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118-494: OM Digital Solutions Corporation ( OMDS ) (OMデジタルソリューションズ) is a Japanese manufacturer of opto - digital products for business and consumer use, branded OM System . The company acquired the camera, audio recorder and binocular product divisions of the manufacturer Olympus in January 2021. On 30 September 2020, Olympus announced that it had entered into an agreement with financial investor Japan Industrial Partners (JIP) to transfer

236-433: A 0 {\displaystyle a_{0}} satisfies a transport equation . The small parameter ε {\displaystyle \varepsilon \,} 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

354-421: A ( t , x ) e i ( k ⋅ x − ω t ) {\displaystyle u(t,x)\approx a(t,x)e^{i(k\cdot x-\omega t)}} where k , ω {\displaystyle k,\omega } satisfy a dispersion relation , and the amplitude a ( t , x ) {\displaystyle a(t,x)} varies slowly. More precisely,

472-736: A (moving) surface in R 3 {\displaystyle \mathbf {R} ^{3}} described by the equation ψ ( x , y , z ) − c t = 0 {\displaystyle \psi (x,y,z)-ct=0} . Then Maxwell's equations in the integral form imply that ψ {\displaystyle \psi } satisfies the eikonal equation : ψ x 2 + ψ y 2 + ψ z 2 = ε μ = n 2 , {\displaystyle \psi _{x}^{2}+\psi _{y}^{2}+\psi _{z}^{2}=\varepsilon \mu =n^{2},} where n {\displaystyle n}

590-1034: A 2D surface moving through space, modelled as level surfaces of ψ {\displaystyle \psi } . (Mathematically ψ {\displaystyle \psi } exists if φ t ≠ 0 {\displaystyle \varphi _{t}\neq 0} by the implicit function theorem .) The above equation written in terms of ψ {\displaystyle \psi } becomes: ‖ ∇ ψ ‖ 2 = ε μ c 2 ( − c ) 2 = ε μ = n 2 {\displaystyle \|\nabla \psi \|^{2}={\varepsilon \mu \over c^{2}}\,(-c)^{2}=\varepsilon \mu =n^{2}} i.e., ψ x 2 + ψ y 2 + ψ z 2 = n 2 {\displaystyle \psi _{x}^{2}+\psi _{y}^{2}+\psi _{z}^{2}=n^{2}} which

708-430: A broad band, or extremely low reflectivity at a single wavelength. Constructive interference in thin films can create a strong reflection of light in a range of wavelengths, which can be narrow or broad depending on the design of the coating. These films are used to make dielectric mirrors , interference filters , heat reflectors , and filters for colour separation in colour television cameras. This interference effect

826-619: A changing index of refraction; this principle allows for lenses and the focusing of light. The simplest case of refraction occurs when there is an interface between a uniform medium with index of refraction n 1 and another medium with index of refraction n 2 . In such situations, Snell's Law describes the resulting deflection of the light ray: n 1 sin ⁡ θ 1 = n 2 sin ⁡ θ 2 {\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}} where θ 1 and θ 2 are

944-399: A converging lens has positive focal length, while a diverging lens has negative focal length. Smaller focal length indicates that the lens has a stronger converging or diverging effect. The focal length of a simple lens in air is given by the lensmaker's equation . Ray tracing can be used to show how images are formed by a lens. For a thin lens in air, the location of the image is given by

1062-409: A fiber optic cable, they undergo 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

1180-1027: A finite discontinuity. Then at each point of the hypersurface the following formulas hold: ∇ φ ⋅ [ ε E ] = 0 ∇ φ ⋅ [ μ H ] = 0 ∇ φ × [ E ] + 1 c φ t [ μ H ] = 0 ∇ φ × [ H ] − 1 c φ t [ ε E ] = 0 {\displaystyle {\begin{aligned}\nabla \varphi \cdot [\varepsilon \mathbf {E} ]&=0\\[1ex]\nabla \varphi \cdot [\mu \mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {E} ]+{\frac {1}{c}}\,\varphi _{t}\,[\mu \mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {H} ]-{\frac {1}{c}}\,\varphi _{t}\,[\varepsilon \mathbf {E} ]&=0\end{aligned}}} where

1298-422: A light ray follows from Fermat's principle , which states that the path taken between two points by a ray of light is the path that can be traversed 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

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1416-410: A mathematical study, geometrical optics emerges as a short- wavelength limit for solutions to hyperbolic partial differential equations (Sommerfeld–Runge method) or as a property of propagation of field discontinuities according to Maxwell's equations (Luneburg method). In this short-wavelength limit, it is possible to approximate the solution locally by u ( t , x ) ≈

1534-410: A prism is famously 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

1652-470: A simple equation that determines the location of the images given a particular focal length ( f {\displaystyle f} ) and object distance ( S 1 {\displaystyle S_{1}} ): 1 S 1 + 1 S 2 = 1 f {\displaystyle {\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}={\frac {1}{f}}} where S 2 {\displaystyle S_{2}}

1770-477: A single scalar quantity to represent the electric field of the light wave, rather than using a vector model with orthogonal electric and magnetic vectors. The Huygens–Fresnel equation is one such model. This was derived empirically by Fresnel in 1815, based on Huygens' hypothesis that each point on a wavefront generates a secondary spherical wavefront, which Fresnel combined with the principle of superposition of waves. The Kirchhoff diffraction equation , which

1888-522: A single point on the image, while chromatic aberration occurs because the index of refraction of the lens varies with the wavelength of the light. In physical optics, light is considered to propagate as waves. This model predicts phenomena such as interference and diffraction, which are not explained by geometric optics. The speed of light waves in air is approximately 3.0×10  m/s (exactly 299,792,458 m/s in vacuum ). The wavelength of visible light waves varies between 400 and 700 nm, but

2006-437: A spectrum. The discovery of this phenomenon when passing light through a prism is famously attributed to Isaac Newton. Some media have an index of refraction which varies gradually with position and, therefore, light rays in the medium are curved. This effect is responsible for mirages seen on hot days: a change in index of refraction air with height causes light rays to bend, creating the appearance of specular reflections in

2124-465: A thickness of one-fourth the wavelength of incident light. The reflected wave from the top of the film and the reflected wave from the film/material interface are then exactly 180° out of phase, causing destructive interference. The waves are only exactly out of phase for one wavelength, which would typically be chosen to be near the centre of the visible spectrum, around 550 nm. More complex designs using multiple layers can achieve low reflectivity over

2242-476: A variety of technologies and everyday objects, including mirrors , lenses , telescopes , microscopes , lasers , and fibre optics . Optics began with the development of lenses by the ancient Egyptians and Mesopotamians . The earliest known lenses, made from polished crystal , often quartz , date from as early as 2000 BC from Crete (Archaeological Museum of Heraclion, Greece). Lenses from Rhodes date around 700 BC, as do Assyrian lenses such as

2360-464: 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. Likewise, the magnification of a lens is given by M = − S 2 S 1 = f f − S 1 {\displaystyle M=-{\frac {S_{2}}{S_{1}}}={\frac {f}{f-S_{1}}}} where

2478-525: A wide range of scientific topics, and discussed light from four different perspectives: an epistemology of light, a metaphysics or cosmogony of light, an etiology or physics of light, and a theology of light, basing it on the works of Aristotle and Platonism. Grosseteste's most famous disciple, Roger Bacon , wrote works citing a wide range of recently translated optical and philosophical works, including those of Alhazen, Aristotle, Avicenna , Averroes , Euclid, al-Kindi, Ptolemy, Tideus, and Constantine

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2596-681: Is a reference to the Olympus OM system . Initially, the company continued production of products that were launched under Olympus ownership. In February 2022, OMDS introduced its first new product, the OM System OM-1 Micro Four Thirds camera to celebrate the 50th anniversary of the original Olympus OM-1 . Optics Most optical phenomena can be accounted for by using the classical electromagnetic description of light, however complete electromagnetic descriptions of light are often difficult to apply in practice. Practical optics

2714-427: Is a simple paraxial physical optics model for the propagation of coherent radiation such as laser beams. This technique partially accounts for diffraction, allowing accurate calculations of the rate at which a laser beam expands with distance, and the minimum size to which the beam can be focused. Gaussian beam propagation thus bridges the gap between geometric and physical optics. In the absence of nonlinear effects,

2832-613: Is also what causes the colourful rainbow patterns seen in oil slicks. Geometrical optics Geometrical optics , or ray optics , is a model of optics that describes light propagation in terms of rays . The ray in geometrical optics is an abstraction useful for approximating the paths along which light propagates under certain circumstances. The simplifying assumptions of geometrical optics include that light rays: Geometrical optics does not account for certain optical effects such as diffraction and interference , which are considered in physical optics . This simplification

2950-906: Is based on investigating how Maxwell's equations govern the propagation of discontinuities of solutions. The basic technical lemma is as follows: A technical lemma. Let φ ( x , y , z , t ) = 0 {\displaystyle \varphi (x,y,z,t)=0} be a hypersurface (a 3-dimensional manifold) in spacetime R 4 {\displaystyle \mathbf {R} ^{4}} on which one or more of: E ( x , y , z , t ) {\displaystyle \mathbf {E} (x,y,z,t)} , H ( x , y , z , t ) {\displaystyle \mathbf {H} (x,y,z,t)} , ε ( x , y , z ) {\displaystyle \varepsilon (x,y,z)} , μ ( x , y , z ) {\displaystyle \mu (x,y,z)} , have

3068-486: Is considered to travel in straight lines, while in physical optics, light is considered as an electromagnetic wave. Geometrical optics can be viewed as an approximation of physical optics that applies when the wavelength of the light used is much smaller than the size of the optical elements in the system being modelled. Geometrical optics , or ray optics , describes the propagation of light in terms of "rays" which travel in straight lines, and whose paths are governed by

3186-484: Is derived using Maxwell's equations, puts the Huygens-Fresnel equation on a firmer physical foundation. Examples of the application of Huygens–Fresnel principle can be found in the articles on diffraction and Fraunhofer diffraction . More rigorous models, involving the modelling of both electric and magnetic fields of the light wave, are required when dealing with materials whose electric and magnetic properties affect

3304-620: Is known as Brewster's angle . Snell's Law can be used to predict the deflection of light rays as they pass through "linear media" as long as the indexes of refraction and the geometry of the media are known. For example, the propagation of light through a prism results in the light ray being deflected depending on the shape and orientation of the prism. Additionally, since different frequencies of light have slightly different indexes of refraction in most materials, refraction can be used to produce dispersion spectra that appear as rainbows. The discovery of this phenomenon when passing light through

3422-463: Is known as a lens . Thin lenses produce focal points on either side that can be modeled using the lensmaker's 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

3540-791: Is known as the eikonal equation , which determines the eikonal S ( r ) {\displaystyle S(\mathbf {r} )} is a Hamilton–Jacobi equation , written for example in Cartesian coordinates becomes ( ∂ S ∂ x ) 2 + ( ∂ S ∂ y ) 2 + ( ∂ S ∂ z ) 2 = n 2 . {\displaystyle \left({\frac {\partial S}{\partial x}}\right)^{2}+\left({\frac {\partial S}{\partial y}}\right)^{2}+\left({\frac {\partial S}{\partial z}}\right)^{2}=n^{2}.} The remaining equations determine

3658-691: Is satisfied by e.g. plane waves but is not additive. The main conclusion of Luneburg's approach is the following: Theorem. Suppose the fields E ( x , y , z , t ) {\displaystyle \mathbf {E} (x,y,z,t)} and H ( x , y , z , t ) {\displaystyle \mathbf {H} (x,y,z,t)} (in a linear isotropic medium described by dielectric constants ε ( x , y , z ) {\displaystyle \varepsilon (x,y,z)} and μ ( x , y , z ) {\displaystyle \mu (x,y,z)} ) have finite discontinuities along

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3776-456: Is the distance associated with the image and is considered by convention to be negative if on the same side of the lens as the object and positive if on the opposite side of the lens. The focal length f is considered 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

3894-569: Is the eikonal equation and it holds for all x {\displaystyle x} , y {\displaystyle y} , z {\displaystyle z} , since the variable t {\displaystyle t} is absent. Other laws of optics like Snell's law and Fresnel formulae can be similarly obtained by considering discontinuities in ε {\displaystyle \varepsilon } and μ {\displaystyle \mu } . In four-vector notation used in special relativity ,

4012-1008: Is the index of refraction of the medium (Gaussian units). An example of such surface of discontinuity is the initial wave front emanating from a source that starts radiating at a certain instant of time. The surfaces of field discontinuity thus become geometrical optics wave fronts with the corresponding geometrical optics fields defined as: E ∗ ( x , y , z ) = E ( x , y , z , ψ ( x , y , z ) / c ) H ∗ ( x , y , z ) = H ( x , y , z , ψ ( x , y , z ) / c ) {\displaystyle {\begin{aligned}\mathbf {E} ^{*}(x,y,z)&=\mathbf {E} (x,y,z,\psi (x,y,z)/c)\\[1ex]\mathbf {H} ^{*}(x,y,z)&=\mathbf {H} (x,y,z,\psi (x,y,z)/c)\end{aligned}}} Those fields obey transport equations consistent with

4130-434: Is the projection of the outward unit normal ( x N , y N , z N , t N ) {\displaystyle (x_{N},y_{N},z_{N},t_{N})} of Γ {\displaystyle \Gamma } onto the 3D slice t = c o n s t {\displaystyle t={\rm {const}}} , and d S {\displaystyle dS}

4248-1473: Is the volume 3-form on Γ {\displaystyle \Gamma } . Similarly, one establishes the following from the remaining Maxwell's equations: ∮ Γ ( M ⋅ μ H ) d S = 0 ∮ Γ ( M × E + μ c t N H ) d S = 0 ∮ Γ ( M × H − ε c t N E ) d S = 0 {\displaystyle {\begin{aligned}\oint _{\Gamma }\left(\mathbf {M} \cdot \mu \mathbf {H} \right)dS&=0\\[1.55ex]\oint _{\Gamma }\left(\mathbf {M} \times \mathbf {E} +{\frac {\mu }{c}}\,t_{N}\,\mathbf {H} \right)dS&=0\\[1.55ex]\oint _{\Gamma }\left(\mathbf {M} \times \mathbf {H} -{\frac {\varepsilon }{c}}\,t_{N}\,\mathbf {E} \right)dS&=0\end{aligned}}} Now by considering arbitrary small sub-surfaces Γ 0 {\displaystyle \Gamma _{0}} of Γ {\displaystyle \Gamma } and setting up small neighbourhoods surrounding Γ 0 {\displaystyle \Gamma _{0}} in R 4 {\displaystyle \mathbf {R} ^{4}} , and subtracting

4366-409: Is to the lens, the further the image is from the lens. With diverging lenses, incoming parallel rays diverge after going through the lens, in such a way that they seem to have originated at a spot one focal length in front of the lens. This is the lens's front focal point. Rays from an object at a finite distance are associated with a virtual image that is closer to the lens than the focal point, and on

4484-454: Is useful in practice; it is an excellent approximation when the wavelength is small compared to the size of structures with which the light interacts. The techniques are particularly useful in describing geometrical aspects of imaging , including optical aberrations . A light ray is a line or curve that is perpendicular to the light's wavefronts (and is therefore collinear with the wave vector ). A slightly more rigorous definition of

4602-406: Is usually done using simplified models. The most common of these, geometric optics , treats light as a collection of rays that travel in straight lines and bend when they pass through or reflect from surfaces. Physical optics is a more comprehensive model of light, which includes wave effects such as diffraction and interference that cannot be accounted for in geometric optics. Historically,

4720-478: The ∇ {\displaystyle \nabla } operator acts in the x y z {\displaystyle xyz} -space (for every fixed t {\displaystyle t} ) and the square brackets denote the difference in values on both sides of the discontinuity surface (set up according to an arbitrary but fixed convention, e.g. the gradient ∇ φ {\displaystyle \nabla \varphi } pointing in

4838-476: The Book of Optics ( Kitab al-manazir ) in which he explored reflection and refraction and proposed a new system for explaining vision and light based on observation and experiment. He rejected the "emission theory" of Ptolemaic optics with its rays being emitted by the eye, and instead put forward the idea that light reflected in all directions in straight lines from all points of the objects being viewed and then entered

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4956-607: The Nimrud lens . The ancient Romans and Greeks filled glass spheres with water to make lenses. These practical developments were followed by the development of theories of light and vision by ancient Greek and Indian philosophers, and the development of geometrical optics in the Greco-Roman world . The word optics comes from the ancient Greek word ὀπτική , optikē ' appearance, look ' . Greek philosophy on optics broke down into two opposing theories on how vision worked,

5074-449: The emission theory , the idea that visual perception is accomplished by rays emitted by the eyes. He also commented on the parity reversal of mirrors in Timaeus . Some hundred years later, Euclid (4th–3rd century BC) wrote a treatise entitled Optics where he linked vision to geometry , creating geometrical optics . He based his work on Plato's emission theory wherein he described

5192-468: The intromission theory and the emission theory . The intromission approach saw vision as coming from objects casting off copies of themselves (called eidola) that were captured by the eye. With many propagators including Democritus , Epicurus , Aristotle and their followers, this theory seems to have some contact with modern theories of what vision really is, but it remained only speculation lacking any experimental foundation. Plato first articulated

5310-717: The leading order solution takes the form a 0 ( t , x ) e i φ ( t , x ) / ε . {\displaystyle a_{0}(t,x)e^{i\varphi (t,x)/\varepsilon }.} The phase φ ( t , x ) / ε {\displaystyle \varphi (t,x)/\varepsilon } can be linearized to recover large wavenumber k := ∇ x φ {\displaystyle k:=\nabla _{x}\varphi } , and frequency ω := − ∂ t φ {\displaystyle \omega :=-\partial _{t}\varphi } . The amplitude

5428-430: The speed of light in vacuum. Here, n ( r ) {\displaystyle n(\mathbf {r} )} is the refractive index of the medium. Without loss of generality, let us introduce ϕ = A ( k o , r ) e i k o S ( r ) {\displaystyle \phi =A(k_{o},\mathbf {r} )e^{ik_{o}S(\mathbf {r} )}} to convert

5546-448: The superposition principle , which is a wave-like property not predicted by Newton's corpuscle theory. This work led to a theory of diffraction for light and opened an entire area of study in physical optics. Wave optics was successfully unified with electromagnetic theory by James Clerk Maxwell in the 1860s. The next development in optical theory came in 1899 when Max Planck correctly modelled blackbody radiation by assuming that

5664-463: The surface normal , a line perpendicular to the surface at the point where the ray hits. The incident and reflected rays and the normal lie in a single plane, and the angle between the reflected ray and the surface normal is the same as that between the incident ray and the normal. This is known as the Law of Reflection . For flat mirrors , the law of reflection implies that images of objects are upright and

5782-554: The African . Bacon was able to use parts of glass spheres as magnifying glasses to demonstrate that light reflects from objects rather than being released from them. The first wearable eyeglasses were invented in Italy around 1286. This was the start of the optical industry of grinding and polishing lenses for these "spectacles", first in Venice and Florence in the thirteenth century, and later in

5900-548: The Huygens–Fresnel principle states that every point of a wavefront is associated with the production of a new disturbance, it is possible for a wavefront to interfere with itself constructively or destructively at different locations producing bright and dark fringes in regular and predictable patterns. Interferometry is the science of measuring these patterns, usually as a means of making precise determinations of distances or angular resolutions . The Michelson interferometer

6018-568: The Olympus Imaging division to a newly established wholly-owned subsidiary of Olympus. This subsidiary was named OM Digital Solutions. On 1 January 2021, 95% of the shares in OM Digital Solutions were transferred to OJ Holdings, Ltd, a specially established subsidiary of JIP. Olympus retained ownership of the remaining 5%. In 2021, OM Digital Solutions reported that all its products would be rebranded from Olympus to OM System. The new name

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6136-1962: The above integrals accordingly, one obtains: ∫ Γ 0 ( ∇ φ ⋅ [ ε E ] ) d S ‖ ∇ 4 D φ ‖ = 0 ∫ Γ 0 ( ∇ φ ⋅ [ μ H ] ) d S ‖ ∇ 4 D φ ‖ = 0 ∫ Γ 0 ( ∇ φ × [ E ] + 1 c φ t [ μ H ] ) d S ‖ ∇ 4 D φ ‖ = 0 ∫ Γ 0 ( ∇ φ × [ H ] − 1 c φ t [ ε E ] ) d S ‖ ∇ 4 D φ ‖ = 0 {\displaystyle {\begin{aligned}\int _{\Gamma _{0}}(\nabla \varphi \cdot [\varepsilon \mathbf {E} ])\,{dS \over \|\nabla ^{4D}\varphi \|}&=0\\[1ex]\int _{\Gamma _{0}}(\nabla \varphi \cdot [\mu \mathbf {H} ])\,{dS \over \|\nabla ^{4D}\varphi \|}&=0\\[1ex]\int _{\Gamma _{0}}\left(\nabla \varphi \times [\mathbf {E} ]+{1 \over c}\,\varphi _{t}\,[\mu \mathbf {H} ]\right)\,{\frac {dS}{\|\nabla ^{4D}\varphi \|}}&=0\\[1ex]\int _{\Gamma _{0}}\left(\nabla \varphi \times [\mathbf {H} ]-{1 \over c}\,\varphi _{t}\,[\varepsilon \mathbf {E} ]\right)\,{\frac {dS}{\|\nabla ^{4D}\varphi \|}}&=0\end{aligned}}} where ∇ 4 D {\displaystyle \nabla ^{4D}} denotes

6254-647: The amplitude A ( k o , r ) {\displaystyle A(k_{o},\mathbf {r} )} and phase S ( r ) {\displaystyle S(\mathbf {r} )} satisfy the equation lim k 0 → ∞ 1 k 0 ( 1 A ∇ S ⋅ ∇ A + 1 2 ∇ 2 S ) = 0 {\textstyle \lim _{k_{0}\to \infty }{\frac {1}{k_{0}}}\left({\frac {1}{A}}\,\nabla S\cdot \nabla A+{\frac {1}{2}}\nabla ^{2}S\right)=0} . This condition

6372-484: The amplitude of the wave, which for light is associated with a brightening of the waveform in that location. Alternatively, if the two waves of the same wavelength and frequency are out of phase, then the wave crests will align with wave troughs and vice versa. This results in destructive interference and a decrease in the amplitude of the wave, which for light is associated with a dimming of the waveform at that location. See below for an illustration of this effect. Since

6490-542: The angle of incidence. Plutarch (1st–2nd century AD) described multiple reflections on spherical mirrors and discussed the creation of magnified and reduced images, both real and imaginary, including the case of chirality of the images. During the Middle Ages , Greek ideas about optics were resurrected and extended by writers in the Muslim world . One of the earliest of these was Al-Kindi ( c.  801 –873) who wrote on

6608-435: The angles between the normal (to the interface) and the incident and refracted waves, respectively. The index of refraction of a medium is related to the speed, v , of light in that medium by n = c / v , {\displaystyle n=c/v,} where c is the speed of light in vacuum . Snell's Law can be used to predict the deflection of light rays as they pass through linear media as long as

6726-614: 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: v 1 sin ⁡ θ 2   = v 2 sin ⁡ θ 1 {\displaystyle v_{1}\sin \theta _{2}\ =v_{2}\sin \theta _{1}} where v 1 {\displaystyle v_{1}} and v 2 {\displaystyle v_{2}} are

6844-622: The components of electric or magnetic field and hence the function ϕ {\displaystyle \phi } satisfy the wave equation ∇ 2 ϕ + k o 2 n ( r ) 2 ϕ = 0 {\displaystyle \nabla ^{2}\phi +k_{o}^{2}n(\mathbf {r} )^{2}\phi =0} where k o = ω / c = 2 π / λ o {\displaystyle k_{o}=\omega /c=2\pi /\lambda _{o}} with c {\displaystyle c} being

6962-1149: The cross product of the second equation with ∇ φ {\displaystyle \nabla \varphi } and substituting the first yields: ∇ φ × ( ∇ φ × [ H ] ) − ε c φ t ( ∇ φ × [ E ] ) = ( ∇ φ ⋅ [ H ] ) ∇ φ − ‖ ∇ φ ‖ 2 [ H ] + ε μ c 2 φ t 2 [ H ] = 0 {\displaystyle \nabla \varphi \times (\nabla \varphi \times [\mathbf {H} ])-{\varepsilon \over c}\,\varphi _{t}\,(\nabla \varphi \times [\mathbf {E} ])=(\nabla \varphi \cdot [\mathbf {H} ])\,\nabla \varphi -\|\nabla \varphi \|^{2}\,[\mathbf {H} ]+{\varepsilon \mu \over c^{2}}\varphi _{t}^{2}\,[\mathbf {H} ]=0} The continuity of μ {\displaystyle \mu } and

7080-434: 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 . The method of obtaining equations of geometrical optics by taking

7198-946: The direction of the quantities being subtracted from ). Sketch of proof. Start with Maxwell's equations away from the sources (Gaussian units): ∇ ⋅ ε E = 0 ∇ ⋅ μ H = 0 ∇ × E + μ c H t = 0 ∇ × H − ε c E t = 0 {\displaystyle {\begin{aligned}\nabla \cdot \varepsilon \mathbf {E} =0\\[1ex]\nabla \cdot \mu \mathbf {H} =0\\[1ex]\nabla \times \mathbf {E} +{\tfrac {\mu }{c}}\,\mathbf {H} _{t}=0\\[1ex]\nabla \times \mathbf {H} -{\tfrac {\varepsilon }{c}}\,\mathbf {E} _{t}=0\end{aligned}}} Using Stokes' theorem in R 4 {\displaystyle \mathbf {R} ^{4}} one can conclude from

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7316-424: The direction of the reflected ray is determined by the angle the incident ray makes with the surface normal , a line perpendicular to the surface at the point where the ray hits. The incident and reflected rays lie in a single plane, and the angle between the reflected ray and the surface normal is the same as that between the incident ray and the normal. This is known as the Law of Reflection . For flat mirrors ,

7434-483: The discontinuities of E {\displaystyle \mathbf {E} } and H {\displaystyle \mathbf {H} } satisfy: [ ε E ] = ε [ E ] {\displaystyle [\varepsilon \mathbf {E} ]=\varepsilon [\mathbf {E} ]} and [ μ H ] = μ [ H ] {\displaystyle [\mu \mathbf {H} ]=\mu [\mathbf {H} ]} . In this case

7552-410: 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 device which produces converging or diverging light rays due to refraction

7670-449: The distance (as if on the surface of a pool of water). Optical materials with varying indexes of refraction are called gradient-index (GRIN) materials. Such materials are used to make gradient-index optics . For light rays travelling from a material with a high index of refraction to a material with a low index of refraction, Snell's law predicts that there is no θ 2 when θ 1 is large. In this case, no transmission occurs; all

7788-493: The equation to − k o 2 A [ ( ∇ S ) 2 − n 2 ] + 2 i k o ( ∇ S ⋅ ∇ A ) + i k o A ∇ 2 S + ∇ 2 A = 0. {\displaystyle -k_{o}^{2}A[(\nabla S)^{2}-n^{2}]+2ik_{o}(\nabla S\cdot \nabla A)+ik_{o}A\nabla ^{2}S+\nabla ^{2}A=0.} Since

7906-426: The exchange of energy between light and matter only occurred in discrete amounts he called quanta . In 1905, Albert Einstein published the theory of the photoelectric effect that firmly established the quantization of light itself. In 1913, Niels Bohr showed that atoms could only emit discrete amounts of energy, thus explaining the discrete lines seen in emission and absorption spectra . The understanding of

8024-574: The eye, although he was unable to correctly explain how the eye captured the rays. Alhazen's work was largely ignored in the Arabic world but it was anonymously translated into Latin around 1200 A.D. and further summarised and expanded on by the Polish monk Witelo making it a standard text on optics in Europe for the next 400 years. In the 13th century in medieval Europe, English bishop Robert Grosseteste wrote on

8142-535: The feud between the two lasted until Hooke's death. In 1704, Newton published Opticks and, at the time, partly because of his success in other areas of physics, he was generally considered to be the victor in the debate over the nature of light. Newtonian optics was generally accepted until the early 19th century when Thomas Young and Augustin-Jean Fresnel conducted experiments on the interference of light that firmly established light's wave nature. Young's famous double slit experiment showed that light followed

8260-684: The first of the above equations that for any domain D {\displaystyle D} in R 4 {\displaystyle \mathbf {R} ^{4}} with a piecewise smooth (3-dimensional) boundary Γ {\displaystyle \Gamma } the following is true: ∮ Γ ( M ⋅ ε E ) d S = 0 {\displaystyle \oint _{\Gamma }(\mathbf {M} \cdot \varepsilon \mathbf {E} )\,dS=0} where M = ( x N , y N , z N ) {\displaystyle \mathbf {M} =(x_{N},y_{N},z_{N})}

8378-474: The focus to be smeared out in space. In particular, spherical mirrors exhibit spherical aberration . Curved mirrors can form images with a magnification greater than or less than one, and the magnification can be negative, indicating that the image is inverted. An upright image formed by reflection in a mirror is always virtual, while an inverted image is real and can be projected onto a screen. Refraction occurs when light travels through an area of space that has

8496-484: The functions A m ( r ) {\displaystyle A_{m}(\mathbf {r} )} . The method of obtaining equations of geometrical optics by analysing surfaces of discontinuities of solutions to Maxwell's equations was first described by Rudolf Karl Luneburg in 1944. It does not restrict the electromagnetic field to have a special form required by the Sommerfeld-Runge method which assumes

8614-411: The gloss of surfaces such as mirrors, which reflect light in a simple, predictable way. This allows for the production of reflected images that can be associated with an actual ( real ) or extrapolated ( virtual ) location in space. Diffuse reflection describes non-glossy materials, such as paper or rock. The reflections from these surfaces can only be described statistically, with the exact distribution of

8732-546: The gradient in the 4D x y z t {\displaystyle xyzt} -space. And since Γ 0 {\displaystyle \Gamma _{0}} is arbitrary, the integrands must be equal to 0 which proves the lemma. It's now easy to show that as they propagate through a continuous medium, the discontinuity surfaces obey the eikonal equation. Specifically, if ε {\displaystyle \varepsilon } and μ {\displaystyle \mu } are continuous, then

8850-611: The image can be upright or inverted. An upright image formed by reflection in a mirror is always virtual, while an inverted image is real and can be projected onto a screen. Refraction occurs when light travels through an area of space that has a changing index of refraction. The simplest case of refraction occurs when there is an interface between a uniform medium with index of refraction n 1 {\displaystyle n_{1}} and another medium with index of refraction n 2 {\displaystyle n_{2}} . In such situations, Snell's Law describes

8968-416: The incident rays came. This is called retroreflection . Mirrors with curved surfaces can be modelled by ray tracing and using the law of reflection at each point on the surface. For mirrors with parabolic surfaces , parallel rays incident on the mirror produce reflected rays that converge at a common focus . Other curved surfaces may also focus light, but with aberrations due to the diverging shape causing

9086-418: The indexes of refraction and the geometry of the media are known. For example, the propagation of light through a prism results in the light ray being deflected depending on the shape and orientation of the prism. In most materials, the index of refraction varies with the frequency of the light, known as dispersion . Taking this into account, Snell's Law can be used to predict how a prism will disperse light into

9204-436: The interaction between light and matter that followed from these developments not only formed the basis of quantum optics but also was crucial for the development of quantum mechanics as a whole. The ultimate culmination, the theory of quantum electrodynamics , explains all optics and electromagnetic processes in general as the result of the exchange of real and virtual photons. Quantum optics gained practical importance with

9322-426: The interaction of light with the material. For instance, the behaviour of a light wave interacting with a metal surface is quite different from what happens when it interacts with a dielectric material. A vector model must also be used to model polarised light. Numerical modeling techniques such as the finite element method , the boundary element method and the transmission-line matrix method can be used to model

9440-477: The invention of the compound optical microscope around 1595, and the refracting telescope in 1608, both of which appeared in the spectacle making centres in the Netherlands. In the early 17th century, Johannes Kepler expanded on geometric optics in his writings, covering lenses, reflection by flat and curved mirrors, the principles of pinhole cameras , inverse-square law governing the intensity of light, and

9558-491: The inventions of the maser in 1953 and of the laser in 1960. Following the work of Paul Dirac in quantum field theory , George Sudarshan , Roy J. Glauber , and Leonard Mandel applied quantum theory to the electromagnetic field in the 1950s and 1960s to gain a more detailed understanding of photodetection and the statistics of light. Classical optics is divided into two main branches: geometrical (or ray) optics and physical (or wave) optics. In geometrical optics, light

9676-670: The last two equations of the lemma can be written as: ∇ φ × [ E ] + μ c φ t [ H ] = 0 ∇ φ × [ H ] − ε c φ t [ E ] = 0 {\displaystyle {\begin{aligned}\nabla \varphi \times [\mathbf {E} ]+{\mu \over c}\,\varphi _{t}\,[\mathbf {H} ]&=0\\[1ex]\nabla \varphi \times [\mathbf {H} ]-{\varepsilon \over c}\,\varphi _{t}\,[\mathbf {E} ]&=0\end{aligned}}} Taking

9794-473: The law of reflection at each point on the surface. For mirrors with parabolic surfaces , parallel rays incident on the mirror produce reflected rays that converge at a common focus . Other curved surfaces may also focus light, but with aberrations due to the diverging shape causing the focus to be smeared out in space. In particular, spherical mirrors exhibit spherical aberration . Curved mirrors can form images with magnification greater than or less than one, and

9912-436: 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 equal to one.) The law also implies that mirror images are parity inverted , which is perceived as a left-right inversion. Mirrors with curved surfaces can be modeled by ray tracing and using

10030-504: The laws of reflection and refraction at interfaces between different media. These laws were discovered empirically as far back as 984 AD and have been used in the design of optical components and instruments from then until the present day. They can be summarised as follows: When a ray of light hits the boundary between two transparent materials, it is divided into a reflected and a refracted ray. The laws of reflection and refraction can be derived from Fermat's principle which states that

10148-434: The lens than the focal distance; the closer the object is to the lens, the further the image is from the lens. With concave 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

10266-449: The light is modelled as a collection of particles called " photons ". Quantum optics deals with the application of quantum mechanics to optical systems. Optical science is relevant to and studied in many related disciplines including astronomy , various engineering fields, photography , and medicine (particularly ophthalmology and optometry , in which it is called physiological optics). Practical applications of optics are found in

10384-422: The light is reflected. This phenomenon is called total internal reflection and allows for fibre optics technology. As light travels down an optical fibre, it undergoes total internal reflection allowing for essentially no light to be lost over the length of the cable. A device that produces converging or diverging light rays due to refraction is known as a lens . Lenses are characterized by their focal length :

10502-487: The limit of zero wavelength was first described by Arnold Sommerfeld and J. Runge in 1911. Their derivation was based on an oral remark by Peter Debye . Consider a monochromatic scalar field ψ ( r , t ) = ϕ ( r ) e i ω t {\displaystyle \psi (\mathbf {r} ,t)=\phi (\mathbf {r} )e^{i\omega t}} , where ψ {\displaystyle \psi } could be any of

10620-443: The mathematical rules of perspective and described the effects of refraction qualitatively, although he questioned that a beam of light from the eye could instantaneously light up the stars every time someone blinked. Euclid stated the principle of shortest trajectory of light, and considered multiple reflections on flat and spherical mirrors. Ptolemy , in his treatise Optics , held an extramission-intromission theory of vision:

10738-489: The merits of Aristotelian and Euclidean ideas of optics, favouring the emission theory since it could better quantify optical phenomena. In 984, the Persian mathematician Ibn Sahl wrote the treatise "On burning mirrors and lenses", correctly describing a law of refraction equivalent to Snell's law. He used this law to compute optimum shapes for lenses and curved mirrors . In the early 11th century, Alhazen (Ibn al-Haytham) wrote

10856-477: 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 suffer from aberrations that distort images and focal points. These are due to both to geometrical imperfections and due to the changing index of refraction for different wavelengths of light ( chromatic aberration ). As

10974-405: The object and image distances are positive if the object and image are on opposite sides of the lens. Incoming parallel rays are focused by a converging lens onto a spot one focal length from the lens, on the far side of the lens. This is called the rear focal point of the lens. Rays from an object at a finite distance are focused further from the lens than the focal distance; the closer the object

11092-401: The optical explanations of astronomical phenomena such as lunar and solar eclipses and astronomical parallax . He was also able to correctly deduce the role of the retina as the actual organ that recorded images, finally being able to scientifically quantify the effects of different types of lenses that spectacle makers had been observing over the previous 300 years. After the invention of

11210-676: The path taken between two points by a ray of light is the path that can be traversed in the least time. Geometric optics is often simplified by making the paraxial approximation , or "small angle approximation". The mathematical behaviour 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 . Reflections can be divided into two types: specular reflection and diffuse reflection . Specular reflection describes

11328-486: The presence of the discontinuity is essential in this step as we'd be dividing by zero otherwise.) Because of the physical considerations one can assume without loss of generality that φ {\displaystyle \varphi } is of the following form: φ ( x , y , z , t ) = ψ ( x , y , z ) − c t {\displaystyle \varphi (x,y,z,t)=\psi (x,y,z)-ct} , i.e.

11446-511: The propagation of light in systems which cannot be solved analytically. Such models are computationally demanding and are normally only used to solve small-scale problems that require accuracy beyond that which can be achieved with analytical solutions. All of the results from geometrical optics can be recovered using the techniques of Fourier optics which apply many of the same mathematical and analytical techniques used in acoustic engineering and signal processing . Gaussian beam propagation

11564-416: The ray-based model of light was developed first, followed by the wave model of light. Progress in electromagnetic theory in the 19th century led to the discovery that light waves were in fact electromagnetic radiation. Some phenomena depend on light having both wave-like and particle-like properties . Explanation of these effects requires quantum mechanics . When considering light's particle-like properties,

11682-416: The rays (or flux) from the eye formed a cone, the vertex being within the eye, and the base defining the visual field. The rays were sensitive, and conveyed information back to the observer's intellect about the distance and orientation of surfaces. He summarized much of Euclid and went on to describe a way to measure the angle of refraction , though he failed to notice the empirical relationship between it and

11800-423: The reflected light depending on the microscopic structure of the material. Many diffuse reflectors are described or can be approximated by Lambert's cosine law , which describes surfaces that have equal luminance when viewed from any angle. Glossy surfaces can give both specular and diffuse reflection. In specular reflection, the direction of the reflected ray is determined by the angle the incident ray makes with

11918-425: The resulting deflection of the light ray: n 1 sin ⁡ θ 1 = n 2 sin ⁡ θ 2 {\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}} where θ 1 {\displaystyle \theta _{1}} and θ 2 {\displaystyle \theta _{2}} are

12036-415: 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 law also implies that mirror images are parity inverted, which we perceive as a left-right inversion. Images formed from reflection in two (or any even number of) mirrors are not parity inverted. Corner reflectors produce reflected rays that travel back in the direction from which

12154-407: 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. As with mirrors, upright images produced by a single lens are virtual, while inverted images are real. Lenses suffer from aberrations that distort images. Monochromatic aberrations occur because the geometry of the lens does not perfectly direct rays from each object point to

12272-585: The second equation of the lemma imply: ∇ φ ⋅ [ H ] = 0 {\displaystyle \nabla \varphi \cdot [\mathbf {H} ]=0} , hence, for points lying on the surface φ = 0 {\displaystyle \varphi =0} only : ‖ ∇ φ ‖ 2 = ε μ c 2 φ t 2 {\displaystyle \|\nabla \varphi \|^{2}={\varepsilon \mu \over c^{2}}\varphi _{t}^{2}} (Notice

12390-1672: The series diverges, and one has to be careful in keeping only appropriate first few terms. For each value of k o {\displaystyle k_{o}} , one can find an optimum number of terms to be kept and adding more terms than the optimum number might result in a poorer approximation. Substituting the series into the equation and collecting terms of different orders, one finds O ( k o 2 ) : ( ∇ S ) 2 = n 2 , O ( k o ) : 2 ∇ S ⋅ ∇ A 0 + A 0 ∇ 2 S = 0 , O ( 1 ) : 2 ∇ S ⋅ ∇ A 1 + A 1 ∇ 2 S = − ∇ 2 A 0 , {\displaystyle {\begin{aligned}O(k_{o}^{2}):&\quad (\nabla S)^{2}=n^{2},\\[1ex]O(k_{o}):&\quad 2\nabla S\cdot \nabla A_{0}+A_{0}\nabla ^{2}S=0,\\[1ex]O(1):&\quad 2\nabla S\cdot \nabla A_{1}+A_{1}\nabla ^{2}S=-\nabla ^{2}A_{0},\end{aligned}}} in general, O ( k o 1 − m ) : 2 ∇ S ⋅ ∇ A m + A m ∇ 2 S = − ∇ 2 A m − 1 . {\displaystyle O(k_{o}^{1-m}):\quad 2\nabla S\cdot \nabla A_{m}+A_{m}\nabla ^{2}S=-\nabla ^{2}A_{m-1}.} The first equation

12508-405: The simple equation 1 S 1 + 1 S 2 = 1 f , {\displaystyle {\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}={\frac {1}{f}},} where S 1 is the distance from the object to the lens, θ 2 is the distance from the lens to the image, and f is the focal length of the lens. In the sign convention used here,

12626-464: The spectacle making centres in both the Netherlands and Germany. Spectacle makers created improved types of lenses for the correction of vision based more on empirical knowledge gained from observing the effects of the lenses rather than using the rudimentary optical theory of the day (theory which for the most part could not even adequately explain how spectacles worked). This practical development, mastery, and experimentation with lenses led directly to

12744-444: The superposition principle can be used to predict the shape of interacting waveforms through the simple addition of the disturbances. This interaction of waves to produce a resulting pattern is generally termed "interference" and can result in a variety of outcomes. If two waves of the same wavelength and frequency are in phase , both the wave crests and wave troughs align. This results in constructive interference and an increase in

12862-441: 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 . 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. With such surfaces,

12980-467: The telescope, Kepler set out the theoretical basis on how they worked and described an improved version, known as the Keplerian telescope , using two convex lenses to produce higher magnification. Optical theory progressed in the mid-17th century with treatises written by philosopher René Descartes , which explained a variety of optical phenomena including reflection and refraction by assuming that light

13098-440: The term "light" is also often applied to infrared (0.7–300 μm) and ultraviolet radiation (10–400 nm). The wave model can be used to make predictions about how an optical system will behave without requiring an explanation of what is "waving" in what medium. Until the middle of the 19th century, most physicists believed in an "ethereal" medium in which the light disturbance propagated. The existence of electromagnetic waves

13216-451: The transport equations of the Sommerfeld-Runge approach. Light rays in Luneburg's theory are defined as trajectories orthogonal to the discontinuity surfaces and can be shown to obey Fermat's principle of least time thus establishing the identity of those rays with light rays of standard optics. The above developments can be generalised to anisotropic media. The proof of Luneburg's theorem

13334-711: The underlying principle of geometrical optics lies in the limit λ o ∼ k o − 1 → 0 {\displaystyle \lambda _{o}\sim k_{o}^{-1}\rightarrow 0} , the following asymptotic series is assumed, A ( k o , r ) = ∑ m = 0 ∞ A m ( r ) ( i k o ) m {\displaystyle A(k_{o},\mathbf {r} )=\sum _{m=0}^{\infty }{\frac {A_{m}(\mathbf {r} )}{(ik_{o})^{m}}}} For large but finite value of k o {\displaystyle k_{o}} ,

13452-1446: The wave equation can be written as ∂ 2 ψ ∂ x i ∂ x i = 0 {\displaystyle {\frac {\partial ^{2}\psi }{\partial x_{i}\partial x^{i}}}=0} and the substitution ψ = A e i S / ε {\displaystyle \psi =Ae^{iS/\varepsilon }} leads to − A ε 2 ∂ S ∂ x i ∂ S ∂ x i + 2 i ε ∂ A ∂ x i ∂ S ∂ x i + i A ε ∂ 2 S ∂ x i ∂ x i + ∂ 2 A ∂ x i ∂ x i = 0. {\displaystyle -{\frac {A}{\varepsilon ^{2}}}{\frac {\partial S}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}+{\frac {2i}{\varepsilon }}{\frac {\partial A}{\partial x_{i}}}{\frac {\partial S}{\partial x^{i}}}+{\frac {iA}{\varepsilon }}{\frac {\partial ^{2}S}{\partial x_{i}\partial x^{i}}}+{\frac {\partial ^{2}A}{\partial x_{i}\partial x^{i}}}=0.} Therefore,

13570-440: The wave velocities through the respective media. Various consequences of Snell's 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

13688-400: Was a famous instrument which used interference effects to accurately measure the speed of light. The appearance of thin films and coatings is directly affected by interference effects. Antireflective coatings use destructive interference to reduce the reflectivity of the surfaces they coat, and can be used to minimise glare and unwanted reflections. The simplest case is a single layer with

13806-540: Was emitted by objects which produced it. This differed substantively from the ancient Greek emission theory. In the late 1660s and early 1670s, Isaac Newton expanded Descartes's ideas into a corpuscle theory of light , famously determining that white light was a mix of colours that can be separated into its component parts with a prism . In 1690, Christiaan Huygens proposed a wave theory for light based on suggestions that had been made by Robert Hooke in 1664. Hooke himself publicly criticised Newton's theories of light and

13924-467: Was predicted in 1865 by Maxwell's equations . These waves propagate at the speed of light and have varying electric and magnetic fields which are orthogonal to one another, and also to the direction of propagation of the waves. Light waves are now generally treated as electromagnetic waves except when quantum mechanical effects have to be considered. Many simplified approximations are available for analysing and designing optical systems. Most of these use

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