## Snells Law and Glass Dispersion

The first proof that the propagation of light is with finite velocity was obtained by Romer (1676), who observed that the time intervals between the epochs of eclipses of Jupiter's first satellite were varying over an annual period. From his measures he derived the first determination of the velocity of light. In 1728, Bradley noticed the annual elliptic variations of the position of the stars which were not at the ecliptic pole - the so-called stellar aberration (20.5 arcsec) - and thus derived a new measure of velocity. In 1975, the Fifteenth General Conference on Weights and Measures and Bipm7 defined the velocity of light in a vacuum as c = 299,792,458 m/s. (1.3)

The light is the visible part of the electromagnetic spectrum which covers wavelengths from gamma rays to radio waves that all propagate at velocity c in a vacuum. The propagation of the electric and magnetic vectors E and H is represented by Maxwell's equations of electromagnetism (cf. Born and Wolf [17]). In an homogeneous medium (as all media considered hereafter), these equations reduce to the wave equations

c2 dt2 c2 dt2

where £ and 1 are the dielectric constant (or permittivity) and the magnetic permeability of the medium. From these standard equations of wave motion, the velocity v of a wave is c v = < c, (1.5)

where 1 = 1 for all non-magnetic media and £ > 1 for a transparent medium (£ = 1 in vacuum). Thus, the velocity of light is smaller than c in air or glass; this conclusion was first demonstrated from experiment by propagation of light in water in 1850 by Foucault and Fizeau.

In order to introduce the concept of geometrical wavefront, let us assume points in an isotropic medium: a ray from a point source O propagates in a straight line through another point P during a delay At. During the same delay, the light propagating from O, via a neighboring ray of P, stands on a wavefront surface X passing through P. Thus, a wavefront surface contains the notion of both ray package, i.e. optical pencil or optical beam, and constant phase in time over all its front. A wavefront or constant phase surface is generally an asphere, i.e. a non-spherical surface because of the geometrical aberrations that are generated from refraction or reflection at the boundary of successive media. However, the wavefront concept is only a useful approximation at large distances from a focus or an obstruction edge. Near the focusing regions, where a caustic is the envelope of light rays, the effect of diffraction requires using the electromagnetic theory to correctly describe the light distribution of phase and intensity.

Let us consider a ray or a plane wave X that propagates in a medium of refractive index n and reaching at incident angle i a plane surface separating a second medium of refractive index n' in which the ray is refracted at angle i' so that the refracted plane wave is X' (Fig. 1.13). The sine refraction law, first clearly established from experiment by Snell in 1621 and published by Descartes in 1637 (see Sect. 1.1), or Snell's law is

7 Bureau International des Poids et Mesures

 Medium (n) y V' \ E' £ ' VX