## 5 SBS in a DOF
### Numerical Optimization of Longitudinal Fiber Profile for SBS Mitigation
The above analysis can also be extended to include the optimization of the profile with respect to the W-ratio of the fiber preform to reduce the acousto-optic overlap. In this investigation we fixed the value of the W-ratio to 2.3. Furthermore, the SBS threshold was evaluated as a function the dispersion oscillation period and the modulation depth which is the ratio of \(\bar{\beta}_{2}\) and \(\overline{\beta}_{2}\) [see Figure 14a]. Figure 14a shows that an increase in the modulation depth leads to an elevation of the SBS threshold. This is an expected trend as we reiterate from Equation (5) that the variation in the dispersion profile is only significant when the modulation amplitude is of the same order of magnitude or larger than the average dispersion. On the other hand, the dependence of SBS threshold on the dispersion oscillation wavelength was found to be feeble, albeit present. The inhomogenously separated bands that appear at different modulation wavelengths arise due to the variation of the phase-matching criterion, and is reminiscent of the band structure of modulation instability spectrum in a DOF [11]. As discussed, the Brillouin frequency shift for a uniform fiber represents a singular value that corresponds to the precise phase-matching condition necessary for the Brillouin process to occur. However, in the presence of perturbations or variations along the fiber length, the Brillouin gain can be dispersed over a range of frequencies. This dispersion of the gain implies that the phase-matching condition is satisfied for different frequencies at distinct locations along the length of the fiber. As a result, the Brillouin scattering process is not confined to a single, fixed frequency but is distributed over a spectrum of frequencies. As illustrated in Figure 15, for the system discussed in Figure 14, the periodic variations in the fiber's properties leads to a distribution of Brillouin frequencies throughout the fiber, effectively demonstrating the impact of modulation on the Brillouin process. In essence, the act of elevating the SBS threshold brings about a notable consequence--the flattening of the Brillouin gain spectrum. To gauge this phenomenon, we employ a quantitative measure: the 3 dB bandwidth, signifying the range over which the SBS threshold dwindles to precisely half its peak value. As demonstrated in Figure 14b, we graphically depict the relationship between this bandwidth and the corresponding threshold power for different values of modulation depth of three arbitrarily chosen modulation periods: \(z_{\text{mod}}=181\) m (red filled circles), 281 m (blue filled triangles) and 377 m (magenta filled stars). A clear trend emerges in the plot, showcasing that broader bandwidths align with higher threshold power values. In the context of a uniform fiber, this full Width at half maximum (FWHM) relates to the acoustic line-width, often denoted as \(\Gamma_{B}\), which itself is the reciprocal of the phonon lifetime \(\tau_{b}\). Remarkably, as shown in the figure, the introduction of dispersion oscillations magnifies this bandwidth manifold, surging from an initial 45 MHz to a substantial 250 MHz. This dramatic expansion in bandwidth equates  
Figure 14: **(a)** False color plot of SBS threshold power vs. dispersion oscillation wavelength \(z_{\text{mod}}\) vs. modulation depth \(\bar{\beta}_{2}/\bar{\beta}_{2}\). **(b)** SBS threshold power vs. 3 dB bandwidth of reflected Stokes wave for different modulation depths [along the grey dashed lines in (a)], considering different dispersion modulation wavelength: \(z_{\text{mod}}=181\) m (red filled circles), 281 m (blue filled triangles) and 377 m (magenta filled stars). These values of \(z_{\text{mod}}\) are demarcated with dashed lines in (**a**). Parameters of the DOF are provided in the text. to a staggering 5 dB augmentation in the threshold power.