NationMaster - Encyclopedia: Fourier optics

Fourier optics is one of the three major viewpoints for understanding classical optics, the other two being the diffraction integral viewpoint and geometrical optics. Fourier optics has its origins in the plane wave spectrum or spectral domain technique borrowed from the broader context of general electromagnetic theory (Scott [1989]). The plane wave spectrum stems from the fact that in source-free regions (and virtually all of classical optics pertains to source-free regions), electromagnetic fields may be expressed in terms of a spectrum of propagating and evanescent plane waves. More specifically, Fourier optics refers to optical technologies which arise when the plane wave spectrum viewpoint (section 2) is combined with the Fourier transforming property of quadratic lenses (section 3), to yield 2D image processing devices (section 4) analogous to the 1D signal processing devices common in electronic signal processing. The hallmark of Fourier optics is the use of the spatial frequency domain (k_x, k_y) as the conjugate of the spatial (x,y) domain, and the use of terms and concepts from 1D signal processing, such as: transform theory, spectrum, bandwidth, window functions, sampling, etc. Wave Refraction in the manner of Huygens. ... See also list of optical topics. ... An evanescent wave is an electromagnetic wave that decays exponentially with distance. ... In the physics of wave propagation (especially electromagnetic waves), a plane wave (also spelled planewave) is a constant-frequency wave whose wavefronts (surfaces of constant amplitude and phase) are infinite parallel planes normal to the propagation direction. ... UPIICSA IPN - Binary image Image processing is any form of information processing for which the input is an image, such as photographs or frames of video; the output is not necessarily an image, but can be for instance a set of features of the image. ... Analog signal processing is any signal processing conducted on analog signals by analog means. ...

1 Electromagnetic Wave Propagation
2 The Plane Wave Spectrum: The Basic Foundation of Fourier Optics
3 Fourier Transforming Property of Lenses
4 4F Correlator
5 Afterword: Plane Wave Spectrum Within the Broader Context of Functional Decomposition
6 Applications
7 See also
8 References

Electromagnetic Wave Propagation

The Wave Equation

Fourier optics begins with the homogeneous, scalar wave equation: The wave equation is an important partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves. ...

$left(nabla^2-frac{1}{c^2}frac{partial^2}{partial{t}^2}right)u(mathbf{r},t)=0.$

where u(r,t) is a real-valued, scalar representation of an electromagnetic wave propagating through free space. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

The Helmholtz Equation

If we next assume that the solution of this equation takes a time-harmonic form, or in other words, This article is about the components of sound. ...

$u(mathbf{r},t) = mathrm{Re} left{ psi(mathbf{r}) e^{jomega t} right}$

and substitute this expression into the wave equation, we derive the time-independent form of the wave equation, also known as the Helmholtz equation: The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation where is the Laplacian, is a constant, and the unknown function is defined on three-dimensional Euclidean space R3. ...

$left(nabla^2+ k^2 right) psi (mathbf{r})=0.$

where

$k = { omega over c} = { 2 pi over lambda }$

is the wave number, j is the imaginary unit, and ψ(r) is the time-independent, complex-valued amplitude of the propagating wave. In mathematics, the imaginary unit (or sometimes the Latin or the Greek iota, see below) allows the real number system to be extended to the complex number system . ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...

The paraxial approximation

We can simplify the complex wave amplitude further by a simple change of variable:

$psi(mathbf{r}) = A(mathbf{r}) e^{-j mathbf{k} cdot mathbf{r}}$

where

$mathbf{k} = k_x mathbf{x} + k_y mathbf{y} + k_zmathbf{z}$

is the wave vector, and A wave vector is a vector that represents two properties of a wave: the magnitude of the vector represents wavenumber (inversely related to wavelength), and the vector points in the direction of wave propagation. ...

$k = |mathbf{k}| = sqrt{k_x^2 + k_y^2 + k_z^2} = {omega over c}$

is the wave number. Next, using the paraxial approximation, we assume that In geometric optics, the paraxial approximation is an approximation used in ray tracing of light through an optical system (such as a lens). ...

$k_x^2 + k_y^2 ll k_z^2$

or equivalently,

$sin theta approx theta$

where θ is the angle between the wave vector k and the z-axis.

As a result,

$k approx k_z$

and

$psi(mathbf{r}) approx A(mathbf{r}) e^{-jkz}$

The paraxial wave equation

Substituting this expression into the Helmholtz equation, we derive the paraxial wave equation:

$nabla_T^2 A - 2jk { partial A over partial z} = 0$

where

$nabla_T^2 = {partial^2 over partial x^2} + {partial^2 over partial y^2}$

is the transverse Laplacian operator. In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ...

The Plane Wave Spectrum: The Basic Foundation of Fourier Optics

The plane wave spectrum concept is the basic foundation of Fourier Optics. For readers already familiar with ray optics, the plane wave spectrum concept might seem unsettling at first. This is because a plane wave spectrum is not as easily visualized in the mind's eye as a light ray is. Almost anyone can mentally picture a curved optical phasefront, with the ray propagation direction being normal to the constant-phase surface. Some might try to imagine the plane wave spectrum as a locally-plane approximation to the curved optical phasefront and, in the far field, this is an appropriate way to visualize what the plane wave spectrum is. The plane wave spectrum is a continuous spectrum of uniform plane waves, and there is one plane wave component in the spectrum for every tangent point on the far-field phase front. The amplitude of that plane wave component would be the amplitude of the optical field at that tangent point. Again, this is true only in the far field, defined as: Range = 2 D² / λ where D is the maximum linear extent of the optical sources and λ is the wavelength. The plane wave spectrum is often regarded as being discrete for certain types of periodic gratings, though in reality, the spectra from gratings are continuous as well. In the physics of wave propagation (especially electromagnetic waves), a plane wave (also spelled planewave) is a constant-frequency wave whose wavefronts (surfaces of constant amplitude and phase) are infinite parallel planes normal to the propagation direction. ...

Readers already familiar with Fourier analysis of electrical signals will find a direct analogy here with the plane wave spectrum representation of optical fields. For example, the plane wave component propagating parallel to the optic axis is analogous to the DC component of an electrical signal. Bandwidth in electrical signals relates to the difference between the highest and lowest frequencies present in the spectrum of the signal. For optical systems, bandwidth is a measure of how far a plane wave is tilted away from the optic axis, so for this reason, this type of bandwidth is often referred to as angular bandwidth or spatial bandwidth. It takes more frequency bandwidth to produce a short pulse in an electrical circuit, and more angular (or, spatial frequency) bandwidth to produce a sharp spot in an optical system (see discussion related to Point spread function). Image formation in a confocal microscope: central longitudinal (XZ) slice. ...

The plane wave spectrum arises naturally as the eigenfunction solution to the homogeneous electromagnetic wave equation in rectangular coordinates (see also Electromagnetic radiation, which derives the wave equation from Maxwell's equations in source-free media, or Scott [1998]). In the frequency domain, the homogeneous electromagnetic wave equation assumes the form: Lasers used for visual effects during a musical performance. ... This box: Electromagnetic (EM) radiation is a self-propagating wave in space with electric and magnetic components. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ... The Helmholtz equation, named for Hermann von Helmholtz, is the elliptic partial differential equation where is the Laplacian, is a constant, and the unknown function is defined on three-dimensional Euclidean space R3. ...

$nabla^2 E_u + k^2E_u = 0$

where u = x, y, z and k = 2π/λ, the wavenumber of the medium. We may readily find solutions to this equation in rectangular coordinates by using the principle of separation of variables for partial differential equations. This principle says that in separable orthogonal coordinates, we may construct a so-called elementary product solution to this wave equation of the following form: Wavenumber in most physical sciences is a wave property inversely related to wavelength, having SI units of reciprocal meters (mâˆ’1). ... Fig. ... In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to re-write an equation so that each of two variables occurs on a different side of the equation. ... In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ... In mathematics, orthogonal coordinates are defined as a set of coordinates that have no off-diagonal terms in their metric tensor, i. ...

$E_u(x, y, z) = f_x(x) times f_y(y) times f_z(z)$

i.e., a solution which is expressed as the product of a function of x, times a function of y, times a function of z. If we now plug this elementary product solution into the wave equation, using the scalar Laplacian (aka, Laplace operator) in rectangular coordinates In mathematics and physics, the Laplace operator or Laplacian, denoted by Î”, is a differential operator, specifically an important case of an elliptic operator, with many applications. ...

$nabla^2 E_u = frac{partial^2 E_u}{partial x^2} + frac{partial^2 E_u}{partial y^2} + frac{partial^2 E_u}{partial z^2}$

we obtain

f'' x (x) f y (y) f z (z) + f x (x) f'' y (y) f z (z) + f x (x) f y (y) f'' z (z) + k 2 f x (x) f y (y) f z (z) = 0

which may be rearranged into the form:

$frac{f''_x(x)}{f_x(x)}+ frac{f''_y(y)}{f_y(y)} + frac{f''_z(z)}{f_z(z)} + k^2=0$

We may now argue that each of the quotients in the equation above must, of necessity, be constant. For, say the first quotient is not constant, and is a function of x. None of the other terms in the equation has any dependence on the variable x. Therefore, the first term may not have any x-dependence either; it must be constant. Let's call that constant -k_x². Reasoning in a similar way for the y and z quotients, we now obtain three ordinary differential equations for the f_x, f_y and f_z, along with one separation condition:

$frac{d^2}{dx^2}f_x(x) + k_x^2 f_x(x)=0$

$frac{d^2}{dy^2}f_y(y) + k_y^2 f_y(y)=0$

$frac{d^2}{dz^2}f_z(z) + k_z^2 f_z(z)=0$

$k_x^2+k_y^2+k_z^2= k^2$

Each of these 3 differential equations has the same solution, a complex exponential, so that the elementary product solution for E_u is:

$E_u(x,y,z)=e^{j(k_x x + k_y y)} e^{pm j sqrt{k^2-k_x^2-k_y^2}z}$

which represents a propagating or exponentially decaying uniform plane wave solution to the homogeneous wave equation. The - sign is used for a wave propagating/decaying in the +z direction and the + sign is used for a wave propagating/decaying in the -z direction (this follows the engineering time convention, which assumes an e^jωt time dependence). This field represents a propagating plane wave when the quantity under the radical is positive, and an exponentially decaying wave when it is negative (in passive media, we always choose the root with a negative imaginary part, to represent decay, not amplification).

A general solution to the homogeneous electromagnetic wave equation in rectangular coordinates is formed as a weighted superposition of elementary plane wave solutions as:

$E_u(x,y,z)=int!!!int E_u(k_x,k_y) ~ e^{j(k_x x + k_y y)} ~ e^{pm j sqrt{k^2-k_x^2-k_y^2}z} ~ dk_x dk_y$ (2.1)

where the integrals extend from minus infinity to infinity.

This plane wave spectrum representation of the electromagnetic field is the basic foundation of Fourier Optics (this point cannot be emphasized strongly enough), because we see that when z=0, the equation above simply becomes a Fourier transform (FT) relationship between the field and its plane wave content (hence the name, "Fourier optics").

Fourier optics cannot be understood from the viewpoint of ray optics, because ray optics is the asymptotic, zero wavelength limit of wave optics (it's like trying to understand relativity through Newtonian mechanics). The concept of a ray only exists as the far-field, zero-wavelength limit of the plane wave spectrum. Ray optics is a simplification of wave optics and as such, is a tiny subset of wave optics and therefore, cannot be used to understand it. That is why it can be difficult to try and understand Fourier optics without bringing to bear the broader viewpoint of general electromagnetic theory (i.e., Maxwell's equations). Unfortunately, humans first began to understand optical phenomena through ray-based models, and that approach is still perpetuated in optics instruction today, even though it is now well known (Sommerfeld, Stratton, Born, Mie, Zernike, Airy) that optical phenomena are more accurately understood via the framework of Maxwell's equations, with ray optics being a limiting case (Eikonal equation, Asymptotic expansion). Arnold Johannes Wilhelm Sommerfeld (December 5, 1868 in KÃ¶nigsberg, East Prussia â€“ April 26, 1951 in Munich, Germany) was a German physicist who introduced the fine-structure constant in 1919. ... Julius Adams Stratton (1901 - 1994) was a U.S. educator. ... Max Born (December 11, 1882 â€“ January 5, 1970) was a German physicist and mathematician. ... Gustav Adolf Feodor Wilhelm Ludwig Mie (September 29, 1869 Rostock â€“ February 13, 1957 Freiburg im Breisgau) was a German physicist. ... Frederik Zernike (Amsterdam, July 16, 1888 â€“ March 10, 1966) was a Dutch physicist and winner of the Nobel prize for physics in 1953 for his invention of the phase contrast microscope, an instrument that permits the study of internal cell structure without the need to stain and thus kill the... George Biddell Airy Sir George Biddell Airy FRS (July 27, 1801â€“January 2, 1892) was an English mathematician and astronomer, Astronomer Royal from 1835 to 1881. ... The eikonal equation is a non-linear partial differential equation of the form subject to , where is an open set in with well-behaved boundary, is a function with positive values, ; denotes the gradient and |Â·| is the Euclidean norm. ... In mathematics an asymptotic expansion, asymptotic series or PoincarÃ© expansion (after Henri PoincarÃ©) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular...

All spatial dependence of the individual plane wave components is described explicitly via the exponential functions. The coefficients of the exponentials are only functions of spatial wavenumber k_x, k_y, just as in ordinary Fourier analysis and Fourier transforms. Fourier analysis, named after Joseph Fouriers introduction of the Fourier series, is the decomposition of a function in terms of a sum of sinusoidal basis functions (vs. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

The equation above may be evaluated asymptotically in the far field (using the principle of stationary phase) to show that the field at the point (x,y,z) is indeed solely due to the plane wave component (k_x, k_y, k_z) which propagates parallel to the vector (x,y,z), and whose plane is tangent to the phasefront at (x,y,z). The mathematical details of this process may be found in Scott [1998] or Scott [1990]. The result of performing a stationary phase integration on the expression above is the following expression, In mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to oscillatory integrals taken over n-dimensional space Rn. ...

$E_u(r,theta,phi)~=~2 pi j~ (k~costheta)~ frac{e^{-jkr}}{r}~ E_u(k~sintheta~cosphi,k~sintheta~sinphi)$

which clearly indicates that the field at (x,y,z) is directly proportional to the spectral component in the direction of (x,y,z), where,

$x = r ~ sin theta ~ cos phi$

$y = r ~ sin theta ~ sin phi$

$z = r ~ cos theta ~$

Stated another way, the radiation pattern of any planar field distribution is the FT of that source distribution (see Huygens-Fresnel principle, wherein the same equation is developed using a Green's function approach). Note that this is NOT a plane wave, as many might think. The $frac{e^{-jkr}}{r}$ radial dependence is a spherical wave - both in magnitude and phase - whose local amplitude is the FT of the source plane distribution at that far field angle. The plane wave spectrum has nothing to do with saying that the field behaves something like a plane wave for far distances. Wave Refraction in the manner of Huygens. ...

In addition, we may determine the image plane distribution of an object plane distribution by tracing the progress of the individual plane wave components through the imaging system, and then re-assembling them in the image plane, each with its own particular magnitude and phase. If we were to consider the action of an optical system on each plane wave component in that fashion, then we'd be interested in such "angular frequency domain" figures-of-merit as the optical transfer function of the system. The Optical Transfer Function (OTF) describes the spatial (angular) variation as a function of spatial (angular) frequency. ...

The separation condition,

$k_x^2+k_y^2+k_z^2=k^2$

which so closely resembles the equation for the length of a vector in terms of its rectangular components, suggests the notion of k-vector, or wave vector, defined (for propagating plane waves) in rectangular coordinates as A wave vector is a vector that represents two properties of a wave: the magnitude of the vector represents wavenumber (inversely related to wavelength), and the vector points in the direction of wave propagation. ...

$bold k = k_x hat bold x + k_y hat bold y + k_z hat bold z$

and in the spherical coordinate system as A point plotted using the spherical coordinate system In mathematics, the spherical coordinate system is a coordinate system for representing geometric figures in three dimensions using three coordinates: the radial distance of a point from a fixed origin, the zenith angle from the positive z-axis, and the azimuth angle...

$k_x = k ~ sin theta ~ cos phi$

$k_y = k ~ sin theta ~ sin phi$

$k_z = k ~ cos theta ~$

We'll make use of these spherical coordinate system relations in the next section.

Fourier Transforming Property of Lenses

If a transmissive object is placed one focal length in front of a lens, then its Fourier transform will be formed one focal length behind the lens. We may show this using what we now know about the plane wave spectrum representation of the transmittance function in the front focal plane. Consider the figure to the right (click to enlarge) This article is about the optical device. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

On the Fourier transforming property of lenses

In this figure, we assume a plane wave incident from the left. The transmittance function in the front focal plane (i.e., Plane 1) spatially modulates this incident plane wave in magnitude and phase, like on the left-hand side of eqn. (2.1), and in so doing, produces a spectrum of plane waves corresponding to the FT of the transmittnace function, like on the right-hand side of eqn. (2.1). The various plane wave components propagate at different tilt angles with respect to the optic axis of the lens (i.e., the horizontal axis). The finer the features in the transparency, the broader the bandwidth of the plane wave spectrum. We'll consider one such plane wave component, propagating at angle θ with respect to the optic axis. We'll assume θ is small (paraxial approximation), so that Image File history File links Size of this preview: 800 Ã— 329 pixelsFull resolution (934 Ã— 384 pixel, file size: 30 KB, MIME type: image/jpeg) I, the copyright holder of this work, hereby release it into the public domain. ... Image File history File links Size of this preview: 800 Ã— 329 pixelsFull resolution (934 Ã— 384 pixel, file size: 30 KB, MIME type: image/jpeg) I, the copyright holder of this work, hereby release it into the public domain. ... In geometric optics, the paraxial approximation is an approximation used in ray tracing of light through an optical system (such as a lens). ...

$frac{k_x}{k} = sin theta cong theta$

and

$frac{k_z}{k} = cos theta cong 1 - frac{theta^2}{2}$

and

$frac{1}{cos theta} cong frac{1}{1 - frac{theta^2}{2}} cong 1 + frac{theta^2}{2}$

In the figure, the plane wave phase, moving horizontally from the front focal plane to the lens plane, is

$e^{j k f cos theta} ,$

and the spherical wave phase from the lens to the spot in the back focal plane is:

$e^{j k f / cos theta} ,$

and the sum of the two path lengths is f (1 + θ² + 1 - θ²) = 2f i.e., it is a constant value, independent of tilt angle, θ, for paraxial plane waves. Each paraxial plane wave component of the field in the front focal plane appears as a PSF spot in the back focal plane, with an intensity and phase equal to the intensity and phase of the original plane wave component in the front focal plane. In other words, the field in the back focal plane is the Fourier transform of the field in the front focal plane. Image formation in a confocal microscope: central longitudinal (XZ) slice. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ...

The beauty of performing a Fourier transform optically is that all FT components are computed simultaneously - in parallel - at the speed of light. As an example, light travels at a speed of roughly 1 ft. / ns, so if a lens has a 1 ft. focal length, an entire 2D FT can be computed in about 2 ns (2 x 10^-9 seconds). If the focal length is 1 in., then the time is under 200 ps. No electronic computer can compete with these kinds of numbers, or perhaps ever hope to. The disadvantage is that, as the derivation shows, the FT relationship only holds for paraxial plane waves, so this FT "computer" is inherently bandlimited. On the other hand, since the wavelength of visible light is so minute in relation to even the smallest visible feature dimensions in the image i.e.,

$k^2 >> k_x ^2 + k_y ^2$

(for all k_x, k_y within the spatial bandwidth of the image, so that k_z is nearly equal to k), the paraxial approximation is not terribly limiting in practice. And, of course, this is an analog - not a digital - computer, so precision is limited. Also, phase can be challenging to extract; often it is inferred interferometrically.

As a side note, we should recognize that the spatially modulated electric field, shown on the left-hand side of eqn. (2.1), typically only occupies a finite (usually rectangular) aperture in the x,y plane. The rectangular aperture function acts like a 2D square-top pulse function, and we usually assume the field to be zero outside this 2D rectangle. So, our spatial domain integrals for calculating the FT coefficients on the right-hand side of eqn. (2.1) are truncated at the boundary of this aperture. This step truncation can introduce inaccuracies in both theoretical calculations and measured values of the plane wave coefficients on the RHS of eqn. (2.1). Whenever a function is discontinuously truncated in one FT domain, broadening and rippling are introduced in the other FT domain. A perfect example from optics is in connection with the Point spread function, which for on-axis plane wave illumination of a quadratic lens (with circular aperture), is an Airy function, J₁(x)/x. Literally, the point source has been "spread out" (with ripples added), to form the Airy point spread function. This source of error is known as Gibbs phenomenon and it may be mitigated by simply ensuring that all significant content lies near the center of the transparency, or through the use of window functions which smoothly taper the field to zero at the frame boundaries. By the convolution theorem, the FT of an arbitrary transparency function - multiplied (or truncated) by an aperture function - is equal to the FT of the non-truncated transparency function convolved against the FT of the aperture function, which in this case becomes a type of "Greens function" or "impulse response function" in the spectral domain. The FT of a circular aperture function is J₁(x)/x and the FT of a rectangular aperture function is a product of sinc functions, sin x/x. Image formation in a confocal microscope: central longitudinal (XZ) slice. ... Approximation of square wave in 5 steps Approximation of square wave in 25 steps Approximation of square wave in 125 steps In mathematics, the Gibbs phenomenon, named after the American physicist J. Willard Gibbs, (also known as ringing artifacts) is the peculiar manner in which the Fourier series of a... In signal processing, a window function (or apodization function) is a function that is zero-valued outside of some chosen interval. ...

4F Correlator

As stated in the introduction, when the plane wave spectrum representation of the electric field (section 2) is combined with the Fourier transforming property of quadratic lenses (section 3), it leads naturally to the development of numerous 2D image processing devices. One of the primary applications of Fourier Optics is in the mathematical operations of cross-correlation and convolution, and these have historically been done with a device known as a 4F correlator, shown in the figure below (click to enlarge). In statistics, the term cross-correlation is sometimes used to refer to the covariance cov(X, Y) between two random vectors X and Y, in order to distinguish that concept from the covariance of a random vector X, which is understood to be the matrix of covariances between the scalar... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ...

4F Correlator

The 4F correlator is based on the convolution theorem from Fourier transform theory, which states that convolution in the spatial (x,y) domain is equivalent to direct multiplication in the spatial frequency (k_x, k_y) domain. Once again, a plane wave is assumed incident from the left and a transparency containing one 2D function, f(x,y), is placed in the input plane of the correlator, located one focal length in front of the first lens. The transparency spatially modulates the incident plane wave in magnitude and phase, like on the left-hand side of eqn. (2.1), and in so doing, produces a spectrum of plane waves corresponding to the FT of the transmittnace function, like on the right-hand side of eqn. (2.1). That spectrum is then formed as an "image" one focal length behind the first lens, as shown. A transmission mask containing the FT of the second function, g(x,y), is placed in this same plane, one focal length behind the first lens, causing the transmission through the mask to be equal to the product, F(k_x,k_y) x G(k_x,k_y). This product now lies in the "input plane" of the second lens (one focal length in front), so that the FT of this product (i.e., the convolution of f(x,y) and g(x,y)), is formed in the back focal plane of the second lens. Image File history File links Size of this preview: 800 Ã— 516 pixelsFull resolution (944 Ã— 609 pixel, file size: 54 KB, MIME type: image/jpeg) I, the copyright holder of this work, hereby release it into the public domain. ... Image File history File links Size of this preview: 800 Ã— 516 pixelsFull resolution (944 Ã— 609 pixel, file size: 54 KB, MIME type: image/jpeg) I, the copyright holder of this work, hereby release it into the public domain. ... In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution is the point-wise product of Fourier transforms. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ... In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ...

Afterword: Plane Wave Spectrum Within the Broader Context of Functional Decomposition

Electrical fields are really just particular types of mathematical functions and, as such, may often be represented in many different ways. In the Huygens-Fresnel or Stratton-Chu viewpoints, we represent the electric field as a superposition of point sources, each one of which gives rise to a Green's function field. The total field is then the weighted sum of all of the individual Greens function fields. That seems to be the most natural way of viewing the electric field for most people - no doubt because most of us have, at one time or another, drawn out the circles with protractor and paper, much the same way Thomas Young did in his classic paper. However, it is by no means the only way to represent the electric field. As we have seen herein, the field may also be represented as a spectrum of sinusoidally varying plane waves. In addition, Frits Zernike proposed still another functional decomposition based on his Zernike polynomials, defined on the unit disc. All of these functional decompositions have utility in different circumstances. The optical scientist having access to these various different representational forms has available a richer insight to the nature of these marvelous fields and their properties. The reader is encouraged to embrace these different ways of looking at the field, rather than viewing them as being in any way conflicting or contradictory. Then the true beauty of optics begins to unfold. Wave Refraction in the manner of Huygens. ... Slit experiment redirects here. ... Functional decomposition of engineering is a method for analyzing engineered systems. ... In mathematics the Zernike polynomials, named after Frits Zernike, are a sequence of orthogonal polynomials which play an important role in geometrical optics. ...

Applications

Fourier optics is used in the field of optical information processing, the staple of which is the classical 4F processor.

The Fourier transform properties of a lens provide numerous applications in optical signal processing such as spatial filtering, optical correlation and computer generated holograms. In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... This article is about the optical device. ... A spatial filter is an optical device which uses the principles of Fourier optics to alter the structure of a beam of coherent light or other electromagnetic radiation. ...

Fourier optical theory is used in interferometers, optical tweezers, atom traps, and quantum computing. Concepts of Fourier optics are used to reconstruct the phase of light intensity in the spatial frequency plane (see adaptive-additive algorithm). It has been suggested that Optical interferometry be merged into this article or section. ... An optical tweezer is a scientific instrument that uses a focused laser beam to provide an attractive or repulsive force, depending on the index mismatch (typically on the order of piconewtons) to physically hold and move microscopic dielectric objects. ... Molecule of alanine used in NMR implementation of error correction. ... This article is about a portion of a periodic process. ... 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. ...

References

Goodman, Joseph (2005). Introduction to Fourier Optics, 3rd ed,, Roberts & Co Publishers. ISBN 0974707724. or online here
Hecht, Eugene (1987). Optics, 2nd ed., Addison Wesley. ISBN 0-201-11609-X.
Wilson, Raymond (1995). Fourier Series and Optical Transform Techniques in Contemporary Optics. Wiley. ISBN 0471303577.
Scott, Craig (1998). Introduction to Optics and Optical Imaging. Wiley. ISBN 0-7803-3440-X.
Scott, Craig (1990). Modern Methods of Reflector Antenna Analysis and Design. Artech House. ISBN 0-89006-419-9.
Scott, Craig (1989). The Spectral Domain Method in Electromagnetics. Artech House. ISBN ---.

Categories: Optics | Physical optics | Fourier analysis

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Contents

Electromagnetic Wave Propagation GA_googleFillSlot("encyclopedia_square");

The Wave Equation

The Helmholtz Equation

The paraxial approximation

The paraxial wave equation

The Plane Wave Spectrum: The Basic Foundation of Fourier Optics

Fourier Transforming Property of Lenses

4F Correlator

Afterword: Plane Wave Spectrum Within the Broader Context of Functional Decomposition

Applications

See also

References

Electromagnetic Wave Propagation