optical path

Interference of two types

Interference by wave-front and amplitude splitting.

Optics Series Lecture, Lecture – XIV, XV, XVI.

Topics covered in this lecture

A. Color of thin films

B. Newton’s rings

C. Lloyd’s mirror and

D. Phase changes during reflection

We have previously discussed what is interference and what are wave-front splitting and amplitude splitting interference. We have also discussed in much details two wave-front splitting interference viz.

a. Young’s double slit interference, here — Lecture — IX, and

b. Fresnel’s bi-prism, here —  Lecture – XI.

Today we will discuss one more wave-front splitting interference namely Lloyd’s mirror interference before moving onto the amplitude splitting interference of the Newton’s rings.

Also we will discuss two interesting and related concepts;

i. Phase change on reflection and

ii. Color of thin films

Today we will discuss one more wave-front splitting interference namely Lloyd’s mirror interference before moving onto the amplitude splitting interference of the Newton’s Rings.

Also we will discuss two interesting and related concepts;

i. Phase change on reflection and

ii. Color of thin films

Fermat’s Principle, a lecture in optics

Optics series lecture, Lecture-III

“Geometrical Optics and Fermat’s Principle”.

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, article 1 — a detailed, historical and kind of long article, and article 2 — a conceptual but a short article.

Before Fermat, Hero of Alexandria, who lived sometime between 150 BC and 250 AD explained reflection of light. ( Read the more extensive history in the already linked article, article 1 above ) 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.

Optical Path and Fermat’s Principle.

Mirror and Fermat’s principle: We can see ourselves in the mirror and take our mirror reflected selfie as a consequence of Fermat’s Principle, the topic of discussion of the blog.

Snell’s Law governs refraction which is adjustment of optical paths in in-homogeneous media because light can no more travel at its speed in free-space.

Snell’s Law and Refraction.
The above expression comes from Fermat’s optical theorem, called as “Fermat’s least time principle” which can in turn follow either from;

a. Huygens’s Principle; that light travels like spherical wave-fronts hence satisfies geometric rules or,

b. Principle of least or stationary action ( or Hamilton’s action principle ).

So in the beginning of our understanding we thought “Light travels a path which is shortest, or the least-path”. This is due to Heron of Alexandria, who lived between 10 AD and 70 AD. Then this path which is traversed by light was redefined to something called optical path, which led us to our understanding that “light travels the path of least time”.

All physical laws are derivable mathematically –with appropriate physical understanding, from Hamilton’s Action variation or Principle of Action, made to provide the least or stationary time. Then time is replaced by definition of action as the most general formulation of the law.

So all in all, path/distance >> Time >> Action . That is crudeness goes towards abstract physical understanding. This can be recognized as an attribute of unification and tells us why Mathematics reigns supreme in Physics.

In the last few weeks I am trying to understand why light traverses straight lines and why it refracts. The other day, I saw a little mug, floating inside a bucket full of water. Inside water any object would look shortened, this is known by a phenomenon called as refraction.

Lets envisage the phenomenon or observation, via this quickly reproducible trick. Lets dip our favorite pencil in a glass of water and another in a glass filled with air. This is how it looks, I quickly made this arrangement to which my 3 year old niece just glued with rapt attention.