Optics series lecture, Lecture-III
“Geometrical Optics and Fermat’s Principle”.
This lecture was delivered on 30th Jan 2017.
Geometric Optics: When the size of objects that a wave of light interacts with are large compared to the wavelength of light λ, λ can be neglected for practical purposes and the light waves behave like rays of light.
Rays of light are geometric line segments from one point of incidence of light to another. Study of optics under the limit of negligible wavelength — λ → 0, is called Geometric Optics.
Geometric Optics can be studied using Fermat’s Principle, much like motion of objects in the realm of classical mechanics are studied using Newton’s laws of motion. To know the basic grounding of Fermat’s Principle follow the links to read two articles which expound the subject matter of Fermat’s Principle, art1 — detailed, historical and long, art2 — conceptual but short.
Before Fermat, Hero of Alexandria, who lived sometime between 150 BCE and 250 AD explained reflection of light. (Read the more extensive history in the linked article) His formulation is stated as Principle of shortest path.
Since reflection occurs in only one medium (homogeneous medium) light indeed travels a geometric shortest path; this is the straight line path between any two points — or coordinate of the ray. For homogeneous medium optical path and physical and geometrical path are merely either proportional to each other or equal.
In the modern times Fermat reformulated Hero’s principle of shortest path — to its equivalent form of shortest optical path. This entailed the principle to be applicable to both reflection and refraction and any other possible optical phenomena which could be explained by virtue of Fermat’s principle in general.
In its original — shortest path form the principle could not explain refraction, because the latter involves traversal of light rays in in-homogeneous media, that is different media are traversed at different speeds and optical path and geometric or physical path are no more equivalents. We will soon see this in detail.
The new formulation of Fermat which is based on improvement of the earlier Hero’s principle for reflection is called as Fermat’s principle of least time. It states that “a ray of light travels through those coordinates of the ray in a given system of media of varying refractive indices for which the amount of time taken is least .”
This can successfully explain both reflection and refraction. But it can still be generalized and the modern form is in terms of the shortest optical path which is different from how it was originally formulated. Before we study the modern form lets discuss its original form.
According to Fermat “The ray of light will correspond to that path for which time taken is an extremum in comparison to nearby paths.” Mathematically extremum implies time for a particular path can be minimum, maximum or stationary for a given neighborhood of paths. If n(x, y, z) is the refractive index as a function of path or position (x, y, z) then;