Energy-Gap Topology for Mapping Coupling Architectures in Lanthanide Co-Doped Photonic Materials

Lanthanide co-doped photonic materials are commonly interpreted through selected resonant transitions between assigned 4f multiplet states. Here, we introduce an energy-gap topology framework that compares Ln3+ ion pairs from the complete set of internal separations within their 4f manifolds. Reported multiplet-centre energies for a common LaF3 spectroscopic reference were reduced to energy-only manifolds, from which normalized gap distributions were constructed. A symmetric descriptor, CA−B, quantifies global similarity between complete gap distributions, whereas a directional descriptor, DA→B, measures the inclusion of the gaps of ion A within the occupied support of ion B. When applied to 11 trivalent lanthanide ions, this reveals distinct coupling architectures. Tb3+–Ho3+, Sm3+–Dy3+, Dy3+–Ho3+, Nd3+–Ho3+, and Er3+–Ho3+ exhibit high symmetric overlap, defining a broad manifold–manifold topology signature. In contrast, Yb3+-containing pairs show low global compatibility but high directional inclusion of the Yb3+ gap within the landscapes of candidate acceptor ions, such as Er3+, Tm3+, and Nd3+, consistent with sparse-to-rich sensitizer-like architectures. The pairwise organization remains stable for bin widths between 250 and 1000 cm−1. The descriptors provide a pre-spectroscopic screening map of energetic architecture; they do not predict the transfer efficiency, dominant mechanism, or final optical performance and should not be interpreted as host-independent constants.