Photon Motion and Shadows of Rotating Black Holes with Nonlinear Electromagnetic and Anisotropic Matter Fields

This paper investigates the effects of the nonlinear electromagnetic field and the anisotropic matter field on photon motion, shadow structures, and the energy emission rate of a rotating black hole (BH). Using the Hamilton-Jacobi formalism, we derive the photon motion equations and analyze the distribution and stability of photon regions. The results show that the anisotropic matter field parameters affect the size and shape of the photon region outside the event horizon more significantly than the nonlinear electromagnetic field parameter. As the anisotropic matter field parameter $K$ decreases, the unstable photon region outside the BH gradually expands and becomes increasingly flattened. Furthermore, we construct the BH shadow in terms of the celestial coordinates and obtain the corresponding shadow images by backward ray tracing. Several shadow observables, including the shadow radius, distortion parameter, shadow area, and oblateness, are also analyzed. The results indicate that the anisotropic matter field affects the shadow size more strongly than the shadow shape. Specifically, the shadow radius and area both decrease significantly as the parameter $K$ decreases, but increase markedly as the anisotropic matter state parameter $\omega$ increases. In addition, we analyze the energy emission rate of the BH and find that decreasing $K$ or increasing magnetic charge $Q$ suppresses its peak value, while the influence of $\omega$ remains comparatively mild. These results provide a useful reference for understanding the effects of nonlinear electromagnetic and anisotropic matter fields on rotating BH shadows and related observational signatures.