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# Fourier optics

Fourier optics is the study of classical optics using Fourier transforms (FTs), in which the wave is regarded as a superposition of plane waves that are not related to any identifiable sources; instead they are the natural modes of the propagation medium itself. 

94 relations: Abbe sine condition, Adaptive-additive algorithm, Addison-Wesley, Airy disk, Airy function, Artech House, Bloch wave, Bloch wave – MoM method, Cartesian coordinate system, Causality, Chebyshev polynomials, Color, Coma (optics), Complex number, Computer-generated holography, Convolution, Convolution theorem, Cross-correlation, Cylindrical coordinate system, Diffraction-limited system, Digital image processing, Dirac delta function, Double-slit experiment, Eigenfunction, Eigenvalues and eigenvectors, Electromagnetic radiation, Electromagnetic wave equation, Euclidean distance, Evanescent field, Fourier analysis, Fourier transform, Fraunhofer diffraction, Frequency, Frequency domain, Fresnel diffraction, Frits Zernike, Functional decomposition, Gibbs phenomenon, Green's function, Harmonic, Hartley transform, Helmholtz equation, Hermite polynomials, Hilbert space, Huygens–Fresnel principle, IBM Roadrunner, Imaginary unit, Interferometry, John Wiley & Sons, Julius Adams Stratton, ... Expand index (44 more) »

## Abbe sine condition

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.

In the studies of Fourier optics, sound synthesis, stellar interferometry, optical tweezers, and diffractive optical elements (DOEs) it is often important to know the spatial frequency phase of an observed wave source.

Addison-Wesley is a publisher of textbooks and computer literature.

## Airy disk

In optics, the Airy disk (or Airy disc) and Airy pattern are descriptions of the best focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light.

## Airy function

In the physical sciences, the Airy function (or Airy function of the first kind) Ai(x) is a special function named after the British astronomer George Biddell Airy (1801–92).

## Artech House

Artech House (a.k.a. Artech House Publishers) is a publisher of professional scientific and engineering books.

## Bloch wave

A Bloch wave (also called Bloch state or Bloch function or Bloch wavefunction), named after Swiss physicist Felix Bloch, is a type of wavefunction for a particle in a periodically-repeating environment, most commonly an electron in a crystal.

## Bloch wave – MoM method

Bloch wave – MoM is a first principles technique for determining the photonic band structure of triply-periodic electromagnetic media such as photonic crystals.

## Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

## Causality

Causality (also referred to as causation, or cause and effect) is what connects one process (the cause) with another process or state (the effect), where the first is partly responsible for the second, and the second is partly dependent on the first.

## Chebyshev polynomials

In mathematics the Chebyshev polynomials, named after Pafnuty Chebyshev, are a sequence of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively.

## Color

Color (American English) or colour (Commonwealth English) is the characteristic of human visual perception described through color categories, with names such as red, orange, yellow, green, blue, or purple.

## Coma (optics)

In optics (especially telescopes), the coma, or comatic aberration, in an optical system refers to aberration inherent to certain optical designs or due to imperfection in the lens or other components that results in off-axis point sources such as stars appearing distorted, appearing to have a tail (coma) like a comet.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Computer-generated holography

Computer-generated holography (CGH) is the method of digitally generating holographic interference patterns.

## Convolution

In mathematics (and, in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated.

## Convolution theorem

In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the pointwise product of Fourier transforms.

## Cross-correlation

In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other.

## Cylindrical coordinate system

A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis.

## Diffraction-limited system

The resolution of an optical imaging system a microscope, telescope, or camera can be limited by factors such as imperfections in the lenses or misalignment.

## Digital image processing

In computer science, Digital image processing is the use of computer algorithms to perform image processing on digital images.

## Dirac delta function

In mathematics, the Dirac delta function (function) is a generalized function or distribution introduced by the physicist Paul Dirac.

## Double-slit experiment

In modern physics, the double-slit experiment is a demonstration that light and matter can display characteristics of both classically defined waves and particles; moreover, it displays the fundamentally probabilistic nature of quantum mechanical phenomena.

## Eigenfunction

In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.

## Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.

In physics, electromagnetic radiation (EM radiation or EMR) refers to the waves (or their quanta, photons) of the electromagnetic field, propagating (radiating) through space-time, carrying electromagnetic radiant energy.

## Electromagnetic wave equation

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum.

## Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.

## Evanescent field

In electromagnetics, an evanescent field, or evanescent wave, is an oscillating electric and/or magnetic field that does not propagate as an electromagnetic wave but whose energy is spatially concentrated in the vicinity of the source (oscillating charges and currents).

## Fourier analysis

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

## Fourier transform

The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.

## Fraunhofer diffraction

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.

## Frequency

Frequency is the number of occurrences of a repeating event per unit of time.

## Frequency domain

In electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time.

## Fresnel diffraction

In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near field.

## Frits Zernike

Frits Zernike (16 July 1888 – 10 March 1966) was a Dutch physicist and winner of the Nobel Prize for physics in 1953 for his invention of the phase-contrast microscope.

## Functional decomposition

In mathematics, functional decomposition is the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed (i.e., recomposed) from those parts by function composition.

## Gibbs phenomenon

In mathematics, the Gibbs phenomenon, discovered by Available on-line at: and rediscovered by, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity.

## Green's function

In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential equation defined on a domain, with specified initial conditions or boundary conditions.

## Harmonic

A harmonic is any member of the harmonic series, a divergent infinite series.

## Hartley transform

In mathematics, the Hartley transform (HT) is an integral transform closely related to the Fourier transform (FT), but which transforms real-valued functions to real-valued functions.

## Helmholtz equation

In mathematics & physics, the Helmholtz equation, named for Hermann von Helmholtz, is the partial differential equation where &nabla;2 is the Laplacian, k is the wavenumber, and A is the amplitude.

## Hermite polynomials

In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.

## Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

## Huygens–Fresnel principle

The Huygens–Fresnel principle (named after Dutch physicist Christiaan Huygens and French physicist Augustin-Jean Fresnel) is a method of analysis applied to problems of wave propagation both in the far-field limit and in near-field diffraction.

Roadrunner was a supercomputer built by IBM for the Los Alamos National Laboratory in New Mexico, USA.

## Imaginary unit

The imaginary unit or unit imaginary number is a solution to the quadratic equation.

## Interferometry

Interferometry is a family of techniques in which waves, usually electromagnetic waves, are superimposed causing the phenomenon of interference in order to extract information.

## John Wiley & Sons

John Wiley & Sons, Inc., also referred to as Wiley, is a global publishing company that specializes in academic publishing.

## Julius Adams Stratton

Julius Adams Stratton (May 18, 1901 &ndash; June 22, 1994) was a U.S. electrical engineer and university administrator.

## Laplace operator

In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.

## Legendre polynomials

In mathematics, Legendre functions are solutions to Legendre's differential equation: They are named after Adrien-Marie Legendre.

## Lens (optics)

A lens is a transmissive optical device that focuses or disperses a light beam by means of refraction.

## Low-pass filter

A low-pass filter (LPF) is a filter that passes signals with a frequency lower than a certain cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency.

## Magnetic trap (atoms)

A magnetic trap is an apparatus which uses a magnetic field gradient to trap neutral particles with magnetic moments.

## Nyquist–Shannon sampling theorem

In the field of digital signal processing, the sampling theorem is a fundamental bridge between continuous-time signals (often called "analog signals") and discrete-time signals (often called "digital signals").

## Optical computing

Optical or photonic computing uses photons produced by lasers or diodes for computation.

## Optical correlator

An optical correlator is a device for comparing two signals by utilising the Fourier transforming properties of a lens.

## Optical fiber

An optical fiber or optical fibre is a flexible, transparent fiber made by drawing glass (silica) or plastic to a diameter slightly thicker than that of a human hair.

## Optical transfer function

The optical transfer function (OTF) of an optical system such as a camera, microscope, human eye, or projector specifies how different spatial frequencies are handled by the system.

## Optical tweezers

Optical tweezers (originally called "single-beam gradient force trap") are scientific instruments that use a highly focused laser beam to provide an attractive or repulsive force (typically on the order of piconewtons), depending on the relative refractive index between particle and surrounding medium, to physically hold and move microscopic objects similar to tweezers.

## 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.

## Orthogonal coordinates

In mathematics, orthogonal coordinates are defined as a set of d coordinates q.

## Orthogonal functions

In mathematics, orthogonal functions belong to a function space which is a vector space that has a bilinear form.

## Paraxial approximation

In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens).

## Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

## Phase (waves)

Phase is the position of a point in time (an instant) on a waveform cycle.

## Phase-contrast microscopy

Phase-contrast microscopy is an optical microscopy technique that converts phase shifts in light passing through a transparent specimen to brightness changes in the image.

## Plane wave

In the physics of wave propagation, a plane wave (also spelled planewave) is a wave whose wavefronts (surfaces of constant phase) are infinite parallel planes.

## Point spread function

The point spread function (PSF) describes the response of an imaging system to a point source or point object.

## Quantum computing

Quantum computing is computing using quantum-mechanical phenomena, such as superposition and entanglement.

## Quantum optics

Quantum optics (QO) is a field of research that uses semi-classical and quantum-mechanical physics to investigate phenomena involving light and its interactions with matter at submicroscopic levels.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Separation of variables

In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.

## Signal processing

Signal processing concerns the analysis, synthesis, and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound, images, and biological measurements.

## Sinc function

In mathematics, physics and engineering, the cardinal sine function or sinc function, denoted by, has two slightly different definitions.

## Soliton (optics)

In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium.

## Spatial filter

A spatial filter is an optical device which uses the principles of Fourier optics to alter the structure of a beam of light or other electromagnetic radiation, typically coherent laser light.

## Spatial frequency

In mathematics, physics, and engineering, spatial frequency is a characteristic of any structure that is periodic across position in space.

## Spherical coordinate system

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.

## Stationary phase approximation

In mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to oscillatory integrals taken over n-dimensional space ℝn where i is the imaginary unit.

## Tacoma Narrows Bridge (1940)

The 1940 Tacoma Narrows Bridge, the first Tacoma Narrows Bridge, was a suspension bridge in the U.S. state of Washington that spanned the Tacoma Narrows strait of Puget Sound between Tacoma and the Kitsap Peninsula.

## Time in physics

Time in physics is defined by its measurement: time is what a clock reads.

## Wave

In physics, a wave is a disturbance that transfers energy through matter or space, with little or no associated mass transport.

## Wave equation

The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves.

## Wave function

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system.

## Wave vector

In physics, a wave vector (also spelled wavevector) is a vector which helps describe a wave.

## Waveguide (electromagnetism)

In electromagnetics and communications engineering, the term waveguide may refer to any linear structure that conveys electromagnetic waves between its endpoints.

## Wavelength

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.

## Wavelet

A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero.

## Wavenumber

In the physical sciences, the wavenumber (also wave number or repetency) is the spatial frequency of a wave, measured in cycles per unit distance or radians per unit distance.

## Whittaker–Shannon interpolation formula

The Whittaker–Shannon interpolation formula or sinc interpolation is a method to construct a continuous-time bandlimited function from a sequence of real numbers.

## Window function

In signal processing, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen interval.

## Zernike polynomials

In mathematics, the Zernike polynomials are a sequence of polynomials that are orthogonal on the unit disk.

## References

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