Optical Path and Fermat’s Principle.

This article at the moment is well prepared although it may still be improved. The article is partly technical and partly historical, its not intended to be given as a lecture by me although its a comprehensive discourse.

A separate lecture is available for Honors students, see below. Happy read.

( Honors lecture on Fermat’s principle ) click to learn about Fermat’s principle and optical path in a honors course

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; .

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.

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’ have 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 someone says density, this is perhaps a misconception I have carried even 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 lead to a higher optical density in line with matter density, perhaps?

Also lets realize the fact that things might not necessarily look shortened right-away, eg as shown in the above picture. But one can see for oneself an object immersed in a bucket 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 the 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.E. 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.

Note that here the shortest path is synonymous with least time, since during reflection light ray traverses a homogeneous medium, i.e., there is only one speed with which the ray travels, hence least amount of time correspond to shortest path.

The fact to be noted here to avoid future “I am in a pickle” moment is to realize, reflection necessarily means light remains in a homogeneous medium. That means light travels with the same speed, depending on the refractive index of the given medium. This necessarily means minimizing time and minimizing corresponding distance, are the same thing.

Lets give it a 3rd  shot to understand, this time mathematically: there are 3 variables, speed, distance and time, if speed is fixed, given that there already exists a relation between distance and time, in terms of speed, changing one among distance and time necessarily changes the other among these two, the same way.

In simpler words having speed constant, distance and time change in the same manner, since speed is the ratio of distance and time, hence minimizing distance is same as minimizing time. Hence Heron could not catch the subtlety here, namely for refraction minimal time is not same as minimal distance.

Hence Heron’s principle of least path or for him least time, would not suffice to explain refraction, which is no more simply, a principle of least path, although still a principle of least time, satisfactorily.

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

Also note that Ibn Sahl although a noted optical physicist of his time, was not the most celebrated scientist, Ibn al-Haytham or Alhazen who wrote book of optics — “Kitab al Manazir” was. Although it shall also be noted that al-Hayatham was influenced by Ibn Sahl, even though al-Hayatham took optics to a far higher echelon, than the former.

Ibn Sahl had given the refinement of law of reflection by including refraction, to be following the shortest-time path as well as the principle 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 Huygens’s principle, a geometrical attribute known to be valid for light as waves for centuries now. Note that optics is primarily ray optics or geometrical optics apart from another exclusive regime known as wave optics.

In ray optics waves are assumed to be following straight-line motion because of negligible size of wavelength in a given situation. So ray optics must follow from wave-optics. Fermat’s principle is the most important edifice of ray or geometrical optics. Ray and wave optics are together known as classical optics as opposed to modern optics which is where the quantum mechanical views of light production and propagation becomes important.

Also note that classical optics is essentially studied as a first order theory and goes by the name paraxial optics. Any higher order corrections are going to bring into light the precise nature of light and there are a flurry of worrisome deviations from how simply we have formulated the various principles based eg on Fermat’s principle. These deviations are known as Aberrations.

( Honors lecture on Aberrations ) click to learn more about optical aberration

Nonetheless Fermat’s principle is still valid and governs these phenomena only hesitation we have is the paraxial assumption and we must let it be corrected for a better understanding of the underlying phenomena.

Three words of caution in the continuity of our discussion in the penultimate paragraphs.

a. 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 fundamental validity. Only in specific cases the action is least and only in specific cases the time, — of flight, is fundamental and is least.

b. There could be other physical situations; when action itself can be either flat (stationary)  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.

c. Fermat’s principle as we understand today and formally stated; 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, the solution 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 or versions classical mechanics, theory of relativity or new classical mechanics and quantum mechanics.

But Huygens’s principle is a principle on the wave nature of objects eg electromagnetic waves. The Huygens’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, given at the top of this article, a statement of the fact that speed of travel of light must depend upon the angle at which light is traveling, in a given medium, wrt another.

That’s 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 any 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 fashion the dictum of a “principle of stationary action” as opposed to the principle of stationary time. First, time correlated with speed, is a general step towards time correlated with energy or energy in a form known as Lagrangian, this correlation of time with Lagrangian is 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 they are called principle of extremum. So light travels that path, which when suffers a little variation in time, or in the energy, action, speed or path, does not jump abruptly, rather changes to almost null, a condition known as stationary.

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

The above principle of flatness is applicable in general to all things physical, which is a 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. An optical path is then defined as the refractive-index times 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 given as . Refractive index is always greater than 1.0, satisfying 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, given by; .

Then light takes that path where a variation of the optical path is zero, a condition called as extremum. That would mean the optical path would be shortest, longest or stationary or 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; , where the incidence is occurring in medium 1 at angle i, given by refractive index n₁ and r is refraction angle, occurring in medium 2, given by refractive index n₂. In other words the speeds reduce in ratio of the refractive indices, of optical media.

( Snell’s law from Fermat’s principle ) click to learn more about how refraction and Snell’s law are derivable from Fermat’s principle

Here is a summary of the optical path as it pertains to optical phenomena from simpler to the more involved, and why the understanding took the historical perspective that it did. I represent a simple tabulation of the values or ideas as it occurred to me while teaching the Optical path and Fermat’s principle to my honors class.

The article was originally written in October 2013 but my lectures to honors students that I just mentioned was in 2016-17. So the idea is just an addendum.

Phenomena	Geometrical path (distance)	Physical path (wrt time)	Optical path (wrt free-space)
Reflection	Reflection occurs in homogeneous medium; minimization is same
Refraction	These two are same	Differs

Rough Patch ahead, just sharing, needs review.

I tried to derive the conditions mathematically, from geometric considerations, 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 the latter 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 Huygens’s Principle which makes spherical waves propagation.

But I haven’t made any more progress, I was simply trying to apply algebra; 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 or conserved”. Here the phase of original ray and the phase of refracted ray . Instead of energy or momentum, I started with the more fundamental quantity, to the problem; the phase.

So I have two approaches;

a. Geometric: it leads to trigonometry, I need some more tinkering with that.

b. 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: ; I get . Where being the 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 or new classical and modern or quantum mechanical ideas. Think this: if etc then by realizing that A, B and C are equivalents, if one must simply be zero it 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 saying no mass is the correct way, 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 ( a strict equality follows only in case of non-zero mass particles ).