Snell\'s law - Citizendia

In optics and physics, Snell's law (also known as Descartes' law or the law of diffraction), is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves, passing through a boundary between two different isotropic media, such as water and glass. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Mathematics and in the Sciences a formula (plural formulae, formulæ or formulas) is a concise way of expressing information A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. Isotropy is uniformity in all directions Precise definitions depend on the subject area An optical medium is material through which Electromagnetic waves propagate The law says that the ratio of the sines of the angles of incidence and of refraction is a constant that depends on the media.

In optics, the law is used in ray tracing to compute the angles of incidence or refraction, and in experimental optics to find the refractive index of a material. In physics ray tracing is a method for calculating the path of Waves or Particles through a system with regions of varying propagation Velocity, absorption Angle of incidence is a measure of deviation of something from "straight on" for example in the approach of a ray to a surface or the angle In Optics and Physics, Snell's law (also known as Descartes' law or the law of refraction) is a formula used to describe the relationship The refractive index (or index of Refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves is reduced inside the medium

$Refraction of light at the interface between two media of different refractive indices, with n2 > n1. Since the velocity is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.$

Refraction of light at the interface between two media of different refractive indices, with n₂ > n₁. Refraction is the change in direction of a Wave due to a change in its Speed. The refractive index (or index of Refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves is reduced inside the medium Since the velocity is lower in the second medium (v₂ < v₁), the angle of refraction θ₂ is less than the angle of incidence θ₁; that is, the ray in the higher-index medium is closer to the normal.

Named after Dutch mathematician Willebrord Snellius, one of its discoverers, Snell's law states that the ratio of the sines of the angles of incidence and refraction is equivalent to the ratio of velocities in the two media, or equivalent to the opposite ratio of the indices of refraction:

$\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1}$

$n_1\sin\theta_1 = n_2\sin\theta_2\ .$

Snell's law follows from Fermat's principle of least time, which in turn follows from the propagation of light as waves. “Snellius” redirects here For the lunar crater named Snellius see Snellius (crater. In Physics, velocity is defined as the rate of change of Position. Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the In Optics, Fermat's principle or the principle of least time is the idea that the path taken between two points by a ray of light is the path that can be

1 History
2 Explanation
- 2.1 Total internal reflection and critical angle
- 2.2 Derivations
3 Vector form
4 Dispersion
5 See also
6 References
7 External links

History

Ptolemy, of ancient Greece, had, through experiment, found a relationship regarding refraction angles, but which was inaccurate for angles that were not small. Claudius Ptolemaeus ( Greek: Klaúdios Ptolemaîos; after 83 &ndash ca Ptolemy was confident he had found an accurate empirical law, partially as a result of fudging his data to fit theory (see: confirmation bias). In Psychology and Cognitive science, confirmation bias is a tendency to search for or interpret new information in a way that confirms one's preconceptions and avoids ^[1]

An 1837 view of the history of "the Law of the Sines"^[2]

Snell's law was first described in a formal manuscript in a 984 writing by Ibn Sahl,^[3]^[4] who used it to work out the shapes of lenses that focus light with no geometric aberrations, known as anaclastic lenses. Events By Place Asia Emperor Kazan succeeds Emperor En'yū on the throne of Japan. This article is about the physicist For the physician see Ali ibn Sahl Rabban al-Tabari. An aspheric lens or asphere is a lens whose surfaces have a profile that is neither a portion of a Sphere nor of a circular cylinder.

It was described again by Thomas Harriot in 1602,^[5] who did not publish his work. Thomas Harriot ( c 1560 – 2 July 1621) was an English astronomer, Mathematician, Ethnographer, and Translator

In 1621, Willebrord Snellius (Snel) derived a mathematically equivalent form, that remained unpublished during his lifetime. René Descartes independently derived the law using heuristic momentum conservation arguments in terms of sines in his 1637 treatise Discourse on Method, and used it to solve a range of optical problems. Organization How to think correctly The Method of Science Morals Maxims deduced from this Method Proof of God and the Soul Physics the heart Rejecting Descartes' solution, Pierre de Fermat arrived at the same solution based solely on his principle of least time.

According to Dijksterhuis^[6], "In De natura lucis et proprietate (1662) Isaac Vossius said that Descartes had seen Snell's paper and concocted his own proof. We now know this charge to be undeserved but it has been adopted many times since. " Both Fermat and Huygens repeated this accusation that Descartes had copied Snell.

In French, Snell's Law is called "la loi de Descartes" or "loi de Snell-Descartes. French ( français,) is a Romance language spoken around the world by 118 million people as a native language and by about 180 to 260 million people "

Huygens's construction

In his 1678 Traité de la Lumiere, Christiaan Huygens showed how Snell's law of sines could be explained by, or derived from, the wave nature of light, using what we have come to call the Huygens–Fresnel principle. Christiaan Huygens (ˈhaɪgənz in English ˈhœyɣəns in Dutch) ( April 14, 1629 &ndash July 8, 1695) was a Dutch The Huygens–Fresnel principle (named for Dutch Physicist Christiaan Huygens, and French physicist Augustin-Jean Fresnel

Although he spelled his name "Snel", as noted above, it has conventionally been spelled "Snell", apparently by misinterpreting the Latin form of his name, "Snellius". ^[7]

Explanation

Snell's law is used to determine the direction of light rays through refractive media with varying indices of refraction. The indices of refraction of the media, labeled $n 1, n 2$ and so on, are used to represent the factor by which a light ray's speed decreases when traveling through a refractive medium, such as glass or water, as opposed ^[8] to its velocity in a vacuum.

As light passes the border between media, depending upon the relative refractive indices of the two media, the light will either be refracted to a lesser angle, or a greater one. These angles are measured with respect to the normal line, represented perpendicular to the boundary. In the case of light traveling from air into water, light would be refracted towards the normal line, because the light is slowed down in water; light traveling from water to air would refract away from the normal line.

Refraction between two surfaces is also referred to as reversible because if all conditions were identical, the angles would be the same for light propagating in the opposite direction.

Snell's law is generally true only for isotropic or specular media (such as glass). Glass in the common sense refers to a Hard, Brittle, transparent Solid, such as that used for Windows many In anisotropic media such as some crystals, birefringence may split the refracted ray into two rays, the ordinary or o-ray which follows Snell's law, and the other extraordinary or e-ray which may not be co-planar with the incident ray. Anisotropy (pronounced with stress on the third syllable ˌænaɪˈsɒtrəpi is the property of being directionally dependent as opposed to Isotropy, which means homogeneity In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating Birefringence, or double refraction, is the decomposition of a ray of Light into two rays (the ordinary ray and the extraordinary ray

When the light or other wave involved is monochromatic, that is, of a single frequency, Snell's law can also be expressed in terms of a ratio of wavelengths in the two media, λ₁ and λ₂:

$\frac{\sin\theta_1}{\sin\theta_2} = \frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2}$

Total internal reflection and critical angle

An example of the angles involved within total internal reflection.

Main article: Total internal reflection

When light moves from a dense to a less dense medium, such as from water to air, Snell's law cannot be used to calculate the refracted angle when the resolved sine value is higher than 1. At this point, light is reflected in the incident medium, known as internal reflection. Before the ray totally internally reflects, the light refracts at the critical angle; it travels directly along the surface between the two refractive media, without a change in phases like in other forms of optical phenomena.

As an example, a ray of light is incident at $50 o$ towards a water–air boundary. If the angle is calculated using Snell's Law, then the resulting sine value will not invert, and thus the refracted angle cannot be calculated by Snell's law, due to the absence of a refracted outgoing ray:

$\theta_2 = \sin^{-1} \left(\frac{n_1}{n_2}\sin\theta_1\right) = \sin^{-1} \left(\frac{1.333}{1.000}0.766\right) = \sin^{-1} 1.021$

In order to calculate the critical angle, let $θ 2 = 90 o$ and solve for $θ crit$ :

$\theta_{\mathrm{crit}} = \sin^{-1} \left( \frac{n_2}{n_1} \right)$

When θ₁ > θ_crit, no refracted ray appears, and the incident ray undergoes total internal reflection from the interface medium.

Derivations

$Wavefronts due to a point source in the context of Snell's law (the region below the gray line has a higher index of refraction than the region above it).$

Wavefronts due to a point source in the context of Snell's law (the region below the gray line has a higher index of refraction than the region above it). In Optics and Physics, a wavefront is the locus (a line, or in a Wave propagating in 3 dimensions a Surface) of A point source is a single identifiable localized source of something

Snell's law may be derived from Fermat's principle, which states that the light travels the path which takes the least time. In Optics, Fermat's principle or the principle of least time is the idea that the path taken between two points by a ray of light is the path that can be By taking the derivative of the optical path length, the stationary point is found giving the path taken by the light (though it should be noted that the result does not show light taking the least time path, but rather one that is stationary with respect to small variations as there are cases where light actually takes the greatest time path, as in a spherical mirror). In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Optics, optical path length (OPL is the product of the geometric length of the path light follows through the system and the Index of refraction of the medium In Mathematics, particularly in Calculus, a stationary point is an input to a function where the Derivative is zero (equivalently the In a classic analogy by Richard Feynman, the area of lower refractive index is replaced by a beach, the area of higher refractive index by the sea, and the fastest way for a rescuer on the beach to get to a drowning person in the sea is to run along a path that follows Snell's law. Richard Phillips Feynman (ˈfaɪnmən May 11 1918 – February 15 1988 was an American Physicist known for the Path integral formulation of quantum Drowning is Death as caused by suffocation when a liquid causes interruption of the body's absorption of oxygen from the air leading to Asphyxia.

Alternatively, Snell's law can be derived using interference of all possible paths of light wave from source to observer—it results in destructive interference everywhere except extrema of phase (where interference is constructive)—which become actual paths.

Another way to derive Snell’s Law involves an application of the general boundary conditions of Maxwell equations for electromagnetic radiation. In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter.

Vector form

Given a normalized light vector l (pointing from the light source toward the surface) and a normalized plane normal vector n, one can work out the normalized reflected and refracted rays:

$\cos\theta_1=\mathbf{n}\cdot(-\mathbf{l})$

$\cos\theta_2=\sqrt{1-\left(\frac{n_1}{n_2}\right)^2\left(1-\left(\cos\theta_1\right)^2\right)}$

$\mathbf{v}_{\mathrm{reflect}}=\mathbf{l}-\left(2\cos\theta_1\right)\mathbf{n}$

$\mathbf{v}_{\mathrm{refract}}=\left(\frac{n_1}{n_2}\right)\mathbf{l} + \left( \frac{n_1}{n_2}\cos\theta_1 - \cos\theta_2\right)\mathbf{n}$

Note: $\mathbf{n}\cdot(-\mathbf{l})$ must be positive. Otherwise, use

$\mathbf{v}_{\mathrm{refract}}=\left(\frac{n_1}{n_2}\right)\mathbf{l} + \left(\frac{n_1}{n_2}\cos\theta_1 + \cos\theta_2\right)\mathbf{n}.$

Example:

$\mathbf{l}=\{0.707107, -0.707107\},~\mathbf{n}=\{0,1\},~\frac{n_1}{n_2}=1.1$

$\mathbf{~}\cos\theta_1=0.707107,~\cos\theta_2=0.62849$

$\mathbf{v}_{\mathrm{reflect}}=\{0.707107, 0.707107\} ,~\mathbf{v}_{\mathrm{refract}}=\{0.777817, -0.62849\}$

The cosines may be recycled and used in the Fresnel equations for working out the intensity of the resulting rays. During total internal reflection an evanescent wave is produced, which rapidly decays from the surface into the second medium. An evanescent wave is a nearfield standing Wave exhibiting Exponential decay with distance A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value Conservation of energy is maintained by the circulation of energy across the boundary, averaging to zero net energy transmission.

Dispersion

Main article: Dispersion (optics)

In many wave-propagation media, wave velocity changes with frequency or wavelength of the waves; this is true of light propagation in most transparent substances other than a vacuum. In Optics, dispersion is the phenomenon in which the Phase velocity of a wave depends on its frequency These media are called dispersive. The result is that the angles determined by Snell's law also depend on frequency or wavelength, so that a ray of mixed wavelengths, such as white light, will spread or disperse. Such dispersion of light in glass or water underlies the origin of rainbows, in which different wavelengths appear as different colors. A rainbow is an optical and meteorological phenomenon that causes a spectrum of Light to appear in the Sky when the Sun

In optical instruments, dispersion leads to chromatic aberration, a color-dependent blurring that sometimes is the resolution-limiting effect. In Optics, chromatic aberration is caused by a lens having a different Refractive index for different Wavelengths of Light This was especially true in refracting telescopes, before the invention of achromatic objective lenses. A refracting or refractor telescope is a dioptric Telescope that uses a lens as its objective to form an image An achromatic lens or achromat is a lens that is designed to limit the effects of chromatic and Spherical aberration.

References

^ Ptolemy (ca. 100-ca. 170). An evanescent wave is a nearfield standing Wave exhibiting Exponential decay with distance Reflection is the change in direction of a Wave front at an interface between two different media so that the wave front returns into the medium from which Refraction is the change in direction of a Wave due to a change in its Speed. The refractive index (or index of Refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves is reduced inside the medium Snell's window is a phenomenon by which an underwater viewer sees everything above the surface through a cone of light of width of about 96 degrees Eric Weinstein's World of Scientific Biography.
^ William Whewell, History of the Inductive Science from the Earliest to the Present Times, London: John H. Parker, 1837.
^ Wolf, K. B. (1995), "Geometry and dynamics in refracting systems", European Journal of Physics 16: 14-20.
^ Rashed, Roshdi (1990). "A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses". Isis 81: 464–491. Isis is an Academic journal published by The University of Chicago Press devoted to the History of science, History of medicine doi:10.1086/355456. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ Kwan, A. , Dudley, J. , and Lantz, E. (2002). "Who really discovered Snell's law?". Physics World 15 (4): 64. Physics World is the membership magazine of the Institute of Physics, one of the largest physical societies in the world
^ Fokko Jan Dijksterhuis (2004). Lenses and Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century. Springer. ISBN 1402026978.
^ George Sarton (1955). The Appreciation of Ancient and Medieval Science During the Renaissance. University of Pennsylvania Press, p. xiii.
^ Snell's Law

External links

Discovery of the law of refraction
Snell's Law of Refraction (Wave Fronts) by Todd Rowland, The Wolfram Demonstrations Project.

Dictionary

Snell's law

-noun

(optics) the law that, for a ray incident on the interface of two media, the sine of the angle of incidence times the index of the refraction of the first medium is equal to the sine of the angle of refraction times the index of refraction of the second medium

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