# Optical Path 3

Snell’s Law (v1/v2 = sin theta1 / sin theta2 = lambda1 / lamda2) !

Snell’s Law (v1/v2 = sin theta1 / sin theta2 = lambda1 / lamda2) !
It comes from Fermat’s optical theorem called as “Fermat’s least time principle” which can in turn follow either from
1. Huygen’s Principle; that light is spherical wave-fronts hence satisfies geometric rules.
or
2. Principle of least or stationary action or Hamilton’s action principle.
So first we thought Light travels least-path. (Heron of Alexandria)
Then path was redefined to optical path, so we understood light travels path of least time.
Then like all physical laws are derivable mathematically (with appropriate physical understanding) from Hamilton’s Action Variation (or Principle of Action, variationized to least/stationary) TIME was replaced by definition of action.
So all in all, path (distance) >> Time >> Action.
That is crudeness goes towards abstract physical understanding.
This is unification and tells you 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. Dip your 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. What are you doing kaka? I am like I am playing. She didn’t want to leave me, she was so thrilled with this unusual thing happening around her. Kids are always curious, until its completely wiped out of them by social apathy.

So the shortening of the image is a direct consequence of shortening of the path that light ‘rays’ has to travel inside water or any such optically denser media.

Yes, optical density is totally different from mass density of materials that we so fervently talk about or imagine when anyone says density, perhaps a misconception I have even carried until a year ago, when I confirmed both densities are different, but I certainly think they can be correlated, more matter in a small volume can also determine an optical density in line with matter density, perhaps.

Also realize the fact that things might not necessarily look shortened right-away, eg as shown in the above picture. But you can see for yourself an object immersed in a buck full of water, which was a bit harder for me to produce here. Big experiments need bigger funding. Having a big bucket full of water might make my mother cringe about it harder, domestic etiquette, you know, I resisted.

This shortening is because, light rays would “bend” inside water, towards a direction where they have to take a shorter path, as their speed must reduce, given to the fact, that, in the same time in a rarer medium, light would have traveled a longer distance, in the exact same time, which is no longer possible, due to the crowd of molecules and subatomic ghetto that light meets along its way.

Lets look at the process a bit deeper. Lets look at historical understanding of it first.

More than 2000 years ago, scientists of those time, knew that, light would take a path of shortest time. It is a law of reflection. Heron of Alexandria in C. 60, had noted that; Light reflections, to any number, from a flat mirror surface would traverse that path, which would be shortest among all paths nearby, in the same media.

About a 1000 years ago, in Middle-east, Ibn Sahl propounded the law of “least time” path of light although its accounted to the much later day scientist and polymath Fermat who gave the general formulation of such.

Ibn had given the refinement of law of reflection by including the refraction, to be following the shortest time path as well as it does for reflection. Ibn’s work was based on work on refraction, from Ptolemy.

Fermat’s principle of least time which is the most important basis of Optics, the Physics of light, and from there Quantum Electrodynamics, Physics of light and electrons, is a specific form of the more general Principle of least action known as Hamilton’s Principle. Fermat’s Principle is also derivable from Huygen’s Principle.

Three words of caution.

• While Fermat himself stated his principle in terms of least time, the essence of his principle was not that of least time. Its to be understood that in general its not least nor it is time which has any priority of validity. Only in specific cases the action is least and only in specific cases the time, — of flight, is least.
• There are other physical situations, when action itself can be either flat or maximum. Also same can be said about time. Time of flight can be maximum or flat among a slew of possible paths for the physical situation under consideration.
• Fermat’s principle as we understand today and formally state it is based on the idea of optical path and not time or action, and this will be soon discussed.

The principle of least action or more appropriately stationary action, is the most important principle in all of Physics. Its the principle that can lead to any principles of Physics, conceptually, formally and fundamentally. Thats because its the generator of whats called as “Equations of motion”.

An equation of motion is a set of differential equation, that is, its an equation that involves differentials of motion and when solved gives a solution to the equation of motion known as trajectory or path of the physical objects, under consideration.

Needless to say the Principle of stationary or least action contains in it both waves and particles and in formalism covers in its various avatars-/-versions classical mechanics, Relativity (or new classical mechanics) and Quantum Mechanics.

But Huygen’s principle is a principle of the wave nature of objects eg electromagnetic waves. The Huygen’s Principle elucidates the spherical propagation of waves. Hence this can easily give rise to the law of refraction or law of sines known as Snell’s Law, a statement of the fact that speed of travel of light must depend upon the angle at which light is traveling in a given media wrt another.

Thats so because of the fact that angles are ratio of length segments, here the length being what is known as “Optical path”. Optical path is a physical quantity that differentiates from the Physical path by taking into account the idea that speed of light must reduce, when it travels in a denser media.

A geometrical path is a pure length segment, a physical path is one which takes “amount of time taken” included into a geometric consideration and an optical path is the additional inclusion of speed (of light).

So far so good we are only taking in a general matter the dictum of a Principle of Stationary Action as opposed to the Principle of a stationary time. First time correlated with speed, is a general step towards time correlated with ENERGY or Lagrangian known as Action.

Action is a correlation of time with energy of a system, rather than just time being a determinant of the fate of a system. All generalizations in Physics must produce the Principle of Stationary Action.

When we say least we mean: the shortest. In general the light path could also be the longest and/or stationary. Together its called a principle of extremum. So light travels that path, which when suffers a little variation in time, the energy-/-action-/-time-/-speed-/-path does not jump abruptly rather changes to almost null, a condition known as stationary.

Its like your car moving on highway. If there is a slight hitch you don’t fall into oblivion. You catch on almost smoothly, hence you keep on moving. Thats also a condition known as flat. And at the infinitesimal everything’s gotta be smooth, flat, hardly changing then only can you move.

(Your mother’s love is as good as your aunt’s NO , but even if your mother’s love doesn’t sound like its flat, the slight variations of Monday through Saturday ain’t causing much of a heartburn, you can sail smoothly through her, the principle of stationary love)

The above principle of flatness is applicable in general to all which is the beauty of Physics. Even if life of Physicists are not mean or easy by any means it is not a bug in jaggery.  (or is it?) By studying little variations in these flatness we come to know about new phenomena  and new forces and mysteries of nature. One challenge in such task is how to deal with any prevalent inconsistencies. **

So light travels the shortest path. A shortest path is that where time taken is the shortest among different neighboring path, if we are considering homogeneous or isotropic medium. I am not sure if such is also valid for the contrary.

An optical path is then defined as the refractive-index*physical-path where physical path is a geometric path taken in a given time interval. Refractive index given as n is the amount by which speed of light REDUCES in a medium from what it is in free-space.

In free-space under all conditions of arbitrariness, such as changing frames of reference, changing wavelength, frequency, energy and so on, speed of light is a fixed value of c = 2.99 x 10^8 m/s. Refractive index is always greater than 1.0, reflecting a condition that speed is always less than c, and density of the medium is higher than that in free-space. As mentioned in the beginning density is optical density which is different from matter density, m/v.

Then light takes that path where a variation of optical path is zero, a condition called as extremum. That would mean the optical path would be shortest, longest or stationary-/-flat. Stationary or flat means all nearby paths are almost same wrt a given quantity such as energy.

This extremum principle is known as Fermat’s Principle. This leads to the law of refraction known as Snell’s law: given as; n1/n2 = v1/v2 = lambda1/lambda2 = sin i / sin r, where the incidence is occuring in media 1, given by n1 and r is refraction occuring in medium 2, given by n2.

In other words the speeds reduce in ratio of the refractive indices of optical media.

I tried to derive the conditions mathematically from geometric consideration, although I am half-way. The diagram that occurred to me, turns out to be same as the one Ibn’s manuscript possesses. Although I didn’t look at it ever before, I am stuck by the fact why should I consider perpendiculars in the original and refracted path of the light ray, as opposed to any other possible intercepts, that’s when I started researching more when I understood the fact known as Huygen’s Principle which makes spherical waves propagation.

But I haven’t made any more progress, I was simply trying to apply algebra (kx + wt = k’x’ + w’t’) as the condition of phase, where one must realize the conditions of symmetry in Physics are nothing but a simple application of this algebraic equation which states “given to a change in some conditions there must be a quantity that must be the same/conserved”.

Here phase of original ray = kx + wt and phase of refracted ray = k’x’ + w’t’. Instead of energy or momentum, I started with the more fundamental quantity, to the problem; phase.

So I have two approaches,

* geometric: it leads to trigonometry, I need some more tinkering with that.

* Algebraic: two methods, in one method if I apply conservation of momentum without thinking exactly what its doing, I get Snell’s law, 2nd method: the one I am doing here: (kx + wt = k’x’ + w’t’); I get v/v’ = x/x’. Where x, x’ being physical location of the wave, give the distance light is traveling in two cases, original ray vs refracted ray, but how to go from there.

The idea of mine is to show that energy conservation is a natural idea built into refraction, because this is the condition that gives a particular speed given a particular angle of path of light.

** Talking about inconsistencies >> Relativity of Einstein is a beautiful application of Algebra into Geometry, to remove inconsistencies from existing theories of waves and particles just as Re-normalization is an application of various mathematical techniques (such as calculus and group theory) to remove inconsistencies from Quantum Mechanics, which was already a refined theory of wave and particle mechanics, known as classical mechanics.

I just realized how this is so, when I was trying to understand the refraction of light as a process that involves classical, relativistic (new classical) and quantum mechanical ideas.

Think this: if Alpha = A+B+C+D etc then by realizing that A, B and C are equivalents one must simply be zero and must not be included in the denominator in order to make our theory consistent. eg Photon has no mass. It is wrong to say rest mass of photon, just no mass, when it has it only has energy and mass and energy are equivalents given to the right unit-dimension and if A=mass and B=energy we must see that A does never come in the denominator simply because its zero, so there are two distinct forms of energy as Physics has been formulated, one is mass and the other is other forms of energy which are again equivalents among each other.