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Fresnel Diffraction
In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff's diffraction formula, Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near and far field, near field. It is used to calculate the diffraction pattern created by waves passing through an aperture or around an object, when viewed from relatively close to the object. In contrast the diffraction pattern in the near and far field, far field region is given by the Fraunhofer diffraction equation. The near field can be specified by the Fresnel number, , of the optical arrangement. When F \ll 1 the diffracted wave is considered to be in the Fraunhofer field. However, the validity of the Fresnel diffraction integral is deduced by the approximations derived below. Specifically, the phase terms of third order and higher must be negligible, a condition that may be written as \frac \ll 1, where \theta is the maximal angle described by \theta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffraction
Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation, propagating wave. Diffraction is the same physical effect as Wave interference, interference, but interference is typically applied to superposition of a few waves and the term diffraction is used when many waves are superposed. Italian scientist Francesco Maria Grimaldi coined the word ''diffraction'' and was the first to record accurate observations of the phenomenon in 1660 in science, 1660. In classical physics, the diffraction phenomenon is described by the Huygens–Fresnel principle that treats each point in a propagating wavefront as a collection of individual spherical wavelets. The characteristic pattern is most pronounced when a wave from a Coherence (physics), coherent source (such as a laser) encounters a slit/aperture tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shear Mapping
In plane geometry, a shear mapping is an affine transformation that displaces each point in a fixed direction by an amount proportional to its signed distance function, signed distance from a given straight line, line parallel (geometry), parallel to that direction. This type of mapping is also called shear transformation, transvection, or just shearing. The transformations can be applied with a shear matrix or transvection, an elementary matrix that represents the Elementary row operations#Row-addition transformations, addition of a multiple of one row or column to another. Such a matrix (mathematics), matrix may be derived by taking the identity matrix and replacing one of the zero elements with a non-zero value. An example is the linear map that takes any point with Cartesian coordinates, coordinates (x,y) to the point (x + 2y,y). In this case, the displacement is horizontal by a factor of 2 where the fixed line is the -axis, and the signed distance is the -coordinate. Not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Canonical Transformation
In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the action of the special linear group SL2(C) on the time–frequency plane (domain). As this defines the original function up to a sign, this translates into an action of its double cover on the original function space. The LCT generalizes the Fourier, fractional Fourier, Laplace, Gauss–Weierstrass, Bargmann and the Fresnel transforms as particular cases. The name "linear canonical transformation" is from canonical transformation, a map that preserves the symplectic structure, as SL2(R) can also be interpreted as the symplectic group Sp2, and thus LCTs are the linear maps of the time–frequency domain which preserve the symplectic form, and their action on the Hilbert space is given by the Metaplectic group. The basic properties of ... [...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] |
Impulse Response
In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reaction of any dynamic system in response to some external change. In both cases, the impulse response describes the reaction of the system as a function of time (or possibly as a function of some other independent variable that parameterizes the dynamic behavior of the system). In all these cases, the dynamic system and its impulse response may be actual physical objects, or may be mathematical systems of equations describing such objects. Since the impulse function contains all frequencies (see the Fourier transform of the Dirac delta function, showing infinite frequency bandwidth that the Dirac delta function has), the impulse response defines the response of a linear time-invariant system for all frequencies. Mathematical considerat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The term ''convolution'' refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see #Properties, commutativity). Graphically, it expresses how the 'shape' of one function is modified by the other. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution f*g differs from cross-correlation f \star g only in that either f(x) or g(x) is reflected about the y-axis in convolution; thus i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase (waves)
In physics and mathematics, the phase (symbol φ or ϕ) of a wave or other periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is expressed in such a scale that it varies by one full turn as the variable t goes through each period (and F(t) goes through each complete cycle). It may be measured in any angular unit such as degrees or radians, thus increasing by 360° or 2\pi as the variable t completes a full period. This convention is especially appropriate for a sinusoidal function, since its value at any argument t then can be expressed as \varphi(t), the sine of the phase, multiplied by some factor (the amplitude of the sinusoid). (The cosine may be used instead of sine, depending on where one considers each period to start.) Usually, whole turns are ignored when expressing the phase; so that \varphi(t) is also a periodic function, with the same period as F, that repeatedly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavefront
In physics, the wavefront of a time-varying ''wave field (physics), field'' is the set (locus (mathematics), locus) of all point (geometry), points having the same ''phase (waves), phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal frequency (otherwise the phase is not well defined). Wavefronts usually move with time. For waves propagating in a dimension (mathematics), unidimensional medium, the wavefronts are usually single points; they are curves in a two dimensional medium, and surface (mathematics), surfaces in a three-dimensional one. For a sinusoidal plane wave, the wavefronts are planes perpendicular to the direction of propagation, that move in that direction together with the wave. For a sinusoidal spherical wave, the wavefronts are spherical surfaces that expand with it. If the speed of propagation is different at different points of a wavefront, the shape and/or orientation of the wavefro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fresnel To Fraunhofer Transition
Augustin-Jean Fresnel (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton's corpuscular theory, from the late 1830s until the end of the 19th century. He is perhaps better known for inventing the catadioptric (reflective/refractive) Fresnel lens and for pioneering the use of "stepped" lenses to extend the visibility of lighthouses, saving countless lives at sea. The simpler dioptric (purely refractive) stepped lens, first proposed by Count Buffon and independently reinvented by Fresnel, is used in screen magnifiers and in condenser lenses for overhead projectors. By expressing Huygens's principle of secondary waves and Young's principle of interference in quantitative terms, and supposing that simple colors consist of sinusoidal waves, Fresnel gave the first satisfactory explanation of diffraction by straight edges, includin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar Wave Approximation
Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as a real number **Lorentz scalar, a quantity in the theory of relativity which is invariant under a Lorentz transformation **Pseudoscalar, a quantity that behaves like a scalar, except that it changes sign under a parity inversion *Scalar (computing), any non-composite value *Scalar boson, in physics, a boson subatomic particle whose spin equals zero See also *dot product, also known as scalar product *dimensionless quantity, also known as scalar quantity *Inner product space *Scalar field *Scale (music) *Scaler (other) *''Pterophyllum scalare'' (Lichtenstein, 1823), a species of freshwater angelfish * Scala (other) Scala or SCALA may refer to: Automobiles * Renault Scala, multiple automobile models * Škoda Scala, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comparison Sommerfeld-Fresnel-Fraunhofer
Comparison or comparing is the act of evaluating two or more things by determining the relevant, comparable characteristics of each thing, and then determining which characteristics of each are similar to the other, which are different, and to what degree. Where characteristics are different, the differences may then be evaluated to determine which thing is best suited for a particular purpose. The description of similarities and differences found between the two things is also called a comparison. Comparison can take many distinct forms, varying by field: To compare things, they must have characteristics that are similar enough in relevant ways to merit comparison. If two things are too different to compare in a useful way, an attempt to compare them is colloquially referred to in English as "comparing apples and oranges." Comparison is widely used in society, in science and the arts. General usage Comparison is a natural activity, which even animals engage in when decidin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
