Gravitational lensing of charged Ayon-Beato-Garcia black holes and non-linear effects of Maxwell fields

Abstract: Non-singular Ayon-Beato-Garcia (ABG) spherically symmetric static black hole (BH) with charge to mass ratio $q$ is metric solution of Born-Infeld nonlinear Maxwell-Einstein theory. Central region of the BH behaves as (anti-) de Sitter for $(|q|>1)~|q|<1 .$ In case of $|q|=1$ the BH central region behaves as Minkowski flat metric. Nonlinear Electromagnetic (NEM) fields counterpart causes to deviate light geodesics and so light rays will forced to move on effective metric. In this paper we study weak and strong gravitational lensing of light rays by seeking affects of NEM fields counterpart on image locations and corresponding magnification. We set our calculations to experimentally observed Sgr A$^*$ BH. In short we obtained: For large distances the NEM counterpart is negligible reaching to linear Maxwell fields. The NEM makes enlarge the BH photon sphere radius as linearly by raising $|q|>1$ but deceases by raising $|q|\leq1.$ Sign of deflection angle of bending light rays is changed in presence of NEM effects with respect to ones obtained in absence of NEM fields. Absolute value of deflection angle raises by increasing $|q|\to1.$ Weak image locations decreases (increases) by raising $0<|q|<1$ in presence (absence) of NEM fields. By raising the closest distance of the bending light rays weak image locations changes from left (right) to right (left) in absence (presence) of NEM fields. Einstein rings radius and corresponding Magnification centroid become larger (smaller) in presence (absence) of NEM fields in case of weak lensing. Angular separation $s$ between the innermost and outermost relativistic images increases (decreases) by increasing $0<|q|<1$ in absence (presence) of NEM fields. Corresponding magnification $r$ decreases (increases) by raising $0<|q|<1$ in absence (presence) of NEM fields. $s (r)$ raises (decreases) by increasing $|q|>>1.$