Causality Sum Rules in Conventional Scattering Matrices

Conventional scattering matrices are the standard tool for describing photonic and electromagnetic devices, yet they do not explicitly display causality constraints in a way that is directly accessible for design. While passivity is easily verified, causality—which governs fundamental limits on bandwidth, size, and delay—is typically formulated using auxiliary variables or alternative representations like Green functions. This paper seeks to bridge this gap by formulating causality sum rules directly within the conventional scattering matrix.

The authors identify that the conventional scattering matrix contains a geometry-dependent time-advance phase due to the choice of reference surfaces. By defining a domain-delayed scattering matrix that removes this reference-dependent delay, they restore the causal time origin. This transformation ensures the matrix satisfies the mathematical properties of a Schur function, allowing the application of the Cayley-Herglotz construction. This mathematical framework maps the scattering response to a spectral integral relation, which provides the basis for deriving universal causality bounds.

The study establishes two primary causality sum rules:

1. Projected Sum Rule: Constrains the coherent return suppression for channel superpositions within a common-delay subspace, relating the achievable bandwidth and suppression depth to the physical delay budget.
2. Determinant Sum Rule: Provides a basis-independent constraint on the aggregate multichannel attenuation, capturing collective contraction across all singular channels.

These rules recover established scalar limits, such as the Rozanov absorber bound and spherical-multipole scattering bounds, while extending them to measurable quantities like insertion loss and coherent channel suppression. The authors also demonstrate a hybrid AI-human discovery workflow, where the initial theoretical derivation was autonomously explored by an AI research system (Qiushi Engine) and subsequently refined and validated by the human authors.