In optics, Fermat's principle or the principle of least time, named after French mathematician
Pierre de Fermat, is the principle that the path taken between two points by a ray of
light is the path that can be traversed in the least time. This principle is sometimes
taken as the definition of a ray of light. However, this version of the principle is
not general; a more modern statement of the principle is that rays of light traverse the
path of stationary optical length with respect to variations of the path. In other words,
a ray of light prefers the path such that there are other paths, arbitrarily nearby
on either side, along which the ray would take almost exactly the same time to traverse.
Fermat's principle can be used to describe the properties of light rays reflected off
mirrors, refracted through different media, or undergoing total internal reflection. It
follows mathematically from Huygens' principle (at the limit of small wavelength). Fermat's
text Analyse des réfractions exploits the technique of adequality to derive Snell's
law of refraction and the law of reflection. Fermat's principle has the same form as Hamilton's
principle and it is the basis of Hamiltonian optics.
== Modern version == The time T a point of the electromagnetic
wave needs to cover a path between the points A and B is given by:
T =
∫
t
0
t
1
d t
=
1 c
∫
t
0
t
1
c v
d s
d t
=
1 c
∫
A
B
n
d s
{\displaystyle T=\int _{\mathbf {t_{0}} }^{\mathbf {t_{1}} }\,dt={\frac {1}{c}}\int _{\mathbf
{t_{0}} }^{\mathbf {t_{1}} }{\frac {c}{v}}{\frac {ds}{dt}}\,dt={\frac {1}{c}}\int _{\mathbf
{A} }^{\mathbf {B} }n\,ds} c is the speed of light in vacuum, ds an infinitesimal
displacement along the ray, v = ds/dt the speed of light in a medium and n = c/v the
refractive index of that medium,
t
0
{\displaystyle t_{0}} is the starting time (the wave front is in
A),
t
1
{\displaystyle t_{1}} is the arrival time at B. The optical path
length of a ray from a point A to a point B is defined by:
S =
∫
A
B
n
d s
{\displaystyle S=\int _{\mathbf {A} }^{\mathbf {B} }n\,ds}
and it is related to the travel time by S = cT. The optical path length is a purely
geometrical quantity since time is not considered in its calculation. An extremum in the light
travel time between two points A and B is equivalent to an extremum of the optical path
length between those two points. The historical form proposed by Fermat is incomplete. A complete
modern statement of the variational Fermat principle is that the optical length of the
path followed by light between two fixed points, A and B, is an extremum. The optical length
is defined as the physical length multiplied by the refractive index of the material."
In the context of calculus of variations this can be written as
δ S
= δ
∫
A
B
n
d s
= 0
{\displaystyle \delta S=\delta \int _{\mathbf {A} }^{\mathbf {B} }n\,ds=0}
In general, the refractive index is a scalar field of position
in space, that is,
n =
n
(
x
1
,
x
2
,
x
3
)
{\displaystyle n=n\left(x_{1},x_{2},x_{3}\right)} in 3D euclidean space. Assuming now that light
has a component that travels along the x3 axis, the path of a light ray may be parametrized
as
s =
(
x
1
(
x
3
)
,
x
2
(
x
3
)
,
x
3
)
{\displaystyle s=\left(x_{1}\left(x_{3}\right),x_{2}\left(x_{3}\right),x_{3}\right)} and
n d
s =
n
d
x
1
2
+ d
x
2
+ d
x
3
2
d
x
3
d
x
3
= n
1 +
x ˙
1
2
+
x ˙
2
d
x
3
{\displaystyle nds=n{\frac {\sqrt {dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}}{dx_{3}}}dx_{3}=n{\sqrt {1+{\dot {x}}_{1}^{2}+{\dot {x}}_{2}^{2}}}\
dx_{3}} where
x ˙
k
= d
x
k
/
d
x
3
{\displaystyle {\dot {x}}_{k}=dx_{k}/dx_{3}} . The principle of Fermat can now be written
as
δ S
= δ
∫
x
3 A
x
3 B
n
(
x
1
,
x
2
,
x
3
)
1 +
x ˙
1
2
+
x ˙
2
d
x
3
{\displaystyle \delta S=\delta \int _{x_{3A}}^{x_{3B}}n\left(x_{1},x_{2},x_{3}\right){\sqrt {1+{\dot {x}}_{1}^{2}+{\dot {x}}_{2}^{2}}}\,dx_{3}}
= δ
∫
x
3 A
x
3 B
L
(
x
1
(
x
3
)
,
x
2
(
x
3
)
,
x ˙
1
(
x
3
)
,
x ˙
2
(
x
3
)
,
x
3
)
d
x
3
= 0
{\displaystyle =\delta \int _{x_{3A}}^{x_{3B}}L\left(x_{1}\left(x_{3}\right),x_{2}\left(x_{3}\right),{\dot {x}}_{1}\left(x_{3}\right),{\dot {x}}_{2}\left(x_{3}\right),x_{3}\right)\,dx_{3}=0}
which has the same form as Hamilton's principle but in which x3 takes the role of time in
classical mechanics. Function
L
(
x
1
,
x
2
,
x ˙
1
,
x ˙
2
,
x
3
)
{\displaystyle L\left(x_{1},x_{2},{\dot {x}}_{1},{\dot {x}}_{2},x_{3}\right)}
is the optical Lagrangian from which the Lagrangian and Hamiltonian (as in Hamiltonian mechanics)
formulations of geometrical optics may be derived.
== Derivation == Classically, Fermat's principle can be considered
as a mathematical consequence of Huygens' principle. Indeed, of all secondary waves
(along all possible paths) the waves with the extremal (stationary) paths contribute
most due to constructive interference. Suppose that light waves propagate from A to B by
all possible routes ABj, unrestricted initially by rules of geometrical or physical optics.
The various optical paths ABj will vary by amounts greatly in excess of one wavelength,
and so the waves arriving at B will have a large range of phases and will tend to interfere
destructively. But if there is a shortest route AB0, and the optical path varies smoothly
through it, then a considerable number of neighboring routes close to AB0 will have
optical paths differing from AB0 by second-order amounts only and will therefore interfere
constructively. Waves along and close to this shortest route will thus dominate and AB0
will be the route along which the light is seen to travel.Fermat's principle is the main
principle of quantum electrodynamics which states that any particle (e.g. a photon or
an electron) propagates over all available, unobstructed paths and that the interference,
or superposition, of its wavefunction over all those paths at the point of observation
gives the probability of detecting the particle at this point. Thus, because the extremal
paths (shortest, longest, or stationary) cannot be completely canceled out, they contribute
most to this interference. In humans, for example, Fermat's principle can be demonstrated
in a situation when a lifeguard has to find the fastest way to traverse both beach and
water in order to reach a drowning swimmer. The principle has been tested in studies with
ants, in which the ants' nest is on one end of a container and food is on the opposite
end, but the ants choose to follow the path of least time, rather than the most direct
path.In the classic mechanics of waves, Fermat's principle follows from the extremum principle
of mechanics (see variational principle).
== History == Euclid, c. 320 BCE in his Catoptrics (on mirrors,
including spherical mirrors) and Optics, laid the foundations for reflection, which was
repeated by Ptolemy, and then in his more detailed books that have surfaced, Hero of
Alexandria (Heron) (c. 60) described the principle of reflection, which stated that a ray of
light that goes from point A to point B, suffering any number of reflections on flat mirrors
in the same medium, has a smaller path length than any nearby path.Ibn al-Haytham (Alhacen),
in his Book of Optics (1021), expanded the principle to both reflection and refraction,
and expressed an early version of the principle of least time. His experiments were based
on earlier works on refraction carried out by the Greek scientist Ptolemy.
The generalized principle of least time in its modern form was stated by Fermat in a
letter dated January 1, 1662, to Cureau de la Chambre. It was met with objections by
Claude Clerselier in May 1662, an expert in optics and leading spokesman for the Cartesians
at the time. Amongst his objections, Clerselier states:
... The principle which you take as the basis for your proof, namely that Nature always
acts by using the simplest and shortest paths, is merely a moral, and not a physical one.
It is not, and cannot be, the cause of any effect in Nature.
The original French, from Mahoney, is as follows:
Le principe que vous prenez pour fondement de votre démonstration, à savoir que la
nature agit toujours par les voies les plus courtes et les plus simples, n’est qu’un
principe moral et non point physique, qui n’est point et qui ne peut être la cause
d’aucun effet de la nature.
Although Fermat's principle does not hold standing alone, we now know it can be derived
from earlier principles such as Huygens' principle. Historically, Fermat's principle has served
as a guiding principle in the formulation of physical laws with the use of variational
calculus (see Principle of least action).
== See also == Adequality
Eikonal equation Fermat’s and energy variation principles
in field theory Geodesic
Hamilton's principle Huygens' principle
Path integral formulation Pierre de Fermat
Principle of least action Snell's law
== Notes ==