Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the path that can be traversed in the least time. In order to be true in all cases, this statement must be weakened by replacing the "least" time with a time that is "stationary" with respect to variations of the path — so that a deviation in the path causes, at most, a second-order change in the traversal time. To put it loosely, a ray path is surrounded by close paths that can be traversed in very close times. It can be shown that this technical definition corresponds to more intuitive notions of a ray, such as a line of sight or the path of a narrow beam.

First proposed by the French mathematician Pierre de Fermat in 1662, as a means of explaining the ordinary law of refraction of light (Fig. 1), Fermat's principle was initially controversial because it seemed to ascribe knowledge and intent to nature. Not until the 19th century was it understood that nature's ability to test alternative paths is merely a fundamental property of waves. If points A and B are given, a wavefront expanding from A sweeps all possible ray paths radiating from A, whether they pass through B or not. If the wavefront reaches point B, it sweeps not only the ray path(s) from A to B, but also an infinitude of nearby paths with the same endpoints. Fermat's principle describes any ray that happens to reach point B; there is no implication that the ray "knew" the shortest path or "intended" to take that path.

For the purpose of comparing traversal times, the time from one point to the next nominated point is taken as if the first point were a point-source. Without this condition, the traversal time would be ambiguous; for example, if the propagation time from P to P′ were reckoned from an arbitrary wavefront W containing P  (Fig. 2), that time could be made arbitrarily small by suitably angling the wavefront.

Treating a point on the path as a source is the minimum requirement of Huygens' principle, and is part of the explanation of Fermat's principle. But it can also be shown that the geometric construction by which Huygens tried to apply his own principle (as distinct from the principle itself) is simply an invocation of Fermat's principle. Hence all the conclusions that Huygens drew from that construction — including, without limitation, the laws of rectilinear propagation of light, ordinary reflection, ordinary refraction, and the extraordinary refraction of "Iceland crystal" (calcite) — are also consequences of Fermat's principle.

الاستنتاج

Equivalence to Huygens' construction

In this article we distinguish between Huygens' principle, which states that every point crossed by a traveling wave becomes the source of a secondary wave, and Huygens' construction, which is described below.

النسخة الحديثة

الصياغة كدالة في معامل الانكسار

Let a path Γ extend from point A to point B. Let s be the arc length measured along the path from A, and let t be the time taken to traverse that arc length at the ray speed ${\displaystyle v_{\mathrm {r$ (that is, at the radial speed of the local secondary wavefront, for each location and direction on the path). Then the traversal time of the entire path Γ is

${\displaystyle T=\int _{A ^{B \!dt\,=\int _{A ^{B {\frac {ds {\,v_{\mathrm {r \,$

(1)

(where A and B simply denote the endpoints and are not to be construed as values of t or s). The condition for Γ to be a ray path is that the first-order change in T due to a change in Γ is zero; that is,

${\displaystyle \delta T=\,\delta \int _{A ^{B {\frac {ds {\,v_{\mathrm {r \, \,=\,0\,$.

العلاقة بمبدأ هاملتون

If x,y,z are Cartesian coordinates and an overdot denotes differentiation with respect to s , Fermat's principle (2) may be written

${\displaystyle {\begin{aligned \delta S&=\,\delta \int _{A ^{B \!n_{\mathrm {r \,{\sqrt {dx^{2 +dy^{2 +dz^{2 \\&=\,\delta \int _{A ^{B \!n_{\mathrm {r \,{\sqrt {{\dot {x ^{2 +{\dot {y ^{2 +{\dot {z ^{2 ~ds\,=\,0\,.\end{aligned$

In the case of an isotropic medium, we may replace n_r with the normal refractive index  n(x,y,z), which is simply a scalar field. If we then define the optical Lagrangian as

${\displaystyle L(x,y,z,{\dot {x ,{\dot {y ,{\dot {z )\,=\,n(x,y,z)\,{\sqrt {{\dot {x ^{2 +{\dot {y ^{2 +{\dot {z ^{2 \,,$

Fermat's principle becomes

${\displaystyle \delta S=\,\delta \int _{A ^{B \!L(x,y,z,{\dot {x ,{\dot {y ,{\dot {z )\,ds\,=\,0\,$.

If the direction of propagation is always such that we can use z instead of s as the parameter of the path (and the overdot to denote differentiation w.r.t. z instead of s), the optical Lagrangian can instead be written

${\displaystyle L{\big ( x(z),y(z),{\dot {x (z),{\dot {y (z),z{\big ) =n(x,y,z)\,{\sqrt {1+{\dot {x ^{2 +{\dot {y ^{2 \,,$

so that Fermat's principle becomes

${\displaystyle \delta S=\,\delta \int _{A ^{B \!L{\big ( x(z),y(z),{\dot {x (z),{\dot {y (z),z{\big ) \,dz\,=\,0\,$.

This has the form of Hamilton's principle in classical mechanics, except that the time dimension is missing: the third spatial coordinate in optics takes the role of time in mechanics. The optical Lagrangian is the function which, when integrated w.r.t. the parameter of the path, yields the OPL; it is the foundation of Lagrangian and Hamiltonian optics.

التاريخ

فرما لقاء الكارتيزيين

If a ray follows a straight line, it obviously takes the path of least length. هيرون السكندري، في كتابه Catoptrics (1st century CE), showed that the ordinary law of reflection off a plane surface follows from the premise that the total length of the ray path is a minimum. In 1657, Pierre de Fermat received from Marin Cureau de la Chambre a copy of newly published treatise, in which La Chambre noted Hero's principle and complained that it did not work for refraction.

غفلة هويگنز

لاپلاس وينگ وفرنل ولورنتس

• پيير-سيمون لاپلاس (1749–1827)

• توماس ينگ (1773–1829)

• أوگوستان-جان فرنل (1788–1827)

• هندريك لورنتس (1928–1853)

انظر أيضاً

Notes

References

ببليوگرافيا

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• A.J. de Witte, 1959, "Equivalence of Huygens' principle and Fermat's principle in ray geometry", American Journal of Physics, vol. 27, no. 5 (May 1959), pp. 293–301, DOI:10.1119/1.1934839.  Erratum: In Fig. 7(b), each instance of "ray" should be "normal" (noted in vol. 27, no. 6, p. 387).
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للاستزادة

• A. Bhatia (26 March 2014), To save drowning people, ask yourself 'What would light do?', http://nautil.us/blog/to-save-drowning-people-ask-yourself-what-would-light-do, retrieved onسبعة August 2019 .
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