Étendue and radiance conservation in transformation optics establish strict analytical bounds on field enhancement

We establish an intuitive connection between transformation optics (TO) and the classical invariants of étendue and radiance. 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. Combined with Hamiltonian ray dynamics, this implies that Liouville's theorem applies as well to transformation-optical media, enforcing phase-space volume preservation and fundamental constraints on radiance and étendue under passive mappings. To our knowledge, this explicit phase-space formulation has not been systematically developed within the transformation-optics framework. We derive strict analytical bounds on achievable field enhancement. In particular, we show that the maximum average intensity attainable in any passive TO concentrator is limited solely by the geometric area-compression ratio of the underlying coordinate transformation, independent of the specific material realization. We apply this framework to zero-index media, optical-null media, and illusion devices and find the same rules. Consequently, our findings demonstrate that TO redistributes optical intensity without increasing radiance, consistent with a Liouville-type constraint. This result provides a consistent, metric-based explanation for fundamental concentration limits in passive, impedance-matched metamaterials.