Remarks on the light ring images and the optical appearance of hairy black holes in Einstein-Maxwell-dilaton gravity

The behaviors of null geodesics in the spherical symmetric black holes in Einstein-Maxwell-dilaton (EMD) theory with coupling function $f(\Phi)=e^{-2\alpha \Phi}$ are meticulously analyzed. We investigate the effects of coupling constant $\alpha$ on the effective potential of photon trajectories within three ranges, namely $0<\alpha <1$, $\alpha =1$ and $\alpha >1$. We find that the thicknesses of lensing and photon rings are smaller at larger $\alpha$ and fixed electric charge in the unit of mass $q$, whereas they are larger at fixed $\alpha$ and larger $q$. This behavior can be described by using the angular Lyapunov exponent $\gamma$ in the vicinity of the critical curve. Remarkably, the behaviors of photon trajectories are found to be more interesting when $\alpha>1$. Namely, the radius of the black hole shadow $R_\text{s}$ becomes to be smaller than the photon sphere radius $r_\text{ph}$ when $\alpha > 1$ and $q>q^*$. Moreover, $R_\text{s}$ goes to zero as $q$ saturates the extremal limit, beyond which the photon orbit becomes absent. Furthermore, we construct the optical appearance of black holes surrounded by optically and geometrically thin accretion disk with three cases of Gralla-Lupsasca-Marrone (GLM) emission profile. Our results indicate that the observed flux originating from the lensing and photon rings exhibits suppression as $\alpha$ increases, while it undergoes amplification with the increasing parameter $q$.