## How rainbows are created

How Rainbows are created. Optics lecture series – IV

“Primary and Secondary rainbows”, a lecture in Optics.

This lecture was delivered on February 02, 2017.

Sunlight is white in color.

Do you know Newton would have greatly disliked this statement, do you know why? “Color is not the property of light” but a property of an agent making observations assisted by light. We need a separate article for that. Don’t we?

That means it comprises of 7 primary colors. VIBGYOR is an acronym for these basic colors: Violet, Indigo, Blue, Green, Yellow, Orange and Red. Each color of light corresponds to a different wavelength. Violet has the shortest wavelength and Red has the highest wavelength.

Accordingly Violet has the highest intensity or consequently energy and Red has the lowest intensity or energy. In other words Red is the faintest color in the primary visible spectrum.

Different colors or wavelength of light have different refractive indices, this fact is known as dispersion, that is, different wavelengths of light would travel in different directions upon refraction at any optically denser or rarer media.

That means different wavelength or color component of light would travel at different speed and correspondingly different angles, upon incidence on a media whose refractive index differs from the medium from where incidence occurs.

The above phenomena is known as dispersion. Prism is an optical device which shows such dispersion of colors. We say that the angle of deviation of different colors of light is different at refracting interfaces of optical media. This is mathematically given by the Snell’s law of refraction which we write as follows:

Rainbows are caused by dispersion of light. Dispersion is an optical phenomena where wave velocity or phase velocity of light depends upon frequency or wavelength of light. Different colors of light emerge at different directions during refraction in a transparent medium such as water droplets during rainfall. When sunlight falls on small water droplets rainbow manifests as the result of dispersion of the incident rays.

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## 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.

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## Primary aberration, a lecture in optics.

Lecture-II; delivered on 27-1-2017

In our Lecture-I  we discussed the phenomena of aberrations that arise because of a discrepancy of a first order theory and the 3rd order theory as depicted by the Maclaurin series; where we saw that first order theory represents the so called paraxial optical systems.

Please have a look of the linked article to get a basic view of the ground on which we are discussing this topic. At-least going half-way through the lecture and stopping short of the derivation will do well.

We discussed that there are two kind of aberrations. Monochromatic and Chromatic. As the name suggests the monochromatic aberrations are a result of the discrepancy when we considered our incoming ray to be having a single wavelength of light.

The chromatic rays on the other hand can have multiple colors or wavelength of light. The monochromatic aberrations are also called as Seidel or Primary aberrations and we will shed more light on them today.

The chromatic aberrations were dealt in greater detail — eg the derivations pertained to the chromatic aberrations. We did so because the chromatic aberrations are simple to understand.

So lets discuss in detail the 5 types of primary aberrations now.

Primary Aberrations.
1. Spherical Aberration.
When Paraxial Rays  refract after emerging from an object point they meet at a sharp focus.
But when non-paraxial or marginal rays emerge — or appear to emerge, from an axial object point they do not meet at a sharp focus.
Therefore different rays meet at different focal points. The resulting aberration is called as spherical aberration.

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## Aberrations, a lecture in optics.

Optics series Lecture — I
Optical Aberrations
delivered on 24 – 1 – 2017 — all optics series lectures can be accessed here.

This lecture has been delivered in one of the honors class that I am teaching this semester. You will do really well to read the linked article on Optical path and Fermat’s principle which is not not intended as a honors lecture.

Optical systems are studied under two assumptions:

a. Object points do not lie far away from the axis of the optical system.

b. Rays taking part in image formation make a small angle with the axis of the optical system.

The domain of optics where the above two assumptions are valid is called as Paraxial optics. Paraxial systems are highly idealized and in reality they do not perfectly represent the situation. The consequential errors in image reconstruction are known as aberrations.

The paraxial assumption can be represented by truncating at the first term of the polynomial expansion of the sine function by the Maclaurin series.

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## 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.

read more Optical Path and Fermat’s Principle.