Does light always take the shortest-time path?

Not exactly; Fermat's principle requires the optical path to be stationary, which is usually a minimum but can in some geometries, such as reflection from a concave mirror, be a maximum or a saddle point.

What causes total internal reflection?

When light inside a denser medium strikes the boundary with a less dense medium beyond a critical angle, Snell's law has no transmitted solution and all the light is reflected back into the denser medium.

Ray Tracing and Fermat's Principle

Fermat's principle states that light follows the path of stationary optical length, from which the laws of reflection and refraction and the techniques of ray tracing follow.

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Definition

Fermat's principle holds that the optical path taken by light between two points is stationary with respect to small variations of the path; ray tracing is the procedure of following individual rays through a sequence of refracting and reflecting surfaces using the resulting laws.

Scope

This topic covers the variational foundation of geometrical optics in Fermat's principle, the derivation of the laws of reflection and Snell's law of refraction from it, the concept of optical path length, and the systematic tracing of rays through optical systems by exact and paraxial (matrix) methods. It includes total internal reflection, the eikonal equation linking ray and wave descriptions, and the use of ray-transfer (ABCD) matrices for paraxial analysis.

Core questions

Why do the laws of reflection and refraction follow from a single variational principle?
How is a ray propagated through a sequence of optical surfaces?
How does the paraxial ray-transfer matrix summarize an optical system?
Under what conditions does total internal reflection occur?

Key concepts

optical path length
Snell's law
law of reflection
total internal reflection
eikonal equation
ray-transfer matrix
critical angle

Key theories

Fermat's principle of stationary optical path: Light follows the path for which the optical path length, the integral of refractive index over distance, is stationary; both the law of reflection and Snell's law emerge as conditions for this stationarity.
Ray-transfer matrix method: In the paraxial approximation each optical element acts as a 2x2 matrix on a ray's height and angle, so a whole system is represented by the product of its element matrices, enabling systematic tracing and analysis.

Clinical relevance

Ray-tracing methods are used to design and evaluate lenses for cameras, microscopes, and corrective eyewear, and total internal reflection is the operating principle of optical fibres used in telecommunications and endoscopy.

History

Fermat formulated his principle of least time around 1662 to explain refraction, building on Snellius's empirical law of 1621. Hamilton's nineteenth-century work on the characteristic function and the eikonal connected geometrical optics to a variational and ultimately to a wave description, foreshadowing the analogy with classical mechanics.

Key figures

Pierre de Fermat
Willebrord Snellius
William Rowan Hamilton

Seminal works

hecht2017
bornwolf1999

Frequently asked questions

Does light always take the shortest-time path?: Not exactly; Fermat's principle requires the optical path to be stationary, which is usually a minimum but can in some geometries, such as reflection from a concave mirror, be a maximum or a saddle point.
What causes total internal reflection?: When light inside a denser medium strikes the boundary with a less dense medium beyond a critical angle, Snell's law has no transmitted solution and all the light is reflected back into the denser medium.