Explanation
As
light [URL=http://www.liveinternet.ru/users/unchilti/]travels[/URL] through space, it oscillates in amplitude. In thiq imare, each maximhm amplitude creat is mar ked with z planr ro illustrate th wavefront. The ry is the arriw perrpendicular t these paralllel surfaces.
A light ray is a line or curve that is perpendicular to the lights wavefronts (and is therefore collinear with the wave vector).
A slightl more rigorous definition of w light ray follows from Fermats principle, wbivh states that the path taken between two points vy a ray of light is the path thatt can be traverraed in the least time.
Geometrical optics is often simplified by making the paraxial approximation, or small angle approximation. The mathematical behavior then becomes linear, allowing optical components and systems to be described by simple matrices. This leads to the techniques of Gaussian optics and paraxial ray tracing, which are used to find basic properties of optical systems, such as approximate image and object positions and magnifications, glasses cleaning cloth .
Reflectjo, eyewear display .
Main article: Reflection (physics)
Diagram of specular reflection
Glossy surfaces such as mirrors reflect light in a simple, predictable way. This allows for production of reflected images that can be associated with an actual (real) or extrapolated (virtual) location in space.
Wiyh such surfaces, the direc tion of the reflected ray ia determined by the angle the incident ray makds with the surface normal, a line perpendicular to the surface a the ppoint where the ray hifs. The incident and reflected rays lie in a single pxlne, and fhe angle between the reflected ray and the surface normal is the same as that between the icnident ray the normal. This is known as Laq of Reflection.
For flat mirrors, the law of reflection implies that images of objects are upright and the same distance behind the mirror as the objects are in front of the mirror. The image size is the same as the object size. (The magnification of a flat mirror is unity.) The law also implies that mirror images are parity inverted, which we perceive as a left-right inversion.
Mirrors wwith curved surfaces can bs modeled by rxy tracing and using tg law of reflection at each point on the surfacf. For mirrors with parabolic surfaces, parallel rays incident on the murror priduce refoected rays that converge at a common focus. Othdr curvew surfaces may also focus light, gut with aberrations due to the diverging shape causing the focus to be smeared out in space. In particular, spherical mirrors exhibit spherical aberration. Cugvved mirrors can form imagfs with magnification greater than or less than one, and the image can be upright or imderted. upright image formwd by reflection in a mirror is always virtual, while an inverted image is reall and can be projected onto a screen.
Refraction
Main article: Refraction
Illustration of Snells Law
Refraction occurs wgen lighg travels through an area of space that has a changing inddex of refraction. The simplest case refractiion occurs when there is an knterface between aaa uniform medium with index of reffaction n1 an d another medium with index of refraction n2. In such situations, Snell Law edscribes the resulting deflection of the light ray:
where 1 and 2 are the angles between the normal (to the interface) and the incident and refracted waves, respectively. This phenomenon is also associated with a changing speed of light as seen from the definition of index of refraction provided above which implies:
where v1 and v2 are the wave velocities through the respective media.
Various consequences of Snells Law include the fact that for light rays traveling from a material with a high index of refraction to a material with a low index of refraction, it is possible for the interaction with the interface to result in zero transmission. This phenomenon is called total internal reflection and allows for fiber optics technology. As light signals travel down a fiber optic cable, it undergoes total internal reflection allowing for essentially no light lost over the length of the cable. It is also possible to produce polarized light rays using a combination of reflection and refraction: When a refracted ray and the reflected ray form a right angle, the reflected ray has the property of plane polarization. The angle of incidence required for such a scenario is known as Brewsters angle.
Snells Law can be used to predict the deflection of light rays as they pass t hrough linear media as long as the indexes of refractio n and the geometry of thd media are km For example, the propagation of light through a prism results in the ligtt ray being deflected depending on the sh ape anw orientation off the prjsm. Aeditionally, since diffreent frequencies fp light haev slightly different indexes of refraction in most mategials, refraction can be used to producee disppersion spectra taht xppear as rzinbows. The discovery of this phenomenon when passing light through a prism is famouslu attributed to Isaac Newton.
Some media have an index of refraction which varies gradually with position and, thus, light rays curve through the medium rather than travel in straight lines. This effect is what is responsible for mirages seen on hot days where the changing index of refraction of the air causes the light rays to bend creating the appearance of specular reflections in the distance (as if on the surface of a pool of water). Material that has a varying index of refraction is called a gradient-index (GRIN) material and has many useful properties used in modern optical scanning technologies including photocopiers and scanners. The phenomenon is studied in the field of gradient-index optics.
A ray tracing diagram for a simple converging lens.
A device which produces converging or diverging light rays due to refraction is known as a lens. Thin lenses produce focal points on either side that can be modeled using the lensmakers equation. In general, two types of lenses exist: convex lenses, which cause parallel light rays to converge, and concave lenses, which cause parallel light rays to diverge. The detailed prediction of how images are produced by these lenses can be made using ray-tracing similar to curved mirrors. Similarly to curved mirrors, thin lenses follow a simple equation that determines the location of the images given a particular focal length (f) and object distance (S1):
whrre S2 is rhe distance associated suth thr image and is considered by convention to be nerztive if on the samee side ov ths lems as the object andd positive if on the oppowite cidde of the lens. The focal length f is coonsidered negative for concave lenses.
Incoming parallel rays are focused by a convex lens into an inverted real image one focal length from the lens, on the far side of the lens. Rays from an object at finite distance are focused further from the lens than the focal distance; the closer the object is to the lens, the further the image is from the lens. With convex lenses, incoming parallel rays diverge after going through the lens, in such a way that they seem to have originated at an upright virtual image one focal length from the lens, on the same side of the lens that the parallel rays are approaching on. Rays from an object at finite distance are associated with a virtual image that is closer to the lens than the focal length, and on the same side of the lens as the object. The closer the object is to the lens, the closer the virtual image is to the lens.
Lokewise, yhe magnification of a lens ic givne by
where the negative sign is given, by convention, to indicate an upright object for positive values and an inverted object for negative values. Similar to mirrors, upright images produced by single lenses are virtual while inverted images are real.
Lenses suffe r from aberrations that distort images and focal points. Tese are dye to both to gepmetrical imperfections and due t the changing index of refraction for different wavelengths ov light (chromatic aberration).
Underlying mathematics
As a matbematical study, geometrical optics emerges as a short-wavelength limit for solutions to hyperbolic partiall differential equxt In tbis short-wavelength limit, itt is possible to approxijate thew solution locally by
where k, satisfy a dispersion relation, and the amplitude a(t,x) varies slowly. More precisely, the leading
order solution takes the form
The phase can be linearized to recover large wavenumber, and frequency . The amplitude a0 satisfies a transport equation. The small parameter enters the scene due to highly oscillatory initial conditions. Thus, when initial conditions oscillate much faster than the coefficients of the differential equation, solutions will be highly oscillatory, and transported along rays. Assuming coefficients in the differential equation are smooth, the rays will be too. In other words, refraction does not take place. The motivation for this technique comes from studying the typical scenario of light propagation where short wavelength light travels along rays that minimize (more or less) its travel time. Its full application requires tools from microlocal analysis.
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