Fermat's Principle of Least Time

Equivalent forms

\delta t = 0

t = \frac{1}{c}\int_A^B n\,ds

One sentence — light takes the path of stationary time — generates all of ray optics and prefigures the least-action principle of mechanics.

Symbol	Name	SI unit	Dimension	Range
$n1$	Index above Refractive index of the upper medium	dimensionless	1	1 – 2
$n2$	Index below Refractive index of the lower medium	dimensionless	1	1 – 2.5
$x$	Crossing point Horizontal position where the ray crosses the interface	m	L	0 – 10
$t_rel$	Relative travel timeoutput Travel time for the chosen crossing point	ns	T	0 – 100

Unit systems

SI:: n dimensionless, distances in m, time in s
CGS:: distances in cm
Imperial:: distances in ft

Where it holds

Ray (eikonal) limit of wave optics. 'Least' time is really 'stationary' time — it can be a minimum, maximum, or saddle (e.g. at a focus or in a gravitational lens), all of which still satisfy

\delta (OPL) = 0

.

Discovery

Pierre de Fermat · 1662

Fermat proposed that light minimizes travel time, and derived Snell's law from it — controversially, since it implied light slows in dense media (opposite to Newton's later corpuscular claim). It became the seed of the calculus of variations and, via Hamilton, of the action principle in all of physics.

Try this

How does light 'know' to take the fastest path before it sets out?

Light travels between two points along the path whose travel time is stationary. From this one variational statement, the laws of reflection and refraction fall out — a famous problem that reframed all of optics.

Research status: stable

Common misconceptions

It is not strictly 'least' time — at a focus many paths take equal (stationary) time, and some configurations give a maximum. Light does not 'try' paths; the stationary-phase condition emerges from wave interference.

Derivation

Travel time

t = (1/c)

integral n ds.

For refraction, fix endpoints A (height a, in n1) and B (height b, in n2) separated horizontally by d; let the ray cross at x.

t(x) = n_{1} \sqrt

(a^2+x^2)/c + n2 sqrt(b^2+(d-x)^2)/c.

Setting

dt/dx = 0

gives n1 x/sqrt(a^

2+x^2) = n_{2} (d-x)/\sqrt(b^2+(d-x)^2)

, i.e.

n_{1} \sin \theta 1 = n_{2} \sin \theta 2

.

Limiting cases

Reflection (single medium)⟶ Least time gives angle of incidence = angle of reflection

Refraction (two media)⟶ Stationary time gives

n_{1} \sin \theta 1 = n_{2} \sin \theta 2

(Snell)

Concave mirror focus⟶ Many paths share equal time — a stationary maximum, not a minimum

Snell's Law→Law of Reflection→Optical Path Length→Huygens' Principle→