![]() ![]() To put it loosely, a ray path is surrounded by close paths that can be traversed in very close times. 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. 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 traveled in the least time. (The medium is not assumed to be homogeneous or isotropic.) The surface Σ (with unit normal n̂ at P′) is the locus of points that a disturbance at P can reach in the same time that it takes to reach P′ in other words, Σ is the secondary wavefront with radius PP′. ![]() For the purposes of Fermat's principle, the propagation time from P to P′ is taken as for a point-source at P, not (e.g.) for an arbitrary wavefront W passing through P. 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.įig. 2: Two points P and P′ on a path from A to B. 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. 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. Not until the 19th century was it understood that nature's ability to test alternative paths is merely a fundamental property of waves. Fermat's principle states that the path taken by a ray between two given points is the path that can be traveled in the least time.įirst 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. If we seek the required value of x, we find that the angles α and β satisfy Snell's law.įermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. Given an object-point A in the air, and an observation point B in the water, the refraction point P is that which minimizes the time taken by the light to travel the path APB. Perpendicular case: Reflected % and transmitted %.For other theorems named after Pierre de Fermat, see Fermat's theorem.įig. 1: Fermat's principle in the case of refraction of light at a flat surface between (say) air and water. Parallel case: Reflected % and transmitted %. Which applies to both the parallel and perpendicular cases. For further details, see Jenkins and White.Ĭhecking out conservation of energy in this situation leads to the relationship When you take the intensity times the area for both the reflected and refracted beams, the total energy flux must equal that in the incident beam. (For example, try light incident from a medium of n 1=1.5 upon a medium of n 2=1.0 with an angle of incidence of 30°.) But the square of the transmission coefficient gives the transmitted energy flux per unit area (intensity), and the area of the transmitted beam is smaller in the refracted beam than in the incident beam if the index of refraction is less than that of the incident medium. You can choose values of parameters which will give transmission coefficients greater than 1, and that would appear to violate conservation of energy. Note that these coefficients are fractional amplitudes, and must be squared to get fractional intensities for reflection and transmission. For a dielectric medium where Snell's Law can be used to relate the incident and transmitted angles, Fresnel's Equations can be stated in terms of the angles of incidence and transmission.įresnel's equations give the reflection coefficients: That is, they give the reflection and transmission coefficients for waves parallel and perpendicular to the plane of incidence. External Reflection: Fresnell's Equations Reflection and Transmission External Reflection Go to internal reflectionįresnel's equations describe the reflection and transmission of electromagnetic waves at an interface.
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