Photon rings, gravitational lensing, and ISCOs of exotic compact objects in Einstein-scalar-Maxwell theories

In Einstein-scalar-Maxwell theories with a coupling between the scalar field $\phi$ and the electromagnetic field strength $F$ of the form $\mu(\phi) F$, we investigate the existence of exotic compact objects (ECOs) and their observational signatures in photon and massive-particle dynamics. For $\mu(\phi)$ diverging at the origin while all physical quantities remain finite, we demonstrate the existence of electrically charged ECOs with a shell-like structure whose density peaks at an intermediate radius. We compute their mass and radius, together with the scalar and vector field profiles, on a static and spherically symmetric background. We then examine the existence of photon rings and place bounds on a model parameter by requiring the absence of a linearly stable photon ring. Under this condition, photon echoes from ECOs are absent. We also compute the gravitational-lensing deflection angle $\Psi$ and show that it attains a maximum for an impact parameter of the same order as the ECO radius. Finally, we study the parameter space in which innermost stable circular orbits of massive particles exist.