Analyzing deflection angles and photon sphere dynamics of magnetically charged black holes in nonlinear electrodynamic

Abstract In this paper, we investigate the gravitational lensing properties of magnetically charged black holes within the framework of nonlinear electrodynamics. We derive the deflection angle and examine the influence of the nonlinear electrodynamics parameter $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> on light bending. We initially employ a geometric approach based on the Gauss–Bonnet theorem to analyze the gravitational deflection of null and timelike particles. This method encapsulates the global characteristics of the lensing effect in an elegant manner. In the subsequent part of the work, we explore the impact of nonlinear electromagnetic corrections on the black hole shadow. Using numerical techniques, we study the behavior of the photon sphere and demonstrate that a reduction in the photon sphere radius leads to a correspondingly smaller shadow. We compare these results with those for the Schwarzschild and Reissner–Nordström black holes, highlighting the distinctive features introduced by nonlinear electrodynamics. Furthermore, we examine the strong deflection limit for light trajectories near these black holes, focusing on the roles of both the magnetic charge $$Q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> and the nonlinear parameter $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> . Our analysis reveals that the combined effects of $$Q$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Q</mml:mi> </mml:math> and $$\xi $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ξ</mml:mi> </mml:math> enhance the strong deflection angle, resulting in a more pronounced lensing effect than that predicted by the classical Reissner–Nordström solution. These findings suggest that the nonlinear interactions may provide a potential observational signature for identifying NED black holes.