For elastic scattering, the scattering potential can only change the outgoing spherical wave function up to a phase. In the asymptotic limit, far away from the scattering potential, we get for the spherical bessel function $$ j_l(kr) \xrightarrow[]{ r \gg 1} \frac{\sin(kr -l\pi/2)}{kr} = \frac{1}{2ik}\left( \frac{e^{i(kr-l\pi/2)}}{r} - \frac{e^{-i(kr-l\pi/2)}}{r}\right) $$ The outgoing wave will change by a phase shift \( \delta_l \), from which we can define the S-matrix \( S_l(k) = e^{2i\delta_l(k)} \). Thus, we have $$ \frac{e^{i(kr-l\pi/2)}}{r} \xrightarrow[]{\mathrm{phase change}} \frac{S_l(k)e^{i(kr-l\pi/2)}}{r} $$