The photon sphere is the predicted shell around a black hole, at 3/2 event horizon radius, where light theoretically orbits. I say “theoretically” because it’s an extremely unstable orbit. Generally, light travelling tangent to the black hole would quickly fly out or fall in.
If you managed to get your speedster spacecraft arbitrarily close to the speed of light, and orbit close to the photon sphere, this is what it would look like:
You’d also do a really good job of (irreversably) time travelling to the future - a nice place, albeit dizzying, for a restaurant at the end of the universe.
But this really exists only in theory. The main point of the photon sphere is to showcase the difference between Newton and Einstein’s theory of gravity.
Newton’s theory, G=m1*m2/d^2, predicts that a photon - or any generic object travelling at c, would orbit a black hole at half the schwarzchild radius.
It took Einstein’s theory of gravity as spacetime curvature to recalculate the fact that, no, photons only would orbit at 3/2r(s). Photons travelling tangent to the event horizon, outside the event horizon but inside the photon sphere, would fall in. Just outside the event horizon, only a photon travelling directly outward would manage to eventually escape.
This article gives some good derivations of the difference in the two theories.
Now, this may all be theoretical, but it underlines an important point. Modern orbital mechanics understands that Newtonian mechanics falls short at extreme conditions. Those conditions include precise calculations for satellite orbits and signals.