## 5 SBS in a DOF
### Numerical Optimization of Longitudinal Fiber Profile for SBS Mitigation  
In a uniform optical fiber, the Brillouin Stokes wave naturally builds up throughout the fiber at a specific frequency, which is determined by the fiber's modal characteristics. However, when we introduce variations in the transverse profile, whether caused by mechanical stress or engineered during the fiber drawing process, a single Brillouinfrequency shift is no longer sustainable. As a result, it becomes imperative to scan across a frequency range to determine the Brillouin frequency shift corresponding to the maximum Brillouin gain. The extent of this frequency range depends on the range of variation in the modal characteristics of the fiber incurred due to the transverse change in the fiber profile. Thus for a DOF our objective is to identify the optimal perturbation in the longitudinal fiber profile that maximizes the suppression of SBS. To achieve this, first we compute the modal parameters for various values of the fiber diameter. The core radius \(r(z)\), is discretized along the length of the fiber and the SBS threshold power across a range of Brillouin frequencies is computed numerically using the steady-state approach, with a weak (\(10^{-15}\) W in power) seeded Stokes wave. We employ a non-linear optimization approach using the predictor-corrector interior-point method (within the optimization toolbox in _MATLAB_[53]) to find a \(r(z)\) that leads to the highest SBS threshold. This method relies on a gradient-based optimization approach and assumes that the objective function is differentiable. Starting from an initial guess, the algorithm computes subsequent points in the direction of gradient descent. Given the unpredictability of the objective function's variation with respect to a large set of unknown parameters, we also implement both gradient-based and gradient-free pattern search approaches to locate the global optima function for \(r(z)\). The pattern search method, being gradient-free, is suitable for functions that are non-smooth and non-differentiable. This technique calculates the functional value at multiple initial points around an initial guess and proceeds toward the direction of the minimum function value. In our investigation, we observed that the SBS threshold for a sinusoidal variation profile was higher compared to other intuitive designs. Therefore, we adopted a sinusoidal profile as the initial guess for all the optimization algorithms. In Figure 13a, the radius profiles (optimal and standard) are shown. The corresponding threshold power variation with frequency is plotted in Figure 13b. The SBS threshold power for the optimal profiles are about 14 mW, around 3 times the threshold computed for a constant profile of \(r=2\)\(\upmu\)m. In fact the sinusoidal profile, which by the way was the optimal profile for the PSA case, also shows an improvement of a factor of two in terms of its SBS threshold power. While the power-frequency response corresponding to both the optimal profiles are similar, the radial variation of the profiles are considerably different, implying that multiple profiles could correspond to a similar threshold characteristics. This also opens new perspectives in terms of investigating the PSA performances of the SBS-optimized radius profiles. Clearly from Figure 13b, we see that the perturbation in the longitudinal radius profiles lead to a flattening effect on the gain spectrum which is typically a Lorentzian distribution for a uniform profile. Figure 13: (**a**) Variation of core radius along the length of the fiber for different profiles. (**b**) Numerically computed SBS threshold power as a function of the seed frequency for different profiles. Uniform profile in red, linear profile in blue, sinusoidal profile in magenta, computed optimal profiles through interior point method in cyan and through pattern search method in black. Other parameters are provided in the text.