## 2 Laser Optimized Multimode Optical Fibers with 100-\(\upmu\)m Core Diameter
We consider typical silica optical fiber geometry containing fused silica core doped by Germanium and Fluorine bounded by single pure fused silica outer solid cladding. It satisfies weakly guiding optical waveguide approximation. Here, unlike to the most known solutions, we apply stratification method [14] approach to describe complicated graded index refractive index profile. Therefore, in the core region, designed refractive index profile for large core LOMF with equalized selected mode delays is represented in the form of multilayered optical fiber. It contains finite set on \(N\) layers, where refractive index value stays constant. Desired refractive index profile form should satisfy minimization of some objective function \(F\), which is described by the following proposed simple formula:  
\[F=\sum_{j=1}^{M}\Bigl{(}t_{d(j)}-t_{REF}\Bigr{)}^{2} \tag{1}\]  
where \(t_{d(j)}\) is desired value of delay for \(j\)-th guided mode \(LP_{lm(j)}\) been computed at corresponding optimization step; \(t_{REF}\) is some reference value of mode delay, which is utilized for DMD diagram equalization; \(M\) is the total number of selected guided modes transferring laser-excited optical signal with normalized amplitude not less than \(A_{j}>0.1\) and core power (known also as optical confinement factor) not less than \(P_{co}^{(j)}\geq 0.5\). Objective function arguments are an array of local parameters \(h_{k}\), completely describing optical fiber refractive index profile under the stratification method approach:  
\[h_{k}=\frac{n_{max}^{2}-n_{k}^{2}}{n_{max}^{2}-n_{N}^{2}} \tag{2}\]  
where \(n_{k}\) is refractive index of \(k\) layer (\(k=0\dots\)\(N\)); \(n_{max}\) is the maximal core refractive index; \(n_{N}\) is outer cladding refractive index. During objective function (1) minimization, a direct problem of analysis of optical fiber with current refractive index profile should be solved at the each optimization iteration. For this purpose, we utilize earlier on developed modified [15] Gaussian approximation [16], extended for analysis and computation of arbitrary order guided mode transmission parameters in weakly guiding optical fibers with arbitrary axially symmetrical refractive index profile [15; 17; 18] (EMGA--Extended Modified Gaussian Approximation), under the following corresponding modification for 100-\(\upmu\)m core, which is described in detail in the publications mentioned above [10; 11; 12]. In the first step, we utilized LOMF 50/125 refractive index profile, taken from measurement report and further scaled up to 100-\(\upmu\)m core diameter, as the reference profile to define boundary values of the reference mode delay \(t_{REF}\) range by computing of selected guide mode DMD diagram. These measurements of refractive index profile mentioned above were performed by certified optical fiber analyzer for the sample of commercially available LOMF of Cat. OM2+/OM3 [19]. By using EMGA, we performed analysis of the reference 100-\(\upmu\)m core MMF over \"O\"-band, which included computation of mode field radius, propagation constant (with following test on satisfaction to cut-off condition) and optical confinement factor. According to computational results, the reference graded-index 100-\(\upmu\)m core MMF supports propagation of 145 modes \(LP_{lm}\) (\(l=0...24\); \(m=1\dots\) 11) satisfying to the cut-off conditions with optical confinement factor more \(P_{co}>0.5\). By taking into account described above features of laser-excited optical signal propagation over MMF in a few-mode regime, we do not need to equalize mode delays over all guided modes unlike to multimode regime. Here only guided modes \(LP_{lm}\) with normalized amplitudes more \(A_{j}>0.1\) and optical confinement more \(P_{co}^{(j)}\geq 0.5\) should be considered under particular launching conditions.