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refraction

Matrix formulation in geometrical optics

Topics covered in this lecture

Ray tracing
a. Translation matrix

b. Refraction matrix

c. System matrix 

In this lecture, we will discuss about one of the most interesting and powerful methods in Geometrical Optics. As we have discussed here (https://mdashf.org/2017/02/25/fermats-principle-a-lecture-in-optics/), geometrical optics is that segment of optics in which we are limited to a situation when the wavelength of light is negligible eg λ is insignificant compared to the size of the objects light interacts with.

As a consequence light can be considered as rays or geometrical straight lines and the nuances of light as wave undulations can be postponed to a happy hour.

Ray Tracing

Any general optical system has a ray which can be traced through two basic types of traversal of the ray: Translation and Refraction. The law of refraction is thus the central tool for ray-tracing.A ray can be described in an optical system by its coordinates which we will define soon.

Our goal is to find the matrix which governs the displacement of the ray from one coordinate to another coordinate of the ray, as the ray travels from one geometric point to another. This will enable us to study simple as well as much more complicated systems in the most effective and powerful way as we will see.

Lets discuss the basic matrices available for ray tracing when the  ray travels from one coordinate to another in two cases.

I. Translation Matrix for simple straight line motion in a homogeneous medium.

II. Refraction Matrix for refraction at the interfaces of two different media.

In general therefore the total traversals of the ray can constitute of any number of translations or refraction. A reflection would merely be two translations and a general refraction might be construed from refraction as well as translations.

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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; 

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.

Equivalence from simple notions of Geometry ? Yes.

Now it may also be related that light bends in a denser media compared to a rarer media because an additional rotational force is working. In other words, the definition of straight line has to change in the medium, that is of different density, because path of light is changing. Light is the guy who suffers the least when something tries to buzz it, because its inertial property of mass is zero. The curvature of light or the bending or deflection known as Refraction is thus a measure of the sideways force or energy.

Thus speed of light in different media is a measure of this bending or curvature and is known as Snail’s law ( — Pun intended, its actually; Snell’s Law — ) . Automatically when distance and time have to readjust, to produce an angle or bending, known as refraction ( — possible because distance and time can produce an angle if they are equivalents — ) the speed must also change.

All these are inter related. Light refracts and its speed changes, in relative change in density of media because there are rotational or non-inertial effects.

( — or additional energy is available, or a force is acting to bring a curvature in light’s path, perhaps the electromagnetic effects of the molecules? Its not only distance or time that are equivalents, read one article of mine “All You Need To Know About Relativity” to understand; how energy and time and distance and in short all Physical Variables are equivalents of each other — )

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