Wave Foundations of Ray Optics

Chapter 1 Huygens' principle, too often asserted independently, is here developed with Huygens' original inevitability and a modern generality. Fermat's principle, too often stated in isolation and without even specifying the applicable propagation speed, is here developed as a consequence of Huygens' principle and justified in terms of other notions of a ray (line of sight, narrow beam). The equivalence of Huygens' construction and Fermat's principle — recognized by Young, Laplace, and Fresnel, further developed by Lorentz, proven analytically for two dimensions by De Witte, but neglected in textbooks — is here demonstrated in its full generality by elementary geometric arguments. The foregoing principles lead to rectilinear propagation in homogeneous media, the coincidence of rays and wave-normals in isotropic media, the general relation between the ray velocity and the normal velocity, and the general and ordinary laws of refraction and reflection. These laws are restated as conservation of the tangential component of wave slowness, leading to Hamilton's "common perpendicular" rule. The other feature of Hamilton's construction — the relation between the wave-slowness surface and the direction and magnitude of the ray slowness — is derived in two pages, and the construction is then compared with that of Huygens. For isotropic media, Hamilton's construction yields an ironic retrospective explanation of early geometric formulations of the ordinary law of refraction. In the last section, the refractive index is introduced as a normalized wave slowness, and previous results are restated in terms of refractive indices. Historical notes, with sources, are included. Chapter 2 Electromagnetic waves in transparent anisotropic media — from Faraday's law to conical refraction — are treated at the freshman level without assuming sinusoidal time dependence: If B is understood in terms of Faraday's law for an infinitesimal loop, and D and H in terms of applied dipole moments per unit volume, one easily obtains the energy densities. The infinitesimal-loop concept leads naturally to the curl, hence the differential form of Faraday's law and, by analogy, the Maxwell-Ampère law with no conduction current; and it facilitates the introduction of Gauss's laws (with no free charges) and the interface conditions. The magnetic Lorentz force is derived from Faraday's law as applied to a moving loop in a steady field. For plane waves with wave slowness s, the Faraday and Maxwell-Ampère laws become B = s×E and D = −s×H. Consideration of the Minkowski vector D×B, which is found to have the direction of s, shows that the electric and magnetic energy densities are equal. Consideration of dB and dD then shows that the Poynting vector E×H is in the ray direction, allowing the re-expression of the Faraday and Maxwell-Ampère laws in terms of ray velocity, with the Poynting vector as the power density. For magnetically isotropic, electrically anisotropic media, the existence of perpendicular principal axes is shown by a degrees-of-freedom argument, leading to Fresnel's ray-velocity surface, the ray ellipsoid, Biot's dihedral law, external conical refraction, and their duals. The significance of the extreme radii of the cross-section of the ray ellipsoid is explained by analogy with Fermat's principle: the permitted polarizations are those for which the propagation time is stationary w.r.t. variations of the succession of polarization along the path. (To be continued.) Chapter 3 "Digressions" that can be omitted without loss of continuity: The circles of contact between Fresnel's ray-velocity surface and its tangent plane are found by first showing that they lie on a common ellipsoid. From these circles and their counterparts on the wave-slowness/index surface, angles pertaining to conical refraction are derived. The pressure and momentum density of light, and the proportionality between energy and momentum, are derived in a manner valid for anisotropic media provided that the absorbent surface is parallel to the wavefronts. The density of momentum transferable to objects within the medium is found to be the Minkowski density. (To be continued.