  In this paper, we introduce a new approach to study the behavior of the
photon sphere and shadow radius. Our method uses extended gravitational
decoupling and reveals two important analytic results. First, the additional
matter field alters the photon sphere radius: it increases if
$g'(r_{ph}^{(0)})>0$ and decreases if $g'(r_{ph}^{(0)})<0$ (where $g'$
represents the derivative of a specific metric function evaluated at the
original photon sphere radius). Second, the presence of the matter field can
modify the black hole shadow size. If $g\left(r_{ph}^{(0)}\right)>0$, the
shadow shrinks, while it grows for $g\left(r_{ph}^{(0)}\right)<0$. These
findings provide a deeper insight into how matter distribution influences the
characteristics of black holes and their observable features. Through a
systematic framework and various illustrative examples, our investigation not
only clarifies these fundamental aspects but also significantly enhances the
theoretical framework of black hole astrophysics.
