Fermat's Principle

Fermat’s Principle of Least Time, also known as the law of least time, is a fundamental principle in optics proposed by Pierre de Fermat in 1662. The principle provides a means of determining the path taken by light when it travels from one point to another.

Principle Statement §

Fermat’s Principle states that:

Of all the possible paths that light could take to travel from one point to another, it takes the path which requires the least time.

This principle explains why light travels in a straight line in a homogeneous medium, where the speed of light is constant. It also accounts for the bending of light (refraction) when it passes from one medium to another with a different refractive index, as the speed of light changes, causing it to take a path that minimizes travel time.

Application’s of Fermat’s Theorem §

Proving the law of reflection ($\theta ={\theta }^{\prime }$) §

Fermat’s Principle and its Application to the Law of Reflection.excalidraw

Using the set up above, where red plots a trajectory that obeys the law of reflection ($\theta ={\theta }^{\prime }$) and blue plots a trajectory that directly contradicts the law of reflection ($\varphi \ne {\varphi }^{\prime }$), we can clearly see that $|A^{\prime }PB|<|A^{\prime }{P}^{\prime }B|⟹|APB|<|A{P}^{\prime }B|$ : if we assume that the medium the light is travelling through is of equal density (no differentiation in temperature, for example) then the only factor affecting its time from $A$ to $B$ is the distance of its route.

Thus, since Fermat’s Theorem states that ”of all the possible paths that light could take to travel from one point to another, it takes the path which requires the least time.”, the path that light will take will be the shortest, so, $|APB|$ (the red path) - proving the law of reflection. However, instead of just measuring the distance, we can algebraically prove that $|A^{\prime }PB|$ is the shortest distance by proving that it’s a straight line:

$$A^{\prime }\hat{P}B=O\hat{P}{A}^{\prime }+O\hat{P}A+\theta +{\theta }^{\prime }=2(O\hat{P}A+\theta )=2(90^{\circ })=180^{\circ }⟹A^{\prime }PB\text{ is a straight line}⟹|APB|<|A{P}^{\prime }B|■$$

Pierre de Fermat §

Pierre de Fermat, a lawyer by profession and a mathematician by passion, lived from 1607 to 1665. While he’s perhaps best known for Fermat’s Last Theorem, his principle of least time is a foundational concept in the field of optics.

Connection to Ibn Al-Haytham §

It’s worth noting that an early version of this principle was proposed by the Arab physicist Ibn Al-Haytham in the 11th century. Ibn Al-Haytham’s work on optics, including his early formulation of the law of least time, greatly influenced later scientists, including Fermat.

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In conclusion, Fermat’s Principle of Least Time is a key postulate in the study of light and its behaviour. It provides a simple yet powerful explanation for why light travels the way it does, whether in straight lines or bending when it enters a new medium.