Is the path always the shortest time?

Stationary can mean minimum, maximum, or saddle in exotic layouts; reflection and standard refraction cases are typically minima.

How does this relate to Huygens' construction?

Wavefronts evolve so that equal-phase surfaces remain consistent with constructive interference; rays are the normals to those fronts. Fermat's principle picks the same rays in the short-wavelength limit.

Why can't Fermat explain a single-slit diffraction pattern?

Diffraction needs wave superposition across the aperture. Geometric ray limits miss interference structure when λ is not negligible.

Does general relativity change Fermat's principle?

In curved spacetime light follows null geodesics; a generalized Fermat principle still describes stationary arrival times in an effective optical metric.

Fermat's Principle

Fermat's principle of extremal optical path length states that light rays between two points follow paths for which the travel time is stationary—usually a minimum—compared to neighboring paths. In homogeneous media this yields straight rays; at interfaces it yields Snell's law when refraction lowers overall time; in graded-index media rays bend toward regions of higher refractive index where phase velocity is lower. The simulator visualizes comparing candidate paths and accumulated optical path length n·s or time ∫ (n/c) ds, reinforcing the link between wavefronts (Huygens) and rays (Fermat). Idealizations use geometric optics, ignoring diffraction when apertures are comparable to wavelength, and assuming isotropic n. Students reconcile Fermat with reversibility of rays and connect the principle to variational formulations used across physics.

Who it's for: Undergraduate optics students after Snell's law who want a unifying variational viewpoint before wave optics dominates.

Key terms

Fermat's principle
Optical path length
Snell's law
Refraction
Stationary time
Ray optics
Reversibility
Index of refraction

Live graphs

How it works

Between two fixed points in different homogeneous media separated by a plane interface, the physically realized ray is the one that makes the optical path length n₁ℓ₁ + n₂ℓ₂ stationary (a minimum here). Varying the breakpoint P on the interface traces a smooth OPL curve whose minimum coincides with Snell’s law.

Key equations

OPL = n₁ |AP| + n₂ |PB| → min over P on interface

⇒ n₁ sin θᵢ = n₂ sin θₜ

Frequently asked questions

Is the path always the shortest time?: Stationary can mean minimum, maximum, or saddle in exotic layouts; reflection and standard refraction cases are typically minima.
How does this relate to Huygens' construction?: Wavefronts evolve so that equal-phase surfaces remain consistent with constructive interference; rays are the normals to those fronts. Fermat's principle picks the same rays in the short-wavelength limit.
Why can't Fermat explain a single-slit diffraction pattern?: Diffraction needs wave superposition across the aperture. Geometric ray limits miss interference structure when λ is not negligible.
Does general relativity change Fermat's principle?: In curved spacetime light follows null geodesics; a generalized Fermat principle still describes stationary arrival times in an effective optical metric.

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Fermat's Principle

Live graphs

How it works

Key equations

Frequently asked questions

More from Optics & Light

Chromatic Aberration

Diffraction

Color Mixing

Polarization (Malus)

Birefringence (Calcite sketch)

Holography (Recording principle)

Fermat's Principle

Live graphs

How it works

Key equations

Frequently asked questions