1.
Wavefront
–
In physics, a wavefront is the locus of points characterized by propagation of position of the same phase, a propagation of a line in 1D, a curve in 2D or a surface for a wave in 3D. Additionally, most optical systems and detectors are indifferent to polarization, at radio wavelengths, the polarization becomes more important, and receivers are usually phase-sensitive. Many audio detectors are also phase-sensitive, Optical systems can be described with Maxwells equations, and linear propagating waves such as sound or electron beams have similar wave equations. However, given the above simplifications, Huygens principle provides a method to predict the propagation of a wavefront through, for example. The construction is as follows, Let every point on the wavefront be considered a new point source, by calculating the total effect from every point source, the resulting field at new points can be computed. Computational algorithms are based on this approach. Specific cases for simple wavefronts can be computed directly, for example, a spherical wavefront will remain spherical as the energy of the wave is carried away equally in all directions. Such directions of flow, which are always perpendicular to the wavefront, are called rays creating multiple wavefronts. The simplest form of a wavefront is the wave, where the rays are parallel to one another. The light from this type of wave is referred to as collimated light, for many purposes, such a wavefront can be considered planar. Wavefront travel with the speed of light in all directions in an isotropic medium, methods utilizing wavefront measurements or predictions can be considered an advanced approach to lens optics, where a single focal distance may not exist due to lens thickness or imperfections. Note also that for manufacturing reasons, a lens has a spherical surface shape though, theoretically. Shortcomings such as these in an optical system cause what are called optical aberrations, the best-known aberrations include spherical aberration and coma. However there may be more complex sources of such as in a large telescope due to spatial variations in the index of refraction of the atmosphere. The deviation of a wavefront in a system from a desired perfect planar wavefront is called the wavefront aberration. Wavefront aberrations are usually described as either an image or a collection of two-dimensional polynomial terms. Minimization of these aberrations is considered desirable for applications in optical systems. A wavefront sensor is a device which measures the wavefront aberration in a coherent signal to describe the quality or lack thereof in an optical system
Wavefront
–
The wavefronts of a
plane wave are
planes.
2.
Optics
–
Optics is the branch of physics which involves the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light, because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties. Most optical phenomena can be accounted for using the classical description of light. Complete electromagnetic descriptions of light are, however, 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, 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, the 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 waves were in fact electromagnetic radiation. Some phenomena depend on the fact that light has both wave-like and particle-like properties, explanation of these effects requires quantum mechanics. When considering lights 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, photography, practical applications of optics are found in 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 700 BC for Assyrian lenses such as the Layard/Nimrud lens. The ancient Romans and Greeks filled glass spheres with water to make lenses, 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 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, plato first articulated 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 wrote a treatise entitled Optics where he linked vision to geometry, creating geometrical optics. Ptolemy, in his treatise Optics, held a theory of vision, the rays from the eye formed a cone, the vertex being within the eye. The rays were sensitive, and conveyed back to the observer’s intellect about the distance. He summarised much of Euclid and went on to describe a way to measure the angle of refraction, during the Middle Ages, Greek ideas about optics were resurrected and extended by writers in the Muslim world
Optics
–
Optics includes study of
dispersion of light.
Optics
–
The Nimrud lens
Optics
–
Reproduction of a page of
Ibn Sahl 's manuscript showing his knowledge of the law of refraction, now known as
Snell's law
Optics
–
Cover of the first edition of Newton's Opticks
3.
Light beam
–
A light beam or beam of light is a directional projection of light energy radiating from a light source. Sunlight forms a light beam when filtered through media such as clouds, foliage, to artificially produce a light beam, a lamp and a parabolic reflector is used in many lighting devices such as spotlights, car headlights, PAR Cans and LED housings. Light from certain types of laser has the smallest possible beam divergence. From the side, a beam of light is visible if part of the light is scattered by objects, tiny particles like dust, water droplets, hail, snow, or smoke. If there are objects in the light path, then it appears as a continuous beam. In any case, this scattering of light from a beam, flashlight, beam directed by hand Headlight, forward beam, the lamp is mounted in a vehicle, or on the forehead of a person, e. g. The difference between the two is that the fog itself is also a visual effect, Laser lighting display- Laser beams are often used for visual effects, often in combination with music. This also used to be done for movie premieres, the waving searchlight beams are still to be seen as an element in the logo of the 20th Century Fox movie studio
Light beam
–
A Symphony of Lights in
Victoria Harbour,
Hong Kong
Light beam
–
Light beams were used to symbolize the missing towers of the
World Trade Center as part of the
Tribute in Light.
Light beam
–
A natural lightbeam in the
Majlis al-Jinn (literally 'Meeting place of the
jinn ') cave in
Oman
Light beam
–
Beams shining through water-based haze in a photo studio setting.
4.
Electromagnetic radiation
–
In physics, electromagnetic radiation refers to the waves of the electromagnetic field, propagating through space carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, light, ultraviolet, X-, classically, electromagnetic radiation consists of electromagnetic waves, which are synchronized oscillations of electric and magnetic fields that propagate at the speed of light through a vacuum. The oscillations of the two fields are perpendicular to other and perpendicular to the direction of energy and wave propagation. The wavefront of electromagnetic waves emitted from a point source is a sphere, the position of an electromagnetic wave within the electromagnetic spectrum can be characterized by either its frequency of oscillation or its wavelength. Electromagnetic waves are produced whenever charged particles are accelerated, and these waves can interact with other charged particles. EM waves carry energy, momentum and angular momentum away from their source particle, quanta of EM waves are called photons, whose rest mass is zero, but whose energy, or equivalent total mass, is not zero so they are still affected by gravity. Thus, EMR is sometimes referred to as the far field, in this language, the near field refers to EM fields near the charges and current that directly produced them, specifically, electromagnetic induction and electrostatic induction phenomena. In the quantum theory of electromagnetism, EMR consists of photons, quantum effects provide additional sources of EMR, such as the transition of electrons to lower energy levels in an atom and black-body radiation. The energy of a photon is quantized and is greater for photons of higher frequency. This relationship is given by Plancks equation E = hν, where E is the energy per photon, ν is the frequency of the photon, a single gamma ray photon, for example, might carry ~100,000 times the energy of a single photon of visible light. The effects of EMR upon chemical compounds and biological organisms depend both upon the power and its frequency. EMR of visible or lower frequencies is called non-ionizing radiation, because its photons do not individually have enough energy to ionize atoms or molecules, the effects of these radiations on chemical systems and living tissue are caused primarily by heating effects from the combined energy transfer of many photons. In contrast, high ultraviolet, X-rays and gamma rays are called ionizing radiation since individual photons of high frequency have enough energy to ionize molecules or break chemical bonds. These radiations have the ability to cause chemical reactions and damage living cells beyond that resulting from simple heating, Maxwell derived a wave form of the electric and magnetic equations, thus uncovering the wave-like nature of electric and magnetic fields and their symmetry. Because the speed of EM waves predicted by the wave equation coincided with the speed of light. Maxwell’s equations were confirmed by Heinrich Hertz through experiments with radio waves, according to Maxwells equations, a spatially varying electric field is always associated with a magnetic field that changes over time. Likewise, a varying magnetic field is associated with specific changes over time in the electric field. In an electromagnetic wave, the changes in the field are always accompanied by a wave in the magnetic field in one direction
Electromagnetic radiation
5.
Electric field
–
An electric field is a vector field that associates to each point in space the Coulomb force that would be experienced per unit of electric charge, by an infinitesimal test charge at that point. Electric fields are created by electric charges and can be induced by time-varying magnetic fields, the electric field combines with the magnetic field to form the electromagnetic field. The electric field, E, at a point is defined as the force, F. A particle of charge q would be subject to a force F = q E and its SI units are newtons per coulomb or, equivalently, volts per metre, which in terms of SI base units are kg⋅m⋅s−3⋅A−1. Electric fields are caused by electric charges or varying magnetic fields, in the special case of a steady state, the Maxwell-Faraday inductive effect disappears. The resulting two equations, taken together, are equivalent to Coulombs law, written as E =14 π ε0 ∫ d r ′ ρ r − r ′ | r − r ′ |3 for a charge density ρ. Notice that ε0, the permittivity of vacuum, must be substituted if charges are considered in non-empty media, the equations of electromagnetism are best described in a continuous description. A charge q located at r 0 can be described mathematically as a charge density ρ = q δ, conversely, a charge distribution can be approximated by many small point charges. Electric fields satisfy the principle, because Maxwells equations are linear. This principle is useful to calculate the field created by point charges. Q n are stationary in space at r 1, r 2, in that case, Coulombs law fully describes the field. If a system is static, such that magnetic fields are not time-varying, then by Faradays law, in this case, one can define an electric potential, that is, a function Φ such that E = − ∇ Φ. This is analogous to the gravitational potential, Coulombs law, which describes the interaction of electric charges, F = q = q E is similar to Newtons law of universal gravitation, F = m = m g. This suggests similarities between the electric field E and the gravitational field g, or their associated potentials, mass is sometimes called gravitational charge because of that similarity. Electrostatic and gravitational forces both are central, conservative and obey an inverse-square law, a uniform field is one in which the electric field is constant at every point. It can be approximated by placing two conducting plates parallel to other and maintaining a voltage between them, it is only an approximation because of boundary effects. Assuming infinite planes, the magnitude of the electric field E is, electrodynamic fields are E-fields which do change with time, for instance when charges are in motion. The electric field cannot be described independently of the field in that case
Electric field
–
Electric field lines emanating from a point positive
electric charge suspended over an infinite sheet of conducting material.
6.
Gaussian function
–
In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form, f = a e −22 c 2 for arbitrary real constants a, b and c. It is named after the mathematician Carl Friedrich Gauss, the graph of a Gaussian is a characteristic symmetric bell curve shape. The parameter a is the height of the peak, b is the position of the center of the peak. Gaussian functions arise by composing the exponential function with a quadratic function. The Gaussian functions are thus those functions whose logarithm is a quadratic function. The parameter c is related to the width at half maximum of the peak according to F W H M =22 ln 2 c ≈2.35482 c. The full width at tenth of maximum for a Gaussian could be of interest and is F W T M =22 ln 10 c ≈4.29193 c, Gaussian functions are analytic, and their limit as x → ∞ is 0. Gaussian functions are among those functions that are elementary but lack elementary antiderivatives and these Gaussians are plotted in the accompanying figure. Gaussian functions centered at zero minimize the Fourier uncertainty principle. The product of two Gaussian functions is a Gaussian, and the convolution of two Gaussian functions is also a Gaussian, with variance being the sum of the original variances, the product of two Gaussian probability density functions, though, is not in general a Gaussian PDF. Taking the Fourier transform of a Gaussian function with parameters a =1, b =0 and c yields another Gaussian function, so in particular the Gaussian functions with b =0 and c =1 are kept fixed by the Fourier transform. A physical realization is that of the pattern, for example. The integral ∫ − ∞ ∞ a e −2 /2 c 2 d x for some real constants a, b, c >0 can be calculated by putting it into the form of a Gaussian integral. First, the constant a can simply be factored out of the integral, next, the variable of integration is changed from x to y = x - b. Consequently, the sets of the Gaussian will always be ellipses. A particular example of a two-dimensional Gaussian function is f = A exp , here the coefficient A is the amplitude, xo, yo is the center and σx, σy are the x and y spreads of the blob. The figure on the right was created using A =1, xo =0, yo =0, σx = σy =1. The volume under the Gaussian function is given by V = ∫ − ∞ ∞ ∫ − ∞ ∞ f d x d y =2 π A σ x σ y
Gaussian function
–
Normalized Gaussian curves with
expected value μ and
variance σ 2. The corresponding parameters are, b = μ, and c = σ.
7.
Transverse mode
–
A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular to the propagation direction of the beam. Transverse modes occur in waves and microwaves confined to a waveguide. Transverse modes occur because of conditions imposed on the wave by the waveguide. For this reason, the supported by a waveguide are quantized. The allowed modes can be found by solving Maxwells equations for the conditions of a given waveguide. Unguided electromagnetic waves in space, or in a bulk isotropic dielectric, can be described as a superposition of plane waves. However in any sort of waveguide where boundary conditions are imposed by a physical structure and these modes generally follow different propagation constants. When two or more modes have a propagation constant along the waveguide, then there is more than one modal decomposition possible in order to describe a wave with that propagation constant. Modes in waveguides can be classified as follows, Transverse electromagnetic modes. Transverse electric modes, no field in the direction of propagation. These are sometimes called H modes because there is only a field along the direction of propagation. Transverse magnetic modes, no field in the direction of propagation. These are sometimes called E modes because there is only a field along the direction of propagation. Hybrid modes, non-zero electric and magnetic fields in the direction of propagation, hollow metallic waveguides filled with a homogeneous, isotropic material support TE and TM modes but not the TEM mode. In coaxial cable energy is transported in the fundamental TEM mode. The TEM mode is usually assumed for most other electrical conductor line formats as well. In an optical fiber or other dielectric waveguide, modes are generally of the hybrid type, in circular waveguides, circular modes exist and here m is the number of full-wave patterns along the circumference and n is the number of half-wave patterns along the diameter. In a laser with cylindrical symmetry, the transverse mode patterns are described by a combination of a Gaussian beam profile with a Laguerre polynomial, the modes are denoted TEMpl where p and l are integers labeling the radial and angular mode orders, respectively
Transverse mode
–
Cylindrical transverse mode patterns TEM(pl)
8.
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 piece of transparent material, while a compound lens consists of several simple lenses. Lenses are made from such as glass or plastic, and are ground. A lens can focus light to form an image, unlike a prism, devices that similarly focus or disperse waves and radiation other than visible light are also called lenses, such as microwave lenses, electron lenses, acoustic lenses, or explosive lenses. The word lens comes from the Latin name of the lentil, the genus of the lentil plant is Lens, and the most commonly eaten species is Lens culinaris. The lentil plant also gives its name to a geometric figure, the variant spelling lense is sometimes seen. While it is listed as a spelling in some dictionaries. The oldest lens artifact is the Nimrud lens, dating back 2700 years to ancient Assyria, david Brewster proposed that it may have been used as a magnifying glass, or as a burning-glass to start fires by concentrating sunlight. Another early reference to magnification dates back to ancient Egyptian hieroglyphs in the 8th century BC, the earliest written records of lenses date to Ancient Greece, with Aristophanes play The Clouds mentioning a burning-glass. Some scholars argue that the evidence indicates that there was widespread use of lenses in antiquity. Such lenses were used by artisans for fine work, and for authenticating seal impressions, both Pliny and Seneca the Younger described the magnifying effect of a glass globe filled with water. The Viking lenses were capable of concentrating enough sunlight to ignite fires, between the 11th and 13th century reading stones were invented. Often used by monks to assist in illuminating manuscripts, these were primitive plano-convex lenses initially made by cutting a sphere in half. As the stones were experimented with, it was understood that shallower lenses magnified more effectively. Lenses came into use in Europe with the invention of spectacles. Spectacle makers created improved types of lenses for the correction of vision based more on knowledge gained from observing the effects of the lenses. With the invention of the telescope and microscope there was a deal of experimentation with lens shapes in the 17th. Opticians tried to construct lenses of varying forms of curvature, wrongly assuming errors arose from defects in the figure of their surfaces
Lens (optics)
–
A biconvex lens.
Lens (optics)
–
The
Nimrud lens
Lens (optics)
Lens (optics)
9.
Wavelength
–
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the waves shape repeats, and thus the inverse of the spatial frequency. Wavelength is commonly designated by the Greek letter lambda, the concept can also be applied to periodic waves of non-sinusoidal shape. The term wavelength is also applied to modulated waves. Wavelength depends on the medium that a wave travels through, examples of wave-like phenomena are sound waves, light, water waves and periodic electrical signals in a conductor. A sound wave is a variation in air pressure, while in light and other electromagnetic radiation the strength of the electric, water waves are variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary, wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in waves over deep water a particle near the waters surface moves in a circle of the same diameter as the wave height. The range of wavelengths or frequencies for wave phenomena is called a spectrum, the name originated with the visible light spectrum but now can be applied to the entire electromagnetic spectrum as well as to a sound spectrum or vibration spectrum. In linear media, any pattern can be described in terms of the independent propagation of sinusoidal components. The wavelength λ of a sinusoidal waveform traveling at constant speed v is given by λ = v f, in a dispersive medium, the phase speed itself depends upon the frequency of the wave, making the relationship between wavelength and frequency nonlinear. In the case of electromagnetic radiation—such as light—in free space, the speed is the speed of light. Thus the wavelength of a 100 MHz electromagnetic wave is about, the wavelength of visible light ranges from deep red, roughly 700 nm, to violet, roughly 400 nm. For sound waves in air, the speed of sound is 343 m/s, the wavelengths of sound frequencies audible to the human ear are thus between approximately 17 m and 17 mm, respectively. Note that the wavelengths in audible sound are much longer than those in visible light, a standing wave is an undulatory motion that stays in one place. A sinusoidal standing wave includes stationary points of no motion, called nodes, the upper figure shows three standing waves in a box. The walls of the box are considered to require the wave to have nodes at the walls of the box determining which wavelengths are allowed, the stationary wave can be viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities. Consequently, wavelength, period, and wave velocity are related just as for a traveling wave, for example, the speed of light can be determined from observation of standing waves in a metal box containing an ideal vacuum. In that case, the k, the magnitude of k, is still in the same relationship with wavelength as shown above
Wavelength
–
Wavelength is decreased in a medium with slower propagation.
Wavelength
–
Wavelength of a
sine wave, λ, can be measured between any two points with the same
phase, such as between crests, or troughs, or corresponding
zero crossings as shown.
Wavelength
–
Various local wavelengths on a crest-to-crest basis in an ocean wave approaching shore
Wavelength
–
A wave on a line of atoms can be interpreted according to a variety of wavelengths.
10.
Polarization (waves)
–
Polarization is a parameter applying to transverse waves that specifies the geometrical orientation of the oscillations. In a transverse wave the direction of the oscillation is transverse to the direction of motion of the wave, a simple example of a polarized transverse wave is vibrations traveling along a taut string for example in a musical instrument like a guitar string. Depending on how the string is plucked, the vibrations can be in a direction, horizontal direction. Transverse waves that exhibit polarization include electromagnetic waves such as light and radio waves, gravitational waves, in some types of transverse waves the wave displacement is limited to a single direction, so these also dont exhibit polarization. For example, in waves in liquids the wave displacement of the particles is always in a vertical plane. In linear polarization the fields oscillate in a single direction, in circular or elliptical polarization the fields rotate at a constant rate in a plane as the wave travels. Polarized light can be produced by passing unpolarized light through a polarizing filter, some of these are used to make polarizing filters. Light is also partially polarized when it reflects from a surface, according to quantum mechanics, electromagnetic waves consist of particles called photons. When viewed in this way, the polarization of a wave is determined by a quantum mechanical property of photons called their spin. A photon has one of two spins, it can either spin in a right hand sense or a left hand sense about its direction of travel. Circularly polarized electromagnetic waves are composed of photons with one type of spin. Linearly polarized waves consist of numbers of right and left hand spinning photons. Polarization is an important parameter in areas of science dealing with transverse waves, such as optics, seismology, radio, especially impacted are technologies such as lasers, wireless and optical fiber telecommunications, and radar. And incoherent states can be modeled stochastically as a combination of such uncorrelated waves with some distribution of frequencies, phases. Considering a monochromatic wave of optical frequency f, let us take the direction of propagation as the z axis. Being a transverse wave the E and H fields must then contain components only in the x and y directions whereas Ez=Hz=0. Using complex notation, we understand the instantaneous electric and magnetic fields to be given by the real parts of the complex quantities occurring in the following equations. Here ex, ey, hx, and hy are complex numbers, in the second more compact form, as these equations are customarily expressed, these factors are described using the wavenumber k =2 π n / λ and angular frequency ω =2 π f
Polarization (waves)
–
Circular polarization on rubber thread, converted to linear polarization
Polarization (waves)
–
Color pattern of a plastic box possessing
stress induced birefringence when placed in between two crossed
polarizers.
Polarization (waves)
–
Stress in plastic glasses
Polarization (waves)
–
Photomicrograph of a
volcanic sand grain; upper picture is plane-polarized light, bottom picture is cross-polarized light, scale box at left-center is 0.25 millimeter.
11.
Cartesian coordinates
–
Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half
Cartesian coordinates
–
The
right hand rule.
Cartesian coordinates
–
Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: (2,3) in green, (−3,1) in red, (−1.5,−2.5) in blue, and the origin (0,0) in purple.
Cartesian coordinates
–
3D Cartesian Coordinate Handedness
12.
Cylindrical coordinates
–
The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point. The origin of the system is the point where all three coordinates can be given as zero and this is the intersection between the reference plane and the axis. The third coordinate may be called the height or altitude, longitudinal position and they are sometimes called cylindrical polar coordinates and polar cylindrical coordinates, and are sometimes used to specify the position of stars in a galaxy. The three coordinates of a point P are defined as, The radial distance ρ is the Euclidean distance from the z-axis to the point P. The azimuth φ is the angle between the direction on the chosen plane and the line from the origin to the projection of P on the plane. The height z is the distance from the chosen plane to the point P. As in polar coordinates, the point with cylindrical coordinates has infinitely many equivalent coordinates, namely and. Moreover, if the radius ρ is zero, the azimuth is arbitrary, in situations where someone wants a unique set of coordinates for each point, one may restrict the radius to be non-negative and the azimuth φ to lie in a specific interval spanning 360°, such as. The notation for cylindrical coordinates is not uniform, the ISO standard 31-11 recommends, where ρ is the radial coordinate, φ the azimuth, and z the height. However, the radius is often denoted r or s, the azimuth by θ or t. In concrete situations, and in many illustrations, a positive angular coordinate is measured counterclockwise as seen from any point with positive height. The cylindrical coordinate system is one of many coordinate systems. The following formulae may be used to convert between them, the arcsin function is the inverse of the sine function, and is assumed to return an angle in the range =. These formulas yield an azimuth φ in the range, for other formulas, see the polar coordinate article. Many modern programming languages provide a function that will compute the correct azimuth φ, in the range, given x and y, for example, this function is called by atan2 in the C programming language, and atan in Common Lisp. In many problems involving cylindrical polar coordinates, it is useful to know the line and volume elements, the line element is d r = d ρ ρ ^ + ρ d φ φ ^ + d z z ^. The volume element is d V = ρ d ρ d φ d z, the surface element in a surface of constant radius ρ is d S ρ = ρ d φ d z. The surface element in a surface of constant azimuth φ is d S φ = d ρ d z, the surface element in a surface of constant height z is d S z = ρ d ρ d φ
Cylindrical coordinates
–
A cylindrical coordinate system with origin O, polar axis A, and longitudinal axis L. The dot is the point with radial distance ρ = 4, angular coordinate φ = 130°, and height z = 4.
13.
Superposition principle
–
So that if input A produces response X and input B produces response Y then input produces response. The homogeneity and additivity properties together are called the superposition principle, a linear function is one that satisfies the properties of superposition. It is defined as F = F + F Additivity F = a F Homogeneity for scalar a and this principle has many applications in physics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a system where the input stimulus is the load on the beam. Because physical systems are only approximately linear, the superposition principle is only an approximation of the true physical behaviour. The superposition principle applies to any system, including algebraic equations, linear differential equations. The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, note that when vectors or vector fields are involved, a superposition is interpreted as a vector sum. By writing a very general stimulus as the superposition of stimuli of a specific, simple form, for example, in Fourier analysis, the stimulus is written as the superposition of infinitely many sinusoids. Due to the principle, each of these sinusoids can be analyzed separately. According to the principle, the response to the original stimulus is the sum of all the individual sinusoidal responses. Fourier analysis is common for waves. For example, in theory, ordinary light is described as a superposition of plane waves. As long as the principle holds, the behavior of any light wave can be understood as a superposition of the behavior of these simpler plane waves. Waves are usually described by variations in some parameter space and time—for example, height in a water wave, pressure in a sound wave. The value of this parameter is called the amplitude of the wave, in any system with waves, the waveform at a given time is a function of the sources and initial conditions of the system. In many cases, the equation describing the wave is linear, when this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side, with regard to wave superposition, Richard Feynman wrote, No-one has ever been able to define the difference between interference and diffraction satisfactorily
Superposition principle
–
Superposition of almost
plane waves (diagonal lines) from a distant source and waves from the
wake of the
ducks.
Linearity holds only approximately in water and only for waves with small amplitudes relative to their wavelengths.
14.
Laser resonator
–
An optical cavity, resonating cavity or optical resonator is an arrangement of mirrors that forms a standing wave cavity resonator for light waves. Optical cavities are a component of lasers, surrounding the gain medium. They are also used in optical parametric oscillators and some interferometers, light confined in the cavity reflects multiple times producing standing waves for certain resonance frequencies. Different resonator types are distinguished by the lengths of the two mirrors and the distance between them. The geometry must be chosen so that the beam remains stable, resonator types are also designed to meet other criteria such as minimum beam waist or having no focal point inside the cavity. Optical cavities are designed to have a large Q factor, a beam will reflect a large number of times with little attenuation. Therefore, the line width of the beam is very small indeed compared to the frequency of the laser. In general, radiation patterns which are reproduced on every round-trip of the light through the resonator are the most stable, the basic, or fundamental transverse mode of a resonator is a Gaussian beam. The most common types of optical cavities consist of two facing plane or spherical mirrors, the simplest of these is the plane-parallel or Fabry–Pérot cavity, consisting of two opposing flat mirrors. However, this problem is reduced for very short cavities with a small mirror separation distance. Plane-parallel resonators are therefore used in microchip and microcavity lasers. In these cases, rather than using separate mirrors, an optical coating may be directly applied to the laser medium itself. The plane-parallel resonator is also the basis of the Fabry–Pérot interferometer, for a resonator with two mirrors with radii of curvature R1 and R2, there are a number of common cavity configurations. If the two curvatures are equal to half the cavity length, a concentric or spherical resonator results and this type of cavity produces a diffraction-limited beam waist in the centre of the cavity, with large beam diameters at the mirrors, filling the whole mirror aperture. Similar to this is the cavity, with one plane mirror. A common and important design is the confocal resonator, with equal curvature mirrors equal to the cavity length. This design produces the smallest possible beam diameter at the cavity mirrors for a given cavity length, a concave-convex cavity has one convex mirror with a negative radius of curvature. A transparent dielectric sphere, such as a liquid droplet, also forms an optical cavity
Laser resonator
–
A glass nanoparticle is suspended in an optical cavity
15.
Transverse electromagnetic mode
–
A transverse mode of electromagnetic radiation is a particular electromagnetic field pattern of radiation measured in a plane perpendicular to the propagation direction of the beam. Transverse modes occur in waves and microwaves confined to a waveguide. Transverse modes occur because of conditions imposed on the wave by the waveguide. For this reason, the supported by a waveguide are quantized. The allowed modes can be found by solving Maxwells equations for the conditions of a given waveguide. Unguided electromagnetic waves in space, or in a bulk isotropic dielectric, can be described as a superposition of plane waves. However in any sort of waveguide where boundary conditions are imposed by a physical structure and these modes generally follow different propagation constants. When two or more modes have a propagation constant along the waveguide, then there is more than one modal decomposition possible in order to describe a wave with that propagation constant. Modes in waveguides can be classified as follows, Transverse electromagnetic modes. Transverse electric modes, no field in the direction of propagation. These are sometimes called H modes because there is only a field along the direction of propagation. Transverse magnetic modes, no field in the direction of propagation. These are sometimes called E modes because there is only a field along the direction of propagation. Hybrid modes, non-zero electric and magnetic fields in the direction of propagation, hollow metallic waveguides filled with a homogeneous, isotropic material support TE and TM modes but not the TEM mode. In coaxial cable energy is transported in the fundamental TEM mode. The TEM mode is usually assumed for most other electrical conductor line formats as well. In an optical fiber or other dielectric waveguide, modes are generally of the hybrid type, in circular waveguides, circular modes exist and here m is the number of full-wave patterns along the circumference and n is the number of half-wave patterns along the diameter. In a laser with cylindrical symmetry, the transverse mode patterns are described by a combination of a Gaussian beam profile with a Laguerre polynomial, the modes are denoted TEMpl where p and l are integers labeling the radial and angular mode orders, respectively
Transverse electromagnetic mode
–
Field patterns of some commonly used waveguide modes
16.
Helmholtz equation
–
In mathematics, the Helmholtz equation, named for Hermann von Helmholtz, is the partial differential equation ∇2 A + k 2 A =0 where ∇2 is the Laplacian, k is the wavenumber, and A is the amplitude. The Helmholtz equation often arises in the study of problems involving partial differential equations in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, for example, consider the wave equation u =0. Separation of variables begins by assuming that the function u is in fact separable. Substituting this form into the equation, and then simplifying, we obtain the following equation. Notice the expression on the left-hand side depends only on r, as a result, this equation is valid in the general case if and only if both sides of the equation are equal to a constant value. Rearranging the first equation, we obtain the Helmholtz equation, ∇2 A + k 2 A = A =0 and we now have Helmholtzs equation for the spatial variable r and a second-order ordinary differential equation in time. The solution in time will be a combination of sine and cosine functions, with angular frequency of ω. Alternatively, integral transforms, such as the Laplace or Fourier transform, are used to transform a hyperbolic PDE into a form of the Helmholtz equation. Because of its relationship to the equation, the Helmholtz equation arises in problems in such areas of physics as the study of electromagnetic radiation, seismology. The solution to the spatial Helmholtz equation A =0 can be obtained for simple geometries using separation of variables, the two-dimensional analogue of the vibrating string is the vibrating membrane, with the edges clamped to be motionless. The elliptical drumhead was studied by Émile Mathieu, leading to Mathieus differential equation, the solvable shapes all correspond to shapes whose dynamical billiard table is integrable, that is, not chaotic. When the motion on a billiard table is chaotic, then no closed form solutions to the Helmholtz equation are known. The study of systems is known as quantum chaos, as the Helmholtz equation. An interesting situation happens with a shape where about half of the solutions are integrable, a simple shape where this happens is with the regular hexagon. Another simple shape where this happens is with an L shape made by reflecting a square down, if the domain is a circle of radius a, then it is appropriate to introduce polar coordinates r and θ. The Helmholtz equation takes the form A r r +1 r A r +1 r 2 A θ θ + k 2 A =0 and we may impose the boundary condition that A vanish if r = a, thus A =0. The method of separation of variables leads to solutions of the form A = R Θ
Helmholtz equation
–
Two sources of radiation in the plane, given mathematically by a function ƒ which is zero in the blue region.
17.
Phasor
–
In physics and engineering, a phasor, is a complex number representing a sinusoidal function whose amplitude, angular frequency, and initial phase are time-invariant. The complex constant, which encapsulates amplitude and phase dependence, is known as phasor, complex amplitude, a common situation in electrical networks is the existence of multiple sinusoids all with the same frequency, but different amplitudes and phases. The only difference in their analytic representations is the complex amplitude, a linear combination of such functions can be factored into the product of a linear combination of phasors and the time/frequency dependent factor that they all have in common. The origin of the term phasor rightfully suggests that a somewhat similar to that possible for vectors is possible for phasors as well. The originator of the phasor transform was Charles Proteus Steinmetz working at General Electric in the late 19th century, however, the Laplace transform is mathematically more difficult to apply and the effort may be unjustified if only steady state analysis is required. The function A ⋅ e i is the representation of A ⋅ cos . Figure 2 depicts it as a vector in a complex plane. It is sometimes convenient to refer to the function as a phasor. But the term usually implies just the static vector, A e i θ. An even more compact representation of a phasor is the angle notation, multiplication of the phasor A e i θ e i ω t by a complex constant, B e i ϕ, produces another phasor. That means its only effect is to change the amplitude and phase of the underlying sinusoid, Re = Re = A B cos In electronics, B e i ϕ would represent an impedance, in particular it is not the shorthand notation for another phasor. Multiplying a phasor current by an impedance produces a phasor voltage, but the product of two phasors would represent the product of two sinusoids, which is a non-linear operation that produces new frequency components. Phasor notation can only represent systems with one frequency, such as a linear system stimulated by a sinusoid, the time derivative or integral of a phasor produces another phasor. For example, Re = Re = Re = Re = ω A ⋅ cos Therefore, in phasor representation, similarly, integrating a phasor corresponds to multiplication by 1 i ω = e − i π /2 ω. The time-dependent factor, e i ω t, is unaffected, when we solve a linear differential equation with phasor arithmetic, we are merely factoring e i ω t out of all terms of the equation, and reinserting it into the answer. In polar coordinate form, it is,11 +2 ⋅ e − i ϕ, Therefore, v C =11 +2 ⋅ V P cos The sum of multiple phasors produces another phasor. A key point is that A3 and θ3 do not depend on ω or t, the time and frequency dependence can be suppressed and re-inserted into the outcome as long as the only operations used in between are ones that produce another phasor. In angle notation, the operation shown above is written, A1 ∠ θ1 + A2 ∠ θ2 = A3 ∠ θ3, another way to view addition is that two vectors with coordinates and are added vectorially to produce a resultant vector with coordinates
Phasor
–
An example of series
RLC circuit and respective phasor diagram for a specific ω
18.
Imaginary unit
–
The imaginary unit or unit imaginary number is a solution to the quadratic equation x2 +1 =0. The term imaginary is used there is no real number having a negative square. There are two square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero, which has one double square root. In contexts where i is ambiguous or problematic, j or the Greek ι is sometimes used, in the disciplines of electrical engineering and control systems engineering, the imaginary unit is normally denoted by j instead of i, because i is commonly used to denote electric current. For the history of the unit, see Complex number § History. The imaginary number i is defined solely by the property that its square is −1, with i defined this way, it follows directly from algebra that i and −i are both square roots of −1. In polar form, i is represented as 1eiπ/2, having a value of 1. In the complex plane, i is the point located one unit from the origin along the imaginary axis, more precisely, once a solution i of the equation has been fixed, the value −i, which is distinct from i, is also a solution. Since the equation is the definition of i, it appears that the definition is ambiguous. However, no ambiguity results as long as one or other of the solutions is chosen and labelled as i and this is because, although −i and i are not quantitatively equivalent, there is no algebraic difference between i and −i. Both imaginary numbers have equal claim to being the number whose square is −1, the issue can be a subtle one. See also Complex conjugate and Galois group, a more precise explanation is to say that the automorphism group of the special orthogonal group SO has exactly two elements — the identity and the automorphism which exchanges CW and CCW rotations. All these ambiguities can be solved by adopting a rigorous definition of complex number. For example, the pair, in the usual construction of the complex numbers with two-dimensional vectors. The imaginary unit is sometimes written √−1 in advanced mathematics contexts, however, great care needs to be taken when manipulating formulas involving radicals. The radical sign notation is reserved either for the square root function. Similarly,1 i =1 −1 =1 −1 = −11 = −1 = i, the calculation rules a ⋅ b = a ⋅ b and a b = a b are only valid for real, non-negative values of a and b. These problems are avoided by writing and manipulating expressions like i√7, for a more thorough discussion, see Square root and Branch point
Imaginary unit
–
i in the
complex or
cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis
19.
Radian
–
The radian is the standard unit of angular measure, used in many areas of mathematics. The length of an arc of a circle is numerically equal to the measurement in radians of the angle that it subtends. The unit was formerly an SI supplementary unit, but this category was abolished in 1995, separately, the SI unit of solid angle measurement is the steradian. The radian is represented by the symbol rad, so for example, a value of 1.2 radians could be written as 1.2 rad,1.2 r,1. 2rad, or 1. 2c. Radian describes the angle subtended by a circular arc as the length of the arc divided by the radius of the arc. One radian is the angle subtended at the center of a circle by an arc that is equal in length to the radius of the circle. Conversely, the length of the arc is equal to the radius multiplied by the magnitude of the angle in radians. As the ratio of two lengths, the radian is a number that needs no unit symbol, and in mathematical writing the symbol rad is almost always omitted. When quantifying an angle in the absence of any symbol, radians are assumed, and it follows that the magnitude in radians of one complete revolution is the length of the entire circumference divided by the radius, or 2πr / r, or 2π. Thus 2π radians is equal to 360 degrees, meaning that one radian is equal to 180/π degrees, the concept of radian measure, as opposed to the degree of an angle, is normally credited to Roger Cotes in 1714. He described the radian in everything but name, and he recognized its naturalness as a unit of angular measure, the idea of measuring angles by the length of the arc was already in use by other mathematicians. For example, al-Kashi used so-called diameter parts as units where one part was 1/60 radian. The term radian first appeared in print on 5 June 1873, in examination questions set by James Thomson at Queens College, Belfast. He had used the term as early as 1871, while in 1869, Thomas Muir, then of the University of St Andrews, in 1874, after a consultation with James Thomson, Muir adopted radian. As stated, one radian is equal to 180/π degrees, thus, to convert from radians to degrees, multiply by 180/π. The length of circumference of a circle is given by 2 π r, so, to convert from radians to gradians multiply by 200 / π, and to convert from gradians to radians multiply by π /200. This is because radians have a mathematical naturalness that leads to a more elegant formulation of a number of important results, most notably, results in analysis involving trigonometric functions are simple and elegant when the functions arguments are expressed in radians. Because of these and other properties, the trigonometric functions appear in solutions to problems that are not obviously related to the functions geometrical meanings
Radian
–
A chart to convert between degrees and radians
Radian
–
An arc of a
circle with the same length as the
radius of that circle corresponds to an angle of 1 radian. A full circle corresponds to an angle of 2
π radians.
20.
Real part
–
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0
Real part
–
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an
Argand diagram, representing the
complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i is the
imaginary unit which satisfies i 2 = −1.
21.
Magnetic field
–
A magnetic field is the magnetic effect of electric currents and magnetic materials. The magnetic field at any point is specified by both a direction and a magnitude, as such it is represented by a vector field. The term is used for two distinct but closely related fields denoted by the symbols B and H, where H is measured in units of amperes per meter in the SI, B is measured in teslas and newtons per meter per ampere in the SI. B is most commonly defined in terms of the Lorentz force it exerts on moving electric charges, Magnetic fields can be produced by moving electric charges and the intrinsic magnetic moments of elementary particles associated with a fundamental quantum property, their spin. In quantum physics, the field is quantized and electromagnetic interactions result from the exchange of photons. Magnetic fields are used throughout modern technology, particularly in electrical engineering. The Earth produces its own field, which is important in navigation. Rotating magnetic fields are used in electric motors and generators. Magnetic forces give information about the carriers in a material through the Hall effect. The interaction of magnetic fields in electric devices such as transformers is studied in the discipline of magnetic circuits, noting that the resulting field lines crossed at two points he named those points poles in analogy to Earths poles. He also clearly articulated the principle that magnets always have both a north and south pole, no matter how finely one slices them, almost three centuries later, William Gilbert of Colchester replicated Petrus Peregrinus work and was the first to state explicitly that Earth is a magnet. Published in 1600, Gilberts work, De Magnete, helped to establish magnetism as a science, in 1750, John Michell stated that magnetic poles attract and repel in accordance with an inverse square law. Charles-Augustin de Coulomb experimentally verified this in 1785 and stated explicitly that the north and south poles cannot be separated, building on this force between poles, Siméon Denis Poisson created the first successful model of the magnetic field, which he presented in 1824. In this model, a magnetic H-field is produced by magnetic poles, three discoveries challenged this foundation of magnetism, though. First, in 1819, Hans Christian Ørsted discovered that an electric current generates a magnetic field encircling it, then in 1820, André-Marie Ampère showed that parallel wires having currents in the same direction attract one another. Finally, Jean-Baptiste Biot and Félix Savart discovered the Biot–Savart law in 1820, extending these experiments, Ampère published his own successful model of magnetism in 1825. This has the benefit of explaining why magnetic charge can not be isolated. Also in this work, Ampère introduced the term electrodynamics to describe the relationship between electricity and magnetism, in 1831, Michael Faraday discovered electromagnetic induction when he found that a changing magnetic field generates an encircling electric field
Magnetic field
–
One of the first drawings of a magnetic field, by
René Descartes, 1644. It illustrated his theory that magnetism was caused by the circulation of tiny helical particles, "threaded parts", through threaded pores in magnets.
Magnetic field
–
Magnetic field of an ideal cylindrical
magnet with its axis of symmetry inside the image plane. The magnetic field is represented by
magnetic field lines, which show the direction of the field at different points.
Magnetic field
–
Hans Christian Ørsted, Der Geist in der Natur, 1854
22.
Poynting vector
–
In physics, the Poynting vector represents the directional energy flux density of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre and it is named after its discoverer John Henry Poynting who first derived it in 1884. Oliver Heaviside and Nikolay Umov also independently discovered the Poynting vector and this expression is often called the Abraham form. The Poynting vector is denoted by S or N. In the microscopic version of Maxwells equations, this definition must be replaced by a definition in terms of the electric field E and the magnetic field B. It is also possible to combine the electric displacement field D with the magnetic field B to get the Minkowski form of the Poynting vector, or use D and H to construct yet another version. The choice has been controversial, Pfeifer et al. summarize, the Poynting vector represents the particular case of an energy flux vector for electromagnetic energy. However, any type of energy has its direction of movement in space, as well as its density, so energy flux vectors can be defined for types of energy as well. The Umov–Poynting vector discovered by Nikolay Umov in 1874 describes energy flux in liquid, in this definition, bound electrical currents are not included in this term, and instead contribute to S and u. For linear, nondispersive and isotropic materials, the relations can be written as D = ε E, H =1 μ B. Here ε and μ are scalar, real-valued constants independent of position, direction, in principle, this limits Poyntings theorem in this form to fields in vacuum and nondispersive linear materials. A generalization to dispersive materials is possible under certain circumstances at the cost of additional terms, the microscopic version of Maxwells equations admits only the fundamental fields E and B, without a built-in model of material media. Only the vacuum permittivity and permeability are used, and there is no D or H. When this model is used, the Poynting vector is defined as S =1 μ0 E × B and it can be derived directly from Maxwells equations in terms of total charge and current and the Lorentz force law only. The two alternative definitions of the Poynting vector are equal in vacuum or in non-magnetic materials, where B = μ0H and this is especially true for the electromagnetic energy density, in contrast to the macroscopic form E × H. The above form for the Poynting vector represents the power flow due to instantaneous electric and magnetic fields. More commonly, problems in electromagnetics are solved in terms of sinusoidally varying fields at a specified frequency, the results can then be applied more generally, for instance, by representing incoherent radiation as a superposition of such waves at different frequencies and with fluctuating amplitudes. We would thus not be considering the instantaneous E and H used above, note that these complex amplitude vectors are not functions of time, as they are understood to refer to oscillations over all time
Poynting vector
Poynting vector
–
Dipole radiation of a dipole vertically in the page showing electric field strength (colour) and Poynting vector (arrows) in the plane of the page.
23.
Complex conjugate
–
In mathematics, the complex conjugate of a complex number is the number with equal real part and imaginary part equal in magnitude but opposite in sign. For example, the conjugate of 3 + 4i is 3 − 4i. In polar form, the conjugate of ρ e i ϕ is ρ e − i ϕ and this can be shown using Eulers formula. Complex conjugates are important for finding roots of polynomials, according to the complex conjugate root theorem, if a complex number is a root to a polynomial in one variable with real coefficients, so is its conjugate. The complex conjugate of a number z is written as z ¯ or z ∗. The first notation avoids confusion with the notation for the transpose of a matrix. The second is preferred in physics, where dagger is used for the conjugate transpose, If a complex number is represented as a 2×2 matrix, the notations are identical. In some texts, the conjugate of a previous known number is abbreviated as c. c. A significant property of the conjugate is that a complex number is equal to its complex conjugate if its imaginary part is zero. The conjugate of the conjugate of a number z is z. The ultimate relation is the method of choice to compute the inverse of a number if it is given in rectangular coordinates. Exp = exp ¯ log = log ¯ if z is non-zero If p is a polynomial with real coefficients, thus, non-real roots of real polynomials occur in complex conjugate pairs. In general, if ϕ is a function whose restriction to the real numbers is real-valued. The map σ = z ¯ from C to C is a homeomorphism and antilinear, even though it appears to be a well-behaved function, it is not holomorphic, it reverses orientation whereas holomorphic functions locally preserve orientation. It is bijective and compatible with the operations, and hence is a field automorphism. As it keeps the real numbers fixed, it is an element of the Galois group of the field extension C / R and this Galois group has only two elements, σ and the identity on C. Thus the only two field automorphisms of C that leave the real numbers fixed are the identity map and complex conjugation. Similarly, for a fixed complex unit u = exp, the equation z − z 0 z ¯ − z 0 ¯ = u determines the line through z 0 in the direction of u
Complex conjugate
–
Geometric representation of z and its conjugate z̅ in the complex plane. The complex conjugate is found by
reflecting z across the real axis.
24.
FWHM
–
In other words, it is the width of a spectrum curve measured between those points on the y-axis which are half the maximum amplitude. Half width at half maximum is half of the FWHM, FWHM is applied to such phenomena as the duration of pulse waveforms and the spectral width of sources used for optical communications and the resolution of spectrometers. The term full duration at half maximum is preferred when the independent variable is time, in signal processing terms, this is at most −3 dB of attenuation, called half power point.355 σ. The width does not depend on the expected value x0, it is invariant under translations, in spectroscopy half the width at half maximum, HWHM, is in common use. For example, a Lorentzian/Cauchy distribution of height 1/πγ can be defined by f =1 π γ and F W H M =2 γ, another important distribution function, related to solitons in optics, is the hyperbolic secant, f = sech . Any translating element was omitted, since it does not affect the FWHM, for this impulse we have, F W H M =2 arsech X =2 ln X ≈2.634 X where arsech is the inverse hyperbolic secant. Gaussian function Cutoff frequency This article incorporates public domain material from the General Services Administration document Federal Standard 1037C
FWHM
–
full width at half maximum
25.
Rayleigh range
–
A related parameter is the confocal parameter, b, which is twice the Rayleigh length. The Rayleigh length is important when beams are modeled as Gaussian beams. This equation and those that assume that the waist is not extraordinarily small. The radius of the beam at a distance z from the waist is w = w 01 +2, the minimum value of w occurs at w = w 0, by definition. At distance z R from the beam waist, the radius is increased by a factor 2. The total angular spread of a Gaussian beam in radians is related to the Rayleigh length by Θ d i v ≃2 w 0 z R. The diameter of the beam at its waist is given by D =2 w 0 ≃4 λ π Θ d i v and these equations are valid within the limits of the paraxial approximation. For beams with much larger divergence the Gaussian beam model is no longer accurate and a physical optics analysis is required
Rayleigh range
–
Gaussian beam width as a function of the axial distance.: beam waist;: confocal parameter;: Rayleigh length;: total angular spread
26.
Full width at half maximum
–
In other words, it is the width of a spectrum curve measured between those points on the y-axis which are half the maximum amplitude. Half width at half maximum is half of the FWHM, FWHM is applied to such phenomena as the duration of pulse waveforms and the spectral width of sources used for optical communications and the resolution of spectrometers. The term full duration at half maximum is preferred when the independent variable is time, in signal processing terms, this is at most −3 dB of attenuation, called half power point.355 σ. The width does not depend on the expected value x0, it is invariant under translations, in spectroscopy half the width at half maximum, HWHM, is in common use. For example, a Lorentzian/Cauchy distribution of height 1/πγ can be defined by f =1 π γ and F W H M =2 γ, another important distribution function, related to solitons in optics, is the hyperbolic secant, f = sech . Any translating element was omitted, since it does not affect the FWHM, for this impulse we have, F W H M =2 arsech X =2 ln X ≈2.634 X where arsech is the inverse hyperbolic secant. Gaussian function Cutoff frequency This article incorporates public domain material from the General Services Administration document Federal Standard 1037C
Full width at half maximum
–
full width at half maximum
27.
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 in either along or decentered from the local optical axis. The vertex of the 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, thus when viewing a biconvex lens from the side, the left surface radius of curvature is positive, and the right surface has a negative radius of curvature. Note however that in areas of other than design, other sign conventions are sometimes used. In particular, many physics textbooks use an alternate sign convention in which convex surfaces of lenses are always positive. Care should be taken when using formulas taken from different sources, optical surfaces with non-spherical profiles, such as the surfaces of aspheric lenses, also have a radius of curvature. If α1 and α2 are zero, then 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
Radius of curvature (optics)
–
Radius of curvature sign convention for optical design
28.
Louis Georges Gouy
–
Louis Georges Gouy was a French physicist who was born at Vals-les-Bains, Ardèche in 1854 and died January 271926. He is the namesake of the Gouy balance, the Gouy-Chapman electric double layer model and he became a correspondent of the Académie des sciences in 1901, and a member in 1913. His principal scientific work was related to the subjects, The propagation velocity of light waves in dispersive media. Propagation of spherical waves of small radius, G. Gouy La Nature n°2708 du 27 février 1926 A Sella, Gouys Balance, Chemistry World, December 2010
Louis Georges Gouy
–
Louis Georges Gouy
Louis Georges Gouy
–
Solvay conference of 1913. Louis Georges Gouy is in the first row on the right
29.
Diffraction
–
Diffraction refers to various phenomena that occur when a wave encounters an obstacle or a slit. It is defined as the bending of light around the corners of an obstacle or aperture into the region of shadow of the obstacle. In classical physics, the phenomenon is described as the interference of waves according to the Huygens–Fresnel principle. These characteristic behaviors are exhibited when a wave encounters an obstacle or a slit that is comparable in size to its wavelength. Similar effects occur when a wave travels through a medium with a varying refractive index. Diffraction occurs with all waves, including sound waves, water waves, since physical objects have wave-like properties, diffraction also occurs with matter and can be studied according to the principles of quantum mechanics. Italian scientist Francesco Maria Grimaldi coined the word diffraction and was the first to record observations of the phenomenon in 1660. If the obstructing object provides multiple, closely spaced openings, a pattern of varying intensity can result. This is due to the addition, or interference, of different parts of a wave that travel to the observer by different paths, the formalism of diffraction can also describe the way in which waves of finite extent propagate in free space. For example, the profile of a laser beam, the beam shape of a radar antenna. The effects of diffraction are often seen in everyday life and this principle can be extended to engineer a grating with a structure such that it will produce any diffraction pattern desired, the hologram on a credit card is an example. Diffraction in the atmosphere by small particles can cause a ring to be visible around a bright light source like the sun or the moon. A shadow of an object, using light from a compact source. The speckle pattern which is observed when laser light falls on a rough surface is also a diffraction phenomenon. When deli meat appears to be iridescent, that is diffraction off the meat fibers, all these effects are a consequence of the fact that light propagates as a wave. Diffraction can occur with any kind of wave, ocean waves diffract around jetties and other obstacles. Sound waves can diffract around objects, which is why one can hear someone calling even when hiding behind a tree. Diffraction can also be a concern in some applications, it sets a fundamental limit to the resolution of a camera, telescope
Diffraction
–
Diffraction pattern of red
laser beam made on a plate after passing a small circular hole in another plate
Diffraction
–
Solar glory at the
steam from
hot springs. A glory is an optical phenomenon produced by light backscattered (a combination of diffraction,
reflection and
refraction) towards its source by a cloud of uniformly sized water droplets.
Diffraction
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Photograph of single-slit diffraction in a circular
ripple tank
Diffraction
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2-slit (top) and 5-slit diffraction of red laser light
30.
Fourier transform
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The Fourier transform decomposes a function of time into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies of its constituent notes. The Fourier transform is called the frequency domain representation of the original signal, the term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. The Fourier transform is not limited to functions of time, but in order to have a unified language, linear operations performed in one domain have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the domain corresponds to multiplication by the frequency. Also, convolution in the domain corresponds to ordinary multiplication in the frequency domain. Concretely, this means that any linear time-invariant system, such as a filter applied to a signal, after performing the desired operations, transformation of the result can be made back to the time domain. Functions that are localized in the domain have Fourier transforms that are spread out across the frequency domain and vice versa. The Fourier transform of a Gaussian function is another Gaussian function, Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. The Fourier transform can also be generalized to functions of variables on Euclidean space. In general, functions to which Fourier methods are applicable are complex-valued, the latter is routinely employed to handle periodic functions. The fast Fourier transform is an algorithm for computing the DFT, the Fourier transform of the function f is traditionally denoted by adding a circumflex, f ^. There are several conventions for defining the Fourier transform of an integrable function f, ℝ → ℂ. Here we will use the definition, f ^ = ∫ − ∞ ∞ f e −2 π i x ξ d x. When the independent variable x represents time, the transform variable ξ represents frequency. Under suitable conditions, f is determined by f ^ via the inverse transform, f = ∫ − ∞ ∞ f ^ e 2 π i ξ x d ξ, the functions f and f ^ often are referred to as a Fourier integral pair or Fourier transform pair. For other common conventions and notations, including using the angular frequency ω instead of the frequency ξ, see Other conventions, the Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum. Many other characterizations of the Fourier transform exist, for example, one uses the Stone–von Neumann theorem, the Fourier transform is the unique unitary intertwiner for the symplectic and Euclidean Schrödinger representations of the Heisenberg group. In 1822, Joseph Fourier showed that some functions could be written as an sum of harmonics
Fourier transform