Spherical mirrors in the paraxial limit

At what ray height does exact Snell's-law tracing through a spherical surface depart measurably from the paraxial focal prediction? Useful evidence includes exact and paraxial ray traces, focus position versus ray height, transverse spot size, step-size convergence, an index-variation case, and a quantified validity limit.

This GioPhysics course map is an adaptable learning sequence, not academic credit, accreditation, or a universal university syllabus. Third-semester content is the least standardised of the introductory sequence: departments place oscillations, waves, optics, and thermal physics differently, and the modern-physics units here are a bounded survey rather than a complete course in relativity, quantum mechanics, atomic, nuclear, or particle physics. Follow your institution's published scope, notation, laboratory programme, and assessment rules. A result should be checked against units, signs, limiting cases, and the conditions under which its model was derived.