Deflection angle with electromagnetic interaction and gravitational-electromagnetic dual lensing

The trajectory deflection and gravitational-electromagnetic dual lensing (GEL) of charged signal in general static spherically symmetric spacetimes are considered this work. We showed that the perturbative approach previously developed for neutral particles can be extended to electromagnetic interaction case. angle still takes a (quasi-)series form finite distance effect both source observer taken into account. Comparing pure gravitational case, apparent angles images GEL, their magnifications time delay all receive corrections starting from first non-trivial order. sign relative size leading determined by $\sim \frac{Q}{M}\frac{q}{E}$ where $M,~Q,~q,~E$ spacetime mass charge, particle charge energy respectively. It is found $qQ>0$ (or $<0$), will decrease increase) angle, GEL impact parameters, angles, total travel each image. increased small $\beta$ $qQ>0$, otherwise always regardless $qQ$. results then applied protons cosmic rays Reissner-Nordstrom, dilaton Horndeski spacetimes.