  We have studied the spacetime of a Kerr black hole immersed in Melvin
magnetic field, and found not only unstable light rings could exist, but also
stable light rings could exist. Both the prograde and retrograde unstable light
rings radiuses increase with the magnetic field parameter $B$, but it is the
opposite for stable light rings. The existence of unstable, stable light rings
depend on both the rotation parameter $a$ and the magnetic field parameter $B$.
For a certain $a$, there are both the prograde and retroprade unstable (stable)
light rings when $B$ is less than a critical value $B_{c}$ of retrograde light
ring. In this case, the shadows of Melvin-Kerr black hole have two gray regions
on both sides of the middle main shadow, which correspond to the prograde and
retrograde stable photon orbits. The photons in stable orbits are always moving
around Melvin-Kerr black hole, they can't enter the black hole or escape to
infinity. As $B$ continues to increase, there is only the prograde unstable
(stable) light ring. In this case, the gray region only emerges in the life of
the main shadow, which corresponds to the prograde stable photon orbits. The
absence of the retrograde unstable (stable) light rings makes the Melvin-Kerr
black hole shadow an half-panoramic (equatorial) shadow. When $B$ is bigger
than $B_{C}$ of prograde light ring, neither prograde nor retroprade unstable
(stable) light rings exist. In this case, the shadow of Melvin-Kerr black hole
has no gray region for stable photon orbits, and becomes a panoramic
(equatorial) shadow. In addition, there also exist some self-similar fractal
structures in the shadow of Melvin-Kerr black hole arising from the chaotic
motion of photon.
