A Unified Perspective on Causality and One-Sided System Responses in Time and Space Across Physical and Fourier Domains

The principle of causality has long been mathematically associated with the frameworks of Titchmarsh's theorem and the Kramers--Kronig relations.

While these relations arise naturally in the context of temporal system responses -- ensuring that the effect of an applied field or force does not precede its cause -- they have recently been shown to provide a pathway for realizing one-sided system responses in a variety of physical settings.

In particular, one-sided frequency responses and one-sided wavevector responses have been successfully studied and engineered, enabling the prospect of numerous applications based on the complete suppression of backward scattering.

In this work, we present a brief review of causality and its connection to these Fourier-domain analogs.

We then turn our attention to the only remaining setting in which a one-sided system response may be explored: one-sided spatial nonlocality.

We specifically investigate the possibility of realizing a one-sided spatial response within the widely used framework of nonlocal flat optics, where we uncover fundamental obstacles that hinder the achievement of such functionality in these structures.

This, in turn, raises an intriguing open question: is one-sided spatial nonlocal response merely incompatible with the specific platform of nonlocal flat optics, or is it fundamentally forbidden by nature itself?