14668:
14065:
14663:{\displaystyle {\begin{aligned}{\mathcal {I}}\left(x_{2},y_{2}\right)&=\left|r_{o}\exp \left(-j2\pi \alpha y_{2}\right)+{\frac {1}{\lambda f}}H\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\right|^{2}\\&=r_{o}^{2}+{\frac {1}{\lambda ^{2}f^{2}}}\left|H\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\right|^{2}+{\frac {r_{o}}{\lambda f}}H\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\exp \left(j2\pi \alpha y_{2}\right)+{\frac {r_{o}}{\lambda f}}H^{*}\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\exp \left(-j2\pi \alpha y_{2}\right)\end{aligned}}}
12666:), Then, the lens passes - from the object plane over onto the image plane - only that portion of the radiated spherical wave which lies inside the edge angle of the lens. In this far-field case, truncation of the radiated spherical wave is equivalent to truncation of the plane wave spectrum of the small source. So, the plane wave components in this far-field spherical wave, which lie beyond the edge angle of the lens, are not captured by the lens and are not transferred over to the image plane. Note: this logic is valid only for small sources, such that the lens is in the far field region of the source, according to the 2
11347:. This is where the convolution equation above comes from. The convolution equation is useful because it is often much easier to find the response of a system to a delta function input - and then perform the convolution above to find the response to an arbitrary input - than it is to try to find the response to the arbitrary input directly. Also, the impulse response (in either time or frequency domains) usually yields insight to relevant figures of merit of the system. In the case of most lenses, the point spread function (PSF) is a pretty common figure of merit for evaluation purposes.
9139:, it is often a case that the elements of a matrix will be very complicated functions of frequency and wavenumber, and the matrix will be non-singular (I.e., it has the inverse matrix.) for most combinations of frequency and wavenumber, but will also be singular (I.e., it does not have the inverse matrix.) for certain specific combinations. By finding which combinations of frequency and wavenumber drive the determinant of the matrix to zero, the propagation characteristics of the medium may be determined. Relations of this type, between frequency and wavenumber, are known as
12701:(engineering) time dependence, coherent (laser) light is implicitly assumed, which has a delta function dependence in the frequency domain. Light at different (delta function) frequencies will "spray" the plane wave spectrum out at different angles, and as a result these plane wave components will be focused at different places in the output plane. The Fourier transforming property of lenses works best with coherent light, unless there is some special reason to combine light of different frequencies, to achieve some special purpose.
16219:. All of these functional decompositions have utility in different circumstances. The optical scientist having access to these various representational forms has available a richer insight to the nature of these marvelous fields and their properties. These different ways of looking at the field are not conflicting or contradictory, rather, by exploring their connections, one can often gain deeper insight into the nature of wave fields.
10235:
necessary to have an ideal point source in order to determine an exact impulse response. This is because any source bandwidth which lies outside the bandwidth of the optical system under consideration won't matter anyway (since it cannot even be captured by the optical system), so therefore it's not necessary in determining the impulse response. The source only needs to have at least as much (angular) bandwidth as the optical system.
13830:
12590:
Airy function. This is somewhat like the point spread function, except now we're really looking at it as a kind of input-to-output plane transfer function (like MTF), and not so much in absolute terms, relative to a perfect point. Similarly, Gaussian wavelets, which would correspond to the waist of a propagating
Gaussian beam, could also potentially be used in still another functional decomposition of the object plane field.
12747:
11194:, that is to say that the output of the system due to two different inputs (possibly at two different times) is the sum of the individual outputs of the system to the two inputs, when introduced individually. Thus the optical system may contain no nonlinear materials nor active devices (except possibly, extremely linear active devices). The output of the system, for a single delta function input is defined as the
10239:
imaging system) is desired to be a high-quality reproduction of an optical field in the object plane (the input plane of the imaging system). The impulse response function of an optical imaging system is desired to approximate a 2D delta function, at the location (or a linearly scaled location) in the output plane corresponding to the location of the impulse (an ideal point source) in the input plane. The
2587:
15315:
5427:. The imaging is the reconstruction of a wave on the object plane (having information about a pattern on the object plane to be imaged) on the image plane via the proper wave propagation from the object to the image planes, (E.g., think about the imaging of an image in an aerial space.) and the wave on the object plane, that fully follows the pattern to be imaged, is in principle, described by the
51:, in which the wavefront is regarded as being made up of a combination of spherical wavefronts (also called phasefronts) whose sum is the wavefront being studied. A key difference is that Fourier optics considers the plane waves to be natural modes of the propagation medium, as opposed to Huygens–Fresnel, where the spherical waves originate in the physical medium.
11893:
3004:
12660:). Consider a "small" light source located on-axis in the object plane of the lens. It is assumed that the source is small enough that, by the far-field criterion, the lens is in the far field of the "small" source. Then, the field radiated by the small source is a spherical wave which is modulated by the FT of the source distribution, as in eqn. (
13648:
2202:
14959:
12731:, this device - 4 focal lengths long - actually serves a wide variety of image processing operations that go well beyond what its name implies. A diagram of a typical 4F correlator is shown in the figure below (click to enlarge). This device may be readily understood by combining the plane wave spectrum representation of the electric field (
15980:
8410:(on the other), in the far field. Note that the term "far field" usually means we're talking about a converging or diverging spherical wave with a pretty well defined phase center. The connection between spatial and angular bandwidth in the far field is essential in understanding the low pass filtering property of thin lenses. See the
11811:. Thus, the input-plane plane wave spectrum is transformed into the output-plane plane wave spectrum through the multiplicative action of the system transfer function. It is at this stage of understanding that the previous background on the plane wave spectrum becomes invaluable to the conceptualization of Fourier optical systems.
12681:). Bandwidth truncation causes a (fictitious, mathematical, ideal) point source in the object plane to be blurred (or, spread out) in the image plane, giving rise to the term, "point spread function." Whenever bandwidth is expanded or contracted, image size is typically contracted or expanded accordingly, in such a way that the
3995:
11936:> 0). 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 angular bandwidth of the plane wave spectrum. We'll consider one such plane wave component, propagating at angle
8908:. an Infinite homogeneous media admits the rectangular, circular and spherical harmonic solutions to the Helmholtz equation, depending on the coordinate system under consideration. The propagating plane waves that we'll study in this article are perhaps the simplest type of propagating waves found in any type of media.
12419:), typically only occupies a finite (usually rectangular) aperture in the x,y plane. The rectangular aperture function acts like a 2D square-top filter, where the field is assumed to be zero outside this 2D rectangle. The spatial domain integrals for calculating the FT coefficients on the right-hand side of eqn. (
1669:
12517:-direction. It is of course, very tempting to think that if a plane wave emanating from the finite aperture of the transparency is tilted too far from horizontal, it will somehow "miss" the lens altogether but again, since the uniform plane wave extends infinitely far in all directions in the transverse (
14952:
12598:
In the figure above, illustrating the
Fourier transforming property of lenses, the lens is in the near field of the object plane transparency, therefore the object plane field at the lens may be regarded as a superposition of plane waves, each one of which propagates at some angle with respect to the
12308:
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 (0.30 m) per nanosecond, so if a lens has a 1 ft (0.30 m) focal length, an entire 2D FT can be computed in about 2 ns (2 × 10 seconds).
12528:
This issue brings up perhaps the predominant difficulty with
Fourier analysis, namely that the input-plane function, defined over a finite support (i.e., over its own finite aperture), is being approximated with other functions (sinusoids) which have infinite support (i.e., they are defined over the
9143:
and some physical systems may admit many different kinds of dispersion relations. An example from electromagnetics is an ordinary waveguide, which may admit numerous dispersion relations, each associated with a unique propagation mode of the waveguide. Each propagation mode of the waveguide is known
1948:
54:
A curved phasefront may be synthesized from an infinite number of these "natural modes" i.e., from plane wave phasefronts oriented in different directions in space. When an expanding spherical wave is far from its sources, it is locally tangent to a planar phase front (a single plane wave out of the
16190:
field. The total field is then the weighted sum of all of the individual Green's 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
12895:
If an ideal, mathematical point source of light is placed on-axis in the input plane of the first lens, then there will be a uniform, collimated field produced in the output plane of the first lens. When this uniform, collimated field is multiplied by the FT plane mask, and then
Fourier transformed
12434:
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
12313:
may actually prove faster than optics, as improbable as that may seem. However, their speed is obtained by combining numerous computers which, individually, are still slower than optics. The disadvantage of the optical FT is that, as the derivation shows, the FT relationship only holds for paraxial
10254:
The second type is optical image processing systems, in which a significant feature in the input plane optical field is to be located and isolated. In this case, the impulse response of such a system is desired to be a close replica (picture) of that feature which is being searched for in the input
10238:
Optical systems typically fall into one of two different categories. The first is ordinary focused optical imaging systems (e.g., cameras), wherein the input plane is called the object plane and the output plane is called the image plane. An optical field in the image plane (the output plane of the
9080:
are linear operators on their respective functions / vector spaces. (The minus sign in this matrix equation is, for all intents and purposes, immaterial. However, the plus sign in the
Helmholtz equation is significant.) It is perhaps worthwhile to note that the eigenfunction solutions / eigenvector
8630:
plane, and therefore is analogous to the (constant) 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 a signal, practically with a criterion to cut off high and low frequency edges of
8498:
to be "imaged away" while the fictitious magnetic currents are obtained from twice the aperture electric field (see Scott ). Then the radiated electric field is calculated from the magnetic currents using an equation similar to the equation for the magnetic field radiated by an electric current. In
12585:
correspond to converging or diverging spherical waves, and therefore could potentially be implemented as a whole new functional decomposition of the object plane function, thereby leading to another point of view similar in nature to
Fourier optics. This would basically be the same as conventional
8417:
Once the concept of angular bandwidth is understood, the optical scientist can "jump back and forth" between the spatial and spectral domains to quickly gain insights which would ordinarily not be so readily available just through spatial domain or ray optics considerations alone. For example, any
12589:
Perhaps a lens figure-of-merit in this "point spread function" viewpoint would be to ask how well a lens transforms an Airy function in the object plane into an Airy function in the image plane, as a function of radial distance from the optic axis, or as a function of the size of the object plane
10751:
in focused imaging systems). However, high quality optical systems are often "shift invariant enough" over certain regions of the input plane that we may regard the impulse response as being a function of only the difference between input and output plane coordinates, and thereby use the equation
9148:
solution (or eigenmode solution) to
Maxwell's equations in the waveguide. Free space also admits eigenmode (natural mode) solutions (known more commonly as plane waves), but with the distinction that for any given frequency, free space admits a continuous modal spectrum, whereas waveguides have a
8635:
systems, bandwidth also relates to spatial frequency content (spatial bandwidth), but it also has a secondary meaning. It also measures how far from the optic axis the corresponding plane waves are tilted, and so this type of bandwidth is often referred to also as angular bandwidth. It takes more
8383:
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. A plane wave spectrum does not necessarily mean that the field as the superposition of the plane wave components in that spectrum behaves
2676:
8601:
Likely to electrical signals, bandwidth in optics is a measure of how finely detailed an image is; the finer the detail, the greater the bandwidth required to represent it. A DC (Direct
Current) electrical signal is constant and has no oscillations; a plane wave propagating parallel to the optic
8518:
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, roughly defined as the
7071:
6594:
12541:
were conceived, that is to represent a function (defined on a finite interval or area) in terms of oscillatory functions which are also defined over finite intervals or areas. Thus, instead of getting the frequency content of the entire image all at once (along with the frequency content of the
10234:
The impulse response of an optical imaging system is the output plane field which is produced when an ideal mathematical optical field point source of light, that is an impulse input to the system, is placed in the input plane (usually on-axis, i.e., on the optical axis). In practice, it is not
12458:
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
5640:
8597:
is the wavelength (Scott ). 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, since no physical device can have the infinite extent required to produce a true line spectrum.
13393:
4928:
All spatial dependence of each plane wave component is described explicitly by an exponential function. The coefficient of the exponential is a function of only two components of the wave vector for each plane wave (since other remained component can be determined via the above mentioned
2137:
10057:
7919:
7388:
9085:(i.e., form a basis set for) the function space / vector space under consideration. The interested reader may investigate other functional linear operators (so for different equations than the Helmholtz equation) which give rise to different kinds of orthogonal eigenfunctions such as
4694:
15714:
10947:
10742:
The alert reader will note that the integral above tacitly assumes that the impulse response is NOT a function of the position (x',y') of the impulse of light in the input plane (if this were not the case, this type of convolution would not be possible). This property is known as
5230:
12459:
function, which in this case becomes a type of "Greens function" or "impulse response function" in the spectral domain. Therefore, the image of a circular lens is equal to the object plane function convolved against the Airy function (the FT of a circular aperture function is
10724:
3729:
12175:
8421:
As a side note, electromagnetics scientists have devised an alternative means to calculate an electric field in a far zone which does not involve stationary phase integration. They have devised a concept known as "fictitious magnetic currents" usually denoted by
409:
2582:{\displaystyle {\begin{aligned}{\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}\end{aligned}}}
15310:{\displaystyle I\left(x_{2},y_{2}\right)=r_{o}^{2}+{\frac {1}{\lambda ^{2}f^{2}}}S^{2}\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)+{\frac {2r_{o}}{\lambda f}}S\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\cos \left.}
9815:
16066:
1530:
16169:
3402:
14056:
6850:
14735:
9422:
12924:) will be some type of feature which must be identified and located within the input plane field (see Scott ). In military applications, this feature may be a tank, ship or airplane which must be quickly identified within some more complex scene.
8870:
In the case of differential equations, as in the case of matrix equations, whenever the right-hand side of an equation is zero (For example, a forcing function, forcing vector, or the source of a force is zero.), the equation may still admit a
1070:
750:
6298:
183:
Light can be described as a waveform propagating through a free space (vacuum) or a material medium (such as air or glass). Mathematically, a real-valued component of a vector field describing a wave is represented by a scalar wave function
12402:), 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.
6935:
6488:
12450:. Literally, the point source has been "spread out" (with ripples added), to form the Airy point spread function (as the result of truncation of the plane wave spectrum by the finite aperture of the lens). This source of error is known as
6403:
59:
pattern is created, which emanates from a single spherical wave phase center. In the near field, no single well-defined spherical wave phase center exists, so the wavefront isn't locally tangent to a spherical ball. In this case, a
9857:
7729:
6188:
10255:
plane field, so that a convolution of the impulse response (an image of the desired feature) against the input plane field will produce a bright spot at the feature location in the output plane. It is this latter type of optical
8488:
4496:
11186:
12804:), 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. (
12073:
3469:
11317:
9042:
5434:
4924:
932:
4817:
2207:
1674:
7278:
2999:{\displaystyle {\begin{aligned}\psi (x,y,z)&=Ae^{ik_{x}x}e^{ik_{y}y}e^{ik_{z}z}\\&=Ae^{i(k_{x}x+k_{y}y)}e^{ik_{z}z}\\&=Ae^{i(k_{x}x+k_{y}y)}e^{\pm iz{\sqrt {k^{2}-k_{x}^{2}-k_{y}^{2}}}}\end{aligned}}}
12314:
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.,
12300:
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
1377:
11660:
7448:
6930:
16239:
which will span that domain. Depending on the operator and the dimensionality (and shape, and boundary conditions) of its domain, many different types of functional decompositions are, in principle, possible.
8879:
solution, in physics as a "natural mode" solution, and in electrical circuit theory as the "zero-input response." This is a concept that spans a wide range of physical disciplines. Common physical examples of
12078:
15678:
5032:
10747:(Scott ). No optical system is perfectly shift invariant: as the ideal, mathematical point of light is scanned away from the optic axis, aberrations will eventually degrade the impulse response (known as a
300:
13643:{\displaystyle H\left(f_{X},f_{Y}\right)={\frac {S^{*}\left(f_{X},f_{Y}\right)}{\left|S\left(f_{X},f_{Y}\right)\right|^{2}+{\frac {\Phi _{n}\left(f_{X},f_{Y}\right)}{\Phi _{o}\left(f_{X},f_{Y}\right)}}}},}
11998:
6701:
68:
source, consisting of a distribution of (physically identifiable) spherical wave sources in space. In the near field, a full spectrum of plane waves is necessary to represent the
Fresnel near-field wave,
13837:
Ragnarsson proposed a method to realize Wiener restoration filters optically by holographic technique like setup shown in the figure. The derivation of the function of the setup is described as follows.
12425:) 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. (
1953:
12723:
in hardware using only 2 identical lenses and a transparency plate - the 4F correlator. Although one important application of this device would certainly be to implement the mathematical operations of
9605:
The notion of k-space is central to many disciplines in engineering and physics, especially in the study of periodic volumes, such as in crystallography and the band theory of semiconductor materials.
9230:
5393:
4170:
1121:
9654:
8490:
In this equation, it is assumed that the unit vector in the z-direction points into the half-space where the far field calculations will be made. These equivalent magnetic currents are obtained using
15988:
12566:
plane) correspond to plane wave functions in three dimensions. However, the FTs of most wavelets are well known and could possibly be shown to be equivalent to some useful type of propagating field.
11429:
5841:
15975:{\displaystyle \Delta I\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)={\frac {2r_{o}}{\lambda f}}S\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\cos \left.}
11210:). And, by our linearity assumption (i.e., that the output of system to a pulse train input is the sum of the outputs due to each individual pulse), we can now say that the general input function
1505:
16235:, both briefly alluded to here, are not completely independent. The eigenfunction expansions to certain linear operators defined over a given domain, will often yield a countably infinite set of
16073:
12586:
ray optics, but with diffraction effects included. In this case, each point spread function would be a type of "smooth pixel," in much the same way that a soliton on a fiber is a "smooth pulse."
10769:
6103:
on a wafer. As a result, machines realizing such an optical lithography have become more and more complex and expensive, significantly increasing the cost of the electronic component production.
14070:
13382:
9311:
8636:
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
6493:
5315:
4491:
4092:
2681:
808:
15373:), the recorded wave from the lower portion should be much stronger than that from the upper portion because the wave passing through the lower path is focused, which leads to the relationship
13271:
8977:
6706:
13821:
13761:
12991:). And, as mentioned above, the impulse response of the correlator is just a picture of the feature we're trying to find in the input image. In the 4F correlator, the system transfer function
5669:
takes an infinite range of real numbers. It means that, for a given light frequency, only a part of the full feature of the pattern can be imaged because of the above-mentioned constraints on
9324:
6479:
1303:
10143:, for focused optical systems). The impulse response function uniquely defines the input-output behavior of the optical system. By convention, the optical axis of the system is taken as the
12619:
is on the order of 10 m, or hundreds of meters. On the other hand, the far field distance from a PSF spot is on the order of λ. This is because D for the spot is on the order of λ, so that
10559:
16207:, defined on the unit disc. The third-order (and lower) Zernike polynomials correspond to the normal lens aberrations. And still another functional decomposition could be made in terms of
10460:
10359:
8765:
81:", all of which could (when they collide with something such as a rock in the way) scatter independently of one other. These mathematical simplifications and calculations are the realm of
15416:
12375:
9544:
9486:
8278:
8220:
989:
666:
13035:
Image blurring by a point spread function is studied extensively in optical information processing, one way to alleviate the blurring is to adopt Wiener Filter. For example, assume that
7721:. The mathematical details of this process may be found in Scott or Scott . The result of performing a stationary phase integration on the expression above is the following expression,
6219:
2141:
It may now be argued that each quotient in the equation above must, of necessity, be constant. To justify this, let's say that the first quotient is not a constant, and is a function of
15569:
15371:
3990:{\displaystyle \psi (x,y,z)=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }\Psi _{0}(k_{x},k_{y})~e^{i(k_{x}x+k_{y}y)}~e^{\pm iz{\sqrt {k^{2}-k_{x}^{2}-k_{y}^{2}}}}~dk_{x}dk_{y}}
13701:
12550:
plane, over which the image has zero value), the result is instead the frequency content of different parts of the image, which is usually much simpler. Unfortunately, wavelets in the
13955:
8115:
8064:
7521:
85:– together, they can describe what happens when light passes through various slits, lenses or mirrors that are curved one way or the other, or is fully or partially reflected.
12405:
Optical processing is especially useful in real time applications where rapid processing of massive amounts of 2D data is required, particularly in relation to pattern recognition.
10549:
4258:
8381:
12674:, Scott ), each of which has its spectrum truncated in this fashion, then every point of the entire object plane transparency suffers the same effects of this low pass filtering.
12577:- which are not only the point spread functions of rectangular and circular apertures, respectively, but are also cardinal functions commonly used for functional decomposition in
12274:
10230:
3304:
275:
12223:
11079:
Obtaining the convolution representation of the system response requires representing the input signal as a weighted superposition over a train of impulse functions by using the
8418:
source bandwidth which lies past the edge angle to the first lens (This edge angle sets the bandwidth of the optical system.) will not be captured by the system to be processed.
7643:
6303:
5733:
4317:
1172:
229:
13967:
9593:
8327:
3208:) is a real number here since waves in a source-free medium has been assumed so each plane wave is not decayed or amplified as it propagates in the medium. The negative sign of
16340:
Ragnarsson, SI. "Physica
Scripta A new Holographic Method of Generating a High Efficiency, Extended Range Spatial Filter with Application to Restoration of Defocussed Images".
10107:
1256:
8856:
8499:
this way, a vector equation is obtained for the radiated electric field in terms of the aperture electric field, and the derivation requires no use of stationary phase ideas.
13317:
11760:
8555:
7483:
11809:
8157:
6119:
5425:
4202:
860:
603:
10117:
In a high level overview, an optical system consists of three parts; an input plane, and output plane, and a set of components between these planes that transform an image
8690:
571:
8429:
7128:
6210:
2671:
2630:
9074:
7155:
16195:. However, it is by no means the only way to represent the electric field, which may also be represented as a spectrum of sinusoidally varying plane waves. In addition,
15709:
11691:
11523:
11492:
11461:
5925:
4999:
Let's consider an imaging system where the z-axis is the optical axis of the system and the object plane (to be imaged on the image plane of the system) is the plane at
1664:{\displaystyle \nabla ^{2}\psi ={\frac {\partial ^{2}\psi }{\partial x^{2}}}+{\frac {\partial ^{2}\psi }{\partial y^{2}}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}},}
637:
12309:
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, although
12003:
9122:
8595:
1141:
8819:
8011:
7973:
7719:
7681:
7584:
7102:
6616:
1216:
15478:
13173:
13138:
13103:
13068:
9852:
9649:
7271:
7218:
7188:
15524:
15501:
14730:
6070:
6003:
5976:
5868:
5760:
5694:
5667:
5342:
4981:
4954:
4398:
4371:
4344:
4119:
3582:
3233:
3140:
3109:
3082:
3055:
976:
657:
8990:
14947:{\displaystyle H\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)=S\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)\exp \left,}
5023:
4822:
3628:
3515:
3259:
3166:
7241:
865:
15592:
15443:
14710:
14690:
13887:
13867:
9179:
8620:
8575:
6097:
6043:
6023:
5945:
5888:
3715:
3668:
3648:
3602:
3555:
3535:
3489:
3299:
3279:
3206:
3186:
3024:
1196:
956:
828:
770:
429:
12816:). 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,
6045:
axis does not physically make sense if there is no amplification material between the object and image planes, and this is a usual case.) so waves with such
4725:
13275:
The goal of image restoration is to find a linear restoration filter that minimize the mean-squared error between the true distribution and the estimation
13027:) of the input function, to produce the spectrum of the output function. This is how electrical signal processing systems operate on 1D temporal signals.
11090:
7066:{\displaystyle \nabla _{T}^{2}=\nabla ^{2}-{\partial ^{2} \over \partial z^{2}}={\partial ^{2} \over \partial x^{2}}+{\partial ^{2} \over \partial y^{2}}}
6589:{\displaystyle {\begin{aligned}\sin \theta &\approx \theta \\\tan \theta &\approx \theta \\\cos \theta &\approx 1-\theta ^{2}/2\end{aligned}}}
1312:
11540:
6862:
6116:
A solution to the Helmholtz equation as the spatial part of a complex-valued Cartesian component of a single frequency wave is assumed to take the form:
12630:
Since the lens is in the far field of any PSF spot, the field incident on the lens from the spot may be regarded as being a spherical wave, as in eqn. (
9132:
infinite number of eigenvalues/eigenfunctions (in confined regions) or uncountably infinite (continuous) spectra of solutions, as in unbounded regions.
9124:
may be found by setting the determinant of the matrix equal to zero, i.e. finding where the matrix has no inverse. (Such a square matrix is said to be
5635:{\textstyle \psi _{0,{\text{unc}}}(x,y)=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }\Psi _{0}(k_{x},k_{y})~e^{i(k_{x}x+k_{y}y)}~dk_{x}dk_{y}}
15600:
3407:
11221:
2591:
Each of these 3 differential equations has the same solution form: sines, cosines or complex exponentials. We'll go with the complex exponential as
12810:), and in so doing, produces a spectrum of plane waves corresponding to the FT of the transmittance function, like on the right-hand side of eqn. (
11951:
6628:
10052:{\displaystyle u(x,y)={\frac {1}{(2\pi )^{2}}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }U(k_{x},k_{y})e^{i(k_{x}x+k_{y}y)}dk_{x}dk_{y}}
7914:{\displaystyle E_{u}(r,\theta ,\phi )~=~2\pi i~(k~\cos \theta )~{\frac {e^{-ikr}}{r}}~E_{u}(k~\sin \theta ~\cos \phi ,k~\sin \theta ~\sin \phi )}
7383:{\textstyle \left|{\frac {\partial ^{2}}{\partial z^{2}}}A(\mathbf {r} )\right|\ll \left|k{\frac {\partial }{\partial z}}A(\mathbf {r} )\right|}
1075:
12828:), 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,
11369:
10247:, whose radius is on the order of the wavelength of the light used. The impulse response function in this case is typically referred to as a
4689:{\displaystyle \psi _{0}(x,y)=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }\Psi _{0}(k_{x},k_{y})~e^{i(k_{x}x+k_{y}y)}~dk_{x}dk_{y}}
1409:
8332:
Stated another way, the radiation pattern of any planar field distribution is the FT (Fourier Transform) of that source distribution (see
6079:
of electronic components, these (1) and (2) are the reasons why light of a higher frequency (smaller wavelength, thus larger magnitude of
5225:{\textstyle \psi _{0}(x,y)=\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }\Psi _{0}(k_{x},k_{y})~e^{i(k_{x}x+k_{y}y)}~dk_{x}dk_{y}}
16212:
13322:
12671:
12578:
9235:
4406:
13178:
12864:). 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
12454:
and it may be mitigated by simply ensuring that all significant content lies near the center of the transparency, or through the use of
9081:
solutions to the Helmholtz equation / the matrix equation, often yield an orthogonal set of the eigenfunctions / the eigenvectors which
8920:
7393:
1309:, is mostly concerned as treating a complex-valued function is often much easier than treating the corresponding real-valued function.
16330:
The equation 2.3 below suggests that u in this equation is such as u = x, y, or z. Need to confirm if this is the right understanding.
6416:
11358:
of the system. So the spatial domain operation of a linear optical system is analogous in this way to the Huygens–Fresnel principle.
11900:
In this figure, a plane wave incident from the left is assumed. The transmittance function in the front focal plane (i.e., Plane 1)
10383:
10282:
8884:
natural modes would include the resonant vibrational modes of stringed instruments (1D), percussion instruments (2D) or the former
8704:
12317:
9184:
5347:
4124:
1943:{\displaystyle 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}
138:) domain. Terms and concepts such as transform theory, spectrum, bandwidth, window functions and sampling from one-dimensional
1218:
are linearly related to one another, a typical characteristic of transverse electromagnetic (TEM) waves in homogeneous media.
16366:
7451:
5765:
12670:/λ criterion mentioned previously. If an object plane transparency is imagined as a summation over small sources (as in the
10471:
149:
in which a pattern on a reticle to be imaged on wafers for semiconductor chip production is so dense such that light (e.g.,
16216:
12648:) - the full plane wave spectrum - accurately represents the field incident on the lens from that larger, extended source.
16186:-Chu viewpoints, the electric field is represented as a superposition of point sources, each one of which gives rise to a
12927:
The 4F correlator is an excellent device for illustrating the "systems" aspects of optical instruments, alluded to in the
12232:
12170:{\displaystyle {\frac {1}{\cos \theta }}\simeq {\frac {1}{1-{\frac {1}{2}}\theta ^{2}}}\simeq 1+{\frac {1}{2}}\theta ^{2}}
11819:
Fourier optics is used in the field of optical information processing, the staple of which is the classical 4F processor.
10251:, since the mathematical point of light in the object plane has been spread out into an Airy function in the image plane.
10158:
234:
12186:
5239:
4016:
779:
191:
12558:
plane don't correspond to any known type of propagating wave function, in the same way that Fourier's sinusoids (in the
986:
Substituting this expression into the scalar wave equation above yields the time-independent form of the wave equation,
404:{\displaystyle \left(\nabla ^{2}-{\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial {t}^{2}}}\right)u(\mathbf {r} ,t)=0.}
13766:
13706:
6481:
so, up to the second order approximation of trigonometric functions (that is, taking only up to the second term in the
10147:-axis. As a result, the two images and the impulse response function are all functions of the transverse coordinates,
1261:
16532:
16509:
16486:
16463:
16440:
16411:
5978:
are allowed (by an unknown interaction between light and the object plane pattern that is usually a solid material),
157:) emanated from the reticle is diffracted and each diffracted light may correspond to a different spatial frequency (
5951:) imaging system that is expensive and difficult to build. For (1), even if complex-valued longitudinal wavenumbers
4699:
The plane wave spectrum representation of a general electromagnetic field (e.g., a spherical wave) in the equation (
2132:{\displaystyle {\frac {f_{x}''(x)}{f_{x}(x)}}+{\frac {f_{y}''(y)}{f_{y}(y)}}+{\frac {f_{z}''(z)}{f_{z}(z)}}+k^{2}=0}
833:
12296:, for paraxial plane waves. Each paraxial plane wave component of the field in the front focal plane appears as a
9810:{\displaystyle U(k_{x},k_{y})=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }u(x,y)e^{-i(k_{x}x+k_{y}y)}dxdy}
16061:{\displaystyle \Delta t\propto \Delta \phi \propto \Delta D\propto \Delta (\log I)\approx {\frac {\Delta I}{I}}.}
15376:
12719:
is somewhat abstract. However, there is one very well known device which implements the system transfer function
9492:
9434:
8226:
8168:
15529:
15322:
12611:
of the transparency is on the order of cm (10 m) and the wavelength of light is on the order of 10 m, therefore
77:
moving forward (like an expanding ocean wave coming toward the shore) can be regarded as an infinite number of "
13653:
7545:
17:
13908:
12677:
Loss of the high (spatial) frequency content causes blurring and loss of sharpness (see discussion related to
9128:.) Finite matrices have only a finite number of eigenvalues/eigenvectors, whereas linear operators can have a
16590:
16265:
16179:
16164:{\displaystyle \Delta t\propto {\frac {S^{\ast }}{r_{0}^{2}\lambda ^{2}f^{2}+\left\vert S\right\vert ^{2}}}.}
12792:). Once again, a plane wave is assumed incident from the left and a transparency containing one 2D function,
11351:
10942:{\displaystyle g(x,y)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }h_{M}(x-Mx',y-My')~f(x',y')~dx'dy'}
8648:
8333:
8070:
8019:
7488:
3674:
3397:{\displaystyle \mathbf {k} =k_{x}{\hat {\mathbf {x} }}+k_{y}{\hat {\mathbf {y} }}+k_{z}{\hat {\mathbf {z} }}}
1395:
48:
12928:
12740:
12716:
12642:). On the other hand, the lens is in the near field of the entire input plane transparency, therefore eqn. (
9599:
4207:
14051:{\displaystyle {\frac {1}{\lambda f}}H{\left({\frac {x_{2}}{\lambda f}},{\frac {y_{2}}{\lambda f}}\right)}}
11843:
10263:
presents one hardware implementation of the optical image processing operations described in this section.
10109:
is present whenever angular frequency (radians) is used, but not when ordinary frequency (cycles) is used.
8980:
8893:
8885:
8343:
6845:{\displaystyle \psi (\mathbf {r} )\approx A(\mathbf {r} )e^{-i(k_{x}x+k_{y}y)}e^{ikz\theta ^{2}/2}e^{-ikz}}
540:(as from a single-mode laser) is assumed, then, based on the engineering time convention, which assumes an
10112:
2157:, then there is no way to make the left hand side of this equation be zero.) This constant is denoted as -
16260:
11870:
9425:
9417:{\displaystyle \mathbf {k} ~=~k_{x}\mathbf {\hat {x}} +k_{y}\mathbf {\hat {y}} +k_{z}\mathbf {\hat {z}} }
9136:
7589:
7078:
4263:
3718:
3678:
3112:
1387:
1146:
517:
278:
16173:
11889:
will be formed at one focal length behind the lens. Consider the figure to the right (click to enlarge)
5699:
5696:; (1) a fine feature which representation in the inverse Fourier transform requires spatial frequencies
13105:
is the intensity distribution of its image which is blurred by a space-invariant point-spread function
10719:{\displaystyle g(x,y)=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }h(x-x',y-y')~f(x',y')~dx'dy'}
10064:
9550:
9321:
in a three-dimensional "k-space", defined (for propagating plane waves) in rectangular coordinates as:
8284:
1224:
8824:
16639:
16305:
16275:
13278:
11831:
11713:
11694:
8652:
8522:
7457:
6410:
1065:{\displaystyle \operatorname {Re} \left\{\left(\nabla ^{2}+k^{2}\right)\psi (\mathbf {r} )\right\}=0}
745:{\displaystyle u(\mathbf {r} ,t)=\operatorname {Re} \left\{\psi (\mathbf {r} )e^{i\omega t}\right\}.}
11765:
11048:
exists in the time domain, but not in the spatial domain. Causality means that the impulse response
8655:, which derives the wave equation from Maxwell's equations in source-free media, or Scott ). In the
6293:{\displaystyle \mathbf {k} \cdot \mathbf {r} =k_{x}\mathbf {x} +k_{y}\mathbf {y} +k_{z}\mathbf {z} }
5398:
4175:
576:
16634:
16232:
16200:
15574:
The phase shift of the transparency after bleaching is linearly proportional to the silver density
13901:
3, forming a virtual point source generating a spherical wave. The wave is then collimated by lens
8872:
8662:
8121:
543:
7107:
6193:
2635:
2594:
92:, as well as applications where information needs to be extracted from optical sources such as in
16250:
13841:
Assume there is a transparency as the recording plane and an impulse emitted from a point source
12682:
9052:
7133:
6859:
Substituting this expression into the Helmholtz equation, the paraxial wave equation is derived:
1391:
43:
15685:
11667:
11499:
11468:
11437:
10133:
by convolving the input image with the optical impulse response function of the optical system,
6099:) or a higher NA imaging system is required to image finer features of integrated circuits on a
5897:
55:
infinite spectrum), which is transverse to the radial direction of propagation. In this case, a
16381:
16280:
16192:
11945:
10260:
7533:
6406:
1399:
608:
56:
9107:
8580:
3697:
A general solution to the homogeneous electromagnetic wave equation at a fixed time frequency
1126:
16270:
16183:
12704:
12678:
12513:-direction, from Plane 1 to Plane 2) must be considered, and not the phase transverse to the
12310:
12297:
11858:
10464:
10248:
10139:
9090:
8786:
8637:
8491:
7978:
7940:
7686:
7648:
7551:
7087:
6601:
3721:
may be formed as a weighted superposition of all possible elementary plane wave solutions as
3682:
1201:
16356:
15448:
13143:
13108:
13073:
13038:
9822:
9619:
7246:
7193:
7163:
6398:{\displaystyle k=\|\mathbf {k} \|={\sqrt {k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}}={\omega \over c}}
16478:
16455:
16389:
16310:
16236:
15506:
15486:
14715:
11084:
9086:
6048:
5981:
5954:
5846:
5738:
5672:
5645:
5320:
4959:
4932:
4376:
4349:
4322:
4097:
3560:
3211:
3118:
3087:
3060:
3033:
961:
642:
12607:
is the maximum linear extent of the optical sources and λ is the wavelength (Scott ). The
12537:
plane). This is unbelievably inefficient computationally, and is the principal reason why
7081:. In the derivation of the paraxial wave equation, the following approximations are used.
6072:
may not reach the image plane that is usually sufficiently far way from the object plane.
41:(FTs), in which the waveform being considered is regarded as made up of a combination, or
8:
16285:
16255:
16204:
16187:
13823:
are the power spectral densities of the point-spread function, the object and the noise.
12755:
12686:
11839:
11704:
11355:
9140:
9094:
8337:
5002:
3686:
3607:
3494:
3238:
3145:
172:
154:
61:
10755:
Also, this equation assumes unit magnification. If magnification is present, then eqn. (
7223:
16300:
16290:
15577:
15428:
14695:
14675:
13872:
13852:
12710:
12498:
plane (Plane 1), the uniform plane waves comprising the plane wave spectrum occupy the
10113:
Optical systems: general overview and analogy with electrical signal processing systems
9164:
8693:
8605:
8560:
6183:{\displaystyle \psi (\mathbf {r} )=A(\mathbf {r} )e^{-i\mathbf {k} \cdot \mathbf {r} }}
6082:
6028:
6008:
5948:
5930:
5873:
4260:
is the weight factor or the amplitude of the plane wave component with the wave vector
3700:
3653:
3633:
3587:
3540:
3520:
3474:
3284:
3264:
3191:
3171:
3009:
1306:
1181:
941:
813:
755:
451:
Cartesian component of an electromagnetic wave propagating through a free space (e.g.,
414:
145:
Fourier optics plays an important role for high-precision optical applications such as
12509:
plane, which is why (for this purpose) only the longitudinal plane wave phase (in the
11762:
is associated with the coefficient of the plane wave whose transverse wavenumbers are
16613:
16577:
16528:
16505:
16482:
16459:
16436:
16407:
16397:
16362:
16174:
Afterword: Plane wave spectrum within the broader context of functional decomposition
12949:) is the system transfer function of the correlator, which we'd in general denote as
12759:
12724:
12302:
11886:
11862:
11835:
11823:
9610:
9125:
8483:{\displaystyle \mathbf {M} ~=~2\mathbf {E} ^{\text{aper}}\times \mathbf {\hat {z}} .}
5947:
so higher wave outgoing angles with respect to the optical axis, requires a high NA (
5891:
4988:
139:
113:
109:
38:
11354:, or Stratton-Chu formulation, wherein the "impulse response" is referred to as the
175:
analysis may not provide the details of how light is diffracted from each reticle.
16605:
16567:
16178:
Electrical fields can be represented mathematically in many different ways. In the
12451:
11854:
9314:
9047:
8901:
8656:
7074:
6076:
4984:
3689:(with the remaining separable coordinate systems being used much less frequently).
2145:. Since none of the other terms in the equation has any dependence on the variable
1524:
1178:
Cartesian component of an electromagnetic wave. Note that the propagation constant
146:
89:
82:
8865:
16401:
12657:
12474:
and the FT of a rectangular aperture function is a product of sinc functions, sin
12455:
12068:{\displaystyle {\frac {k_{z}}{k}}=\cos \theta \simeq 1-{\frac {1}{2}}\theta ^{2}}
4713:
Fourier transform (FT) relationship between the field and its plane wave contents
3464:{\textstyle \left|\mathbf {k} \right|=k={\omega \over c}={2\pi \over \lambda }}
104:
12599:
z-axis. In this regard, the far-field criterion is loosely defined as: Range = 2
8494:
which, in the case of an infinite planar interface, allow any electric currents
16432:
11850:
9149:
discrete mode spectrum. In this case, the dispersion relation is linear, as in
5026:
3027:
1175:
935:
773:
432:
93:
16541:
1258:
of the scalar wave equation can be simply obtained by taking the real part of
16628:
16617:
16581:
16295:
16228:
16208:
16196:
13387:
12574:
12570:
12413:
The spatially modulated electric field, shown on the left-hand side of eqn. (
11882:
11866:
11827:
11331:) is the (impulse) response of the linear system to the delta function input
10748:
9145:
9129:
9037:{\displaystyle \left(\mathbf {A} -\lambda \mathbf {I} \right)\mathbf {x} =0,}
8897:
8876:
8692:, the homogeneous electromagnetic wave equation becomes what is known as the
8644:
6482:
294:
78:
16070:
Finally, we get a amplitude transmittance with the form of a Wiener filter:
12183:
phase, moving horizontally from the front focal plane to the lens plane, is
8502:
4919:{\displaystyle \psi _{0}(x,y)={\mathcal {F}}^{-1}\{\Psi _{0}(k_{x},k_{y})\}}
16524:
16501:
12705:
Hardware implementation of the system transfer function: The 4F correlator
11044:
The extension to two dimensions is trivial, except for the difference that
6111:
4994:
927:{\displaystyle \psi (\mathbf {r} )=a(\mathbf {r} )e^{i\phi (\mathbf {r} )}}
16609:
16572:
16555:
12774:) domain is equivalent to direct multiplication in the spatial frequency (
10465:
The 2D convolution of input function against the impulse response function
6622:
and the z-axis as the optical axis of an optical system under discussion.
15421:
In Ragnarsson' s work, this method is based on the following postulates:
13893:
The upper portion is first focused (i.e., Fourier transformed) by a lens
13829:
12865:
12763:
12728:
12490:
Even though the input transparency only occupies a finite portion of the
9318:
9082:
6213:
6100:
3673:
Product solutions to the Helmholtz equation are also readily obtained in
448:
178:
150:
12967:), and it is the FT of the impulse response function of the correlator,
2153:-dependence; it must be a constant. (If the first term is a function of
862:
is the angular frequency (in radians per unit time) of light waves, and
9598:
Use will be made of these spherical coordinate system relations in the
8905:
8859:
8509:
7975:
is directly proportional to the spectral component in the direction of
4812:{\displaystyle \Psi _{0}(k_{x},k_{y})={\mathcal {F}}\{\psi _{0}(x,y)\}}
2632:
to be a complex function. As a result, the elementary product solution
1671:
then the following equation for the 3 individual functions is obtained
533:
16452:
Fourier Series and Optical Transform Techniques in Contemporary Optics
12896:
by the second lens, the output plane field (which in this case is the
12732:
10243:
impulse response function of an imaging system typically resembles an
7586:
is indeed due solely to the plane wave component with the wave vector
2172:
quotients, three ordinary differential equations are obtained for the
15480:, that has recorded the known impulse response of the blurred system.
11881:
If a transmissive object is placed at one focal length in front of a
11045:
10244:
9616:
A spectrum analysis equation (calculating the spectrum of a function
9317:
in a three-dimensional configuration space, suggests the notion of a
529:
98:
12739:) to yield the optical image processing operations described in the
11710:
This equation takes on its real meaning when the Fourier transform,
11181:{\displaystyle f(t)=\int _{-\infty }^{\infty }\delta (t-t')f(t')dt'}
7544:) above may be evaluated asymptotically in the far field (using the
3692:
1372:{\displaystyle \left(\nabla ^{2}+k^{2}\right)\psi (\mathbf {r} )=0.}
11655:{\displaystyle G(k_{x},k_{y})~=~H(k_{x},k_{y})\cdot F(k_{x},k_{y})}
7443:{\textstyle {\frac {\partial ^{2}}{\partial z^{2}}}A(\mathbf {r} )}
6925:{\displaystyle \nabla _{T}^{2}A-2ik{\partial A \over \partial z}=0}
660:
15597:
The density is linearly proportional to the logarithm of exposure
3471:) means that the wave propagation direction vector has a positive
1143:
in vacuum, is the wave number (also called propagation constant),
15425:
Assume there is a transparency, with its amplitude transmittance
12538:
10368:
The output plane is defined as the locus of all points such that
96:. To put it in a slightly complex way, similar to the concept of
15673:{\displaystyle I=r_{o}^{2}+{\frac {1}{\lambda ^{2}f^{2}}}S^{2}.}
12746:
12615:/λ for the whole transparency is on the order of 10. This times
12485:
10271:
The input plane is defined as the locus of all points such that
1406:
to this wave equation may be constructed of the following form:
11366:
If the last equation above is Fourier transformed, it becomes:
8911:
There is a striking similarity between the Helmholtz equation (
8866:
Eigenfunction (natural mode) solutions: background and overview
171:). Due to generally non-uniform patterns on reticles, a simple
34:
16222:
12735:) with the Fourier transforming property of quadratic lenses (
11312:{\displaystyle g(t)=\int _{-\infty }^{\infty }h(t-t')f(t')dt'}
8411:
4707:(this point cannot be emphasized strongly enough), because at
11993:{\displaystyle {\frac {k_{x}}{k}}=\sin \theta \simeq \theta }
11060:) of an electrical system, due to an impulse applied at time
10997:
will be a magnified version of the impulse response function
7485:
is slowly varying compared with the major period of the wave
6696:{\displaystyle k_{z}=k\cos \theta \approx k(1-\theta ^{2}/2)}
537:
12736:
12593:
12229:
phase from the lens to the spot in the back focal plane is:
11814:
10125:
formed in the output plane. The optical system output image
12292:; i.e., it is a constant value, independent of tilt angle,
11190:
It is then presumed that the system under consideration is
8512:
spectrum concept is the basic foundation of Fourier Optics.
573:
time dependence in wave solutions at the angular frequency
277:
represents a position in a three dimensional space (in the
74:
12892:)), is formed in the back focal plane of the second lens.
12715:
The theory on optical transfer functions presented in the
12525:) plane, the planar wave components cannot miss the lens.
11892:
11876:
11693:. In optical imaging this function is better known as the
10965:
which basically translates the impulse response function,
9232:
for a homogeneous vacuum space, the separation condition,
9225:{\displaystyle k={\omega \over c}={2\pi \over \lambda }}
5388:{\displaystyle k={\omega \over c}={2\pi \over \lambda }}
4165:{\displaystyle k={\omega \over c}={2\pi \over \lambda }}
1116:{\displaystyle k={\omega \over c}={2\pi \over \lambda }}
15985:
By these postulates, we have the following relationship:
15319:
Note that for the point at the origin of the film plane (
13070:
is the intensity distribution from an incoherent object,
12408:
11865:. Concepts of Fourier optics are used to reconstruct the
11039:
8503:
The plane wave spectrum: the foundation of Fourier optics
13849:
1, forming a distribution equal to the impulse response
12685:
remains constant, by Heisenberg's principle (Scott and
9608:
8577:
is the maximum linear extent of the optical sources and
6112:
Paraxial wave propagation (optic axis assumed as z axis)
4995:
Connection between Fourier optics and imaging resolution
16498:
Modern Methods of Reflector Antenna Analysis and Design
12697:
While working in the frequency domain, with an assumed
11922:
corresponding to the FT of the transmittance function,
11869:
of light intensity in the spatial frequency plane (see
11424:{\displaystyle G(\omega )~=~H(\omega )\cdot F(\omega )}
5836:{\displaystyle k_{T}^{2}=k_{x}^{2}+k_{y}^{2}\geq k^{2}}
3111:
as the electric field component along each axis in the
16191:
same way Thomas Young did in his classic paper on the
15379:
12656:
A lens is basically a low-pass plane wave filter (see
11703:
Once again it may be noted from the discussion on the
9553:
9495:
9437:
8287:
8229:
8171:
8124:
8073:
8022:
7491:
7396:
7281:
5702:
5437:
5035:
3410:
179:
Propagation of light in homogeneous, source-free media
16076:
15991:
15717:
15688:
15603:
15580:
15532:
15509:
15489:
15451:
15431:
15325:
14962:
14738:
14718:
14698:
14678:
14068:
13970:
13911:
13875:
13855:
13769:
13709:
13656:
13396:
13325:
13281:
13181:
13146:
13111:
13076:
13041:
12900:
of the correlator) is just our correlating function,
12692:
12320:
12235:
12189:
12081:
12006:
11954:
11768:
11716:
11670:
11543:
11502:
11471:
11440:
11372:
11224:
11093:
10772:
10562:
10474:
10386:
10285:
10161:
10067:
9860:
9825:
9657:
9622:
9327:
9238:
9187:
9167:
9110:
9055:
8993:
8923:
8827:
8789:
8707:
8665:
8631:
the spectrum to represent bandwidth in a number. For
8608:
8583:
8563:
8525:
8432:
8346:
7981:
7943:
7732:
7689:
7651:
7592:
7554:
7460:
7249:
7226:
7196:
7166:
7136:
7110:
7090:
6938:
6865:
6709:
6631:
6604:
6491:
6419:
6306:
6222:
6196:
6122:
6085:
6051:
6031:
6011:
5984:
5957:
5933:
5900:
5876:
5849:
5768:
5741:
5675:
5648:
5401:
5350:
5323:
5242:
5005:
4962:
4935:
4825:
4728:
4499:
4409:
4379:
4352:
4325:
4266:
4210:
4178:
4127:
4100:
4019:
3732:
3703:
3656:
3636:
3610:
3590:
3563:
3543:
3523:
3497:
3477:
3307:
3287:
3267:
3241:
3214:
3194:
3174:
3148:
3121:
3090:
3063:
3036:
3012:
2679:
2638:
2597:
2205:
1956:
1677:
1533:
1500:{\displaystyle \psi (x,y,z)=f_{x}(x)f_{y}(y)f_{z}(z)}
1412:
1315:
1264:
1227:
1204:
1184:
1149:
1129:
1078:
992:
964:
944:
868:
836:
816:
782:
758:
669:
645:
611:
579:
546:
417:
303:
237:
194:
16386:
The Fourier Transform and its Applications to Optics
15503:
shift introduced by the filter is much smaller than
8514:
The plane wave spectrum is a continuous spectrum of
13377:{\displaystyle \varepsilon ^{2}=|o-{\hat {o}}|^{2}}
12391:within the spatial bandwidth of the image, so that
11940:with respect to the optic axis. It is assumed that
9306:{\displaystyle k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=k^{2}}
8387:
8340:approach). Note that this is NOT a plane wave. The
5310:{\displaystyle k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=k^{2}}
4486:{\displaystyle \psi _{0}(x,y)=\psi (x,y,z)|_{z=0}.}
4087:{\displaystyle k_{x}^{2}+k_{y}^{2}+k_{z}^{2}=k^{2}}
803:{\displaystyle \operatorname {Re} \left\{x\right\}}
293:Fourier optics begins with the homogeneous, scalar
16163:
16060:
15974:
15703:
15672:
15586:
15563:
15518:
15495:
15472:
15437:
15410:
15365:
15309:
14946:
14724:
14704:
14684:
14662:
14050:
13949:
13881:
13861:
13815:
13755:
13695:
13642:
13376:
13311:
13266:{\displaystyle i(x,y)=o(x,y)\otimes s(x,y)+n(x,y)}
13265:
13167:
13132:
13097:
13062:
12369:
12268:
12217:
12169:
12067:
11992:
11803:
11754:
11685:
11654:
11517:
11486:
11455:
11423:
11311:
11180:
10941:
10718:
10543:
10454:
10353:
10224:
10101:
10051:
9846:
9819:A synthesis equation (reconstructing the function
9809:
9643:
9587:
9538:
9480:
9416:
9305:
9224:
9173:
9116:
9068:
9036:
8972:{\displaystyle \left(\nabla ^{2}+k^{2}\right)f=0,}
8971:
8850:
8813:
8759:
8684:
8614:
8589:
8569:
8549:
8482:
8375:
8321:
8272:
8214:
8151:
8109:
8058:
8005:
7967:
7913:
7713:
7683:, and whose plane is tangent to the phasefront at
7675:
7637:
7578:
7515:
7477:
7442:
7382:
7265:
7235:
7212:
7182:
7149:
7122:
7096:
7065:
6924:
6844:
6695:
6610:
6588:
6473:
6397:
6292:
6204:
6182:
6091:
6064:
6037:
6017:
5997:
5970:
5939:
5919:
5882:
5862:
5835:
5754:
5727:
5688:
5661:
5634:
5419:
5387:
5336:
5309:
5224:
5029:Cartesian component of a wave is, as shown above,
5017:
4975:
4948:
4918:
4811:
4688:
4485:
4392:
4365:
4338:
4311:
4252:
4196:
4164:
4113:
4086:
3989:
3709:
3662:
3642:
3622:
3596:
3576:
3549:
3529:
3509:
3483:
3463:
3396:
3293:
3273:
3253:
3227:
3200:
3180:
3160:
3134:
3103:
3076:
3049:
3018:
2998:
2665:
2624:
2581:
2131:
1942:
1663:
1499:
1371:
1297:
1250:
1221:Since the originally desired real-valued solution
1210:
1190:
1166:
1135:
1115:
1064:
970:
950:
926:
854:
822:
802:
764:
744:
651:
631:
597:
565:
423:
403:
269:
223:
27:Study of classical optics using Fourier transforms
13960:The lower portion is directly collimated by lens
13905:3 and produces a tilted plane wave with the form
13816:{\displaystyle \Phi _{n}\left(f_{X},f_{Y}\right)}
13756:{\displaystyle \Phi _{o}\left(f_{X},f_{Y}\right)}
11707:, that this equation assumes unit magnification.
10121:formed in the input plane into a different image
9408:
9383:
9358:
8471:
8414:for the condition defining the far field region.
8336:, wherein the same equation is developed using a
7454:, means that the amplitude or envelope of a wave
6618:is the angle (in radian) between the wave vector
3693:The complete solution: the superposition integral
1523:is substituted into the wave equation, using the
1381:
64:pattern would be created, which emanates from an
16626:
16406:(3 ed.). Roberts & Company Publishers.
10259:system that is the subject of this section. The
8643:The plane wave spectrum arises naturally as the
6474:{\displaystyle k_{x}^{2}+k_{y}^{2}\ll k_{z}^{2}}
5843:, can not be fully imaged since waves with such
1298:{\displaystyle \psi (\mathbf {r} )e^{i\omega t}}
14062:Therefore, the total intensity distribution is
11920:in so doing, produces a spectrum of plane waves
9135:In certain physics applications such as in the
1305:, solving the following equation, known as the
88:Fourier optics forms much of the theory behind
47:, of plane waves. It has some parallels to the
16604:(1). American Physical Society (APS): 99–107.
16521:The Spectral Domain Method in Electromagnetics
13009:) is directly multiplied against the spectrum
11896:On the Fourier transforming property of lenses
11350:The same logic is used in connection with the
8647:or "natural mode" solution to the homogeneous
8384:something like a plane wave at far distances.
7527:
16591:"Diffraction Theory of Electromagnetic Waves"
16542:Intro to Fourier Optics and the 4F correlator
13386:The solution of this optimization problem is
12636:), not as a plane wave spectrum, as in eqn. (
12486:Fourier analysis and functional decomposition
10455:{\displaystyle g(x,y)=U(x,y,z){\big |}_{z=d}}
10435:
10354:{\displaystyle f(x,y)=U(x,y,z){\big |}_{z=0}}
10334:
8875:solution, known in applied mathematics as an
8760:{\displaystyle \nabla ^{2}E_{u}+k^{2}E_{u}=0}
6854:
6106:
5025:. On the object plane, the spatial part of a
15411:{\textstyle r_{0}\ll {\frac {1}{\lambda f}}}
13897:2 to a spot in the front focal plan of lens
13845:. The wave of impulse is collimated by lens
12651:
12370:{\displaystyle k^{2}\gg k_{x}^{2}+k_{y}^{2}}
9539:{\textstyle k_{y}=k~\sin \theta ~\sin \phi }
9481:{\textstyle k_{x}=k~\sin \theta ~\cos \phi }
8273:{\textstyle k_{y}=k~\sin \theta ~\sin \phi }
8215:{\textstyle k_{x}=k~\sin \theta ~\cos \phi }
7548:) to show that the field at a distant point
6321:
6313:
4913:
4871:
4806:
4778:
16588:
16223:Functional decomposition and eigenfunctions
11902:spatially modulates the incident plane wave
11361:
9313:which is identical to the equation for the
9150:
6485:expansion of each trigonometric function),
4705:) is the basic foundation of Fourier optics
2149:, so the first term also must not have any
1950:which is readily rearranged into the form:
1386:Solutions to the Helmholtz equation in the
16589:Stratton, J. A.; Chu, L. J. (1939-07-01).
16475:Introduction to Optics and Optical Imaging
15564:{\displaystyle e^{j\phi }\approx 1+j\phi }
15366:{\displaystyle (x_{2},y_{2})=\mathbf {0} }
14956:then we can rewrite intensity as follows:
11064:, must of necessity be zero for all times
11001:of a similar, unmagnified system, so that
7937:which clearly indicates that the field at
1390:may readily be found via the principle of
523:
16571:
13696:{\displaystyle S\left(f_{X},f_{Y}\right)}
12979:) which is just our correlating function
11815:Applications of Fourier optics principles
9137:computation of bands in a periodic volume
3026:. This solution is the spatial part of a
981:
16556:"Optical Computing: A 60-Year Adventure"
16380:
13950:{\displaystyle r_{0}e^{-j2\pi \alpha y}}
13828:
12745:
12623:/λ is on the order of unity; this times
12594:Far-field range and the 2D / λ criterion
11891:
8110:{\textstyle y=r~\sin \theta ~\sin \phi }
8059:{\textstyle x=r~\sin \theta ~\cos \phi }
7645:which propagates parallel to the vector
7516:{\textstyle \lambda ={\frac {2\pi }{k}}}
3557:)-component, while the positive sign of
1398:. This principle says that in separable
810:is the operator taking the real part of
659:is a time period of the waves, the time-
16396:
16354:
16213:Whittaker–Shannon interpolation formula
15711:is much stronger than varying exposure
12672:Whittaker–Shannon interpolation formula
12627:(i.e., λ) is on the order of λ (10 m).
12305:of the field in the front focal plane.
12276:and the sum of the two path lengths is
11877:Fourier transforming property of lenses
11535:) may be Fourier transformed to yield:
5762:are transverse wave numbers satisfying
14:
16627:
16449:
16339:
12409:Object truncation and Gibbs phenomenon
11040:Derivation of the convolution equation
10544:{\displaystyle g(x,y)~=~h(x,y)*f(x,y)}
4253:{\displaystyle \Psi _{0}(k_{x},k_{y})}
1507:i.e., as the product of a function of
958:in non-negative real number and phase
663:form of the optical field is given as
16518:
16495:
16472:
16426:
13964:, yielding an amplitude distribution
13175:introduced in the detection process:
9100:In the matrix equation case in which
8659:, with an assumed time convention of
8651:in rectangular coordinates (see also
8376:{\displaystyle {\frac {e^{-ikr}}{r}}}
7452:slowly varying envelope approximation
5894:.), and (2) spatial frequencies with
2164:. Reasoning in a similar way for the
510:-axis component of an electric field
188:that depends on both space and time:
16553:
13030:
12269:{\displaystyle e^{ikf/\cos \theta }}
11463:is the spectrum of the output signal
10763:
10553:
10225:{\displaystyle g(x,y)=h(x,y)*f(x,y)}
8979:and the usual equation form for the
8698:
7723:
6025:axis (Light amplification along the
5870:do not exist for the given light of
3723:
1527:in the Cartesian coordinates system
288:
270:{\displaystyle \mathbf {r} =(x,y,z)}
12931:above. The FT plane mask function,
12218:{\displaystyle e^{ikf\cos \theta }}
11924:like on the right-hand side of eqn.
11525:is the spectrum of the input signal
7638:{\displaystyle (k_{x},k_{y},k_{z})}
6005:give rise to light decay along the
5728:{\textstyle {\frac {k_{T}}{2\pi }}}
4711:= 0, the equation simply becomes a
4312:{\displaystyle (k_{x},k_{y},k_{z})}
1167:{\displaystyle \psi (\mathbf {r} )}
224:{\displaystyle u=u(\mathbf {r} ,t)}
130:) as the conjugate of the spatial (
24:
16361:. Roberts and Company Publishers.
16077:
16043:
16019:
16010:
16001:
15992:
15718:
14075:
13771:
13711:
13589:
13544:
12754:The 4F correlator is based on the
12693:Coherence and Fourier transforming
11906:like on the left-hand side of eqn.
11849:Fourier optical theory is used in
11253:
11248:
11122:
11117:
10825:
10820:
10807:
10802:
10615:
10610:
10597:
10592:
9941:
9936:
9923:
9918:
9724:
9719:
9706:
9701:
9588:{\textstyle k_{z}=k~\cos \theta ~}
9057:
8930:
8709:
8322:{\textstyle k_{z}=k~\cos \theta ~}
7410:
7400:
7352:
7348:
7300:
7290:
7220:are much smaller than a term with
7047:
7037:
7015:
7005:
6983:
6973:
6958:
6940:
6907:
6899:
6867:
6405:is the wave number. Next, use the
5517:
5511:
5503:
5490:
5482:
5107:
5101:
5093:
5080:
5072:
4875:
4857:
4773:
4730:
4571:
4565:
4557:
4544:
4536:
4212:
3803:
3797:
3789:
3776:
3768:
1642:
1628:
1605:
1591:
1568:
1554:
1535:
1322:
1010:
938:quantity, with separate amplitude
352:
342:
310:
112:, Fourier optics makes use of the
25:
16651:
16547:
11830:provide numerous applications in
10102:{\displaystyle {1}/{(2\pi )^{2}}}
8622:) axis has constant value in any
8402:to making the connection between
7273:) so these two terms are ignored.
5890:(This phenomenon is known as the
1251:{\displaystyle u(\mathbf {r} ,t)}
16560:Advances in Optical Technologies
16217:Nyquist–Shannon sampling theorem
15359:
9405:
9380:
9355:
9329:
9104:is a square matrix, eigenvalues
9021:
9011:
9000:
8851:{\displaystyle k=2\pi /\lambda }
8468:
8452:
8434:
8388:Spatial versus angular bandwidth
7468:
7433:
7368:
7323:
6734:
6717:
6317:
6286:
6268:
6250:
6232:
6224:
6198:
6174:
6166:
6147:
6130:
3416:
3384:
3357:
3330:
3309:
3006:with a generally complex number
1356:
1272:
1235:
1157:
1044:
915:
893:
876:
711:
677:
382:
297:(valid in source-free regions):
239:
208:
13312:{\displaystyle {\hat {o}}(x,y)}
11755:{\displaystyle ~G(k_{x},k_{y})}
11494:is the system transfer function
10363:
8550:{\displaystyle 2D^{2}/\lambda }
7478:{\displaystyle A(\mathbf {r} )}
3115:) of a propagating plane wave.
16403:Introduction to Fourier Optics
16358:Introduction to Fourier Optics
16348:
16333:
16324:
16211:and Airy functions, as in the
16034:
16022:
15695:
15467:
15455:
15352:
15326:
14692:has an amplitude distribution
13889:is then split into two parts:
13364:
13356:
13340:
13306:
13294:
13288:
13260:
13248:
13239:
13227:
13218:
13206:
13197:
13185:
13162:
13150:
13127:
13115:
13092:
13080:
13057:
13045:
12912:). In practical applications,
11804:{\displaystyle ~(k_{x},k_{y})}
11798:
11772:
11749:
11723:
11680:
11674:
11664:The system transfer function,
11649:
11623:
11614:
11588:
11573:
11547:
11512:
11506:
11481:
11475:
11450:
11444:
11418:
11412:
11403:
11397:
11382:
11376:
11295:
11284:
11278:
11261:
11234:
11228:
11164:
11153:
11147:
11130:
11103:
11097:
10911:
10889:
10880:
10840:
10788:
10776:
10688:
10666:
10657:
10623:
10578:
10566:
10538:
10526:
10517:
10505:
10490:
10478:
10429:
10411:
10402:
10390:
10328:
10310:
10301:
10289:
10266:
10219:
10207:
10198:
10186:
10177:
10165:
10129:is related to the input image
10089:
10079:
10018:
9986:
9975:
9949:
9898:
9888:
9876:
9864:
9841:
9829:
9790:
9758:
9744:
9732:
9687:
9661:
9638:
9626:
8917:) above, which may be written
8152:{\textstyle z=r~\cos \theta ~}
8000:
7982:
7962:
7944:
7908:
7848:
7803:
7785:
7761:
7743:
7708:
7690:
7670:
7652:
7632:
7593:
7573:
7555:
7472:
7464:
7437:
7429:
7372:
7364:
7327:
7319:
6784:
6752:
6738:
6730:
6721:
6713:
6690:
6663:
6151:
6143:
6134:
6126:
5598:
5566:
5552:
5526:
5468:
5456:
5420:{\displaystyle \omega =2\pi f}
5188:
5156:
5142:
5116:
5058:
5046:
4910:
4884:
4848:
4836:
4803:
4791:
4765:
4739:
4652:
4620:
4606:
4580:
4522:
4510:
4464:
4459:
4441:
4432:
4420:
4306:
4267:
4247:
4221:
4197:{\displaystyle \omega =2\pi f}
3884:
3852:
3838:
3812:
3754:
3736:
3388:
3361:
3334:
2921:
2889:
2840:
2808:
2705:
2687:
2660:
2642:
2619:
2601:
2490:
2484:
2453:
2447:
2391:
2385:
2354:
2348:
2292:
2286:
2255:
2249:
2104:
2098:
2083:
2077:
2052:
2046:
2031:
2025:
2000:
1994:
1979:
1973:
1931:
1925:
1912:
1906:
1893:
1887:
1861:
1855:
1839:
1833:
1820:
1814:
1798:
1792:
1779:
1773:
1757:
1751:
1735:
1729:
1716:
1710:
1697:
1691:
1494:
1488:
1475:
1469:
1456:
1450:
1434:
1416:
1396:partial differential equations
1382:Solving the Helmholtz equation
1360:
1352:
1276:
1268:
1245:
1231:
1161:
1153:
1048:
1040:
919:
911:
897:
889:
880:
872:
855:{\displaystyle \omega =2\pi f}
715:
707:
687:
673:
598:{\displaystyle \omega =2\pi f}
392:
378:
264:
246:
218:
204:
83:Fourier analysis and synthesis
13:
1:
16317:
12812:
12806:
12662:
12644:
12638:
12632:
12579:interpolation/sampling theory
12427:
12421:
12415:
11928:
11910:
11531:
10986:
10955:
10757:
10732:
8913:
8773:
8685:{\displaystyle e^{i\omega t}}
8649:electromagnetic wave equation
8394:
7927:
7540:
4701:
4400:by the mentioned constraint.
4003:
566:{\displaystyle e^{i\omega t}}
11844:computer generated holograms
9046:particularly since both the
8892:natural modes would include
7123:{\displaystyle \theta \ll 1}
7079:Cartesian coordinates system
6205:{\displaystyle \mathbf {k} }
3670:)-component of that vector.
2666:{\displaystyle \psi (x,y,z)}
2625:{\displaystyle \psi (x,y,z)}
7:
16355:Goodman, Joseph W. (2005).
16261:Adaptive-additive algorithm
16243:
11871:adaptive-additive algorithm
9426:spherical coordinate system
9069:{\displaystyle \nabla ^{2}}
7528:The far field approximation
7150:{\displaystyle \theta ^{2}}
3719:Cartesian coordinate system
3113:Cartesian coordinate system
3030:Cartesian component (e.g.,
1521:elementary product solution
1404:elementary product solution
1388:Cartesian coordinate system
518:Cartesian coordinate system
279:Cartesian coordinate system
90:image processing techniques
10:
16656:
15704:{\displaystyle {\bar {I}}}
12762:theory, which states that
12708:
11686:{\displaystyle H(\omega )}
11518:{\displaystyle F(\omega )}
11487:{\displaystyle H(\omega )}
11456:{\displaystyle G(\omega )}
10061:The normalizing factor of
9156:
8981:eigenvalues / eigenvectors
7531:
6855:The paraxial wave equation
6107:The paraxial approximation
5920:{\displaystyle k_{T}<k}
5431:inverse Fourier transform
4929:constraints), for example
4346:is determined in terms of
1198:and the angular frequency
33:is the study of classical
16566:. Hindawi Limited: 1–15.
16306:Optical Hartley transform
16276:Phase contrast microscopy
16266:Huygens–Fresnel principle
15594:present before bleaching.
14712:and a phase distribution
12652:Lens as a low-pass filter
11832:optical signal processing
11695:optical transfer function
11352:Huygens–Fresnel principle
8653:Electromagnetic radiation
8334:Huygens–Fresnel principle
6411:small-angle approximation
4204:. In this superposition,
1174:is the spatial part of a
632:{\displaystyle f=1/\tau }
49:Huygens–Fresnel principle
16450:Wilson, Raymond (1995).
16233:functional decomposition
16201:functional decomposition
13869:. Then the distribution
11904:in magnitude and phase,
11362:System transfer function
9117:{\displaystyle \lambda }
8590:{\displaystyle \lambda }
4013:with the constraints of
1136:{\displaystyle \lambda }
110:Fourier transform theory
16382:Duffieux, Pierre-Michel
16251:Angular spectrum method
16199:proposed still another
13957:at the recording plane.
13319:. That is, to minimize
12683:space-bandwidth product
11529:In like fashion, eqn. (
11218:) produces the output:
8814:{\displaystyle u=x,y,z}
8006:{\displaystyle (x,y,z)}
7968:{\displaystyle (x,y,z)}
7714:{\displaystyle (x,y,z)}
7676:{\displaystyle (x,y,z)}
7579:{\displaystyle (x,y,z)}
7546:stationary phase method
7097:{\displaystyle \theta }
6611:{\displaystyle \theta }
1392:separation of variables
1211:{\displaystyle \omega }
524:Sinusoidal steady state
16427:Hecht, Eugene (1987).
16281:Fraunhofer diffraction
16193:double-slit experiment
16165:
16062:
15976:
15705:
15674:
15588:
15565:
15520:
15497:
15474:
15473:{\displaystyle s(x,y)}
15439:
15412:
15367:
15311:
14948:
14726:
14706:
14686:
14664:
14052:
13951:
13883:
13863:
13834:
13833:The recording geometry
13817:
13757:
13697:
13644:
13378:
13313:
13267:
13169:
13168:{\displaystyle n(x,y)}
13134:
13133:{\displaystyle s(x,y)}
13099:
13098:{\displaystyle i(x,y)}
13064:
13063:{\displaystyle o(x,y)}
12751:
12371:
12270:
12219:
12171:
12069:
11994:
11946:paraxial approximation
11897:
11805:
11756:
11687:
11656:
11519:
11488:
11457:
11425:
11313:
11182:
10943:
10720:
10545:
10456:
10355:
10226:
10103:
10053:
9848:
9847:{\displaystyle u(x,y)}
9811:
9645:
9644:{\displaystyle u(x,y)}
9589:
9540:
9482:
9418:
9307:
9226:
9175:
9118:
9070:
9038:
8973:
8852:
8815:
8761:
8686:
8616:
8591:
8571:
8551:
8492:equivalence principles
8484:
8406:(on the one hand) and
8377:
8323:
8274:
8216:
8153:
8111:
8060:
8007:
7969:
7915:
7715:
7677:
7639:
7580:
7534:Fraunhofer diffraction
7517:
7479:
7450:is ignored. It is the
7444:
7384:
7267:
7266:{\displaystyle 2k_{z}}
7237:
7214:
7213:{\displaystyle 2k_{y}}
7184:
7183:{\displaystyle 2k_{x}}
7151:
7124:
7098:
7067:
6926:
6846:
6697:
6612:
6590:
6475:
6407:paraxial approximation
6399:
6294:
6206:
6184:
6093:
6066:
6039:
6019:
5999:
5972:
5941:
5921:
5884:
5864:
5837:
5756:
5729:
5690:
5663:
5636:
5421:
5389:
5344:as a real number, and
5338:
5311:
5226:
5019:
4983:, just as in ordinary
4977:
4950:
4920:
4813:
4690:
4487:
4394:
4367:
4340:
4313:
4254:
4198:
4166:
4121:as a real number, and
4115:
4088:
3991:
3711:
3664:
3644:
3624:
3598:
3578:
3551:
3531:
3511:
3485:
3465:
3398:
3295:
3275:
3255:
3229:
3202:
3182:
3162:
3136:
3105:
3078:
3051:
3020:
3000:
2667:
2626:
2583:
2133:
1944:
1665:
1515:, times a function of
1511:, times a function of
1501:
1400:orthogonal coordinates
1373:
1299:
1252:
1212:
1192:
1168:
1137:
1117:
1066:
982:The Helmholtz equation
972:
952:
928:
856:
824:
804:
766:
746:
653:
633:
599:
567:
425:
405:
271:
225:
57:Fraunhofer diffraction
16610:10.1103/physrev.56.99
16554:Ambs, Pierre (2010).
16519:Scott, Craig (1989).
16496:Scott, Craig (1990).
16479:John Wiley & Sons
16473:Scott, Craig (1998).
16456:John Wiley & Sons
16390:John Wiley & Sons
16271:Point spread function
16227:The twin subjects of
16166:
16063:
15977:
15706:
15682:The average exposure
15675:
15589:
15566:
15521:
15519:{\displaystyle 2\pi }
15498:
15496:{\displaystyle \phi }
15475:
15440:
15413:
15368:
15312:
14949:
14727:
14725:{\displaystyle \psi }
14707:
14687:
14665:
14053:
13952:
13884:
13864:
13832:
13818:
13758:
13698:
13645:
13379:
13314:
13268:
13170:
13135:
13100:
13065:
12749:
12679:point spread function
12372:
12298:point spread function
12271:
12220:
12172:
12070:
11995:
11895:
11806:
11757:
11688:
11657:
11520:
11489:
11458:
11426:
11314:
11183:
11085:Dirac delta functions
10944:
10752:above with impunity.
10721:
10546:
10457:
10356:
10275:= 0. The input image
10249:point spread function
10227:
10140:point-spread function
10104:
10054:
9849:
9812:
9646:
9590:
9541:
9483:
9419:
9308:
9227:
9176:
9119:
9091:Chebyshev polynomials
9071:
9039:
8974:
8886:Tacoma Narrows Bridge
8853:
8816:
8762:
8687:
8638:Point spread function
8617:
8592:
8572:
8552:
8485:
8378:
8324:
8275:
8217:
8154:
8112:
8061:
8008:
7970:
7916:
7716:
7678:
7640:
7581:
7518:
7480:
7445:
7385:
7268:
7238:
7215:
7185:
7152:
7125:
7099:
7068:
6927:
6847:
6698:
6613:
6591:
6476:
6400:
6295:
6207:
6185:
6094:
6067:
6065:{\displaystyle k_{z}}
6040:
6020:
6000:
5998:{\displaystyle k_{z}}
5973:
5971:{\displaystyle k_{z}}
5942:
5922:
5885:
5865:
5863:{\displaystyle k_{T}}
5838:
5757:
5755:{\displaystyle k_{T}}
5730:
5691:
5689:{\displaystyle k_{i}}
5664:
5662:{\displaystyle k_{i}}
5637:
5422:
5390:
5339:
5337:{\displaystyle k_{i}}
5312:
5227:
5020:
4978:
4976:{\displaystyle k_{y}}
4951:
4949:{\displaystyle k_{x}}
4921:
4814:
4691:
4488:
4395:
4393:{\displaystyle k_{y}}
4368:
4366:{\displaystyle k_{x}}
4341:
4339:{\displaystyle k_{z}}
4314:
4255:
4199:
4167:
4116:
4114:{\displaystyle k_{i}}
4089:
3992:
3712:
3679:spherical coordinates
3665:
3645:
3625:
3599:
3579:
3577:{\displaystyle k_{i}}
3552:
3532:
3512:
3486:
3466:
3399:
3296:
3276:
3256:
3230:
3228:{\displaystyle k_{i}}
3203:
3183:
3163:
3137:
3135:{\displaystyle k_{i}}
3106:
3104:{\displaystyle E_{z}}
3079:
3077:{\displaystyle E_{y}}
3052:
3050:{\displaystyle E_{x}}
3021:
3001:
2668:
2627:
2584:
2134:
1945:
1666:
1502:
1374:
1300:
1253:
1213:
1193:
1169:
1138:
1118:
1067:
973:
971:{\displaystyle \phi }
953:
929:
857:
825:
805:
767:
747:
654:
652:{\displaystyle \tau }
634:
600:
568:
426:
406:
272:
226:
16311:Wave field synthesis
16237:orthogonal functions
16074:
15989:
15715:
15686:
15601:
15578:
15530:
15507:
15487:
15449:
15429:
15377:
15323:
14960:
14736:
14716:
14696:
14676:
14066:
13968:
13909:
13873:
13853:
13767:
13707:
13654:
13394:
13323:
13279:
13179:
13144:
13109:
13074:
13039:
12318:
12233:
12187:
12079:
12004:
11952:
11766:
11714:
11668:
11541:
11500:
11469:
11438:
11370:
11222:
11091:
10770:
10560:
10472:
10384:
10283:
10159:
10065:
9858:
9854:from its spectrum):
9823:
9655:
9620:
9609:The two-dimensional
9551:
9493:
9435:
9325:
9236:
9185:
9165:
9141:dispersion relations
9108:
9087:Legendre polynomials
9053:
8991:
8921:
8825:
8787:
8705:
8663:
8606:
8581:
8561:
8523:
8430:
8344:
8285:
8227:
8169:
8122:
8071:
8020:
7979:
7941:
7730:
7687:
7649:
7590:
7552:
7489:
7458:
7394:
7279:
7247:
7224:
7194:
7164:
7134:
7108:
7088:
6936:
6863:
6707:
6629:
6602:
6489:
6417:
6304:
6220:
6194:
6120:
6083:
6049:
6029:
6009:
5982:
5955:
5931:
5898:
5874:
5847:
5766:
5739:
5700:
5673:
5646:
5435:
5399:
5348:
5321:
5240:
5033:
5003:
4960:
4933:
4823:
4726:
4497:
4407:
4377:
4350:
4323:
4264:
4208:
4176:
4125:
4098:
4017:
3730:
3701:
3654:
3634:
3608:
3588:
3561:
3541:
3521:
3495:
3475:
3408:
3305:
3285:
3265:
3239:
3212:
3192:
3172:
3146:
3119:
3088:
3061:
3034:
3010:
2677:
2636:
2595:
2203:
2195:separation condition
1954:
1675:
1531:
1410:
1313:
1262:
1225:
1202:
1182:
1147:
1127:
1123:with the wavelength
1076:
990:
962:
942:
866:
834:
814:
780:
756:
667:
643:
609:
577:
544:
528:If light of a fixed
415:
301:
235:
192:
108:used in traditional
16573:10.1155/2010/372652
16286:Fresnel diffraction
16256:Abbe sine condition
16205:Zernike polynomials
16113:
15624:
15016:
14290:
12756:convolution theorem
12687:Abbe sine condition
12569:On the other hand,
12542:entire rest of the
12398:is nearly equal to
12366:
12348:
12179:In the figure, the
11840:optical correlation
11705:Abbe sine condition
11343:), applied at time
11257:
11126:
10829:
10811:
10619:
10601:
10376:. The output image
9945:
9927:
9728:
9710:
9289:
9271:
9253:
9095:Hermite polynomials
8983:of a square matrix
8696:and takes the form
6953:
6880:
6470:
6452:
6434:
6379:
6361:
6343:
6075:In connection with
5819:
5801:
5783:
5515:
5494:
5293:
5275:
5257:
5105:
5084:
5018:{\displaystyle z=0}
4569:
4548:
4070:
4052:
4034:
3953:
3935:
3801:
3780:
3687:spherical harmonics
3623:{\displaystyle i=x}
3510:{\displaystyle i=x}
3301:) in a wave vector
3254:{\displaystyle i=x}
3161:{\displaystyle i=x}
2987:
2969:
2557:
2539:
2521:
2473:
2374:
2275:
2076:
2024:
1972:
1854:
1772:
1690:
173:diffraction grating
142:are commonly used.
62:Fresnel diffraction
16301:Optical correlator
16291:Geometrical optics
16161:
16099:
16058:
15972:
15701:
15670:
15610:
15584:
15561:
15516:
15493:
15483:The maximum phase
15470:
15435:
15408:
15363:
15307:
15002:
14944:
14722:
14702:
14682:
14660:
14658:
14276:
14048:
13947:
13879:
13859:
13835:
13813:
13753:
13693:
13640:
13374:
13309:
13263:
13165:
13130:
13095:
13060:
12752:
12711:Optical correlator
12435:an Airy function,
12367:
12352:
12334:
12266:
12215:
12167:
12065:
11990:
11898:
11801:
11752:
11683:
11652:
11515:
11484:
11453:
11421:
11309:
11240:
11178:
11109:
10939:
10812:
10794:
10716:
10602:
10584:
10541:
10452:
10351:
10222:
10099:
10049:
9928:
9910:
9844:
9807:
9711:
9693:
9641:
9585:
9536:
9478:
9414:
9303:
9275:
9257:
9239:
9222:
9171:
9114:
9066:
9034:
8969:
8888:(3D). Examples of
8848:
8811:
8757:
8694:Helmholtz equation
8682:
8612:
8587:
8567:
8547:
8480:
8373:
8319:
8270:
8212:
8149:
8107:
8056:
8003:
7965:
7911:
7711:
7673:
7635:
7576:
7513:
7475:
7440:
7380:
7263:
7236:{\displaystyle 2k}
7233:
7210:
7180:
7147:
7120:
7094:
7073:is the transverse
7063:
6939:
6922:
6866:
6842:
6693:
6608:
6586:
6584:
6471:
6456:
6438:
6420:
6395:
6365:
6347:
6329:
6290:
6202:
6180:
6089:
6062:
6035:
6015:
5995:
5968:
5949:Numerical Aperture
5937:
5917:
5880:
5860:
5833:
5805:
5787:
5769:
5752:
5725:
5686:
5659:
5632:
5495:
5474:
5417:
5385:
5334:
5307:
5279:
5261:
5243:
5222:
5085:
5064:
5015:
4989:Fourier transforms
4973:
4946:
4916:
4809:
4686:
4549:
4528:
4483:
4390:
4363:
4336:
4309:
4250:
4194:
4162:
4111:
4084:
4056:
4038:
4020:
3987:
3939:
3921:
3781:
3760:
3707:
3660:
3640:
3620:
3594:
3574:
3547:
3527:
3507:
3481:
3461:
3394:
3291:
3271:
3251:
3225:
3198:
3178:
3158:
3132:
3101:
3074:
3047:
3016:
2996:
2994:
2973:
2955:
2663:
2622:
2579:
2577:
2543:
2525:
2507:
2459:
2360:
2261:
2129:
2064:
2012:
1960:
1940:
1842:
1760:
1678:
1661:
1497:
1369:
1307:Helmholtz equation
1295:
1248:
1208:
1188:
1164:
1133:
1113:
1062:
968:
948:
934:is, in general, a
924:
852:
820:
800:
762:
742:
649:
629:
595:
563:
421:
401:
267:
221:
39:Fourier transforms
16388:. New York, USA:
16368:978-0-9747077-2-3
16156:
16053:
15957:
15932:
15864:
15839:
15809:
15774:
15749:
15698:
15655:
15587:{\displaystyle D}
15438:{\displaystyle t}
15406:
15292:
15267:
15199:
15174:
15144:
15109:
15084:
15047:
14929:
14904:
14855:
14830:
14792:
14767:
14705:{\displaystyle A}
14685:{\displaystyle H}
14608:
14583:
14546:
14478:
14453:
14423:
14382:
14357:
14321:
14248:
14223:
14193:
14040:
14015:
13984:
13882:{\displaystyle h}
13862:{\displaystyle h}
13635:
13632:
13359:
13291:
13031:Image restoration
12760:Fourier transform
12725:cross-correlation
12303:Fourier transform
12155:
12136:
12123:
12098:
12053:
12022:
11970:
11887:Fourier transform
11863:quantum computing
11836:spatial filtering
11824:Fourier transform
11771:
11719:
11584:
11578:
11393:
11387:
10963:
10962:
10916:
10885:
10740:
10739:
10693:
10662:
10501:
10495:
9908:
9611:Fourier transform
9584:
9572:
9526:
9514:
9468:
9456:
9411:
9386:
9361:
9341:
9335:
9220:
9202:
9174:{\displaystyle k}
8781:
8780:
8615:{\displaystyle z}
8570:{\displaystyle D}
8474:
8459:
8446:
8440:
8426:, and defined as
8408:angular bandwidth
8404:spatial bandwidth
8371:
8318:
8306:
8260:
8248:
8202:
8190:
8148:
8136:
8097:
8085:
8046:
8034:
7935:
7934:
7898:
7886:
7868:
7856:
7837:
7833:
7808:
7793:
7784:
7772:
7766:
7511:
7424:
7359:
7314:
7130:) so a term with
7061:
7029:
6997:
6914:
6393:
6380:
6092:{\displaystyle k}
6038:{\displaystyle z}
6018:{\displaystyle z}
5940:{\displaystyle k}
5892:diffraction limit
5883:{\displaystyle k}
5723:
5605:
5557:
5452:
5383:
5365:
5195:
5147:
4715:(hence the name,
4659:
4611:
4160:
4142:
4011:
4010:
3960:
3954:
3891:
3843:
3710:{\displaystyle f}
3663:{\displaystyle z}
3643:{\displaystyle y}
3597:{\displaystyle i}
3584:means a negative
3550:{\displaystyle z}
3530:{\displaystyle y}
3484:{\displaystyle i}
3459:
3441:
3391:
3364:
3337:
3294:{\displaystyle z}
3274:{\displaystyle y}
3201:{\displaystyle z}
3181:{\displaystyle y}
3019:{\displaystyle A}
2988:
2435:
2336:
2237:
2193:, along with one
2108:
2056:
2004:
1656:
1619:
1582:
1191:{\displaystyle k}
1111:
1093:
951:{\displaystyle a}
823:{\displaystyle x}
765:{\displaystyle i}
530:frequency in time
424:{\displaystyle c}
368:
337:
289:The wave equation
285:represents time.
140:signal processing
114:spatial frequency
16:(Redirected from
16647:
16640:Fourier analysis
16621:
16595:
16585:
16575:
16538:
16515:
16492:
16469:
16446:
16423:
16421:
16420:
16393:
16373:
16372:
16352:
16346:
16345:
16337:
16331:
16328:
16188:Green's function
16170:
16168:
16167:
16162:
16157:
16155:
16154:
16153:
16148:
16133:
16132:
16123:
16122:
16112:
16107:
16097:
16096:
16087:
16067:
16065:
16064:
16059:
16054:
16049:
16041:
15981:
15979:
15978:
15973:
15968:
15964:
15963:
15959:
15958:
15956:
15948:
15947:
15938:
15933:
15931:
15923:
15922:
15913:
15900:
15899:
15870:
15866:
15865:
15863:
15855:
15854:
15845:
15840:
15838:
15830:
15829:
15820:
15810:
15808:
15800:
15799:
15798:
15785:
15780:
15776:
15775:
15773:
15765:
15764:
15755:
15750:
15748:
15740:
15739:
15730:
15710:
15708:
15707:
15702:
15700:
15699:
15691:
15679:
15677:
15676:
15671:
15666:
15665:
15656:
15654:
15653:
15652:
15643:
15642:
15629:
15623:
15618:
15593:
15591:
15590:
15585:
15570:
15568:
15567:
15562:
15545:
15544:
15526:radians so that
15525:
15523:
15522:
15517:
15502:
15500:
15499:
15494:
15479:
15477:
15476:
15471:
15445:proportional to
15444:
15442:
15441:
15436:
15417:
15415:
15414:
15409:
15407:
15405:
15394:
15389:
15388:
15372:
15370:
15369:
15364:
15362:
15351:
15350:
15338:
15337:
15316:
15314:
15313:
15308:
15303:
15299:
15298:
15294:
15293:
15291:
15283:
15282:
15273:
15268:
15266:
15258:
15257:
15248:
15235:
15234:
15205:
15201:
15200:
15198:
15190:
15189:
15180:
15175:
15173:
15165:
15164:
15155:
15145:
15143:
15135:
15134:
15133:
15120:
15115:
15111:
15110:
15108:
15100:
15099:
15090:
15085:
15083:
15075:
15074:
15065:
15058:
15057:
15048:
15046:
15045:
15044:
15035:
15034:
15021:
15015:
15010:
14998:
14994:
14993:
14992:
14980:
14979:
14953:
14951:
14950:
14945:
14940:
14936:
14935:
14931:
14930:
14928:
14920:
14919:
14910:
14905:
14903:
14895:
14894:
14885:
14861:
14857:
14856:
14854:
14846:
14845:
14836:
14831:
14829:
14821:
14820:
14811:
14798:
14794:
14793:
14791:
14783:
14782:
14773:
14768:
14766:
14758:
14757:
14748:
14731:
14729:
14728:
14723:
14711:
14709:
14708:
14703:
14691:
14689:
14688:
14683:
14669:
14667:
14666:
14661:
14659:
14655:
14651:
14650:
14649:
14614:
14610:
14609:
14607:
14599:
14598:
14589:
14584:
14582:
14574:
14573:
14564:
14557:
14556:
14547:
14545:
14537:
14536:
14527:
14522:
14518:
14517:
14516:
14484:
14480:
14479:
14477:
14469:
14468:
14459:
14454:
14452:
14444:
14443:
14434:
14424:
14422:
14414:
14413:
14404:
14399:
14398:
14393:
14389:
14388:
14384:
14383:
14381:
14373:
14372:
14363:
14358:
14356:
14348:
14347:
14338:
14322:
14320:
14319:
14318:
14309:
14308:
14295:
14289:
14284:
14269:
14265:
14264:
14259:
14255:
14254:
14250:
14249:
14247:
14239:
14238:
14229:
14224:
14222:
14214:
14213:
14204:
14194:
14192:
14181:
14176:
14172:
14171:
14170:
14135:
14134:
14112:
14108:
14107:
14106:
14094:
14093:
14079:
14078:
14057:
14055:
14054:
14049:
14047:
14046:
14042:
14041:
14039:
14031:
14030:
14021:
14016:
14014:
14006:
14005:
13996:
13985:
13983:
13972:
13956:
13954:
13953:
13948:
13946:
13945:
13921:
13920:
13888:
13886:
13885:
13880:
13868:
13866:
13865:
13860:
13822:
13820:
13819:
13814:
13812:
13808:
13807:
13806:
13794:
13793:
13779:
13778:
13762:
13760:
13759:
13754:
13752:
13748:
13747:
13746:
13734:
13733:
13719:
13718:
13702:
13700:
13699:
13694:
13692:
13688:
13687:
13686:
13674:
13673:
13649:
13647:
13646:
13641:
13636:
13634:
13633:
13631:
13630:
13626:
13625:
13624:
13612:
13611:
13597:
13596:
13586:
13585:
13581:
13580:
13579:
13567:
13566:
13552:
13551:
13541:
13536:
13535:
13530:
13526:
13525:
13521:
13520:
13519:
13507:
13506:
13482:
13481:
13477:
13476:
13475:
13463:
13462:
13448:
13447:
13437:
13432:
13428:
13427:
13426:
13414:
13413:
13383:
13381:
13380:
13375:
13373:
13372:
13367:
13361:
13360:
13352:
13343:
13335:
13334:
13318:
13316:
13315:
13310:
13293:
13292:
13284:
13272:
13270:
13269:
13264:
13174:
13172:
13171:
13166:
13139:
13137:
13136:
13131:
13104:
13102:
13101:
13096:
13069:
13067:
13066:
13061:
12898:impulse response
12766:in the spatial (
12529:entire infinite
12456:window functions
12452:Gibbs phenomenon
12376:
12374:
12373:
12368:
12365:
12360:
12347:
12342:
12330:
12329:
12275:
12273:
12272:
12267:
12265:
12264:
12254:
12224:
12222:
12221:
12216:
12214:
12213:
12176:
12174:
12173:
12168:
12166:
12165:
12156:
12148:
12137:
12135:
12134:
12133:
12124:
12116:
12104:
12099:
12097:
12083:
12074:
12072:
12071:
12066:
12064:
12063:
12054:
12046:
12023:
12018:
12017:
12008:
11999:
11997:
11996:
11991:
11971:
11966:
11965:
11956:
11914:) (specified to
11855:optical tweezers
11826:properties of a
11810:
11808:
11807:
11802:
11797:
11796:
11784:
11783:
11769:
11761:
11759:
11758:
11753:
11748:
11747:
11735:
11734:
11717:
11692:
11690:
11689:
11684:
11661:
11659:
11658:
11653:
11648:
11647:
11635:
11634:
11613:
11612:
11600:
11599:
11582:
11576:
11572:
11571:
11559:
11558:
11524:
11522:
11521:
11516:
11493:
11491:
11490:
11485:
11462:
11460:
11459:
11454:
11430:
11428:
11427:
11422:
11391:
11385:
11356:Green's function
11318:
11316:
11315:
11310:
11308:
11294:
11277:
11256:
11251:
11196:impulse response
11187:
11185:
11184:
11179:
11177:
11163:
11146:
11125:
11120:
11081:sifting property
10957:
10948:
10946:
10945:
10940:
10938:
10927:
10914:
10910:
10899:
10883:
10879:
10859:
10839:
10838:
10828:
10823:
10810:
10805:
10764:
10745:shift invariance
10734:
10725:
10723:
10722:
10717:
10715:
10704:
10691:
10687:
10676:
10660:
10656:
10639:
10618:
10613:
10600:
10595:
10554:
10550:
10548:
10547:
10542:
10499:
10493:
10461:
10459:
10458:
10453:
10451:
10450:
10439:
10438:
10360:
10358:
10357:
10352:
10350:
10349:
10338:
10337:
10257:image processing
10231:
10229:
10228:
10223:
10108:
10106:
10105:
10100:
10098:
10097:
10096:
10077:
10072:
10058:
10056:
10055:
10050:
10048:
10047:
10035:
10034:
10022:
10021:
10014:
10013:
9998:
9997:
9974:
9973:
9961:
9960:
9944:
9939:
9926:
9921:
9909:
9907:
9906:
9905:
9883:
9853:
9851:
9850:
9845:
9816:
9814:
9813:
9808:
9794:
9793:
9786:
9785:
9770:
9769:
9727:
9722:
9709:
9704:
9686:
9685:
9673:
9672:
9650:
9648:
9647:
9642:
9594:
9592:
9591:
9586:
9582:
9570:
9563:
9562:
9545:
9543:
9542:
9537:
9524:
9512:
9505:
9504:
9487:
9485:
9484:
9479:
9466:
9454:
9447:
9446:
9423:
9421:
9420:
9415:
9413:
9412:
9404:
9401:
9400:
9388:
9387:
9379:
9376:
9375:
9363:
9362:
9354:
9351:
9350:
9339:
9333:
9332:
9315:Euclidean metric
9312:
9310:
9309:
9304:
9302:
9301:
9288:
9283:
9270:
9265:
9252:
9247:
9231:
9229:
9228:
9223:
9221:
9216:
9208:
9203:
9195:
9180:
9178:
9177:
9172:
9123:
9121:
9120:
9115:
9075:
9073:
9072:
9067:
9065:
9064:
9048:scalar Laplacian
9043:
9041:
9040:
9035:
9024:
9019:
9015:
9014:
9003:
8978:
8976:
8975:
8970:
8956:
8952:
8951:
8950:
8938:
8937:
8857:
8855:
8854:
8849:
8844:
8820:
8818:
8817:
8812:
8775:
8766:
8764:
8763:
8758:
8750:
8749:
8740:
8739:
8727:
8726:
8717:
8716:
8699:
8691:
8689:
8688:
8683:
8681:
8680:
8657:frequency domain
8621:
8619:
8618:
8613:
8596:
8594:
8593:
8588:
8576:
8574:
8573:
8568:
8556:
8554:
8553:
8548:
8543:
8538:
8537:
8489:
8487:
8486:
8481:
8476:
8475:
8467:
8461:
8460:
8457:
8455:
8444:
8438:
8437:
8382:
8380:
8379:
8374:
8372:
8367:
8366:
8348:
8338:Green's function
8328:
8326:
8325:
8320:
8316:
8304:
8297:
8296:
8279:
8277:
8276:
8271:
8258:
8246:
8239:
8238:
8221:
8219:
8218:
8213:
8200:
8188:
8181:
8180:
8158:
8156:
8155:
8150:
8146:
8134:
8116:
8114:
8113:
8108:
8095:
8083:
8065:
8063:
8062:
8057:
8044:
8032:
8012:
8010:
8009:
8004:
7974:
7972:
7971:
7966:
7929:
7920:
7918:
7917:
7912:
7896:
7884:
7866:
7854:
7847:
7846:
7835:
7834:
7829:
7828:
7810:
7806:
7791:
7782:
7770:
7764:
7742:
7741:
7724:
7720:
7718:
7717:
7712:
7682:
7680:
7679:
7674:
7644:
7642:
7641:
7636:
7631:
7630:
7618:
7617:
7605:
7604:
7585:
7583:
7582:
7577:
7522:
7520:
7519:
7514:
7512:
7507:
7499:
7484:
7482:
7481:
7476:
7471:
7449:
7447:
7446:
7441:
7436:
7425:
7423:
7422:
7421:
7408:
7407:
7398:
7389:
7387:
7386:
7381:
7379:
7375:
7371:
7360:
7358:
7347:
7334:
7330:
7326:
7315:
7313:
7312:
7311:
7298:
7297:
7288:
7272:
7270:
7269:
7264:
7262:
7261:
7242:
7240:
7239:
7234:
7219:
7217:
7216:
7211:
7209:
7208:
7189:
7187:
7186:
7181:
7179:
7178:
7156:
7154:
7153:
7148:
7146:
7145:
7129:
7127:
7126:
7121:
7103:
7101:
7100:
7095:
7075:Laplace operator
7072:
7070:
7069:
7064:
7062:
7060:
7059:
7058:
7045:
7044:
7035:
7030:
7028:
7027:
7026:
7013:
7012:
7003:
6998:
6996:
6995:
6994:
6981:
6980:
6971:
6966:
6965:
6952:
6947:
6931:
6929:
6928:
6923:
6915:
6913:
6905:
6897:
6879:
6874:
6851:
6849:
6848:
6843:
6841:
6840:
6822:
6821:
6817:
6812:
6811:
6788:
6787:
6780:
6779:
6764:
6763:
6737:
6720:
6702:
6700:
6699:
6694:
6686:
6681:
6680:
6641:
6640:
6617:
6615:
6614:
6609:
6595:
6593:
6592:
6587:
6585:
6578:
6573:
6572:
6480:
6478:
6477:
6472:
6469:
6464:
6451:
6446:
6433:
6428:
6404:
6402:
6401:
6396:
6394:
6386:
6381:
6378:
6373:
6360:
6355:
6342:
6337:
6328:
6320:
6299:
6297:
6296:
6291:
6289:
6284:
6283:
6271:
6266:
6265:
6253:
6248:
6247:
6235:
6227:
6211:
6209:
6208:
6203:
6201:
6189:
6187:
6186:
6181:
6179:
6178:
6177:
6169:
6150:
6133:
6098:
6096:
6095:
6090:
6077:photolithography
6071:
6069:
6068:
6063:
6061:
6060:
6044:
6042:
6041:
6036:
6024:
6022:
6021:
6016:
6004:
6002:
6001:
5996:
5994:
5993:
5977:
5975:
5974:
5969:
5967:
5966:
5946:
5944:
5943:
5938:
5926:
5924:
5923:
5918:
5910:
5909:
5889:
5887:
5886:
5881:
5869:
5867:
5866:
5861:
5859:
5858:
5842:
5840:
5839:
5834:
5832:
5831:
5818:
5813:
5800:
5795:
5782:
5777:
5761:
5759:
5758:
5753:
5751:
5750:
5734:
5732:
5731:
5726:
5724:
5722:
5714:
5713:
5704:
5695:
5693:
5692:
5687:
5685:
5684:
5668:
5666:
5665:
5660:
5658:
5657:
5641:
5639:
5638:
5633:
5631:
5630:
5618:
5617:
5603:
5602:
5601:
5594:
5593:
5578:
5577:
5555:
5551:
5550:
5538:
5537:
5525:
5524:
5514:
5506:
5493:
5485:
5455:
5454:
5453:
5450:
5426:
5424:
5423:
5418:
5394:
5392:
5391:
5386:
5384:
5379:
5371:
5366:
5358:
5343:
5341:
5340:
5335:
5333:
5332:
5316:
5314:
5313:
5308:
5306:
5305:
5292:
5287:
5274:
5269:
5256:
5251:
5231:
5229:
5228:
5223:
5221:
5220:
5208:
5207:
5193:
5192:
5191:
5184:
5183:
5168:
5167:
5145:
5141:
5140:
5128:
5127:
5115:
5114:
5104:
5096:
5083:
5075:
5045:
5044:
5024:
5022:
5021:
5016:
4985:Fourier analysis
4982:
4980:
4979:
4974:
4972:
4971:
4955:
4953:
4952:
4947:
4945:
4944:
4925:
4923:
4922:
4917:
4909:
4908:
4896:
4895:
4883:
4882:
4870:
4869:
4861:
4860:
4835:
4834:
4818:
4816:
4815:
4810:
4790:
4789:
4777:
4776:
4764:
4763:
4751:
4750:
4738:
4737:
4695:
4693:
4692:
4687:
4685:
4684:
4672:
4671:
4657:
4656:
4655:
4648:
4647:
4632:
4631:
4609:
4605:
4604:
4592:
4591:
4579:
4578:
4568:
4560:
4547:
4539:
4509:
4508:
4492:
4490:
4489:
4484:
4479:
4478:
4467:
4419:
4418:
4399:
4397:
4396:
4391:
4389:
4388:
4372:
4370:
4369:
4364:
4362:
4361:
4345:
4343:
4342:
4337:
4335:
4334:
4318:
4316:
4315:
4310:
4305:
4304:
4292:
4291:
4279:
4278:
4259:
4257:
4256:
4251:
4246:
4245:
4233:
4232:
4220:
4219:
4203:
4201:
4200:
4195:
4171:
4169:
4168:
4163:
4161:
4156:
4148:
4143:
4135:
4120:
4118:
4117:
4112:
4110:
4109:
4093:
4091:
4090:
4085:
4083:
4082:
4069:
4064:
4051:
4046:
4033:
4028:
4005:
3996:
3994:
3993:
3988:
3986:
3985:
3973:
3972:
3958:
3957:
3956:
3955:
3952:
3947:
3934:
3929:
3917:
3916:
3907:
3889:
3888:
3887:
3880:
3879:
3864:
3863:
3841:
3837:
3836:
3824:
3823:
3811:
3810:
3800:
3792:
3779:
3771:
3724:
3716:
3714:
3713:
3708:
3669:
3667:
3666:
3661:
3649:
3647:
3646:
3641:
3629:
3627:
3626:
3621:
3603:
3601:
3600:
3595:
3583:
3581:
3580:
3575:
3573:
3572:
3556:
3554:
3553:
3548:
3536:
3534:
3533:
3528:
3516:
3514:
3513:
3508:
3490:
3488:
3487:
3482:
3470:
3468:
3467:
3462:
3460:
3455:
3447:
3442:
3434:
3423:
3419:
3403:
3401:
3400:
3395:
3393:
3392:
3387:
3382:
3379:
3378:
3366:
3365:
3360:
3355:
3352:
3351:
3339:
3338:
3333:
3328:
3325:
3324:
3312:
3300:
3298:
3297:
3292:
3280:
3278:
3277:
3272:
3260:
3258:
3257:
3252:
3234:
3232:
3231:
3226:
3224:
3223:
3207:
3205:
3204:
3199:
3187:
3185:
3184:
3179:
3167:
3165:
3164:
3159:
3141:
3139:
3138:
3133:
3131:
3130:
3110:
3108:
3107:
3102:
3100:
3099:
3083:
3081:
3080:
3075:
3073:
3072:
3056:
3054:
3053:
3048:
3046:
3045:
3025:
3023:
3022:
3017:
3005:
3003:
3002:
2997:
2995:
2991:
2990:
2989:
2986:
2981:
2968:
2963:
2951:
2950:
2941:
2925:
2924:
2917:
2916:
2901:
2900:
2871:
2867:
2866:
2862:
2861:
2844:
2843:
2836:
2835:
2820:
2819:
2790:
2786:
2785:
2781:
2780:
2763:
2762:
2758:
2757:
2740:
2739:
2735:
2734:
2672:
2670:
2669:
2664:
2631:
2629:
2628:
2623:
2588:
2586:
2585:
2580:
2578:
2574:
2573:
2556:
2551:
2538:
2533:
2520:
2515:
2483:
2482:
2472:
2467:
2446:
2445:
2436:
2434:
2433:
2432:
2419:
2418:
2409:
2384:
2383:
2373:
2368:
2347:
2346:
2337:
2335:
2334:
2333:
2320:
2319:
2310:
2285:
2284:
2274:
2269:
2248:
2247:
2238:
2236:
2235:
2234:
2221:
2220:
2211:
2138:
2136:
2135:
2130:
2122:
2121:
2109:
2107:
2097:
2096:
2086:
2072:
2062:
2057:
2055:
2045:
2044:
2034:
2020:
2010:
2005:
2003:
1993:
1992:
1982:
1968:
1958:
1949:
1947:
1946:
1941:
1924:
1923:
1905:
1904:
1886:
1885:
1876:
1875:
1850:
1832:
1831:
1813:
1812:
1791:
1790:
1768:
1750:
1749:
1728:
1727:
1709:
1708:
1686:
1670:
1668:
1667:
1662:
1657:
1655:
1654:
1653:
1640:
1636:
1635:
1625:
1620:
1618:
1617:
1616:
1603:
1599:
1598:
1588:
1583:
1581:
1580:
1579:
1566:
1562:
1561:
1551:
1543:
1542:
1525:scalar Laplacian
1506:
1504:
1503:
1498:
1487:
1486:
1468:
1467:
1449:
1448:
1378:
1376:
1375:
1370:
1359:
1348:
1344:
1343:
1342:
1330:
1329:
1304:
1302:
1301:
1296:
1294:
1293:
1275:
1257:
1255:
1254:
1249:
1238:
1217:
1215:
1214:
1209:
1197:
1195:
1194:
1189:
1173:
1171:
1170:
1165:
1160:
1142:
1140:
1139:
1134:
1122:
1120:
1119:
1114:
1112:
1107:
1099:
1094:
1086:
1071:
1069:
1068:
1063:
1055:
1051:
1047:
1036:
1032:
1031:
1030:
1018:
1017:
977:
975:
974:
969:
957:
955:
954:
949:
933:
931:
930:
925:
923:
922:
918:
896:
879:
861:
859:
858:
853:
829:
827:
826:
821:
809:
807:
806:
801:
799:
771:
769:
768:
763:
751:
749:
748:
743:
738:
734:
733:
732:
714:
680:
658:
656:
655:
650:
638:
636:
635:
630:
625:
604:
602:
601:
596:
572:
570:
569:
564:
562:
561:
498:
480:
430:
428:
427:
422:
410:
408:
407:
402:
385:
374:
370:
369:
367:
366:
365:
360:
350:
349:
340:
338:
336:
335:
323:
318:
317:
276:
274:
273:
268:
242:
230:
228:
227:
222:
211:
147:photolithography
79:plane wave modes
21:
16655:
16654:
16650:
16649:
16648:
16646:
16645:
16644:
16635:Physical optics
16625:
16624:
16598:Physical Review
16593:
16550:
16535:
16512:
16489:
16466:
16443:
16418:
16416:
16414:
16398:Goodman, Joseph
16377:
16376:
16369:
16353:
16349:
16342:Physica Scripta
16338:
16334:
16329:
16325:
16320:
16315:
16246:
16231:expansions and
16225:
16180:Huygens–Fresnel
16176:
16149:
16138:
16137:
16128:
16124:
16118:
16114:
16108:
16103:
16098:
16092:
16088:
16086:
16075:
16072:
16071:
16042:
16040:
15990:
15987:
15986:
15949:
15943:
15939:
15937:
15924:
15918:
15914:
15912:
15911:
15907:
15895:
15891:
15881:
15877:
15856:
15850:
15846:
15844:
15831:
15825:
15821:
15819:
15818:
15814:
15801:
15794:
15790:
15786:
15784:
15766:
15760:
15756:
15754:
15741:
15735:
15731:
15729:
15728:
15724:
15716:
15713:
15712:
15690:
15689:
15687:
15684:
15683:
15661:
15657:
15648:
15644:
15638:
15634:
15633:
15628:
15619:
15614:
15602:
15599:
15598:
15579:
15576:
15575:
15537:
15533:
15531:
15528:
15527:
15508:
15505:
15504:
15488:
15485:
15484:
15450:
15447:
15446:
15430:
15427:
15426:
15398:
15393:
15384:
15380:
15378:
15375:
15374:
15358:
15346:
15342:
15333:
15329:
15324:
15321:
15320:
15284:
15278:
15274:
15272:
15259:
15253:
15249:
15247:
15246:
15242:
15230:
15226:
15216:
15212:
15191:
15185:
15181:
15179:
15166:
15160:
15156:
15154:
15153:
15149:
15136:
15129:
15125:
15121:
15119:
15101:
15095:
15091:
15089:
15076:
15070:
15066:
15064:
15063:
15059:
15053:
15049:
15040:
15036:
15030:
15026:
15025:
15020:
15011:
15006:
14988:
14984:
14975:
14971:
14970:
14966:
14961:
14958:
14957:
14921:
14915:
14911:
14909:
14896:
14890:
14886:
14884:
14883:
14879:
14872:
14868:
14847:
14841:
14837:
14835:
14822:
14816:
14812:
14810:
14809:
14805:
14784:
14778:
14774:
14772:
14759:
14753:
14749:
14747:
14746:
14742:
14737:
14734:
14733:
14717:
14714:
14713:
14697:
14694:
14693:
14677:
14674:
14673:
14657:
14656:
14645:
14641:
14625:
14621:
14600:
14594:
14590:
14588:
14575:
14569:
14565:
14563:
14562:
14558:
14552:
14548:
14538:
14532:
14528:
14526:
14512:
14508:
14495:
14491:
14470:
14464:
14460:
14458:
14445:
14439:
14435:
14433:
14432:
14428:
14415:
14409:
14405:
14403:
14394:
14374:
14368:
14364:
14362:
14349:
14343:
14339:
14337:
14336:
14332:
14328:
14324:
14323:
14314:
14310:
14304:
14300:
14299:
14294:
14285:
14280:
14267:
14266:
14260:
14240:
14234:
14230:
14228:
14215:
14209:
14205:
14203:
14202:
14198:
14185:
14180:
14166:
14162:
14146:
14142:
14130:
14126:
14125:
14121:
14120:
14113:
14102:
14098:
14089:
14085:
14084:
14080:
14074:
14073:
14069:
14067:
14064:
14063:
14032:
14026:
14022:
14020:
14007:
14001:
13997:
13995:
13994:
13990:
13989:
13976:
13971:
13969:
13966:
13965:
13926:
13922:
13916:
13912:
13910:
13907:
13906:
13874:
13871:
13870:
13854:
13851:
13850:
13826:
13802:
13798:
13789:
13785:
13784:
13780:
13774:
13770:
13768:
13765:
13764:
13742:
13738:
13729:
13725:
13724:
13720:
13714:
13710:
13708:
13705:
13704:
13682:
13678:
13669:
13665:
13664:
13660:
13655:
13652:
13651:
13620:
13616:
13607:
13603:
13602:
13598:
13592:
13588:
13587:
13575:
13571:
13562:
13558:
13557:
13553:
13547:
13543:
13542:
13540:
13531:
13515:
13511:
13502:
13498:
13497:
13493:
13489:
13485:
13484:
13483:
13471:
13467:
13458:
13454:
13453:
13449:
13443:
13439:
13438:
13436:
13422:
13418:
13409:
13405:
13404:
13400:
13395:
13392:
13391:
13368:
13363:
13362:
13351:
13350:
13339:
13330:
13326:
13324:
13321:
13320:
13283:
13282:
13280:
13277:
13276:
13180:
13177:
13176:
13145:
13142:
13141:
13110:
13107:
13106:
13075:
13072:
13071:
13040:
13037:
13036:
13033:
13026:
13019:
13008:
13001:
12966:
12959:
12948:
12941:
12863:
12856:
12845:
12838:
12790:spectral domain
12788:) domain (aka:
12787:
12780:
12713:
12707:
12695:
12658:Low-pass filter
12654:
12596:
12488:
12465:
12441:
12411:
12396:
12389:
12382:
12361:
12356:
12343:
12338:
12325:
12321:
12319:
12316:
12315:
12250:
12240:
12236:
12234:
12231:
12230:
12194:
12190:
12188:
12185:
12184:
12161:
12157:
12147:
12129:
12125:
12115:
12108:
12103:
12087:
12082:
12080:
12077:
12076:
12059:
12055:
12045:
12013:
12009:
12007:
12005:
12002:
12001:
11961:
11957:
11955:
11953:
11950:
11949:
11879:
11817:
11792:
11788:
11779:
11775:
11767:
11764:
11763:
11743:
11739:
11730:
11726:
11715:
11712:
11711:
11669:
11666:
11665:
11643:
11639:
11630:
11626:
11608:
11604:
11595:
11591:
11567:
11563:
11554:
11550:
11542:
11539:
11538:
11501:
11498:
11497:
11470:
11467:
11466:
11439:
11436:
11435:
11371:
11368:
11367:
11364:
11301:
11287:
11270:
11252:
11244:
11223:
11220:
11219:
11198:of the system,
11170:
11156:
11139:
11121:
11113:
11092:
11089:
11088:
11042:
11006:
10995:
10970:
10931:
10920:
10903:
10892:
10872:
10852:
10834:
10830:
10824:
10816:
10806:
10798:
10771:
10768:
10767:
10708:
10697:
10680:
10669:
10649:
10632:
10614:
10606:
10596:
10588:
10561:
10558:
10557:
10473:
10470:
10469:
10467:
10440:
10434:
10433:
10432:
10385:
10382:
10381:
10366:
10339:
10333:
10332:
10331:
10284:
10281:
10280:
10269:
10160:
10157:
10156:
10115:
10092:
10088:
10078:
10073:
10068:
10066:
10063:
10062:
10043:
10039:
10030:
10026:
10009:
10005:
9993:
9989:
9982:
9978:
9969:
9965:
9956:
9952:
9940:
9932:
9922:
9914:
9901:
9897:
9887:
9882:
9859:
9856:
9855:
9824:
9821:
9820:
9781:
9777:
9765:
9761:
9751:
9747:
9723:
9715:
9705:
9697:
9681:
9677:
9668:
9664:
9656:
9653:
9652:
9621:
9618:
9617:
9614:
9558:
9554:
9552:
9549:
9548:
9500:
9496:
9494:
9491:
9490:
9442:
9438:
9436:
9433:
9432:
9403:
9402:
9396:
9392:
9378:
9377:
9371:
9367:
9353:
9352:
9346:
9342:
9328:
9326:
9323:
9322:
9297:
9293:
9284:
9279:
9266:
9261:
9248:
9243:
9237:
9234:
9233:
9209:
9207:
9194:
9186:
9183:
9182:
9166:
9163:
9162:
9159:
9109:
9106:
9105:
9076:and the matrix
9060:
9056:
9054:
9051:
9050:
9020:
9010:
8999:
8998:
8994:
8992:
8989:
8988:
8946:
8942:
8933:
8929:
8928:
8924:
8922:
8919:
8918:
8868:
8862:of the medium.
8840:
8826:
8823:
8822:
8788:
8785:
8784:
8745:
8741:
8735:
8731:
8722:
8718:
8712:
8708:
8706:
8703:
8702:
8670:
8666:
8664:
8661:
8660:
8607:
8604:
8603:
8582:
8579:
8578:
8562:
8559:
8558:
8539:
8533:
8529:
8524:
8521:
8520:
8505:
8466:
8465:
8456:
8451:
8450:
8433:
8431:
8428:
8427:
8390:
8353:
8349:
8347:
8345:
8342:
8341:
8292:
8288:
8286:
8283:
8282:
8234:
8230:
8228:
8225:
8224:
8176:
8172:
8170:
8167:
8166:
8123:
8120:
8119:
8072:
8069:
8068:
8021:
8018:
8017:
7980:
7977:
7976:
7942:
7939:
7938:
7842:
7838:
7815:
7811:
7809:
7737:
7733:
7731:
7728:
7727:
7688:
7685:
7684:
7650:
7647:
7646:
7626:
7622:
7613:
7609:
7600:
7596:
7591:
7588:
7587:
7553:
7550:
7549:
7536:
7530:
7500:
7498:
7490:
7487:
7486:
7467:
7459:
7456:
7455:
7432:
7417:
7413:
7409:
7403:
7399:
7397:
7395:
7392:
7391:
7390:so a term with
7367:
7351:
7346:
7342:
7338:
7322:
7307:
7303:
7299:
7293:
7289:
7287:
7286:
7282:
7280:
7277:
7276:
7257:
7253:
7248:
7245:
7244:
7225:
7222:
7221:
7204:
7200:
7195:
7192:
7191:
7174:
7170:
7165:
7162:
7161:
7141:
7137:
7135:
7132:
7131:
7109:
7106:
7105:
7089:
7086:
7085:
7054:
7050:
7046:
7040:
7036:
7034:
7022:
7018:
7014:
7008:
7004:
7002:
6990:
6986:
6982:
6976:
6972:
6970:
6961:
6957:
6948:
6943:
6937:
6934:
6933:
6906:
6898:
6896:
6875:
6870:
6864:
6861:
6860:
6857:
6827:
6823:
6813:
6807:
6803:
6793:
6789:
6775:
6771:
6759:
6755:
6745:
6741:
6733:
6716:
6708:
6705:
6704:
6682:
6676:
6672:
6636:
6632:
6630:
6627:
6626:
6603:
6600:
6599:
6583:
6582:
6574:
6568:
6564:
6551:
6539:
6538:
6528:
6516:
6515:
6505:
6492:
6490:
6487:
6486:
6465:
6460:
6447:
6442:
6429:
6424:
6418:
6415:
6414:
6385:
6374:
6369:
6356:
6351:
6338:
6333:
6327:
6316:
6305:
6302:
6301:
6285:
6279:
6275:
6267:
6261:
6257:
6249:
6243:
6239:
6231:
6223:
6221:
6218:
6217:
6197:
6195:
6192:
6191:
6173:
6165:
6158:
6154:
6146:
6129:
6121:
6118:
6117:
6114:
6109:
6084:
6081:
6080:
6056:
6052:
6050:
6047:
6046:
6030:
6027:
6026:
6010:
6007:
6006:
5989:
5985:
5983:
5980:
5979:
5962:
5958:
5956:
5953:
5952:
5932:
5929:
5928:
5905:
5901:
5899:
5896:
5895:
5875:
5872:
5871:
5854:
5850:
5848:
5845:
5844:
5827:
5823:
5814:
5809:
5796:
5791:
5778:
5773:
5767:
5764:
5763:
5746:
5742:
5740:
5737:
5736:
5715:
5709:
5705:
5703:
5701:
5698:
5697:
5680:
5676:
5674:
5671:
5670:
5653:
5649:
5647:
5644:
5643:
5626:
5622:
5613:
5609:
5589:
5585:
5573:
5569:
5562:
5558:
5546:
5542:
5533:
5529:
5520:
5516:
5507:
5499:
5486:
5478:
5449:
5442:
5438:
5436:
5433:
5432:
5400:
5397:
5396:
5372:
5370:
5357:
5349:
5346:
5345:
5328:
5324:
5322:
5319:
5318:
5301:
5297:
5288:
5283:
5270:
5265:
5252:
5247:
5241:
5238:
5237:
5216:
5212:
5203:
5199:
5179:
5175:
5163:
5159:
5152:
5148:
5136:
5132:
5123:
5119:
5110:
5106:
5097:
5089:
5076:
5068:
5040:
5036:
5034:
5031:
5030:
5004:
5001:
5000:
4997:
4967:
4963:
4961:
4958:
4957:
4940:
4936:
4934:
4931:
4930:
4904:
4900:
4891:
4887:
4878:
4874:
4862:
4856:
4855:
4854:
4830:
4826:
4824:
4821:
4820:
4785:
4781:
4772:
4771:
4759:
4755:
4746:
4742:
4733:
4729:
4727:
4724:
4723:
4680:
4676:
4667:
4663:
4643:
4639:
4627:
4623:
4616:
4612:
4600:
4596:
4587:
4583:
4574:
4570:
4561:
4553:
4540:
4532:
4504:
4500:
4498:
4495:
4494:
4468:
4463:
4462:
4414:
4410:
4408:
4405:
4404:
4384:
4380:
4378:
4375:
4374:
4357:
4353:
4351:
4348:
4347:
4330:
4326:
4324:
4321:
4320:
4300:
4296:
4287:
4283:
4274:
4270:
4265:
4262:
4261:
4241:
4237:
4228:
4224:
4215:
4211:
4209:
4206:
4205:
4177:
4174:
4173:
4149:
4147:
4134:
4126:
4123:
4122:
4105:
4101:
4099:
4096:
4095:
4078:
4074:
4065:
4060:
4047:
4042:
4029:
4024:
4018:
4015:
4014:
3981:
3977:
3968:
3964:
3948:
3943:
3930:
3925:
3912:
3908:
3906:
3896:
3892:
3875:
3871:
3859:
3855:
3848:
3844:
3832:
3828:
3819:
3815:
3806:
3802:
3793:
3785:
3772:
3764:
3731:
3728:
3727:
3702:
3699:
3698:
3695:
3655:
3652:
3651:
3635:
3632:
3631:
3609:
3606:
3605:
3589:
3586:
3585:
3568:
3564:
3562:
3559:
3558:
3542:
3539:
3538:
3522:
3519:
3518:
3496:
3493:
3492:
3476:
3473:
3472:
3448:
3446:
3433:
3415:
3411:
3409:
3406:
3405:
3383:
3381:
3380:
3374:
3370:
3356:
3354:
3353:
3347:
3343:
3329:
3327:
3326:
3320:
3316:
3308:
3306:
3303:
3302:
3286:
3283:
3282:
3266:
3263:
3262:
3240:
3237:
3236:
3219:
3215:
3213:
3210:
3209:
3193:
3190:
3189:
3173:
3170:
3169:
3147:
3144:
3143:
3126:
3122:
3120:
3117:
3116:
3095:
3091:
3089:
3086:
3085:
3068:
3064:
3062:
3059:
3058:
3041:
3037:
3035:
3032:
3031:
3011:
3008:
3007:
2993:
2992:
2982:
2977:
2964:
2959:
2946:
2942:
2940:
2930:
2926:
2912:
2908:
2896:
2892:
2885:
2881:
2869:
2868:
2857:
2853:
2849:
2845:
2831:
2827:
2815:
2811:
2804:
2800:
2788:
2787:
2776:
2772:
2768:
2764:
2753:
2749:
2745:
2741:
2730:
2726:
2722:
2718:
2708:
2680:
2678:
2675:
2674:
2637:
2634:
2633:
2596:
2593:
2592:
2576:
2575:
2569:
2565:
2558:
2552:
2547:
2534:
2529:
2516:
2511:
2504:
2503:
2493:
2478:
2474:
2468:
2463:
2441:
2437:
2428:
2424:
2420:
2414:
2410:
2408:
2405:
2404:
2394:
2379:
2375:
2369:
2364:
2342:
2338:
2329:
2325:
2321:
2315:
2311:
2309:
2306:
2305:
2295:
2280:
2276:
2270:
2265:
2243:
2239:
2230:
2226:
2222:
2216:
2212:
2210:
2206:
2204:
2201:
2200:
2192:
2185:
2178:
2163:
2117:
2113:
2092:
2088:
2087:
2068:
2063:
2061:
2040:
2036:
2035:
2016:
2011:
2009:
1988:
1984:
1983:
1964:
1959:
1957:
1955:
1952:
1951:
1919:
1915:
1900:
1896:
1881:
1877:
1871:
1867:
1846:
1827:
1823:
1808:
1804:
1786:
1782:
1764:
1745:
1741:
1723:
1719:
1704:
1700:
1682:
1676:
1673:
1672:
1649:
1645:
1641:
1631:
1627:
1626:
1624:
1612:
1608:
1604:
1594:
1590:
1589:
1587:
1575:
1571:
1567:
1557:
1553:
1552:
1550:
1538:
1534:
1532:
1529:
1528:
1482:
1478:
1463:
1459:
1444:
1440:
1411:
1408:
1407:
1384:
1355:
1338:
1334:
1325:
1321:
1320:
1316:
1314:
1311:
1310:
1283:
1279:
1271:
1263:
1260:
1259:
1234:
1226:
1223:
1222:
1203:
1200:
1199:
1183:
1180:
1179:
1156:
1148:
1145:
1144:
1128:
1125:
1124:
1100:
1098:
1085:
1077:
1074:
1073:
1043:
1026:
1022:
1013:
1009:
1008:
1004:
1003:
999:
991:
988:
987:
984:
963:
960:
959:
943:
940:
939:
914:
904:
900:
892:
875:
867:
864:
863:
835:
832:
831:
815:
812:
811:
789:
781:
778:
777:
757:
754:
753:
722:
718:
710:
703:
699:
676:
668:
665:
664:
644:
641:
640:
621:
610:
607:
606:
578:
575:
574:
551:
547:
545:
542:
541:
526:
504:
482:
469:
452:
416:
413:
412:
381:
361:
356:
355:
351:
345:
341:
339:
331:
327:
322:
313:
309:
308:
304:
302:
299:
298:
291:
238:
236:
233:
232:
207:
193:
190:
189:
181:
169:
162:
128:
121:
28:
23:
22:
15:
12:
11:
5:
16653:
16643:
16642:
16637:
16623:
16622:
16586:
16549:
16548:External links
16546:
16545:
16544:
16539:
16533:
16516:
16510:
16493:
16487:
16470:
16464:
16447:
16441:
16433:Addison Wesley
16431:(2 ed.).
16424:
16412:
16394:
16375:
16374:
16367:
16347:
16332:
16322:
16321:
16319:
16316:
16314:
16313:
16308:
16303:
16298:
16293:
16288:
16283:
16278:
16273:
16268:
16263:
16258:
16253:
16247:
16245:
16242:
16224:
16221:
16209:Sinc functions
16175:
16172:
16160:
16152:
16147:
16144:
16141:
16136:
16131:
16127:
16121:
16117:
16111:
16106:
16102:
16095:
16091:
16085:
16082:
16079:
16057:
16052:
16048:
16045:
16039:
16036:
16033:
16030:
16027:
16024:
16021:
16018:
16015:
16012:
16009:
16006:
16003:
16000:
15997:
15994:
15983:
15982:
15971:
15967:
15962:
15955:
15952:
15946:
15942:
15936:
15930:
15927:
15921:
15917:
15910:
15906:
15903:
15898:
15894:
15890:
15887:
15884:
15880:
15876:
15873:
15869:
15862:
15859:
15853:
15849:
15843:
15837:
15834:
15828:
15824:
15817:
15813:
15807:
15804:
15797:
15793:
15789:
15783:
15779:
15772:
15769:
15763:
15759:
15753:
15747:
15744:
15738:
15734:
15727:
15723:
15720:
15697:
15694:
15680:
15669:
15664:
15660:
15651:
15647:
15641:
15637:
15632:
15627:
15622:
15617:
15613:
15609:
15606:
15595:
15583:
15572:
15560:
15557:
15554:
15551:
15548:
15543:
15540:
15536:
15515:
15512:
15492:
15481:
15469:
15466:
15463:
15460:
15457:
15454:
15434:
15404:
15401:
15397:
15392:
15387:
15383:
15361:
15357:
15354:
15349:
15345:
15341:
15336:
15332:
15328:
15306:
15302:
15297:
15290:
15287:
15281:
15277:
15271:
15265:
15262:
15256:
15252:
15245:
15241:
15238:
15233:
15229:
15225:
15222:
15219:
15215:
15211:
15208:
15204:
15197:
15194:
15188:
15184:
15178:
15172:
15169:
15163:
15159:
15152:
15148:
15142:
15139:
15132:
15128:
15124:
15118:
15114:
15107:
15104:
15098:
15094:
15088:
15082:
15079:
15073:
15069:
15062:
15056:
15052:
15043:
15039:
15033:
15029:
15024:
15019:
15014:
15009:
15005:
15001:
14997:
14991:
14987:
14983:
14978:
14974:
14969:
14965:
14943:
14939:
14934:
14927:
14924:
14918:
14914:
14908:
14902:
14899:
14893:
14889:
14882:
14878:
14875:
14871:
14867:
14864:
14860:
14853:
14850:
14844:
14840:
14834:
14828:
14825:
14819:
14815:
14808:
14804:
14801:
14797:
14790:
14787:
14781:
14777:
14771:
14765:
14762:
14756:
14752:
14745:
14741:
14721:
14701:
14681:
14654:
14648:
14644:
14640:
14637:
14634:
14631:
14628:
14624:
14620:
14617:
14613:
14606:
14603:
14597:
14593:
14587:
14581:
14578:
14572:
14568:
14561:
14555:
14551:
14544:
14541:
14535:
14531:
14525:
14521:
14515:
14511:
14507:
14504:
14501:
14498:
14494:
14490:
14487:
14483:
14476:
14473:
14467:
14463:
14457:
14451:
14448:
14442:
14438:
14431:
14427:
14421:
14418:
14412:
14408:
14402:
14397:
14392:
14387:
14380:
14377:
14371:
14367:
14361:
14355:
14352:
14346:
14342:
14335:
14331:
14327:
14317:
14313:
14307:
14303:
14298:
14293:
14288:
14283:
14279:
14275:
14272:
14270:
14268:
14263:
14258:
14253:
14246:
14243:
14237:
14233:
14227:
14221:
14218:
14212:
14208:
14201:
14197:
14191:
14188:
14184:
14179:
14175:
14169:
14165:
14161:
14158:
14155:
14152:
14149:
14145:
14141:
14138:
14133:
14129:
14124:
14119:
14116:
14114:
14111:
14105:
14101:
14097:
14092:
14088:
14083:
14077:
14072:
14071:
14060:
14059:
14045:
14038:
14035:
14029:
14025:
14019:
14013:
14010:
14004:
14000:
13993:
13988:
13982:
13979:
13975:
13958:
13944:
13941:
13938:
13935:
13932:
13929:
13925:
13919:
13915:
13878:
13858:
13811:
13805:
13801:
13797:
13792:
13788:
13783:
13777:
13773:
13751:
13745:
13741:
13737:
13732:
13728:
13723:
13717:
13713:
13691:
13685:
13681:
13677:
13672:
13668:
13663:
13659:
13639:
13629:
13623:
13619:
13615:
13610:
13606:
13601:
13595:
13591:
13584:
13578:
13574:
13570:
13565:
13561:
13556:
13550:
13546:
13539:
13534:
13529:
13524:
13518:
13514:
13510:
13505:
13501:
13496:
13492:
13488:
13480:
13474:
13470:
13466:
13461:
13457:
13452:
13446:
13442:
13435:
13431:
13425:
13421:
13417:
13412:
13408:
13403:
13399:
13371:
13366:
13358:
13355:
13349:
13346:
13342:
13338:
13333:
13329:
13308:
13305:
13302:
13299:
13296:
13290:
13287:
13262:
13259:
13256:
13253:
13250:
13247:
13244:
13241:
13238:
13235:
13232:
13229:
13226:
13223:
13220:
13217:
13214:
13211:
13208:
13205:
13202:
13199:
13196:
13193:
13190:
13187:
13184:
13164:
13161:
13158:
13155:
13152:
13149:
13129:
13126:
13123:
13120:
13117:
13114:
13094:
13091:
13088:
13085:
13082:
13079:
13059:
13056:
13053:
13050:
13047:
13044:
13032:
13029:
13024:
13017:
13006:
12999:
12964:
12957:
12946:
12939:
12861:
12854:
12843:
12836:
12785:
12778:
12709:Main article:
12706:
12703:
12694:
12691:
12653:
12650:
12595:
12592:
12575:Airy functions
12571:sinc functions
12487:
12484:
12463:
12439:
12410:
12407:
12394:
12387:
12380:
12364:
12359:
12355:
12351:
12346:
12341:
12337:
12333:
12328:
12324:
12311:supercomputers
12263:
12260:
12257:
12253:
12249:
12246:
12243:
12239:
12227:spherical wave
12212:
12209:
12206:
12203:
12200:
12197:
12193:
12164:
12160:
12154:
12151:
12146:
12143:
12140:
12132:
12128:
12122:
12119:
12114:
12111:
12107:
12102:
12096:
12093:
12090:
12086:
12062:
12058:
12052:
12049:
12044:
12041:
12038:
12035:
12032:
12029:
12026:
12021:
12016:
12012:
11989:
11986:
11983:
11980:
11977:
11974:
11969:
11964:
11960:
11878:
11875:
11851:interferometry
11816:
11813:
11800:
11795:
11791:
11787:
11782:
11778:
11774:
11751:
11746:
11742:
11738:
11733:
11729:
11725:
11722:
11682:
11679:
11676:
11673:
11651:
11646:
11642:
11638:
11633:
11629:
11625:
11622:
11619:
11616:
11611:
11607:
11603:
11598:
11594:
11590:
11587:
11581:
11575:
11570:
11566:
11562:
11557:
11553:
11549:
11546:
11527:
11526:
11514:
11511:
11508:
11505:
11495:
11483:
11480:
11477:
11474:
11464:
11452:
11449:
11446:
11443:
11420:
11417:
11414:
11411:
11408:
11405:
11402:
11399:
11396:
11390:
11384:
11381:
11378:
11375:
11363:
11360:
11307:
11304:
11300:
11297:
11293:
11290:
11286:
11283:
11280:
11276:
11273:
11269:
11266:
11263:
11260:
11255:
11250:
11247:
11243:
11239:
11236:
11233:
11230:
11227:
11176:
11173:
11169:
11166:
11162:
11159:
11155:
11152:
11149:
11145:
11142:
11138:
11135:
11132:
11129:
11124:
11119:
11116:
11112:
11108:
11105:
11102:
11099:
11096:
11041:
11038:
11004:
10993:
10968:
10961:
10960:
10951:
10949:
10937:
10934:
10930:
10926:
10923:
10919:
10913:
10909:
10906:
10902:
10898:
10895:
10891:
10888:
10882:
10878:
10875:
10871:
10868:
10865:
10862:
10858:
10855:
10851:
10848:
10845:
10842:
10837:
10833:
10827:
10822:
10819:
10815:
10809:
10804:
10801:
10797:
10793:
10790:
10787:
10784:
10781:
10778:
10775:
10738:
10737:
10728:
10726:
10714:
10711:
10707:
10703:
10700:
10696:
10690:
10686:
10683:
10679:
10675:
10672:
10668:
10665:
10659:
10655:
10652:
10648:
10645:
10642:
10638:
10635:
10631:
10628:
10625:
10622:
10617:
10612:
10609:
10605:
10599:
10594:
10591:
10587:
10583:
10580:
10577:
10574:
10571:
10568:
10565:
10540:
10537:
10534:
10531:
10528:
10525:
10522:
10519:
10516:
10513:
10510:
10507:
10504:
10498:
10492:
10489:
10486:
10483:
10480:
10477:
10466:
10463:
10449:
10446:
10443:
10437:
10431:
10428:
10425:
10422:
10419:
10416:
10413:
10410:
10407:
10404:
10401:
10398:
10395:
10392:
10389:
10365:
10362:
10348:
10345:
10342:
10336:
10330:
10327:
10324:
10321:
10318:
10315:
10312:
10309:
10306:
10303:
10300:
10297:
10294:
10291:
10288:
10268:
10265:
10221:
10218:
10215:
10212:
10209:
10206:
10203:
10200:
10197:
10194:
10191:
10188:
10185:
10182:
10179:
10176:
10173:
10170:
10167:
10164:
10137:(known as the
10114:
10111:
10095:
10091:
10087:
10084:
10081:
10076:
10071:
10046:
10042:
10038:
10033:
10029:
10025:
10020:
10017:
10012:
10008:
10004:
10001:
9996:
9992:
9988:
9985:
9981:
9977:
9972:
9968:
9964:
9959:
9955:
9951:
9948:
9943:
9938:
9935:
9931:
9925:
9920:
9917:
9913:
9904:
9900:
9896:
9893:
9890:
9886:
9881:
9878:
9875:
9872:
9869:
9866:
9863:
9843:
9840:
9837:
9834:
9831:
9828:
9806:
9803:
9800:
9797:
9792:
9789:
9784:
9780:
9776:
9773:
9768:
9764:
9760:
9757:
9754:
9750:
9746:
9743:
9740:
9737:
9734:
9731:
9726:
9721:
9718:
9714:
9708:
9703:
9700:
9696:
9692:
9689:
9684:
9680:
9676:
9671:
9667:
9663:
9660:
9640:
9637:
9634:
9631:
9628:
9625:
9613:
9607:
9596:
9595:
9581:
9578:
9575:
9569:
9566:
9561:
9557:
9546:
9535:
9532:
9529:
9523:
9520:
9517:
9511:
9508:
9503:
9499:
9488:
9477:
9474:
9471:
9465:
9462:
9459:
9453:
9450:
9445:
9441:
9410:
9407:
9399:
9395:
9391:
9385:
9382:
9374:
9370:
9366:
9360:
9357:
9349:
9345:
9338:
9331:
9300:
9296:
9292:
9287:
9282:
9278:
9274:
9269:
9264:
9260:
9256:
9251:
9246:
9242:
9219:
9215:
9212:
9206:
9201:
9198:
9193:
9190:
9170:
9158:
9155:
9113:
9063:
9059:
9033:
9030:
9027:
9023:
9018:
9013:
9009:
9006:
9002:
8997:
8968:
8965:
8962:
8959:
8955:
8949:
8945:
8941:
8936:
8932:
8927:
8867:
8864:
8847:
8843:
8839:
8836:
8833:
8830:
8810:
8807:
8804:
8801:
8798:
8795:
8792:
8779:
8778:
8769:
8767:
8756:
8753:
8748:
8744:
8738:
8734:
8730:
8725:
8721:
8715:
8711:
8679:
8676:
8673:
8669:
8611:
8586:
8566:
8546:
8542:
8536:
8532:
8528:
8504:
8501:
8479:
8473:
8470:
8464:
8454:
8449:
8443:
8436:
8392:The equation (
8389:
8386:
8370:
8365:
8362:
8359:
8356:
8352:
8330:
8329:
8315:
8312:
8309:
8303:
8300:
8295:
8291:
8280:
8269:
8266:
8263:
8257:
8254:
8251:
8245:
8242:
8237:
8233:
8222:
8211:
8208:
8205:
8199:
8196:
8193:
8187:
8184:
8179:
8175:
8160:
8159:
8145:
8142:
8139:
8133:
8130:
8127:
8117:
8106:
8103:
8100:
8094:
8091:
8088:
8082:
8079:
8076:
8066:
8055:
8052:
8049:
8043:
8040:
8037:
8031:
8028:
8025:
8002:
7999:
7996:
7993:
7990:
7987:
7984:
7964:
7961:
7958:
7955:
7952:
7949:
7946:
7933:
7932:
7923:
7921:
7910:
7907:
7904:
7901:
7895:
7892:
7889:
7883:
7880:
7877:
7874:
7871:
7865:
7862:
7859:
7853:
7850:
7845:
7841:
7832:
7827:
7824:
7821:
7818:
7814:
7805:
7802:
7799:
7796:
7790:
7787:
7781:
7778:
7775:
7769:
7763:
7760:
7757:
7754:
7751:
7748:
7745:
7740:
7736:
7710:
7707:
7704:
7701:
7698:
7695:
7692:
7672:
7669:
7666:
7663:
7660:
7657:
7654:
7634:
7629:
7625:
7621:
7616:
7612:
7608:
7603:
7599:
7595:
7575:
7572:
7569:
7566:
7563:
7560:
7557:
7538:The equation (
7532:Main article:
7529:
7526:
7525:
7524:
7510:
7506:
7503:
7497:
7494:
7474:
7470:
7466:
7463:
7439:
7435:
7431:
7428:
7420:
7416:
7412:
7406:
7402:
7378:
7374:
7370:
7366:
7363:
7357:
7354:
7350:
7345:
7341:
7337:
7333:
7329:
7325:
7321:
7318:
7310:
7306:
7302:
7296:
7292:
7285:
7274:
7260:
7256:
7252:
7232:
7229:
7207:
7203:
7199:
7177:
7173:
7169:
7158:
7144:
7140:
7119:
7116:
7113:
7093:
7057:
7053:
7049:
7043:
7039:
7033:
7025:
7021:
7017:
7011:
7007:
7001:
6993:
6989:
6985:
6979:
6975:
6969:
6964:
6960:
6956:
6951:
6946:
6942:
6921:
6918:
6912:
6909:
6904:
6901:
6895:
6892:
6889:
6886:
6883:
6878:
6873:
6869:
6856:
6853:
6839:
6836:
6833:
6830:
6826:
6820:
6816:
6810:
6806:
6802:
6799:
6796:
6792:
6786:
6783:
6778:
6774:
6770:
6767:
6762:
6758:
6754:
6751:
6748:
6744:
6740:
6736:
6732:
6729:
6726:
6723:
6719:
6715:
6712:
6692:
6689:
6685:
6679:
6675:
6671:
6668:
6665:
6662:
6659:
6656:
6653:
6650:
6647:
6644:
6639:
6635:
6607:
6581:
6577:
6571:
6567:
6563:
6560:
6557:
6554:
6552:
6550:
6547:
6544:
6541:
6540:
6537:
6534:
6531:
6529:
6527:
6524:
6521:
6518:
6517:
6514:
6511:
6508:
6506:
6504:
6501:
6498:
6495:
6494:
6468:
6463:
6459:
6455:
6450:
6445:
6441:
6437:
6432:
6427:
6423:
6392:
6389:
6384:
6377:
6372:
6368:
6364:
6359:
6354:
6350:
6346:
6341:
6336:
6332:
6326:
6323:
6319:
6315:
6312:
6309:
6288:
6282:
6278:
6274:
6270:
6264:
6260:
6256:
6252:
6246:
6242:
6238:
6234:
6230:
6226:
6200:
6176:
6172:
6168:
6164:
6161:
6157:
6153:
6149:
6145:
6142:
6139:
6136:
6132:
6128:
6125:
6113:
6110:
6108:
6105:
6088:
6059:
6055:
6034:
6014:
5992:
5988:
5965:
5961:
5936:
5916:
5913:
5908:
5904:
5879:
5857:
5853:
5830:
5826:
5822:
5817:
5812:
5808:
5804:
5799:
5794:
5790:
5786:
5781:
5776:
5772:
5749:
5745:
5721:
5718:
5712:
5708:
5683:
5679:
5656:
5652:
5629:
5625:
5621:
5616:
5612:
5608:
5600:
5597:
5592:
5588:
5584:
5581:
5576:
5572:
5568:
5565:
5561:
5554:
5549:
5545:
5541:
5536:
5532:
5528:
5523:
5519:
5513:
5510:
5505:
5502:
5498:
5492:
5489:
5484:
5481:
5477:
5473:
5470:
5467:
5464:
5461:
5458:
5448:
5445:
5441:
5416:
5413:
5410:
5407:
5404:
5382:
5378:
5375:
5369:
5364:
5361:
5356:
5353:
5331:
5327:
5304:
5300:
5296:
5291:
5286:
5282:
5278:
5273:
5268:
5264:
5260:
5255:
5250:
5246:
5219:
5215:
5211:
5206:
5202:
5198:
5190:
5187:
5182:
5178:
5174:
5171:
5166:
5162:
5158:
5155:
5151:
5144:
5139:
5135:
5131:
5126:
5122:
5118:
5113:
5109:
5103:
5100:
5095:
5092:
5088:
5082:
5079:
5074:
5071:
5067:
5063:
5060:
5057:
5054:
5051:
5048:
5043:
5039:
5027:complex-valued
5014:
5011:
5008:
4996:
4993:
4970:
4966:
4943:
4939:
4915:
4912:
4907:
4903:
4899:
4894:
4890:
4886:
4881:
4877:
4873:
4868:
4865:
4859:
4853:
4850:
4847:
4844:
4841:
4838:
4833:
4829:
4808:
4805:
4802:
4799:
4796:
4793:
4788:
4784:
4780:
4775:
4770:
4767:
4762:
4758:
4754:
4749:
4745:
4741:
4736:
4732:
4717:Fourier optics
4683:
4679:
4675:
4670:
4666:
4662:
4654:
4651:
4646:
4642:
4638:
4635:
4630:
4626:
4622:
4619:
4615:
4608:
4603:
4599:
4595:
4590:
4586:
4582:
4577:
4573:
4567:
4564:
4559:
4556:
4552:
4546:
4543:
4538:
4535:
4531:
4527:
4524:
4521:
4518:
4515:
4512:
4507:
4503:
4482:
4477:
4474:
4471:
4466:
4461:
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4434:
4431:
4428:
4425:
4422:
4417:
4413:
4387:
4383:
4360:
4356:
4333:
4329:
4308:
4303:
4299:
4295:
4290:
4286:
4282:
4277:
4273:
4269:
4249:
4244:
4240:
4236:
4231:
4227:
4223:
4218:
4214:
4193:
4190:
4187:
4184:
4181:
4159:
4155:
4152:
4146:
4141:
4138:
4133:
4130:
4108:
4104:
4081:
4077:
4073:
4068:
4063:
4059:
4055:
4050:
4045:
4041:
4037:
4032:
4027:
4023:
4009:
4008:
3999:
3997:
3984:
3980:
3976:
3971:
3967:
3963:
3951:
3946:
3942:
3938:
3933:
3928:
3924:
3920:
3915:
3911:
3905:
3902:
3899:
3895:
3886:
3883:
3878:
3874:
3870:
3867:
3862:
3858:
3854:
3851:
3847:
3840:
3835:
3831:
3827:
3822:
3818:
3814:
3809:
3805:
3799:
3796:
3791:
3788:
3784:
3778:
3775:
3770:
3767:
3763:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3706:
3694:
3691:
3659:
3639:
3619:
3616:
3613:
3593:
3571:
3567:
3546:
3526:
3506:
3503:
3500:
3480:
3458:
3454:
3451:
3445:
3440:
3437:
3432:
3429:
3426:
3422:
3418:
3414:
3390:
3386:
3377:
3373:
3369:
3363:
3359:
3350:
3346:
3342:
3336:
3332:
3323:
3319:
3315:
3311:
3290:
3270:
3250:
3247:
3244:
3222:
3218:
3197:
3177:
3157:
3154:
3151:
3129:
3125:
3098:
3094:
3071:
3067:
3044:
3040:
3028:complex-valued
3015:
2985:
2980:
2976:
2972:
2967:
2962:
2958:
2954:
2949:
2945:
2939:
2936:
2933:
2929:
2923:
2920:
2915:
2911:
2907:
2904:
2899:
2895:
2891:
2888:
2884:
2880:
2877:
2874:
2872:
2870:
2865:
2860:
2856:
2852:
2848:
2842:
2839:
2834:
2830:
2826:
2823:
2818:
2814:
2810:
2807:
2803:
2799:
2796:
2793:
2791:
2789:
2784:
2779:
2775:
2771:
2767:
2761:
2756:
2752:
2748:
2744:
2738:
2733:
2729:
2725:
2721:
2717:
2714:
2711:
2709:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2682:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2621:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2572:
2568:
2564:
2561:
2559:
2555:
2550:
2546:
2542:
2537:
2532:
2528:
2524:
2519:
2514:
2510:
2506:
2505:
2502:
2499:
2496:
2494:
2492:
2489:
2486:
2481:
2477:
2471:
2466:
2462:
2458:
2455:
2452:
2449:
2444:
2440:
2431:
2427:
2423:
2417:
2413:
2407:
2406:
2403:
2400:
2397:
2395:
2393:
2390:
2387:
2382:
2378:
2372:
2367:
2363:
2359:
2356:
2353:
2350:
2345:
2341:
2332:
2328:
2324:
2318:
2314:
2308:
2307:
2304:
2301:
2298:
2296:
2294:
2291:
2288:
2283:
2279:
2273:
2268:
2264:
2260:
2257:
2254:
2251:
2246:
2242:
2233:
2229:
2225:
2219:
2215:
2209:
2208:
2190:
2183:
2176:
2161:
2128:
2125:
2120:
2116:
2112:
2106:
2103:
2100:
2095:
2091:
2085:
2082:
2079:
2075:
2071:
2067:
2060:
2054:
2051:
2048:
2043:
2039:
2033:
2030:
2027:
2023:
2019:
2015:
2008:
2002:
1999:
1996:
1991:
1987:
1981:
1978:
1975:
1971:
1967:
1963:
1939:
1936:
1933:
1930:
1927:
1922:
1918:
1914:
1911:
1908:
1903:
1899:
1895:
1892:
1889:
1884:
1880:
1874:
1870:
1866:
1863:
1860:
1857:
1853:
1849:
1845:
1841:
1838:
1835:
1830:
1826:
1822:
1819:
1816:
1811:
1807:
1803:
1800:
1797:
1794:
1789:
1785:
1781:
1778:
1775:
1771:
1767:
1763:
1759:
1756:
1753:
1748:
1744:
1740:
1737:
1734:
1731:
1726:
1722:
1718:
1715:
1712:
1707:
1703:
1699:
1696:
1693:
1689:
1685:
1681:
1660:
1652:
1648:
1644:
1639:
1634:
1630:
1623:
1615:
1611:
1607:
1602:
1597:
1593:
1586:
1578:
1574:
1570:
1565:
1560:
1556:
1549:
1546:
1541:
1537:
1496:
1493:
1490:
1485:
1481:
1477:
1474:
1471:
1466:
1462:
1458:
1455:
1452:
1447:
1443:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1383:
1380:
1368:
1365:
1362:
1358:
1354:
1351:
1347:
1341:
1337:
1333:
1328:
1324:
1319:
1292:
1289:
1286:
1282:
1278:
1274:
1270:
1267:
1247:
1244:
1241:
1237:
1233:
1230:
1207:
1187:
1176:complex-valued
1163:
1159:
1155:
1152:
1132:
1110:
1106:
1103:
1097:
1092:
1089:
1084:
1081:
1061:
1058:
1054:
1050:
1046:
1042:
1039:
1035:
1029:
1025:
1021:
1016:
1012:
1007:
1002:
998:
995:
983:
980:
967:
947:
921:
917:
913:
910:
907:
903:
899:
895:
891:
888:
885:
882:
878:
874:
871:
851:
848:
845:
842:
839:
819:
798:
795:
792:
788:
785:
774:imaginary unit
761:
741:
737:
731:
728:
725:
721:
717:
713:
709:
706:
702:
698:
695:
692:
689:
686:
683:
679:
675:
672:
648:
628:
624:
620:
617:
614:
594:
591:
588:
585:
582:
560:
557:
554:
550:
525:
522:
502:
467:
433:speed of light
420:
400:
397:
394:
391:
388:
384:
380:
377:
373:
364:
359:
354:
348:
344:
334:
330:
326:
321:
316:
312:
307:
290:
287:
266:
263:
260:
257:
254:
251:
248:
245:
241:
220:
217:
214:
210:
206:
203:
200:
197:
180:
177:
167:
160:
126:
119:
94:quantum optics
31:Fourier optics
26:
18:Fourier Optics
9:
6:
4:
3:
2:
16652:
16641:
16638:
16636:
16633:
16632:
16630:
16619:
16615:
16611:
16607:
16603:
16599:
16592:
16587:
16583:
16579:
16574:
16569:
16565:
16561:
16557:
16552:
16551:
16543:
16540:
16536:
16534:0-89006-349-4
16530:
16526:
16522:
16517:
16513:
16511:0-89006-419-9
16507:
16503:
16499:
16494:
16490:
16488:0-7803-3440-X
16484:
16480:
16476:
16471:
16467:
16465:0-471-30357-7
16461:
16457:
16453:
16448:
16444:
16442:0-201-11609-X
16438:
16434:
16430:
16425:
16415:
16413:0-9747077-2-4
16409:
16405:
16404:
16399:
16395:
16391:
16387:
16383:
16379:
16378:
16370:
16364:
16360:
16359:
16351:
16343:
16336:
16327:
16323:
16312:
16309:
16307:
16304:
16302:
16299:
16297:
16296:Hilbert space
16294:
16292:
16289:
16287:
16284:
16282:
16279:
16277:
16274:
16272:
16269:
16267:
16264:
16262:
16259:
16257:
16254:
16252:
16249:
16248:
16241:
16238:
16234:
16230:
16229:eigenfunction
16220:
16218:
16214:
16210:
16206:
16203:based on his
16202:
16198:
16197:Frits Zernike
16194:
16189:
16185:
16181:
16171:
16158:
16150:
16145:
16142:
16139:
16134:
16129:
16125:
16119:
16115:
16109:
16104:
16100:
16093:
16089:
16083:
16080:
16068:
16055:
16050:
16046:
16037:
16031:
16028:
16025:
16016:
16013:
16007:
16004:
15998:
15995:
15969:
15965:
15960:
15953:
15950:
15944:
15940:
15934:
15928:
15925:
15919:
15915:
15908:
15904:
15901:
15896:
15892:
15888:
15885:
15882:
15878:
15874:
15871:
15867:
15860:
15857:
15851:
15847:
15841:
15835:
15832:
15826:
15822:
15815:
15811:
15805:
15802:
15795:
15791:
15787:
15781:
15777:
15770:
15767:
15761:
15757:
15751:
15745:
15742:
15736:
15732:
15725:
15721:
15692:
15681:
15667:
15662:
15658:
15649:
15645:
15639:
15635:
15630:
15625:
15620:
15615:
15611:
15607:
15604:
15596:
15581:
15573:
15558:
15555:
15552:
15549:
15546:
15541:
15538:
15534:
15513:
15510:
15490:
15482:
15464:
15461:
15458:
15452:
15432:
15424:
15423:
15422:
15419:
15402:
15399:
15395:
15390:
15385:
15381:
15355:
15347:
15343:
15339:
15334:
15330:
15317:
15304:
15300:
15295:
15288:
15285:
15279:
15275:
15269:
15263:
15260:
15254:
15250:
15243:
15239:
15236:
15231:
15227:
15223:
15220:
15217:
15213:
15209:
15206:
15202:
15195:
15192:
15186:
15182:
15176:
15170:
15167:
15161:
15157:
15150:
15146:
15140:
15137:
15130:
15126:
15122:
15116:
15112:
15105:
15102:
15096:
15092:
15086:
15080:
15077:
15071:
15067:
15060:
15054:
15050:
15041:
15037:
15031:
15027:
15022:
15017:
15012:
15007:
15003:
14999:
14995:
14989:
14985:
14981:
14976:
14972:
14967:
14963:
14954:
14941:
14937:
14932:
14925:
14922:
14916:
14912:
14906:
14900:
14897:
14891:
14887:
14880:
14876:
14873:
14869:
14865:
14862:
14858:
14851:
14848:
14842:
14838:
14832:
14826:
14823:
14817:
14813:
14806:
14802:
14799:
14795:
14788:
14785:
14779:
14775:
14769:
14763:
14760:
14754:
14750:
14743:
14739:
14719:
14699:
14679:
14670:
14652:
14646:
14642:
14638:
14635:
14632:
14629:
14626:
14622:
14618:
14615:
14611:
14604:
14601:
14595:
14591:
14585:
14579:
14576:
14570:
14566:
14559:
14553:
14549:
14542:
14539:
14533:
14529:
14523:
14519:
14513:
14509:
14505:
14502:
14499:
14496:
14492:
14488:
14485:
14481:
14474:
14471:
14465:
14461:
14455:
14449:
14446:
14440:
14436:
14429:
14425:
14419:
14416:
14410:
14406:
14400:
14395:
14390:
14385:
14378:
14375:
14369:
14365:
14359:
14353:
14350:
14344:
14340:
14333:
14329:
14325:
14315:
14311:
14305:
14301:
14296:
14291:
14286:
14281:
14277:
14273:
14271:
14261:
14256:
14251:
14244:
14241:
14235:
14231:
14225:
14219:
14216:
14210:
14206:
14199:
14195:
14189:
14186:
14182:
14177:
14173:
14167:
14163:
14159:
14156:
14153:
14150:
14147:
14143:
14139:
14136:
14131:
14127:
14122:
14117:
14115:
14109:
14103:
14099:
14095:
14090:
14086:
14081:
14043:
14036:
14033:
14027:
14023:
14017:
14011:
14008:
14002:
13998:
13991:
13986:
13980:
13977:
13973:
13963:
13959:
13942:
13939:
13936:
13933:
13930:
13927:
13923:
13917:
13913:
13904:
13900:
13896:
13892:
13891:
13890:
13876:
13856:
13848:
13844:
13839:
13831:
13827:
13824:
13809:
13803:
13799:
13795:
13790:
13786:
13781:
13775:
13749:
13743:
13739:
13735:
13730:
13726:
13721:
13715:
13689:
13683:
13679:
13675:
13670:
13666:
13661:
13657:
13637:
13627:
13621:
13617:
13613:
13608:
13604:
13599:
13593:
13582:
13576:
13572:
13568:
13563:
13559:
13554:
13548:
13537:
13532:
13527:
13522:
13516:
13512:
13508:
13503:
13499:
13494:
13490:
13486:
13478:
13472:
13468:
13464:
13459:
13455:
13450:
13444:
13440:
13433:
13429:
13423:
13419:
13415:
13410:
13406:
13401:
13397:
13389:
13388:Wiener filter
13384:
13369:
13353:
13347:
13344:
13336:
13331:
13327:
13303:
13300:
13297:
13285:
13273:
13257:
13254:
13251:
13245:
13242:
13236:
13233:
13230:
13224:
13221:
13215:
13212:
13209:
13203:
13200:
13194:
13191:
13188:
13182:
13159:
13156:
13153:
13147:
13124:
13121:
13118:
13112:
13089:
13086:
13083:
13077:
13054:
13051:
13048:
13042:
13028:
13023:
13016:
13012:
13005:
12998:
12994:
12990:
12986:
12982:
12978:
12974:
12970:
12963:
12956:
12952:
12945:
12938:
12934:
12930:
12925:
12923:
12919:
12915:
12911:
12907:
12903:
12899:
12893:
12891:
12887:
12883:
12879:
12875:
12871:
12867:
12860:
12853:
12849:
12842:
12835:
12831:
12827:
12823:
12819:
12815:
12814:
12809:
12808:
12803:
12799:
12795:
12791:
12784:
12777:
12773:
12769:
12765:
12761:
12757:
12750:4F Correlator
12748:
12744:
12742:
12738:
12734:
12730:
12726:
12722:
12718:
12712:
12702:
12700:
12690:
12688:
12684:
12680:
12675:
12673:
12669:
12665:
12664:
12659:
12649:
12647:
12646:
12641:
12640:
12635:
12634:
12628:
12626:
12622:
12618:
12614:
12610:
12606:
12602:
12591:
12587:
12584:
12580:
12576:
12572:
12567:
12565:
12561:
12557:
12553:
12549:
12545:
12540:
12536:
12532:
12526:
12524:
12520:
12516:
12512:
12508:
12504:
12501:
12497:
12493:
12483:
12481:
12477:
12473:
12469:
12462:
12457:
12453:
12449:
12445:
12438:
12432:
12430:
12429:
12424:
12423:
12418:
12417:
12406:
12403:
12401:
12397:
12390:
12383:
12362:
12357:
12353:
12349:
12344:
12339:
12335:
12331:
12326:
12322:
12312:
12306:
12304:
12299:
12295:
12291:
12287:
12283:
12279:
12261:
12258:
12255:
12251:
12247:
12244:
12241:
12237:
12228:
12210:
12207:
12204:
12201:
12198:
12195:
12191:
12182:
12177:
12162:
12158:
12152:
12149:
12144:
12141:
12138:
12130:
12126:
12120:
12117:
12112:
12109:
12105:
12100:
12094:
12091:
12088:
12084:
12060:
12056:
12050:
12047:
12042:
12039:
12036:
12033:
12030:
12027:
12024:
12019:
12014:
12010:
11987:
11984:
11981:
11978:
11975:
11972:
11967:
11962:
11958:
11947:
11943:
11939:
11935:
11931:
11930:
11925:
11921:
11917:
11913:
11912:
11907:
11903:
11894:
11890:
11888:
11884:
11874:
11872:
11868:
11864:
11860:
11856:
11852:
11847:
11845:
11841:
11837:
11833:
11829:
11825:
11820:
11812:
11793:
11789:
11785:
11780:
11776:
11744:
11740:
11736:
11731:
11727:
11720:
11708:
11706:
11701:
11699:
11696:
11677:
11671:
11662:
11644:
11640:
11636:
11631:
11627:
11620:
11617:
11609:
11605:
11601:
11596:
11592:
11585:
11579:
11568:
11564:
11560:
11555:
11551:
11544:
11536:
11534:
11533:
11509:
11503:
11496:
11478:
11472:
11465:
11447:
11441:
11434:
11433:
11432:
11415:
11409:
11406:
11400:
11394:
11388:
11379:
11373:
11359:
11357:
11353:
11348:
11346:
11342:
11338:
11334:
11330:
11326:
11322:
11305:
11302:
11298:
11291:
11288:
11281:
11274:
11271:
11267:
11264:
11258:
11245:
11241:
11237:
11231:
11225:
11217:
11213:
11209:
11205:
11201:
11197:
11193:
11188:
11174:
11171:
11167:
11160:
11157:
11150:
11143:
11140:
11136:
11133:
11127:
11114:
11110:
11106:
11100:
11094:
11086:
11082:
11077:
11075:
11071:
11067:
11063:
11059:
11055:
11051:
11047:
11037:
11035:
11031:
11027:
11023:
11019:
11015:
11011:
11007:
11000:
10996:
10989:
10988:
10983:
10979:
10975:
10971:
10959:
10952:
10950:
10935:
10932:
10928:
10924:
10921:
10917:
10907:
10904:
10900:
10896:
10893:
10886:
10876:
10873:
10869:
10866:
10863:
10860:
10856:
10853:
10849:
10846:
10843:
10835:
10831:
10817:
10813:
10799:
10795:
10791:
10785:
10782:
10779:
10773:
10766:
10765:
10762:
10760:
10759:
10753:
10750:
10746:
10736:
10729:
10727:
10712:
10709:
10705:
10701:
10698:
10694:
10684:
10681:
10677:
10673:
10670:
10663:
10653:
10650:
10646:
10643:
10640:
10636:
10633:
10629:
10626:
10620:
10607:
10603:
10589:
10585:
10581:
10575:
10572:
10569:
10563:
10556:
10555:
10552:
10535:
10532:
10529:
10523:
10520:
10514:
10511:
10508:
10502:
10496:
10487:
10484:
10481:
10475:
10462:
10447:
10444:
10441:
10426:
10423:
10420:
10417:
10414:
10408:
10405:
10399:
10396:
10393:
10387:
10380:is therefore
10379:
10375:
10371:
10361:
10346:
10343:
10340:
10325:
10322:
10319:
10316:
10313:
10307:
10304:
10298:
10295:
10292:
10286:
10279:is therefore
10278:
10274:
10264:
10262:
10258:
10252:
10250:
10246:
10245:Airy function
10242:
10236:
10232:
10216:
10213:
10210:
10204:
10201:
10195:
10192:
10189:
10183:
10180:
10174:
10171:
10168:
10162:
10154:
10150:
10146:
10142:
10141:
10136:
10132:
10128:
10124:
10120:
10110:
10093:
10085:
10082:
10074:
10069:
10059:
10044:
10040:
10036:
10031:
10027:
10023:
10015:
10010:
10006:
10002:
9999:
9994:
9990:
9983:
9979:
9970:
9966:
9962:
9957:
9953:
9946:
9933:
9929:
9915:
9911:
9902:
9894:
9891:
9884:
9879:
9873:
9870:
9867:
9861:
9838:
9835:
9832:
9826:
9817:
9804:
9801:
9798:
9795:
9787:
9782:
9778:
9774:
9771:
9766:
9762:
9755:
9752:
9748:
9741:
9738:
9735:
9729:
9716:
9712:
9698:
9694:
9690:
9682:
9678:
9674:
9669:
9665:
9658:
9635:
9632:
9629:
9623:
9612:
9606:
9603:
9601:
9579:
9576:
9573:
9567:
9564:
9559:
9555:
9547:
9533:
9530:
9527:
9521:
9518:
9515:
9509:
9506:
9501:
9497:
9489:
9475:
9472:
9469:
9463:
9460:
9457:
9451:
9448:
9443:
9439:
9431:
9430:
9429:
9427:
9397:
9393:
9389:
9372:
9368:
9364:
9347:
9343:
9336:
9320:
9316:
9298:
9294:
9290:
9285:
9280:
9276:
9272:
9267:
9262:
9258:
9254:
9249:
9244:
9240:
9217:
9213:
9210:
9204:
9199:
9196:
9191:
9188:
9168:
9154:
9152:
9147:
9146:eigenfunction
9142:
9138:
9133:
9131:
9127:
9111:
9103:
9098:
9096:
9092:
9088:
9084:
9079:
9061:
9049:
9044:
9031:
9028:
9025:
9016:
9007:
9004:
8995:
8986:
8982:
8966:
8963:
8960:
8957:
8953:
8947:
8943:
8939:
8934:
8925:
8916:
8915:
8909:
8907:
8903:
8899:
8898:optical fiber
8895:
8891:
8887:
8883:
8878:
8877:eigenfunction
8874:
8863:
8861:
8845:
8841:
8837:
8834:
8831:
8828:
8808:
8805:
8802:
8799:
8796:
8793:
8790:
8777:
8770:
8768:
8754:
8751:
8746:
8742:
8736:
8732:
8728:
8723:
8719:
8713:
8701:
8700:
8697:
8695:
8677:
8674:
8671:
8667:
8658:
8654:
8650:
8646:
8645:eigenfunction
8641:
8639:
8634:
8629:
8625:
8609:
8599:
8584:
8564:
8544:
8540:
8534:
8530:
8526:
8519:range beyond
8517:
8513:
8511:
8500:
8497:
8493:
8477:
8462:
8447:
8441:
8425:
8419:
8415:
8413:
8412:section 6.1.3
8409:
8405:
8401:
8397:
8396:
8385:
8368:
8363:
8360:
8357:
8354:
8350:
8339:
8335:
8313:
8310:
8307:
8301:
8298:
8293:
8289:
8281:
8267:
8264:
8261:
8255:
8252:
8249:
8243:
8240:
8235:
8231:
8223:
8209:
8206:
8203:
8197:
8194:
8191:
8185:
8182:
8177:
8173:
8165:
8164:
8163:
8143:
8140:
8137:
8131:
8128:
8125:
8118:
8104:
8101:
8098:
8092:
8089:
8086:
8080:
8077:
8074:
8067:
8053:
8050:
8047:
8041:
8038:
8035:
8029:
8026:
8023:
8016:
8015:
8014:
7997:
7994:
7991:
7988:
7985:
7959:
7956:
7953:
7950:
7947:
7931:
7924:
7922:
7905:
7902:
7899:
7893:
7890:
7887:
7881:
7878:
7875:
7872:
7869:
7863:
7860:
7857:
7851:
7843:
7839:
7830:
7825:
7822:
7819:
7816:
7812:
7800:
7797:
7794:
7788:
7779:
7776:
7773:
7767:
7758:
7755:
7752:
7749:
7746:
7738:
7734:
7726:
7725:
7722:
7705:
7702:
7699:
7696:
7693:
7667:
7664:
7661:
7658:
7655:
7627:
7623:
7619:
7614:
7610:
7606:
7601:
7597:
7570:
7567:
7564:
7561:
7558:
7547:
7543:
7542:
7535:
7508:
7504:
7501:
7495:
7492:
7461:
7453:
7426:
7418:
7414:
7404:
7376:
7361:
7355:
7343:
7339:
7335:
7331:
7316:
7308:
7304:
7294:
7283:
7275:
7258:
7254:
7250:
7230:
7227:
7205:
7201:
7197:
7175:
7171:
7167:
7159:
7142:
7138:
7117:
7114:
7111:
7091:
7084:
7083:
7082:
7080:
7076:
7055:
7051:
7041:
7031:
7023:
7019:
7009:
6999:
6991:
6987:
6977:
6967:
6962:
6954:
6949:
6944:
6919:
6916:
6910:
6902:
6893:
6890:
6887:
6884:
6881:
6876:
6871:
6852:
6837:
6834:
6831:
6828:
6824:
6818:
6814:
6808:
6804:
6800:
6797:
6794:
6790:
6781:
6776:
6772:
6768:
6765:
6760:
6756:
6749:
6746:
6742:
6727:
6724:
6710:
6687:
6683:
6677:
6673:
6669:
6666:
6660:
6657:
6654:
6651:
6648:
6645:
6642:
6637:
6633:
6625:As a result,
6623:
6621:
6605:
6596:
6579:
6575:
6569:
6565:
6561:
6558:
6555:
6553:
6548:
6545:
6542:
6535:
6532:
6530:
6525:
6522:
6519:
6512:
6509:
6507:
6502:
6499:
6496:
6484:
6483:Taylor series
6466:
6461:
6457:
6453:
6448:
6443:
6439:
6435:
6430:
6425:
6421:
6412:
6408:
6390:
6387:
6382:
6375:
6370:
6366:
6362:
6357:
6352:
6348:
6344:
6339:
6334:
6330:
6324:
6310:
6307:
6280:
6276:
6272:
6262:
6258:
6254:
6244:
6240:
6236:
6228:
6215:
6170:
6162:
6159:
6155:
6140:
6137:
6123:
6104:
6102:
6086:
6078:
6073:
6057:
6053:
6032:
6012:
5990:
5986:
5963:
5959:
5950:
5934:
5927:but close to
5914:
5911:
5906:
5902:
5893:
5877:
5855:
5851:
5828:
5824:
5820:
5815:
5810:
5806:
5802:
5797:
5792:
5788:
5784:
5779:
5774:
5770:
5747:
5743:
5719:
5716:
5710:
5706:
5681:
5677:
5654:
5650:
5627:
5623:
5619:
5614:
5610:
5606:
5595:
5590:
5586:
5582:
5579:
5574:
5570:
5563:
5559:
5547:
5543:
5539:
5534:
5530:
5521:
5508:
5500:
5496:
5487:
5479:
5475:
5471:
5465:
5462:
5459:
5446:
5443:
5439:
5430:
5429:unconstrained
5414:
5411:
5408:
5405:
5402:
5380:
5376:
5373:
5367:
5362:
5359:
5354:
5351:
5329:
5325:
5302:
5298:
5294:
5289:
5284:
5280:
5276:
5271:
5266:
5262:
5258:
5253:
5248:
5244:
5235:
5217:
5213:
5209:
5204:
5200:
5196:
5185:
5180:
5176:
5172:
5169:
5164:
5160:
5153:
5149:
5137:
5133:
5129:
5124:
5120:
5111:
5098:
5090:
5086:
5077:
5069:
5065:
5061:
5055:
5052:
5049:
5041:
5037:
5028:
5012:
5009:
5006:
4992:
4990:
4986:
4968:
4964:
4941:
4937:
4926:
4905:
4901:
4897:
4892:
4888:
4879:
4866:
4863:
4851:
4845:
4842:
4839:
4831:
4827:
4800:
4797:
4794:
4786:
4782:
4768:
4760:
4756:
4752:
4747:
4743:
4734:
4720:
4718:
4714:
4710:
4706:
4703:
4700:
4696:
4681:
4677:
4673:
4668:
4664:
4660:
4649:
4644:
4640:
4636:
4633:
4628:
4624:
4617:
4613:
4601:
4597:
4593:
4588:
4584:
4575:
4562:
4554:
4550:
4541:
4533:
4529:
4525:
4519:
4516:
4513:
4505:
4501:
4480:
4475:
4472:
4469:
4456:
4453:
4450:
4447:
4444:
4438:
4435:
4429:
4426:
4423:
4415:
4411:
4401:
4385:
4381:
4358:
4354:
4331:
4327:
4301:
4297:
4293:
4288:
4284:
4280:
4275:
4271:
4242:
4238:
4234:
4229:
4225:
4216:
4191:
4188:
4185:
4182:
4179:
4157:
4153:
4150:
4144:
4139:
4136:
4131:
4128:
4106:
4102:
4079:
4075:
4071:
4066:
4061:
4057:
4053:
4048:
4043:
4039:
4035:
4030:
4025:
4021:
4007:
4000:
3998:
3982:
3978:
3974:
3969:
3965:
3961:
3949:
3944:
3940:
3936:
3931:
3926:
3922:
3918:
3913:
3909:
3903:
3900:
3897:
3893:
3881:
3876:
3872:
3868:
3865:
3860:
3856:
3849:
3845:
3833:
3829:
3825:
3820:
3816:
3807:
3794:
3786:
3782:
3773:
3765:
3761:
3757:
3751:
3748:
3745:
3742:
3739:
3733:
3726:
3725:
3722:
3720:
3704:
3690:
3688:
3684:
3680:
3676:
3671:
3657:
3637:
3617:
3614:
3611:
3591:
3569:
3565:
3544:
3524:
3504:
3501:
3498:
3478:
3456:
3452:
3449:
3443:
3438:
3435:
3430:
3427:
3424:
3420:
3412:
3375:
3371:
3367:
3348:
3344:
3340:
3321:
3317:
3313:
3288:
3268:
3248:
3245:
3242:
3220:
3216:
3195:
3175:
3155:
3152:
3149:
3127:
3123:
3114:
3096:
3092:
3069:
3065:
3042:
3038:
3029:
3013:
2983:
2978:
2974:
2970:
2965:
2960:
2956:
2952:
2947:
2943:
2937:
2934:
2931:
2927:
2918:
2913:
2909:
2905:
2902:
2897:
2893:
2886:
2882:
2878:
2875:
2873:
2863:
2858:
2854:
2850:
2846:
2837:
2832:
2828:
2824:
2821:
2816:
2812:
2805:
2801:
2797:
2794:
2792:
2782:
2777:
2773:
2769:
2765:
2759:
2754:
2750:
2746:
2742:
2736:
2731:
2727:
2723:
2719:
2715:
2712:
2710:
2702:
2699:
2696:
2693:
2690:
2684:
2657:
2654:
2651:
2648:
2645:
2639:
2616:
2613:
2610:
2607:
2604:
2598:
2589:
2570:
2566:
2562:
2560:
2553:
2548:
2544:
2540:
2535:
2530:
2526:
2522:
2517:
2512:
2508:
2500:
2497:
2495:
2487:
2479:
2475:
2469:
2464:
2460:
2456:
2450:
2442:
2438:
2429:
2425:
2421:
2415:
2411:
2401:
2398:
2396:
2388:
2380:
2376:
2370:
2365:
2361:
2357:
2351:
2343:
2339:
2330:
2326:
2322:
2316:
2312:
2302:
2299:
2297:
2289:
2281:
2277:
2271:
2266:
2262:
2258:
2252:
2244:
2240:
2231:
2227:
2223:
2217:
2213:
2198:
2196:
2189:
2182:
2175:
2171:
2167:
2160:
2156:
2152:
2148:
2144:
2139:
2126:
2123:
2118:
2114:
2110:
2101:
2093:
2089:
2080:
2073:
2069:
2065:
2058:
2049:
2041:
2037:
2028:
2021:
2017:
2013:
2006:
1997:
1989:
1985:
1976:
1969:
1965:
1961:
1937:
1934:
1928:
1920:
1916:
1909:
1901:
1897:
1890:
1882:
1878:
1872:
1868:
1864:
1858:
1851:
1847:
1843:
1836:
1828:
1824:
1817:
1809:
1805:
1801:
1795:
1787:
1783:
1776:
1769:
1765:
1761:
1754:
1746:
1742:
1738:
1732:
1724:
1720:
1713:
1705:
1701:
1694:
1687:
1683:
1679:
1658:
1650:
1646:
1637:
1632:
1621:
1613:
1609:
1600:
1595:
1584:
1576:
1572:
1563:
1558:
1547:
1544:
1539:
1526:
1522:
1518:
1514:
1510:
1491:
1483:
1479:
1472:
1464:
1460:
1453:
1445:
1441:
1437:
1431:
1428:
1425:
1422:
1419:
1413:
1405:
1401:
1397:
1393:
1389:
1379:
1366:
1363:
1349:
1345:
1339:
1335:
1331:
1326:
1317:
1308:
1290:
1287:
1284:
1280:
1265:
1242:
1239:
1228:
1219:
1205:
1185:
1177:
1150:
1130:
1108:
1104:
1101:
1095:
1090:
1087:
1082:
1079:
1059:
1056:
1052:
1037:
1033:
1027:
1023:
1019:
1014:
1005:
1000:
996:
993:
979:
965:
945:
937:
908:
905:
901:
886:
883:
869:
849:
846:
843:
840:
837:
817:
796:
793:
790:
786:
783:
775:
759:
739:
735:
729:
726:
723:
719:
704:
700:
696:
693:
690:
684:
681:
670:
662:
646:
626:
622:
618:
615:
612:
592:
589:
586:
583:
580:
558:
555:
552:
548:
539:
535:
531:
521:
519:
515:
514:
509:
505:
497:
493:
489:
485:
478:
474:
470:
463:
459:
455:
450:
446:
442:
438:
434:
418:
398:
395:
389:
386:
375:
371:
362:
357:
346:
332:
328:
324:
319:
314:
305:
296:
295:wave equation
286:
284:
280:
261:
258:
255:
252:
249:
243:
215:
212:
201:
198:
195:
187:
176:
174:
170:
163:
156:
152:
148:
143:
141:
137:
133:
129:
122:
115:
111:
107:
106:
101:
100:
95:
91:
86:
84:
80:
76:
72:
67:
63:
58:
52:
50:
46:
45:
44:superposition
40:
36:
32:
19:
16601:
16597:
16563:
16559:
16525:Artech House
16520:
16502:Artech House
16497:
16474:
16451:
16428:
16417:. Retrieved
16402:
16385:
16357:
16350:
16341:
16335:
16326:
16226:
16177:
16069:
15984:
15420:
15318:
14955:
14671:
14061:
13961:
13902:
13898:
13894:
13846:
13842:
13840:
13836:
13825:
13385:
13274:
13140:and a noise
13034:
13021:
13014:
13010:
13003:
12996:
12992:
12988:
12984:
12980:
12976:
12972:
12968:
12961:
12954:
12950:
12943:
12936:
12932:
12926:
12921:
12917:
12913:
12909:
12905:
12901:
12897:
12894:
12889:
12885:
12881:
12877:
12873:
12869:
12858:
12851:
12847:
12840:
12833:
12829:
12825:
12821:
12817:
12811:
12805:
12801:
12797:
12793:
12789:
12782:
12775:
12771:
12767:
12753:
12720:
12714:
12698:
12696:
12676:
12667:
12661:
12655:
12643:
12637:
12631:
12629:
12624:
12620:
12616:
12612:
12608:
12604:
12600:
12597:
12588:
12582:
12568:
12563:
12559:
12555:
12551:
12547:
12543:
12534:
12530:
12527:
12522:
12518:
12514:
12510:
12506:
12502:
12499:
12495:
12491:
12489:
12479:
12475:
12471:
12467:
12460:
12447:
12443:
12436:
12433:
12426:
12420:
12414:
12412:
12404:
12399:
12392:
12385:
12378:
12307:
12293:
12289:
12285:
12281:
12277:
12226:
12180:
12178:
11941:
11937:
11933:
11927:
11923:
11919:
11915:
11909:
11905:
11901:
11899:
11880:
11848:
11821:
11818:
11709:
11702:
11697:
11663:
11537:
11530:
11528:
11365:
11349:
11344:
11340:
11336:
11332:
11328:
11324:
11320:
11215:
11211:
11207:
11203:
11199:
11195:
11191:
11189:
11080:
11078:
11073:
11069:
11065:
11061:
11057:
11053:
11049:
11043:
11033:
11029:
11025:
11021:
11017:
11013:
11009:
11002:
10998:
10991:
10985:
10981:
10977:
10973:
10966:
10964:
10953:
10756:
10754:
10744:
10741:
10730:
10468:
10377:
10373:
10369:
10367:
10364:Output plane
10276:
10272:
10270:
10256:
10253:
10240:
10237:
10233:
10152:
10148:
10144:
10138:
10134:
10130:
10126:
10122:
10118:
10116:
10060:
9818:
9615:
9604:
9600:next section
9597:
9161:For a given
9160:
9134:
9101:
9099:
9077:
9045:
8984:
8912:
8910:
8889:
8881:
8869:
8782:
8771:
8642:
8632:
8627:
8623:
8600:
8515:
8507:
8506:
8495:
8423:
8420:
8416:
8407:
8403:
8399:
8393:
8391:
8331:
8161:
7936:
7925:
7539:
7537:
6858:
6624:
6619:
6597:
6409:, that is a
6115:
6074:
5428:
5233:
4998:
4927:
4721:
4716:
4712:
4708:
4704:
4698:
4697:
4402:
4012:
4001:
3696:
3672:
2590:
2199:
2194:
2187:
2180:
2173:
2169:
2165:
2158:
2154:
2150:
2146:
2142:
2140:
1520:
1516:
1512:
1508:
1403:
1385:
1220:
985:
527:
512:
511:
507:
500:
495:
491:
487:
483:
476:
472:
465:
461:
457:
453:
444:
440:
436:
292:
282:
185:
182:
165:
158:
144:
135:
131:
124:
117:
103:
97:
87:
71:even locally
70:
65:
53:
42:
30:
29:
12866:convolution
12764:convolution
12737:section 6.1
12733:section 1.5
12729:convolution
11948:), so that
11885:, then its
10984:. In eqn. (
10267:Input plane
10261:section 6.2
9424:and in the
9151:section 1.3
8906:Bloch waves
8890:propagating
8398:) above is
7160:Terms with
7157:is ignored.
6214:wave vector
6101:photoresist
5234:constraints
3683:cylindrical
3681:, yielding
3675:cylindrical
449:real-valued
281:here), and
73:. A "wide"
16629:Categories
16419:2017-10-28
16318:References
14732:such that
12181:plane wave
11944:is small (
11918:= 0), and
11859:atom traps
11068:such that
10761:) becomes
8860:wavenumber
8510:plane wave
7104:is small (
6413:such that
4403:Next, let
1519:. If this
534:wavelength
16618:0031-899X
16582:1687-6393
16116:λ
16094:∗
16084:∝
16078:Δ
16044:Δ
16038:≈
16029:
16020:Δ
16017:∝
16011:Δ
16008:∝
16005:ϕ
16002:Δ
15999:∝
15993:Δ
15951:λ
15926:λ
15905:ψ
15889:α
15886:π
15875:
15858:λ
15833:λ
15803:λ
15768:λ
15743:λ
15719:Δ
15696:¯
15636:λ
15559:ϕ
15547:≈
15542:ϕ
15514:π
15491:ϕ
15400:λ
15391:≪
15286:λ
15261:λ
15240:ψ
15224:α
15221:π
15210:
15193:λ
15168:λ
15138:λ
15103:λ
15078:λ
15028:λ
14923:λ
14898:λ
14877:ψ
14866:
14849:λ
14824:λ
14786:λ
14761:λ
14720:ψ
14639:α
14636:π
14627:−
14619:
14602:λ
14577:λ
14554:∗
14540:λ
14506:α
14503:π
14489:
14472:λ
14447:λ
14417:λ
14376:λ
14351:λ
14302:λ
14242:λ
14217:λ
14187:λ
14160:α
14157:π
14148:−
14140:
14034:λ
14009:λ
13978:λ
13940:α
13937:π
13928:−
13772:Φ
13712:Φ
13590:Φ
13545:Φ
13445:∗
13357:^
13348:−
13328:ε
13289:^
13222:⊗
12929:section 5
12846:) ×
12741:section 5
12717:section 5
12603:/λ where
12377:(for all
12332:≫
12284:/2 + 1 −
12262:θ
12259:
12211:θ
12208:
12159:θ
12139:≃
12127:θ
12113:−
12101:≃
12095:θ
12092:
12057:θ
12043:−
12037:≃
12034:θ
12031:
11988:θ
11985:≃
11982:θ
11979:
11698:(Goodman)
11678:ω
11618:⋅
11510:ω
11479:ω
11448:ω
11416:ω
11407:⋅
11401:ω
11380:ω
11268:−
11254:∞
11249:∞
11246:−
11242:∫
11137:−
11128:δ
11123:∞
11118:∞
11115:−
11111:∫
11046:causality
10982:Mx′
10972:(), from
10867:−
10847:−
10826:∞
10821:∞
10818:−
10814:∫
10808:∞
10803:∞
10800:−
10796:∫
10647:−
10630:−
10616:∞
10611:∞
10608:−
10604:∫
10598:∞
10593:∞
10590:−
10586:∫
10521:∗
10202:∗
10086:π
9942:∞
9937:∞
9934:−
9930:∫
9924:∞
9919:∞
9916:−
9912:∫
9895:π
9753:−
9725:∞
9720:∞
9717:−
9713:∫
9707:∞
9702:∞
9699:−
9695:∫
9580:θ
9577:
9534:ϕ
9531:
9522:θ
9519:
9476:ϕ
9473:
9464:θ
9461:
9409:^
9384:^
9359:^
9218:λ
9214:π
9197:ω
9130:countably
9112:λ
9058:∇
9008:λ
9005:−
8931:∇
8894:waveguide
8846:λ
8838:π
8710:∇
8675:ω
8585:λ
8545:λ
8472:^
8463:×
8355:−
8314:θ
8311:
8268:ϕ
8265:
8256:θ
8253:
8210:ϕ
8207:
8198:θ
8195:
8144:θ
8141:
8105:ϕ
8102:
8093:θ
8090:
8054:ϕ
8051:
8042:θ
8039:
8013:, where,
7906:ϕ
7903:
7894:θ
7891:
7876:ϕ
7873:
7864:θ
7861:
7817:−
7801:θ
7798:
7777:π
7759:ϕ
7753:θ
7505:π
7493:λ
7411:∂
7401:∂
7353:∂
7349:∂
7336:≪
7301:∂
7291:∂
7139:θ
7115:≪
7112:θ
7092:θ
7048:∂
7038:∂
7016:∂
7006:∂
6984:∂
6974:∂
6968:−
6959:∇
6941:∇
6908:∂
6900:∂
6885:−
6868:∇
6829:−
6805:θ
6747:−
6725:≈
6711:ψ
6674:θ
6670:−
6658:≈
6655:θ
6652:
6606:θ
6566:θ
6562:−
6556:≈
6549:θ
6546:
6536:θ
6533:≈
6526:θ
6523:
6513:θ
6510:≈
6503:θ
6500:
6454:≪
6388:ω
6322:‖
6314:‖
6229:⋅
6171:⋅
6160:−
6124:ψ
5821:≥
5720:π
5518:Ψ
5512:∞
5504:∞
5501:−
5497:∫
5491:∞
5483:∞
5480:−
5476:∫
5440:ψ
5412:π
5403:ω
5381:λ
5377:π
5360:ω
5232:with the
5108:Ψ
5102:∞
5094:∞
5091:−
5087:∫
5081:∞
5073:∞
5070:−
5066:∫
5038:ψ
4876:Ψ
4864:−
4828:ψ
4783:ψ
4731:Ψ
4572:Ψ
4566:∞
4558:∞
4555:−
4551:∫
4545:∞
4537:∞
4534:−
4530:∫
4502:ψ
4439:ψ
4412:ψ
4213:Ψ
4189:π
4180:ω
4158:λ
4154:π
4137:ω
3937:−
3919:−
3898:±
3804:Ψ
3798:∞
3790:∞
3787:−
3783:∫
3777:∞
3769:∞
3766:−
3762:∫
3734:ψ
3457:λ
3453:π
3436:ω
3389:^
3362:^
3335:^
2971:−
2953:−
2932:±
2685:ψ
2640:ψ
2599:ψ
1643:∂
1638:ψ
1629:∂
1606:∂
1601:ψ
1592:∂
1569:∂
1564:ψ
1555:∂
1545:ψ
1536:∇
1414:ψ
1350:ψ
1323:∇
1288:ω
1266:ψ
1206:ω
1151:ψ
1131:λ
1109:λ
1105:π
1088:ω
1038:ψ
1011:∇
997:
966:ϕ
909:ϕ
870:ψ
847:π
838:ω
787:
727:ω
705:ψ
697:
647:τ
627:τ
590:π
581:ω
556:ω
353:∂
343:∂
320:−
311:∇
99:frequency
16400:(2005).
16384:(1983).
16244:See also
16215:and the
16184:Stratton
12539:wavelets
12225:and the
11834:such as
11341:t′
11329:t′
11306:′
11292:′
11275:′
11208:t′
11175:′
11161:′
11144:′
11076:< 0.
11074:t′
11058:t′
10974:x′
10936:′
10925:′
10908:′
10897:′
10877:′
10857:′
10713:′
10702:′
10685:′
10674:′
10654:′
10637:′
9319:k-vector
9181:such as
9126:singular
8902:solitons
8882:resonant
8400:critical
5735:, where
2074:″
2022:″
1970:″
1852:″
1770:″
1688:″
661:harmonic
116:domain (
66:extended
14672:Assume
12288:/2) = 2
11932:) (for
9157:K-space
8900:modes,
8896:modes,
8873:trivial
8858:is the
8633:optical
8516:uniform
7077:in the
6212:is the
5317:, each
4094:, each
3717:in the
3404:(where
1072:where
936:complex
772:is the
516:in the
506:is the
447:) is a
431:is the
16616:
16580:
16531:
16508:
16485:
16462:
16439:
16429:Optics
16410:
16365:
13650:where
12880:) and
12500:entire
11861:, and
11770:
11718:
11583:
11577:
11431:where
11392:
11386:
11319:where
11192:linear
10915:
10884:
10692:
10661:
10551:i.e.,
10500:
10494:
10241:actual
9583:
9571:
9525:
9513:
9467:
9455:
9340:
9334:
9144:as an
8783:where
8557:where
8445:
8439:
8317:
8305:
8259:
8247:
8201:
8189:
8147:
8135:
8096:
8084:
8045:
8033:
7897:
7885:
7867:
7855:
7836:
7807:
7792:
7783:
7771:
7765:
6932:where
6598:where
6216:, and
6190:where
5642:where
5604:
5556:
5395:where
5194:
5146:
4722:Thus:
4658:
4610:
4493:Then:
4319:where
4172:where
3959:
3890:
3842:
752:where
639:where
499:where
411:where
231:where
37:using
35:optics
16594:(PDF)
12758:from
12280:(1 +
11867:phase
3650:, or
3537:, or
3281:, or
3188:, or
3084:, or
1402:, an
605:with
538:color
494:, or
16614:ISSN
16578:ISSN
16564:2010
16529:ISBN
16506:ISBN
16483:ISBN
16460:ISBN
16437:ISBN
16408:ISBN
16363:ISBN
12727:and
12573:and
12075:and
12000:and
11883:lens
11842:and
11828:lens
11822:The
11016:) =
10749:coma
10151:and
9093:and
9083:span
8904:and
8871:non-
8821:and
8508:The
8458:aper
8162:and
7243:(or
7190:and
6703:and
6300:and
5912:<
4987:and
4956:and
4819:and
4373:and
3685:and
3677:and
2186:and
2168:and
1394:for
481:for
464:) =
435:and
105:time
102:and
75:wave
16606:doi
16568:doi
16182:or
16026:log
15872:cos
15207:cos
14863:exp
14616:exp
14486:exp
14137:exp
12868:of
12813:2.1
12807:2.1
12689:).
12663:2.2
12645:2.1
12639:2.1
12633:2.2
12482:).
12431:).
12428:2.1
12422:2.1
12416:2.1
12256:cos
12205:cos
12089:cos
12028:cos
11976:sin
11929:2.1
11911:2.1
11873:).
11532:4.1
11083:of
11036:).
10990:),
10987:4.2
10976:to
10956:4.2
10758:4.1
10733:4.1
9651:):
9574:cos
9528:sin
9516:sin
9470:cos
9458:sin
9428:as
8914:2.3
8774:2.3
8640:).
8395:2.2
8308:cos
8262:sin
8250:sin
8204:cos
8192:sin
8138:cos
8099:sin
8087:sin
8048:cos
8036:sin
7928:2.2
7900:sin
7888:sin
7870:cos
7858:sin
7795:cos
7541:2.1
6649:cos
6543:cos
6520:tan
6497:sin
5451:unc
5236:of
4719:).
4702:2.1
4004:2.1
2673:is
830:,
520:).
155:EUV
153:or
151:DUV
16631::
16612:.
16602:56
16600:.
16596:.
16576:.
16562:.
16558:.
16527:.
16523:.
16504:.
16500:.
16481:.
16477:.
16458:.
16454:.
16435:.
15418:.
13962:L3
13763:,
13703:,
13390::
12781:,
12743:.
12583:do
12581:-
12470:)/
12446:)/
12384:,
11857:,
11853:,
11846:.
11838:,
11700:.
11345:t'
11339:−
11327:−
11206:−
11087:.
11072:−
11062:t'
11056:−
10980:=
10372:=
10155:.
9602:.
9153:.
9097:.
9089:,
8987:,
4991:.
3630:,
3517:,
3261:,
3168:,
3057:,
2197::
2179:,
1367:0.
994:Re
978:.
784:Re
776:,
694:Re
490:,
486:=
475:,
460:,
399:0.
164:,
134:,
123:,
16620:.
16608::
16584:.
16570::
16537:.
16514:.
16491:.
16468:.
16445:.
16422:.
16392:.
16371:.
16344:.
16159:.
16151:2
16146:|
16143:S
16140:|
16135:+
16130:2
16126:f
16120:2
16110:2
16105:0
16101:r
16090:S
16081:t
16056:.
16051:I
16047:I
16035:)
16032:I
16023:(
16014:D
15996:t
15970:.
15966:]
15961:)
15954:f
15945:2
15941:y
15935:,
15929:f
15920:2
15916:x
15909:(
15902:+
15897:2
15893:y
15883:2
15879:[
15868:)
15861:f
15852:2
15848:y
15842:,
15836:f
15827:2
15823:x
15816:(
15812:S
15806:f
15796:o
15792:r
15788:2
15782:=
15778:)
15771:f
15762:2
15758:y
15752:,
15746:f
15737:2
15733:x
15726:(
15722:I
15693:I
15668:.
15663:2
15659:S
15650:2
15646:f
15640:2
15631:1
15626:+
15621:2
15616:o
15612:r
15608:=
15605:I
15582:D
15571:.
15556:j
15553:+
15550:1
15539:j
15535:e
15511:2
15468:)
15465:y
15462:,
15459:x
15456:(
15453:s
15433:t
15403:f
15396:1
15386:0
15382:r
15360:0
15356:=
15353:)
15348:2
15344:y
15340:,
15335:2
15331:x
15327:(
15305:.
15301:]
15296:)
15289:f
15280:2
15276:y
15270:,
15264:f
15255:2
15251:x
15244:(
15237:+
15232:2
15228:y
15218:2
15214:[
15203:)
15196:f
15187:2
15183:y
15177:,
15171:f
15162:2
15158:x
15151:(
15147:S
15141:f
15131:o
15127:r
15123:2
15117:+
15113:)
15106:f
15097:2
15093:y
15087:,
15081:f
15072:2
15068:x
15061:(
15055:2
15051:S
15042:2
15038:f
15032:2
15023:1
15018:+
15013:2
15008:o
15004:r
15000:=
14996:)
14990:2
14986:y
14982:,
14977:2
14973:x
14968:(
14964:I
14942:,
14938:]
14933:)
14926:f
14917:2
14913:y
14907:,
14901:f
14892:2
14888:x
14881:(
14874:j
14870:[
14859:)
14852:f
14843:2
14839:y
14833:,
14827:f
14818:2
14814:x
14807:(
14803:S
14800:=
14796:)
14789:f
14780:2
14776:y
14770:,
14764:f
14755:2
14751:x
14744:(
14740:H
14700:A
14680:H
14653:)
14647:2
14643:y
14633:2
14630:j
14623:(
14612:)
14605:f
14596:2
14592:y
14586:,
14580:f
14571:2
14567:x
14560:(
14550:H
14543:f
14534:o
14530:r
14524:+
14520:)
14514:2
14510:y
14500:2
14497:j
14493:(
14482:)
14475:f
14466:2
14462:y
14456:,
14450:f
14441:2
14437:x
14430:(
14426:H
14420:f
14411:o
14407:r
14401:+
14396:2
14391:|
14386:)
14379:f
14370:2
14366:y
14360:,
14354:f
14345:2
14341:x
14334:(
14330:H
14326:|
14316:2
14312:f
14306:2
14297:1
14292:+
14287:2
14282:o
14278:r
14274:=
14262:2
14257:|
14252:)
14245:f
14236:2
14232:y
14226:,
14220:f
14211:2
14207:x
14200:(
14196:H
14190:f
14183:1
14178:+
14174:)
14168:2
14164:y
14154:2
14151:j
14144:(
14132:o
14128:r
14123:|
14118:=
14110:)
14104:2
14100:y
14096:,
14091:2
14087:x
14082:(
14076:I
14058:.
14044:)
14037:f
14028:2
14024:y
14018:,
14012:f
14003:2
13999:x
13992:(
13987:H
13981:f
13974:1
13943:y
13934:2
13931:j
13924:e
13918:0
13914:r
13903:L
13899:L
13895:L
13877:h
13857:h
13847:L
13843:S
13810:)
13804:Y
13800:f
13796:,
13791:X
13787:f
13782:(
13776:n
13750:)
13744:Y
13740:f
13736:,
13731:X
13727:f
13722:(
13716:o
13690:)
13684:Y
13680:f
13676:,
13671:X
13667:f
13662:(
13658:S
13638:,
13628:)
13622:Y
13618:f
13614:,
13609:X
13605:f
13600:(
13594:o
13583:)
13577:Y
13573:f
13569:,
13564:X
13560:f
13555:(
13549:n
13538:+
13533:2
13528:|
13523:)
13517:Y
13513:f
13509:,
13504:X
13500:f
13495:(
13491:S
13487:|
13479:)
13473:Y
13469:f
13465:,
13460:X
13456:f
13451:(
13441:S
13434:=
13430:)
13424:Y
13420:f
13416:,
13411:X
13407:f
13402:(
13398:H
13370:2
13365:|
13354:o
13345:o
13341:|
13337:=
13332:2
13307:)
13304:y
13301:,
13298:x
13295:(
13286:o
13261:)
13258:y
13255:,
13252:x
13249:(
13246:n
13243:+
13240:)
13237:y
13234:,
13231:x
13228:(
13225:s
13219:)
13216:y
13213:,
13210:x
13207:(
13204:o
13201:=
13198:)
13195:y
13192:,
13189:x
13186:(
13183:i
13163:)
13160:y
13157:,
13154:x
13151:(
13148:n
13128:)
13125:y
13122:,
13119:x
13116:(
13113:s
13093:)
13090:y
13087:,
13084:x
13081:(
13078:i
13058:)
13055:y
13052:,
13049:x
13046:(
13043:o
13025:y
13022:k
13020:,
13018:x
13015:k
13013:(
13011:F
13007:y
13004:k
13002:,
13000:x
12997:k
12995:(
12993:H
12989:y
12987:,
12985:x
12983:(
12981:g
12977:y
12975:,
12973:x
12971:(
12969:h
12965:y
12962:k
12960:,
12958:x
12955:k
12953:(
12951:H
12947:y
12944:k
12942:,
12940:x
12937:k
12935:(
12933:G
12922:y
12920:,
12918:x
12916:(
12914:g
12910:y
12908:,
12906:x
12904:(
12902:g
12890:y
12888:,
12886:x
12884:(
12882:g
12878:y
12876:,
12874:x
12872:(
12870:f
12862:y
12859:k
12857:,
12855:x
12852:k
12850:(
12848:G
12844:y
12841:k
12839:,
12837:x
12834:k
12832:(
12830:F
12826:y
12824:,
12822:x
12820:(
12818:g
12802:y
12800:,
12798:x
12796:(
12794:f
12786:y
12783:k
12779:x
12776:k
12772:y
12770:,
12768:x
12721:H
12699:e
12668:D
12625:D
12621:D
12617:D
12613:D
12609:D
12605:D
12601:D
12564:y
12562:-
12560:x
12556:y
12554:-
12552:x
12548:y
12546:-
12544:x
12535:y
12533:-
12531:x
12523:y
12521:-
12519:x
12515:z
12511:z
12507:y
12505:-
12503:x
12496:y
12494:-
12492:x
12480:x
12478:/
12476:x
12472:x
12468:x
12466:(
12464:1
12461:J
12448:x
12444:x
12442:(
12440:1
12437:J
12400:k
12395:z
12393:k
12388:y
12386:k
12381:x
12379:k
12363:2
12358:y
12354:k
12350:+
12345:2
12340:x
12336:k
12327:2
12323:k
12294:θ
12290:f
12286:θ
12282:θ
12278:f
12252:/
12248:f
12245:k
12242:i
12238:e
12202:f
12199:k
12196:i
12192:e
12163:2
12153:2
12150:1
12145:+
12142:1
12131:2
12121:2
12118:1
12110:1
12106:1
12085:1
12061:2
12051:2
12048:1
12040:1
12025:=
12020:k
12015:z
12011:k
11973:=
11968:k
11963:x
11959:k
11942:θ
11938:θ
11934:z
11926:(
11916:z
11908:(
11799:)
11794:y
11790:k
11786:,
11781:x
11777:k
11773:(
11750:)
11745:y
11741:k
11737:,
11732:x
11728:k
11724:(
11721:G
11681:)
11675:(
11672:H
11650:)
11645:y
11641:k
11637:,
11632:x
11628:k
11624:(
11621:F
11615:)
11610:y
11606:k
11602:,
11597:x
11593:k
11589:(
11586:H
11580:=
11574:)
11569:y
11565:k
11561:,
11556:x
11552:k
11548:(
11545:G
11513:)
11507:(
11504:F
11482:)
11476:(
11473:H
11451:)
11445:(
11442:G
11419:)
11413:(
11410:F
11404:)
11398:(
11395:H
11389:=
11383:)
11377:(
11374:G
11337:t
11335:(
11333:δ
11325:t
11323:(
11321:h
11303:t
11299:d
11296:)
11289:t
11285:(
11282:f
11279:)
11272:t
11265:t
11262:(
11259:h
11238:=
11235:)
11232:t
11229:(
11226:g
11216:t
11214:(
11212:f
11204:t
11202:(
11200:h
11172:t
11168:d
11165:)
11158:t
11154:(
11151:f
11148:)
11141:t
11134:t
11131:(
11107:=
11104:)
11101:t
11098:(
11095:f
11070:t
11066:t
11054:t
11052:(
11050:h
11034:M
11032:/
11030:y
11028:,
11026:M
11024:/
11022:x
11020:(
11018:h
11014:y
11012:,
11010:x
11008:(
11005:M
11003:h
10999:h
10994:M
10992:h
10978:x
10969:M
10967:h
10958:)
10954:(
10933:y
10929:d
10922:x
10918:d
10912:)
10905:y
10901:,
10894:x
10890:(
10887:f
10881:)
10874:y
10870:M
10864:y
10861:,
10854:x
10850:M
10844:x
10841:(
10836:M
10832:h
10792:=
10789:)
10786:y
10783:,
10780:x
10777:(
10774:g
10735:)
10731:(
10710:y
10706:d
10699:x
10695:d
10689:)
10682:y
10678:,
10671:x
10667:(
10664:f
10658:)
10651:y
10644:y
10641:,
10634:x
10627:x
10624:(
10621:h
10582:=
10579:)
10576:y
10573:,
10570:x
10567:(
10564:g
10539:)
10536:y
10533:,
10530:x
10527:(
10524:f
10518:)
10515:y
10512:,
10509:x
10506:(
10503:h
10497:=
10491:)
10488:y
10485:,
10482:x
10479:(
10476:g
10448:d
10445:=
10442:z
10436:|
10430:)
10427:z
10424:,
10421:y
10418:,
10415:x
10412:(
10409:U
10406:=
10403:)
10400:y
10397:,
10394:x
10391:(
10388:g
10378:g
10374:d
10370:z
10347:0
10344:=
10341:z
10335:|
10329:)
10326:z
10323:,
10320:y
10317:,
10314:x
10311:(
10308:U
10305:=
10302:)
10299:y
10296:,
10293:x
10290:(
10287:f
10277:f
10273:z
10220:)
10217:y
10214:,
10211:x
10208:(
10205:f
10199:)
10196:y
10193:,
10190:x
10187:(
10184:h
10181:=
10178:)
10175:y
10172:,
10169:x
10166:(
10163:g
10153:y
10149:x
10145:z
10135:h
10131:f
10127:g
10123:g
10119:f
10094:2
10090:)
10083:2
10080:(
10075:/
10070:1
10045:y
10041:k
10037:d
10032:x
10028:k
10024:d
10019:)
10016:y
10011:y
10007:k
10003:+
10000:x
9995:x
9991:k
9987:(
9984:i
9980:e
9976:)
9971:y
9967:k
9963:,
9958:x
9954:k
9950:(
9947:U
9903:2
9899:)
9892:2
9889:(
9885:1
9880:=
9877:)
9874:y
9871:,
9868:x
9865:(
9862:u
9842:)
9839:y
9836:,
9833:x
9830:(
9827:u
9805:y
9802:d
9799:x
9796:d
9791:)
9788:y
9783:y
9779:k
9775:+
9772:x
9767:x
9763:k
9759:(
9756:i
9749:e
9745:)
9742:y
9739:,
9736:x
9733:(
9730:u
9691:=
9688:)
9683:y
9679:k
9675:,
9670:x
9666:k
9662:(
9659:U
9639:)
9636:y
9633:,
9630:x
9627:(
9624:u
9568:k
9565:=
9560:z
9556:k
9510:k
9507:=
9502:y
9498:k
9452:k
9449:=
9444:x
9440:k
9406:z
9398:z
9394:k
9390:+
9381:y
9373:y
9369:k
9365:+
9356:x
9348:x
9344:k
9337:=
9330:k
9299:2
9295:k
9291:=
9286:2
9281:z
9277:k
9273:+
9268:2
9263:y
9259:k
9255:+
9250:2
9245:x
9241:k
9211:2
9205:=
9200:c
9192:=
9189:k
9169:k
9102:A
9078:A
9062:2
9032:,
9029:0
9026:=
9022:x
9017:)
9012:I
9001:A
8996:(
8985:A
8967:,
8964:0
8961:=
8958:f
8954:)
8948:2
8944:k
8940:+
8935:2
8926:(
8842:/
8835:2
8832:=
8829:k
8809:z
8806:,
8803:y
8800:,
8797:x
8794:=
8791:u
8776:)
8772:(
8755:0
8752:=
8747:u
8743:E
8737:2
8733:k
8729:+
8724:u
8720:E
8714:2
8678:t
8672:i
8668:e
8628:y
8626:-
8624:x
8610:z
8602:(
8565:D
8541:/
8535:2
8531:D
8527:2
8496:J
8478:.
8469:z
8453:E
8448:2
8442:=
8435:M
8424:M
8369:r
8364:r
8361:k
8358:i
8351:e
8302:k
8299:=
8294:z
8290:k
8244:k
8241:=
8236:y
8232:k
8186:k
8183:=
8178:x
8174:k
8132:r
8129:=
8126:z
8081:r
8078:=
8075:y
8030:r
8027:=
8024:x
8001:)
7998:z
7995:,
7992:y
7989:,
7986:x
7983:(
7963:)
7960:z
7957:,
7954:y
7951:,
7948:x
7945:(
7930:)
7926:(
7909:)
7882:k
7879:,
7852:k
7849:(
7844:u
7840:E
7831:r
7826:r
7823:k
7820:i
7813:e
7804:)
7789:k
7786:(
7780:i
7774:2
7768:=
7762:)
7756:,
7750:,
7747:r
7744:(
7739:u
7735:E
7709:)
7706:z
7703:,
7700:y
7697:,
7694:x
7691:(
7671:)
7668:z
7665:,
7662:y
7659:,
7656:x
7653:(
7633:)
7628:z
7624:k
7620:,
7615:y
7611:k
7607:,
7602:x
7598:k
7594:(
7574:)
7571:z
7568:,
7565:y
7562:,
7559:x
7556:(
7523:.
7509:k
7502:2
7496:=
7473:)
7469:r
7465:(
7462:A
7438:)
7434:r
7430:(
7427:A
7419:2
7415:z
7405:2
7377:|
7373:)
7369:r
7365:(
7362:A
7356:z
7344:k
7340:|
7332:|
7328:)
7324:r
7320:(
7317:A
7309:2
7305:z
7295:2
7284:|
7259:z
7255:k
7251:2
7231:k
7228:2
7206:y
7202:k
7198:2
7176:x
7172:k
7168:2
7143:2
7118:1
7056:2
7052:y
7042:2
7032:+
7024:2
7020:x
7010:2
7000:=
6992:2
6988:z
6978:2
6963:2
6955:=
6950:2
6945:T
6920:0
6917:=
6911:z
6903:A
6894:k
6891:i
6888:2
6882:A
6877:2
6872:T
6838:z
6835:k
6832:i
6825:e
6819:2
6815:/
6809:2
6801:z
6798:k
6795:i
6791:e
6785:)
6782:y
6777:y
6773:k
6769:+
6766:x
6761:x
6757:k
6753:(
6750:i
6743:e
6739:)
6735:r
6731:(
6728:A
6722:)
6718:r
6714:(
6691:)
6688:2
6684:/
6678:2
6667:1
6664:(
6661:k
6646:k
6643:=
6638:z
6634:k
6620:k
6580:2
6576:/
6570:2
6559:1
6467:2
6462:z
6458:k
6449:2
6444:y
6440:k
6436:+
6431:2
6426:x
6422:k
6391:c
6383:=
6376:2
6371:z
6367:k
6363:+
6358:2
6353:y
6349:k
6345:+
6340:2
6335:x
6331:k
6325:=
6318:k
6311:=
6308:k
6287:z
6281:z
6277:k
6273:+
6269:y
6263:y
6259:k
6255:+
6251:x
6245:x
6241:k
6237:=
6233:r
6225:k
6199:k
6175:r
6167:k
6163:i
6156:e
6152:)
6148:r
6144:(
6141:A
6138:=
6135:)
6131:r
6127:(
6087:k
6058:z
6054:k
6033:z
6013:z
5991:z
5987:k
5964:z
5960:k
5935:k
5915:k
5907:T
5903:k
5878:k
5856:T
5852:k
5829:2
5825:k
5816:2
5811:y
5807:k
5803:+
5798:2
5793:x
5789:k
5785:=
5780:2
5775:T
5771:k
5748:T
5744:k
5717:2
5711:T
5707:k
5682:i
5678:k
5655:i
5651:k
5628:y
5624:k
5620:d
5615:x
5611:k
5607:d
5599:)
5596:y
5591:y
5587:k
5583:+
5580:x
5575:x
5571:k
5567:(
5564:i
5560:e
5553:)
5548:y
5544:k
5540:,
5535:x
5531:k
5527:(
5522:0
5509:+
5488:+
5472:=
5469:)
5466:y
5463:,
5460:x
5457:(
5447:,
5444:0
5415:f
5409:2
5406:=
5374:2
5368:=
5363:c
5355:=
5352:k
5330:i
5326:k
5303:2
5299:k
5295:=
5290:2
5285:z
5281:k
5277:+
5272:2
5267:y
5263:k
5259:+
5254:2
5249:x
5245:k
5218:y
5214:k
5210:d
5205:x
5201:k
5197:d
5189:)
5186:y
5181:y
5177:k
5173:+
5170:x
5165:x
5161:k
5157:(
5154:i
5150:e
5143:)
5138:y
5134:k
5130:,
5125:x
5121:k
5117:(
5112:0
5099:+
5078:+
5062:=
5059:)
5056:y
5053:,
5050:x
5047:(
5042:0
5013:0
5010:=
5007:z
4969:y
4965:k
4942:x
4938:k
4914:}
4911:)
4906:y
4902:k
4898:,
4893:x
4889:k
4885:(
4880:0
4872:{
4867:1
4858:F
4852:=
4849:)
4846:y
4843:,
4840:x
4837:(
4832:0
4807:}
4804:)
4801:y
4798:,
4795:x
4792:(
4787:0
4779:{
4774:F
4769:=
4766:)
4761:y
4757:k
4753:,
4748:x
4744:k
4740:(
4735:0
4709:z
4682:y
4678:k
4674:d
4669:x
4665:k
4661:d
4653:)
4650:y
4645:y
4641:k
4637:+
4634:x
4629:x
4625:k
4621:(
4618:i
4614:e
4607:)
4602:y
4598:k
4594:,
4589:x
4585:k
4581:(
4576:0
4563:+
4542:+
4526:=
4523:)
4520:y
4517:,
4514:x
4511:(
4506:0
4481:.
4476:0
4473:=
4470:z
4465:|
4460:)
4457:z
4454:,
4451:y
4448:,
4445:x
4442:(
4436:=
4433:)
4430:y
4427:,
4424:x
4421:(
4416:0
4386:y
4382:k
4359:x
4355:k
4332:z
4328:k
4307:)
4302:z
4298:k
4294:,
4289:y
4285:k
4281:,
4276:x
4272:k
4268:(
4248:)
4243:y
4239:k
4235:,
4230:x
4226:k
4222:(
4217:0
4192:f
4186:2
4183:=
4151:2
4145:=
4140:c
4132:=
4129:k
4107:i
4103:k
4080:2
4076:k
4072:=
4067:2
4062:z
4058:k
4054:+
4049:2
4044:y
4040:k
4036:+
4031:2
4026:x
4022:k
4006:)
4002:(
3983:y
3979:k
3975:d
3970:x
3966:k
3962:d
3950:2
3945:y
3941:k
3932:2
3927:x
3923:k
3914:2
3910:k
3904:z
3901:i
3894:e
3885:)
3882:y
3877:y
3873:k
3869:+
3866:x
3861:x
3857:k
3853:(
3850:i
3846:e
3839:)
3834:y
3830:k
3826:,
3821:x
3817:k
3813:(
3808:0
3795:+
3774:+
3758:=
3755:)
3752:z
3749:,
3746:y
3743:,
3740:x
3737:(
3705:f
3658:z
3638:y
3618:x
3615:=
3612:i
3604:(
3592:i
3570:i
3566:k
3545:z
3525:y
3505:x
3502:=
3499:i
3491:(
3479:i
3450:2
3444:=
3439:c
3431:=
3428:k
3425:=
3421:|
3417:k
3413:|
3385:z
3376:z
3372:k
3368:+
3358:y
3349:y
3345:k
3341:+
3331:x
3322:x
3318:k
3314:=
3310:k
3289:z
3269:y
3249:x
3246:=
3243:i
3235:(
3221:i
3217:k
3196:z
3176:y
3156:x
3153:=
3150:i
3142:(
3128:i
3124:k
3097:z
3093:E
3070:y
3066:E
3043:x
3039:E
3014:A
2984:2
2979:y
2975:k
2966:2
2961:x
2957:k
2948:2
2944:k
2938:z
2935:i
2928:e
2922:)
2919:y
2914:y
2910:k
2906:+
2903:x
2898:x
2894:k
2890:(
2887:i
2883:e
2879:A
2876:=
2864:z
2859:z
2855:k
2851:i
2847:e
2841:)
2838:y
2833:y
2829:k
2825:+
2822:x
2817:x
2813:k
2809:(
2806:i
2802:e
2798:A
2795:=
2783:z
2778:z
2774:k
2770:i
2766:e
2760:y
2755:y
2751:k
2747:i
2743:e
2737:x
2732:x
2728:k
2724:i
2720:e
2716:A
2713:=
2706:)
2703:z
2700:,
2697:y
2694:,
2691:x
2688:(
2661:)
2658:z
2655:,
2652:y
2649:,
2646:x
2643:(
2620:)
2617:z
2614:,
2611:y
2608:,
2605:x
2602:(
2571:2
2567:k
2563:=
2554:2
2549:z
2545:k
2541:+
2536:2
2531:y
2527:k
2523:+
2518:2
2513:x
2509:k
2501:0
2498:=
2491:)
2488:z
2485:(
2480:z
2476:f
2470:2
2465:z
2461:k
2457:+
2454:)
2451:z
2448:(
2443:z
2439:f
2430:2
2426:z
2422:d
2416:2
2412:d
2402:0
2399:=
2392:)
2389:y
2386:(
2381:y
2377:f
2371:2
2366:y
2362:k
2358:+
2355:)
2352:y
2349:(
2344:y
2340:f
2331:2
2327:y
2323:d
2317:2
2313:d
2303:0
2300:=
2293:)
2290:x
2287:(
2282:x
2278:f
2272:2
2267:x
2263:k
2259:+
2256:)
2253:x
2250:(
2245:x
2241:f
2232:2
2228:x
2224:d
2218:2
2214:d
2191:z
2188:f
2184:y
2181:f
2177:x
2174:f
2170:z
2166:y
2162:x
2159:k
2155:x
2151:x
2147:x
2143:x
2127:0
2124:=
2119:2
2115:k
2111:+
2105:)
2102:z
2099:(
2094:z
2090:f
2084:)
2081:z
2078:(
2070:z
2066:f
2059:+
2053:)
2050:y
2047:(
2042:y
2038:f
2032:)
2029:y
2026:(
2018:y
2014:f
2007:+
2001:)
1998:x
1995:(
1990:x
1986:f
1980:)
1977:x
1974:(
1966:x
1962:f
1938:0
1935:=
1932:)
1929:z
1926:(
1921:z
1917:f
1913:)
1910:y
1907:(
1902:y
1898:f
1894:)
1891:x
1888:(
1883:x
1879:f
1873:2
1869:k
1865:+
1862:)
1859:z
1856:(
1848:z
1844:f
1840:)
1837:y
1834:(
1829:y
1825:f
1821:)
1818:x
1815:(
1810:x
1806:f
1802:+
1799:)
1796:z
1793:(
1788:z
1784:f
1780:)
1777:y
1774:(
1766:y
1762:f
1758:)
1755:x
1752:(
1747:x
1743:f
1739:+
1736:)
1733:z
1730:(
1725:z
1721:f
1717:)
1714:y
1711:(
1706:y
1702:f
1698:)
1695:x
1692:(
1684:x
1680:f
1659:,
1651:2
1647:z
1633:2
1622:+
1614:2
1610:y
1596:2
1585:+
1577:2
1573:x
1559:2
1548:=
1540:2
1517:z
1513:y
1509:x
1495:)
1492:z
1489:(
1484:z
1480:f
1476:)
1473:y
1470:(
1465:y
1461:f
1457:)
1454:x
1451:(
1446:x
1442:f
1438:=
1435:)
1432:z
1429:,
1426:y
1423:,
1420:x
1417:(
1364:=
1361:)
1357:r
1353:(
1346:)
1340:2
1336:k
1332:+
1327:2
1318:(
1291:t
1285:i
1281:e
1277:)
1273:r
1269:(
1246:)
1243:t
1240:,
1236:r
1232:(
1229:u
1186:k
1162:)
1158:r
1154:(
1102:2
1096:=
1091:c
1083:=
1080:k
1060:0
1057:=
1053:}
1049:)
1045:r
1041:(
1034:)
1028:2
1024:k
1020:+
1015:2
1006:(
1001:{
946:a
920:)
916:r
912:(
906:i
902:e
898:)
894:r
890:(
887:a
884:=
881:)
877:r
873:(
850:f
844:2
841:=
818:x
797:}
794:x
791:{
760:i
740:.
736:}
730:t
724:i
720:e
716:)
712:r
708:(
701:{
691:=
688:)
685:t
682:,
678:r
674:(
671:u
623:/
619:1
616:=
613:f
593:f
587:2
584:=
559:t
553:i
549:e
536:/
532:/
513:E
508:i
503:i
501:E
496:z
492:y
488:x
484:i
479:)
477:t
473:r
471:(
468:i
466:E
462:t
458:r
456:(
454:u
445:t
443:,
441:r
439:(
437:u
419:c
396:=
393:)
390:t
387:,
383:r
379:(
376:u
372:)
363:2
358:t
347:2
333:2
329:c
325:1
315:2
306:(
283:t
265:)
262:z
259:,
256:y
253:,
250:x
247:(
244:=
240:r
219:)
216:t
213:,
209:r
205:(
202:u
199:=
196:u
186:u
168:y
166:k
161:x
159:k
136:y
132:x
127:y
125:k
120:x
118:k
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.