What is Fermat's principle: Definition and 49 Discussions

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 quickest 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.

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  1. lemma28

    B Fermat's principle and refraction -- Possible misunderstanding

    Consider a light starting at A in media 1 and going in and out a media 2 (say shaped as a disk) with relative index of refraction n to arrive at point B (in media 1). Fermat's principle says that the path taken by the ray between points A and B is the path that can be traversed in the least...
  2. F

    Does twisted light violate Fermat's principle?

    The twisted light propagate in curved line, then does twisted light violate Ferma's priciple?And then it violate refraction and refractive laws?
  3. fando1234

    I What does Fermat's principle of least time mean for causality

    I have been reading some fairly mind bending stuff about the principle or least time (and those of least action) raising questions about causality and free will. Can anyone explain this to me? Is this total 'woo woo' psuedo science, or are these philosophical questions widely accepted? Thanks...
  4. A

    I Calculating the resolution limit using Fermat's principle

    Hi, I read the Feynman Lectures Volume 1, Chapter 27, section 27-7, which can be here. In the lecture he describes the fundamental limits of resolution and provides a criterion. Here is the diagram I am referring to, figure 27.-9: There are two light sources, ##P## and ##P'## There is an...
  5. Zubair Ahmad

    I Fermat's Principle: Is Light Travel Time Min?

    Is time taken by light always minimum between two points. If yes then what is wrong with Fermat's original statement?
  6. Clara Chung

    Deriving the Lens Maker formula of a plano-convex lens using Fermat's Principle

    Homework Statement Homework EquationsThe Attempt at a Solution I only need help on part c. I tried to add up t1 and t2 and differentiate it. However what variables should I differentiate with respect to? If I differentiate with respect to f I got f=root(2) * h, if i differentiate with respect...
  7. N

    Proving Fermat's principle without calculus?

    According to me this topic must be raised and discussed how fermat did it without calculus.What problems he faced since calculus was developed afterwards by Newton leibniz. http://aapt.scitation.org/doi/10.1119/1.1514235 Moderator's edit: File substituted by link due to potential copyright...
  8. shihab-kol

    Simple proof of Snell's law without calculus

    Well, I have checked out the ones with calculus but I was just wondering if there was one without calculus I tried it but could not do it I think Fermat's principle can be used to do it but I am not being successful So, anyone please help
  9. KT KIM

    Apply Fermat's principle to a Concave mirror

    I am now taking optics class at my school. Fermat principle can be applied on mirror of course. Then what about Concave mirror? According to the calculus of variation. the optimized path(actual path of the light) should be the shortest path. but in the concave mirror case, it goes through the...
  10. Z

    I How Fermat's principle interprets diffraction and interference

    <Moved from the homework section> 1. Homework Statement I have read several chapters of De Brogile's article "the theory of quanta".The motion of a particle could be analogious to a ray in general optics.This is an analogy between Maupertui's principle and fermat's principle. How to use this...
  11. LLT71

    I Fermat's principle of least time

    while I was studying "fermat's principle of least time" I was (and still I am) ntrigued by the fact that "out of all possible paths that it might take to get from one point to another, light takes the path which requires the shortest time". this question may be a bit philosophical but: "how"...
  12. W

    Deriving back focal length using Fermat's principle

    Homework Statement The plano-convex lens has a diameter of 200 mm and the central thickness of 20 mm. The edge thickness is zero and the refractive index of glass is 1.5. Find the following parameters of the lens: (i) the back focal length Homework Equations N/A The Attempt at a Solution I've...
  13. Roger Dodger

    Fermat's Principle, Snell's Law, and the Prism

    I have a conundrum of sorts that has made me feel like an idiot and I am hoping someone can point out my mistake. Suppose a light source is placed to the left of a prism and a detector is placed on the opposite side. I have seen plenty of pictures of this sort, and they all appear to show the...
  14. P

    Fermat's principle can be derived from Maxwell equations?

    As we know, the Fermat's principle states: Light takes the path of least time. I wonder whether Fermat's principle can be derived from Maxwell equations. If it can, then Fermat's principle is included in Maxwell equations, or Fermat's principle is not an independent postulate.
  15. Anton Alice

    Fermat's principle seems indefinite

    Hello forum, please take a look at the following picture: It's a salt solution, with increasing refractive index, as you go down the solution. How can I explain this with Fermat's principle? Let's set the starting point A to the point, where the laser beam penetrates the left wall of the...
  16. W

    Geometric optics and Fermat's principle

    Homework Statement A ray travels as shown in the image attached below. In this case, Fermat's principle may be written as ##A =\frac{n(1+ay)}{\sqrt{1+(y')^2}}## Where y' is dy/dx, n is the index of refraction and A is a real constant. The trajectory of a ray of light is given by ##y =...
  17. isnainidiah

    Fermat's Principle to explain lenses and mirrors

    How do we use Fermat's principle of least time to explain the required shapes of lenses and mirrors?
  18. hideelo

    Does Fermat's principle ever give the longest time?

    I know that the way we calculate in Fermat's principle in optics is to take the path length (as an integral) and demand that it be stationary to first order. Now this approach is mathematically the same as calculating a geodesic, or finding stationary action, namely we use the calculus of...
  19. F

    Trouble understanding Fermat's principle of least time

    I've been reading about Fermats last principle in Feynmans lectures on physics, and he sort of goes through the derivation of Snell's law by considering a simple refraction and applying a bit of trigonometry (see photo). What I'm having trouble with is this. He states that if you take a two...
  20. N

    Proof of Fermat's principle of least time?

    I've been looking for this proof for months but I wasn't able to find something...
  21. N

    Why does light take an extremum path according to Fermat's Principle?

    According to Fermat's Principle (modified version) a light ray takes an extremum path. Can anyone explain the physical reason behind this? I mean, sure the light can take the least path but how can it take the longest path? And if it can indeed take the longest path, why not the paths shorter...
  22. kq6up

    Fermat's Principle with Calculus of Variations

    Homework Statement This is problem 6.3 in Taylor’s Classical Mechanics. It is in context of the calculus of variations. Consider a ray of light traveling in a vacuum from point P1to P2 by way of the point Q on a plane mirror, as in Figure 6.8. Show that Fermat's principle implies that, on the...
  23. M

    What is the meaning of the delta in Fermat's principle integral?

    I am unable understand this Integral, what does it actually saying? What does that "δ" means here? I haven't learned Calculus of variations, explain me with diagrams with possible.
  24. M

    Why Is the Denominator in Fermat's Principle (c/n) Instead of (cn)?

    The expression gives time taken to reach a a distance ds in a medium. The expression has the dimension of time, But my question is how come the denominator is (c/n)? why it can't be (cn)? explain me.
  25. N

    How does Fermat's Principle follow from Huygen's principle?

    How does Fermat's principle of least time (that if light goes from one point to another, any small deviation in the path will result in a path that takes the same time on the first order) follow from Huygen's wave principle? Everyone always says that Fermat's principle can be derived from...
  26. S

    Fermat's principle to derive lens formula

    1. For the past 1 hour, I'm trying to derive lens maker's equation using fermat's principle, which of course is our homework. Any help would be great regarding the topic. 2. According to Fermat's Principle, optical path length OPL = PA + AQ, here PA and AQ are two rays. Now using this I...
  27. I

    Deriving fermat's principle of least time (lagrangian stuff)

    original question: Consider light passing from one medium with index of refraction n1 into another medium with index of refraction n2. User Fermat's principle to minimize time, and derive snell's law (n1sintheta1=n2sintheta2). here is part of the solution key http://i.imgur.com/S1Pg9jC.png...
  28. zrek

    Does Fermat's Principle Apply in All Special Cases of Light Path Alteration?

    Fermat's principle is the "path of least time principle" or we can say that "he path a light ray takes is a local minimum". Quantum physics also says that the light take (test) all possible paths and only the minimal remains. Is it working in special cases like this too? Let's examine the fig...
  29. Matterwave

    Derivation of Eikonal equations from Fermat's Principle

    So, I'm reading Holm's Geometric Mechanics Part I, and in it he wants to derive the Eikonal equations from Fermat's principle. There's one part of the derivation that I don't understand. He gives the "Optical path" as: A=\int_a^b n(\vec{r}(s))ds Where ds is the infinitessimal arc...
  30. D

    Fermat's Principle Explained by Feynman

    In his QED lectures, Feynman demonstrates, in a way, how Fermat's principle follows from adding up the amplitudes for all possible paths, and then noting that removing the amplitude for a path near the path-of-least-time from the calculation will have a greater effect on the total amplitude than...
  31. P

    How Does Fermat's Principle Integrate with Quantum Physics?

    Fermat's principle is also known as the path of least time principle, and it explains why the angle of incidence is equal to the angle of reflection, as well as why light refracts the way it does when it enters certain materials. I understand it pretty well, but then I read that Fermat's...
  32. B

    Derive spherical mirror formula using Fermat's principle

    Homework Statement Using Fermat’s principle, derive the spherical mirror formula in paraxial approximation: \frac{1}{s_o} + \frac{1}{s_i} = \frac{-2}{R} where so and si are object and image distances, R is the radius of curvature of the sphere. Homework Equations As far as I know you...
  33. N

    A Question about Fermat's Principle

    Fermat's principle is well-known to everybody. If light travels from point 1 to point 2, it will take the path along which \int_{1}^{2}n dl is stationary where \ n is refractive index. When points 1 and 2 are two points on the path of the light ray, there is no problem. However, 1 or 2 are...
  34. S

    Fermat's Principle, Fourier Analysis, Standing waves, Normal Modes

    Can someone provide me a link that explains and provides a proof for the following principles: 1. Fermat's Principle that light always takes the path that minimizes the time taken 2. Solution to a Fourier Series and why all periodic motion can be represented as an infinite sum of sines and...
  35. D

    Polarization and Fermat's principle

    Homework Statement a) A beam of light in the air strikes the surface of a smooth piece of plastic having an index of refraction of 1.55 at an angle with the normal of 60 degrees. The incident light has equal component E-field amplitudes parallel and perpendicular to the plane of incidence...
  36. fluidistic

    Calculus of variations, Fermat's principle

    Homework Statement Fermat's principle establishes that the path taken by a light ray between 2 given points is such that the time that the light takes is the smallest possible. 1)Demonstrate that a light ray which propagates through a medium with a constant refractive index follows a straight...
  37. C

    Fermat's Principle: Minimizing Int. w/ Euler-Lagrange

    Homework Statement Need to minimize \int_{(x_1,y_1)}^{(x_2,y_2)} n(x,y)~ds where n(x,y)=e^y and (x_1,y_1)=(-1,1), (x_2,y_2)=(1,1). Homework Equations Euler-Lagrange equation The Attempt at a Solution \frac{d}{dx}\frac{dF}{dy'} - \frac{dF}{dy}=0 0 - e^y y' = 0 y' = 0 so y = constant or y...
  38. H

    Show that perfect a perfect reflector is a conic section with Fermat's principle

    Homework Statement Show, using Fermat's principle, that perfect reflecting surfaces are conic sections. Homework Equations Equations for the ellipse, parabola and hyperbola The Attempt at a Solution Ok, the ellipse seems easy. All rays coming from one focus reflecting to the other...
  39. N

    Fermat's principle and Snell's law

    Hi, While reading about Fermat's principle, and how to use it to derive Snell's law, I came across something I can't find any information about... I had heard about Fermat's principle before; that it said that the path taken by a lightbeam is the path that takes the least time. Now...
  40. S

    Elementary particles and fermat's principle

    Photons(light) follow the fermat's principle of least time...so do all elementary particles also follow fermat's principle of least time?..say electron,proton etc..
  41. S

    Does light have the ability to choose its path using Fermat's principle?

    I read in feynman's lecture that all rays passing through the converging lens meet at the focal point since they all take the same least time because of the change in the velocity of light while passing through the lens..is there any mathematical proof for this...
  42. N

    How does Fermat's principle explain the shape of a lens in Feynman's derivation?

    I have a problem in understanding Feynman's derivation of a lens' shape using Fermat's principle. Feynman writes that we have to choose such a surface of the lens that all optical ways from the source S to the focal point F will have the same length (so all will be taken by the traveling light)...
  43. C

    Proving Fermat's principle using calculus

    Homework Statement Let v1 be the velocity of light in air and v2 the velocity of light in water. According to Fermat's principle, a ray of light will travel from a point A in the air to a point B in the water by a path ACB that minimizes the time taken. Homework Equations @=angle...
  44. S

    Can Fermat's principle be applied to gravitational lensing?

    Light having to travel through a gravitational field deflects towards the mass and thus increases the length and duration of its journey (traveling through more curved space-time takes more proper time than traveling through less curved space-time) I understand that unlike in refraction, light's...
  45. S

    Refraction, Fermat's Principle and Lawnmowers

    According to Fermat's principle the path taken between two points by a ray of light is the path that can be traversed in the least time. So if the ray is refracted and if you take two points, one from before and one from after the refraction, the path taken between them was the one that takes...
  46. S

    Calculus of Variations - Fermat's Principle

    Light travels in a medium in which the speed of light c(x,y) is a function of position. Fermat's principle states that the time required for light to travel between two points is an extremum relative to all possible paths connecting the two points. 1) Show that the time for the light to travel...
  47. P

    Fermat's principle light and time

    I don't understand this idea. My terxtbook says that Fermat's principle is that light travels by the path that takes the least amount of time. Does that mean that light will go in crazy, curved paths if those are faster? How does it "know" which path will be the fastest? For example, let's...
  48. D

    Reformatting of Fermat's Principle

    The Real Statement The actual path between two points taken by a beam of light is the one which is traversed in the least time. Modern Restructured Statement To travel from one point to another ,Light would choose the path such that all other paths nearby take almost exactly the same...
  49. T

    Fermat's principle of least time and Kepler's second law

    While studying the history of classical mechanics I noticed that the primary motor for most of the early equations is the principle of conservation framed firstly as the law of inertia and then as the principle of least time (Fermat) and then as... But while trying to regain for the principle of...