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Abbe Sine Condition
In optics, the Abbe sine condition is a condition that must be fulfilled by a lens or other optical system in order for it to produce sharp images of off-axis as well as on-axis objects. It was formulated by Ernst Abbe in the context of microscopes. The Abbe sine condition says that the sine of the object-space angle \alpha_\mathrm should be proportional to the sine of the image space angle \alpha_\mathrm Furthermore, the ratio equals the magnification of the system multiplied by the ratio of refractive indices. In mathematical terms this is: \frac = \frac = \frac, M, where the variables (\alpha_\mathrm, \beta_\mathrm) are the angles (relative to the optic axis) of any two rays as they leave the object, and (\alpha_\mathrm, \beta_\mathrm) are the angles of the same rays where they reach the image plane (say, the film plane of a camera). For example, (\alpha_\mathrm, \alpha_\mathrm) might represent a paraxial ray (i.e., a ray nearly parallel with the optic axis), and (\beta_\m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sine Condition
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle \theta, the sine and cosine functions are denoted as \sin(\theta) and \cos(\theta). The definitions of sine and cosine have been extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves, the position and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microscopes
A microscope () is a laboratory instrument used to examine objects that are too small to be seen by the naked eye. Microscopy is the science of investigating small objects and structures using a microscope. Microscopic means being invisible to the eye unless aided by a microscope. There are many types of microscopes, and they may be grouped in different ways. One way is to describe the method an instrument uses to interact with a sample and produce images, either by sending a beam of light or electrons through a sample in its optical path, by detecting photon emissions from a sample, or by scanning across and a short distance from the surface of a sample using a probe. The most common microscope (and the first to be invented) is the optical microscope, which uses lenses to refract visible light that passed through a thinly sectioned sample to produce an observable image. Other major types of microscopes are the fluorescence microscope, electron microscope (both the t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometrical Optics
Geometrical optics, or ray optics, is a model of optics that describes light Wave propagation, propagation in terms of ''ray (optics), rays''. The ray in geometrical optics is an abstract object, abstraction useful for approximating the paths along which light propagates under certain circumstances. The simplifying assumptions of geometrical optics include that light rays: * propagate in straight-line paths as they travel in a Homogeneity (physics), homogeneous medium * bend, and in particular circumstances may split in two, at the Interface (matter), interface between two dissimilar optical medium, media * follow curved paths in a medium in which the refractive index changes * may be absorbed or reflected. Geometrical optics does not account for certain optical effects such as diffraction and Interference (wave propagation), interference, which are considered in physical optics. This simplification is useful in practice; it is an excellent approximation when the wavelength is smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herschel's Condition
In optics, the Herschel's condition is a condition for an optical system to produce sharp images for objects over an extended axial range, i.e. for objects displaced along the optical axis. It was formulated by John Herschel. Mathematical formulation The Herschel's condition in mathematical form is \frac = \frac = \frac, M_T, ^2 where \alpha_o,\beta_o are the object side ray angle, \alpha_i,\beta_i are the image side ray angle. n_o,n_i are the object and image side refractive index, and M_T is the transverse magnification. This condition can be derived by the Fermat's principle. This condition can also be expressed as M_L = \frac=\frac \text M_T = \frac where M_L = \fracM_T^2 is the longitudinal magnification. This condition is in general conflict with the Abbe sine condition, which is the condition for aberration free imaging for objects displaced off-axis. They can be simultaneously satisfied only when the system has magnification equal to the ratio of refractive index , M_T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Invariant
In optics the Lagrange invariant is a measure of the light propagating through an optical system. It is defined by :H = n\overliney - nu\overline, where and are the marginal ray height and angle respectively, and and are the chief ray height and angle. is the ambient refractive index. In order to reduce confusion with other quantities, the symbol may be used in place of . is proportional to the throughput of the optical system (related to étendue). For a given optical system, the Lagrange invariant is a constant throughout all space, that is, it is invariant upon refraction In physics, refraction is the redirection of a wave as it passes from one transmission medium, medium to another. The redirection can be caused by the wave's change in speed or by a change in the medium. Refraction of light is the most commo ... and transfer. The optical invariant is a generalization of the Lagrange invariant which is formed using the ray heights and angles of any two rays ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spherical Coordinate System
In mathematics, a spherical coordinate system specifies a given point in three-dimensional space by using a distance and two angles as its three coordinates. These are * the radial distance along the line connecting the point to a fixed point called the origin; * the polar angle between this radial line and a given ''polar axis''; and * the azimuthal angle , which is the angle of rotation of the radial line around the polar axis. (See graphic regarding the "physics convention".) Once the radius is fixed, the three coordinates (''r'', ''θ'', ''φ''), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle) is called the ''reference plane'' (sometimes '' fundamental plane''). Terminology The radial distance from the fixed point of origin is also called the ''radius'', or ''radial line'', or ''radial coor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavenumber
In the physical sciences, the wavenumber (or wave number), also known as repetency, is the spatial frequency of a wave. Ordinary wavenumber is defined as the number of wave cycles divided by length; it is a physical quantity with dimension of reciprocal length, expressed in SI units of cycles per metre or reciprocal metre (m−1). Angular wavenumber, defined as the wave phase divided by time, is a quantity with dimension of angle per length and SI units of radians per metre. They are analogous to temporal frequency, respectively the '' ordinary frequency'', defined as the number of wave cycles divided by time (in cycles per second or reciprocal seconds), and the ''angular frequency'', defined as the phase angle divided by time (in radians per second). In multidimensional systems, the wavenumber is the magnitude of the '' wave vector''. The space of wave vectors is called ''reciprocal space''. Wave numbers and wave vectors play an essential role in optics and the physics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Magnification
Magnification is the process of enlarging the apparent size, not physical size, of something. This enlargement is quantified by a size ratio called optical magnification. When this number is less than one, it refers to a reduction in size, sometimes called ''de-magnification''. Typically, magnification is related to scaling up visuals or images to be able to see more detail, increasing resolution, using microscope, printing techniques, or digital processing. In all cases, the magnification of the image does not change the perspective of the image. Examples of magnification Some optical instruments provide visual aid by magnifying small or distant subjects. * A magnifying glass, which uses a positive (convex) lens to make things look bigger by allowing the user to hold them closer to their eye. * A telescope, which uses its large objective lens or primary mirror to create an image of a distant object and then allows the user to examine the image closely with a smaller ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image Distortion
In geometric optics, distortion is a deviation from rectilinear projection; a projection in which straight lines in a scene remain straight in an image. It is a form of optical aberration that may be distinguished from other aberrations such as spherical aberration, coma, chromatic aberration, field curvature, and astigmatism in a sense that these impact the image sharpness without changing an object shape or structure in the image (e.g., a straight line in an object is still a straight line in the image although the image sharpness may be degraded by the mentioned aberrations) while distortion can change the object structure in the image (so named as distortion). Radial distortion Although distortion can be irregular or follow many patterns, the most commonly encountered distortions are radially symmetric, or approximately so, arising from the symmetry of a photographic lens. These ''radial distortions'' can usually be classified as either ''barrel'' distortions or '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imaginary Unit
The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of in a complex number is Imaginary numbers are an important mathematical concept; they extend the real number system \mathbb to the complex number system \mathbb, in which at least one Root of a function, root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra). Here, the term ''imaginary'' is used because there is no real number having a negative square (algebra), square. There are two complex square roots of and , just as there are two complex square roots of every real number other than zero (which has one multiple root, double square root). In contexts in which use of the letter is ambiguous or problematic, the le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
