drawing process. However, the spatial distribution of the rods, and hence the spatial distribution of the refractive index, remain essentially unchanged (Fig. 1(b)).
The basis of nanostructurization is the effective medium theory described by the Maxwell-Garnett effective medium approximation (EMA) model [18]. In this approach, the physical properties of a dielectric material composed of discrete elements with different physical properties are assumed to be a weighted average of the properties of the individual components in a local neighborhood. In the case of the refractive index n, it can be written as [18]:
where fi defines the concentration and ni the refractive index of the ith component, respectively, whereby, the condition ∑fi = 1 is satisfied. Importantly, for the material properties to be considered effective, the size criterion must be satisfied. It states that the size of individual elements cannot be larger than λ/2π [18]. This theoretical limitation can be significantly relaxed in real systems where diffusion occurs. As we have verified experimentally in our previous work [32,33], the effective medium condition is satisfied for individual element sizes comparable to λ/3.4. Nanostructurized weakly coupled FMF
Glass properties are the first aspect to determine when using nanostructurization in fiber production. In the current study, we use undoped and germanium (GeO2) doped silica glasses. Such similar glasses provide thermal matching, which is necessary to draw a double-glass fiber on an optical tower. Therefore, it is crucial to determine the required level of GeO2 doping that will provide flexibility in the design of the FMF.
The starting point of the analysis presented below is the fiber structure shown in Fig. 2(a) based on a conventional FMF with three circular symmetry rings [14]. Assuming the geometrical parameters and a constant difference of 2 mol.% for a high GeO2-doped silica ring, the fibers were studied in terms of the effective refractive indices for all guiding modes as a function of the germanium doping level (Fig. 2(b)).
Fig. 2. The initial structure of FMF: (a) geometrical parameters and doping level of the fiber, (b) effective refractive index as a function of doping level X for a wavelength of 1.55 µm. The range of guiding only 4-LP mods is highlighted.
The analysis carried out (Fig. 2(b)) shows that 4-LP modes are guided only within the range of GeO2 doping from X = 4.4 to X = 6.75 mol.%. This corresponds to the doping at the level from 6.4 to 8.75 mol.%, respectively, of the higher-refractive-index ring (marked dark green in Fig. 2(a)). Above this range, further higher-order modes (HOM) occur. The selection of germanium doping above 6.75 mol.% results in the propagation of more modes in the structure. However, for modes higher than LP02 it is possible to reduce the neff below 1.444 (the refractive index of the cladding made of pure silica for λ=1.55 µm) so that HOM cannot be guided in the fiber. For this purpose, low refractive index elements can be added, the area covered by high refractive index elements can be reduced, the geometry of the structure can be changed, or the diameter of the core can be reduced. However, the results obtained here showed an approximate range of dopant concentration in which optimal solutions would have to be found using the nanostructuring technique.
The next step on the way to obtaining a nanostructured FMF is to determine the geometrical parameters of the rods from which the preform will be stacked, according to the scheme shown in Fig. 1(b). The GeO2 doped silica preforms, obtained by MCVD are cylindrical, with the doped region covering the central part of the preform (Fig. 3(a)). In further analysis, we assumed that the ratio of doped to the undoped area is R = 0.58, which corresponds to the available silica preforms doped with GeO2. Since, as shown above, the necessary doping level for guiding 4-LP modes is between 6.4 and 8.75 mol.%, the final dopant concentration, in the central part of the rod, taking into account the ratio R, should be in the range of 11 to 15 mol.%. Finally, a preform of the target fiber is assembled from doped rods and undoped rods of the same diameter (Fig. 3(b)).
Fig. 3. Fabrication of nanostructured preform: (a) preform of a single GeO2 doped rod, (b) preform stacking to the target fiber, (c) refractive index distribution in the final fiber after preform drawing, (d) example fiber core preform composed of 2482 rods arranged in a triangular grid with 63 elements on the diagonal.
Importantly, the target fiber needs to satisfy the effective medium criterion for the designed wavelength λ=1.55 µm, i.e., that the size of a single inclusion should not be larger than λ/2π [18]. Therefore, in further consideration, we assumed a lattice constant of the target fiber Λ=300 nm, which corresponds to a doping inclusion size of d≈230 nm (Fig. 3(c)). As a result, in this study, the target fiber's core preform consisted of 2482 rods arranged in a triangular lattice with 63 elements on the diagonal (Fig. 3(d)). The effective refractive index can be calculated in this case by transforming Eq. (1) to the form:
where nSiO2 and nGeO2 are the refractive indices of pure silica and doped silica, respectively and f is the local filling factor.5. Optimization algorithm
The key to obtaining a nanostructured fiber with given optical parameters is determining the optimal refractive index distribution. Since the fiber preform consists of thousands of elements of two types, it is impossible to test all 22482 combinations. Thus, an optimization process is needed to determine the distribution of individual rods.
The basis of the optimization process is to define a fitting function (FF) that quantifies how well our requirements are fulfilled. The FF can include various parameters that are important to the performance of the target fiber. In the case of the FMF guiding 4-LP modes, we define the FF as follows:
where wi determines the weights with which the quantities D and E are taken into account, whereby the condition w1 + w2 = 1 is met. The parameter D denotes the Δneff between neff of LP01 and LP02 modes:The parameter E defines the deviation from a uniform distribution of neff for all four modes:
The starting point of the optimization procedure is the structure shown in Fig. 3(d). It consists of 2482 rods of two types. The GeO2 doped rods are placed inside the structure forming a circle of 16 um diameter. The other rods are made of undoped glass. The proposed optimization algorithm is shown in Fig. 4. After initialization, where weights w1 and w2 are set, a single rod is randomly selected and replaced with a rod of the other type. The random selection of rods in subsequent optimization steps was used as the most general one and does not assume any a priori knowledge about the final structure. The FF is then determined, indicating whether the change has improved or worsened the fiber properties. If degradation has occurred, the change in rod type is canceled. The procedure of drawing the rod, replacing it, checking the FF, canceling the change, or preserving it is iterated until the fiber parameters reach the expected values or the maximum number of algorithm steps is reached. Whereby, as the criterion for the end of the optimization algorithm, the situation when the LP02 mode stops being guided. At the same time, from the loss point of view, the optimal structure is obtained when the fifth LP31 mode stops being guided. Importantly, the fit function can be supplemented as needed with additional parts, e.g. with a component responsible for minimizing bending losses, component setting limits on the size of effective mode area Aeff, or a component limiting the number of GeO2 doped rods used. Importantly, more efficient Genetic and AI-based optimization algorithms can be used to maximize the fit function (FF). They lead to similar solutions with shorter optimization times.
Fig. 4. The optimization procedure block diagram. The subscript i denotes the ith iteration. The randomly changed rod is highlighted in yellow.
As a mode solver for calculating the supported modes of the waveguide used the Finite Difference Method (FDM) implemented in the BMP-Matlab open-source optical propagation simulation tool [34]. In the simulations, it was assumed that one pixel corresponds to a size of 50 nm. This allows us to obtain results with good accuracy and, in addition, allows us to take into account the diffusion effect that occurs in real drawing nanostructured fibers.
6. FMF structure with forced symmetry
The algorithm for finding the optimal distribution of the rods does not assume any symmetry. However, we have forced a hexagonal symmetry in the first approach to better compare with existing symmetrical solutions. The optimization was performed for different GeO2 doping in the range from 11 to 16 mol.% with a step of 1 mol.%. The highest concentration of 16 mol.%, beyond the framework defined earlier, was added to test the performance of the fiber at high doping levels. The weights present in Eq. (3) were arbitrarily set to the constants w1 = 0.6, w2 = 0.4, which primarily forces increasing Δneff between LP01 and LP02 modes. The obtained optimal structures for different dopings, marked with colored dots are shown in Fig. 5. They show, among others, that the GeO2 doping at the level of 11 mol.% is too small to perform an efficient optimization. Moreover, such fiber is highly similar to a step-index fiber and effectively does not guide 4 modes. For higher doping, the optimization requires more steps and decreases the proportion of GeO2 doped rods with increasing doping (inset in Fig. 5). For doping above 11 mol.%, in the central part of the structure, a region appears where some doped rods have been replaced by undoped rods. This means that it is necessary to reduce the refractive index in the center of the structure to obtain better separation of the modes. Importantly, regardless of the doping level above 11 mol.%, the diameter of the central region with a reduced refractive index is practically the same. As the doping increases, the proportion of undoped rods at large distances from the center of the structure also increases, which can be interpreted as reducing the core diameter to limit the guiding of HOMs. For the highest doping level - 16 mol.%, additionally, the first undoped rods appear within the doped circular area in the center.
Fig. 5. Optimization of nanostructured FMFs with forced hexagonal symmetry leading 4-LP mods for different GeO2 doping concentrations. The fit function (FF) depends on the number of rods type replacements. The colored dots on the main graph indicate the number of optimization steps at which the fiber stops guiding 5 modes and becomes a 4-mode fiber. Inset - dependence of the number of GeO2-doped rods on their doping.
Due to the forced hexagonal symmetry, a similarity can be observed between the obtained nanostructured fibers and traditional FMFs composed of concentric rings (Fig. 6). This similarity is also evident in the details shown in Table 2.
Fig. 6. Comparison (a) of effective refractive index cross-section of the optimal structure (b) composed of rings [14] and nanostructured fiber (c) doped with GeO2 at 15 mol.% with forced hexagonal symmetry.
Table 2. Comparison of basic parameters of FMFs leading 4-LP modes
The dynamics of the optimization process are shown in Fig. 7 for a structure doped at 15 mol.%. At the beginning of the optimization, the investigated fiber guides five modes: LP01, LP11, LP12, LP02, and LP13. As the simulation progresses, the neff of all modes decreases (Fig. 7(a)). This increases the distance between LP01 and LP02 modes (parameter D, Eq. (5)) and to cutting mode LP13 off. The algorithm also tries to modify the rods’ arrangement so that the neff of the remaining modes are uniformly distributed, in accordance with parameter E (Eq. (5)). As a result of the optimization, in line with the criteria established for the current study, the optimal structure was obtained in iteration 106 (Fig. 6(c)). It consisted of 2058 doped and 424 undoped rods.
Fig. 7. Dynamics of the optimization process of nanostructured FMFs with forced hexagonal symmetry for GeO2 doping at 15 mol.% level: (a) change of effective refractive indices of guided mods as a function of the number of replacements of rods types (black dashed lines show a uniform distribution of neff), (b) istribution of the field in each mode for the optimal structure (vertical red line), (c) evolution of the optimized structure in successive optimization steps, the number indicates the optimization step, (d) evolution of Δneff of the neighboring mods, (e) evolution of Aeff. The red dashed vertical lines denote the iteration step of the algorithm in which the structure stops guiding mode LP13.
Two more parameters are important for the practical use of FMFs: differential group delay (DGD) and bending losses. DGD, in the weakly coupled case, increases with the number of guided modes [12,22], and was determined based on the formula:
7. FMF structure without forced symmetry
The optimization process without forced symmetry leads to the structures shown in Fig. 8(b). However, similarly to the simulation with forced symmetry, the obtained distributions of rods show high circular symmetry. An increase in the proportion of undoped rods both in the center and periphery of the structure is also observed. In the first optimization phase, the algorithm increases the Δneff between the LP01 and LP02 modes. This results in the decreasing neff of the LP31 modes until it fully vanishes when it reaches a value equal to the cladding refractive index at 667 iteration. Earlier, at about 500 iteration, it can be seen (Fig. 8(a)) that the neff of the LP21 mode reaches a value that satisfies the criterion of equal distance between modes. This leads to the zeroing of the second component of Eq. (5) and a noticeable change in the rate of change of the Δneff in successive iterations (Fig. 8(c)). The optimal solution for the fixed weights w1 = 0.6 and w2 = 0.4, unchanged during optimization, was obtained for iteration 667. It consisted of 1987 doped and 495 undoped rods. The fiber optimized in such a way is characterized by a slightly higher Δneff of the LP01 and LP02 modes, compared to the earlier fiber (Table 2). The main changes are in the Aeff, which decreased from 139 to 118 µm2 for the LP11 mode and increased from 145 to 168 µm2 for the LP21 mode.
Fig. 8. Dynamics of the optimization process of nanostructured FMFs without forced symmetry for GeO2 doping at 15 mol.% level: (a) change of effective refractive indices of guided mods as a function of the number of replacements of rods types (black dashed lines show a uniform distribution of neff), (b) distribution of the field in each mode for the optimal structure (vertical red line), (c) evolution of the optimized structure in successive optimization steps, the number indicates the optimization step, (d) evolution of Δneff of the neighboring mods, (e) evolution of Aeff. The red dashed vertical lines denote the iteration step of the algorithm in which the structure stops guiding mode LP13.
The DGD parameter reaches values comparable to the reference fiber and the fiber with forced symmetry. On the one hand, the bending loss for the last LP02 module increased to 14.3 dB/turn at a radius of 10 mm. On the other hand, the future structure can be further equipped with a trench of a lower refractive index, which effectively reduces the loss of fiber for all propagating modes [35]. The optimization process can also be stopped a few dozen iterations earlier. Then, the bending losses of all modes will be reduced, and mode LP31 will be eliminated by naturally occurring fiber bends. It is also possible to add appropriate terms to the fitting function (FF) responsible for minimizing bending losses and Aeff.
8. FMF with uniform distribution of modes
In both optimizations presented above, the weights w1 and w2 were time-invariant and favored increasing the Δneff between LP01 and LP02 modes. In the third approach, the weights w1 and w2 were changed once. The structure obtained in the 481st optimization step without forced symmetry was used as a starting point. The choice here is arbitrary and any other structure can be used. However, choosing a structure that is partially optimized in terms of obtaining the largest Δneff provides relatively low crosstalk between modes and reduces the time of the new optimization.
Further optimization was carried out at w1 = 0.1 and w2 = 0.9. This choice of weights is arbitrary but gives slight pressure to increase Δneff (parameter D and weight w1) and strongly force a uniform distribution of neff of all guided modes (parameter E and weight w2). The optimization process results from 400 iterations are shown in Fig. 9. Changing the weights at step 481 induces the neff of all guided modes to increase in subsequent steps until around step 700 (Fig. 9(a)). They then begin to decrease, such that at 881 iterations, mode LP31 is no longer guided. And according to the criteria assumed earlier, this structure is considered optimal for the given optimization parameters. This optimal structure consists of 2180 doped and 302 undoped rods. An analysis of the Δneff of the neighboring modes (Fig. 9(c)) shows that, as expected, the modes distribute more and more uniformly, reaching equal distances at iteration 1084. The optimization also leads to small changes in the Aeff of modes LP11, LP21, and LP02 (Fig. 9(d)). However, the Aeff of the LP01 fundamental mode is significantly reduced to 100 µm2. This is due to strong symmetry breaking, which can be observed in images of structures from selected steps of the optimization procedure (Fig. 9(b)). Importantly, a noticeable shift of the central, a less doped region, visible at about the 700th iteration, is necessary to achieve an equal distance between the modes. The direction of this shift is random and depends on the order in which the rods are chosen for conversion.
Fig. 9. Dynamics of the optimization process of nanostructured FMFs without symmetry, with forced optimization of the equal Δneff between neighboring modes for GeO2 doping at 15 mol.%: (a) change of effective refractive indices of guided mods as a function of the number of replacements of rods types (black dashed lines show a uniform distribution of neff), (b) distribution of the field in each mode for the optimal structure (vertical red line), (c) evolution of the optimized structure in successive optimization steps, the number indicates the optimization step, (d) evolution of Δneff of the neighboring mods, (e) evolution of Aeff. The black and red dashed vertical lines denote the iteration step of the algorithm in which the weights w1 and w2 were changed, and the structure stops guiding mode LP13, respectively.
The obtained results show that forcing equal Δneff for neighboring modes does not improve the properties of FMF. However, they show that nanostructuring allows for optimizing its selected properties.
9. Conclusions
We have presented how using nanostructurzation and numerical optimization techniques can improve the properties of weakly coupled few-mode fibers for MDM systems. The optimization was carried out to obtain the largest possible Δneff between the neighboring LP modes while maintaining a large Aeff, high DGD, and low bending losses. Three approaches were presented: (1) enforcing symmetry of the optimized fiber, (2) assuming no symmetry, and (3) aiming to achieve the same Δneff difference between neighboring guided LP modes. Optimization was performed in terms of the GeO2 doping level and the spatial refractive index distribution for a wavelength of 1.55 µm.
As the initial nanostructure, we chose a core consisting of subwavelength glass rods with a higher refractive index forming a circular distribution based on a hexagonal lattice with a diameter of 16 µm. For forced symmetry, a fiber with a doping level of 15 mol.% consisting of 2058 doped and 424 undoped rods was identified as the optimum. It achieved a minimum Δneff of 1.82 × 10−3 while maintaining an Aeff above 100 µm2 for all four modes. In the case without forced symmetry, a fiber consisting of 1987 doped and 495 undoped rods was optimal. This fiber achieved a minimum Δneff of 1.86 × 10−3 while maintaining all other parameters required for weakly coupled few-mode fibers. In the third approach, the optimization algorithm provided the same Δneff between the neighboring LP modes. In this case, a fiber consisting of 2180 doped and 302 undoped rods was optimal. This fiber met the requirements for weakly coupled few-mode fibers and achieved a minimum Δneff of 1.98 × 10−3.
In all fibers, relatively high bending losses were obtained for the LP02 mode, which can be a potential problem. However, the future structure can be further equipped with a trench with a lower refractive index, which effectively reduces the losses in the fibers for all propagating modes. It is also possible to stop the optimization process a dozen or more iterations earlier. Then the bending losses of all modes will be reduced, and any HOMs occurring will be eliminated by naturally occurring fiber bends. It is also possible to add appropriate terms to the fitting function (FF) responsible for minimizing guiding and bending losses.
Nanostructurization and the optimization method can be successfully used to fabricate free-form fibers without circular symmetry whose properties do not differ from those offered by other methods. Moreover, the optimization procedure allows for tailoring different optical parameters of the fiber. It is also possible to introduce changes during the optimization process as to which fiber or its optical parameters are to be considered, and with what weight at a given stage.
Although the presented optimized FMF is limited to 4-LP modes, the approach can be successfully extended to more spatial modes. In addition, the designed fiber is easy to implement practically into optical fiber systems due to its all-glass structure, which provides easy splicing yet low transmission loss. This opens up new possibilities for shaping the modal characteristics of silica fibers for multimode transmission applications, particularly for MDM systems and future quantum computer networks.
Funding
H2020 Leadership in Enabling and Industrial Technologies (no. 825051); Lukasiewicz Research Network (1/L-IMiF/CL/2021).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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