Fermat’s principle, also known as the Principle of Least Time, states that light follows a path of minimum time when it travels from one point to another. It considers the quickest time to cover the path and not the shortest distance. This principle displays the connection between wave optics and ray optics. Fermat's principle was proposed by the French mathematician and physicist Pierre de Fermat in 1662 and is suitable to study optical devices.
Consider a path AB along the curve I, the actual path of light ray, where t represents the time taken by a light ray to travel along path AB on curve I, and t' be the time taken by a light ray to travel along path AB on curve II. Therefore, for any path AB along curve II, t can be either equal to, greater than, or less than t' based on the principle.
The time taken by light ray to travel a distance dS in the medium is given by,
Where, v is the velocity of light
In inhomogeneous medium, the optical properties of the medium are typically described by the refractive index n(x, y, z).
It is known that, refractive index ‘n’ is
Where c is the velocity of light in vacuum
The total time taken by light ray to travel the path AB along the curve I is given by
According to Fermat’s principle, light ray will choose the path with the shortest travel time among numerous paths connecting two points, such as A and B.
Since ‘c’ is constant, the light ray follows a path for which either is either minimum or maximum or 0.
If the medium is homogeneous, so that n(x, y, z) = n0, then optical path length is the geometrical path length multiplied by the refractive index.
Fermat’s principle holds true for both plane and curved surfaces
For refraction at plane surfaces: When light passes from rarer to denser medium, it changes its speed and direction due to the change in the refractive index of the mediums. The light ray follows the path that minimizes the time taken to travel from the source to the destination point.
The time taken to travel between the two points is determined by dividing the distance in each medium by the speed of light within that specific medium.
In order to minimize the time, set the derivative of time with respect to x equal to zero.
Also, utilize the definition of the sine as opposite side over hypotenuse to relate the lengths to the angles of incidence and reflection.
Thus, law of refraction is confirmed.
At the point of angle of incidence and refraction, light takes a path of quickest time after refraction.
For reflection at plane surfaces: When light hits a plane reflective surface (e.g., a mirror), it obeys the law of reflection, which states that the angle of incidence is equal to the angle of reflection with respect to the normal of the surface. Fermat's principle holds here as the path taken by the light minimizes the time it takes to reach the reflected point.
The time required for the light to travel between the two points is determined by calculating the length of each path and dividing the length by the speed of light.
Also, utilizes the definition of the sine as opposite side over hypotenuse to relate the lengths to the angles of incidence and reflection. Then,
When θ1 and θ2 are small,
Thus, Law of reflection is confirmed.
At this point where angle of incidence equals the angle of reflection, light takes a path of quickest time after reflection.
However, for the spherical surfaces, the time taken by the light ray is either maximum or minimum.
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