Conic sections arising from light reflection, refraction, and diffraction

Astigmatic focal loci govern field-dependent imaging behavior in optical systems. We derive the spatial loci of the local tangential and sagittal focal positions as the object point varies continuously along a meridional field line in a flat object plane. We show that, within the Coddington formulation, these loci range from simple low-order geometries-including lines, planes, and conic sections-to higher-order algebraic curves, depending on the physical interaction at the interface and the object position. For a real object and a concave non-diffractive spherical mirror, the tangential locus is an exact conic whose eccentricity is determined by the normalized object coordinate L/R, with e=R/(2L), where R is the algebraic radius of curvature of the surface and L is the algebraic axial coordinate of the object plane. In contrast, the refractive spherical surface retains a quadratic sagittal locus but generally produces a higher-order tangential locus, while the one-dimensional spherical gratings considered here generically give rise to higher-order field-dependent loci. Under the separate Rowland object-space condition, the tangential focal locus reduces to the same circular form for spherical mirrors, refractive spherical surfaces, and the spherical gratings considered here. These results extend the Coddington framework from a local description of astigmatism to a field-wide geometric characterization of the focal structure.