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Fig. 2.3. Geometrical wavefronts and rays

Fig. 2.3. Geometrical wavefronts and rays wavefront W into the emergent wavefront W1 centered on I .If the wavefront W1, and equally the wavefront W2 at a later instant, are strictly spherical, then the image I is perfect from the point of view of geometrical optics. If, however, the wavefronts are not exactly spherical, the phase error involved is a measure of the geometrical aberrations of the optical system. This concept will be developed in Chap. 3 in detail. At this stage, it is important to note that, in a given medium, the phase error remains constant with the advance of the wavefront: any phase error in W1 will be identical in W2. The time tn> the light takes to travel from I to I is t

where ci is the velocity of light, n the refractive index of the medium and ds an element of the path. The quantity J" n ds (which reduces to ^ nAs for practical systems with a few finite jumps in the refractive index n) is called the optical path length. The wavefronts are surfaces of constant optical path length.

Equation (2.11) defines the refractive index n completely but it is better known as a consequence of Snell's law of refraction at a surface separating two different optical media, which states that the incident and refracted rays are coplanar with the normal and that n sin i = n sin i , (2.12)

where i and i are the angles of incidence and refraction to the normal. We shall see in the next section that this relation is even more powerful and general than the word "refraction" implies, because "reflection" can be handled as a special case of refraction by the same equation.

A more general approach to the geometrical optics of the phenomenon of refraction is provided by Fermat's principle. This approach is dealt with exhaustively in such works as Born-Wolf [2.5] or Schroeder [2.6], to which the reader is referred. 