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Encyclopedia > Snell's law
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.

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 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 air and glass. 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. Image File history File links Snells_law. ... Image File history File links Snells_law. ... The straw seems to be broken, due to refraction of light as it emerges into the air. ... 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. ... Table of Opticks, 1728 Cyclopaedia Optics ( appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. ... Physics (Greek: (phÃºsis), nature and (phusikÃ©), knowledge of nature) is the science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ... A formula is a concise way of expressing information symbolically (as in a mathematical or chemical formula) or a general relationship between quantities. ... This article is about waves in the most general scientific sense. ... Isotropic means independent of direction. Isotropic radiation has the same intensity regardless of the direction of measurement, and an isotropic field exerts the same action regardless of how the test particle is oriented. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

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. A ray traced scene. ... Fig. ... Angle of refraction refers to the angle a wave makes to the line of normal incidence when a wave passes from one medium to another. ... 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. ...

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 equal to the ratio of velocities in the two media, or equivalently to the inverse ratio of the indices of refraction: Willebrord Snellius Willebrord Snellius (Willebrord Snel van Royen) (1580â€“October 30, 1626) was a Dutch astronomer and mathematician, most famous for the law of refraction now known as Snells law. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... This article is about velocity in physics. ...

 $frac{sintheta_1}{sintheta_2} = frac{v_1}{v_2} = frac{n_2}{n_1}$ or $n_1sintheta_1 = n_2sintheta_2$

Snell's law follows from Fermat's principle of least time, which in turn follows from the propagation of light as waves. Pierre de Fermat Pierre de Fermat (August 17, 1601 &#8211; January 12, 1665) was a French lawyer at the Parliament of Toulouse and a mathematician who is given credit for the development of modern calculus. ... Fermats principle assures that the angles given by Snells law always reflect lights quickest path between P and Q. Fermats principle in optics states: This principle was first stated by Pierre de Fermat. ...

An 1837 view of the history of "the Law of the Sines"[1]

Snell's law was first described in a manuscript written in 984 by Ibn Sahl,[2][3] who used it to work out the shapes of lenses that focus light with no geometric aberrations, known as anaclastic lenses. Image File history File links Download high-resolution version (401x687, 61 KB) Page from Rev. ... Image File history File links Download high-resolution version (401x687, 61 KB) Page from Rev. ... Events End of the reign of Emperor Enyu of Japan Emperor Kazan ascends to the throne of Japan Births Deaths Categories: 984 ... Ibn Sahl - Wikipedia, the free encyclopedia /**/ @import /skins-1. ... 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,[4] who did not publish his work. Thomas Harriot (ca. ... This page is about the year. ...

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. Rejecting Descartes' solution, Pierre de Fermat arrived at the same solution based solely on his principle of least time. 1621 was a common year starting on Friday of the Gregorian calendar (or a common year starting on Monday of the 10-day slower Julian calendar). ... RenÃ© Descartes (French IPA: ) (March 31, 1596 â€“ February 11, 1650), also known as Renatus Cartesius (latinized form), was a highly influential French philosopher, mathematician, scientist, and writer. ... Events February 3 - Tulipmania collapses in Netherlands by government order February 15 - Ferdinand III becomes Holy Roman Emperor December 17 - Shimabara Rebellion erupts in Japan Pierre de Fermat makes a marginal claim to have proof of what would become known as Fermats last theorem. ... The Discourse on Method is a philosophical and mathematical treatise published by RenÃ© Descartes in 1637. ...

According to Dijksterhuis[5], "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."

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. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Christiaan Huygens (pronounced in English (IPA): ; in Dutch: ) (April 14, 1629 â€“ July 8, 1695), was a Dutch mathematician, astronomer and physicist; born in The Hague as the son of Constantijn Huygens. ... Wave Refraction in the manner of Huygens. ...

## Explanation

Snell's law is used to determine the direction of light rays though refractive media with varying indices of refraction. The indices of refraction of the media, labeled n1,n2 and so on, are used to represent the factor by which light is "slowed down" within a refractive medium, such as glass or water, compared 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, due to the fact that 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 due to the fact that 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). 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. Glass can be made transparent and flat, or into other shapes and colors as shown in this sphere from the Verrerie of Brehat in Brittany. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... Quartz crystal Synthetic bismuth crystal Insulin crystals Gallium, a metal that easily forms large single crystals A huge monocrystal of potassium dihydrogen phosphate grown from solution by Saint-Gobain for the megajoule laser of CEA. In chemistry and mineralogy, a crystal is a solid in which the constituent atoms, molecules... A calcite crystal laid upon a paper with some letters showing the double refraction Birefringence, or double refraction, is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals, depending on...

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:

$frac{sintheta_1}{sintheta_2} = frac{v_1}{v_2} = frac{lambda_1}{lambda_2}$

### Total internal reflection and critical angle

An example of the angles involved within total internal reflection.

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. Image File history File links Refraction_internal_reflection_diagram. ... Image File history File links Refraction_internal_reflection_diagram. ... The larger the angle to the normal, the smaller is the fraction of light transmitted, until the angle when total internal reflection occurs. ...

As an example, a ray of light is incident at 50o 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:

$theta_2 = sin^{-1} left(frac{n_1}{n_2}sintheta_1right) = sin^{-1} left(frac{1.333}{1.000}0.766right) = sin^{-1} 1.021$

In order to calculate the critical angle, let θ2 = 90o and solve for θcrit:

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

When θ1 > θcrit, no refracted ray appears, and the incident ray undergoes total internal reflection from the interface medium. The larger the angle to the normal, the smaller is the fraction of light transmitted, until the angle when total internal reflection occurs. ...

### Derivations

Snell's law may be derived from Fermat's principle, which states that the light travels the path which takes the least time. 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 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. Fermats principle assures that the angles given by Snells law always reflect lights quickest path between P and Q. Fermats principle in optics states: This principle was first stated by Pierre de Fermat. ... For a non-technical overview of the subject, see Calculus. ... In optics and telecommunication, the term optical path length has the following meanings: In a medium of constant refractive index, n , the product of the geometric distance and the refractive index. ... Stationary points (red pluses) and inflection points (green circles). ... Richard Phillips Feynman (May 11, 1918 â€“ February 15, 1988; surname pronounced ) was an American physicist known for expanding the theory of quantum electrodynamics, the physics of the superfluidity of supercooled liquid helium, and particle theory. ...

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, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ... It has been suggested that this article or section be merged with light. ...

## Uses

### Calculating refractive indices

In the diagram on the right, two media of refractive indices n1 (on the left) and n2 (on the right) meet at a surface or interface (vertical line). n2 > n1, and light has a slower phase velocity within the second medium. The phase velocity of a wave is the rate at which the phase of the wave propagates in space. ...

A light ray PO in the leftmost medium strikes the interface at the point O. From point O, we project a straight line at right angles to the line of the interface; this is known as the normal to the surface (horizontal line). The angle between the normal and the light ray PO is known as the angle of incidence, θ1. A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ...

The ray continues through the interface into the medium on the right; this is shown as the ray OQ. The angle it makes to the normal is known as the angle of refraction, θ2.

n1sinθ1 = n2sinθ2 or $sum_{k=x,y}^N x=n_xsintheta_x,n_y sintheta_y$
$frac{n_1}{n_2} = frac{sintheta_2}{sintheta_1}$

Note that, for the case of θ1 = 0° (i.e., a ray perpendicular to the interface) the solution is θ2 = 0° regardless of the values of n1 and n2-- a ray entering a medium perpendicular to the surface is never bent.

The above is also valid for light going from a dense to a less dense medium; the symmetry of Snell's law shows that the same ray paths are applicable in opposite direction.

A qualitative rule for determining the direction of refraction is that the ray in the denser medium is always closer to the normal. An analogy often used to remember this is done by visualizing the ray as a car crossing the boundary between asphalt (the less dense medium) and mud (the denser medium). Depending on the angle, either the left wheel or the right wheel of the car will cross into the new medium first, causing the car to swerve.

### Vector form

Given a normalized ray vector v and a normalized plane normal vector p, one can work out the normalized reflected and refracted rays:

$costheta_1=mathbf{v}cdotmathbf{p}$
$costheta_2=sqrt{1-left(frac{n_1}{n_2}right)^2left(1-left(costheta_1right)^2right)}$
$mathbf{v}_{mathrm{reflect}}=mathbf{v}-left(2costheta_1right)mathbf{p}$
$mathbf{v}_{mathrm{refract}}=left(frac{n_1}{n_2}right)mathbf{v} + left(costheta_2 - frac{n_1}{n_2}costheta_1right)mathbf{p}$

Note: $mathbf{v}cdotmathbf{p}$ must be positive. Otherwise, use

$mathbf{v}_{mathrm{refract}}=left(frac{n_1}{n_2}right)mathbf{v} - left(costheta_2 + frac{n_1}{n_2}costheta_1right)mathbf{p}$

Example:

$mathbf{v}={0.707107, -0.707107},~mathbf{p}={0,1},~frac{n_1}{n_2}=1.1$
$mathbf{~}costheta_1=-0.707107,~costheta_2=0.62849$
$mathbf{v}_{mathrm{reflect}}={0.707107, 0.707107} ,~mathbf{v}_{mathrm{refract}}={0.777817, -0.62849}$

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. Conservation of energy is maintained by the circulation of energy across the boundary, averaging to zero net energy transmission. The Fresnel equations, deduced by Augustin-Jean Fresnel, describe the behaviour of light when moving between media of differing refractive indices. ... An evanescent wave is an electromagnetic wave that decays exponentially with distance. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ...

## Dispersion

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. 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, since different wavelengths appear as different colors. Full featured double rainbow in Wrangell-St. ...

In optical instruments, dispersion leads to chromatic aberration, a color-dependent blurring that sometimes is the resolution-limiting effect. This was especially true in refracting telescopes, before the invention of achromatic objective lenses. Dispersion of a light beam in a prism. ... Chromatic aberration is caused by the dispersion of the lens material, the variation of its refractive index n with the wavelength of light. ... Image of a refracting telescope from the Cincinnati Observatory in 1848 A refracting or refractor telescope is a dioptric telescope that uses a lens as its objective to form an image. ... Diagram of an achromatic lens (doublet). ...

An evanescent wave is an electromagnetic wave that decays exponentially with distance. ... The Fresnel equations, deduced by Augustin-Jean Fresnel, describe the behaviour of light when moving between media of differing refractive indices. ... The reflection of a bridge in Indianapolis, Indianas Central Canal. ... The straw seems to be broken, due to refraction of light as it emerges into the air. ... 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. ... Snells window is a phenomenon by which an under-water viewer sees everything above the surface through a cone of light of width around 100 degrees. ... The larger the angle to the normal, the smaller is the fraction of light transmitted, until the angle when total internal reflection occurs. ...

## References

1. ^ William Whewell, History of the Inductive Science from the Earliest to the Present Times, London: John H. Parker, 1837.
2. ^ Wolf, K. B. (1995), "Geometry and dynamics in refracting systems", European Journal of Physics 16: 14-20.
3. ^ Rashed, Roshdi (1990). "A pioneer in anaclastics: Ibn Sahl on burning mirrors and lenses". Isis 81: 464–491. DOI:10.1086/355456.
4. ^ Kwan, A., Dudley, J., and Lantz, E. (2002). "Who really discovered Snell's law?". Physics World 15 (4): 64.
5. ^ Fokko Jan Dijksterhuis (2004). Lenses and Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century. Springer. ISBN 1402026978.

Isis is an academic journal published by the University of Chicago devoted to the history of science, history of medicine, and the history of technology, as well as their cultural influences, featuring both original research articles as well as extensive book reviews and review essays. ... A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ... Physics World cover from the February 2005 issue Physics World is the membership magazine of the Institute of Physics, one of the largest physical societies in the world. ...

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