Fermat's principle states that a light ray's path is such that the time to traverse that path is an extremum (a minimum or a maximum) when compared with times for nearby paths. Show that Fermat's principle implies Snell's law by proving that a light ray going from point \(A\) in one medium to point \(B\) in a second medium will take the least time if it obeys Snell's law.

Optics is a branch of physics that involves the study of light and its interactions with matter. From the beautiful phenomena of rainbows and mirages to the critical applications in lenses and optical fibers, optics plays a fundamental role in understanding and utilizing light.

In educational terms, optics explores how light behaves when it encounters different media—whether it reflects off a mirror, bends as it passes through a glass lens, or scatters when hitting a rough surface. Key concepts in optics include reflection, refraction, diffraction, interference, and polarization, each contributing to our comprehension of visual systems, both natural and engineered.

Understanding the principles of optics not only unravels the path taken by light but also enables technological advancements in microscopy, telescopes, photography, and laser technologies among others.

At its core, the physics of light, or 'optical physics', examines the physical properties of light and the laws that govern its behavior. Light is not just the medium through which we see the world, but it is also a form of electromagnetic radiation that can exhibit characteristics of both waves and particles—a duality that has puzzled and fascinated scientists for centuries.

Understanding the speed of light and its interaction with various materials is essential. For instance, light travels at different speeds in different media, an observation that is crucial when exploring phenomena such as refraction. By examining how light moves and interacts, we delve deeper into the fundamental nature of the universe, exploring quantum optics, laser science, and the role of photons in conveying energy and information.

Refraction is the bending of light as it passes from one transparent medium into another. This bending occurs because light travels at different speeds in different substances. For example, light travels slower in water than in air, which is why objects under water look skewed to an observer above the surface.

The principles of refraction are based on two fundamental laws:

Fermat's Principle of Least Time

This principle suggests that the path taken between two points by a ray of light is the path that can be traversed in the least time.

Snell's Law

This law provides a quantitative formula to determine the angle at which light is refracted when entering a new medium, mathematically linking the ratio of the sine of the angles to the ratio of the light speeds in the media.

Geometric optics, often called ray optics, simplifies the study of light by considering its propagation in terms of rays. It is most applicable when the wavelength of the light is small compared to the size of objects and openings it encounters.

Using geometric optics, we can easily describe the interactions of light with lenses, mirrors, and prisms by tracing the paths of rays and applying the laws of reflection and refraction. It is this branch of optics that explains image formation, magnification, and the basic principles behind the working of optical devices such as eyeglasses, cameras, and telescopes.

Understanding geometric optics provides a foundation for exploring more complex optical phenomena and for the design and analysis of a wide range of optical instruments.