Reconciliation of Geometric Optics and Physical Optics
We have seen how the basic principles of geometric optics are consequences of Fermat’s principle. In turn, Fermat’s principle can be understood as a consequence of the wave properties of light. Although a rigorous derivation is beyond the scope of this volume, a heuristic description of the concepts can be instructive.
A beam of light propagates as a wave, oscillating perpendicular to the direction of travel. Two such waves that coincide will interfere with one another. This interference is constructive or destructive, according to whether the peaks and troughs of the waves coincide (see Fig 2-6). If 2 coincident waves of light emanate from a common source, their peaks and troughs coincide whenever the travel time of the 2 waves is identical. In this case, the full strength of the light waves is evident. Conversely, if the travel times of 2 coincident waves differ such that the arrival of their peaks and troughs differ by exactly half a wavelength, the waves cancel each other perfectly, and no light appears to be transmitted.
When small variations in the light path occur, possible paths for which the travel time is essentially unchanged will thus contribute strongly to the resultant image, whereas those paths that vary more strongly in travel time will essentially cancel out. Paths for which small variations (often referred to as “perturbations”) of the light path have little effect on travel time are referred to as extremal paths. The most familiar of these are the paths of shortest travel time, such as the straight-line paths of light through uniform media. The formal derivation of these observations is the concern of the mathematical subject known as the “calculus of variations,” which is beyond the scope of this book.
In rare instances, paths other than those of shortest travel time may also be extremal and are also permitted by the wave properties of light. One important example occurs in fiber-optic light-guides, which are fabricated with a gradient in the refractive index such that the “densest” material is at the center of each fiber. The path down the center of the fiber takes the longest time of all possible nearby light paths, steering the light down the center of each fiber.
Excerpted from BCSC 2020-2021 series : Section 3 - Clinical Optics. For more information and to purchase the entire series, please visit https://www.aao.org/bcsc.