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.

Fermat’s Principle, a lecture in Optics.

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; 

Primary Aberrations, 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.

My analysis of Odisha elections-2017

The concurrent election in Odisha just drew to a close and I did an analysis on the results available tentatively. Notice that there is no visible errors here even if I eg adjusted 854 to 850 and so on. If you add up the % figures they add to 100% perfectly — I simply did the calculation and applied no tricks, that means there is some simple pattern in the data which is the reason I made this post.

Inherent ability = difficulty * accomplishment.

All of Physics is this “Inherent ability = difficulty * accomplishment”. Thats just intuitive but can easily be seen to correspond mathematically with the Principle of least action.

First the edifice: whats the problem? The problem is given you move in straight line when every direction is same around you, which direction will you chose? While you are waiting for a good answer from astrologers intelligent people already give a good hint. Think you have some inherent ability which is fixed.

fixed: which changes only if estimated wrong.

That inherent ability is actually action. Accomplishments are adjusted for difficulties, you waded through a swamp 5 meters you would have accomplished in sand 8 meters with that given inherent ability called action. Because action is abstract we have been sticking to time and path-length, but they are not as fundamental, they are merely specifics.

Aberrations; a lecture in Optics.

Lecture-I; delivered 24-1-2017

This lecture has been delivered in one of the honors class 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:

Object point does not lie far away from the axis of the optical system.

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

The domain of optics where above two assumptions are valid is called as Paraxial optics. Paraxial systems are highly idealized and in reality 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 sin function by the Maclaurin series. 

If instead the 2nd term in the Maclaurin series retained and higher order terms are truncated then we say it’s a 3rd order theory — as opposed to the 1st order theory which we called the paraxial optical assumption. If a single wavelength source of light is considered along with a 3rd order theory the deviations from 1st order theory results thus obtained are summed up as primary or monochromatic aberrations. These aberrations are also known as Seidel aberrations in accordance with the name of the scientist Ludwig Von Seidel who studied them.

Thus these primary aberrations are broadly categorized into 5 types;

Spherical aberration
Petzval field curvature

The first 3 type of aberration lead to a deterioration in the quality of the image making them unclear. The last 2 types cause deforming of the shape or size of the images.

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