Fuck off. Deborah Chiddix, Psy.D. jm...@sonic.net (707) 696-6620
On Nov 17, 2012, at 5:05 AM, Jared Chiddix <j.chid...@me.com> wrote: > Fresnel equations > This article is about the Fresnel equations describing reflection and > refraction of light at uniform planar interfaces. For the diffraction of > light through an aperture, see Fresnel diffraction. For the thin lens and > mirror technology, see Fresnel lens. > > Partial transmission and reflection amplitudes of a wave travelling from a > low to high refractive index medium. > The Fresnel equations (or Fresnel conditions), deduced by Augustin-Jean > Fresnel ( /frɛˈnɛl/), describe the behaviour of light when moving between > media of differing refractive indices. The reflection of light that the > equations predict is known as Fresnel reflection. > > Overview > > When light moves from a medium of a given refractive index n1 into a second > medium with refractive index n2, both reflection and refraction of the light > may occur. The Fresnel equations describe what fraction of the light is > reflected and what fraction is refracted (i.e., transmitted). They also > describe the phase shift of the reflected light. > > The equations assume the interface is flat, planar, and homogeneous, and that > the light is a plane wave. > > ↑Jump back a section > Definitions and power equations > > > Variables used in the Fresnel equations. > In the diagram on the right, an incident light ray IO strikes the interface > between two media of refractive indices n1 and n2 at point O. Part of the ray > is reflected as ray OR and part refracted as ray OT. The angles that the > incident, reflected and refracted rays make to the normal of the interface > are given as θi, θr and θt, respectively. > > The relationship between these angles is given by the law of reflection: > > > and Snell's law: > > > The fraction of the incident power that is reflected from the interface is > given by the reflectance R and the fraction that is refracted is given by the > transmittance T.[1] The media are assumed to be non-magnetic. > > The calculations of R and T depend on polarisation of the incident ray. Two > cases are analyzed: > > The incident light is s-polarized. That means its electric field is in the > plane of the interface (perpendicular to the plane of the diagram above). > The incident light is p-polarized. That means its electric field is in a > perpendicular direction to s-polarized above (in the plane of the diagram > above). > For the s-polarized light, the reflection coefficient is given by > > > > , > where the second form is derived from the first by eliminating θt using > Snell's law and trigonometric identities. > > For the p-polarized light, the R is given by > > > > . > As a consequence of the conservation of energy, the transmission coefficients > are given by [2] > > > and > > > These relationships hold only for power coefficients, not for amplitude > coefficients as defined below. > > If the incident light is unpolarised (containing an equal mix of s- and > p-polarisations), the reflection coefficient is > > > For common glass, the reflection coefficient is about 4%. Note that > reflection by a window is from the front side as well as the back side, and > that some of the light bounces back and forth a number of times between the > two sides. The combined reflection coefficient for this case is 2R/(1 + R), > when interference can be neglected (see below). > > The discussion given here assumes that the permeability μ is equal to the > vacuum permeability μ0 in both media. This is approximately true for most > dielectric materials, but not for some other types of material. The > completely general Fresnel equations are more complicated. > > > > Special angles > > Main articles: Brewster's angle and Total internal reflection > At one particular angle for a given n1 and n2, the value of Rp goes to zero > and a p-polarised incident ray is purely refracted. This angle is known as > Brewster's angle, and is around 56° for a glass medium in air or vacuum. Note > that this statement is only true when the refractive indices of both > materials are real numbers, as is the case for materials like air and glass. > For materials that absorb light, like metals and semiconductors, n is > complex, and Rp does not generally go to zero. > > When moving from a denser medium into a less dense one (i.e., n1 > n2), above > an incidence angle known as the critical angle, all light is reflected and Rs > = Rp = 1. This phenomenon is known as total internal reflection. The critical > angle is approximately 41° for glass in air. > > ↑Jump back a section > Amplitude equations > > Equations for coefficients corresponding to ratios of the electric field > complex-valued amplitudes of the waves (not necessarily real-valued > magnitudes) are also called "Fresnel equations". These take several different > forms, depending on the choice of formalism and sign convention used. The > amplitude coefficients are usually represented by lower case r and t. > > > Amplitude ratios: air to glass > > Amplitude ratios: glass to air > Conventions used here > > In this treatment, the coefficient r is the ratio of the reflected wave's > complex electric field amplitude to that of the incident wave. The > coefficient t is the ratio of the transmitted wave's electric field amplitude > to that of the incident wave. The light is split into s and p polarizations > as defined above. (In the figures to the right, s polarization is denoted "" > and p is denoted "".) > > For s-polarization, a positive r or t means that the electric fields of the > incoming and reflected or transmitted wave are parallel, while negative means > anti-parallel. For p-polarization, a positive r or t means that the magnetic > fields of the waves are parallel, while negative means anti-parallel.[3] > > Formulas > > Using the conventions above,[3] > > > > > > Notice that but .[4] > > Because the reflected and incident waves propagate in the same medium and > make the same angle with the normal to the surface, the amplitude reflection > coefficient is related to the reflectance R by [5] > > . > The transmittance T is generally not equal to |t|2, since the light travels > with different direction and speed in the two media. The transmittance is > related to t by.[6] > > . > The factor of n2/n1 occurs from the ratio of intensities (closely related to > irradiance). The factor of cos θt/cos θi occurs from the change in area of > the pencil of rays, needed since T, the ratio of powers, is equal to the > ratio of (intensity × area). In terms of the ratio of refractive indices, > > , > and the magnification of beam diameter at the interface, m, > > . > ↑Jump back a section > Multiple surfaces > > When light makes multiple reflections between two or more parallel surfaces, > the multiple beams of light generally interfere with one another, resulting > in net transmission and reflection amplitudes that depend on the light's > wavelength. The interference, however, is seen only when the surfaces are at > distances comparable to or smaller than the light's coherence length, which > for ordinary white light is few micrometers; it can be much larger for light > from a laser. > > An example of interference between reflections is the iridescent colours seen > in a soap bubble or in thin oil films on water. Applications include > Fabry–Pérot interferometers, antireflection coatings, and optical filters. A > quantitative analysis of these effects is based on the Fresnel equations, but > with additional calculations to account for interference. > > The transfer-matrix method, or the recursive Rouard method[7] can be used to > solve multiple-surface problems. > > ↑Jump back a section > See also > > Index-matching material > Fresnel rhomb, Fresnel's apparatus to produce circularly polarised light [1] > Specular reflection > Schlick's approximation > Snell's window > ↑Jump back a section > References > > ^ Hecht (1987), p. 100. > ^ Hecht (1987), p. 102. > ^ a b Lecture notes by Bo Sernelius, main site, see especially Lecture 12. > ^ Hecht (2003), p. 116, eq.(4.49)-(4.50). > ^ Hecht (2003), p. 120, eq.(4.56). > ^ Hecht (2003), p. 120, eq.(4.57). > ^ see, e.g. O.S. Heavens, Optical Properties of Thin Films, Academic Press, > 1955, chapt. 4. > Hecht, Eugene (1987). Optics (2nd ed.). Addison Wesley. ISBN 0-201-11609-X. > Hecht, Eugene (2002). Optics (4th ed.). Addison Wesley. ISBN 0-321-18878-0. > ↑Jump back a section > Further reading > > The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University > Press, 2010, ISBN 978-0-521-57507-2. > Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson > Education, Dorling Kindersley, 2007, ISBN 81-7758-293-3 > Light and Matter: Electromagnetism, Optics, Spectroscopy and Lasers, Y.B. > Band, John Wiley & Sons, 2010, ISBN 978-0-471-89931-0 > The Light Fantastic – Introduction to Classic and Quantum Optics, I.R. > Kenyon, Oxford University Press, 2008, ISBN 978-0-19-856646-5 > Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC > publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) > 0-89573-752-3 > McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN > 0-07-051400-3 > ↑Jump back a section > External links > > Fresnel Equations – Wolfram > FreeSnell – Free software computes the optical properties of multilayer > materials > Thinfilm – Web interface for calculating optical properties of thin films and > multilayer materials. (Reflection & transmission coefficients, ellipsometric > parameters Psi & Delta) > Simple web interface for calculating single-interface reflection and > refraction angles and strengths. > Reflection and transmittance for two dielectrics – Mathematica interactive > webpage that shows the relations between index of refraction and reflection. > ↑Jump back a section > Read in another language > > This article is available in 17 languages > > العربية > català > česky > Deutsch > español > français > 한국어 > Bahasa Indonesia > italiano > עברית > magyar > Nederlands > 日本語 > русский > slovenščina > українська > 中文 > http://en.m.wikipedia.org/wiki/Intellectual_honesty > http://en.m.wikipedia.org/wiki/Citations > > Citation > > Broadly, a citation is a reference to a published or unpublished source (not > always the original source). More precisely, a citation is an abbreviated > alphanumeric expression (e.g. [Newell84]) embedded in the body of an > intellectual work that denotes an entry in the bibliographic references > section of the work for the purpose of acknowledging the relevance of the > works of others to the topic of discussion at the spot where the citation > appears. Generally the combination of both the in-body citation and the > bibliographic entry constitutes what is commonly thought of as a citation > (whereas bibliographic entries by themselves are not). 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