Propagation-distance limit for a classical nonlocal optical system

We derive closed-form analog quantum-speed-limit (QSL) bounds for highly nonlocal optical beams whose paraxial propagation is mapped to a reversed (inverted) harmonic-oscillator generator. Treating the longitudinal coordinate z as an evolution parameter (propagation distance), we construct the propagator, evaluate the Bures distance, and obtain analytic Mandelstam–Tamm and Margolus–Levitin bounds that fix a propagation distance limit z_PDL to reach a prescribed mode distinguishability. This distance-domain constraint is the classical optical analogue of the minimal-orthogonality time in quantum mechanics. We then propose a compact self-defocusing PDL beam shaper that achieves strong transverse mode conversion within millimetre scales. We further show that small variations in refractive index, beam power, or temperature shift z_SL with high leverage, enabling SL-based metrology with index sensitivities down to 10⁻⁷ RIU and temperature resolutions of order 1 mK. The results bridge distance-domain QSL geometry and practical photonic applications.