The basic rules of geometric optics (straight-line propagation of light in homogeneous media, the law of refraction, and Snell’s law) are readily interpreted as a manifestation of Fermat’s principle. This interpretation is frequently stated as the observation that the path of a light ray between 2 fixed points is the path that takes the least time. Thus, in a homogeneous medium, where the shortest path between the 2 points obviously corresponds to the path that takes the least time, it follows that light travels in straight lines.
Similarly, a simple geometric construction confirms that, of all reflecting paths connecting 2 points on the same side of a smooth reflecting surface, the path with the angle of incidence equal to the angle of reflection is the shortest. In this sense, the law of refraction is likewise a manifestation of Fermat’s principle (Figure 1-A2A).
The proof of Snell’s law is only slightly more difficult. Consider Figure 1-A2B. Among the paths shown between points A and B, the shortest (straight-line) path is not the path of least travel time, as we can save time by changing from, say, Path 3 to Path 2 to reduce the portion of the travel time in glass, where the speed of light is slower than that in air. Eventually, the trade-off works the other way, as in Path 1. We can determine the optimal path by calculating the distance in air and in glass for each possible inflection point and dividing by the speed of light in each medium.
A, Left: The straight path between points A and B is the shortest and thus takes the least time among all possible paths from A to B. Right: Light reflecting from point E to point F takes the shortest possible path when the extension of the straight path from E to the mirror reaches the mirror-image point F′ along a straight line. That extension locates the point of reflection at G, such that the angle of incidence equals the angle of refraction. Other possible paths, such as that through point J, are longer and take more time. B, Because light travels faster through air than through glass, Path 3 is not the path of shortest travel time. Path 2 saves time overall by spending less time in glass. C, Geometry for demonstration of Snell’s law from Fermat’s principle.
Using the geometry shown in Figure 1-A2C, we denote the distance traveled through a medium 1 and a medium 2 as d1 and d2, respectively. Then
where x is the vertical distance traversed through medium 1 and l is the total vertical distance traversed through both media. We divide by the speed of light in each medium (which is c/n1or c/n2, respectively, where c is the speed of light in a vacuum) and add the 2 values to get the total travel time T:
To minimize the total transit time, we seek the value of x for which the derivative of the transit time, dT/dx, is equal to 0. Dredging up our knowledge of freshman calculus (and canceling the common factor of c), we obtain
But careful inspection of Figure 1-A2C indicates that the fractions in this equation are in fact the sines of the angles of incidence and refraction, respectively—that is,
1 sin θ1 = n2 sin θ2
which is Snell’s law.
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.