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Fraunhofer Diffraction (mathematics)
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens. The equation was named in honour of Joseph von Fraunhofer although he was not actually involved in the development of the theory. This article gives the equation in various mathematical forms, and provides detailed calculations of the Fraunhofer diffraction pattern for several different forms of diffracting apertures, specially for normally incident monochromatic plane wave. A qualitative discussion of Fraunhofer diffraction can be found elsewhere. Definition When a beam of light is partly blocked by an obstacle, some of the light is scattered around the object, and light and dark bands are often seen at the edge of the shadow – this effect is known as diffraction. The Kirchhoff diffraction equation provides an expression, derived fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics 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 by using the classical electromagnetic 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 and bend when they pass through or reflect from surfaces. Physical optics is a more comprehensive model of light, which includes wave effects such as diffraction and interference that cannot be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Vector
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a '' directed line segment'', or graphically as an arrow connecting an ''initial point'' ''A'' with a ''terminal point'' ''B'', and denoted by \overrightarrow . A vector is what is needed to "carry" the point ''A'' to the point ''B''; the Latin word ''vector'' means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from ''A'' to ''B''. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number : e^ = \cos x + i\sin x, where is the base of the natural logarithm, is the imaginary unit, and and are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted ("cosine plus i sine"). The formula is still valid if is a complex number, and so some authors refer to the more general complex version as Euler's formula. Euler's formula is ubiquitous in mathematics, physics, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". When , Euler's formula may be rewritten as , which is known as Euler's identity. History In 1714, the English mathematician Roger Cotes presented a geometrical ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Single Slit Diffraction (english)
Single may refer to: Arts, entertainment, and media * Single (music), a song release Songs * "Single" (Natasha Bedingfield song), 2004 * "Single" (New Kids on the Block and Ne-Yo song), 2008 * "Single" (William Wei song), 2016 * "Single", by Meghan Trainor from the album '' Only 17'' Sports * Single (baseball), the most common type of base hit * Single (cricket), point in cricket * Single (football), Canadian football point * Single-speed bicycle Transportation * Single-cylinder engine, an internal combustion engine design with one cylinder, or a motorcycle using such engine * Single (locomotive), a steam locomotive with a single pair of driving wheels * As a verb: to convert a double-track railway to a single-track railway Other uses * Single (mathematics) (1-tuple), a list or sequence with only one element * Single person, a person who is not in a committed relationship * Single precision, a computer numbering format that occupies one storage location in computer memory at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transforms
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. 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 space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - bi. The complex conjugate of z is often denoted as \overline or z^*. In polar form, the conjugate of r e^ is r e^. This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a^2 + b^2 (or r^2 in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root. Notation The complex conjugate of a complex number z is written as \overline z or z^*. The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where dagger (†) is used for the conjugate tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplitude (see below), which are all functions of the magnitude of the differences between the variable's extreme values. In older texts, the phase of a periodic function is sometimes called the amplitude. Definitions Peak amplitude & semi-amplitude For symmetric periodic waves, like sine waves, square waves or triangle waves ''peak amplitude'' and ''semi amplitude'' are the same. Peak amplitude In audio system measurements, telecommunications and others where the measurand is a signal that swings above and below a reference value but is not sinusoidal, peak amplitude is often used. If the reference is zero, this is the maximum absolute value of the signal; if the reference is a mean value (DC component), the peak amplitude is the maximu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase (waves)
In physics and mathematics, the phase of a 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 denoted \phi(t) and 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 \phi(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 \phi(t) is also a periodic function, with the same period as F, that repeatedly scans the same range of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Number
In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temporal frequency, which is defined as the number of wave cycles per unit time (''ordinary frequency'') or radians per unit time (''angular frequency''). 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 of wave scattering, such as X-ray diffraction, neutron diffraction, electron diffraction, and elementary particle physics. For quantum mechanical waves, the wavenumber multiplied by the reduced Planck's constant is the ''canonical momentum''. Wavenumber can be used to specify quantities other than spatial frequency. For example, in optical spectroscopy, it is often used a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane Wave
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, the value of such a field can be written as :F(\vec x,t) = G(\vec x \cdot \vec n, t), where \vec n is a unit-length vector, and G(d,t) is a function that gives the field's value as dependent on only two real parameters: the time t, and the scalar-valued displacement d = \vec x \cdot \vec n of the point \vec x along the direction \vec n. The displacement is constant over each plane perpendicular to \vec n. The values of the field F may be scalars, vectors, or any other physical or mathematical quantity. They can be complex numbers, as in a complex exponential plane wave. When the values of F are vectors, the wave is said to be a longitudinal wave if the vectors are always collinear with the vector \vec n, and a transverse wave if they ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hankel Transform
In mathematics, the Hankel transform expresses any given function ''f''(''r'') as the weighted sum of an infinite number of Bessel functions of the first kind . The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor ''k'' along the ''r'' axis. The necessary coefficient of each Bessel function in the sum, as a function of the scaling factor ''k'' constitutes the transformed function. The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval is related to the Fourier–Bessel series over a finite interval. Definition The Hankel transform of order \nu of a function ''f''(''r'') is given by : F_\nu(k) = \int_0^\infty f(r) J_\nu(kr) \,r\,\mathrmr, where J_\nu is the Bessel function of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel function is a generalizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
