Fermat’s Principle

There was no simple relationship determined between the angles of refraction and incidence since 1621. Dutch investigator Willebrord Snell in 1621 claimed that it is the sines of the angles of refraction and incidence that uphold a constant ratio. Though the statement claimed was correct; however, the observation has not addressed any cause.  However, Fermat was able to determine the cause of the refractive behavior of light. In 1662, Fermat demonstrated that a beam of light follows a path of the fastest time instead of a smaller distance.  For the refraction and reflection at the plane surfaces, Fermat’s Principle of Least Time holds. It means that a ray of light passing from one point to another chooses a certain path along which the time taken is minimum. However, for the spherical surfaces, the time taken by the light ray is either maximum or minimum. 

Who Proposed the Fermat’s Principle?

Fermat’s principle was stated by Pierre de Fermat in 1662 and is suitable to study optical devices. Initially, the principle was a controversial statement as it appeared to assign knowledge of nature’s intent. Until the 19th century, nobody was aware that alternative paths traveling from one point to another are a fundamental property of waves. 

When two points A and B are specified, and a wave front is growing from point A, then it will sweep all the possible rays radiating from A.  However, if the wave front reaches point B, then it will not only sweep all the ray paths with the same endpoints but also the infinitude of nearby paths. 

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The diagram shows that any light rays passing from one medium to another medium bend depending on the rarer and denser medium. 

Fermat’s principle demonstrates that any ray that occurs from one point to another point will cover the smallest path possible. 

State Fermat's Principle

According to Fermat’s principle, light traveling between two points pursues a path such that the optical length between the points is equal. The principle is the link between wave optics and ray optics. 

One of the great ways to state Fermat’s principle is that the path taken by light rays in traveling between two points involves either a maximum or minimum time. It means that two light rays deviating from a distant object will have the same optical path lengths. According to Fermat’s principle of Least Time, the path traveled by light rays between the given two points is the actual path that can be traversed in the least time. 

An Example of Fermat’s Principle

Consider a beam of light traveling from point A to point B. Here, point A is in the air, and point B is in the glass. Fermat presented that the path of light rays is specified by the Principle of Least Time. Fermat’s Principle of Least Time states that a light ray going from A to B will take a shorter time. 

As the speed of light is identical everywhere along all the possible paths, the shortest path is the one that involves the shortest distance. 

According to Fermat’s principle, the entire path made by light to travel from A to B should satisfy the boundary conditions along with the condition that ray takes the path that requires the least amount of time. 

Let the path taken by ray to travel from A to B is equal to ds. Here, ds refer to the small difference of the first order and it refers to the time difference taken along the path, that is; 

$\frac{dt}{ds} = 0$

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The diagram shows that the ray of light passing from point A that is in medium 1 to point B that is in medium 2 covers the shortest distance possible.

Fermat’s Principle- Applications

There are several observations that can be made using Fermat’s principle. These principles will help us to prove and explore the realm of geometric optics. 

• From Fermat's principle, it is found that in homogenous medium rays of light are rectilinear. It shows that light travels in a straight line in a medium having a constant index.

• The angle of incidence is equal to the angle of reflection, which is often called the Law of Reflection. 

• Fermat’s principle refers to the fundamental law of optics that is used in the derivation of other laws of geometrical optics.

• One can make some useful observations about the conic surfaces based on Fermat’s principle.

For Example, Two conjugate points are chosen that are perfect images of each other. The points are chosen in such a way that the optical path length of all paths connecting them is equal. In the case of an ellipse, the point source is located at one focus and is an image of the point located at another focus. However, in the case of the parabola, the point on one focus should be infinite.

The conic surfaces are very useful in mirror optics. The telescope is the best example of a conic surface and its usefulness in mirror optics. The telescope is designed using two conjugate points. Conjugate points are also known as focal points, these points can join one another roughly by 1 parameter. These points are perfect images of each other and all the rays connect equally at the optical path length of the conjugate points.  

For Example, The north pole and the south pole are connected to each other on the sphere by any meridian. 

Now let us consider an ellipse that has a conic surface. An ellipse is basically a plane curve that is surrounded by two different focal points. The sum of all the focal points on the curve will be constant. 

Similarly, if we were to consider a parabola on a conic surface. The parabola is like the circle with a quadratic relation but unlike the circle, a parabola will not be squared at both x and y. Considering this we can interpret that an aggregate of rays will pass through the focus. The rays will be parallel to one another and also to the axis of the paraboloid after it has been recreated by the paraboloid. 

For Example, The Newtonian Telescope is designed based on the same principle. It collects and focuses lights from distant objects. Hence a conic surface will always have two foci with optical conjugating points. 

Law of Reflection

The relationships of an angle's reflection and refraction are derived from Fermat’s Principle. A light incident upon a surface will partially reflect and partially pass as a refracted ray. 

The law of Reflection that can be derived from Fermat’s Principle is that a light ray incident upon a reflective surface will reflect reflection at an angle equal to the incident angle only. 

The law of reflection gives a familiar reflection of the image in a plane mirror. The image distance in this case has to be the same as the object distance which is in front of the mirror as well as behind the mirror. 

Fermat’s Principle of Stationary Time

Fermat’s principles are also known as the principle of least time. It displays the link between wave and rays optics. Fermat believed that a path that any ray takes between any given two points is the path that can be traveled in the least time and hence it is also called the principle of least time. According to this principle, the ray of light traveling from one point to another actually traverses a path and the time taken is a stationary value. 

Conclusion

There are no proofs and axioms in Physics like in Mathematics. There are conclusions drawn on the basis of principles and experiments. We cannot prove that light follows the path of least time as it states in Fermat’s principle that light always would travel to the least time path instead of the other paths in a given situation. In quantum mechanics, it is believed that light can travel through any possible path and if light takes the shortest path it is basically the most probable one.