Etendue and Radiance Conservation in Transformation Optics: Strict Analytical Bounds on Field Enhancement

We establish an intuitive connection between transformation optics (TO) and the classical invariants of etendue and radiance that hasn't been made before. Through explicit application of the optical metric formulation of TO, we demonstrate that any smooth, passive, impedance-matched transformation performs as a canonical (symplectic) mapping on optical phase space. In combination with Hamiltonian ray dynamics, this indicates that Liouville theorem is as well applicable to TO media: the phase-space volume element is preserved, radiance remains constant along rays, and etendue won't decrease under any passive TO mapping. Based on this novel notion, which has never been mentioned in TO literature before, we develop a radiance-invariant phase-space measure specific to TO media. We then utilize it to find severe analytical limitations on field enhancement. We demonstrate that the maximum attainable average intensity in any passive TO concentrator is exclusively constrained by the area-compression ratio of the coordinate mapping, irrespective of material implementation. Using the same paradigm on zero-index media, optical-null media, and illusion devices reveals the same rules. Consequently, our findings demonstrate that transformation optics (TO) can redistribute intensity but cannot increase radiance, and they give a Liouville-type theorem within this field. This provides a consistent, metric-based elucidation for the intrinsic limitations of concentration in passive metamaterials and extreme-index platforms.