Ellipsoidal and plane-elliptical surfaces are widely used as reflective, point-to-point focusing elements in many optical systems, including X-ray optics. Here the classical optical path function approach of Fermat is applied to derive a closed-form expression for these surfaces that are uniquely described by the object and image distances and the angle of incidence at a point on a mirror surface. A compact description facilitates design, modeling, fabrication, and testing to arbitrary accuracy. Congruent surfaces in two useful coordinate systems — a system centered on the ellipsoid's axes of symmetry and a mirror-centered or `vertex' system with the surface tangent to the xy plane at the mirror's center — are presented. Expressions for the local slope and radii of curvature are derived from the result, and the first several terms of the Maclauren series expansion are provided about the mirror center.
On X-ray beamlines and telescopes, glancing-incidence mirrors with parabolic profiles are used to collimate, focus, and collect light. Here, analytic descriptions for paraboloidal, plane-parabolic, and parabolic cylindrical mirrors in several congruent geometries that are commonly used in fabrication, metrology, and modeling are provided. The exact expressions are derived directly from Fermat's principle, without coordinate transformations, in several mirror-centered coordinate systems, including one with the surface tangent to the central point of intersection. Coefficients for a sixth-order polynomial series approximation are calculated for that coordinate system.
Refraction artifacts in ultrasound imaging can produce the appearance of side-by-side structures, and, in some cases, color Doppler jets. This phenomenon arises from the bending of the ultrasound wave at the interface between tissues with differing propagation speeds. The paths followed by both ultrasound waves and light rays can be described by Snell’s law and Fermat’s principle of least time, both of which are traditionally derived using differential calculus. However, such calculus-based proofs often lack intuitive clarity. In this paper, we present a simple geometric proof that clearly demonstrates the path taken is indeed the shortest—and therefore the fastest.
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