Optics

Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics describes the behaviour of visible and infrared light; because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays and radio waves exhibit similar properties. Most optical phenomena can be accounted for using the classical electromagnetic description of light. Complete electromagnetic descriptions of light are, however difficult to apply in practice. Practical optics is 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; 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 the fact that light has both particle-like properties. Explanation of these effects requires quantum mechanics; when considering light's particle-like properties, 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 and medicine. Practical applications of optics are found in a variety of technologies and everyday objects, including mirrors, telescopes, microscopes and fibre optics. Optics began with the development of lenses by Mesopotamians; the earliest known lenses, made from polished crystal quartz, date from as early as 700 BC for Assyrian lenses such as the Layard/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, the development of geometrical optics in the Greco-Roman world. The word optics comes from the ancient Greek word ὀπτική, meaning "appearance, look". Greek philosophy on optics broke down into two opposing theories on how vision worked, the "intromission theory" and the "emission theory"; the intro-mission approach saw vision as coming from objects casting off copies of themselves that were captured by the eye. With many propagators including Democritus, Epicurus and their followers, this theory seems to have some contact with modern theories of what vision is, but it remained only speculation lacking any experimental foundation. Plato first articulated the emission theory, the idea that visual perception is accomplished by rays emitted by the eyes, he commented on the parity reversal of mirrors in Timaeus. Some hundred years Euclid 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 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. Ptolemy, in his treatise Optics, held an extramission-intromission theory of vision: the rays from the eye formed a cone, the vertex being within the eye, the base defining the visual field; the rays were sensitive, conveyed information back to the observer's intellect about the distance and orientation of surfaces. He summarised 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 the angle of incidence. 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 who wrote on 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" describing a law of refraction equivalent to Snell's law. He used this law to compute optimum shapes for curved mirrors. In the early 11th century, Alhazen wrote the Book of Optics 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, instead put forward the idea that light reflected in all directions in straight lines from all points of the objects being viewed and entered the eye, although he was unable to explain how the eye captured the rays. Alhazen's work was 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 a wide range of scientific topics, discussed light from four different perspectives: an epistemology of light, a metaphysics or cosmogony of light, an etiology or physics of light, a theology of light, basing it on the works Aristotle and Platonism.

Grosseteste's most famous disciple, Roger Bacon, wrote w

Focus (optics)

In geometrical optics, a focus called an image point, is the point where light rays originating from a point on the object converge. Although the focus is conceptually a point, physically the focus has a spatial extent, called the blur circle; this non-ideal focusing may be caused by aberrations of the imaging optics. In the absence of significant aberrations, the smallest possible blur circle is the Airy disc, caused by diffraction from the optical system's aperture. Aberrations tend to get worse as the aperture diameter increases, while the Airy circle is smallest for large apertures. An image, or image point or region, is in focus if light from object points is converged as much as possible in the image, out of focus if light is not well converged; the border between these is sometimes defined using a "circle of confusion" criterion. A principal focus or focal point is a special focus: For a lens, or a spherical or parabolic mirror, it is a point onto which collimated light parallel to the axis is focused.

Since light can pass through a lens in either direction, a lens has two focal points – one on each side. The distance in air from the lens or mirror's principal plane to the focus is called the focal length. Elliptical mirrors have two focal points: light that passes through one of these before striking the mirror is reflected such that it passes through the other; the focus of a hyperbolic mirror is either of two points which have the property that light from one is reflected as if it came from the other. Diverging lenses and convex mirrors do not focus a collimated beam to a point. Instead, the focus is the point from which the light appears to be emanating, after it travels through the lens or reflects from the mirror. A convex parabolic mirror will reflect a beam of collimated light to make it appear as if it were radiating from the focal point, or conversely, reflect rays directed toward the focus as a collimated beam. A convex elliptical mirror will reflect light directed towards one focus as if it were radiating from the other focus, both of which are behind the mirror.

A convex hyperbolic mirror will reflect rays emanating from the focal point in front of the mirror as if they were emanating from the focal point behind the mirror. Conversely, it can focus rays directed at the focal point, behind the mirror towards the focal point, in front of the mirror as in a Cassegrain telescope. Autofocus Cardinal point Defocus aberration Depth of field Depth of focus Far point Focus Fixed focus Bokeh Focus stacking Focal Plane Manual focus

Geometry

Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.

While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.

Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.

The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.

Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.

He studied the sp

Conic section

In mathematics, a conic section is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, the ellipse; the circle is a special case of the ellipse, is of sufficient interest in its own right that it was sometimes called a fourth type of conic section. The conic sections have been studied by the ancient Greek mathematicians with this work culminating around 200 BC, when Apollonius of Perga undertook a systematic study of their properties; the conic sections of the Euclidean plane have various distinguishing properties. Many of these have been used as the basis for a definition of the conic sections. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, some particular line, called a directrix, are in a fixed ratio, called the eccentricity; the type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2.

This equation may be written in matrix form, some geometric properties can be studied as algebraic conditions. In the Euclidean plane, the conic sections appear to be quite different from one another, but share many properties. By extending the geometry to a projective plane this apparent difference vanishes, the commonality becomes evident. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically; the conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in Euclidean geometry. A conic is the curve obtained as the intersection of a plane, called the cutting plane, with the surface of a double cone, it shall be assumed that the cone is a right circular cone for the purpose of easy description, but this is not required. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines; these are called degenerate conics and some authors do not consider them to be conics at all.

Unless otherwise stated, "conic" in this article will refer to a non-degenerate conic. There are three types of conics, the ellipse and hyperbola; the circle is a special kind of ellipse, although it had been considered as a fourth type. The circle and the ellipse arise when the intersection of the plane is a closed curve; the circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone – for a right cone, see diagram, this means that the cutting plane is perpendicular to the symmetry axis of the cone. If the cutting plane is parallel to one generating line of the cone the conic is unbounded and is called a parabola. In the remaining case, the figure is a hyperbola. In this case, the plane will intersect both halves of the cone, producing two separate unbounded curves. A property that the conic sections share is presented as the following definition. A conic section is the locus of all points P whose distance to a fixed point F is a constant multiple of the distance from P to a fixed line L.

For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, for e > 1 a hyperbola. A circle is not defined by a focus and directrix, in the plane; the eccentricity of a circle is defined to be zero and its focus is the center of the circle, but there is no line in the Euclidean plane, its directrix. An ellipse and a hyperbola each have distinct directrices for each of them; the line joining the foci is called the principal axis and the points of intersection of the conic with the principal axis are called the vertices of the conic. The line segment joining the vertices of a conic is called the major axis called transverse axis in the hyperbola; the midpoint of this line segment is called the center of the conic. Let a denote the distance from the center to a vertex of an ellipse or hyperbola; the distance from the center to a directrix is a/e while the distance from the center to a focus is ae. A parabola does not have a center; the eccentricity of an ellipse can be seen as a measure of how far the ellipse deviates from being circular.

If the angle between the surface of the cone and its axis is β and the angle between the cutting plane and the axis is α, the eccentricity is cos α cos β. A proof that the conic sections given by the focus-directrix property are the same as those given by planes intersecting a cone is facilitated by the use of Dandelin spheres. Various parameters are associated with a conic section. Recall that the principal axis is the line joining the foci of an ellipse or hyperbola, the center in these cases is the midpoint of the line segment joining the foci; some of the other common features and/or. The linear eccentricity is the distance between the focus; the latus rectum is the chord parallel to the directrix and passing through the focus. Its length is denoted by 2ℓ; the semi-latus rectum is half of the length of the latus rec

Radius of curvature (optics)

Radius of curvature has specific meaning and sign convention in optical design. A spherical lens or mirror surface has a center of curvature located either along or decentered from the system local optical axis; the vertex of the lens surface is located on the local optical axis. The distance from the vertex to the center of curvature is the radius of curvature of the surface; the sign convention for the optical radius of curvature is as follows: If the vertex lies to the left of the center of curvature, the radius of curvature is positive. If the vertex lies to the right of the center of curvature, the radius of curvature is negative, thus when viewing a biconvex lens from the side, the left surface radius of curvature is positive, the right radius of curvature is negative. Note however that in areas of optics other than design, other sign conventions are sometimes used. In particular, many undergraduate physics textbooks use the Gaussian sign convention in which convex surfaces of lenses are always positive.

Care should be taken. Optical surfaces with non-spherical profiles, such as the surfaces of aspheric lenses have a radius of curvature; these surfaces are designed such that their profile is described by the equation z = r 2 R + α 1 r 2 + α 2 r 4 + α 3 r 6 + ⋯, where the optic axis is presumed to lie in the z direction, z is the sag—the z-component of the displacement of the surface from the vertex, at distance r from the axis. If α 1 and α 2 are zero R is the radius of curvature and K is the conic constant, as measured at the vertex; the coefficients α i describe the deviation of the surface from the axially symmetric quadric surface specified by R and K. Radius of curvature Radius Base curve radius Cardinal point Vergence

Confocal microscopy

Confocal microscopy, most confocal laser scanning microscopy or laser confocal scanning microscopy, is an optical imaging technique for increasing optical resolution and contrast of a micrograph by means of using a spatial pinhole to block out-of-focus light in image formation. Capturing multiple two-dimensional images at different depths in a sample enables the reconstruction of three-dimensional structures within an object; this technique is used extensively in the scientific and industrial communities and typical applications are in life sciences, semiconductor inspection and materials science. Light travels through the sample under a conventional microscope as far into the specimen as it can penetrate, while a confocal microscope only focuses a smaller beam of light at one narrow depth level at a time; the CLSM achieves a controlled and limited depth of focus. The principle of confocal imaging was patented in 1957 by Marvin Minsky and aims to overcome some limitations of traditional wide-field fluorescence microscopes.

In a conventional fluorescence microscope, the entire specimen is flooded evenly in light from a light source. All parts of the specimen in the optical path are excited at the same time and the resulting fluorescence is detected by the microscope's photodetector or camera including a large unfocused background part. In contrast, a confocal microscope uses point illumination and a pinhole in an optically conjugate plane in front of the detector to eliminate out-of-focus signal – the name "confocal" stems from this configuration; as only light produced by fluorescence close to the focal plane can be detected, the image's optical resolution in the sample depth direction, is much better than that of wide-field microscopes. However, as much of the light from sample fluorescence is blocked at the pinhole, this increased resolution is at the cost of decreased signal intensity – so long exposures are required. To offset this drop in signal after the pinhole, the light intensity is detected by a sensitive detector a photomultiplier tube or avalanche photodiode, transforming the light signal into an electrical one, recorded by a computer.

As only one point in the sample is illuminated at a time, 2D or 3D imaging requires scanning over a regular raster in the specimen. The beam is scanned across the sample in the horizontal plane by using one or more oscillating mirrors; this scanning method has a low reaction latency and the scan speed can be varied. Slower scans provide a better signal-to-noise ratio, resulting in better contrast and higher resolution; the achievable thickness of the focal plane is defined by the wavelength of the used light divided by the numerical aperture of the objective lens, but by the optical properties of the specimen. The thin optical sectioning possible makes these types of microscopes good at 3D imaging and surface profiling of samples. Successive slices make up a'z-stack' which can either be processed by certain software to create a 3D image, or it is merged into a 2D stack. Confocal microscopy provides the capacity for direct, serial optical sectioning of intact, living specimens with a minimum of sample preparation as well as a marginal improvement in lateral resolution.

Biological samples are treated with fluorescent dyes to make selected objects visible. However, the actual dye concentration can be low to minimize the disturbance of biological systems: some instruments can track single fluorescent molecules. Transgenic techniques can create organisms that produce their own fluorescent chimeric molecules. Confocal microscopes work on the principle of point excitation in the specimen and point detection of the resulting fluorescent signal. A pinhole at the detector provides a physical barrier. Only the in-focus, or central spot of the airy disk, is recorded. Raster scanning the specimen one point at a time permits thin optical sections to be collected by changing the z-focus; the resulting images can be stacked to produce a 3D image of the specimen. Four types of confocal microscopes are commercially available: Confocal laser scanning microscopes use multiple mirrors to scan the laser across the sample and "descan" the image across a fixed pinhole and detector.

Spinning-disk confocal microscopes use a series of moving pinholes on a disc to scan spots of light. Since a series of pinholes scans an area in parallel, each pinhole is allowed to hover over a specific area for a longer amount of time thereby reducing the excitation energy needed to illuminate a sample when compared to laser scanning microscopes. Decreased excitation energy reduces phototoxicity and photobleaching of a sample making it the preferred system for imaging live cells or organisms. Microlens enhanced or dual spinning-disk confocal microscopes work under the same principles as spinning-disk confocal microscopes except a second spinning-disk containing micro-lenses is placed before the spinning-disk containing the pinholes; every pinhole has an associated microlens. The micro-lenses act to capture a broad band of light and focus it into each pinhole increasing the amount of light directed into each pinhole and reducing the amount of light blocked by the spinning-disk. Microlens enhanced confocal mi

Lens (optics)

A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction. A simple lens consists of a single piece of transparent material, while a compound lens consists of several simple lenses arranged along a common axis. Lenses are made from materials such as glass or plastic, are ground and polished or molded to a desired shape. A lens can focus light to form an image, unlike a prism. Devices that focus or disperse waves and radiation other than visible light are called lenses, such as microwave lenses, electron lenses, acoustic lenses, or explosive lenses; the word lens comes from lēns, the Latin name of the lentil, because a double-convex lens is lentil-shaped. The lentil plant gives its name to a geometric figure; some scholars argue that the archeological evidence indicates that there was widespread use of lenses in antiquity, spanning several millennia. The so-called Nimrud lens is a rock crystal artifact dated to the 7th century BC which may or may not have been used as a magnifying glass, or a burning glass.

Others have suggested that certain Egyptian hieroglyphs depict "simple glass meniscal lenses". The oldest certain reference to the use of lenses is from Aristophanes' play The Clouds mentioning a burning-glass. Pliny the Elder confirms. Pliny has the earliest known reference to the use of a corrective lens when he mentions that Nero was said to watch the gladiatorial games using an emerald. Both Pliny and Seneca the Younger described the magnifying effect of a glass globe filled with water. Ptolemy wrote a book on Optics, which however survives only in the Latin translation of an incomplete and poor Arabic translation; the book was, received, by medieval scholars in the Islamic world, commented upon by Ibn Sahl, in turn improved upon by Alhazen. The Arabic translation of Ptolemy's Optics became available in Latin translation in the 12th century. Between the 11th and 13th century "reading stones" were invented; these were primitive plano-convex lenses made by cutting a glass sphere in half. The medieval rock cystal Visby lenses may not have been intended for use as burning glasses.

Spectacles were invented as an improvement of the "reading stones" of the high medieval period in Northern Italy in the second half of the 13th century. This was the start of the optical industry of grinding and polishing lenses for spectacles, first in Venice and Florence in the late 13th century, in 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; the practical development and experimentation with lenses led to the invention of the compound optical microscope around 1595, the refracting telescope in 1608, both of which appeared in the spectacle-making centres in the Netherlands. With the invention of the telescope and microscope there was a great deal of experimentation with lens shapes in the 17th and early 18th centuries by those trying to correct chromatic errors seen in lenses. Opticians tried to construct lenses of varying forms of curvature, wrongly assuming errors arose from defects in the spherical figure of their surfaces.

Optical theory on refraction and experimentation was showing no single-element lens could bring all colours to a focus. This led to the invention of the compound achromatic lens by Chester Moore Hall in England in 1733, an invention claimed by fellow Englishman John Dollond in a 1758 patent. Most lenses are spherical lenses: their two surfaces are parts of the surfaces of spheres; each surface can be concave, or planar. The line joining the centres of the spheres making up the lens surfaces is called the axis of the lens; the lens axis passes through the physical centre of the lens, because of the way they are manufactured. Lenses may ground after manufacturing to give them a different shape or size; the lens axis may not pass through the physical centre of the lens. Toric or sphero-cylindrical lenses have surfaces with two different radii of curvature in two orthogonal planes, they have a different focal power in different meridians. This forms an astigmatic lens. An example is eyeglass lenses. More complex are aspheric lenses.

These are lenses where one or both surfaces have a shape, neither spherical nor cylindrical. The more complicated shapes allow such lenses to form images with less aberration than standard simple lenses, but they are more difficult and expensive to produce. Lenses are classified by the curvature of the two optical surfaces. A lens is biconvex. If both surfaces have the same radius of curvature, the lens is equiconvex. A lens with two concave surfaces is biconcave. If one of the surfaces is flat, the lens is plano-convex or plano-concave depending on the curvature of the other surface. A lens with one convex and one concave side is meniscus, it is this type of lens, most used in corrective lenses. If the lens is biconvex or plano-convex, a collimated beam of light passing through the lens converges to a spot behind the lens. In this case, the lens is called a