540-degree deflecting lens and its general version.

We demonstrate an isotropic device called 540-degree deflecting lens, which has symmetric refractive index and can deflect parallel beam by 540 degrees. The expression of its gradient refractive index is obtained and generalized. We discover it's an optical absolute instrument with self-imaging characteristic. Using conformal mapping, we deduce its general version in one-dimensional space. We also introduce a combined lens called the generalized inside-out 540-degree deflecting lens similar to the inside-out Eaton lens. Ray tracing and wave simulations are used to demonstrate their characteristics. Our study expands the family of absolute instruments and provides new ideas to design optical systems.