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105-451: 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 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

210-430: 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

315-619: 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

420-478: A continuous distribution of charge is: where ρ ( r ′ ) {\displaystyle \rho (\mathbf {r'} )} is the charge density, and r − r ′ {\displaystyle \mathbf {r} -\mathbf {r'} } is the distance from the volume element d 3 r ′ {\displaystyle \mathrm {d^{3}} \mathbf {r'} } to point in space where φ

525-625: 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 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

630-412: A description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics which is a quantum field theory . Fundamental physical aspects of classical electrodynamics are presented in many textbooks. For

735-407: A distribution of point charges, the forces determined from Coulomb's law may be summed. The result after dividing by q 0 is: where n is the number of charges, q i is the amount of charge associated with the i th charge, r i is the position of the i th charge, r is the position where the electric field is being determined, and ε 0 is the electric constant . If the field

840-585: 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 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

945-408: A mirror in front of the viewer, it reflected his painting of the buildings which had been seen previously, so that the vanishing point was centered from the perspective of the participant. Brunelleschi applied the new system of perspective to his paintings around 1425. This scenario is indicative, but faces several problems, that are still debated. First of all, nothing can be said for certain about

1050-429: A moving point particle. Branches of classical electromagnetism such as optics, electrical and electronic engineering consist of a collection of relevant mathematical models of different degrees of simplification and idealization to enhance the understanding of specific electrodynamics phenomena. An electrodynamics phenomenon is determined by the particular fields, specific densities of electric charges and currents, and

1155-399: 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 the amplitude of the wave, which for light is associated with a brightening of the waveform in that location. Alternatively, if

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1260-472: A series of experiments between 1415 and 1420, which included making drawings of various Florentine buildings in correct perspective. According to Vasari and Antonio Manetti , in about 1420, Brunelleschi demonstrated his discovery by having people look through a hole in the back of a painting he had made. Through it, they would see a building such as the Florence Baptistery . When Brunelleschi lifted

1365-436: 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

1470-465: 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

1575-476: 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

1680-431: 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

1785-524: 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

1890-415: 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 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

1995-591: 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, 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,

2100-399: Is also what causes the colourful rainbow patterns seen in oil slicks. Classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model . It is, therefore, a classical field theory . The theory provides

2205-511: Is an approximate representation, generally on a flat surface, of an image as it is seen by the eye . Perspective drawing is useful for representing a three-dimensional scene in a two-dimensional medium, like paper . It is based on the optical fact that for a person an object looks N times (linearly) smaller if it has been moved N times further from the eye than the original distance was. The most characteristic features of linear perspective are that objects appear smaller as their distance from

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2310-399: 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 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

2415-471: Is based on qualitative judgments, and would need to be faced against the material evaluations that have been conducted on Renaissance perspective paintings. Apart from the paintings of Piero della Francesca , which are a model of the genre, the majority of 15th century works show serious errors in their geometric construction. This is true of Masaccio's Trinity fresco and of many works, including those by renowned artists like Leonardo da Vinci. As shown by

2520-528: Is being determined. The scalar φ will add to other potentials as a scalar. This makes it relatively easy to break complex problems down into simple parts and add their potentials. Taking the definition of φ backwards, we see that the electric field is just the negative gradient (the del operator) of the potential. Or: From this formula it is clear that E can be expressed in V/m (volts per meter). A changing electromagnetic field propagates away from its origin in

2625-669: Is close to an object under observation and directly facing an observer's eyes (i.e., the observer is on a line normal or perpendicular to the plane). Then draw straight lines from the object to the observer. The area on the plane where the drawn lines pass through the plane is a point-projection prospective image resembling what is seen by the observer. Additionally, a central vanishing point can be used (just as with one-point perspective) to indicate frontal (foreshortened) depth. The earliest art paintings and drawings typically sized many objects and characters hierarchically according to their spiritual or thematic importance, not their distance from

2730-486: 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

2835-492: Is defined such that, on a stationary charge: where q 0 is what is known as a test charge and F is the force on that charge. The size of the charge does not really matter, as long as it is small enough not to influence the electric field by its mere presence. What is plain from this definition, though, is that the unit of E is N/C ( newtons per coulomb ). This unit is equal to V/m ( volts per meter); see below. In electrostatics, where charges are not moving, around

2940-726: Is evident in Ancient Greek red-figure pottery . Systematic attempts to evolve a system of perspective are usually considered to have begun around the fifth century BC in the art of ancient Greece , as part of a developing interest in illusionism allied to theatrical scenery. This was detailed within Aristotle 's Poetics as skenographia : using flat panels on a stage to give the illusion of depth. The philosophers Anaxagoras and Democritus worked out geometric theories of perspective for use with skenographia . Alcibiades had paintings in his house designed using skenographia , so this art

3045-520: Is instead produced by a continuous distribution of charge, the summation becomes an integral: where ρ ( r ′ ) {\displaystyle \rho (\mathbf {r'} )} is the charge density and r − r ′ {\displaystyle \mathbf {r} -\mathbf {r'} } is the vector that points from the volume element d 3 r ′ {\displaystyle \mathrm {d^{3}} \mathbf {r'} } to

3150-673: Is not a single occurrence of the word "experiment". Fourth, the conditions listed by Manetti are contradictory with each other. For example, the description of the eyepiece sets a visual field of 15°, much narrower than the visual field resulting from the urban landscape described. Soon after Brunelleschi's demonstrations, nearly every interested artist in Florence and in Italy used geometrical perspective in their paintings and sculpture, notably Donatello , Masaccio , Lorenzo Ghiberti , Masolino da Panicale , Paolo Uccello , and Filippo Lippi . Not only

3255-581: Is not systematically related to the rest of the composition. Medieval artists in Europe, like those in the Islamic world and China, were aware of the general principle of varying the relative size of elements according to distance, but even more than classical art were perfectly ready to override it for other reasons. Buildings were often shown obliquely according to a particular convention. The use and sophistication of attempts to convey distance increased steadily during

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3360-460: 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 is derived using Maxwell's equations, puts the Huygens-Fresnel equation on a firmer physical foundation. Examples of

3465-442: Is relatively simple, having been long ago formulated by Euclid. Alberti was also trained in the science of optics through the school of Padua and under the influence of Biagio Pelacani da Parma who studied Alhazen 's Book of Optics . This book, translated around 1200 into Latin, had laid the mathematical foundation for perspective in Europe. Piero della Francesca elaborated on De pictura in his De Prospectiva pingendi in

3570-461: Is the electric field at the location of the particle, v is the velocity of the particle, B is the magnetic field at the location of the particle. The above equation illustrates that the Lorentz force is the sum of two vectors. One is the cross product of the velocity and magnetic field vectors. Based on the properties of the cross product, this produces a vector that is perpendicular to both

3675-400: 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, 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

3780-408: Is the electric potential, and C is the path over which the integral is being taken. Unfortunately, this definition has a caveat. From Maxwell's equations , it is clear that ∇ × E is not always zero, and hence the scalar potential alone is insufficient to define the electric field exactly. As a result, one must add a correction factor, which is generally done by subtracting the time derivative of

3885-410: Is the manifestation of the electromagnetic interaction between charged particles. As simple and satisfying as Coulomb's equation may be, it is not entirely correct in the context of classical electromagnetism. Problems arise because changes in charge distributions require a non-zero amount of time to be "felt" elsewhere (required by special relativity). For the fields of general charge distributions,

3990-405: Is the position. r q {\displaystyle {\textbf {r}}_{q}} and v q {\displaystyle {\textbf {v}}_{q}} are the position and velocity of the charge, respectively, as a function of retarded time . The vector potential is similar: These can then be differentiated accordingly to obtain the complete field equations for

4095-475: 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

4200-434: The A vector potential described below. Whenever the charges are quasistatic, however, this condition will be essentially met. From the definition of charge, one can easily show that the electric potential of a point charge as a function of position is: where q is the point charge's charge, r is the position at which the potential is being determined, and r i is the position of each point charge. The potential for

4305-605: 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,

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4410-597: The east doors of the Florence Baptistery . Masaccio (d. 1428) achieved an illusionistic effect by placing the vanishing point at the viewer's eye level in his Holy Trinity ( c.  1427 ), and in The Tribute Money , it is placed behind the face of Jesus. In the late 15th century, Melozzo da Forlì first applied the technique of foreshortening (in Rome, Loreto , Forlì and others). This overall story

4515-447: 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

4620-467: 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

4725-446: 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

4830-462: 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

4935-435: The 1470s, making many references to Euclid. Alberti had limited himself to figures on the ground plane and giving an overall basis for perspective. Della Francesca fleshed it out, explicitly covering solids in any area of the picture plane. Della Francesca also started the now common practice of using illustrated figures to explain the mathematical concepts, making his treatise easier to understand than Alberti's. Della Francesca

5040-504: 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

5145-541: 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

5250-434: 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

5355-414: 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 the interaction of light with the material. For instance, the behaviour of a light wave interacting with

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5460-455: The correctness of his perspective construction of the Baptistery of San Giovanni, because Brunelleschi's panel is lost. Second, no other perspective painting or drawing by Brunelleschi is known. (In fact, Brunelleschi was not known to have painted at all.) Third, in the account written by Antonio Manetti in his Vita di Ser Brunellesco at the end of the 15th century on Brunelleschi's panel, there

5565-438: The development of methods to measure voltage , current , capacitance , and resistance . Detailed historical accounts are given by Wolfgang Pauli , E. T. Whittaker , Abraham Pais , and Bruce J. Hunt. The electromagnetic field exerts the following force (often called the Lorentz force) on charged particles: where all boldfaced quantities are vectors : F is the force that a particle with charge q experiences, E

5670-448: 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

5775-429: The exact vantage point used in the calculations relative to the image. When viewed from a different point, this cancels out what would appear to be distortions in the image. For example, a sphere drawn in perspective will be stretched into an ellipse. These apparent distortions are more pronounced away from the center of the image as the angle between a projected ray (from the scene to the eye) becomes more acute relative to

5880-422: 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

5985-522: 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

6090-534: 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

6195-452: The field of optics centuries before light was understood to be an electromagnetic wave. However, the theory of electromagnetism , as it is currently understood, grew out of Michael Faraday 's experiments suggesting the existence of an electromagnetic field and James Clerk Maxwell 's use of differential equations to describe it in his A Treatise on Electricity and Magnetism (1873). The development of electromagnetism in Europe included

6300-580: The first or second century until the 18th century. It is not certain how they came to use the technique; Dubery and Willats (1983) speculate that the Chinese acquired the technique from India, which acquired it from Ancient Rome, while others credit it as an indigenous invention of Ancient China . Oblique projection is also seen in Japanese art, such as in the Ukiyo-e paintings of Torii Kiyonaga (1752–1815). By

6405-473: 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

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6510-400: The form of a wave . These waves travel in vacuum at the speed of light and exist in a wide spectrum of wavelengths . Examples of the dynamic fields of electromagnetic radiation (in order of increasing frequency): radio waves , microwaves , light ( infrared , visible light and ultraviolet ), x-rays and gamma rays . In the field of particle physics this electromagnetic radiation

6615-444: The geometry of the lens does not perfectly direct rays from each object point to 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

6720-411: 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

6825-415: 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

6930-418: 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

7035-435: 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

7140-475: 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

7245-490: 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

7350-529: The later periods of antiquity, artists, especially those in less popular traditions, were well aware that distant objects could be shown smaller than those close at hand for increased realism, but whether this convention was actually used in a work depended on many factors. Some of the paintings found in the ruins of Pompeii show a remarkable realism and perspective for their time. It has been claimed that comprehensive systems of perspective were evolved in antiquity, but most scholars do not accept this. Hardly any of

7455-502: 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

7560-409: 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 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

7665-505: 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 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

7770-449: 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

7875-421: 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 :

7980-527: The many works where such a system would have been used have survived. A passage in Philostratus suggests that classical artists and theorists thought in terms of "circles" at equal distance from the viewer, like a classical semi-circular theatre seen from the stage. The roof beams in rooms in the Vatican Virgil , from about 400 AD, are shown converging, more or less, on a common vanishing point, but this

8085-442: 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:

8190-487: 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

8295-724: The observer increases, and that they are subject to foreshortening , meaning that an object's dimensions parallel to the line of sight appear shorter than its dimensions perpendicular to the line of sight. All objects will recede to points in the distance, usually along the horizon line, but also above and below the horizon line depending on the view used. Italian Renaissance painters and architects including Filippo Brunelleschi , Leon Battista Alberti , Masaccio , Paolo Uccello , Piero della Francesca and Luca Pacioli studied linear perspective, wrote treatises on it, and incorporated it into their artworks. linear or point-projection perspective works by putting an imagery flat plane that

8400-400: 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

8505-424: The particular transmission medium. Since there are infinitely many of them, in modeling there is a need for some typical, representative Perspective (graphical) Linear or point-projection perspective (from Latin perspicere  'to see through') is one of two types of graphical projection perspective in the graphic arts ; the other is parallel projection . Linear perspective

8610-674: 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

8715-477: The period, but without a basis in a systematic theory. Byzantine art was also aware of these principles, but also used the reverse perspective convention for the setting of principal figures. Ambrogio Lorenzetti painted a floor with convergent lines in his Presentation at the Temple (1342), though the rest of the painting lacks perspective elements. It is generally accepted that Filippo Brunelleschi conducted

8820-432: The point in space where E is being determined. Both of the above equations are cumbersome, especially if one wants to determine E as a function of position. A scalar function called the electric potential can help. Electric potential, also called voltage (the units for which are the volt), is defined by the line integral where φ ( r ) {\displaystyle \varphi ({\textbf {r}})}

8925-406: The quick proliferation of accurate perspective paintings in Florence, Brunelleschi likely understood (with help from his friend the mathematician Toscanelli ), but did not publish, the mathematics behind perspective. Decades later, his friend Leon Battista Alberti wrote De pictura ( c.  1435 ), a treatise on proper methods of showing distance in painting. Alberti's primary breakthrough

9030-407: 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, 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

9135-416: 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

9240-423: 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

9345-438: 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 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

9450-458: The retarded potentials can be computed and differentiated accordingly to yield Jefimenko's equations. Retarded potentials can also be derived for point charges, and the equations are known as the Liénard–Wiechert potentials. The scalar potential is: where q {\displaystyle q} is the point charge's charge and r {\displaystyle {\textbf {r}}}

9555-415: 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

9660-461: 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

9765-466: 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

9870-470: 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 the Huygens–Fresnel principle states that every point of

9975-526: The undergraduate level, textbooks like The Feynman Lectures on Physics , Electricity and Magnetism , and Introduction to Electrodynamics are considered as classic references and for the graduate level, textbooks like Classical Electricity and Magnetism , Classical Electrodynamics , and Course of Theoretical Physics are considered as classic references. The physical phenomena that electromagnetism describes have been studied as separate fields since antiquity. For example, there were many advances in

10080-523: The velocity and magnetic field vectors. The other vector is in the same direction as the electric field. The sum of these two vectors is the Lorentz force. Although the equation appears to suggest that the electric and magnetic fields are independent, the equation can be rewritten in term of four-current (instead of charge) and a single electromagnetic tensor that represents the combined field ( F μ ν {\displaystyle F^{\mu \nu }} ): The electric field E

10185-455: The viewer, and did not use foreshortening. The most important figures are often shown as the highest in a composition , also from hieratic motives, leading to the so-called "vertical perspective", common in the art of Ancient Egypt , where a group of "nearer" figures are shown below the larger figure or figures; simple overlapping was also employed to relate distance. Additionally, oblique foreshortening of round elements like shields and wheels

10290-438: 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 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

10395-399: 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

10500-457: Was also the first to accurately draw the Platonic solids as they would appear in perspective. Luca Pacioli 's 1509 Divina proportione ( Divine Proportion ), illustrated by Leonardo da Vinci , summarizes the use of perspective in painting, including much of Della Francesca's treatise. Leonardo applied one-point perspective as well as shallow focus to some of his works. Two-point perspective

10605-455: Was demonstrated as early as 1525 by Albrecht Dürer , who studied perspective by reading Piero and Pacioli's works, in his Unterweisung der Messung ("Instruction of the Measurement"). Perspective images are created with reference to a particular center of vision for the picture plane. In order for the resulting image to appear identical to the original scene, a viewer must view the image from

10710-539: 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

10815-478: Was not confined merely to the stage. Euclid in his Optics ( c.  300 BC ) argues correctly that the perceived size of an object is not related to its distance from the eye by a simple proportion. In the first-century BC frescoes of the Villa of P. Fannius Synistor , multiple vanishing points are used in a systematic but not fully consistent manner. Chinese artists made use of oblique projection from

10920-423: Was not to show the mathematics in terms of conical projections, as it actually appears to the eye. Instead, he formulated the theory based on planar projections, or how the rays of light, passing from the viewer's eye to the landscape, would strike the picture plane (the painting). He was then able to calculate the apparent height of a distant object using two similar triangles. The mathematics behind similar triangles

11025-463: Was perspective a way of showing depth, it was also a new method of creating a composition. Visual art could now depict a single, unified scene, rather than a combination of several. Early examples include Masolino's St. Peter Healing a Cripple and the Raising of Tabitha ( c.  1423 ), Donatello's The Feast of Herod ( c.  1427 ), as well as Ghiberti's Jacob and Esau and other panels from

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